Inverse Problems in Global Flow Diagnostics 3031424735, 9783031424731

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Tianshu Liu Zemin Cai

Inverse Problems in Global Flow Diagnostics

Inverse Problems in Global Flow Diagnostics

Tianshu Liu • Zemin Cai

Inverse Problems in Global Flow Diagnostics

Tianshu Liu Department of Mechanical and Aerospace Engineering, College of Engineering and Applied Sciences Western Michigan University Kalamazoo, MI, USA

Zemin Cai Department of Electronics, School of Engineering Shantou University Shantou, Guangdong, China

ISBN 978-3-031-42473-1 ISBN 978-3-031-42474-8 https://doi.org/10.1007/978-3-031-42474-8

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Dedicated to Ruomei and Ranya – T. L. Weixu, Qiaren and Qiayi – Z. C.

Preface

This book presents a systematical summary of inverse problems related to global flow diagnostics. Global flow diagnostics include diverse image-based measurements of the fundamental physical quantities in fluid mechanics and aerodynamics, including velocity, pressure, temperature, density, concentration of species, skin friction, and surface heat flux. Flow visualization images represent observable quantities of flows in terms of the image intensity, including scattering particle images in particle image velocimetry (PIV), fluorescent images in planar laserinduced fluorescence (PLIF) visualizations, Schlieren and shadowgraph images of density-varying flows, transmittance images (X-ray and neutron radiography images), pressure- and temperature-sensitive-paint (PSP and TSP) images, surface luminescent oil-film images, and multispectral images of clouds and oceans taken by satellites and spacecraft. To extract the required physical quantities from flow visualization images, various data-reduction methods have been developed, which are more suitable for specific techniques. From a general perspective, the determination of a physical quantity from observable quantities is an inverse problem since observed information in global flow measurements is often not sufficient to uniquely infer this physical quantity. To formulate the inverse problems, the mathematical models for various flow visualizations are required to relate the observable quantities to the physical quantities to be determined. For example, to determine a vector field in flows (velocity or skin friction), the models can be typically written in a form of the transport equation in the image plane (the optical flow equation). To solve these inverse problems, a variational method with a suitable smoothness constraint is applied, which leads to the Euler-Lagrange equation for extraction of a field of a physical quantity from the observable quantities in images. Therefore, the inverse problems in global flow diagnostics can be considered in a unified framework of the variational formulations, which are the main theme of this book. The theoretical aspects of this book include the projection of the relevant governing partial differential equations in fluid mechanics onto the image plane leading to the physics-based optical flow equation, the on-wall relations between vii

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Preface

skin friction and other surface quantities, the variational formulations for different inverse problems, and the error analysis. The experimental aspects cover various global flow visualizations in fluid mechanics and aerodynamics laboratories and natural environments. More importantly, this book emphasizes a coherent integration of the theoretical and experimental aspects. The materials in this book are useful for researchers and graduate students in fluid mechanics, aerodynamics, and other relevant fields of engineering and sciences, which also provide some interesting problems for researchers in applied mathematics and image processing. This book has 11 chapters and 2 appendices. The contents in each chapter are self-contained, while Chaps. 2, 3, 4, 5, 6, 7, 8, and 9 are unified generally in a framework of the variational formulations. Chapter 10 describes the inverse heat transfer problem solved by using the inverse Laplace transform and numerical optimization method. Chapter 11 presents a mathematical analysis of the physicsbased optical flow that is related to Chaps. 2, 3, 4, 5, 6, 7, 8, and 9. Chapter 1 provides a concise introduction to inverse problems in the context of global flow diagnostics, in which the data-reduction methods for different imagebased flow measurements are classified into three types of inverse problems. Chapter 2 discusses extraction of high-resolution velocity (or displacement) fields from flow visualization images, focusing on the physical foundations of the optical flow method for global flow diagnostics. The projected motion equations are derived for typical flow visualizations, and they are re-cast to the physics-based optical flow equation in the image plane where the physical meaning of the optical flow is clearly elucidated. Then, the variational formulation is proposed for solving the optical flow equation, and the Euler-Lagrange equation with the Neumann condition is given to determine the optical flow. The applications of the optical flow method in measurements of complex flows are described, including planetary cloud tracking, neutron radiography of two-phase flow, particle image velocimetry (PIV), and backgroundoriented Schlieren technique. Chapter 3 covers the global luminescent oil-film (GLOF) skin friction meter. For GLOF visualizations, the thin-oil-film equation is re-cast to the optical flow equation in the image plane, where the optical flow is proportional to skin friction. The variational method is applied for extraction of a time-averaged skin friction field from a time sequence of GLOF images. The applications of the GLOF method in measurements of complex flows are described, including validation of the PoincareBendixson index formula as a topological constraint and reconstruction of the skin friction topology in a wing-body junction flow. Chapter 4 deals with extraction of a skin friction field from a surface pressure field, which is particularly relevant to global surface pressure diagnostics with pressure sensitive paint (PSP). The exact on-wall relation between skin friction and surface pressure through the boundary enstrophy flux (BEF) is derived from the Navier-Stokes (NS) equations. The variational method is developed to determine a skin friction field from a surface pressure field in the first-order approximation where the BEF in a known base flow is used. This method is applied to reconstruction of the elemental structures of separated flows and study of the skin friction topology in incident and swept shock-wave/boundary-layer interactions.

Preface

ix

Chapter 5 discusses extraction of a skin friction field from a surface temperature field, which is particularly relevant to global surface temperature diagnostics with temperature sensitive paint (TSP). The exact on-wall relation between skin friction and surface temperature is derived from the energy equation. The variational method is developed to determine a skin friction field from a surface temperature field in the first-order approximation where the source term in a known base flow is used. This method is applied to extraction of the skin friction topology of turbulent wedges on a transonic swept wing and evaluation of the classical Reynolds analogy in complex flows. Chapter 6 covers extraction of a skin friction field from surface scalar visualizations with PSP as an oxygen sensor and sublimating coatings. The exact on-wall relation between skin friction and surface scalar concentration is derived from the mass transport equation. The variational method is developed to determine a skin friction field from surface scalar visualization images when the source term in a known base flow is used in the first-order approximation. This method is applied to PSP visualizations of nitrogen impinging jets and surface luminescent dye visualizations of delta wings in water flows. Chapter 7 deals with extraction of a skin friction field from the surface optical flow (SOF) defined as a convection velocity of a surface scalar quantity (temperature, scalar concentration or enstrophy). The evolution equations of these surface quantities are written as a generic form of the optical flow equation where the SOF is proportional to skin friction. Therefore, the SOF can be determined by solving the optical flow problem from a time sequence of fields of a surface quantity. The SOF method is applied to extraction of skin friction fields from the surface enstrophy fields obtained in direct numerical simulation (DNS) in a turbulent channel flow and surface temperature fields obtained in time-resolved TSP measurements in the NACA0015 airfoil flow and impinging jets. Chapter 8 discusses extraction of pressure from velocity in space as an inverse problem. The relation between the total pressure and the velocity-related source term is given as an adapted form of the NS equations. The variational formulation is proposed, and then the Euler-Lagrange equation is derived for the total pressure. This method is evaluated through simulations in the oblique Hiemenz flow and used for extraction of the pressure fields near a freely flying hawkmoth. Chapter 9 covers extraction of surface pressure from skin friction as an inverse problem, focusing on its application to GLOF skin friction measurements in complex flows. The relation between surface pressure and skin friction is used as the foundation of this method. The variational formulation is proposed, leading to the Euler-Lagrange equation for surface pressure. The approximate method with the constant BEF is proposed, which is particularly useful in measurements of complex flows. This method is evaluated through simulations in the Falkner-Skan flow and the flow over a 70°-delta wing and is further applied to experimental skin friction data obtained by using the GLOF method in the flow over a 65°-delta wing and the square junction flow.

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Chapter 10 discusses extraction of the surface heat flux from a time sequence of surface temperatures obtained in time-resolved surface temperature measurements with TSP in high-speed flows. The analytical and numerical inverse heat transfer solutions of the one-dimensional (1D) heat conduction equation for the surface heat flux are given for a polymer coating layer on a finite base. Furthermore, the image deconvolution method is developed for correcting the lateral heat conduction effect on 1D heat flux calculation. The inverse heat transfer methods are applied to cone and wedge models in hypersonic wind tunnels and obliquely impinging sonic jets. Chapter 11 provides a mathematical analysis of the variational optical flow solution to preserve discontinuities in a velocity field extracted from flow visualization images, including the uniqueness and convergence of the solution and numerical algorithm. The numerical algorithm is evaluated in numerical experiments. Appendix A presents some relevant results in differential geometry. Appendix B describes the open-source Matlab programs for the methods developed in this book. We would like to acknowledge the following colleagues who collaborated with us on some relevant problems: K. Asai, A. Baldwin, T. Chen, D. Choi, J. Crafton, D.O. Davis, J. Gregory, K. Hayasaka, N. Husen, T. Kakuta, M. Kameda, J. Lai, Y. Liu, M.H.M. Makhmalbaf, L. Mears, A. Merat, P. Merati, M. Miozzi, T. Misaka, J. Montefort, S. Obayashi, S. Palluconi, S. Risius, N. Rogoshchenkov, D. Salazar, K.M. Sanyanagi, S. Schneider, L. Shen, S. Stanfield, J.P. Sullivan, Y. Tagawa, P. Trtik, B. Wang, A.N. Watkins, M.R. Woike, S. Woodiga, J.-Z. Wu, R. Zboray, and H. Zhong. T. Liu is supported by the Presidential Innovation Professorship and the John O. Hallquist Endowed Professorship at Western Michigan University. Some results used in this book were obtained in Liu’s projects supported by NASA and the US Air Force. Z. Cai was supported in part by the National Natural Science Foundation of China and the Guangdong Natural Science Foundation. Kalamazoo, MI, USA Shantou, Guangdong, China

Tianshu Liu Zemin Cai

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Inverse Problem I: Vector Field from Scalar Field . . . 1.1.2 Inverse Problem II: Scalar Field from Vector Field . . . 1.1.3 Inverse Problem III: Surface Flux from Surface Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Velocity from Flow Visualizations . . . . . . . . . . . . . . . . . . . . . 1.3 Skin Friction from Global Luminescent Oil-Film Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Skin Friction from Visualizations of Surface Scalar Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Skin Friction from Surface Optical Flow . . . . . . . . . . . . . . . . 1.6 Pressure from Velocity and Skin Friction . . . . . . . . . . . . . . . . 1.7 Heat Flux from Surface Temperature . . . . . . . . . . . . . . . . . . . Velocity from Flow Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Geometric and Radiometric Projection . . . . . . . . . . . . . . . . . . 2.2 Projected Motion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Laser Sheet-Induced Fluorescence Image . . . . . . . . . . 2.2.2 Images of Density-Varying Flow . . . . . . . . . . . . . . . . 2.2.3 Transmittance Image Through Scattering Particulate Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Image of Scattering Particulate Flow . . . . . . . . . . . . . 2.2.5 Laser Sheet-Illuminated Particle Image . . . . . . . . . . . 2.2.6 Neutron Radiography Image . . . . . . . . . . . . . . . . . . . 2.3 Optical Flow and Variational Method . . . . . . . . . . . . . . . . . . . 2.3.1 Optical Flow Equation . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Optical Flow and Particle Velocity . . . . . . . . . . . . . .

1 1 2 3 3 4 5 6 8 9 11 13 13 16 16 20 24 28 30 31 35 35 37 39 41 xi

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Contents

2.4

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Planetary Cloud Tracking . . . . . . . . . . . . . . . . . . . . . 2.4.2 Neutron Radiography of Two-Phase Flow . . . . . . . . . 2.4.3 Particle Image Velocimetry . . . . . . . . . . . . . . . . . . . . 2.4.4 Background-Oriented Schlieren Technique . . . . . . . .

42 42 49 53 57

Skin Friction from Global Luminescent Oil-Film Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thin Oil-Film Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Luminescent Oil-Film Skin Friction Meter . . . . . . . . . . 3.2.1 Oil-Film Thickness and Luminescent Intensity . . . . . . 3.2.2 Projection from Surface to Image . . . . . . . . . . . . . . . 3.3 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 3.3.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Averaging of Snapshot Solutions . . . . . . . . . . . . . . . 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Skin Friction Topology in Separated Flows . . . . . . . . 3.4.2 Wing-Body Junction Flow . . . . . . . . . . . . . . . . . . . .

61 61 64 64 65 67 67 68 69 70 70 75

4

Skin Friction from Surface Pressure Visualizations . . . . . . . . . . . . 4.1 Relation Between Skin Friction and Surface Pressure . . . . . . . 4.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 4.2.2 Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Elemental Structures of Separated Flows . . . . . . . . . . 4.3.2 Incident Shock-Wave/Boundary-Layer Interaction . . . 4.3.3 Swept Shock-Wave/Boundary-Layer Interaction . . . . .

83 83 87 87 89 92 92 99 103

5

Skin Friction from Surface Temperature Visualizations . . . . . . . . 5.1 Relation Between Skin Friction and Surface Temperature . . . . 5.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 5.2.2 Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Turbulent Wedges on a Swept Wing in Transonic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Skin Friction and Heat Transfer: Beyond the Reynolds Analogy . . . . . . . . . . . . . . . . . . . . . . .

111 111 117 117 117 119

3

6

Skin Friction from Surface Scalar Visualizations . . . . . . . . . . . . 6.1 Relation Between Skin Friction and Surface Scalar Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 General Consideration . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Surface Scalar Visualization with PSP . . . . . . . . . . . 6.1.3 Visualization with Sublimating Coatings . . . . . . . . .

119 127

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. . . .

135 135 138 139

Contents

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Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 6.2.2 Approximate Method . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Impinging Nitrogen Jets . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Delta Wings in Water Flows . . . . . . . . . . . . . . . . . . .

142 142 143 144 144 148

Skin Friction from Surface Optical Flow . . . . . . . . . . . . . . . . . . . . 7.1 Evolution Equations and Surface Optical Flow . . . . . . . . . . . . 7.1.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Enstrophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Relations Between the SOFs of Different Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Turbulent Channel Flow . . . . . . . . . . . . . . . . . . . . . . 7.3.2 NACA0015 Airfoil Flow . . . . . . . . . . . . . . . . . . . . . 7.3.3 Impinging Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 153 153 156 157 159 160 162 162 166 172

8

Pressure from Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Basic Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Pressure and Velocity . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Pressure and Second Invariant . . . . . . . . . . . . . . . . . . 8.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . 8.3.2 Direct Numerical Algorithm . . . . . . . . . . . . . . . . . . . 8.3.3 Iterative Numerical Algorithm . . . . . . . . . . . . . . . . . . 8.4 Method Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Oblique Hiemenz Flow . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . 8.4.3 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Application: Hawkmoth Flight . . . . . . . . . . . . . . . . . . . . . . . .

177 177 177 179 181 181 182 183 183 185 186 188 188 189 192 195

9

Surface Pressure from Skin Friction . . . . . . . . . . . . . . . . . . . . . . 9.1 Formulation and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Variational Method . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Neumann Condition and Lagrange Multiplier . . . . . . 9.1.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Approximate Method with the Constant BEF . . . . . . 9.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Method Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 201 204 206 208 208 209

6.3

7

. . . . . . . .

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Contents

9.3.1 Falkner-Skan Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 70°-Delta Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 65°-Delta Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Square Junction Flow . . . . . . . . . . . . . . . . . . . . . . . .

209 213 217 217 220

10

Heat Flux from Surface Temperature Visualizations . . . . . . . . . . . 10.1 Analytical Inverse Heat Transfer Solution . . . . . . . . . . . . . . . . 10.1.1 Equations and Laplace Transform . . . . . . . . . . . . . . . 10.1.2 Inverse Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Solution Validation . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Numerical Inverse Heat Transfer Solution . . . . . . . . . . . . . . . 10.2.1 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Solution Validation . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Correcting Lateral Heat Conduction Effect . . . . . . . . . . . . . . . 10.3.1 Analogy Between 1D and 3D Direct Solutions . . . . . . 10.3.2 Convolution-Type Integral Equation . . . . . . . . . . . . . 10.3.3 Solution Validation . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 25°/45° Indented Cone at Mach 11 . . . . . . . . . . . . . . 10.4.2 7°-Half-Angle Cone at Mach-6 . . . . . . . . . . . . . . . . . 10.4.3 Obliquely Impinging Sonic Jet . . . . . . . . . . . . . . . . . 10.4.4 In Situ Calibration for TSP on Finite Base . . . . . . . . .

223 223 223 226 231 232 232 236 239 239 244 246 250 250 253 256 262

11

Analysis of Physics-Based Optical Flow . . . . . . . . . . . . . . . . . . . . . 11.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 BV Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 An Integral Representation of the Measure ∇
ðguÞ . . . . . . . . . . . . . . . . . . . . . . 11.2 Successive Approximation and Convergence . . . . . . . . . . . . . 11.2.1 A Result of Γ-Convergence . . . . . . . . . . . . . . . . . . . 11.2.2 The Half-Quadratic Minimization . . . . . . . . . . . . . . . 11.3 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Flow over a Vortex Pair . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Flow Across a Normal Shock . . . . . . . . . . . . . . . . . . 11.4.3 Quasi-2D Turbulence . . . . . . . . . . . . . . . . . . . . . . . .

267 267 268

Appendix A: Useful Results in Differential Geometry . . . . . . . . . . . . . .

295

Appendix B: Open-Source Programs . . . . . . . . . . . . . . . . . . . . . . . . . .

299

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331

9.4

269 272 273 274 284 288 288 291 291

Chapter 1

Introduction

1.1

Inverse Problems

Global flow diagnostics are essentially image-based measurements of the fundamental physical quantities in fluid mechanics and aerodynamics, including velocity, pressure, temperature, density, concentration of species, skin friction, and surface heat flux. In experiments, these fundamental quantities are related to some observable quantities in flow visualizations. The observable information is presented in digital flow visualization images, including scattering particle images in particle image velocimetry (PIV), fluorescent images in planar laser-induced fluorescence (PLIF) visualizations, Schlieren and shadowgraph images of density-varying flows, transmittance images (X-ray and neutron radiography images), pressure- and temperature-sensitive paint images (PSP and TSP images), surface luminescent oil-film images, and multispectral images of clouds and oceans taken by satellites and spacecraft. From a standpoint of image/data processing, a key problem is how to determine required physical quantities from observable quantities that are represented by the image intensity in flow visualizations. This is an inverse problem that belongs to a large class of ill-posed inverse problems in various scientific and engineering fields (Tikhonov and Arsenin 1977; Ramm 2004). Experiments in fluid mechanics and aerodynamics have two essential related aspects: experimental simulation of physical phenomena and measurements of relevant physical quantities. Interestingly, Panton et al. (2007) considered fluid mechanics experiments as a boundary value problem that is a well-posed positive problem. Indeed, the flow conditions at the boundary in experiments are controlled and specified such that they are the same as those for the boundary value problem to solve the governing equations in fluid mechanics. From this perspective of experimental flow simulation, the statement made by Panton et al. (2007) is correct. On the other hand, as another essential part of experiments, global flow measurements are more suitably considered as inverse problems since the required physical quantities are often inferred from insufficient observable information. Therefore, an optimized © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_1

1

2

1 Introduction

solution of an ill-posed inverse problem is often sought in a variational framework with suitable physical and mathematical constraints. In the context of global flow diagnostics, we consider a generic mathematical model relating a field of a vector quantity denoted by u (velocity or skin friction) to a field of a scalar quantity denoted by g, which is symbolically expressed as M ðu, g, f Þ = 0,

ð1:1Þ

where M denotes a symbolic operator describing a physical process (typically a differential operator) and f is a relevant parameter. For example, when M is the material derivative, g is the scalar concentration, and f is a source term, Eq. (1.1) is the transport equation of the scalar, i.e., ð∂=∂t þ u  ∇Þg = f : For image processing, Eq. (1.1) can be projected onto the image plane through the perspective projection transformation, and therefore a two-dimensional (2D) projected vector u = ðu1 , u2 Þ is considered in the image plane. We consider three generic types of the inverse problems in global flow diagnostics.

1.1.1

Inverse Problem I: Vector Field from Scalar Field

Given a measured field of a scalar quantity g that is represented by the image intensity (such as particle-scattering radiation or luminescent oil-film emission), we want to determine a field of the vector quantity u (velocity or skin friction) when the parameter f is known (either given or approximated). This problem is directly related to extraction of velocity fields from flow visualizations (see Chap. 2) and skin friction fields from surface flow visualizations (see Chaps. 3, 4, 5, 6, and 7). In this case, Eq. (1.1) has a form of the transport equation of the scalar quantity g where M is the material derivative. In the image plane, this equation is recast to the optical flow equation, and therefore the optical flow problem is considered in the context of flow measurements. Since there are the two unknown components of u in the single equation, this inverse problem is ill-posed. Thus, additional constraints are required to close the problem. Following Tikhonov and Arsenin (1977), we have a variational formulation, i.e., J ðuÞ = kM ðu, g, f Þk2 þ α k∇u1 k2 þ k∇u2 k2 → min,

ð1:2Þ

where the first term is the equation term, the second term is a regularization term imposing the smoothness condition on u (the Tikhonov first-order functional), α is a

1.1

Inverse Problems

3

Lagrange multiplier, and kk is the L2-norm. By minimizing J ðuÞ, the EulerLagrange equation for u is derived, where the Neumann condition is imposed on the boundary of a computational domain. The numerical solution of the EulerLagrange equation gives a field of u.

1.1.2

Inverse Problem II: Scalar Field from Vector Field

Given a field of a measured vector quantity u (velocity or skin friction), we want to determine a field of a scalar quantity g (pressure in space or on surface) when the parameter f is known (either given or approximated). This problem is directly related to extraction of a pressure field in a cross section in space from global flow velocity diagnostics (see Chap. 8) and a surface pressure field from global skin friction diagnostics (see Chap. 9). This problem is a reciprocal one of Inverse Problem I. For this problem, we have a variational formulation, i.e., J ðgÞ = kM ðu, g, f Þk2 þ αk∇gk2 → min,

ð1:3Þ

where the first term is the equation term and the second term is a regularization term imposing the smoothness condition on g. By minimizing J ðgÞ, the Euler-Lagrange equation for g is derived, where the Neumann condition is imposed on the boundary of a computational domain. The numerical solution of the Euler-Lagrange equation gives a field of g.

1.1.3

Inverse Problem III: Surface Flux from Surface Scalar

This problem is related to calculation of the surface heat flux from a time history of surface temperature as an inverse heat transfer problem (see Chap. 10). In a special case of heat conduction in a body, Eq. (1.1) is reduced to M ðg, f Þ = 0 for a scalar quantity g (temperature in a heat conduction problem). When M is a linear differential operator, by using the Laplace transform, a solution for the positive (forward) problem can be symbolically expressed as g = M - 1 g∂B , ½∂g=∂n∂B ,

ð1:4Þ

where M - 1 denotes the inversion of the differential operator; g∂B and ½∂g=∂n∂B are the scalar quantity and its flux at a surface element ∂B, respectively; and ∂=∂n is the wall-normal derivative. On the surface, the inverse solution of Eq. (1.4) for the surface flux could be symbolically expressed as ½∂g=∂n∂B = F ðg∂B Þ:

ð1:5Þ

4

1.2

1

Introduction

Velocity from Flow Visualizations

Global velocity diagnostics is of fundamental importance in the study of fluid mechanics in order to understand the physics of complex flows. Particle image velocimetry (PIV) is a widely used global velocity measurement technique (Raffel et al. 2007; Adrian and Westerweed 2011). In PIV, a displacement vector is determined by maximizing the correlation peak in the spatial cross-correlation map obtained from two corresponding interrogation windows in successive particle images. In addition to the correlation method, the optical flow method has been used for the determination of high-resolution velocity fields from various images of continuous patterns (like cloud and ocean images in satellite imagery) particularly in geophysical fluid mechanics (Quenot et al. 1998; Wildes et al. 2000; Ruhnau et al. 2005; Yuan et al. 2007; Corpetti et al. 2002, 2006; Héas et al. 2007). A review on the optical flow method applied to fluid flow measurements was given by Heitz et al. (2010). The key problem is to build the mathematical models to lay a rational physical foundation for the optical flow method applied to various flow visualizations. Based on the projection of the transport equations in the 3D object space onto the image plane, Liu and Shen (2008) derived the projected motion equations for various flow visualizations, including laser sheet-induced fluorescence images, transmittance images of passive scalar transport (X-ray and neutron radiography images), schlieren, shadowgraph and transmittance images of density-varying flows, transmittance and scattering images of particulate flows, and laser sheet-illuminated particle images (see Chap. 2). These equations have the same mathematical form, and thus they can be recast to the physics-based optical flow equation in the image plane, where the optical flow is proportional to the light-ray-path-averaged velocity of fluid (or particles) weighted in a relevant field quantity (such as dye concentration, fluid density, or particle concentration). The physics-based optical flow equation is written as (see Chap. 2) ∂g=∂t þ ∇  ðg uÞ = f ,

ð1:6Þ

where u = ðu1 , u2 Þ is the optical flow, g is the normalized image intensity related to a measured physical quantity mapped onto the image plane, f is a source term related to the net flux of a visualized medium across a control surface, and ∇ is the gradient operator in the image plane. The physical meaning of the optical flow u is clear, that is, the optical flow is proportional to the light-path-averaged velocity of fluid or particles in flow visualizations. The determination of the optical flow u in Eq. (1.6) is Inverse Problem I. Therefore, the variational formulation is 2

J ðuÞ = k∂g=∂ t þ ∇  ðg uÞ - f k þ α k∇u1 k2 þ k∇u2 k2 → min,

ð1:7Þ

1.3

Skin Friction from Global Luminescent Oil-Film Visualizations

5

where the first term is the equation term, the second term is the regularization term, and α is a Lagrange multiplier. The Euler-Lagrange equation for u is derived based on Eq. (1.7) (see Chap. 2), where the Neumann condition is imposed on the boundary of a computational domain. The Euler-Lagrange equation is a parabolictype partial differential equation where the Lagrange multiplier acts as a diffusion coefficient. The numerical solution of the Euler-Lagrange equation gives a field of u when the source term f is modeled (approximated). In practical applications, f is set at zero as the first-order approximation when the net mass flux of a visualized medium across the solid surfaces or control surfaces in a measurement domain can be neglected. The relevant technical aspects of the optical flow are discussed in Chap. 2, including the error analysis and the connection between the optical flow and particle velocity. The mathematical analysis and numerical algorithm of the physics-based optical flow are given in Chap. 11. The optical flow method has been used to study the large-scale planetary vortex structures such as the Great Red Spot (GRS) on Jupiter and the north pole vortex on Saturn (Liu et al. 2012b, 2019a; Liu and Salazar 2022), impinging jets (Liu et al. 2010b, 2015b), two-phase flows visualized by neutron radiography (Liu et al. 2021c), and laser-induced underwater explosion visualized by the background-oriented Schlieren (BOS) technique (Hayasaka et al. 2016) (see Chap. 2).

1.3

Skin Friction from Global Luminescent Oil-Film Visualizations

Oil-film skin friction meter is based on detecting the temporal-spatial evolution of the thickness of a thin oil-film to determine skin friction at discrete locations (Tanner and Blows 1976; Naughton and Sheplak 2002). The thin-oil-film equation describes the response of a thin oil-film on a surface to an externally applied 3D aerodynamic flow (Naughton and Brown 1996; Brown and Naughton 1999). To determine skin friction from the development of the oil-film thickness, the first problem is how to measure the oil-thin thickness. Image-based interferometry was often used to measure the oil-film thickness (Naughton and Sheplak 2002; Naughton and Liu 2007). Liu and Sullivan (1998) used luminescent oil-film to visualize the temporal-spatial development of the oil-film thickness since the luminescent intensity of an optically thin oil-film is proportional to its thickness. Along this line, Husen et al. (2018a, b) developed the ratioed image oil-film thickness meter. The second problem is how to determine skin friction from the measured oil-film thickness. The one-dimensional (1D) local similar solution of the thin-oil-film equation has been used to determine skin friction from the oil-film thickness. However, these methods cannot be applied to global oil-film measurements. Brown and Naughton (1999) attempted to recover a global skin friction field as an inverse problem by solving the thin-oil-film equation in a surface domain. However, since one equation for the oil-film thickness is not sufficient for two unknown components of skin friction, the direction of skin friction must be known to solve the equation for the skin friction magnitude.

6

1

Introduction

Liu et al. (2008) developed a global luminescent oil-film (GLOF) skin friction meter where a high-resolution skin friction field is extracted as an inverse problem from GLOF visualizations. The mathematical model for the GLOF method is derived by projecting the thin-oil-film equation onto the image plane, i.e., ∂g=∂ t þ ∇  ðg τ Þ = f ,

ð1:8Þ

where g is the normalized luminescent intensity mapped onto the image plane, τ = ðτ1 , τ2 Þ is a projected and scaled skin friction vector in the image plane, and the source term f represents the effects of the surface pressure gradient and gravity. Since Eq. (1.8) has the same form as Eq. (1.6), the same variational formulation for Inverse Problem I is adopted, i.e., 2

J ðτ Þ = k∂g=∂ t þ ∇  ðg τ Þ - f k þ α k∇τ1 k2 þ k∇τ2 k2 → min:

ð1:9Þ

The Euler-Lagrange equation for τ is derived based on Eq. (1.9) (see Chap. 3), where the Neumann condition is imposed on the boundary of a computational domain. The numerical solution of the Euler-Lagrange equation gives a field of τ when the source term f is modeled (approximated). In general applications, since f is a higher-order small term compared to the transport term in Eq. (1.8), f is set at zero as the firstorder approximation when the effects of the pressure gradient and gravity are neglected. The relevant technical aspects of the GLOF method are discussed in Chap. 3, including the error analysis and the averaging of snapshot solutions. The GLOF method provides high-resolution skin friction fields in complex separated flows, which allows examination of the skin friction topological features including isolated critical points and separation and attachment lines. The GLOF method has been used to measure skin friction fields in flows over different models, including delta wings (Liu et al. 2008; Woodiga et al. 2009, 2015), low aspect ratio wings (Liu et al. 2011a), circular/square junction flows (Kakuta et al. 2010; Liu 2013, 2019; Lee et al. 2018), wing-body junction (Zhong et al. 2015), ground vehicle models (Woodiga et al. 2018; Salazar et al. 2022), topographic hill model (Husen et al. 2018b), axisymmetric bodies (Tran et al. 2018, 2019), and transonic wing (Costantini et al. 2021).

1.4

Skin Friction from Visualizations of Surface Scalar Quantities

The important surface scalar quantities are pressure, temperature, and scalar concentration, which can be visualized by using pressure sensitive paint (PSP, temperature-sensitive paint (TSP) and sublimation coatings, respectively. To extract skin friction fields from visualization images of these surface quantities, the on-wall relations between skin friction and these quantities have to be established. The

1.4

Skin Friction from Visualizations of Surface Scalar Quantities

7

relation between skin friction and surface pressure on a curved surface was derived by Liu et al. (2016) and Chen et al. (2019) from the Navier-Stokes (NS) equations (see Chap. 4). The relation between skin friction and surface temperature was derived from the energy equation by Liu and Woodiga (2011) for a flat surface and by Chen et al. (2019) on a curved surface (see Chap. 5). The relation between skin friction and surface scalar concentration on a curved surface was derived from the binary mass transport equation by Liu et al. (2014) and Chen et al. (2019) (see Chap. 6). Interestingly, the on-wall relations for these surface quantities have a generic mathematical form, i.e., G þ τ  ∇g = 0,

ð1:10Þ

where g is a measured surface quantity (pressure, temperature, or scalar concentration), τ = ðτ1 , τ2 Þ is a skin friction vector, and G is a source term. When g is the surface pressure, G is mainly proportional to the boundary enstrophy flux (BEF). When g is the surface temperature or scalar concentration, G is proportional to the temporal-spatial evolution of the heat flux or mass flux. The determination of τ from Eq. (1.10) is also Inverse Problem I. Therefore, the variational formulation is J ðτ Þ = kG þ τ  ∇gk2 þ α k∇τ1 k2 þ k∇τ2 k2 → min:

ð1:11Þ

The Euler-Lagrange equation for τ is derived based on Eq. (1.11) (see Chap. 4), where the Neumann condition is imposed on the boundary of a computational domain. The numerical solution of the Euler-Lagrange equation gives a field of τ when the source term G is modeled (approximated). Further, the approximate method uses the known base-flow source term G as the first-order approximation. The relevant technical aspects of this method are discussed in Chaps. 4, 5, and 6, including the error analysis and the approximate method based on the known baseflow source term. For surface pressure visualizations with PSP, the approximate method has been applied to reconstruction of the elemental structures of separated flows (Liu 2018) and extraction of skin friction fields from unsteady PSP images in incident and swept shock-wave/boundary-layer interactions (Liu et al. 2021a, 2022b) (see Chap. 4). For surface temperature visualizations with TSP, this method has been used to understand the separation bubbles on a circular cylinder and a rectangular wing in water flows (Miozzi et al. 2016, 2019; Miozzi and Costantini 2021), study the structures of turbulent wedges on a swept transonic wing (Liu et al. 2021b), and elucidate the relationship between skin friction and heat transfer beyond the Reynolds analogy (Woodworth et al. 2023) (see Chap. 5). For surface masstransfer visualizations with PSP as an oxygen sensor, sublimating coatings and luminescent dye in water flows, this method has been applied to impinging nitrogen jets, fin-generated shock-wave/boundary-layer interaction (Liu et al. 2014) and delta wings (Liu et al. 2014, 2015a) (see Chap. 6).

8

1.5

1

Introduction

Skin Friction from Surface Optical Flow

The quantitative estimation of skin friction from the traces of temperature disturbances on a surface was studied by Miozzi et al. (2020a, b) and Miozzi and Costantini (2021) based on the relationship between the celerity of propagation of temperature disturbances and the friction velocity. This relationship is originally built on the experimental evidences of correlation between the time histories of the friction velocity and the celerity of propagation of velocity disturbances (Eckelmann 1974). The experimental results of the relationship between skin friction and the propagation of other features (velocity, pressure, and vorticity) in near-wall flows were confirmed in direct numerical simulation (DNS) of turbulent boundary layers and channel flows (Kim and Hussain 1993; Del Alamo and Jimenez 2009; Geng et al. 2015). By using the Taylor hypothesis, the ensemble-averaged convection velocity of these events can be determined from the local time and space derivatives of the velocity components, which is found to be proportional to the friction velocity. The relationship between the celerity of temperature disturbances and the friction velocity was investigated by Hetsroni et al. (2004). Based on these observations, Miozzi et al. (2020a, b, 2021) inferred the friction velocity and the friction coefficient by detecting the celerity of propagation of temperature disturbances based on time-resolved TSP measurements in boundary layers in a water tunnel. Further, inspired by Miozzi et al. (2020a, b, 2021), Liu et al. (2022a) studied the relationship between skin friction and the surface optical flow (SOF) defined as a convection velocity of a surface scalar quantity based on the evolution equations of the surface scalar quantities. It is found that the SOF of a scalar quantity (temperature, scalar concentration, or enstrophy) is proportional to skin friction. The evolution equations for the instantaneous surface quantities and the kinetic energies of the corresponding fluctuating quantities have the same mathematical form, which can be written as a generic form of the optical flow equation, i.e., ∂g=∂t þ u  ∇g = f ,

ð1:12Þ

where u = ðu1 , u2 Þ is the SOF that is proportional to skin friction (u / τ ), g is a generic measured surface quantity (temperature, scalar, or enstrophy), and f is a source term that contains higher-order correlation terms. Similar to Eq. (1.6), the determination of u from Eq. (1.12) is Inverse Problem I. Therefore, the variational formulation is 2

J ðuÞ = k∂g=∂ t þ u  ∇g - f k þ α k∇u1 k2 þ k∇u2 k2 → min:

ð1:13Þ

The Euler-Lagrange equation for u is derived based on Eq. (1.13) (see Chap. 7), where the Neumann condition is imposed on the boundary of a computational domain. The numerical solution of the Euler-Lagrange equation gives a field of τ when the source term f is modeled (approximated). In practice, f is set at zero as the

1.6

Pressure from Velocity and Skin Friction

9

first-order approximation. This surface optical flow method has been evaluated based on DNS data of a turbulent channel flow and applied to surface temperature visualizations with TSP in the flow over a NACA0015 airfoil and the impinging jet (Liu et al. 2022a) (see Chap. 7).

1.6

Pressure from Velocity and Skin Friction

Determining fluid mechanic pressure in space from measured velocity fields is an active topic in experimental fluid mechanics since a high-resolution pressure distribution in complex flows is difficult to measure directly. Technically, PIV can provide high-resolution time-resolved velocity data such that it is possible to infer pressure from velocity. This problem is referred to as the problem of pressure from velocity (van Oudheusden 2013). For an incompressible flow, by grouping all the velocity-related terms in the NS equations, the pressure gradient is equal to a velocity-related vector. The central problem is how to determine pressure from velocity. There are mainly two approaches to solve this problem. The first approach is a line integral method (Liu and Katz 2006, 2013; Dabiri et al. 2014). Theoretically, the advantage of this method is that a pressure field can be reconstructed from limited known pressure data on a part of the boundary of a computational domain or even at a single point. The second approach is based on the pressure Poisson equation with the Neumann and Dirichlet conditions on the boundary of a computational domain, which is solved by using the standard numerical methods (de Kat and Ganapathisubramani 2013; de Kat and van Oudheusden 2012; Schneiders et al. 2016; McClure and Yarusevych 2017). This method is popular since more mature Poisson solvers developed in applied mathematics can be directly utilized. The performance of the Poisson equation-based method was evaluated in simulations and real measurements (Charonko et al. 2010; Azijli et al. 2016; van Gent, et al. 2017; McClure and Yarusevych 2017). Surface pressure is a fundamental surface quantity (along with skin friction) in fluid mechanics and aerodynamics. Surface pressure on a model can be measured using distributed surface pressure taps connected to pressure transducers through tubing. This technique is straightforward and accurate, which is commonly used in both large wind tunnels in major aerospace institutions (like NASA) and small facilities in most universities. However, for complex models such as thin wings and blades, the installation of pressure taps with tubing in the interior of the models is laborious and expensive. Furthermore, distributed pressure taps are limited such that a high-resolution surface pressure field cannot be obtained for global flow diagnostics in complex flows. In contrast, PSP offers a unique capability for noncontact full-field surface pressure measurements on complex models with a high spatial resolution (Liu et al. 2021d). As an oxygen-quenching-based luminescent sensor, PSP works well in compressible high-speed flows (the Mach number is larger than 0.3). However, in low-speed flows, the change of the oxygen concentration is so small that the

10

1 Introduction

luminescence measurement of PSP using a photosensor (e.g., camera) has a low signal-to-noise ratio (SNR). This intrinsic limitation of PSP prevents wide applications of global surface pressure diagnostics with PSP in low-speed wind tunnels. Therefore, global surface pressure diagnostics in low-speed complex flows remain challenging. To circumvent these difficulties in surface pressure measurements by pressure taps and PSP in low-speed flows, an interesting question is whether a surface pressure field can be determined from a skin friction field obtained in GLOF skin friction measurements (see Chap. 3). According to the above discussions, we have Problems (a) and (b): (a) Pressure from velocity (b) Surface pressure from skin friction These problems belong to Inverse Problem II, which have some similarities in the variational formulations to determine a pressure field from a measured vector field. The variational formulation is proposed for these problems (Cai et al. 2020, 2022). The relation between the total pressure and velocity is directly given by re-arranging the NS equations (Cai et al. 2020) (see Chap. 8). In contrast, the relation between surface pressure and skin friction is given by projecting the NS equations onto a solid surface (Cai et al. 2022) (see Chap. 9). These relations can be written in a generic form, i.e., u  ∇g = Φ,

ð1:14Þ

where u is a known (measured) vector field, g is a field of a scalar quantity to be determined (the total pressure P or surface pressure p∂B ), and Φ is a source term. In Problem (a), u is a given constant test vector field denoted by m, and Φ is the projection of the velocity-related terms on m (Cai et al. 2020). In Problem (b), u is a measured skin friction field (τ-field) and Φ is mainly proportional to the boundary enstrophy flux (BEF) (Cai et al. 2022). The determination of a field of g from Eq. (1.14) is Inverse Problem II. Therefore, the variational formulation is J ðgÞ = ku  ∇g - Φk2 þ αk∇gk2 → min:

ð1:15Þ

The Euler-Lagrange equation for g is derived based on Eq. (1.15), which is the second-order elliptic-type partial differential equation where the coefficients are constants for Problem (a) and variables depending on the skin friction divergence for Problem (b) (see Chaps. 8 and 9). The numerical solution of the Euler-Lagrange equation gives a field of g when the source term Φ is measured (or approximated). The relevant technical aspects of this method are discussed in Chaps. 8 and 9, including the error analysis, the Neumann condition, the Lagrange multiplier, and the approximate method with the constant BEF. This method has been validated in the oblique Hiemenz flow and the Falkner-Skan flow with the known theoretical solutions and in the flow over a delta wing in comparison with CFD data. It was

1.7

Heat Flux from Surface Temperature

11

applied to extraction of pressure fields in hawkmoth flapping flight based on highresolution velocity fields extracted using the optical flow method from high-speed Schlieren images and surface pressure fields on a delta wing and in a square cylinder junction flow based on GLOF skin friction measurements (Cai et al. 2020, 2022) (see Chaps. 8 and 9).

1.7

Heat Flux from Surface Temperature

Surface heat flux into a body (model) is one of the important aerothermodynamic quantities in the designs of hypersonic vehicles, and therefore surface heat flux measurements are critical for testing in hypersonic wind tunnels (Mathews and Rhudy 1994; Simmons 1995; Wadhams et al. 2008; Hollis et al. 2009; Liu et al. 2019c). Global heat flux measurements have attracted considerable attention since they allow a better understanding of complex flow phenomena such as flow transition, turbulence, shocks, vortices, and separations. Imaged-based surface temperature techniques include temperature-sensitive paint (TSP), thermographic phosphors (TP), thermochromic liquid crystals (TLC), and infrared (IR) thermography. In particular, TSP has been used to measure surface heat flux fields on various bodies in hypersonic wind tunnels (Liu et al. 1995, 2010a, 2011b, 2013, 2021b, d; Hubner et al. 2002; Matsumura et al. 2005; Kurits and Lewis 2009; Ozawa et al. 2015; Peng et al. 2016; Risius et al. 2017). In a certain sense, TSP, TP, TLC, and IR are simply considered as temperature-sensitive coating techniques that pose some intriguing inverse heat transfer problems. The determination of the surface heat flux from a time history of surface temperature measured by using the global techniques is Inverse Problem III. In TSP measurements, a thin sensor layer with a relatively low thermal conductivity is applied on the surface of a body (typically a metallic body). Sometimes, a white polymer base coating is applied to the surface of the body for TSP to enhance light scattering leading to the improved SNR. The sensor layer and base coating layer are collectively represented by a composite polymer layer. This polymer/base structure (two-layer system) is considered for an inverse heat transfer analysis. Since this sensor layer itself would alter the time history of surface temperature, its effect on calculation of the surface heat flux should be accounted (Nagai et al. 2008). Applying the Laplace transform to the 1D unsteady heat conduction equation for a two-layer system, Liu et al. (2010a, 2018) gave the analytical inverse heat transfer solutions for the surface heat flux into a layer on a semi-infinite base and a finite base by incorporating explicitly the effect of the polymer coating. Further, the image deconvolution method was developed for correcting the lateral heat conduction effect by solving a convolution-type integral equation (Liu et al. 2011b). The numerical inverse heat transfer solutions were developed by Cai et al. (2011, 2018) for both the finite and semi-infinite bases for materials with the temperaturedependent thermal properties. The numerical solution for the positive problem gives

12

1 Introduction

the estimated temperature change θ0ps ðqs Þ when the assumed surface heat flux qs is given as a boundary condition. The determination of the surface heat flux is formulated as a constrained optimization problem, i.e., J ðqs Þ = θps - θ0ps ðqs Þ

2

þ αk∇qs k2 → min,

ð1:16Þ

where θps is the TSP-measured temperature change, qs is the surface heat flux to be determined, and α is a Lagrange multiplier. Although this inverse problem is not strictly ill-posed, the regularization term in Eq. (1.16) will increase the robustness of the solution against data noise. The difference between the estimated and TSP-measured temperature changes is minimized by using an iterative algorithm in this optimization problem (see Chap. 10). The inverse heat transfer methods have been used to determine the surface heat flux fields from TSP measurements on cone and wedge models in hypersonic wind tunnels (Liu et al. 2010a, 2011b, 2013, 2018a, 2019c; Cai et al. 2011, 2018; Liu and Risius 2019) and impinging sonic jets (Liu et al. 2019b).

Chapter 2

Velocity from Flow Visualizations

This chapter discusses extraction of high-resolution velocity (or displacement) fields from flow visualization images, focusing on the physical foundations of the optical flow method (OFM) for global flow diagnostics. The projected motion equations are derived for typical flow visualizations based on projection of the transport equations and continuity equation in the 3D object space onto the image plane. Here, flow visualization images include laser sheet-induced fluorescence images, images of density-varying flow (Schlieren images, shadowgraph images, and transmittance images), transmittance images through scattering particulate flow, scattering images toward incident direction from particulate flow, laser sheet-illuminated particle images, and neutron radiography images. The projected motion equations provide the relation between the radiance projected to a digital camera and the light-pathaveraged velocity field weighted in a relevant field quantity. These equations for different flow visualizations have the same mathematical form, which can be recast to the optical flow equation in the image plane where the optical flow is proportional to the light-path-averaged velocity. To determine the optical flow as an inverse problem, a variational formulation is proposed, and the Euler-Lagrange equation with the Neumann condition is given. As examples, OFM is applied to planetary cloud tracking, neutron radiography (NR) of two-phase flow, particle image velocimetry (PIV), and background-oriented Schlieren (BOS) visualization.

2.1

Geometric and Radiometric Projection

Global flow diagnostics are mainly image-based measurements of visualized fluids and surfaces using digital cameras. The light emission from a visualized object is projected onto the image plane of a camera. The light projection has the geometric and radiometric aspects that should be considered in mathematical modeling of flow visualizations. The perspective projection geometry is illustrated in Fig. 2.1. The perspective projection transformation between the 3D object-space coordinates and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_2

13

14

2

Velocity from Flow Visualizations

Fig. 2.1 Projection from visualized fluid flow onto the image plane

image coordinates is described by the collinearity equations in photogrammetry (Mikhail et al. 2001; McGlone 1989; Faugeras and Luong 2001; Liu 2004; Liu et al. 2012a). The orthogonal row vectors m1 , m2 , and m3 in the rotational matrix in the collinearity conditions constitute a special object-space coordinate frame located at the perspective center associated with a camera. The vectors m1 and m2 are the directional cosine vectors parallel to the x1 -axis and x2 -axis in the image plane, respectively. The vector m3 is normal to the image plane, directing to the object along the optical axis of the camera. The vectors m1 , m2 , and m3 are functions of the Euler rotational angels of the camera that can be determined by geometric camera calibration (Liu et al. 2000; Gruen and Huang 2001; Liu et al. 2012a). We denote the object-space coordinates by X = ðX 1 , X 2 , X 3 Þ as the components of the object space position vector of a point from the perspective center in the special object-space frame ðm1 , m2 , m3 Þ associated with a camera. The perspective projection transformation in this frame is expressed as

xβ - xβ,p - δxβ = λ X β

ðβ = 1, 2Þ,

ð2:1Þ

where x = ðx1 , x2 Þ denotes the image coordinates of a point, the scaling factor λ = - f =X 3 is a ratio between the principal distance f and the coordinate - X 3 , xβ,p (β = 1, 2) define the principal point location in the image plane, and δ xβ (β = 1, 2) denote the lens distortion terms.

2.1

Geometric and Radiometric Projection

15

Fig. 2.2 Geometry of light incidence and emission

In the object-space frame ðm1 , m2 , m3 Þ, the image intensity is proportional to the radiative flux projected on the plane ðX 1 , X 2 Þ, i.e., I ðx, t Þ = c

LðX 1 , X 2 , t; θ, ψ Þ cos θ dΩ,

ð2:2Þ

where L is the radiance from an emitting object to an imaging system (e.g., camera), c is a coefficient related to the imaging system, and ðθ, ψ Þ are the polar angle and azimuthal angle, respectively, defining the direction of the infinitesimal solid angle dΩ = sin θ dθ dψ of the camera. The geometry of light incidence and emission is illustrated in Fig. 2.2. In fact, integration over a wavelength band is applied in Eq. (2.2), which is not explicitly expressed for brevity. The image coordinates (x) are related to the object-space coordinates (X ) through the perspective projection transformation. When a camera is sufficiently far from the emitting object such that the solid angle of the camera is small, the image intensity is I ðx, t Þ = c LðX 1 , X 2 , t; θc , ψ c Þ cos θc ΔΩ, where ΔΩ is the solid angle of the camera and the polar angle θc and the azimuthal angle ψ c give the angular position of the camera. In particular, when the radiance is independent of the azimuthal angle, we have

16

2

Velocity from Flow Visualizations

I ðx, t Þ = c 2π LðX 1 , X 2 , t; μc Þμc Δμc , where μc is the directional cosine of camera’s polar angle and Δμc is the directional cosine difference of the polar angle defining the observing solid angle of the camera. The scattering from small particles or emission from luminescent molecules can be considered to be independent of the azimuthal angle. In summary, in typical flow visualizations, the image intensity is proportional to the radiance projected onto a camera with a fixed position and a viewing direction. Therefore, the relative radiance can be replaced by the relative image intensity in the derivations of the projected motion equations for different flow visualizations.

2.2

Projected Motion Equations

2.2.1

Laser Sheet-Induced Fluorescence Image

2.2.1.1

Incompressible Flow

Planar laser-induced fluorescence (PLIF) is an optical diagnostic technique widely used for flow visualizations and quantitative measurements of velocity, concentration, temperature, and pressure (Hanson 1988; Crimald 2008). In PLIF, a thin laser sheet is often used to illuminate the fluorescent dye seeded in flows for visualizations. As illustrated in Fig. 2.1, the laser sheet is confined by two planar control surfaces Γ1 and Γ2 described by X 3 = Γ1 ðX 1 , X 2 Þ and X 3 = Γ2 ðX 1 , X 2 Þ, which are placed immediately outside of the laser sheet. Thus, the excited luminescent dye is confined between Γ1 and Γ2 . The fluorescent emission at a certain wavelength is proportional to the dye density or concentration. For a thin laser sheet in which the excited dye is optically thin, the fluorescent radiance received by a camera is proportional to the integral of the density ψ ðX, t Þ of the excited dye across the sheet, i.e., LðX 1 , X 2 , t Þ = c L0 ðX 1 , X 2 Þ

Γ2 Γ1

ψ ðX, t ÞdX 3 ,

ð2:3Þ

where c is a constant coefficient and L0 is the laser sheet radiance distribution averaged along the coordinate X 3 and projected to the plane ðX 1 , X 2 Þ. The laser sheet radiance for excitation is not uniform due to laser sheet spreading and dye absorption and therefore L0 implicitly depends on the dye density. For heavily dyed fluid, intensity attenuation in the laser sheet can be measured a priori, and thus L0 is considered as a known distribution in the plane ðX 1 , X 2 Þ.

2.2

Projected Motion Equations

17

The transport equation for the density of the scalar (ψ) is ∂ψ þ U  ∇ψ = D ∇2 ψ, ∂t

ð2:4Þ

where U = ð U 1 , U 2 , U 3 Þ is the fluid velocity and D is a diffusion coefficient. This equation (as well as some governing equations in this chapter) is expressed in the object-space frame ðm1 , m2 , m3 Þ associated with a camera and the differential operators are given in the object-space coordinates X = ðX 1 , X 2 , X 3 Þ. For an incompressible flow where U  ∇ψ = ∇  ðψUÞ, differentiating Eq. (2.3) with respect to time and using Eq. (2.4), we have ∂L = c L0 ∂t

Γ2 Γ1

- ∇  ðψ U Þ þ D ∇2 ψ dX 3 :

ð2:5Þ

Furthermore, we have the following relations: Γ2 Γ1

∇  ðψ U Þ dX 3 = ∇12 

Γ2 Γ1

ψ U 12 dX 3 - n  ðψ UÞjΓΓ21 ,

ð2:6Þ

and Γ2 Γ1

2

2

∂ ψ ∂ dX 3 = ∂X β ∂X β ∂X α ∂X α ðβ = 1, 2,

Γ2 Γ1

ψ dX 3 þ B,

ð2:7Þ

α = 1, 2, 3Þ

where ∇12 = ð∂=∂X 1 , ∂=∂X 2 Þ is the gradient operator in the plane ðX 1 , X 2 Þ, U 12 = ðU 1 , U 2 Þ is the velocity vector in the plane ðX 1 , X 2 Þ, and n = ð∂X 3 = ∂X 1 , ∂X 3 =∂X 2 , - 1Þ is the normal vector of the control surface. The boundary term B in Eq. (2.7) is related to ψ on the control surfaces and the derivatives of the surfaces, i.e., B = - n  ∇ψjΓΓ21 - ∇12  ψjΓ2 ∇12 Γ2 þ ψjΓ1 ∇12 Γ1 :

ð2:8Þ

Unlike solid surfaces, for the control surfaces Γ1 and Γ2 of the laser sheet, the boundary terms in Eq. (2.8) do not necessarily vanish. We introduce the light-path-averaged velocity weighted with the dye density ψ, i.e.,

18

2

hU12 iψ =

Γ2 Γ1

ψ U 12 dX 3 =

Γ2 Γ1

Velocity from Flow Visualizations

ψ dX 3 :

ð2:9Þ

Substitution of Eqs. (2.6)–(2.9) into Eq. (2.5) yields a projected motion equation, i.e., ∂g þ ∇12 g hU 12 iψ = D ∇212 g þ D c B þ c n  ðψ UÞjΓΓ21 , ∂t

ð2:10Þ

2

where g = L=L0 is the normalized radiance and ∇212 = ∂ =∂X β ∂X β (β = 1, 2) is the Laplace operator on the projection plane ðX 1 , X 2 Þ. The boundary terms in Eq. (2.10) serve as a source term. The last term in the right-hand side (RHS) of Eq. (2.10) represents the accumulating effect of the dye moving across the boundary surfaces in the illuminated volume. According to the discussions on the radiometric projection in Sect. 2.1, the quantity g can be expressed as the image intensity ratio, i.e., g = L=L0 = I=I 0 .

2.2.1.2

Compressible Flow

PLIF has been used as a global diagnostic tool in compressible flows and flames (Collins and Jacobs 2002; Bhuiyan et al. 2010). For a compressible flow where the continuity equation is ∂ρ=∂t þ ∇  ðρU Þ = 0, we have ∇  U = - ρ - 1 ð∂ρ=∂t þ U  ∇ρÞ,

ð2:11Þ

U  ∇ψ = ∇  ðψ U Þ þ ψ ð∂=∂t þ U  ∇Þ ln ρ:

ð2:12Þ

Further, we have Γ2 Γ1

U  ∇ψ dX 3 = c - 1 ∇12  ðL=L0 ÞhU 12 iψ - n  ðψ U ÞjΓΓ21 þc

-1

:

ð2:13Þ

ðL=L0 ÞF K, ρ, hU 12 iψ , hU 3 iψ

Here, the function F is defined as F K, ρ, hU12 iψ , hU 3 iψ = C 1 ∂K ρ =∂t þ C 2 hU 12 iψ  ∇12 K - ln ρjΓ2 ∇12 Γ2 þ ln ρjΓ1 ∇12 Γ1 , þC 3 hU 3 iψ ln ρjΓ2 - ln ρjΓ1 where the quantity K is

ð2:14Þ

2.2

Projected Motion Equations

19 Γ2

K=

Γ1

ln ρ dX 3 ,

ð2:15Þ

and the correlation coefficients are defined as Γ2

C1 =

Γ2 Γ1

Γ1

ψ dX 3 Γ2

C2 =

Γ2 Γ1

Γ1

Γ2 Γ1

Γ2 Γ1

, ∂ ln ρ=∂t dX 3

ψU12  ∇12 ln ρ dX 3 Γ2

ψU 12 dX 3  Γ2

C3 =

ψ∂ln ρ=∂t dX 3

Γ1

Γ1

, ∇12 ln ρ dX 3

ψU 3 ∂ln ρ=∂X 3 dX 3

ψU 3 dX 3

Γ2 Γ1

: ∂ ln ρ=∂X 3 dX 3

Using the above relations, Eqs. (2.11)–(2.13), we obtain a projected motion equation for a compressible flow ∂g þ ∇12  g hU 12 iψ = D ∇212 g þ D c B þ c n  ðψ U ÞjΓΓ21 ∂t , - g F K, ρ, hU 12 iψ , hU 3 iψ

ð2:16Þ

where g = L=L0 = I=I 0 is the normalized image intensity. The effect of the density variation in a compressible flow is clearly seen in Eq. (2.16). When the compressibility effect vanishes, Eq. (2.16) is reduced to Eq. (2.10) for an incompressible flow. In particular, when the control surfaces are planar, i.e., X 3 = Γ1 = const: and X 3 = Γ2 = const:, the image plane is parallel to the laser sheet. Furthermore, the laser sheet is so thin that the air density ρ is approximately the same between Γ1 and Γ2 , and thus K ≈ H ln ρ, where H is the width between the control surfaces. In this case, Eq. (2.16) becomes

20

2

Velocity from Flow Visualizations

∂g þ ∇12  g hU 12 iψ = D ∇212 g þ D c B þ c n  ðψ UÞjΓΓ21 ∂t : ∂ρ - H ρ - 1 g C1 þ C2 hU12 iψ  ∇12 ρ ∂t

2.2.2

Images of Density-Varying Flow

2.2.2.1

Schlieren Image

ð2:17Þ

The Schlieren technique has been widely used for visualizations of density-varying flows such as compressible aerodynamic flows and natural convection flows (Settles 2001; Settles and Hargather 2017). Figure 2.3 illustrates transmittance of a light ray through a density-varying flow. Here, the coordinate system is set at the center of the test section of a wind tunnel, in which X 2 is in the mean flow direction and X 3 is in the normal direction to the window walls along the light ray. The planar control surfaces are given by X 3 = Γ1 ðX 1 , X 2 Þ = const: and X 3 = Γ2 ðX 1 , X 2 Þ = const:, through which light rays can pass. The image plane is parallel to the planar window walls. The image intensity I in a Schlieren image depends on the gradient of the fluid density ρ by the following relation (Goldstein and Kuehn 1996)

Fig. 2.3 Transmittance of a light ray through a densityvarying flow

2.2

Projected Motion Equations

21 Γ2

I - IK =C IK

Γ1

∂ρ dX 3 , ∂ X2

ð2:18Þ

where I K is the reference image intensity with the knife-edge inserted in the focal plane when no variation of the density exists in the test section and C is a coefficient related to the setting of a Schlieren system. The knife-edge is set to be normal to the gradient of the fluid density ∂ρ=∂ X 2 . Taking partial differentiation with respect to time in Eq. (2.18) and using the continuity equation ∂ρ=∂t þ ∇  ðρ U Þ = 0, we have -

Γ2

∂ ðI=I K - 1Þ =C ∂t

Γ1

∂ ½∇  ðρ UÞ dX 3 ∂ X2

ð2:19Þ

Fundamental theory of calculus leads to Γ2 Γ1

Γ2

∂ ∂ ½∇  ðρ U Þ  dX 3 = ∂ X2 ∂ X2

Γ1

∇  ðρ U Þ dX 3

∂X 3 - ∇  ðρ U Þ ∂ X2

X 3 = Γ2

:

ð2:20Þ

X 3 = Γ1

The integral in the first term in the RHS of Eq. (2.20) can be decomposed into Γ2 Γ1

Γ2

∇  ðρ U Þ dX 3 = ∇12 

Γ1

ρ U 12 dX 3 - n  ðρU ÞjΓΓ21 :

ð2:21Þ

From Eq. (2.18), we have ∂ I - IK = CI K ∂ X2

Γ2 Γ1

ρ dX 3 - ρ

∂X 3 ∂ X2

X 3 = Γ2

ð2:22Þ

, X 3 = Γ1

and further Γ2 Γ1

ρ dX 3 =

1 C þ

X2 X 20 X2 X 20

ðI=I K - 1Þ dX 2

∂X 3 ρ ∂ X2

X 3 = Γ2 X 3 = Γ1

dX 2 þ

Γ2 Γ1

,

ð2:23Þ

ρ dX 3 X 2 = X 20

where X 20 is a reference position in the coordinate X 2 . Define the light-path-averaged velocity weighted with the fluid density ρ as

22

2

hU 12 iρ =

Γ2 Γ1

Γ2

ρ U 12 dX 3 =

Γ1

Velocity from Flow Visualizations

ρ dX 3 :

ð2:24Þ

Substituting the above relations into Eq. (2.19), we obtain a projected motion equation ∂g þ ∇12  ghU12 iρ = f , ∂t

ð2:25Þ

where the measurable quantity is defined as X2

g=

X 20

ðI=I K - 1Þ dX 2 þ C

Γ2 Γ1

ρ dX 3

ð2:26Þ

, X 2 = X 20

and the source term is f =C

∂ ∂t

Γ2 Γ1

þC n 

ρ dX 3

ðρU ÞjΓΓ21

X 2 = X 20 X2

þ ∇12  ghU 12 iρ

X 2 = X 20

:

ð2:27Þ

X 2 = X 20

The flux term in f vanishes for the solid planar control surfaces (such as glass windows) where the zero-flux condition n  ðρU Þ = 0 is imposed. In some applications such as Schlieren imaging of buoyancy-driven flow in an open space and focused Schlieren imaging, the virtual control surfaces can be placed sufficiently far away from a measurement domain such that either the local velocity vanishes (U = 0) or the local velocity is parallel to the control surfaces where n  ðρUÞ = 0. In this case, the flux term vanishes also. The other terms are related to the values of the flow properties at the reference position X 2 = X 20 . When the reference position is at the incoming freestream where the fluid density and velocity are steady and spatially uniform, these terms can be neglected.

2.2.2.2

Shadowgraph Image

The image intensity I in a shadowgraph image depends on the second-order derivative of the fluid density ρ, i.e., (Goldstein and Kuehn 1996) I - IT =C IT

Γ2 Γ1

∇212 ρ dX 3 ,

ð2:28Þ

2.2

Projected Motion Equations

23

where I T is the initial image intensity, C is a coefficient related to the setting of a 2 2 shadowgraph system and ∇212 = ∂ =∂X 21 þ ∂ =∂X 22 is a Laplace operator. When the control surfaces are planar, i.e., X 3 = Γ1 = const: and X 3 = Γ2 = const:, taking partial differentiation with respect to time in Eq. (2.28) and using the continuity equation ∂ρ=∂t þ ∇  ðρ U Þ = 0, we have -

1 ∂ðI=I T - 1Þ = ∇212 ∇12  C ∂t

Γ2 Γ1

ρ U 12 dX 3

:

ð2:29Þ

From Eq. (2.28), we have a Poisson equation for the integral of the fluid density: ∇212

Γ2 Γ1

ρ dX 3 = C - 1 ðI=I T - 1Þ:

ð2:30Þ

The integral of the fluid density can be obtained by solving the Poisson equation ∇212 g = I=I T - 1 with suitable boundary conditions. Thus, a projected motion equation is symbolically expressed as ∂g þ ∇12  ghU 12 iρ = C n  ðρ U ÞjΓΓ21 , ∂t

ð2:31Þ

-2 -2 where g = ∇12 ðI=I T - 1Þ is a symbolic solution of the Poisson equation and ∇12 is an inverse operator of the Poisson equation. The actual form of the projected motion equation is more complicated than the symbolic one since the boundary conditions for the Poisson equation should be naturally and explicitly incorporated into the equation.

2.2.2.3

Transmittance Image

For certain imaging system like a collimated monochromatic X-ray system, the image intensity in a transmittance image depends on the fluid density ρ (Wildes et al. 2000): I - IT =C IT

Γ2 Γ1

ρ dX 3 ,

ð2:32Þ

where I T is the initial image intensity and C is a coefficient related to the setting of an imaging system. For the planar control surfaces, using the relations described before, we obtain

24

2

Velocity from Flow Visualizations

∂g þ ∇12  ghU 12 iρ = C n  ð ρ U ÞjΓΓ21 , ∂t

ð2:33Þ

where g = I=I T - 1 is the normalized image intensity.

2.2.3

Transmittance Image Through Scattering Particulate Flow

There are particulate flows in natural environments and engineering systems such as pyroclastic flows in volcanic eruptions, cloud motions, and smoke motions visualizing air flows in wind tunnels. The disperse phase number equation for particulate flow is (Brennen 2005) ∂N þ ∇  ðN U Þ = 0, ∂t

ð2:34Þ

where U = ð U 1 , U 2 , U 3 Þ is the particle velocity and N is the number of particles per unit total volume which is given by the particle distribution function np ðaÞ of the particle diameter a, i.e., 1

N= 0

np ðaÞ da = np a :

ð2:35Þ

The operator hia denotes an integral over the entire range of particle sizes. The light transmittance/scattering through particles depends on the particle distribution. There are three coefficients that generally describe the particle scattering and absorption properties. The particle scattering coefficient is defined as 1

σ= 0

Csca np ðaÞ da = wsca hCsca ia N,

where Csca is the scattering cross section and wsca is a correlation coefficient. Similarly, the absorption and extinction coefficients of particles are, respectively, given by 1

κ= 0

and

Cabs np ðaÞ da = wabs hC abs ia N,

2.2

Projected Motion Equations

25

Fig. 2.4 Transmittance of a light ray through a scattering particulate flow

1

β= 0

C ext np ðaÞda = wext hC ext ia N,

where C abs and Cext are the absorption and extinction cross sections, respectively, and wabs and wext are the corresponding correlation coefficients for the absorption and extinction. In transmittance of a light ray through scattering particulate flow, as shown in Fig. 2.4, a portion of light transmits through the medium, and some light is absorbed and scattered by particles. For a 1D plane medium, the equation of radiative transfer for the radiance L in the particulate medium along a ray defined by the unit vector s (or polar angle θ) is (Pomraning 1973; Modest 1993) dL þ L = Sðτ, sÞ, dτ

ð2:36Þ

where the source function Sðτ, sÞ is Sðτ, sÞ = ð1 - ωÞLb þ

ω 4π

Lðsi Þ Φðsi , sÞdΩi ,

ð2:37Þ

4π

and the optical coordinate along a ray is s

τ=

β ds:

ð2:38Þ

0

The integral over the solid angle of 4π in the above relation represents the gain of radiative energy by the beam due to radiation incident to particles from all directions in the spherical space that is scattered by particles. The scattering phase function Φðsi , sÞ, which defines a directional distribution, represents the probability that the incident radiation along the direction si will be scattered into a certain another

26

2 Velocity from Flow Visualizations

direction s. In Eq. (2.37), Lb is the blackbody emission intensity. The single scattering albedo is ω=

wsca hCsca ia σ σ , = = β σþκ wext hCext ia

which is independent of the number of particles. The formal solution of Eq. (2.36) gives an integral equation: τ

LðτÞ = L0 expð- τÞ þ

Sðτ0 , sÞ exp½ - ðτ - τ0 Þdτ0 :

ð2:39Þ

0

Due to the fixed-point form of Eq. (2.39), given an initial solution Lð0Þ ðτÞ, the Picard iteration approximation is used, i.e., τ

Lðnþ1Þ ðτÞ = L0 expð- τÞ þ

SðnÞ ðτ0 , sÞ exp½ - ðτ - τ0 Þdτ0 ,

ð2:40Þ

0

where SðnÞ ðτ, sÞ = ð1 - ωÞLb þ

ω 4π

LðnÞ ðsi Þ Φðsi , sÞdΩi : 4π

When the Picard iteration converges, we have LðnÞ ðτÞ → LðτÞ and SðnÞ ðτÞ → Sð1Þ ðτÞ as n → 1: Therefore, the solution of Eq. (2.40) can be symbolically written as τ

LðτÞ = L0 expð- τÞ þ

Sð1Þ ðτ0 , sÞ exp½ - ðτ - τ0 Þdτ0 :

ð2:41Þ

0

The radiance LðτÞ along a certain direction depends on the time and position in a particulate flow described by Eq. (2.34). The source function of isotropic scattering particles can be obtained by solving an integral equation (Modest 1993). We consider the problem in the camera object-space coordinate system ðm1 , m2 , m3 Þ. The particulate flow is confined between the control surfaces X 3 = Γ 1 ðX 1 , X 2 Þ and X 3 = Γ 2 ðX 1 , X 2 Þ: The light ray direction through the particulate flow to an observer is parallel to the optical axis of a camera (the X 3 axis). The camera detects the light through the flow from the opposite side of an illuminating source. In this case, when τ = 0 corresponds to X 3 = Γ1 while τ corresponds to X 3 = Γ2 , the optical depth is given by

2.2

Projected Motion Equations

τH =

Γ2 Γ1

27

β dX 3 = wext hC ext ia

Γ2 Γ1

N ðt, XÞ dX 3 :

ð2:42Þ

The radiance transmitted through the particulate flow is given by τH

LðτH Þ = L0 expð- τH Þ þ

Sð1Þ ðτ0 , sÞ exp½ - ðτH - τ0 Þdτ0 :

ð2:43Þ

0

Differentiation of Eq. (2.43) with respect to time and use of Eq. (2.41) yields ∂LðτH Þ = wext hCext ia Sð1Þ ðτH Þ - LðτH Þ ∂t

Γ2 Γ1

∂N ðt, XÞ dX 3 : ∂t

ð2:44Þ

Further, using the disperse phase number equation for the particulate flow, we have Γ2 Γ1

∂N ðt, X Þ dX 3 = ∂t

Γ2 Γ1

∇  ðN U Þ dX 3

= - ∇12 

Γ2 Γ1

: N U 12 dX 3 þ n 

ð2:45Þ

ðN UÞjΓΓ21

Introducing the light-path-averaged particle velocity in terms of the particle number hU 12 iN =

Γ2 Γ1

N U12 dX 3 =

Γ2 Γ1

N dX 3 ,

ð2:46Þ

we have a projected motion equation relating the radiance to the average particle velocity, i.e., ∂LðτH Þ þ Sð1Þ ðτH Þ - LðτH Þ ∇12  hU 12 iN τH ∂t = wwxt hC ext ia Sð1Þ ðτH Þ - LðτH Þ n  ðN U ÞjΓΓ21 :

ð2:47Þ

Since the optical depth τH can be converted to the radiance through Eq. (2.43), Eq. (2.47) gives a projected motion equation relating the radiance emitted from the particulate flow to the path-averaged particle velocity. In particular, for an optically thin medium (τH L - ð0Þ, g is approximately the normalized image intensity directly responding to the scattered radiance. For many applications where the blackbody emission is important like satellite infrared imaging, the above derivations and results are still valid once Qsca is replaced by Qsca þ Qbb .

2.2.5

Laser Sheet-Illuminated Particle Image

In typical planar PIV, a thin laser sheet is used to illuminate particles seeded in flows, which is viewed by a camera perpendicularly (Adrian 1991; Raffel et al. 2007; Adrian and Westerweed 2011). The particles illuminated by the laser sheet are between the virtual planar control surfaces Γ1 and Γ2 . The scattering radiance from particles is proportional to an integral of the number of particles per volume across the laser sheet, i.e., LðX 1 , X 2 , t Þ = C L0 ðX 1 , X 2 , X 3,m Þ

Γ2 Γ1

N ðX, t ÞdX 3 ,

ð2:56Þ

where C is the scattering cross section, L0 is the mean laser sheet radiance distribution that is known after intensity attenuation due to absorption and scattering is measured a priori, and X 3,m denotes the position of the laser sheet. Following the similar procedures described in Sect. 2.2.1, we have ∂g þ ∇12  ghU12 iN = C n  ðN UÞjΓΓ21 , ∂t

ð2:57Þ

where the normalized image intensity is g = L=L0 = I=I 0 . The boundary term serving as a source in the RHS of Eq. (2.57) represents the contribution from particles that move across the laser sheet boundary surfaces and accumulate within the laser sheet. The particle accumulation in a laser sheet, which is often neglected in planar PIV, has been long recognized as an error source, and its effect on the determination of the velocity is explicitly shown in Eq. (2.57). The radiance projected onto the plane ðX 1 , X 2 Þ from discrete particles in the object space is ideally described by

2.2

Projected Motion Equations

31 M

g=

gi ,

ð2:58Þ

i=1

where the radiance from each particle is modeled by a Gaussian distribution: gi =

ðX 1 - X 1,i Þ2 þ ðX 2 - X 2,i Þ2 1 exp : 2σ 2i 2πσ 2i

The object-space coordinates ðX 1,i , X 2,i Þ give the centroid location of the ith particle that is a function of time, while the standard deviation σ i defines the particle size. Substitution of Eq. (2.58) into Eq. (2.57) yields M

dX 1,i gi ðX 1 - X 1,i Þ - hU 1 iN η dt i=1 i þηi max

σ 2i

þ ðX 2 - X 2,i Þ

∇12  hU 12 iN = max

σ 2i

Cn 

dX 2,i - hU 2 iN dt ,

ð2:59Þ

½NU ΓΓ21

where ηi = σ 2i = max σ 2i . When max σ 2i → 0 while ηi remains finite, gi approaches to a Dirac-delta function, i.e., gi → δðX 1 - X 1,i , X 2 - X 2,i Þ: In this case, a relation between the velocity of an individual particle and the lightpath-averaged velocity field is dX 1,i dX 2,i , = δðX 1 - X 1,i , X 2 - X 2,i Þ hU 12 iN : dt dt

ð2:60Þ

Equation (2.60) manifests a lucid connection between the discrete velocity in PIV and the continuous light-path-averaged velocity field in the differential formulation of the projected motion equation. More relevant discussions are given from a variational perspective in Sect. 2.3.4.

2.2.6

Neutron Radiography Image

Neutron radiography (NR) has emerged as a useful technology for radiation diagnosis to understand complex two-phase flow phenomena (Kardjilov et al. 2018; Strobl et al. 2009). Owing to neutrons’ strong attenuation by hydrogen in hydrogenrich fluids while relative insensitivity to both gas and solid phases, NR provides noninvasive and nondestructive detection of water or other hydrogen-rich fluids in media. Therefore, NR is an ideal imaging visualization technique to study multiphase flows.

32

2

Velocity from Flow Visualizations

Fig. 2.5 Schematic of neutron radiography for two-phase flow medium (object), where two control surfaces and a coordinate system are indicated. (From Liu et al. (2021c))

Liu et al. (2021c) derived a projected motion equation for NR images of the air-water two-phase flow. A NR setup has three main components: (1) neutron beam from a source, (2) object to be radiographed, and (3) detector for the radiation intensity information associated with the neutron beam transmitted through the studied object (Harms and Wyman 1984). Figure 2.5 is a schematic of the three components in NR for an object. The radiative fluxes through the collimator plane, object plane, and image plane are considered, and for simplicity of analysis, these idealized planes are parallel to the coordinate plane ðX, Y Þ as shown in Fig. 2.5. The collimator plane is a virtual plane through which the neutron beam from a collimator device passes, and the neutron flux Φ0 ðX, Y Þ generally represents the spatial distribution of neutrons directing toward the object. The object plane represents the object of radiographic interest, and the associated neutron flux Φt ðX, Y Þ describes the spatial distribution of neutrons after passing through the object. The image plane represents the NR image, and ΨðX, Y Þ describes the secondary radiation from a converter which is detected by a photosensor. Sequential processes Φ0 ðX, Y Þ → Φt ðX, Y Þ → ΨðX, Y Þ are modeled mathematically below. For an ideal uniform collimated neutron beam, Φ0 ðX, Y Þ is a constant, which is commonly called the neutron beam flux (n/m2-s). When a collimated beam of neutrons impinges onto an absorbing object, the main physical processes are neutron absorption and scattering interactions. Therefore, neutrons in the beam will either be absorbed and scattered or continue to pass through the object without collision. As shown in Fig. 2.5, we consider an object confined by two virtual control surfaces: Z = Γ1 ðX, Y Þ and Z = Γ2 ðX, Y Þ: The flux of neutrons passing through the object is

2.2 Projected Motion Equations

33

Φt ðX, Y, t Þ = Φ0 ðX, Y Þ exp -

Γ2 Γ1

βobj ðX, t Þ dZ ,

ð2:61Þ

where Φ0 is the incident flux of neutrons from a collimator device, X = ðX, Y, Z Þ is the spatial coordinates, βobj = βabs þ βsca is the object extinction coefficient, and βabs and βsca are the coefficients for the absorption and scattering processes, respectively. The neutron converter (scintillator) consists of a material layer, as shown in Fig. 2.5, which absorbs the neutron flux, and then through scintillation emits the luminescent radiation (secondary radiation) that can be recorded using an imaging sensor. A photosensor could be placed next to the scintillator that converts the neutrons to visible photons. This is the case when the scintillator is in direct contact with the imaging chip. In other case, the secondary radiation could be detected using a CCD or CMOS imaging detector through a mirror system. Nevertheless, mathematical modeling of radiative transport for these arrangements is the same. The neutron flux in the converter at a location Z is ΦðX, Y, Z, t Þ = Φt ðX, Y Þ expð- βc ðZ - Z 0 ÞÞ, where βc is the extinction coefficient of the homogeneous converter material that is a constant and Z 0 is the position of the front surface of the converter. Thus, the yield rate of the secondary radiation excited by the incident neutrons in the converter is Y ðX, Y, Z, t Þ = ε1 Φt ðX, Y Þ expð- βc ðZ - Z 0 ÞÞ, where ε1 is a coefficient. The total flux of the secondary radiation from the converter is given by an integral through the converter, i.e., ΨðX, Y, t Þ =

Z1 Z0

Y ðX, Y, Z, t ÞPðZ ÞdZ,

ð2:62Þ

where PðZ Þ is the probability that a secondary radiation photon produced at Z reaches to Z 0 . The secondary radiation is isotropic, which generally attenuates exponentially in a material, and thus the probability function is given by PðZ Þ = expð- μðZ - Z 0 ÞÞ, where μ is a constant coefficient for attenuation. Evaluation of Eq. (2.62) yields ΨðX, Y, t Þ = B Φt ðX, Y, t Þ, where the proportional coefficient is expressed as

34

2

Velocity from Flow Visualizations

B = ε1 ðβc þ μÞ - 1 ½1 - expð- ðβc þ μÞðZ 1 - Z 0 ÞÞ: For a photosensor with a linear response function, the image intensity is proportional to the received flux of the secondary radiation, i.e., I = ε2 Ψ, where ε2 is a coefficient related to the light collection and visible light detection efficiency of the photosensor. Therefore, we have I ðX, Y, t Þ = I 00 ðX, Y Þ exp -

Γ2 Γ1

βobj ðX, t Þ dZ ,

ð2:63Þ

where I 00 = ε2 BΦ0 is the image intensity without sample attenuation. In a two-phase flow without mass transfer between the phases, the continuity equation for the density of a specific component is generally given by (Brennen 2005) ∂θ=∂t þ ∇  ðθ U Þ = 0,

ð2:64Þ

where U is the transport velocity of a component, and θ = θi (i = 1, 2) denotes the density for the phase 1 or phase 2. Integration of Eq. (2.62) from Γ1 ðX, Y Þ and Γ2 ðX, Y Þ in the Z-direction yields ∂ ∂t

Γ2 Γ1

θ dZ þ ∇12 

Γ2 Γ1

θ U 12 dZ - n  ðθ U ÞjΓΓ21 = 0:

ð2:65Þ

where ∇12 = ð∂=∂X, ∂=∂Y Þ is the gradient operator in the coordinate plane ðX, Y Þ, U 12 = ðU 1 , U 2 Þ is the vector U in the plane ðX, Y Þ, and n is the normal vector of the control surface. The extinction coefficient βobj is decomposed into two parts, i.e., βobj = γ θ þ β0W , where γ is a coefficient and β0W is the extinction coefficient of a medium without water content. We introduce a quantity defined as g = ln

I 0W =γ I

Γ2 Γ1

θðX, t Þ dZ,

ð2:66Þ

where I 0W is the image intensity for the medium without water content. The physical meaning of g is clear according to Eq. (2.66), which is proportional to the integrated water content density along a neutron beam. Sometimes, g = lnðI 0W =I Þ is called the optical density (Abd et al. 2005). Since I 0W =I ≥ 1, the optical density g = lnðI 0W =I Þ is positive. Using Eqs. (2.63)–(2.66), we obtain a projected motion equation:

2.3

Optical Flow and Variational Method

35

∂g þ ∇12  g hU 12 iθ = γ n  ðθ U ÞjΓΓ21 , ∂t

ð2:67Þ

where the beam-path-averaged weighted velocity is defined as hU 12 iθ =

Γ2 Γ1

θ U 12 dZ=

Γ2 Γ1

θ dZ :

ð2:68Þ

When a neutron beam passes through a mixed two-phase domain, the density is expressed as θ = θi ðxÞ for x 2 Di (i = 1, 2), where Di denotes a domain of the phase i. In this case, hU 12 iθ represents an averaged velocity of the mixed two-phase domain along the neutron beam.

2.3 2.3.1

Optical Flow and Variational Method Optical Flow Equation

The projected motion equations derived in Sects. 2.2.1, 2.2.2, 2.2.3, 2.2.4, 2.2.5, and 2.2.6 for the typical flow visualizations have a generic form of the transport equation, i.e., ∂g þ ∇12  g hU12 iψ = f ðX 1 , X 2 , gÞ, ∂t

ð2:69Þ

where g = gðI Þ is a function of the normalized image intensity that is proportional to the radiance received by a camera, and ∇12 = ð∂=∂X 1 , ∂=∂X 2 Þ is the gradient operator in the plane ðX 1 , X 2 Þ. The specific form of gðI Þ depends on a specific flow visualization technique. The field quantity ψ is the scalar (dye) concentration in flows, fluid density in density-varying flows, or particle number per unit total volume for particulate flows. In Eq. (2.69), the light-path-averaged velocity weighted with a field quantity ψ related to a visualized medium is defined as hU12 iψ =

Γ2 Γ1

ψ U 12 dX 3 =

Γ2 Γ1

ψ dX 3 ,

ð2:70Þ

where U12 = ðU 1 , U 2 Þ is the velocity projected onto the coordinate plane ðX 1 , X 2 Þ of the fluid or particle velocity U = ðU 1 , U 2 , U 3 Þ in the object-space frame ðm1 , m2 , m3 Þ associated with a camera (see Sect. 2.1). As shown in Fig. 2.1, the visualized flow domain is confined by two control surfaces

36

2

Velocity from Flow Visualizations

X 3 = Γ1 ðX 1 , X 2 Þ and X 3 = Γ2 ðX 1 , X 2 Þ that could be virtual or solid. In many cases, the planar control surfaces X 3 = Γ1 = const: and X 3 = Γ2 = const: are considered. The source term f in Eq. (2.69) depends on a specific flow visualization technique. For typical flow visualizations with particles and tracers where the diffusion is neglected, f / n  ðψ UÞjΓΓ21 , which represents the accumulation effect of the quantity ψ in the volume across the control surfaces. For the solid control surfaces (e.g., glass windows), the zero-flux condition n  ðψ U ÞjΓΓ21 = 0 is imposed. For the virtual control surfaces confining a laser sheet, the term n  ðψ U ÞjΓΓ21 represents the rate of accumulation of ψ within a laser sheet. In planar PIV measurements, it is often neglected and treated as an error source. In addition, for a laser sheet, n  ðψ UÞjΓΓ21 can be neglected when the transport of ψ across the thin laser sheet (or the visualized domain) is conserved. Therefore, for laser sheet visualization, a reasonable approximation is that f can be neglected. For light transmittance through a flow in an open space, the virtual control surfaces can be placed sufficiently far away from the flow such that n  ðψ U ÞjΓΓ21 = 0 is satisfied. In general, f = 0 is considered as a first-order approximation for some applications. By using the perspective projection transformation, the gradient operator and Laplace operator can be expressed in the image coordinates, i.e., 2

2

∂=∂X β = λ ∂=∂ xβ and ∂ =∂X β ∂X β = λ2 ∂ =∂xβ ∂xβ

ðβ = 1, 2Þ,

where λ is a scaling factor. The velocity in the image plane, which is referred to as the optical flow, is defined as u = ðu1 , u2 Þ = λhU12 iψ : Therefore, Eq. (2.69) can be written as the physics-based optical flow equation in the image plane (Liu and Shen 2008): ∂g þ ∇  ð g uÞ = f , ∂t

ð2:71Þ

where ∇ = ∂=∂xβ is the gradient operator in the image plane, g is the normalized image intensity related to a measured physical quantity mapped onto the image plane, and f is a source term. The physical meaning of the optical flow u is clear, that is, the optical flow is proportional to the light-path-averaged velocity of fluid or particles in flow visualizations. In a special case where ∇  u = 0 and f = 0, Eq. (2.71) has the same form as the Horn-Schunck brightness constraint equation for the optical flow in computer vision, i.e., (Horn and Schunck 1981)

2.3

Optical Flow and Variational Method

37

∂g þ u  ∇g = 0: ∂t However, for flow velocity measurements, the optical flow u is not divergence-free. According to Eq. (2.70), the definition of the optical flow, u = λhU12 iψ , naturally provides a pair of integral equations for the projected velocity components m1  U and m2  U along a light ray in the object space. In particular, when ψ is approximately treated as a constant, the integral equations only contain the unknown U, while m1 and m2 are known for a calibrated/oriented camera and u is determined. Therefore, the determination of U in a domain in the object space is a tomographic reconstruction problem (a classical inverse problem). In principle, when there are sufficient equations along different rays for multiple cameras with different viewing angles, direct tomographic reconstruction of U in a 3D domain from the optical flow is feasible.

2.3.2

Variational Method

To determine the optical flow in Eq. (2.71), a variational formulation was proposed by Liu and Shen (2008), where the first-order Tikhonov functional was used for ill-posed problems (Tikhonov and Arsenin 1977). Given g and f , we define a functional J ðuÞ =

2

D

½∂g=∂t þ ∇  ðg uÞ - f  dx1 dx2

þα

, D

j∇u1 j2 þ j∇u2 j2

ð2:72Þ

dx1 dx2

where α is a Lagrange multiplier and D is a domain in the image plane. To minimize J ðuÞ (i.e., J ðuÞ → min ), introducing a smooth perturbation test function (variation) v = ðv1 , v2 Þ, we have the optimality condition, i.e., dJ ðu þ εvÞ dε

ε=0

= -2 þ2 -α

D

D

D

α g∇w þ ∇2 u  v dx1 dx2 2 ∇  ðgwvÞdx1 dx2 þ α u  ∇2 v dx1 dx2 = 0

where the equation term is defined as

D

∇2 ðu  vÞ dx1 dx2

ð2:73Þ

38

2

Velocity from Flow Visualizations

w = ∂ g=∂ t þ ∇  ð g uÞ - f , and ε is a small amplitude. The derivation of Eq. (2.73) uses the following equalities: 2∇u1  ∇v1 = ∇2 ðu1 v1 Þ - u1 ∇2 v1 - v1 ∇2 u1 , 2∇u2  ∇v2 = ∇2 ðu2 v2 Þ - u2 ∇2 v2 - v2 ∇2 u2 , w∇  ð gvÞ = ∇  ð wgvÞ - gv  ∇w: According to Green’s theorem, for a vector field A, we have

D

∇  A dx1 dx2 =

∂D

n  A dx1 ,

ð2:74Þ

where n = ðdx2 =dx1 , - 1Þ is a normal vector on a closed boundary ∂D of the image domain D. To evaluate the terms in Eq. (2.73), applying Eq. (2.74), we have

D

∇  ðgwvÞdx1 dx2 =

∂D

n  ðgwvÞdx1 = 0,

ð2:75Þ

since the optical flow equation is satisfied on the boundary ∂D i.e., w = 0. Furthermore, we have

D

∇2 ðu  vÞ dx1 dx2 =

∂D

ðn  ∇uÞ  v dx1 þ

∂D

ðn  ∇vÞ  u dx1 :

ð2:76Þ

The perturbation test field is considered, which satisfies the Laplace equation ∇2 v = 0 with the Neumann condition n  ∇v = 0 on ∂D. Further, the Neumann condition n  ∇u = 0 is imposed on ∂D for the optical flow. The last three terms in the RHS in Eq. (2.73) vanish, and only the first term remains. Since the domain D is arbitrary and v is nonzero, the optimality condition dJ ðu þ εvÞ=dεjε = 0 = 0 leads to the Euler-Lagrange equation: g∇ 2

∂g þ ∇  ðguÞ - f þ α ∇2 u = 0, ∂t

ð2:77Þ

where ∇2 = ∂ =∂xi ∂ xi (i = 1, 2) is the Laplace operator in the image plane. Clearly, the Euler-Lagrange equation with the Neumann condition n  ∇u = 0 gives an approximate optimal solution for the optical flow. The standard finite difference method can be used to solve Eq. (2.77) (Wang et al. 2015). The mathematical analysis of the optical flow was given by Aubert et al. (1999) and Wang et al. (2015). The discussions in this section provide the physical foundations of the optical flow method (OFM) applied to global flow diagnostics. An open-source Matlab program “OpenOpticalFlow” was described by Liu (2017) for optical flow

2.3

Optical Flow and Variational Method

39

computation, where the Horn-Schunck estimator was used for an initial approximation and the Liu-Shen estimator for the refined calculation (see Appendix B). Furthermore, OFM integrated with the cross-correlation method for PIV was developed by Liu et al. (2020), and an open-source Matlab program was described by Liu and Salazar (2021) (see Appendix B).

2.3.3

Error Analysis

The Euler-Lagrange equation allows a systematic error analysis of OFM (Liu and Shen 2008). In a sensitivity analysis, the image intensity and optical flow are decomposed into a basic solution and an error, i.e., g = go þ Δg and u = uo þ Δu, where go and uo satisfy exactly Eq. (2.77), Δu is the resulting error in optical flow computation, and Δg is a variation (or difference) in the image intensity measurement. By substituting g = go þ Δg and u = uo þ Δu into Eq. (2.77) and neglecting higher-order small terms, an error propagation equation can be obtained, where the elemental error sources are Δð∂g=∂t Þ, Δð∇gÞ, Δð∇  uÞ and Δg. To understand the error limitation of OFM, an ideal case is considered, where the elemental errors Δð∇gÞ, Δð∇  uÞ, and Δg vanish, and the optical flow error Δu is mainly produced by Δð∂g=∂t Þ. For the first-order time difference, an estimate is given by Δð∂g=∂t Þ ffi - gtt Δ t=2, 2

where gtt = ∂ g=∂t 2 denotes the second-order time differentiation and Δ t is a time interval between two consecutive images. In this case, an estimate for the error of the optical flow is given by g∇ - k∇gk - 1 Δt gtt =2 þ k∇gk - 1 ∇g  ðΔuÞ þαk∇gk - 1 ∇2 ðΔuÞ = 0

,

ð2:78Þ

where k∇gk is a characteristic intensity gradient magnitude for normalization and kk is the L2 -norm in a given domain. The term k∇gk - 1 Δt gtt that represents an elemental error in the time differentiation is particularly interesting. Since Δ t cannot be zero and k∇gk cannot be infinitely large, the product k∇gk - 1 Δ t must be finite, i.e.,

40

2

Velocity from Flow Visualizations

Δt k∇gk - 1 = δ,

ð2:79Þ

where δ is a small positive constant. Hence, according to Eq. (2.79), a finite optical flow error Δu always exists, which imposes an intrinsic limit on OFM. In general, a smaller time interval and a larger-intensity gradient would lead to a smaller error in optical flow computation. Furthermore, the presence of the factor αk∇gk - 1 in Eq. (2.78) indicates that for images with a small intensity gradient magnitude k∇gk, the Lagrange multiplier α must be small in order to control the error of the optical flow. There is no rigorous theory for determining the Lagrange multiplier in the variational formulation of the optical flow equation. The Lagrange multiplier acts like a diffusion coefficient in Eq. (2.77). Therefore, a larger Lagrange multiplier tends to smooth out finer flow structures. In general, the smallest Lagrange multiplier that still leads to a well-posed solution is selected. However, within a considerable range of α, the solution is not significantly sensitive to its selection. Simulations based on a synthetic velocity field for a specific measurement are usually carried out to determine the Lagrange multiplier by using an optimization scheme. In addition to the above consideration of the error propagation, a general constraint is related to the intrinsic error of the finite difference approximation: uðt, xÞ = Δx=Δt þ RðΔt, xÞ, where RðΔt, xÞ ≈ 0:5ðdu=dt ÞΔt is the remainder. Using an approximation kdu=dt k ≈ k∇uk kuk, we have an estimate kRk ≈ 0:5Δ t kuk k∇uk ≈ 0:5 kΔxk k∇uk, where kΔxk is the characteristic displacement of flow structures in the image plane. The time interval is estimated as Δt  kΔxk=kuk. Combining the above estimated errors, we have an estimate for the total error of optical flow computation, i.e., (Liu et al. 2015b) ε = kΔxk

c1 þ c2 k∇uk2 , 2 2 k∇gk kuk

ð2:80Þ

where c1 and c2 are coefficients to be determined. According to Eq. (2.80), the main parameters related to the error of the optical flow are the displacement kΔxk, the image intensity gradient k∇gk, the velocity gradient k∇uk, and the velocity magnitude kuk. Eq. (2.80) indicates that the error is proportional to kΔxk, and small displacements are generally required for a good accuracy of OFM. The error in optical flow computation is a function of location depending on k∇gk and k∇uk. The error would be larger in the regions where k∇uk is large and k∇gk is smaller.

2.3

Optical Flow and Variational Method

2.3.4

41

Optical Flow and Particle Velocity

For the application of OFM to PIV images, it is necessary to further explore the mathematical connection between the optical flow and the particle velocity in the image plane. The intensity distribution of an image of M discrete particles can be ideally described by the linear superposition of many particles, i.e., M

gi ,

g=

ð2:81Þ

i=1

where the intensity of the ith particle is modeled by a Gaussian distribution gi =

Ii exp 2πσ 2i

-

x1 - x1,pðiÞ

2

þ x2 - x2,pðiÞ 2σ 2i

2

:

Here, the image coordinates xpðiÞ = x1,pðiÞ , x2,pðiÞ give the centroid location of the ith particle that is a function of time, the standard deviation σ i defines the particle size in the image, and I i is its intensity coefficient. The time derivative ∂g=∂t is given by ∂g = ∂t

M i=1

∂gi = ∂t

M i=1

gi x - xpðiÞ  upðiÞ , σ 2i

where upðiÞ = dxpðiÞ =dt is the particle velocity in the image plane. When σ i → 0, gi approaches to the Dirac-delta function, i.e., gi → I i δ x - xpðiÞ , and thus the intensity distribution of a particle image becomes very spiky and nonsmooth. Thus, in the case of PIV, the optical flow equation Eq. (2.71) is expressed as M

Gi - F = 0,

ð2:82Þ

i=1

where Gi =

gi x - xpðiÞ  upðiÞ - u , ηi M

F = max σ 2i

gi ∇  u - f , i=1

and ηi = σ 2i = max σ 2i represents the relative or normalized cross-section area of the ith particle.

42

2

Velocity from Flow Visualizations

For simplicity, the following unconstrained variational problem is considered, i.e., 2

M

J ðuÞ =

Gi - F

D

dx1 dx2 → min:

ð2:83Þ

i=1

Furthermore, we consider a limiting case where max σ 2i → 0, while ηi remains constant. In this case, when the image domain D is decomposed into N subdomains (or PIV interrogation windows), i.e., N

D=

Dk , k=1

by using the Cauchy-Schwarz inequality, the variational problem can be equivalently expressed as N

upðiÞ - u

J ðu Þ = k=1

2 Dk

→ min,

ð2:84Þ

where the special L2 -norm is defined as 1=2

M

upðiÞ - u

Dk

=

Dk

H ðDk Þ i=1

G2i dx1 dx2

,

and H ðDk Þ is the Heaviside function. A mathematically trivial but physically meaningful solution for Eq. (2.83) is upðiÞ - u

2 Dk

= 0 ðk = 1, 2, 3, . . . N Þ

ð2:85Þ

Equation (2.85) indicates that the particle velocity equals the optical flow in terms of the L2 -norm in the interrogation windows.

2.4 2.4.1

Applications Planetary Cloud Tracking

The atmospheres of planets (including Jupiter, Saturn, and Earth) have dynamic flow structures visualized by their clouds, which play important roles in physical processes related to heat, temperature, pressure, composition, and radiation (Ingersoll

2.4

Applications

43

et al. 2004; Ingersoll 2020). Because clouds provide passive tracers of winds, cloud tracking has been the primary method of measuring wind speeds in planetary atmospheres through Earth- and space-based remote sensing. It is highly desirable to develop robust cloud-tracking methods in planetary sciences to deduce wind velocity fields from trackable cloud features in time sequences of images obtained in planetary mission-based imaging science campaigns. OFM is particularly suitable for extraction of high-resolution wind velocity fields from cloud images, which has been used to study the structures of Jupiter’s Great Red Spot (GRS) (Liu et al. 2012b) and Saturn’s north polar vortex (Liu et al. 2019a). The critical evaluation of OFM compared with other cloud-tracking methods was given by Liu and Salazar (2022).

2.4.1.1

Jupiter’s Great Red Spot

Using OFM, Liu et al. (2012b) obtained high-resolution velocity fields from the Galileo 1996 (the first orbit of NASA’s Galileo spacecraft denoted by G1) cloud images of the GRS and then studied the intrinsic flow structures of the GRS. Figure 2.6 shows the two images of the GRS with a time interval of 1.2 h (4320 s) between them. A velocity field of 231 × 302 vectors is extracted from this pair of cloud images. Figure 2.7a shows velocity vectors of the GRS, revealing the counterclockwise-rotating near-elliptical high-speed jet called the high-speed collar. The velocity vectors in the inner region enclosed by the high-speed collar are shown in Fig. 2.7b, exhibiting low-speed near-2D turbulence in the orthographically projected plane. The zonal velocity profile across the GRS during the G1 orbit is shown in Fig. 2.8a. For comparison, Figure 2.8a also includes the data obtained by Vasavada et al. (1998) using both the manual tracking and correlation-based methods and by Choi et al. (2007) using the correlation-based method. The zonal velocity profile given by OFM is consistent with those obtained by using other methods. The

Fig. 2.6 Mosaics of the GRS taken by the Galileo spacecraft in 1996 (G1), where the time interval between the images (a) and (b) is 4320 s. (From Liu et al. (2012b))

44

2

Velocity from Flow Visualizations

Fig. 2.7 Velocity fields of the GRS extracted by using OFM: (a) global view and (b) view of the inner region. Here, the velocity fields are downsampled for clear illustration. (From Liu et al. (2012b))

Fig. 2.8 Profiles of (a) the zonal and (b) meridional velocities averaged over a 2° strip along the minor and major axes across the GRS, respectively. The zonal data for comparisons are obtained by Vasavada et al. (1998) using manual tracking and correlation method 1 and by Choi et al. (2007) using the correlation method 2. The meridional data for comparison are given by Choi et al. (2007) using the correlation method and Asay-Davis et al. (2009) using ACCIV. (From Liu et al. (2012b))

measured maximum tangential velocity is about 150 m/s in both the north and south sections of the collar at the latitudes of -16° S and -24° S, respectively. In particular, the optical flow estimation confirms the existence of the intriguing cyclonic (clockwise) rotation near the center of the GRS. Figure 2.8b shows the meridional velocity profiles across the GRS near the latitude of -20° S, where the data obtained by Choi et al. (2007) using the correlation-based method and by Asay-Davis et al. (2009) using the Advection Corrected Correlation Imaging Velocimetry (ACCIV) are also included for comparison.

2.4

Applications

45

Fig. 2.9 Streamlines of the Gaussian-filtered velocity field of the GRS in (a) the high-speed collar, (b) the inner region, and (c) the neighborhood of the cyclonic source node N 1 . (From Liu et al. (2012b))

Figure 2.9a shows near-elliptical streamlines in the high-speed collar mainly confined in the two virtual elliptical boundaries. The major and minor axes of the inner ellipse between the collar and the inner region are 7:9 × 106 m and 3:6 × 106 m, respectively. The major and minor axes of the outer ellipse are 13:7 × 106 m and 8:59 × 106 m, respectively. In an average sense, as marked in Fig. 2.9a, the collar can be further divided into the inner and outer rings by a dividing ellipse where the maximum velocity magnitude is locally attained. The major and minor axes of the dividing ellipse with the maximum velocity magnitude are 11:3 × 106 m and 6:44 × 106 m, respectively. The flow structures of the inner region of the GRS are intriguing. To identify the major coherent structures in the inner region, the Gaussian-filtered velocity field is obtained, as shown in Fig. 2.9b. The coarse-grained flow topology in the inner region is revealed, where four nodes denoted by N 1 , N 2 , N 3 , and N 4 are identified. The nodes N 2 , N 3 , and N 4 are sink nodes at which streamlines spiral inward anticyclonically. A simple topological constraint exists on the surface velocity field in

46

2 Velocity from Flow Visualizations

the inner region of the GRS. Applying the Poincare-Bendixson index formula to a flow in a region enclosed by a penetrable boundary, Liu et al. (2012b) gave a topological rule for the inner region of the GRS, i.e., #N - #S = 1, where #N and #S are the numbers of nodes and saddles in the inner region, respectively. Clearly, the velocity field in Fig. 2.9b satisfies this topological constraint since there are four nodes and three saddles. A consequence of this topological constraint is that there exists at least one node in the inner region of the GRS. This node in the inner region must coexist with the near-elliptical collar, and therefore this node must be long-lived. In Fig. 2.9b, the cyclonic source node N 1 is a credible candidate for such a seed node associated with the cyclonic rotation near the center. This provides a possible explanation for the persistent presence of the observed cyclonic rotation near the center of the GRS. The topological analysis indicates the necessary presence of the node N 1 in the GRS. The physical origin of N 1 is an interesting problem. As shown in the zoomedin view in Fig. 2.9c, N 1 is a source since streamlines originated from it spiral outward cyclonically, inducing the eastward flow in the north of the node and the westward flow in the south of the node. As a result, the mean zonal velocity profile across this node exhibits the zigzag behavior as indicated in Fig. 2.8a. Since streamlines originating from N 1 spiral outward and the 2D velocity divergence are positive there, vertical flow toward the surface (upwelling) could exist beneath it. The connection between the cyclonic source node N 1 and the convection instability was discussed by Liu et al. (2012b). It was inferred that the convection instability could be responsible to the generation of the source node N 1 . Further, the convection-induced stretching of the planetary vorticity could intensify the positive relative vorticity (clockwise rotation) at the source nodeN 1 .

2.4.1.2

Saturn’s North Pole Vortex

A swirling flow pattern with wind speeds peaking at about 100 m/s was revealed by the polar-projected images of Saturn’s north pole vortex (NPV) captured by the Narrow-Angle Camera onboard the Cassini spacecraft over a period of 5 hours and 19 min. on November 27, 2012 (Sayanagi et al. 2017). By applying OFM to 14 successive pre-rotated cloud images (1024 × 1024 pixels) near Saturn’s north pole (NP), the motions of these clouds were analyzed by Liu et al. (2019a) to measure the wind speeds in the NP region. The top visible cloud layer that is about 100 km below the top of the troposphere is made of ammonia clouds. The interval between consecutive images varies from 20.5 to 29.1 min. Figure 2.10 shows the first pair of 14 NPV images. Figure 2.11 shows the time-averaged velocity vector field and streamlines of Saturn’s NPV, representing the average cloud motion over the whole image sequence. The 1024 × 1024 velocity vectors are obtained at a spatial resolution of

2.4

Applications

47

Fig. 2.10 The first pair of cloud images of Saturn’s north-pole vortex (NPV): (a) first image and (b) second image, where the time intervals between them is 21.9 min. (From Liu et al. (2019a))

Fig. 2.11 Time-averaged flow fields of Saturn’s NPV: (a) velocity vectors and (b) streamlines. (From Liu et al. (2019a))

one vector per pixel. Figure 2.12a shows the time-averaged velocity magnitude field, illustrating the overall flow structure in the cyclonic inner core of the NPV, where the velocity magnitude increases with the radial distance from the NP (decreasing latitude) until reaching a maximum speed of ~155 m/s at 88.95°N. Figure 2.12b–d show maps of the time-averaged relative vorticity, divergence, and second invariant of the strain rate tensor. As shown in Fig. 2.12b, the relative vorticity is generally positive, indicating the cyclonic rotation, and its magnitude increases toward the NP. There are near-circular band patterns with distinct fine vorticity patches in the NPV, which could be generated by shear layer instabilities. As shown in Fig. 2.12c, the divergence in the NPV region is non-homogenous,

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Fig. 2.12 Time-averaged flow fields of the NPV: (a) velocity magnitude, (b) relative vorticity, (c) divergence, and (d) second invariant. (From Liu et al. (2019a))

which varies between positive (divðuÞ > 0) and negative (divðuÞ < 0). The region with divðuÞ > 0 has an upwelling motion below. On the other hand, when divðuÞ < 0, the underlying layer has a downwelling motion. Figure 2.12d indicates that regions with the positive second invariant are concentrated near the NP, highlighting a strong rotational motion. Figure 2.13 shows the profiles of the azimuthally averaged zonal velocity and meridional velocity as functions of the planetocentric latitude. The results extracted by OFM are consistent with those given by Sayanagi et al. (2017) using the correlation image velocimetry (CIV). The peak velocity of 150 m/s given by Sayanagi et al. (2017) is about 17% lower than that calculated by OFM. The location of the peak zonal wind is at the latitude of 88:95o , which is consistent with the value given by Sayanagi et al. (2017) and Antuñano et al. (2018). The variation bounds indicated in Fig. 2.13 mainly represent the temporal-spatial changes of the velocity structures in ensemble averaging and averaging over a polar angle range of 0 - 2π.

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49

Fig. 2.13 Mean velocity profiles of the NPV: (a) zonal (circumferential) velocity and (b) meridional velocity. (From Liu et al. (2019a))

2.4.2

Neutron Radiography of Two-Phase Flow

Time sequences of neutron radiography (NR) images visualizing the dynamics of bubbles in air-water two-phase flows were obtained by Zboray and Trtik (2018, 2019) using high-speed NR up to 800 frames per second (fps). They demonstrated dynamic NR of two-phase flows up to 800 fps by using a high-efficiency (relatively thick) scintillator screen in combination with the highest available flux on a continuous spallation source and a high-speed scientific complementary metal oxide semiconductor (sCMOS) camera. This combination achieved the sufficient spatial resolution for bubbles of the size down to about 1.0 mm at a very high frame rate using pixel binning to improve counting statistics for short exposure times. The flat bubbler (with internal dimensions of 300 mm in height, 98 mm in width but of only 5 mm depth in beam direction) filled with water was used in experiments. The flat bubbler was made of aluminum of 4 mm wall thickness. The two-phase flow was produced by injecting air through an air inlet (4 mm in diameter) placed at 95 mm below the field of view in the center line of the bubbler. Due to the force applied by the air bubbles, internal water recirculation was generated. The flat bubbler was imaged directly in the front of the detector screen. Each radiograph was corrected for open beam and dark current images, and next the scarcely occurring gamma (white spots) were suppressed. The noise in the resulting radiographs was then suppressed using anisotropic diffusion filtering procedure. Liu et al. (2021c) applied OFM to a sequence of 100 NR images of the air-water two-phase flow in a period of 1 s obtained by Zboray and Trtik (2018) to determine high-resolution velocity fields of the two-phase flow. From the raw NR images, a sequence of 100 g -images is obtained, where g is the optical density defined in Eq. (2.66). Figure 2.14 shows a sequence of g-images at 0.05, 0.08, 0.11, and 0.14 s to show the evolution of the two-phase flow (e.g., bubbly flow). The first large bubble appears at the bottom of the domain at 0.05–0.08 s, and the second large

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Fig. 2.14 Time sequence of g-images for showing the evolution of air bubbles: (a) 0.05 s, (b) 0.08 s, (c) 0.11 s, and (d) 0.14 s. (From Liu et al. (2021c))

bubble emerges as the first one moves up at 0.11 s. The second bubble follows and catches up the first one at 0.14 s, and then the two bubbles start to coalesce. The optical flow equation derived for g -images is described in Sect. 2.2.6. Figure 2.15 shows the extracted velocity vector and magnitude fields by using OFM at 0.05, 0.08, 0.11, and 0.14 s. The motion of the second bubble is induced by the wake of the first one. Figure 2.16 shows streamlines at the sequential moments, indicating the motion of the large bubbles and the induced circulatory motion by them in the water domain. Figure 2.17 shows the time-averaged velocity vector field and streamlines superposed on the normalized velocity magnitude field, i.e.,

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Applications

51

Fig. 2.15 Time sequence of velocity vectors superposed on the normalized velocity magnitude fields: (a) 0.05 s, (b) 0.08 s, (c) 0.11 s, and (d) 0.14 s. (From Liu et al. (2021c))

hjujiT = max hjujiT , where hiT is a time-averaging operator. The profile of the time-averaged vertical velocity uy T is approximately symmetrical with respective to the middle location x/d = 0, while the profile of the time-averaged horizontal velocity hux iT is approximately anti-symmetrical. The flow is upward in the central region of the measurement domain, while the flow is downward near the right and left sides of the domain. In the shear layers between these opposite flows, the Kelvin-Helmholtz instability would lead to the formation of large vortices as observed in Figs. 2.15 and 2.16. Clearly, OFM applied to high-speed NR provides a useful tool to determine the dynamics of unsteady two-phase flows, including instantaneous high-resolution velocity fields of the two phases and tine histories and statistics of the relevant quantities (such as phase fractions, size, and shape).

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Fig. 2.16 Time sequence of streamlines superposed on the normalized vorticity fields: (a) 0.05 s, (b) 0.08 s, (c) 0.11 s, and (d) 0.14 s. (From Liu et al. (2021c))

Fig. 2.17 Time-averaged flow field: (a) velocity vectors on the normalized velocity magnitude field and (b) streamlines on the normalized vorticity field, where the maximum values are used for normalization. (From Liu et al. (2021c))

2.4

Applications

2.4.3

53

Particle Image Velocimetry

PIV is a widely used global velocity measurement technique in experimental fluid mechanics (Raffel et al. 2007; Adrian and Westerweed 2011). The cross-correlation method is a key element in PIV processing, where velocity vectors are obtained by maximizing the correlation peak in the spatial cross-correlation map from two corresponding interrogation windows in successive PIV images. In contrast to the cross-correlation method that is a local integral approach, the variational OFM is a differential approach that depends on the accurate evaluation of the temporal and spatial differentiations of the image intensity to solve the Euler-Lagrange equation. From this perspective, OFM is better suited to images of continuous patterns. However, since PIV images are basically nonsmooth spatial random noise distributions, they are probably the worst case for OFM. A question is how well OFM works for PIV images under certain constraints. This problem was investigated by Liu et al. (2015b). The error estimate Eq. (2.80) for OFM can be extended by including the effects of the particle image diameter and the particle image density. Evaluating the relevant terms in Eq. (2.80) for PIV images, Liu et al. (2015b) gave an estimate for the total error of OFM applied to PIV images, i.e.,

ε = Δxp

c1 d2p up

þ c2 ∇up 2

2

þ

c3 c þ 4m d2p N p

Δ m xp Δ xp

2 2

,

ð2:86Þ

where c1 , c2 , c3 , and c4 are coefficients to be determined. According to Eq. (2.86), the main parameters are: Particle displacement Δ xp , Particle image diameter d p , Particle velocity gradient ∇up , Particle image density N p . In general, the particle displacement Δ xp should be small in optical flow computation for PIV images, particularly in regions of large velocity gradients. The particle image diameter d p could have the optimal value for optical flow computation since the terms of  d 2p and  dp- 2 in Eq. (2.86) have the opposite trends as dp increases. The particle image density N p should be suitably large. Nevertheless, in the limiting case where N p → 1, PIV images would become uniform and the image intensity gradient becomes small. In this case, accurate extraction of the optical flow is not possible. Therefore, there would be the optimal value of N p . In principle, the above error estimate for OFM is also applicable to the cross-correlation method. To evaluate the performance of OFM compared to the cross-correlation method, simulations were conducted on PIV images in a synthetic canonical flow: an Oseen vortex pair in a uniform flow. A sample PIV image with the size of 250 × 250 pixels

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Fig. 2.18 A typical pair of PIV images for simulations: (a) Image #1 and (b) Image #2. (From Liu et al. (2015b))

Fig. 2.19 Velocity vectors of an Oseen vortex pair in a uniform flow extracted from the particle images by using (a) OFM and (b) the correlation method, where the seeding particle image density is 0.04 1/pixel2. (From Liu et al. (2015b))

and the 8-bit dynamic range is generated, where 10,000 particles with a Gaussian intensity distribution with the standard deviation of 2 pixels are uniformly distributed. The second PIV image is generated by applying an image-shifting (imagewarping) algorithm that uses a translation transformation for large displacements and the discretized optical flow equation for sub-pixel correction. Figure 2.18 shows a typical pair of the generated PIV image used for simulations, where the mean characteristic image diameter of particles is d p = 4 pixels and the particle image density is 0.04 1/pixel2.

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55

Fig. 2.20 Streamlines and vorticity fields of an Oseen vortex pair in a uniform flow extracted from the particle images by using (a) OFM and (b) the correlation method. (From Liu et al. (2015b))

Figure 2.19 shows the velocity vectors extracted from the PIV images where OFM gives 500 × 500 vectors, and the cross-correlation method with the two passes gives 62 × 62 vectors. Figure 2.20 shows the corresponding streamlines and vorticity fields. The Matlab program ‘OpenOpticalFlow’ is used for optical flow computation. The cross-correlation method used for comparison is embedded in LaVision’s DaVis (7.2) software package. The overall flow fields extracted by both the methods are consistent except that OFM yields the results with much higher spatial resolution. The results from both the methods compared well with the true distribution except near the vortex cores. The error of OFM is smaller than that of the cross-correlation method in the vortex core. This difference is not surprising since the crosscorrelation computation in interrogation windows tends to smooth out the velocity gradient in the vortex core region where the velocity changes drastically in its magnitude and direction. The root-mean-square (RMS) error in the whole field is calculated as a function of the maximum displacement in images. As shown in Fig. 2.21a, the error in both OFM and the cross-correlation method increases approximately proportionally with the maximum displacement max Δ xp . When max Δ xp =dp < 1:5, OFM gives more accurate results than the cross-correlation method, and the relative error is less than 2%. When with the Lagrange multipliers are (50, 5000) for the Horn-Schunck estimator and the Liu-Shen estimator, respectively, the error of OFM exceeds that of the cross-correlation method when max Δ xp is larger than about 4–5 pixels. When the larger Lagrange multipliers (200, 20,000) are used, OFM yields the smaller error. Figure 2.21b shows the total RMS error as a function of the velocity gradient, which is essentially the same as that in Fig. 2.21a since the velocity gradient is directly proportional to the displacement. Particle image density is another relevant parameter that affects the accuracy of extracted velocity fields. The total RMS error is shown in Fig. 2.22a as a function of the particle image density. For OFM, there is a shallow valley in the particle image density of 0.015–0.04 1/pixel2, where the error reaches the minimum. This range approximately corresponds to 15–40 particles in a 32 × 32 pixel2 window which is also more suitable for the cross-correlation method.

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Fig. 2.21 RMS error as a function of (a) the maximum displacement and (b) the velocity gradient for an Oseen vortex pair in a uniform flow. (From Liu et al. (2015b))

Fig. 2.22 RMS error as a function of (a) the particle image density and (b) the particle image diameter for an Oseen vortex pair in a uniform flow. (From Liu et al. (2015b))

The RMS error is shown in Fig. 2.22b as a function of the particle image diameter. As indicated in Eq. (2.86), as the particle image diameter increases, the error of OFM deceases for d p ≤ 4 pixels, and increases thereafter due to the elevated error associated with the decay of the image intensity gradient. Thus, there is a shallow valley near the particle image diameter of 4 pixel that corresponds to max Δ xp =dp = 0:62, where the error of OFM reaches the minimum. Interestingly, the error in the cross-correlation method has the similar dependency on the particle image diameter (Raffel et al. 2007).

2.4

Applications

2.4.4

57

Background-Oriented Schlieren Technique

The background-oriented Schlieren (BOS) technique can provide global non-contact diagnostics for measuring the displacement field induced by the density change in fluid, which is used to infer the density and pressure fields (Raffel et al. 2000; Meier 2002; Venkatakrishnan and Meier 2004; Raffel 2015). BOS is an extension of the classical Schlieren technique by utilizing digital cameras and image processing. The experimental setup of a BOS system is much simpler than its classical counterpart with complex arrangements of optics. The basic idea of BOS is straightforward, as illustrated in Fig. 2.23. In general, a background pattern plate is placed behind a measurement domain of fluid, and its images taken by a camera in the cases with and without the fluid density gradient are considered as a reference non-disturbed (flowoff) image and a disturbed (flow-on) image. The disturbed background pattern image is one altered by the density gradient deflecting the light rays radiated from the background pattern plate. The displacement field in the disturbed image relative to the non-disturbed image is related to the path integral of the density gradient. When the displacement field is measured, the density and pressure fields can be inferred. To determine the displacement field from the flow-on and flow-off images, the cross-correlation method in PIV has been applied to aerodynamic flows and gas flows (Raffel 2015). However, BOS is more difficult to use for liquids since the density change in liquids is much smaller than that in gases. Therefore, highresolution measurements of small displacements in BOS images are required particularly for further calculation of the fluid pressure from the fluid density. As a differential approach, OFM is more suitable to BOS to obtain high-resolution displacement fields. Hayasaka et al. (2016) applied OFM to BOS measurements of a shock wave induced by a pulsed laser in water and obtained the displacement field with the high spatial resolution. Here, BOS based on OFM is called OF-BOS, and BOS based on the cross-correlation method in PIV is called PIV-BOS. In OF-BOS, the open-source Matlab program “OpenOpticalFlow” was used. In PIV-BOS, the open-source Matlab PIV program “PIVlab” was used (Thielicke and Stamhuis 2014). To generate an underwater shock wave propagating spherically, Hayasaka et al. (2016) used a laser pulse with the wavelength of 532 nm and pulse width of 6 ns, which was focused through a lens to a point inside a distilled water-filled glass tank (450 × 300 × 300 mm3). The background pattern was illuminated by a laser stroboscope with the pulse width of 20 ns. The pulsed laser, camera, and light source

Fig. 2.23 Illustration of the principle of the BOS technique. (From Hayasaka et al. (2016))

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Fig. 2.24 Typical images obtained in the BOS setup: (i) a shadowgraph snapshot of the laserinduced shock wave, (ii) non-disturbed image in BOS, and (iii) disturbed image in BOS. (From Hayasaka et al. (2016))

were synchronized by using a delay function generator. A hydrophone was used to validate the results obtained by using OF-BOS and PIV-BOS. Figure 2.24 shows the typical images obtained in their experimental setup. Figure 2.24(i) shows a shadowgraph snapshot visualizing a shock wave propagating spherically around a laser-induced bubble, where the high-pressure region is expected at a shock front. Figure 2.24(ii) and (iii) show a non-disturbed reference image and the corresponding disturbed image, respectively. The zoomed-in images of the background particle pattern are shown as well. Without image processing, it is difficult to see directly the shock wave in Fig. 2.24(iii) since the change of the refractive index of water is so small. Figure 2.25 shows the displacement magnitude fields obtained by using OF-BOS giving 2008 × 1085 vectors and PIV-BOS giving 250 × 138 vectors. Both OF-BOS and PIV-BOS capture the sharp change induced by the shock wave in the displacement fields. As expected, the cross-correlation computations in PIV-BOS tend to smooth out the sharp features associated with the shock wave. The profiles of the measured displacement along a ray by both the methods are consistent with the data given by the hydrophone. It is indicated below that the high-resolution displacement field given by OF-BOS is a key for more accurate reconstruction of a density field and a pressure field. In principle, a field of the fluid density ρ can be determined from a displacement field by solving the Poisson equation: ∇2 ρ = S,

ð2:87Þ

where the source term related to the displacement vector w0 on the background plane is S=

2ð1 þ Kρ0 Þ ∇  w0 : cK ðc þ 2l0 Þ

ð2:88Þ

In Eq. (2.88), ρ0 is the fluid density under hydrostatic pressure, c is the thickness of the density gradient domain, l0 is the distance from the density gradient domain to

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Applications

59

Fig. 2.25 Displacement magnitude fields in the half domain obtained by using (i) OF-BOS and (ii) PIV-BOS. (From Hayasaka et al. (2016))

the background plate, and K is the Gladstone-Dale constant. The displacement vector w0 in the background plane is related to that in the image plane by w0 = λw, where λ is a scaling factor and w is a displacement vector in the image plane. Further, a pressure field can be determined by applying the Tait equation: ρ pþB = ρ0 p0 þ B

α

,

ð2:89Þ

where p0 is the hydrostatic pressure, B is a constant of 314 MPa, and the exponent α is 7 (Brujan 2010; Yamamoto et al. 2015). In general, a quantity in the image plane (such as w or ∇  w) is the quantity integrated and projected along a light path in the 3D object space. To determine the quantity in the object space from the corresponding one in the image plane, a tomographic reconstruction problem should be solved. Hayasaka et al. (2016) selected the displacement divergence ∇  w for tomographic reconstruction using the Radon transform. The field of the fluid density is obtained by solving the axisymmetric Poisson equation for a given source term on the plane of symmetry and the corresponding pressure field is calculated by using Eq. (2.89). Figure 2.26 shows the reconstructed pressure fields based on the data obtained by using OF-BOS (2008 × 1085 vectors) and PIV-BOS (250 × 138 vectors). OF-BOS captures correctly the sharp pressure change across the shock wave, and in contrast PIV-BOS has a

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Fig. 2.26 Reconstructed pressure fields obtained by using (i) OF-BOS and (ii) PIV-BOS. (From Hayasaka et al. (2016))

Fig. 2.27 Comparisons between the pressure profiles in the radial direction obtained by using OF-BOS, PIV-BOS, and the hydrophone, where R is the radius of the shock wave. (From Hayasaka et al. (2016))

larger error in the reconstructed pressure field due to its much lower spatial resolution. As shown in Fig. 2.27, the pressure profile given by OF-BOS is consistent with that given by the hydrophone along a ray aligned. Clearly, OF-BOS provides the more accurate reconstruction of the pressure field of the shock wave.

Chapter 3

Skin Friction from Global Luminescent Oil-Film Visualizations

This chapter describes the global luminescent oil-film (GLOF) skin friction meter that is simply called GLOF. The thin oil-film equation is discussed to elucidate the evolution of an oil-film on a surface in an aerodynamic flow. The thin oil-film equation is recast to a form related to the luminescent intensity of an optically thin luminescent oil-film, and it is further projected onto the image plane. The projected thin oil-film equation in the image plane has the same mathematical form of the physics-based optical flow equation, where the equivalent skin friction is represented as the optical flow. Therefore, the variational method used for the optical flow problem is applied to extraction of relative skin friction fields from GLOF images. An error analysis for GLOF is given, and averaging of snapshot solutions is proposed for reconstruction of a skin friction field. One example of the application of GLOF is a low aspect ratio wing at a high angle of attack (AoA), focusing on the complex skin friction topology on the upper surface and the Poincare-Bendixson index formula as a topological constraint in global flow diagnostics. Another example is a wing-body junction where the complex skin friction topology is reconstructed from GLOF images obtained from different views.

3.1

Thin Oil-Film Equation

Oil-film skin friction meter is based on detecting the temporal-spatial evolution of the thickness of a thin oil-film to determine skin friction at discrete locations (Tanner and Blows 1976; Naughton and Sheplak 2002). The thin oil-film equation, which describes the response of a thin oil-film on a surface to an externally applied 3D aerodynamic flow, is given in the summation convention, i.e., (Naughton and Brown 1996; Brown and Naughton 1999)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_3

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∂h ∂ τ i h2 ∂p - ρo ai þ ∂t ∂X i 2μo ∂X i

h3 = 0, ði = 1, 2Þ 3μo

ð3:1Þ

where h is the oil-film thickness, τ = ðτ1 , τ2 Þ is skin friction at the oil/wall interface, p is the pressure, μo is the oil viscosity, ρo is the oil density, ða1 , a2 Þ is the gravity vector, and ðX 1 , X 2 Þ are the surface coordinates. Equation (3.1) is derived from the mass conservation equation with the Taylor series expansion solution of the NavierStokes (NS) equations for the thin oil-film (see Sect. 4.3.1). Here, τ is equal to that at the air/oil interface for a very thin oil-film with the linear velocity profile. Most image-based interferometric oil-film meters have been used for points, lines and regions of surface based on the 1D solution of the thin oil-film equation that provides a local relation between skin friction and oil-film thickness on a surface. In the simplest case where skin friction is constant and the effects of the pressure gradient and gravity are neglected, skin friction along a skin friction line given by Tanner and Blows (1976) is τs = -

μo ∂h ∂h = , h ∂t ∂s

ð3:2Þ

where s is the coordinate along a skin friction line. In order to keep τs independent of time, the oil-film thickness should have a similarity form h / s=t. An analysis of the thin oil-film equation is given to shed some insight to the oil-film evolution in a viscous aerodynamic flow. When the effects of the pressure gradient and gravity are neglected as higher-order small terms, Eq. (3.1) becomes ∂h ∂ τ i h2 = 0: ði = 1, 2Þ þ ∂t ∂X i 2μo

ð3:3Þ

Projection of Eq. (3.3) onto a skin fraction line yields ∂s τ h2 ∂h ∂ τs h2 = 0, þ s κ ni þ 2μo ∂X i ∂t ∂s 2μo

ð3:4Þ

where s is the arc length along a skin friction line, τs = s  τ is the skin friction along the unit tangential vector s of the skin friction line, and κ and ni are the curvature and normal vector of the skin friction line, respectively. To obtain Eq. (3.4), the FrenetSerret formula ∂sj =∂ X i = κ nj ∂s=∂X i is used. When the effect of the curvature is neglected, Eq. (3.4) becomes ∂h ∂ τs h2 þ = 0: ∂t ∂s 2μo By using the similarity variables

ð3:5Þ

3.1

Thin Oil-Film Equation

63

h=h0 = f ðξÞ and ξ = μo s=ðτs t h0 Þ, Equation (3.5) is transformed to a semi-similar equation, i.e., f ðξÞ = ξ þ ðt=t 1 Þ ξ - ðt=t 2 Þ f 2 =f 0 þ ðs=s1 Þ f ,

ð3:6Þ

where the temporal and spatial scales of the oil-film evolution are defined as τs , ∂τs =∂t μo , t2 = h0 ∂τs =∂s τs s1 = : ∂τs =∂s t1 =

In the steady localized case where ∂τs =∂s = 0, ∂τs =∂t = 0, and τs ≠ 0, a selfsimilar solution f ðξÞ = ξ is obtained. In general, only a semi-similar solution can be obtained for the given nondimensional times and location. The thin oil-film evolution deviates from the self-similar state, depending on the two timescales (t1 and t 2 ) and the spatial scale (s1 ). Substitution of the self-similar solution f ðξÞ = ξ into the RHS of Eq. (3.6) leads to a first-order approximate solution f ðξÞ = ξ þ ðt=t1 Þ ξ - ðt=t 2 Þ ξ2 þ ðs=s1 Þ ξ:

ð3:7Þ

Differentiating Eq. (3.6) with respect to s and t yields τ s = μo

∂ ∂h ∂t ∂s

-1

∂h ∂ þ t τs ∂s ∂t

-t -1

∂ s ∂h ∂τs - μo ∂t s1 ∂s ∂t ∂G ∂s

-1

,

ð3:8Þ

where G = h0 f ðξÞ. In an ideal starting process where τs / H ðt Þ and ∂τs =∂t / δðt Þ, a special solution after a transient stage is τ s = μo

∂ ∂h ∂t ∂s

-1

1-

s ∂τs =∂s , τs

ð3:9Þ

where H ðt Þ is the Heaviside function and δðt Þ is the Dirac delta function. The above explicit expressions for skin friction are useful for image-based interferometric oil-film meters based on oil droplets on a surface since the semisimilar solution is applicable to the 1D evolution of an oil droplet along a skin friction line. The similarity analysis elucidates the effects of the temporal and spatial

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scales on the local dynamics of oil-film, providing a rationale for the averaging of snapshot solutions for reconstruction of a complete steady-state skin friction field (see Sect. 3.3.3). However, for a developing continuous oil-film in a large region on a surface in complex flows, the local semi-similarity of a developing oil-film cannot be attained globally. Therefore, to determine a skin friction field from oil-film thickness measurements, Eq. (3.1) should be solved as an inverse problem in a framework of the variational formulations.

3.2 3.2.1

Global Luminescent Oil-Film Skin Friction Meter Oil-Film Thickness and Luminescent Intensity

Liu et al. (2008) recognized that extraction of a high-resolution skin friction field from oil-film thickness measurements could be treated as an inverse problem in a variational framework. This method is called the global luminescent oil-film (GLOF) skin friction meter (for simplicity it is called GLOF hereafter). Figure 3.1 illustrates a generic experimental setup for GLOF visualizations. A thin luminescent oil-film is brushed or sprayed on a surface of a body in the region of interest before starting a wind tunnel, and an illumination field for exciting the luminescent molecules is provided by light sources with a suitable wavelength such as ultraviolet (UV) lights. Usually, the surface is covered by a white Mylar sheet or a white paint to enhance the luminescent emission of the oil on it due to stronger surface scattering of light. Therefore, measurement of the oil-film thickness distribution is converted to measurement of the luminescent intensity distribution by using a digital camera with a suitable optical filter. After the flow is turned on, a time sequence of GLOF images is acquired for image processing. This experimental setup is not only simple but also compatible to other luminescence-based surface flow visualizations such as PSP and TSP.

Fig. 3.1 Generic experimental setup for global luminescent oil-film (GLOF) measurement

3.2

Global Luminescent Oil-Film Skin Friction Meter

65

When a luminescent oil-film applied to a surface is optically thin, the luminescent intensity (I ) of the oil-film under excitation by a suitable illumination light is proportional to the oil-film thickness, i.e., (Liu and Sullivan 1998) I ðX 1 , X 2 Þ = β I ex ðX 1 , X 2 Þ hðX 1 , X 2 Þ,

ð3:10Þ

where I ex ðX 1 , X 2 Þ is the intensity of the excitation light on the surface and β is a coefficient proportional to the quantum efficiency of seeded luminescent molecules and dye concentration. Substitution of h = β - 1 ðI=I ex Þ into Eq. (3.1) yields ∂g ∂ τ i g2 ∂p - ρo ai þ ∂t ∂X i 2μo β ∂X i

g3 =0 3μo β2

ði = 1, 2Þ

ð3:11Þ

where g = I=I ex is the normalized luminescent intensity of the oil-film that eliminates the effect of nonuniform illumination. When the response function of a camera is linear, g is the normalized image intensity. Eq. (3.11) describes the relationship between a skin friction field and a normalized luminescent oil-film intensity field on a surface in the object space.

3.2.2

Projection from Surface to Image

From a standpoint of image processing, it is more convenient to carry out computations in the image plane for extraction of skin friction vectors from GLOF images. Therefore, Eq. (3.11) should be projected onto the image plane. As shown in Fig. 3.2, a special object-space coordinate system X 1 , X 2 , X 3 is considered, in which the plane X 1 , X 2 is parallel to the image plane ðx1 , x2 Þ, and therefore a relation is ∂=∂X i = λ ∂=∂xi , where λ is a scaling constant. An image is the scaled orthographic projection of the surface onto the plane X 1 , X 2 . If the one-to-one transformation between the coordinates X 1 , X 2 and the surface coordinates ðX 1 , X 2 Þ is given by X 1 = F 1 ðX 1 , X 2 Þ and X 2 = F 2 ðX 1 , X 2 Þ, a differential transformation is ∂=∂X i = H ji ∂=∂X j = λ H ji ∂=∂xj

ði, j = 1, 2Þ,

where H ji = ∂F j =∂X i depends on the geometric properties of the surface.

66

3

Skin Friction from Global Luminescent Oil-Film Visualizations

Fig. 3.2 Surface, object space, and image coordinates

By introducing the equivalent skin friction vector τj = τi H ji gðλ=2μo βÞ ,

ð3:12Þ

Equation (3.11) can be re-written in the image coordinates, i.e., ∂g þ ∇  ðg τ Þ = f , ∂t

ð3:13Þ

where τ = ðτ1 , τ2 Þ represents a projected and scaled skin friction in the image plane (called the equivalent skin friction), and the RHS source term is f = λH ji

∂ ∂xj

λH ki

∂p - ρo ai ∂xk

λτ g2 ∂H ji g3 : þ i 2 2μo β ∂xj 3μo β

ði = 1, 2Þ

ð3:14Þ

Here, g is the normalized luminescent intensity mapped onto the image plane when the radiometric responsive function of a camera is linear. Geometrically, τi H ji is the projected skin friction vector on a surface in the object space onto the image plane. Since the determinant is jH kij ≠ 0 in the local affine transformation, the topological structure of a skin friction field in the image plane uniquely corresponds to that on the surface. Interestingly, Eq. (3.13) has the same form as the physicsbased optical flow equation Eq. (2.71), where τ is interpreted as the optical flow. The first term in Eq. (3.14) represents the effects of the pressure gradient and gravity. This term is in the order of λh3 0, indicating an increase of pressure in the wall-normal direction. The sink and source in a skin friction field change the instantaneous velocity profile, inducing the ejection or the sweep events, respectively.

5.3

Applications

127

5.3.2

Skin Friction and Heat Transfer: Beyond the Reynolds Analogy

5.3.2.1

Theoretical Account

This chapter focuses on the exact relation between skin friction and surface temperature through the source term related to the heat flux and its application to extraction of a skin friction field from a surface temperature field. Naturally, a question is how this exact on-wall relation sheds a new insight to the classical Reynolds analogy in viscous flows (Woodworth et al. 2023). The underlying theoretical assumption for the Reynolds analogy is the similarity between distributions of skin friction and heat flux, which is valid only for certain 2D flows with such the similarity. However, the Reynolds analogy does not have a general and universal form for 3D complex flows, and in particular, it cannot give the directional field of skin friction vectors. The Reynolds analogy is generally referred to as a proportional relation between skin friction and local heat flux for certain flows (Schlichting and Gersten 2017). For a 2D laminar boundary layer, when the distributions of the normalized velocity and temperature are similar, there is a nondimensional relation between the local Nusselt number Nu and the skin friction coefficient cf , i.e., Nu =

q∂B l 1 = c Re f ðx=l, PrÞ, kðT ∂B - T ref Þ 2 f

ð5:33Þ

where q∂B is the local heat flux, cf = τ=0:5ρU 21 is the skin friction coefficient, Re = U 1 lρ=μ is the Reynolds number, Pr = μcp =k is the Prandtl number, U 1 is the freestream velocity, x=l is the streamwise surface coordinate normalized by a length scale l, T ∂B is the surface (wall) temperature, T ref is a reference temperature, k is the thermal conductivity of fluid, ρ is the density of fluid, and μ is the dynamic viscosity of fluid. In Eq. (5.33), f ðx=l, PrÞ is a nondimensional function depending on a specific flow, which does not have a universal form for a large class of flows since it is generally a function of the location. Only for the similar flows such as the Falkner-Skan flow, f ðx=l, PrÞ becomes position-independent (Burmeister 1993). Furthermore, some empirical forms of f ðx=l, PrÞ could be found for other laminar and turbulent flows with similar forms of both velocity and temperature based on measurement and CFD data. Based on the exact relation Eq. (5.19), the Reynolds analogy is discussed to understand the underlying assumptions behind it. In the steady-state case, the surface heat flux is modeled as q∂B = q∂B0 H ðt Þ, where qw0 is a steady-state heat flux depending on a location, and H ðt Þ is the Heaviside function defined as H ðt Þ = 1 for t > 0 and H ðt Þ = 0 for t ≤ 0. Applying a time-averaging operator over a period T 1

128

5

Skin Friction from Surface Temperature Visualizations

hiT = T 1- 1

T1

 dt

ð5:34Þ

0

to Eq. (5.23), we have hGiT þ τi ∂hT ∂B iT =∂xi = 0, ði = 1, 2Þ

ð5:35Þ

where τi = Ci h τi iT are the components of the time-averaged skin friction vector τ = ðτ1 , τ2 Þ and C i (i = 1, 2) are the correlation coefficients defined as C i = hτi ∂T ∂B =∂xi iT =hτi iT ∂hT ∂B iT =∂xi : In Eq. (5.35), after the dissipation term, the curvature term and the third-order normal derivative of temperature are neglected, the time-averaged source term is hGiT = - μ f Q

T

= - cp- 1 Pr T 1- 1 - a∇2 q∂B0 ,

ð5:36Þ

where Pr = cp μ=k is the Prandtl number. When the lateral diffusion of the surface heat flux (a∇2 q∂B0 ) can be neglected, Eq. (5.35) is rewritten in a 1D form, i.e., τN = cp- 1 T 1- 1 Pr qw0 = ∇hT w iT ,

ð5:37Þ

where the skin friction component projected in the surface temperature gradient direction is τN = τ  ΝT = sgnðτ  ΝT ÞjτN j, the unit surface temperature gradient vector is N T = ∇hT ∂B iT = ∇hT ∂B iT , and T 1 is a time scale. Eq. (5.37) is a relation along a line with the surface temperature gradient. The sign function is defined as sgnðξÞ = 1 for ξ > 0 and sgnðξÞ = - 1 for ξ < 0. Further, Eq. (5.37) is recast to a nondimensional form similar to the Reynolds analogy, i.e., Nu =

q∂B0 l 1 = sgnðτ  N T Þ cf ,N Reref QT ðx=lÞ, 2 kðhT ∂B iT - T ref Þ

ð5:38Þ

5.3

Applications

129

where this sign function represents the directional effect of the interaction between a skin friction field and a surface temperature gradient field and U ref and l are the reference velocity and length, respectively. In Eq. (5.38), the skin friction coefficient is cf ,N =

jτ N j , ð1=2ÞρU 2ref

ð5:39Þ

ρU 2ref T 1 , μ

ð5:40Þ

the Reynolds number is Re ref =

and the nondimensional surface temperature gradient magnitude is QT ðx=lÞ =

l ∇hT ∂B iT : hT ∂B iT - T ref

ð5:41Þ

In this sense, the Reynolds analogy is a special form of the exact on-wall relation along the temperature gradient direction in the steady-state case when the lateral diffusion of the heat flux is neglected. Equation (5.38) gives a specific form of Eq. (5.33), where the non-specified nondimensional function f in Eq. (5.33) is defined as the nondimensional surface temperature gradient magnitude QT . It is noted that the classical Reynolds analogy for boundary layers is derived for a constant surface temperature (Schlichting and Gersten 2017). In contrast, Eq. (5.38) is given for a flow with a spatially varying surface temperature distribution that is included in the term QT . Strictly, the Reynolds analogy implies that the Nusselt number Nu can be expressed as a function of the skin friction coefficient only where the position coordinate does not explicitly appear. To meet this condition in Eq. (5.38), from a viewpoint of functional equations, Nu, cf ,N , and QT should be power-law functions of the position coordinate in the direction of the surface temperature gradient (an example is the Falkner-Skan flow). To elucidate this point, we assume cf ,N / xβ and QT / xγ , where x is the surface coordinate along a line with the surface temperature gradient and β and γ are constant exponents. Therefore, Eq. (5.38) becomes a power-law relation Nu0 / jτN jλ ,

ð5:42Þ

130

5

Skin Friction from Surface Temperature Visualizations

where λ = 1 þ γ =β is an exponent to be determined from measurements in a specific flow. From this perspective, the Reynolds analogy holds when two of the variables Nu, cf ,N , and QT exhibit the power-law functions of the position coordinate. Therefore, the Reynolds analogy is applicable to not only self-similar laminar flows but also a class of attached flows as long as the power-law forms of these variables appear in some regions. This provides an explanation on the observations that the Reynolds analogy exists in some more complex flows beyond the exact similar flows. However, the Reynolds analogy is limited to a 1D case along a line with the surface temperature gradient. In 3D complex flows such as separated flows, a skin friction field is a vector field exhibiting complex topology. In these flows, the relationship between heat transfer and skin friction should be studied using the variational method discussed in this chapter, which is considered as a global variational form of the Reynolds analogy.

5.3.2.2

Experimental Evidences

To demonstrate this point, the skin friction fields were extracted by Woodworth et al. (2023) using the approximate method from TSP measurements in impinging jets, revealing complex flow structures in the impingement region where the Reynolds analogy does not hold. A low-speed air jet from a circular converging nozzle driven by compressed air impinged on a 0.03-mm-thick, 100-mm-wide, 180-mm-long steel sheet that was heated by passing electric current through it. The nozzle had a length of 84 mm, exit diameter D of 3.9 mm, and a contraction ratio of 245. In experiments, the nozzles were set at the impingement angles of 90°, where the impingement angle is defined as the angle between the symmetrical axis of the nozzle and the planar impingement surface. The exit velocity was 18 m/s in TSP measurements. To measure the surface temperature fields, EuTTA-based TSP was coated on the impingement side of the electrically heated steel-heated steel sheet where the generated heat flux was 183 W/m2. For a low-speed jet impinging on a uniformly heated steel sheet with q∂B0 = const:, when the dissipation term and the third-order normal derivative term are assumed to be small, the source term in Eq. (5.35) is hGiT = - μ q∂B0 =k = const: in the application of the approximate method. According to the sign definition of q∂B in this case, hGiT is a negative constant since heat enters into fluid from the heated surface. In addition, there is no base flow in this case unlike a boundary-layer flow. Figure 5.8a, b show the typical fields of the normalized temperature difference θ and the normalized Nusselt number Nu= maxðNuÞ at the impingement angle of 90° and the height-to-diameter ratio h/D = 8 for ReD = 4520, respectively. The normalized surface temperature difference is defined as

5.3

Applications

131

Fig. 5.8 (a) Normalized temperature difference θ and (b) normalized Nusselt number Nu= maxðNuÞ in the 90° impinging jet from a nozzle with D = 3.9 mm at h/D = 8 for ReD = 4520. The surface plots are on the right side. (From Woodworth et al. (2023))

θ=

T ∂B - T ref : minjT ∂B - T ref j

ð5:43Þ

Similarly, the Nusselt number is defined as Nu =

q∂B D : k ðT ∂B - T ref Þ

ð5:44Þ

where k is the thermal conductivity of air, q∂B is the heat flux to the flow from the steel sheet, T ∂B is the surface temperature of the heated steel sheet, T ref is the ambient temperature (297 K), and the characteristic length is the nozzle exit diameter D. The skin friction field (621 × 621 vectors) was extracted from the surface temperature fields by using the approximate method when the Lagrange multiplier is set at 10-5. Figure 5.9 shows the skin friction field near the impingement region in

132

5

Skin Friction from Surface Temperature Visualizations

Fig. 5.9 Skin friction field extracted from the surface temperature field in the 90° impinging jet from a nozzle with D = 3.9 mm at h/D = 8 for ReD = 4520: (a) skin friction vectors on the normalized magnitude field and (b) skin friction magnitude surface. (From Woodworth et al. (2023))

the 90° impinging jet from a nozzle with D = 3.9 mm at h/D = 8, where only downsampled vectors are shown for clarity. Direct comparison between the Nusselt number and the normalized skin friction magnitude distributions indicates that the Reynolds analogy does not hold in the impingement region (x/D < 4) where the flow is deflected by the surface and the velocity direction is drastically changed. The flow similarity condition for the Reynolds analogy does not exist in such a flow in the impingement region. It is conjectured that the power-law forms of skin friction and the Nusselt number could be developed asymptotically in the wall-jet region as the flow spreads outwardly in the radial direction. To examine this conjecture, the profiles of the Nusselt number and skin friction magnitude normalized by their maximum values in the radial coordinate x=D in the normally impinging jet are presented in log-log plots in Fig. 5.10a, b, respectively. Indeed, in the wall-jet region (x=D > 2), the asymptotic power-law forms are Nu= maxðNuÞ / ðx=DÞ - 1:5 , jτN j= maxðjτN jÞ / ðx=DÞ - 3 : As indicated in Fig. 5.10c, the parameter Nu=jτN j0:5 as a function of x=D, approximately approaches a constant for x=D > 2. Therefore, the Reynolds analogy is asymptotically valid in the wall-jet region. This result approximately corresponds to the Falkner-Skan flow with m = 1 since the power-law relation for this similar flow is Nu / τð1þmÞ=ð1þ3mÞ .

5.3 Applications

133

Fig. 5.10 Profiles of the relevant parameters in the normalized radial coordinate in the 90° impinging jet: (a) Nu= maxðNuÞ, (b) jτN j= maxðjτN jÞ, and (c) Nu=jτN j0:5 . The dashed straight line in (a) and (b) indicates the slope of the linear portion in the log-log plots, and the numbers indicate the slopes (the power-law exponents). (From Woodworth et al. (2023))

Chapter 6

Skin Friction from Surface Scalar Visualizations

This chapter describes extraction of a skin friction field from surface scalar visualizations with pressure sensitive paint (PSP) as an oxygen sensor and sublimating coatings. First, the exact relation between skin friction and surface scalar concentration is derived from the binary mass transport equation for an incompressible flow on a curved surface, and further the specific forms of this on-wall relation are formulated for its applications in several surface scalar visualizations. Then, the variational method is developed to determine a skin friction field from surface scalar visualization images when the source term is modeled (approximated). As examples, the approximate method is applied to PSP visualizations of nitrogen impinging jets and surface luminescent dye visualizations of delta wings in water flows.

6.1 6.1.1

Relation Between Skin Friction and Surface Scalar Concentration General Consideration

Since there is a formal analogy between heat transfer and mass transfer in the physical aspects of scalar transport in flows, global skin friction diagnostics could be made based on surface mass-transfer visualizations with PSP as an oxygen sensor and sublimating coatings. Earlier attempt was made on developing local electrochemical mass-transfer sensors for skin friction measurements since a local solution of the mass transport equation is available for data reduction (Hanratty and Campbell 1996). The key for developing a global method is to establish the on-wall relation between skin friction and scalar concentration on a surface.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_6

135

136

6

Skin Friction from Surface Scalar Visualizations

Similar to the energy equation Eq. (5.11), the on-wall relation between skin friction and surface scalar concentration can be derived from the binary mass transport equation with a source term for an incompressible flow, i.e., (Burmeister 1993) ∂ϕ1 =∂t þ u  ∇ϕ1 = D12 ∇2 ϕ1 þ Qs ,

ð6:1Þ

where ϕ1 = ρ1 =ρ is the relative concentration (density) of the species 1, ρ = ρ1 þ ρ2 is the total density of the binary gas, D12 is the diffusivity of a binary system, and Qs is the material source term. Since Eq. (6.1) has the same mathematical structure as the energy equation, following the same procedure described in Chap. 5, we obtain the on-wall relation between skin friction and surface scalar concentration on a curved surface, i.e., (Liu et al. 2014; Chen et al. 2019) τ  ∇∂B ϕ1∂B = μf M ,

ð6:2Þ

where the source term is written as fM =

∂ 1 - D12 ∇2∂B m_ 1∂B þ Rm , D12 ρ∂B ∂t

ð6:3Þ

and the effects of the surface curvature, the material source term and the third-order normal derivative term are contained in 2

3

∂ ϕ1 ∂ ϕ1 þ D12 þ ½K : K m_ 1∂B =ρ∂B 2 , ∂x ∂x33 ∂B ∂B 3 ∂B þ2D12 Trð∇∂B ∇∂B ϕ∂B  K Þ þ D12 ð∇∂B  K Þ  ð∇∂B ϕ∂B Þ

Rm = þ

∂Qs ∂x3

- 2D12 H mean

ð6:4Þ and the diffusive flux of species 1 on the surface is m_ 1∂B = - D12 ρ∂B ½∂ϕ1 =∂x3 ∂B :

ð6:5Þ

Here, x3 is the wall-normal coordinate and ðx1 , x2 Þ are the surface coordinates. The term ∂=∂t - D12 ∇2∂B m_ 1∂B in Eq. (6.3) is interpreted as a source term in the formal diffusion process of the mass flux on the surface. The time derivative of the mass-transfer flux in Eq. (6.3) is difficult to measure globally. To simplify the problem, we apply a time-average operator hiT = ΔT - 1

T1 T0

 dt,

ð6:6Þ

6.1

Relation Between Skin Friction and Surface Scalar Concentration

137

where ΔT = T 1 - T 0 is the time span of a time interval ½T 0 , T 1 . The time-averaged version of Eq. (6.2) is hGiT þ C ii hτi iT ∂hϕ1∂B iT =∂xi = 0, ði = 1, 2Þ

ð6:7Þ

where G = - μf M and C i are the correlation coefficients defined as C i = hτi ∂ϕ1w =∂xi iT =hτi iT ∂hϕ1w iT =∂xi (no summation convention is applied here). Further, when a snapshot of a time-dependent flow is considered in the time interval ½T 0 , T 1 , we have the following expressions: τi = τi,s W ðt; T 0 , T 1 Þ, ϕ1∂B = ϕ1,0 þ ϕ1∂B,s W ðt; T 0 , T 1 Þ: m_ 1∂B = m_ 1∂B,s W ðt; T 0 , T 1 Þ, where τi,s , ϕ1∂B,s , and m_ 1∂B,s are the steady-state (or short-time-averaged) fields of the skin friction, relative density, and mass-transfer rate of the species 1 in ½T 0 , T 1 , respectively, and ϕ1,0 is a constant initial field. The window function is defined as W ðt; T 0 , T 1 Þ = H ðt - T 0 Þ - H ðt - T 1 Þ, where the Heaviside function is defined as H ðt Þ = 1 for t > 0 and H ðt Þ = 0 for t ≤ 0. In a short time interval, since τi = hτi iT = τi,s , hϕ1∂B iT = ϕ1,0 þ ϕ1∂B,s , hm_ 1w iT = m_ 1w,s , and C j = 1 in ½T 0 , T 1 , after the reminder term Rm is neglected, Eq. (6.7) becomes hGiT þ τi ∂ϕ1∂B,s =∂xi = 0, ði = 1, 2Þ

ð6:8Þ

where hGiT =

μ ρ∂B D12 ΔT

2

- m_ 1∂B,s þ D12 ΔT

∂ m_ 1∂B,s : ∂xi ∂xi

ð6:9Þ

At this stage, the species 1 has not been specified yet, which depends on a specific surface scalar visualization technique used in experiments (see Sects. 6.1.2 and 6.1.3). Furthermore, the scalar concentration ϕ1∂B,s should be converted to the image intensity measured by a specific visualization technique (PSP or sublimating coating).

138

6.1.2

6

Skin Friction from Surface Scalar Visualizations

Surface Scalar Visualization with PSP

PSP is actually an oxygen sensor that could be used for surface mass-transfer visualizations in low-speed flows (Liu et al. 2014, 2021d). If nitrogen is added in flow as the species 1 that diffuses into a PSP polymer layer to purge oxygen (the species 2) in the layer, PSP will emit the stronger luminescence due to the reduced oxygen quenching. In the other case where oxygen is injected to flow, the luminescent emission from PSP will be quenched. According to the solution of the 1D steady-state mass-transfer equation with the zero-flux condition at the PSP-solid interface, the mass-transfer flux into a thin PSP layer at the gas-PSP interface is expressed as m_ 1∂B = - ρ∂B γ m ϕ1∂B - ϕ1∂B,00 ,

ð6:10Þ

where ϕ1∂B and ϕ1∂B,00 are the values of ϕ1 at the gas-PSP interface and the PSP-solid interface, respectively, γ m = 2D1p =h, D1p is the diffusivity of the species 1 in the polymer, and h is the coating thickness. The luminescent emission of PSP depends on the relative oxygen density ϕ2 = ρ2 =ρ (the species 2) in PSP due to the oxygen quenching. The Stern-Volmer relation between the luminescent intensity of PSP and the relative density of oxygen is I 0 =I = 1 þ Kϕ2 ,

ð6:11Þ

where I and I 0 are the intensities in the testing and zero-oxygen conditions, respectively, and K is the Stern-Volmer coefficient (Liu et al. 2021d). Since the zero-oxygen condition cannot be easily applied to fluid mechanics experiments, the wind-off reference condition is usually used. In this case, the Stern-Volmer relation is rewritten as I ref =I = A þ Bðϕ2 =ϕ2ref Þ ϕ2 = ðϕ2ref =BÞðI ref =I Þ - A ϕ2ref =B, where I and I ref are the luminescent intensities of PSP in the wind-on and wind-off reference conditions, respectively, A = ð1 þ Kϕ2ref Þ - 1 and B = Kϕ2ref A are the alternative Stern-Volmer coefficients for aerodynamics applications, and ϕ2ref is the value of ϕ2 at the wind-off reference condition. When nitrogen is added in flow for surface mass-transfer visualizations with PSP, by using the relation ϕ1 = 1 - ϕ2 for the relative density of nitrogen (the species 1) in PSP, Eq. (6.8) becomes

6.1

Relation Between Skin Friction and Surface Scalar Concentration

139

2

g-

I ref ∂ g D Δ Tτi ∂g - D12 ΔT þ 12 = 0, ði = 1, 2Þ I 00 μγ m ∂xi ∂xi ∂xi

ð6:12Þ

where g = I ref =I is the normalized luminescent intensity and I 00 is the luminescent intensity of PSP that is corresponding to ϕ1,00 at the PSP-solid interface. In image-based measurements, Eq. (6.12) is projected onto the image plane in the orthographical projection, and therefore in the image plane, it is expressed as G þ ^τi ∂g=∂xi = 0, ði = 1, 2Þ

ð6:13Þ

where the relative skin friction vector projected onto the image plane is ^τi = ðD12 ΔT=γ m μÞλτi , and other terms are G = g - I ref =I 00 þ ε0 , 2

ε0 = - D12 ΔT λ2 ∂ g=∂xi ∂xi : Here, the time span ΔT is arbitrary, and it does not change the normalized skin friction since it is cancelled out in normalization by dividing the reference value at a selected location. The scaling constant λ in the orthographical projection is approximately the ratio between the focal length of a camera and the distance between the camera and the surface. Therefore, since λ is a small parameter, ε0  λ2 is a higherorder small term. For PSP in a gas flow added with nitrogen (the species 1), since I > I 00 , G < 0 due to the smaller density of nitrogen at the PSP-solid interface. In contrast, when oxygen could be added as the species 1 for surface visualization with PSP, although Eq. (6.13) remains the same, G > 0 since I < I 00 . In applications, a problem is that the luminescent intensity I 00 at the PSP-solid interface is unknown, which is directly related to ϕ1,00 in the Stern-Volmer relation. Since ω1,00 depends on the masstransfer flux distribution, I 00 is generally position-dependent and unknown. Due to this difficulty, the term I ref =I 00 - ε0 is approximately treated as a constant free parameter that is around one since I ref =I 00 is around one and ε0 is small.

6.1.3

Visualization with Sublimating Coatings

6.1.3.1

Coating with Changing Density

Mass-transfer visualizations on a delta wing in a wind tunnel were conducted by Bouvier et al. (2001) using a pyrene luminescent coating (PSP), where pyrene was evaporated in a long run time, while the coating thickness remained unchanged.

140

6 Skin Friction from Surface Scalar Visualizations

When a flow starts suddenly at t = T 0 , for a thin sublimating coating on a wall, the time rate of the mass of the species 1 per area is ∂ðρ10 ϕ1∂B hÞ=∂t = - m_ 1∂B,s W ðt; T 0 , T 1 Þ,

ð6:14Þ

where m_ 1∂B,s is the steady-state mass diffusion flux, ρ10 is the density of a sublimating material in the solid state (or the saturated vapor state), h is the coating thickness that is not dependent of time, and the window function is defined as W ðt; T 0 , T 1 Þ = H ðt - T 0 Þ - H ðt - T 1 Þ, and H ðt Þ is the Heaviside function. Integration of Eq. (6.14) in a time interval ½T 0 , T 1  gives the relation m_ 1∂B,s = ΔT - 1 ðρ10 ϕ1∂B hÞref 1 - ϕ1∂B h=ω1∂B,ref href , where ðρ10 ϕ1∂B hÞref is the reference value at t = T 0 in the wind-off reference condition and ΔT = T 1 - T 0 is the time span. For an optically thin sublimating coating, the relative luminescent intensity of the coating is proportionally related to the relative density ϕ1∂B by I=I ref = ϕ1∂B h=ϕ1∂B,ref href , where I ref and ϕ1∂B,ref are the luminescent intensity and the relative density of the species 1 at the wind-off condition, respectively. Therefore, Eq. (6.8) becomes g - 1 - D12 ΔT

2 D ΔT 2 ρ∂B τi ∂ href ∂ g þ 12 g μ href ρ10 ∂xi h ∂xi ∂xi

þ

D12 ΔT 2 ρ∂B τi ∂g =0, μ hρ10 ∂xi ð6:15Þ

where g = I=I ref is the normalized luminescent intensity. Equation (6.15) can be expressed in the image plane as G þ ^τi ∂g=∂xi = 0, ði = 1, 2Þ where the relative skin friction vector projected onto the image plane is ^τi = D12 ΔT 2 ρ∂B =h μ ρ10 λ τi , and other terms are G = g - 1 þ ε1 þ ε 2 ,

ð6:16Þ

6.1

Relation Between Skin Friction and Surface Scalar Concentration

141

2

ε1 = - D12 ΔT λ2 ∂ g=∂xi ∂xi , ε2 = D12 ΔT 2 ρ∂B =href μ ρ10 λgτi ∂ðhref =hÞ=∂xi : Since λ 550 nm) was mounted on the lens to filter the light captured by the camera allowing only detection of the luminescent emission of TSP. The model was tested at AoAs of 5° and 10° at a freestream velocity of 15 m/s. The Reynolds number based on the chord length was 1.27 × 105. To visualize surface temperature patterns with TSP, the conductive layer on the model was heated by supplying the power of 480 Watts keeping the heating layer temperature constant at about 30 °C higher than the freestream temperature before a run. The model surface was maintained continuous heating in the tests. Figure 7.9 shows the fields of the normalized time-averaged surface temperature hT w i= maxhT w i and surface temperature fluctuation energy eT = max eT on the upper surface of the model at AoA of 5°, where h
i denotes a time-averaging operator (99 instantaneous fields were averaged here). In Fig. 7.9 and other relevant figures, x is the chordwise coordinate with the origin at the leading edge of the airfoil and c is the chord length. The separation bubble near the leading edge is visualized as a region of the higher hT w i= maxhT w i and lower eT = max eT along the spanwise direction. Figure 7.10 shows the fields of hT w i= maxhT w i and eT = max eT on the upper surface of the model at AoA of 10°. Figures 7.11 and 7.12 show the spanwiseaveraged profiles of hT w i= maxhT w i and eT = max eT at AoAs of 5° and 10°, respectively. The location of the separation bubble moves toward the leading edge as AoA increases from 5° to 10°. The peaks of eT in both the cases occur after flow attachment due to the Görtler-like instability.

7.3

Applications

169

Fig. 7.9 Normalized time-averaged surface temperature field (upper) and surface temperature fluctuation energy field (lower) on a NACA0015 airfoil model at AoA of 5°. (From Liu et al. (2022a))

Fig. 7.10 Normalized time-averaged surface temperature field (upper) and surface temperature fluctuation energy field (lower) on a NACA0015 airfoil model at AoA of 10°. (From Liu et al. (2022a))

The SOF of temperature uT is calculated from a pair of sequential eT -images, and thus a sequence of uT -fields is obtained. Figure 7.13 shows the fields of the magnitude and vectors of the SOF on the upper surface of the NACA0015 airfoil model at AoAs of 5° and 10°. Figures 7.14 and 7.15 show the spanwise-averaged profiles of the normalized streamwise component of the SOF at AoAs of 5° and 10°, respectively. The separation bubble is clearly identified as a region with the negative streamwise skin friction component that is characterized by the separation and attachment lines. As expected, the location of the separation bubble approaches to the leading edge at AoA of 10°. After the flow attachment, the skin friction

170

7

Skin Friction from Surface Optical Flow

Fig. 7.11 Profiles of the time-averaged surface temperature on a NACA0015 airfoil model at AoAs of 5° and 10°. (From Liu et al. (2022a))

Fig. 7.12 Profiles of the time-averaged surface temperature fluctuation energy on a NACA0015 airfoil model at AoAs of 5° and 10°. (From Liu et al. (2022a))

magnitude rapidly increases and reaches to the maximum at a downstream location (x/c = 0.55 for AoA of 5° and 0.4 for AoA of 10°). The rapid increase of the skin friction magnitude corresponds to that of the time-averaged temperature fluctuation energy shown in Fig. 7.12. This indicates that the elevated skin friction after the attachment is caused by enhanced turbulence.

7.3

Applications

171

Fig. 7.13 Vectors of the SOF on the SOF magnitude fields on a NACA0015 airfoil model at AoAs of 5° and 10°. (From Liu et al. (2022a))

Fig. 7.14 Spanwise-averaged profiles of the normalized streamwise component of the SOF on a NACA0015 airfoil model at AoA of 5°. The normalized skin friction profile obtained by using the GLOF method is also plotted for comparison. (From Liu et al. (2022a))

To validate the results of the SOF of temperature, GLOF skin friction measurements on the same NACA0015 airfoil model was carried out in the same flow conditions. The principle of the GLOF method is described in Chap. 3. Figure 7.16 shows typical GLOF images on the NACA0015 airfoil model at AoAs of 5° and 10°. Figure 7.17 shows the normalized skin friction fields obtained by the GLOF method. The normalized spanwise-averaged profiles of the streamwise skin friction component are plotted in Figs. 7.14 and 7.15 for comparison with those of the normalized streamwise component of the SOF. The results of the SOF and skin friction are consistent.

172

7

Skin Friction from Surface Optical Flow

Fig. 7.15 Spanwise-averaged profiles of the normalized streamwise component of the SOF on a NACA0015 airfoil model at AoA of 10°. The normalized skin friction profile obtained by using the GLOF method is also plotted for comparison. (From Liu et al. (2022a))

Fig. 7.16 GLOF images on a NACA0015 airfoil model at AoAs of 5° and 10°. The freestream flow is from left to right. (From Liu et al. (2022a))

7.3.3

Impinging Jet

The fields of the SOF of temperature were determined from time-resolved TSP measurements in an impinging jet with different impingement angles. The impinging jet experimental setup consisted of a primary rectangular settling chamber

7.3

Applications

173

Fig. 7.17 Skin friction vectors on the normalized skin friction magnitude fields obtained by using the GLOF method on a NACA0015 airfoil model at AoAs of 5°. and 10°. (From Liu et al. (2022a))

(203 mm × 304 mm × 304 mm) connected to a secondary 147-mm-long cylindrical chamber with a 50.8 mm diameter. A contoured nozzle with an exit diameter (D) of 2 mm, a length of 35 mm, and a contraction ratio of 5.6 was placed at the end of the cylindrical chamber. Using compressed air introduced in the primary settling chamber, the nozzle was able to produce an air jet flow. The exit jet velocity was set at U o = 50 m=s, and the corresponding Reynolds number based on the nozzle exit diameter was Re D = U o D=ν = 6:2 × 103 . The ratio between the nozzle-to-surface distance and the nozzle exit diameter was H=D = 6. The impingement angle was variable, which was defined as the angle between axisymmetric axis and the impingement surface. To measure surface temperature fields, the jet impinged on a 177.8-mm-wide, 228.6-mm-long, and 0.0254-mm-thick stainless steel sheet heated electrically. Two aluminum rods with diameters of 38.1 mm were used as electrodes. The sheet was held in place using four tension coil springs (two in each side). The stainless steel sheet was heated to a surface temperature of 70 °C by supplying the total power of 600 Watts across the sheet using a power generator. EuTTA-dope TSP was used in the experiments, which was applied using an airbrush on a white Mylar film on the backside of the sheet to form a 10-μm-thick TSP layer. During testing, TSP was excited using two UV lamps. TSP images were acquired using a high-speed camera at a frame rate of 480 fps with a resolution of 730 × 570 pixels. The camera was equipped with a long-pass optical filter (>570 nm).

174

7

Skin Friction from Surface Optical Flow

Fig. 7.18 Time-averaged fields in a normally impinging jet: (a) surface temperature difference, (b) normalized surface temperature fluctuation energy, and (c) the SOF vectors on the normalized SOF magnitude field. (From Liu et al. (2022a))

Figure 7.18a shows the time-averaged field of the surface temperature difference hT ∂B - T ref i on the impingement surface when the impingement angle is 90° (normal impingement), where T ref is the ambient temperature. Figure 7.18b shows the normalized time-averaged field of the surface temperature fluctuation energy eT = max eT . The SOF of temperature is calculated from a pair of sequential eT -images with a time interval of 10 ms between them, and thus a sequence of uT -fields is obtained. Figure 7.18c shows the time-averaged uT -field over 100 snapshot fields and the uT -magnitude field normalized by its maximum value of the ring structure.

7.3

Applications

175

Fig. 7.19 Profile of the normalized SOF magnitude in comparison with the hot-film skin friction data in a normally impinging jet. (From Liu et al. (2022a))

Figure 7.19 shows the transverse profile of the time-averaged SOF magnitude normalized the maximum value of skin friction in comparison with the hot-film skin friction data obtained by Liu and Woodiga (2011) in a normally impinging jet from a circular tube in the similar conditions. The profile of the SOF and hot-film skin friction data are consistent particularly in the impingement region. Figure 7.20 shows the SOF vectors on the normalized SOF magnitude fields in the impinging jet at the impingement angles of 75°, 45°, and 30°. The SOF field becomes more asymmetrical relative to the x-axis (the transverse axis across the impingement point) as the impingement angle decreases.

176

7 Skin Friction from Surface Optical Flow

Fig. 7.20 The SOF vectors on the normalized SOF magnitude fields in an impinging jet at different impingement angles: (a) 75°, (b) 45°, and (c) 30°. (From Liu et al. (2022a))

Chapter 8

Pressure from Velocity

This chapter describes extraction of static pressure from velocity as an inverse problem. The nondimensional relation between the total pressure and the velocityrelated source term is given as an adapted form of the Navier-Stokes (NS) equations. Projection of the gradient of the total pressure on a given test vector field leads to a first-order partial differential equation. For a constant test vector field, the method of characteristics is described to calculate a pressure field by using a line integral along a given characteristic line, and then an error analysis for this line integral method is given. Further, a global variational formulation is proposed and the Euler-Lagrange equation is derived for the total pressure. In particular, for a constant test vector field, a second-order elliptical-type particle differential equation with constant coefficients is obtained, where the Neumann condition is imposed on the boundary of a domain. To solve this Euler-Lagrange equation with suitable boundary conditions for a field of the total pressure, the Direct-QR method and the iterative method are described. The accuracy of these methods is evaluated through simulations in the oblique Hiemenz flow. As an example, the pressure fields near a freely flying hawkmoth are obtained from high-resolution velocity fields extracted from Schlieren visualization images by the optical flow method.

8.1 8.1.1

Basic Relation Pressure and Velocity

In general, extraction of a pressure field from a measured velocity field is an inverse problem for inferring a cause from an observable effect since pressure is a driving force in flows. Cai et al. (2020) proposed a variational formulation for the problem of pressure from velocity. To solve this problem, the relation between pressure and velocity is required. Based on the NS equations for an incompressible flow, the gradient of the nondimensional total pressure P is given by © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_8

177

178

8

∇P = Φ,

Pressure from Velocity

ð8:1Þ

where the RHS of Eq. (8.1) is a source term defined as Φ = - St

∂u þ Re - 1 ∇2 u - ω × u: ∂t

ð8:2Þ

In Eqs. (8.1) and (8.2), the nondimensional variables and operators are defined as ~j2 =2 =ρU 2ref ðtotal pressureÞ, P = ~p þ ρju ~=U ref ðvelocityÞ, u=u ~ lref =U ref ðvorticityÞ, ω=ω x = ~x=lref ðcoordinatesÞ, t = ~t f ref ðtimeÞ, ~ ðgradient operatorÞ, ∇ = lref ∇ ~ 2 ðLaplace operatorÞ, ∇2 = l2ref ∇ Re = U ref lref =ν ðReynolds numberÞ, St = f ref lref =U ref ðStrouhal numberÞ, where the tilde “~” denotes the dimensional quantities, ρ is the density of fluid, and ν is the kinematical viscosity, and lref , U ref , and f ref are the reference length, velocity and frequency, respectively. The advantage of using the total pressure P rather than the static pressure p in Eq. (8.1) is that Bernoulli’s integral P = const: is automatically a reduced case of Eq. (8.1) for a steady, irrotational, and inviscid flow. The first, second, and third terms in Eq. (8.2) represent the local acceleration, viscous diffusion, and Lamb vector, respectively, which directly contribute the deviation from Bernoulli’s integral when it is applied to a viscous-flow domain. From this perspective, Bernoulli’s integral can be considered as an initial approximation, and then a refined solution of Eq. (8.1) can be sought for P when Φ is known. For a 2D flow, taking the dot product between Eq. (8.1) and a nondimensional test vector field m = mx , my , we have a first-order partial differential equation: m  ∇P = mx

∂P ∂P þ my = Φm ðx, yÞ ∂x ∂y

where the projection of Φ on m is a source term Φm = m  Φ:

ð8:3Þ

8.1

Basic Relation

179

For a steady, inviscid irrotational flow with Φm = 0, since m is arbitrarily selected, Eq. (8.3) gives P = const in the whole flow field, which is Bernoulli’s integral. Therefore, the deviation from Bernoulli’s integral is caused by the source term Φm representing the effects of the viscous diffusion, vorticity, and unsteadiness of flow. To solve this 2D problem, the method of characteristics and the variational method are described by Cai et al. (2020). The method of characteristics is a straightforward line integral from a single boundary of a domain where the value of P is given (see Sect. 8.2). In contrast, the variational method is a global method to minimize a constrained functional based on Eq. (8.3), which leads to the Euler-Lagrange equation for P with the boundary condition on a closed domain (see Sect. 8.3).

8.1.2

Pressure and Second Invariant

To elucidate the pressure signature of vortex (or vorticity) structures, we discuss the relationship between the static pressure and the second invariant of the strain rate tensor through the pressure Poisson equation. For notational simplicity in the derivations in this subsection, we drop the tilde “~” from the dimensional variables used in Sect. 8.1.1. In an incompressible flow, taking the divergence of the NS equations, we have ∇2 p = - ρ∇  ðu  ∇uÞ = - ρeα 

∂u ∂∇u  ∇u þ u  ∂xα ∂xα

ð8:4Þ

where eα (α = 1, 2, 3) are the unit base vectors in the orthogonal coordinates. The second term in the RHS of Eq. (8.4) is - ρu 

∂∇u ∂u  eα = - ρu  ∇ ∂xα ∂xα

 eα = - ρu  ∇

∂uα = - ρu  ∇ð∇  uÞ = 0 ∂xα

ð8:5Þ where the incompressibility condition ∇  u = 0 is applied. The first term in the RHS of Eq. (8.4) is - ρeα 

∂u ∂u  ∇u = - ρ α  ∇u  eα = - ρ∇u : ð∇uÞT = - ρTr ð∇uÞ2 ∂xα ∂x ð8:6Þ

where “:” denotes the double dot product of tensors and Tr denotes the trace of a tensor. For further evaluation of Eq. (8.6), the Cauchy-Helmholtz decomposition is used. At a small neighborhood of any point in a velocity field that is a differentiable smooth function, the velocity field uðr, t Þ can be expanded as

180

8

Pressure from Velocity

uðr þ ΔrÞ - uðrÞ = u  ∇  Δr þ OðΔrÞ, where r is a positional vector of a point in flow, Δr is a change of the position, and u  ∇ = ð∇  uÞT denotes the transposition of a tensor field. For generality, the tensor product notation  is used here, and, for example, the relevant terms are u  ð∇  T Þ  u  ∇T, u  ð∇  uÞ  u  ∇u, u  u  uu, where T is a scalar field. The velocity gradient can be decomposed into a symmetrical tenor and an antisymmetric part, i.e., u  ∇=S þ Ω

ð8:7Þ

where the symmetrical part S = ðu  ∇ þ ∇  uÞ=2 is the strain rate tensor representing the deformation of a fluid element and the antisymmetric part Ω = ðu  ∇ - ∇  uÞ=2 is the rotation tensor representing an infinitesimal rotation of a fluid element. Using the relations ð∇uÞT = D þ Ω and ∇u = S - Ω, we have - ρTr ð∇uÞ2 = - ρðS - ΩÞ : ðS þ ΩÞ = - ρðS : S - Ω : ΩÞ = 2ρQ

ð8:8Þ

where the second invariant of the strain rate tensor is Q = Ω2 - S2 =2: The use of Eqs. (8.4), (8.5), (8.6), and (8.8) leads to a compact form of the pressure Poisson equation ∇2 p = 2ρQ:

ð8:9Þ

Equation (8.9) clearly indicates the relationship between the static pressure (a dynamical quantity) and the second invariant (a kinematical quantity) as a source term. To illustrate the effect of Q on a pressure change, we consider a domain V in which the flow with Q is enclosed by the boundary ∂V. The pressure difference Δp = p - p1 satisfies ∇2 ðΔpÞ = 2ρQ,

8.2

Method of Characteristics

181

where p1 is the constant freestream pressure. The boundary conditions are Δp = f 1 and ∂ðΔpÞ=∂n = f 2 on ∂V. A formal solution of the Poisson equation is (Sobolev 1989) Δpðx0 Þ = -

ρ 2π

V

Q r - 1 dV þ

1 4π

∂ð1=rÞ 1 f 1 dS 4π ∂n ∂V

∂V

r - 1 f 2 dS ð8:10Þ

where r = jx - x0 j is a distance. The first term in Eq. (8.10) is referred to as the Newtonian potential of generating the pressure variation, while the second and third terms are the potentials of the double layer and single layer, respectively. When f 1 and f 2 approach to zero sufficiently fast as the boundary ∂V goes to infinity, Eq. (8.10) is reduced to Δpðx0 Þ = -

ρ 2π

Q r - 1 dV V

ð8:11Þ

According to Eq. (8.10), the positive Q corresponds to a decrease of pressure. Therefore, it is inferred that a region with the positive Q (such as a vortex) corresponds to a low-pressure region. Hunt et al. (1988) defined as a “vortex” as a compact region with the positive Q without discussing the associated pressure signature. The above exploration adds the dynamical meaning of the kinematical definition of a “vortex” given by Hunt et al. (1988).

8.2 8.2.1

Method of Characteristics Line Integral

From a geometrical perspective, when the coordinates ðx, yÞ are expressed as a function of a single parameter s (e.g., the arc length), the coordinates describe a curve. Thus, from Eq. (8.3), the total derivative along the curve is dP dx ∂P dy ∂P þ = Φm ðx, yÞ: = ds ds ∂x ds ∂y

ð8:12Þ

The characteristic lines are determined by dx dy = mx and = my ds ds with the initial conditions xðs = 0Þ = 0 and yðs = 0Þ = c1 , where c1 is a parameter. In particular, when m is a constant vector, the characteristic line can be determined by solving Eq. (8.12), which is given by

182

8

Pressure from Velocity

y - y0 = my =mx x or x - x0 = mx =my y, where y0 and x0 are the coordinates of the initial position. Thus, Eq. (8.12) can be written as the following forms: dP = mx- 1 Φm ½x, ðmy =mx Þx þ y0  dx

ð8:13aÞ

dP = my- 1 Φm ½ðmx =my Þy þ x0 , y: dy

ð8:13bÞ

and

In the special cases with mx , my = ð1, 0Þ and mx , my = ð0, 1Þ, integration of Eqs. (8.13a) and (8.13b) yields Pðx, y0 Þ =

x x0

Φm x0 , y0 dx0 þ P0 ,

ð8:14aÞ

and Pðx0 , yÞ =

y y0

Φm ðx0 , y0 Þdy0 þ P0 ,

ð8:14bÞ

where P0 = Pðx0 , y0 Þ is the value at a reference location. Equations (8.14a) and (8.14b) are the integrals along the x and y coordinates, respectively, which can be conveniently applied to velocity data in the image plane by changing either the reference position coordinatey0 or x0 (scanning through the image plane in either the vertical or horizontal direction). In measurements of flow over a body, the value of Pðx0 , y0 Þ at a reference location is usually obtained in the outer flow where Pðx0 , y0 Þ = const and Bernoulli’s equation is naturally recovered from Eqs. (8.14a) and (8.14b).

8.2.2

Error Analysis

Without loss of generality, we consider the numerical integration in Eq. (8.14b), i.e., y y0

Φm ðx0 , y0 Þdy =

N

w k Φ m ð x0 , y k Þ h þ E N , k=1

ð8:15Þ

8.3

Variational Method

183

where wk (k = 1, ⋯, N ) are the weight coefficients related to an interpolating polynomial, E N is a residual term, h = ðy - y0 Þ=N is the spatial step, and N is the number of nodes. Therefore, the error of P is N

wk Δ Φm h þ E N þ ΔP0 ,

ΔP =

ð8:16Þ

k=1

where Δ Φm is the error in calculation of Φm and ΔP0 is the error of the value of P at a reference location. For the second-order numerical differentiation scheme, an error estimate is ΔΦm = K 1 h2 þ ε, where K 1 is a coefficient and ε is the measurement error. For the numerical integration, an error estimate is EN = K 2 hn N, where the exponent n and coefficient K 2 depend on a particular integration method used (Hildebrand 1974). For example, for the trapezoid rule, K 2 is proportional to 2 ∂ Φm =∂y2 at a certain point in the integral interval. Therefore, Eq. (8.16) gives ΔP = K 1 hwih3 N þ K 2 hn N þ ε hN hwi þ ΔP0 ,

ð8:17Þ

where the mean weight coefficient is hw i = N - 1

N

wk : k=1

The first term in the RHS of Eq. (8.17) is the error related to the numerical differentiation and integration, the second term is the measurement error accumulated in the integration, and the third term is the error of the initial value. The dependency of the error on the node number N is approximately described by a power-law relation ΔP / N - β , where β is an exponent.

8.3 8.3.1

Variational Method Euler-Lagrange Equation

In principle, solving Eq. (8.3) is not an ill-posed problem since the direct line integral can be applied. Nevertheless, we introduce a smoothness constraint for improving the robustness of computation against noise. Therefore, to solve Eq. (8.3) for P, a constrained variational formulation was proposed by Cai et al. (2020). Introducing the functional J ðPÞ =

D

ðm  ∇P - Φm Þ2 dxdy þ α

D

j∇Pj2 dxdy,

ð8:18Þ

184

8

Pressure from Velocity

we consider the minimization problem J ðPÞ → min , where D is a domain and α is a Lagrange multiplier. The first and second terms in Eq. (8.18) are the equation term and regularization term, respectively. The role of the regularization term is to impose a smoothness constraint. We consider a perturbed quantity P þ εv, where v is a variation (perturbation) and ε is a small amplitude. The optimality condition for J ðPÞ → min is d J ðP þ εvÞ dε

ε=0

=

D

ð∇  ðvf Þ - v∇  f Þdxdy =

∂D

^ dl vf  n

D

v∇  f dxdy = 0 ,

ð8:19Þ ^ is the normal vector on ∂D, and the vector f is where ∂D is the boundary of D, n defined as f = 2ðm  ∇P - Φm Þm þ 2α∇P

ð8:20Þ

In the derivation of Eq. (8.19), Green’s theorem is applied. Further, it is assumed ^  ∇P = 0 hold on that the equation m  ∇P - Φm = 0 and the Neumann condition n ∂D, such that the first term in the RHS of Eq. (8.19) vanishes. Since the domain D is arbitrary, Eq. (8.19) gives ∇  f = 0: Therefore, we obtain the Euler-Lagrange equation: m  ∇ðm  ∇P - Φm Þ þ ðm  ∇P - Φm Þ∇  m þ α ∇2 P = 0:

ð8:21Þ

When m is a constant vector, Eq. (8.21) can be written as 2

LðPÞ = A

2

2

∂ ∂ ∂ þ 2B þ C 2 P = F, ∂x2 ∂x∂y ∂y

where A = m2x þ α, B = mx my C = m2y þ α, F = m  ∇Φm : The determinant of Eq. (8.22) is Δ = B2 - AC = - α m2x þ m2y þ α :

ð8:22Þ

8.3

Variational Method

185

Thus, for α > 0, Δ < 0, indicating that Eq. (8.22) is an elliptic-type partial differential equation (Sobolev 1989). The Neumann condition is imposed on ∂D. The Dirichlet condition can be also adopted in a narrow extended region near ∂D such that the Neumann condition can be satisfied naturally.

8.3.2

Direct Numerical Algorithm

A discrete scheme is developed to solve Eq. (8.22) for P. We denote δ2x , δ2y , and δxy as 2 2 2 discrete approximations of ∂ =∂x2 , ∂ =∂y2 , and ∂ =∂x∂y, respectively, i.e., 2

∂ P ∂x2

i, j

≈ δ2x P

2

∂ P ∂y2

i, j

≈ δ2y P

x

ð8:23aÞ

y

ð8:23bÞ

= h - 2 Pi, j - 2h - 2 Pi, j ,

i, j

i, j

= h - 2 Pi, j - 2h - 2 Pi, j ,

2

∂ P ∂x∂y

i, j

≈ δxy P

x

i, j

=

signðtqÞPiþt, jþq = 4h2 ,

ð8:23cÞ

t , q2f - 1, 1g y

where Pi, j = Piþ1, j þ Pi - 1, j and Pi, j = Pi, jþ1 þ Pi, j - 1 . Similarly, we have ∂Φm ∂x

i, j

∂Φm ∂y

i, j

iþ1, j ≈ ðδx Φm Þi, j = ð2hÞ - 1 Φm - Φim- 1, j

ð8:24aÞ

i, jþ1 i, j - 1 = ð2hÞ - 1 Φm - Φm

ð8:24bÞ

≈ δy Φm

i, j

Therefore, a finite difference form of Eq. (8.22) is written as A δ2x P

i, j

þ 2B δxy P

i, j

þ C δ2y P

i, j

= mx ðδx Φm Þi, j þ my δy Φm

i, j

ð8:25Þ

with the Dirichlet boundary condition Pi, j = gi, j on ∂D, where 0 ≤ i ≤ W, 0 ≤ j ≤ H, and W and H denote the width and height of the given domain D, respectively. Figure 8.1 shows the gridding of the computational domain D. Equation (8.25) generates a linear system of ðH - 1Þ × ðW - 1Þ equations EP = b,

ð8:26Þ

where the ðH - 1Þ × ðW - 1Þ unknowns are contained in a vector T

P = P1,1 , P1,2 , ⋯, P1,ðH - 1Þ , P2,1 , ⋯, Pi, j , ⋯, PðW - 1Þ,ðH - 1Þ ,

ð8:27Þ

186

8

Pressure from Velocity

Fig. 8.1 Gridding of the computational domain D

E is a coefficient matrix with the size N 2 , and b is a vector with the given N = ðH - 1Þ × ðW - 1Þ elements. Most elements in E are zero, except for some nonzero elements with a regular distribution near the diagonal line. The structure of the sparse matrix E and the RHS vector b in Eq. (8.26) are given by Cai et al. (2020). The QR factorization with column pivoting is used as a direct method to solve Eq. (8.26), which is called Direct-QR method. If the dimension of a velocity field is H × W, the system of linear equations, Eq. (8.26), has H × W equations. Hence, it is a large-scale problem of linear equations in normal computations in a laptop computer. Downsampling and upsampling processing are necessary.

8.3.3

Iterative Numerical Algorithm

Although the Direct-QR method achieves better results than the characteristic method, a large-scale system of linear equations, Eq. (8.26), often brings high computational cost and requires a huge storage space. When the size of a velocity field is H × W, then the size of the system of equations increases to H 2 × W 2 . A practical approach to handle the problem is to downsample the original velocity field. However, a low downsampling rate (such as less than 8%) could lead to a larger computational error. To overcome the problem, from Eq. (8.25), an iterative scheme is proposed, i.e., ðk Þ

ðkþ1Þ Pi, j

=

ðk Þ

ðk Þ

Ag1 =2 þ Bg2 =4 þ Cg3 - h2 =2

mx  ðδx Φm Þi, j þ my δy Φm

i, j

AþC ð8:28Þ

8.3

Variational Method

187

where ðk Þ

ðk Þ

ðk Þ

g1 = Piþ1, j þ Pi - 1, j , ðk Þ

ðk Þ

ðk Þ

ðk Þ

ð8:29aÞ ðk Þ

g2 = Piþ1, jþ1 - Piþ1, j - 1 - Pi - 1, jþ1 þ Pi - 1, j - 1 , ðk Þ

ðk Þ

ð8:29bÞ

ðk Þ

g3 = Pi, jþ1 þ Pi, j - 1 :

ð8:29cÞ

After the initialization of pressure Pð0Þ is given, Eq. (8.28) can be used to calculate ðk Þ the total pressure through iterations. In order to achieve faster convergence, Pi - 1, j in ðk Þ

ðkþ1Þ

ðk Þ

ðk Þ

ðkþ1Þ

ðk Þ

ðk Þ

g1 is replaced by Pi - 1, j , Pi - 1, j - 1 in g2 is replaced by Pi - 1, j - 1 , and Pi, j - 1 in g3 ðkþ1Þ

is replaced by Pi, j - 1 . In addition, a successive over-relaxation (SOR) strategy is applied in iteration, as shown in Algorithm 8.1, where the characteristic method is used to estimate the boundary conditions. Algorithm 8.1 Input: The velocity field ux , uy . Initialization: Given Pð0Þ . Boundary conditions estimated by the characteristic method. Initialize error tolerance ε. SOR factor ω. k = 0. While T err > ε, repeat Step 1. Obtain Pðkþ1Þ from the iterative formula given in Eq. (8.28). Step 2. Pðkþ1Þ = ωPðkþ1Þ þ ð1 - ωÞPðkÞ . Step 3. Update T err = Pðkþ1Þ - PðkÞ Step 4. k ← k þ 1. End Output: The solution Pðkþ1Þ .

2 . 2

By comparing with the matrix representation given in Eq. (8.26), the transformation given in Eq. (8.28) is equivalent to a decomposition of the coefficient matrix E = M - N, where the diagonal matrix with elements 2Ah - 2 þ 2Ch - 2 is M = 2Ah - 2 þ 2Ch - 2 : × I, and N is the residual matrix. We have ðM - NÞP = b and then P = M - 1 ðNP þ bÞ: By defining B = M - 1 N and f = M - 1 b, the iterative procedure given in Eq. (8.28) can be expressed as Pðkþ1Þ = BPðkÞ þ f

ð8:30Þ

188

8 Pressure from Velocity

We have kBk = M - 1 N ≤ M - 1 kN k = kN k=kM k, where kk denotes the matrix norm. Based on the definition of matrix M and properties of the diagonally dominant matrix, the norm of the iterative matrix B is less than 1. Finally, we have ρðBÞ ≤ kBk < 1, where ρðÞ is the spectral radius. Hence, the introduced iterative algorithm is convergent (Strang 2006).

8.4 8.4.1

Method Validation Oblique Hiemenz Flow

The accuracy of the method of characteristics, the Direct-QR method, and the iterative method is evaluated through simulation in the 2D oblique Hiemenz flow, including the effects of the relevant parameters on the relative error, including the node number, the Lagrange multiplier, and the downsampling rate. In the oblique Hiemenz flow, the stream function of the outer inviscid flow is given by ψ = α y2 cosðθÞ=2 þ xy sinðθÞ , where α is a constant representing the characteristic strain rate, θ is the characteristic impingement angle of fluid, and x and y are the horizontal and vertical coordinates, respectively. The geometrical impingement angle γ, which is defined as the angle between the inviscid zero streamline and the flat impingement surface, is related to θ by 2 tan θ = tan γ. The origin of the coordinate system is located at the intersection between the inviscid zero streamline and the flat impingement surface. This flow is an exact solution of the NS equations (Dorrepaal 1986; Liu 1992), which is briefly recapitulated here. The nondimensional coordinates are introduced as ξ = ax=ls η = ay=ls , ls = ðν=αÞ1=2 , a = ðsin θÞ1=2 : The nondimensional velocity components are defined as u = ðls a=νÞ u, v = ðls a=νÞ v: The nondimensional velocity components have the similar forms: u = cos θ H ðηÞ þ a2 ξF 0 ðηÞ, v = - a2 F ðηÞ:

ð8:31Þ

8.4

Method Validation

189

where the similarity functions F ðηÞ and H ðηÞ are the similarity functions. The similarity equations F 000 þ FF 00 - F02 þ 1 = 0

ð8:32Þ

where the boundary conditions are F ð0Þ = 0, F 0 ð0Þ = 0 and F 0 ð1Þ = 1, and H 00 þ FH 0 - F 0 H þ A = 0

ð8:33Þ

where the boundary conditions are H ð0Þ = 0 and H 0 ð1Þ = 1, and A = 0:6479. The prime in Eqs. (8.32) and (8.33) denotes the derivative with respect to η. Equations (8.32) and (8.33) have been solved for F ðηÞ and H ðηÞ (Dorrepaal 1986; Liu 1992). For the 2D oblique Hiemenz flow, the nondimensional relation between the total pressure and the source term is ∇P = Φ

ð8:34Þ

where P = p þ a - 2 k , Φ = ∇2 u - a - 2 ω × u, p = p=μα, u = u ls a=ν, k = juj2 =2, 2

ω = ω=α, ∇ = ∇ ls , ∇2 = ∇ l2s : The field of the nondimensional pressure difference is given by Δp = p00 - pðξ, ηÞ = a2 F 2 ðηÞ=2 þ F 0 ðηÞ þ 0:5ξ a2 ξ þ 2A cos θ

ð8:35Þ

where p00 = pð0, 0Þ is the pressure at the coordinate origin or the inviscid impingement point.

8.4.2

Method of Characteristics

The fields of the nondimensional pressure difference Δp for three geometrical impingement angles (γ = 60°, 75°, and 90°) are extracted from the corresponding velocity fields by using the method of characteristics, which are shown in Fig. 8.2. The line integral is calculated from the top boundary of the domain to the wall. At the top boundary, the flow is basically inviscid such that Bernoulli’s equation is valid, i.e., Pðx0 , y0 Þ = const. Figure 8.3 shows comparisons between the extracted pressure profiles and the exact solution at several locations from the surface for the node number of 1000. It is found that the error of the extracted results is relatively large near the wall, and the

190

8

Pressure from Velocity

Fig. 8.2 Extracted pressure fields in the oblique Hiemenz flow by using the method of characteristics for three geometrical impingement angles: (a) 60°, (b) 75°, and (c) 90°, where streamlines are superposed. (From Cai et al. (2020))

error decreases as the location moves farther from the surface. By comparing with the exact solution, the root-mean-square (RMS) error normalized by the maximum pressure difference in the domain is calculated as a measure of the accuracy of the extracted results. The RMS error of pressure is defined as

p kp - ptrue k= N , where kk denotes a L2 norm and N is the node number of the velocity field. Figure 8.4 shows the relative error of the extracted pressure as a function of the nondimensional vertical coordinate η for the node number of 1000. As η decreases to zero, the relative error increases in the three cases since the numerical error in calculation of spatial differentiations becomes larger in the near-wall boundary layer (in several viscous length scales). As the impingement angle decreases, the relative error increases since the boundary layer thickness that is proportional to

8.4

Method Validation

191

Fig. 8.3 Comparisons between the extracted pressure profiles by using the method of characteristics and the exact solution: (a) 60°, (b) 75°, and 90°. Lines and symbols denote the exact solution and extracted data, respectively. (From Cai et al. (2020)) Fig. 8.4 Relative error of the extracted pressure by using the method of characteristics as a function of the nondimensional vertical coordinate for the node number of 1000. (From Cai et al. (2020))

ls =ðsin θÞ1=2 increases as θ decreases. Figure 8.5a shows the log-log plots of the relative error as a function of the node number N, which decays as N - 0:5 . This result is consistent with the error analysis for the method of characteristics in Sect. 8.2.2. The relative error increases as the impingement angle decreases. To simulate the effect of the fixed-pattern error in a velocity field, a velocity variation

192

8

Pressure from Velocity

Fig. 8.5 Relative error of the extracted pressure by using the method of characteristics as a function of (a) node number and (b) relative amplitude of velocity error for the node number of 1000. (From Cai et al. (2020))

δu= maxðjujÞ = Am sinð2πx=LÞ sinð2πy=LÞ is superposed on the velocity field, where Am is the relative amplitude and L is the wavelength of the pattern. Figure 8.5b shows the relative error of the extracted pressure as a function of Am for the node number of 1000, which increases moderately as Am increases in all the three cases.

8.4.3

Variational Method

These cases of the oblique Hiemenz flow are also used to evaluate the Direct-QR method for calculating pressure from velocity by solving the Euler-Lagrange equation. For computational complexity reasons, the velocity field is downsampled in computations, and then the extracted pressure field is upsampled to the original size using nearest-neighbor interpolation for comparison. Figure 8.6 shows the exact and extracted pressure fields and the difference between them for the node number of 1000. Figure 8.7 shows comparisons between the extracted pressure profiles and the exact solution at several locations from the surface. Figures 8.6 and 8.7 indicate that the Direct-QR method achieves more accurate results than the method of characteristics particularly near the wall since the boundary condition is imposed at the wall. The RMS error normalized by the maximum pressure difference in the domain is calculated as a measure of the accuracy of the extracted results. Figure 8.8a shows the relative error distributions of the extracted pressure near the wall at η = 0:009 in the horizontal coordinate ξ. Figure 8.8b shows the relative error distributions of the extracted pressure in the vertical coordinate η.

8.4

Method Validation

193

Fig. 8.6 Exact pressure fields, extracted pressure fields by using the Direct-QR method, and absolute error in the oblique Hiemenz flow at (a) 60°, (b) 75°, and (c) 90°. (From Cai et al. (2020))

For the global stability of the Direct-QR method, the coefficient matrix E in Eq. (8.26) should have a good condition number (Higham 2002). Since mx and my satisfy a constraint m2x þ m2y = 1, the absolute values of both mx and my are small. Hence, the Lagrange multiplier α in Eq. (8.26) should be large relatively. Figure 8.9 shows the relative RMS error of the extracted pressure using the Direct-QR method as a function of the Lagrange multiplier α for γ = 60°, 75°, and 90°. As indicated in Fig. 8.9, the bigger value of α achieves better performance. When α is bigger than 32, the result becomes stable asymptotically. The nondimensional pressure fields in the same test cases are calculated by the iterative algorithm. The pressure data on the bottom and right boundaries are estimated using the method of characteristics as the Dirichlet boundary condition, where the line integrals are calculated from the given top and left boundaries respectively. The relative RMS error of the extracted pressure is given in Fig. 8.10

194

8

Pressure from Velocity

Fig. 8.7 Comparisons between the extracted pressure profiles by using the Direct-QR method and the exact solution: (a) 60°, (b) 75°, and 90°. Lines and symbols denote the exact solution and extracted data, respectively. (From Cai et al. (2020))

Fig. 8.8 Relative error of the extracted pressure by using the Direct-QR method as a function of (a) the horizontal coordinate near the wall (at η = 0:009) and (b) the nondimensional vertical coordinate for the node number of 1000. (From Cai et al. (2020))

as a function of the nondimensional vertical coordinated η. In the three cases, the relative error increases while η decreases due to the error of the boundary value estimation. To evaluate the stability of the iterative method, a spatial velocity variation is superposed on the velocity field. Figure 8.11 shows the RMS error normalized by the maximum pressure difference in the domain as a function of the relative magnitude Am of the velocity variation.

8.5

Application: Hawkmoth Flight

195

Fig. 8.9 Relative error of the extracted pressure by using the Direct-QR method as a function of the Lagrange multiplier α for the node number of 1000. (From Cai et al. (2020))

Fig. 8.10 Relative error of the extracted pressure by using the iterative method as a function of the nondimensional vertical coordinate. (From Cai et al. (2020))

8.5

Application: Hawkmoth Flight

The methods of pressure from velocity were applied to high-resolution velocity fields around a freely flying hawkmoth (Manduca) to obtain pressure fields. Liu et al. (2018b) visualized the unsteady flow around a freely flying Manduca with alcohol

196

8

Pressure from Velocity

Fig. 8.11 Relative error of the extracted pressure by using the iterative method as a function of the relative amplitude of the velocity variation. (From Cai et al. (2020))

vapor as tracers through a Schlieren system. The Schlieren system is illustrated in Fig. 8.12. The system consisted of two single mirror, double-pass, Schlieren set-ups with two high-speed cameras, imaging from two orthogonal views. In each Schlieren system, a white LED light was projected through a 1 mm pinhole onto a 10-inch diameter optical spherical mirror. A portion of the reflected light was then redirected toward the high-speed camera using a 50/50 beam splitter for capture. At the focal point of the reflected light, a razor blade was used as the knife edge producing the Schlieren images. To visualize the vortical flow produced by the flapping Manduca wings, warm isopropyl alcohol (91% isopropyl alcohol heated to 38 °C) was brushed onto the surface of the wings at the beginning of each test. The Manduca was then released to fly freely in the flow observation domain. When the wings flapped, enhanced vaporization of the alcohol from the wings caused the nonuniform density distribution of a mixture of the alcohol vapor and air in the flow, visualizing the flow structures in Schlieren images. Figure 8.13 shows two typical consecutive Schlieren images with a time interval of 1 ms between them, which were captured on a nearly hovering Manduca with an averaged flying speed of 0.3 m/s. The rectangular domains in the images are selected for extracting a velocity field. To obtain high-resolution velocity fields, OFM was applied to this pair of Schlieren images (Liu and Shen 2008; Wang et al. 2015) (see Chap. 2). In this example, an open source optical flow program was used (Liu 2017). The velocity

8.5

Application: Hawkmoth Flight

197

Fig. 8.12 Schematic of a Schlieren imaging system for a freely flying hawkmoth. (From Liu et al. (2018a, b))

Fig. 8.13 Typical Schlieren image pair visualizing the wingtip flow structures. The sub-images in the rectangle tracking domain are used for extracting a velocity field. (From Cai et al. (2020))

198

8

Pressure from Velocity

Fig. 8.14 Velocity vectors extracted from a pair of the Schlieren images by using OFM, where the vorticity and second invariant fields are superposed. (From Cai et al. (2020))

extracted from Schlieren images is the light-path-averaged velocity along a light ray through the measurement domain. The extracted velocity field and vorticity and second invariant fields are shown in Fig. 8.14, where the quantities are normalized by their maximum magnitudes and the coordinates are normalized by Manduca’s body length. There are 149 × 76 velocity vectors in the domain. The second invariant Q represents a balance between the vorticity magnitude and shear strain. When Q is positive, the rotational motion locally prevails over the shearing motion. Thus, the second invariant field in Fig. 8.14b shows the vortical structures generated by the flapping wing. The Direct-QR method and the iterative method were used to extract pressure fields from the velocity field in Fig. 8.14. Figure 8.15 shows the normalized pressure fields extracted by using the Direct-QR method and the iterative method, where the normalized pressure is defined as

ðp - pa Þ= maxðjp - pa jÞ, and pa is the atmospheric pressure. The results given by both the methods are consistent in capturing the tip vortical features. Pressure extracted from Schlieren

8.5

Application: Hawkmoth Flight

199

Fig. 8.15 Pressure fields near the hawkmoth wingtip obtained by using (a) Direct-QR method and (b) iterative method, where the second invariant field is superposed. (From Cai et al. (2020))

images is interpreted as the light-path-averaged pressure through the measurement domain. By comparing Figs. 8.14b and 8.15, it is found that the patterns of the second invariant field and pressure field are well correlated, and the region with the positive Q (the vortex region) corresponds to the low-pressure region. This observed correspondence between the second invariant and pressure is consistent with the discussions in Sect. 8.1.2. It is noted that the method of characteristics does not give satisfactory result in this case due to the accumulated error in line integration.

Chapter 9

Surface Pressure from Skin Friction

This chapter describes extraction of surface pressure from skin friction in complex flows as an inverse problem, focusing on its application to global luminescent oil-film (GLOF) skin friction measurements. The relation between surface pressure and skin friction is the foundation of this method. A variational formulation is given, leading to the Euler-Lagrange equation that is a second-order elliptic-type partial differential equation with the variable coefficients where the Neumann condition is imposed. The numerical algorithm for solving the Euler-Lagrange equation is described, and then the approximate method with a constant boundary enstrophy flux (BEF) is proposed, which is particularly useful for its application to complex flows. The accuracy of this method is evaluated in the Falkner-Skan flow and the separated flow over a 70°-delta wing in simulations. Further, to obtain the normalized surface pressure fields, the approximate method is applied to the skin friction fields obtained by GLOF measurements in the separated flow over a 65° delta wing and the square junction flow.

9.1 9.1.1

Formulation and Analysis Variational Method

Cai et al. (2022) studied the problem of surface pressure from skin friction to resolve the difficulties in global surface pressure measurements in low-speed complex flows, which is related to the application of the GLOF method (see Chap. 3). The foundation of this method is the on-wall relation between skin friction (τ ) and surface pressure (p) given in Eq. (4.11). For convenience of reading, this on-wall relation is recapitulated, i.e.,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_9

201

202

9

Surface Pressure from Skin Friction

τ
∇p = Φ,

ð9:1Þ

where Φ = μf Ω is a source term and f Ω is given in Eq. (4.12). Here, for notational simplicity, the subscript ∂B in Eq. (4.11) is dropped. In contrast to the problem of skin friction from surface pressure discussed in Chap. 4, the present problem is a reciprocal one, that is, extraction of surface pressure from skin friction. Given τ and Φ, to solve Eq. (9.1) for p, a variational formulation is proposed, i.e., J ðpÞ =

D

ðτ
∇p - ΦÞ2 dxdy þ α

D

j∇pj2 dxdy → min,

ð9:2Þ

where D is a domain on a surface (or the image plane after projection) and α is a Lagrange multiplier. The first and second terms in Eq. (9.2) are the equation term and regularization term as a smoothness constraint, respectively. This problem is formally similar to the problem of pressure from velocity considered by Cai et al. (2020) (see Chap. 8). To solve the minimization problem J ðpÞ → min , we consider a perturbed quantity p þ εv, where v is a variation and ε is a small amplitude. The optimality condition is d J ðp þ εvÞ dε

ε=0

= =

D

f
∇v dxdy =

∂D

v f
n dl -

D

D

ð∇
ðv f Þ - v∇
f Þdxdy ,

ð9:3Þ

v∇
f dxdy = 0

where ∂D is the boundary of the domain D, n is the normal vector on ∂D, and the vector f is defined as f = 2ðτ
∇p - ΦÞτ þ 2α∇p:

ð9:4Þ

In the derivation of Eq. (9.3), Green’s theorem is applied, i.e.,

D

∇
ðvf Þdxdy =

∂D

vf
n dl:

ð9:5Þ

On the boundary ∂D, the basic relation, Eq. (9.1), holds on ∂D. Further, it is assumed that the Neumann condition on ∂D holds, i.e., n
∇p = ∂p=∂n = 0: Therefore, the first term in the RHS of Eq. (9.3) vanishes. Since the domain D is arbitrary, Eq. (9.3) gives

9.1

Formulation and Analysis

203

∇
f = 0: Thus, we have the Euler-Lagrange equation τ
∇ðτ
∇p - ΦÞ þ ðτ
∇p - ΦÞ∇
τ þ α∇2 p = 0:

ð9:6Þ

Further, Eq. (9.6) can be written as a typical second-order partial differential equation: LðpÞ = F,

ð9:7Þ

where the partial differential operator is 2

L=A

2

2

∂ ∂ ∂ ∂ ∂ þC 2þD þE , þ 2B ∂x2 ∂x∂y ∂y ∂x ∂y

ð9:8Þ

and the coefficients are A = τx 2 þ α,

ð9:9aÞ

B = τx τy ,

ð9:9bÞ

C = τy þ α,

ð9:9cÞ

D = τx ∇
τ þ τ
∇τx ,

ð9:9dÞ

E = τy ∇
τ þ τ
∇τy ,

ð9:9eÞ

F = Φ∇
τ þ τ
∇Φ:

ð9:9fÞ

2

The determinant of Eq. (9.7) is Δ = B2 - AC = - α jτ j2 þ α : Thus, for α > 0, the determinant is Δ < 0, indicating that Eq. (9.7) is an elliptictype partial differential equation (Sobolev 1989). The Neumann condition ∂p=∂n = 0 on ∂D is imposed for Eq. (9.7). It is noted that Eq. (8.22) in the problem of pressure from velocity has a special form of Eq. (9.7), where the coefficients A, B, and C are constants and D and E are zero. Interestingly, the coefficients D, E and F contain the skin friction divergence ∇
τ that characterizes the skin friction topology and plays an important role in determining near-wall flow structures (Chen et al. 2021a, b; Chen and Liu 2023). The dynamical role of ∇
τ is explicitly expressed in the Taylor series expansion solution of the NS equations for the near-wall velocity and pressure. Essentially, Eq. (9.7) provides the relationship between surface pressure and skin friction structures in the variational framework.

204

9

Surface Pressure from Skin Friction

In particular, for a stationary curved surface, there is an exact on-wall relation derived from the NS equations, i.e., ∂p ∂n

∂B

= - ∇
τ,

elucidating the relationship between the wall-normal pressure derivative on the surface and the skin friction divergence ∇
τ (Chen et al. 2021a). In a skin friction field, a τ -source with ∇
τ > 0 and a τ -sink with ∇
τ < 0 correspond to the negative ½∂p=∂n∂B and positive ½∂p=∂n∂B , respectively. Further, an attachment line with ∇
τ > 0 corresponds to the negative ½∂p=∂n∂B and in contrast a separation line with ∇
τ < 0 corresponds to the positive ½∂p=∂n∂B . Here, an attachment line is defined as a skin friction line from which neighboring skin friction lines diverge, while a separation line is defined as a skin friction line to which neighboring skin friction lines converge. Since the skin friction topology in complex flows is characterized by ∇
τ, the method is particularly applicable to topologically complex flows where k∇
τ k is dominant.

9.1.2

Neumann Condition and Lagrange Multiplier

Neumann Condition The accuracy of the Neumann condition imposed on the problem of surface pressure from skin friction is examined. As indicated before, the physically real surface quantities τ r , pr , and Φr satisfy the exact on-wall relation: τ r
∇pr = Φr ,

ð9:10Þ

where the subscript “r” denotes the real (true) quantities. In addition, from the NS equations, the pressure gradient on a surface is given by 2

∇½pr ∂B = - n × σ = μ

∂ uπ ∂n2

ð9:11Þ

∂B

where σ = μ½∂ω=∂n∂B is the boundary vorticity flux (BVF) (Wu et al. 2000, 2006), n is the unit wall-normal vector of the surface, and uπ is the velocity parallel to the surface. The BVF represents the boundary vorticity creation rate. On the boundary curve (contour) ∂D on the surface, from Eq. (9.10), we have the pressure derivative normal to the boundary ∂D, i.e., ∂½pr ∂B ∂ðω × nÞ = -n
n×σ =n
μ ∂n ∂n

2

∂B

=μ

∂ ðuπ
nÞ ∂n2

ð9:12Þ ∂B

9.1

Formulation and Analysis

205

where n is the outward-pointing normal vector of ∂D. According to Eq. (9.12), the normal derivative of the physically real surface pressure at ∂D (e.g., ∂½pr ∂B =∂n) is determined by the boundary-normal diffusion of the surface-tangential velocity component uπ
n. In general, the RHS term of Eq. (9.12) does not vanish automatically. The Neumann condition ∂½pr ∂B =∂n ≈ 0 on ∂D holds as an effective approximation for kn
n × σ k ≪ 1. This could be more reasonable when the computational domain is sufficiently large. For image processing, the computational domain is usually much larger than the region of interest, and the effect of the Neumann condition as an approximation on the result is small. Lagrange Multiplier The requirements on the Lagrange multiplier are discussed. In the variational framework, the Euler-Lagrange equation for p is used, i.e., τ
∇ðτ
∇p - ΦÞ þ ðτ
∇p - ΦÞ∇
τ þ α∇2 p = 0

ð9:13Þ

with the Neumann condition ∂p=∂n = 0 on ∂D. The perturbed surface pressure is expressed as p = pr þ Δp, where pr is the physically real surface pressure (the exact solution of the NS equations) and Δp is the error introduced by the modeling (or approximation) in the variational framework. Similarly, the similar expression is Φ = Φr þ Δ Φ, where Φr and Δ Φ represent the physically real source term and the corresponding error, respectively. For simplicity, it is assumed that skin friction is accurate, i.e., τ = τ r . In this case, when δp → 0 and Δ Φ → 0, the Lagrange multiplier α should be sufficiently small such that the effect of the surface pressure Laplacian ∇2 p in Eq. (9.13) can be minimized. Then, the limiting behavior of Eq. (9.13) is consistent with Eq. (9.12). On the other hand, we consider the pressure Poisson equation: 2

∇ 2 pr þ

∂ pr 1 = ρ ω2 - S : S 2 ∂n2

ð9:14Þ

where S is the strain rate tensor. Applying the pressure Poisson equation on the surface, we have 2

∇2 p r = -

∂ pr ∂n2

ð9:15Þ

∂B

For most wall regions, according to Eq. (9.11), the first-order wall-normal pressure derivative ½∂pr =∂n∂B is determined by the skin friction divergence ∇
τ r that is directly related to the skin friction topology. In contrast, the effect of the 2 is relatively weak. For second-order wall-normal pressure derivative ∂ pr =∂n2 ∂B

example, in unidirectional pressure-driven Poiseuille flow, this term is exactly zero.

206

9

Surface Pressure from Skin Friction

A simple analysis from Eq. (9.15) indicates that the error Δp in most surface regions should satisfy a Poisson equation, i.e., ∇2 Δp ≈ - α - 1 ½τ r
∇ðτ r
∇Δp - Δ ΦÞ þ ðτ r
∇Δp - Δ ΦÞ∇
τ r 

ð9:16Þ

Therefore, increasing the value of α tends to attenuate the source term in Eq. (9.16), thereby reducing the magnitude of the error Δp and leading to a smoother surface pressure distribution. Near the separation and attachment lines dominated by the skin friction divergence ∇
τ r , Eq. (9.16) is reduced to ∇2 Δp ≈ α - 1 Δ Φ∇
τ r

ð9:17Þ

According to this estimate, the reduction of the pressure error could be achieved by more physical modeling of the BEF and a suitable Lagrange multiplier. In summary, to meet the above conflicting requirements on the Lagrange multiplier, there is an optimal value of the Lagrange multiplier.

9.1.3

Error Analysis

We consider the decompositions: p = p0 þ Δp, A = A0 þ ΔA, B = B0 þ ΔB, C = C 0 þ ΔC D = D0 þ ΔD, E = E0 þ ΔE, F = F 0 þ ΔF

ð9:18Þ

where the subscript “0” denotes a base solution satisfying exactly Eq. (9.7) and Δ denotes a variation from the base solution (or an error). Similarly, the operator is decomposed into two parts, i.e., L = L0 þ ΔL,

ð9:19Þ

where 2

L0 = A0 2

ΔL = ΔA

2

2

∂ ∂ ∂ ∂ ∂ þ 2B0 þ C 0 2 þ D0 þ E0 , 2 ∂x ∂x∂y ∂y ∂x ∂y 2

ð9:20Þ

2

∂ ∂ ∂ ∂ ∂ þ 2ΔB þ ΔC 2 þ ΔD þ ΔE : ∂x ∂y ∂x2 ∂x∂y ∂y

By using Eqs. (9.18)–(9.21), an error propagation equation is given as

ð9:21Þ

9.1

Formulation and Analysis

207

L0 ðΔpÞ = ΔF - ΔLðp0 Þ:

ð9:22Þ

Symbolically, the error is expressed as Δp = L0- 1 ðΔF Þ - L0- 1 ½ΔLðp0 Þ,

ð9:23Þ

where the inverse operator is given by L0- 1 ð
Þ =

D

Gðx, y; x0 , y0 Þð
Þdx0 dy0 ,

ð9:24Þ

and Gðx, y; x0 , y0 Þ is a Green’s function. According to the Cauchy-Schwarz inequality, an estimate of the error upper bound is kΔpk ≤ L0- 1 ðΔF Þ þ L0- 1 ½ΔLðp0 Þ ≤

D

Gðx, y; x0 , y0 Þdx0 dy0

hΔF iD þ hΔLðp0 ÞiD D ,

ð9:25Þ where the domain-averaging operator is defined as h
iD = D - 1

D

ð
Þdxdy:

ð9:26Þ

To compare the terms in the RHS of Eq. (9.25), we use the decompositions of the quantities τ = τ 0 þ Δτ and Φ = Φ0 þ ΔΦ, where Δτ and Δ Φ are the perturbations of these quantities and Δ denotes a difference operator. Then, we give the following order estimates: hΔLðp0 ÞiD  OðkτkkΔτkÞ þ OðkτkkΔ ð∇
τÞkÞ, hΔF iD  OðkτkkΔð∇Δ ΦÞkÞ þ OðkΔτkk∇Δ ΦkÞ,

ð9:27aÞ ð9:27bÞ

where Oð
Þ denotes the order of a quantity. According to Eq. (9.27), we have an estimate: hΔF iD ≪ hΔLðp0 ÞiD : Therefore, the solution of Eq. (9.7) is not very sensitive to an approximation of the source term F. This estimate provides a foundation for the approximation of the constant BEF where the effect of the variation of the BEF is neglected as a smaller term in this problem.

208

9.1.4

9

Surface Pressure from Skin Friction

Approximate Method with the Constant BEF

The above method is based on the solution of the Euler-Lagrange equation, Eq. (9.7), when the source term Φ = μf Ω is given. However, in practical measurements in complex flows, the BEF (f Ω ) is not known a priori. Therefore, an approximation is required to close this open problem. The simplest approximation is to use a negative constant source term, i.e., Φ = const (f Ω = constÞ, since the BEF is generally negative. Thus, a relative (or normalized) surface pressure field is obtained by using the approximate method, where a proportional constant is to be determined by in-situ calibration using pressure tap data at several locations. According to Eq. (9.9f), the effect of Φ on the solution of Eq. (9.7) is solely contained in the coefficient F as a source term. This constant BEF approximation is reasonable when the condition kΦ∇
τ k ≫ kτ
∇Φk is satisfied. The significance of the skin friction divergence ∇
τ in this condition is obvious. The approximate method is applicable to topologically complex flows in which k∇
τ k is dominant. Although Φ could be an arbitrary negative constant, simulations indicate that the absolute value of Φ should be a fraction of kτ
∇pk to achieve the good accuracy. For a specific measurement, trial-and-error tests are used to select a suitable value of Φ. In a certain range of Φ, the extracted surface pressure field normalized by a reference value at a location is not very sensitive to the value of Φ.

9.2

Numerical Scheme

A discrete scheme is given to solve the second-order partial differential equation Eq. (9.7) for the surface pressure (Strang 2014; Buhmiler et al. 2018; Roul et al. 2019). The simple notations δx , δy , δ2x , δ2y , and δxy are used for finite difference 2 2 2 approximations of ∂=∂x, ∂=∂y, ∂ =∂x2 , ∂ =∂y2 , and ∂ =∂x∂y, respectively, i.e., ∂p ∂x ∂p ∂y

i,j

i,j

≈ ðδx pÞi,j = 2h - 1 piþ1,j - pi - 1,j ,

ð9:28aÞ

= 2h - 1 pi,jþ1 - pi,j - 1 ,

ð9:28bÞ

= h - 2 pxi,j - 2h - 2 pi,j ,

ð9:28cÞ

= h - 2 pyi,j - 2h - 2 pi,j ,

ð9:28dÞ

≈ δy p

i,j

2

∂ p ∂x2

i,j

≈ δ2x p

i,j

2

∂ p ∂y2

i,j

≈ δ2y p

i,j

9.3

Method Validation

209

2

∂ p ∂x∂y

i,j

≈ δxy p

i,j

=

signðabÞpiþt,jþq = 4h2 ,

ð9:28eÞ

a, b2f - 1, 1g

where pxi,j = piþ1,j þ pi - 1,j and pyi,j = pi,jþ1 þ pi,j - 1 . Similarly, we can define finite difference approximations for ∂τx =∂x, ∂τy =∂y, ∂ Φ=∂x, and ∂Φ=∂y which were considered in the coefficients D, E, and F in Eq. (9.7), i.e., ∂τx ∂x ∂τy ∂y

i,j

∂Φ ∂x

i,j

∂Φ ∂y

≈ ðδx τx Þi,j = 2h - 1 τxiþ1,j - τix- 1,j ,

ð9:29aÞ

-1 = 2h - 1 τyi,jþ1 - τi,j , y

ð9:29bÞ

≈ ðδx ΦÞi,j = 2h - 1 Φiþ1,j - Φi - 1,j ,

ð9:29cÞ

= 2h - 1 Φi,jþ1 - Φi,j - 1 :

ð9:29dÞ

i,j

i,j

≈ δy τy

≈ δy Φ

i,j

i,j

Therefore, for a given domain D with the width W and the height H, a finite difference form of Eq. (9.7) can be written as A δ2x p

i,j

þ 2B δxy p

i,j

þ C δ2y p

= ðΦÞi,j ðδx τx Þi,j þ δy τy

i,j

i,j

þ Dðδx pÞi,j þ E δy p

þ ðτx Þi,j ðδx ΦÞi,j þ τy

i,j

i,j

δy Φ

,

ð9:30Þ

i,j

with the Neumann boundary condition on the boundary ∂D, where 0 ≤ i ≤ W and 0 ≤ j ≤ H. A system of linear equations with a sparse coefficient matrix is generated from Eq. (9.30), and the QR factorization with column pivoting is used to solve this system.

9.3 9.3.1

Method Validation Falkner-Skan Flow

To evaluate the accuracy of the method of surface pressure from skin friction, we first consider the Falkner-Skan flow (the wedge flow) with the external velocity U ðxÞ = axm , where x is the coordinate on the wedge surface from the wedge leading edge, m is a power-law exponent, and a is a positive constant (Schlichting and Gersten 2017). The surface pressure and pressure gradient are given by

210

9

Surface Pressure from Skin Friction

pðxÞ = p0 - 0:5pa2 x2m ,

ð9:31Þ

where p0 is the total pressure. The pressure gradient is ∂p=∂x = - ρa2 mx2m - 1 . Skin friction is given by τ=

3m - 1 m þ 1 1=2 3=2 00 ρv a f ð0Þx 2 , 2

ð9:32Þ

where v is the kinematic viscosity of fluid. The BEF is given by fΩ =μ

∂Ω mþ1 a = - μβ 2 v ∂y

3=2

a2 f 00 ð0Þx

7m - 3 2

ð9:33Þ

Here, f ðηÞ is the similar boundary-layer velocity profile with the similarity variable η. The value of the second derivative f 00 at the wall is approximately expressed as the piecewise functions (Ishak et al. 2007), i.e., f 00 ð0Þ = 0:749m0:5049 þ 0:4696, 0 ≤ m ≤ 1, f 00 ð0Þ = 1:696m þ 0:4741: - 0:06 ≤ m ≤ 0:2 In simulation, for v = 1:5 × 10 - 5 m2 =s, ρ = 1:2 kg=m3 , and a = 1, the images (2D fields) of the surface pressure, skin friction, and BEF are generated. Figure 9.1 shows the skin friction field (vectors and normalized magnitude) and the normalized BEF field (f Ω = maxjf Ω j) in the Falkner-Skan flow with a constant pressure gradient (m = 1=2 ). The corresponding normalized surface pressure field is shown in Fig. 9.2a. These fields are normalized by their maximum absolute values. The extracted surface pressure field by using the introduced method from the skin friction data is given in Fig. 9.2b when the source term Φ is exactly given, where the Lagrange multiplier is α = 0:3. The extracted surface pressure profile is in excellent agreement with the ground truth exhibiting a linear decay of surface pressure in the streamwise coordinate, as indicated in Fig. 9.3. The relative RMS error is 3.5 × 10-4,

Fig. 9.1 (a) Skin friction vectors on the normalized skin friction magnitude field and (b) the BEF field in the Falkner-Skan flow with m = 1=2. (From Cai et al. (2022))

9.3

Method Validation

211

Fig. 9.2 Normalized surface pressure fields: (a) ground truth, (b) extracted result from the exact Φ, and (c) extracted result from Φ = 10 - 5 in the Falkner-Skan flow with m = 1=2. (From Cai et al. (2022))

Fig. 9.3 Normalized surface pressure profiles in the Falkner-Skan flow with m = 1=2, where the ground truth and extracted results from the exact Φ and different constant values of Φ are compared. (From Cai et al. (2022))

212

9

Surface Pressure from Skin Friction

Fig. 9.4 Relative RMS error of the extracted surface pressure field as a function of the constant Φ in the Falkner-Skan flow with m = 1=2. (From Cai et al. (2022))

where the error is normalized by the maximum pressure. This error is mainly caused by the numerical error in computation. The extracted surface pressure field by using the approximate method with the constant Φ = - 10 - 5 is shown in Fig. 9.2c. Figure 9.3 shows the extracted surface pressure profiles normalized by their maximum values compared with the ground truth, where the extracted results overlap with the ground truth except for Φ = - 10 - 2 . Even in the case of Φ = - 10 - 2 where a significant shift is observed due to the normalization by the surface pressure at the wedge leading edge, the result still exhibits the linear behavior. This shift could be largely corrected by normalization using a value at a suitable location. The dependency of the relative RMS error on Φ is shown in Fig. 9.4, indicating that the relative RMS error is small in a range of Φ = - 10 - 2 to Φ = - 10 - 5 . Figure 9.5 shows the relative RMS error of the extracted surface pressure field as a function of the Lagrange multiplier α for Φ = - 10 - 4 . The power-law exponent m describes the behavior of the surface pressure. For the nonlinear surface pressure distributions with m = 1=3 and m = 2=3, the relative RMS errors are about 0.026 and 0.015, respectively, in a range of Φ = - 10 - 2 to Φ = - 10 - 5 . In these cases, the extracted surface pressure fields have small relative RMS errors in a range from Φ = - 10 - 2 to Φ = - 10 - 5 and a range from α = 0:05 to α = 0:5. There is no rigorous theory to determine the values of α and Φ. However, simulations in the Falkner-Skan flow give useful reference to select α and Φ and the ranges of the relative values: jΦj= maxjτ
∇pj = 10-4-0.4 and α= max jτj2 = 0.07–6.

9.3

Method Validation

213

Fig. 9.5 Relative RMS error of the extracted surface pressure field as a function of the Lagrange multiplier α in the Falkner-Skan flow with m = 1=2 for Φ = - 10 - 4 . (From Cai et al. (2022))

9.3.2

70°-Delta Wing

To further investigate the method of surface pressure from skin friction applied to complex flows, a 70°-delta wing is considered as a typical case at AoA of 20°, Mach number of 0.55, and total pressure of 100 kPa. The normalized surface pressure and skin friction fields obtained in CFD simulations are used to reconstruct the surface pressure field for validation of the method (Liu et al. 2016). Figure 9.6 shows the skin friction and BEF fields on the delta wing, where the coordinates are normalized by the root-chord length (c). Here, the BEF field is normalized by its maximum absolute value, i.e., Φ= max j Φ j = f Ω = maxjf Ω j: The original fields of skin friction and surface pressure have 1000 × 1000 pixels. The separation lines are located approximately at the ray lines of 75.7° and 104.3° swept angles near the leading edges, while the attachment line is at the centerline of the delta wing. The surface pressure field normalized the maximum value is shown in Fig. 9.7a. Since the size of the original skin friction data is too large (1000 × 1000 pixels), downsampling is necessary in computation using this algorithm. Numerical experiments indicate that the accuracy of the extracted result is improved with a higher sampling rate N. However, in practical computation, the sampling rate must be not larger than 0.1 (10% downsampling). For example, for N = 0:2, the size of the

214

9

Surface Pressure from Skin Friction

Fig. 9.6 Skin friction and BEF fields on a 70°-delta wing at AoA of 20°: (a) skin friction vectors, (b) skin friction lines, (c) normalized BEF. (From Cai et al. (2022))

constructed coefficient matrix will be 51282 × 51282, which is beyond the limit of the memory in a laptop computer used in this study. In this validation case, it is found that the algorithm can achieve the satisfactory performance for N = 0:1, and thus this sampling rate is used. The extracted surface pressure field from the downsampled skin friction data (N = 0:1) is given in Fig. 9.7b, where the Lagrange multiplier is α = 0:017 in computation. For comparison, the true and extracted surface pressure profiles are shown in Fig. 9.8, indicating that the extracted result is reasonably good.

9.3

Method Validation

215

Fig. 9.7 Surface pressure fields normalized by the maximum value on a 70°-delta wing at AoA of 20°: (a) ground truth, and (b) extracted field. In computation, the Lagrange multiplier is α = 0:017. (From Cai et al. (2022)) Fig. 9.8 Spanwise surface pressure profiles normalized by the maximum value on a 70°-delta wing at AoA of 20° and comparison between the ground truth and extracted result. In computation, the Lagrange multiplier is α = 0:017. (From Cai et al. (2022))

216

9

Surface Pressure from Skin Friction

Fig. 9.9 Relative RMS error of the extracted surface pressure as a function of the Lagrange multiplier α. (From Cai et al. (2022))

The effect of the Lagrange multiplier α on the accuracy of the extracted result is evaluated. In general, the extracted surface pressure fields become smoother with higher Lagrange multiplier, indicating that the developed method is stable in a certain range of the Lagrange multiplier. The relative RMS errors (normalized by the maximum pressure) are calculated, and the relative RMS error in the case of no noise is shown in Fig. 9.9 as a function of the Lagrange multiplier. It is found that α = 0:08 leads to the smallest RMS error of the solution. The stability of the extracted result from the noise-corrupted skin friction data is studied. To simulate the effect of image-based skin friction measurement uncertainty, a Gaussian white noise of the variance σ 2 is superposed on both the skin friction components τ x and τ y . The noise-corrupted skin friction data are employed to examine the stability of surface pressure determination from skin friction. The RMS errors of the extracted pressure from the clean and noisy skin friction data are shown in Fig. 9.9, indicating that the method can achieve the satisfactory results even when the skin friction data are considerably contaminated. The approximation method is applied to this problem, where Φ is set as a negative constant. The extracted surface pressure field is approximately proportional to the ground truth. Figure 9.10a, b show the extracted surface pressure fields and profiles normalized by the maximum value from the downsampled skin friction data with N = 0:1 for Φ = - 0:1, respectively. It is found that this approximation of a negative constant Φ leads to the reasonable result in this case. The relative RMS error of the extracted surface is about 0.03. The extracted skin friction field in this flow is not sensitive to Φ and α in their ranges.

9.4

Applications

217

Fig. 9.10 Surface pressure field normalized by the maximum value on a 70°-delta wing at AoA of 20°: (a) extracted field and (b) spanwise profiles. In computation, the source term is Φ = - 0:1, and the Lagrange multiplier is α = 0:1. (From Cai et al. (2022))

9.4 9.4.1

Applications 65°-Delta Wing

To examine the feasibility of measuring surface pressure from skin friction, GLOF measurements were carried out using a 65°-delta wing in a low-speed wind tunnel with a test section of 0.4 m × 0.4 m, where the freestream turbulence intensity was about 0.2%. The delta wing model was 3D-printed using polylactide. The model had a chord length of 190 mm, a span of 177 mm, and a thickness of 5 mm. Skin friction measurement were conducted at a freestream velocity of 20 m/s, corresponding to a chord-based Reynolds number of 2.57 × 105. Luminescent oil was made by blending an oil-based UV dye with silicone oil with the viscosity of 350 cs. Luminescent oil was brushed onto a model surface to ensure uniform oil-film application. The resulting luminescent oil emitted the radiation at a longer wavelength (about 550–620 nm) when excited by UV illumination. Two UV lamps were positioned to ensure a uniform illumination field in the test section. A long-pass filter (>550 nm) was used for the detection of the luminescent emission centered at approximately 590 nm. The wind tunnel was run in a dark environment, and images were captured by a CCD camera at 16 fps with a resolution of 1024 × 1024 pixels. Figure 9.11 shows the skin friction field on the upper surface of the delta wing obtained using the GLOF method at AoA of 10°. A typical GLOF image and the normalized skin friction magnitude field are shown in Fig. 9.11a, b as backgrounds

218

9

Surface Pressure from Skin Friction

Fig. 9.11 Skin friction field obtained by using the GLOF method on a 65°-delta wing at AoA of 10°: (a) skin friction vectors on a typical GLOF image and (b) skin friction lines on the normalized skin friction magnitude field. (From Cai et al. (2022))

Fig. 9.12 Normalized surface pressure field extracted from the skin friction field on a 65°-delta wing at AoA of 10°: (a) surface pressure and (b) skin friction lines on the surface pressure field. (From Cai et al. (2022))

on which skin friction vectors and lines are superposed, respectively. The primary attachment line at the centerline of the delta wing and the two secondary separation lines near the leading edges are observed. The normalized surface pressure field is extracted from the skin friction field when the source term Φ is set as a negative constant, as shown in Fig. 9.12a. Computational tests indicate that the extracted result is not sensitive to the selection of the constant source term Φ and the Lagrange multiplier α when Φ is in a range from -0.1 to -0.01 and α is in a range from 0.01 to 0.5. The correspondence between the surface pressure and skin friction lines is shown in Fig. 9.12b.

9.4

Applications

219

Fig. 9.13 Surface pressure profile in the symmetrical axis on a 65°-delta wing at AoA of 10° and comparison between the extracted result and the pressure tap data obtained at Mach 0.4 and Reynolds number based on the chord length of 6 × 106. (From Cai et al. (2022))

Figure 9.13 shows the extracted surface pressure profile normalized by the maximum value in the symmetrical axis on the delta wing in comparison the pressure tap data obtained by Chu and Luckring (1996) at Mach 0.4 and Reynolds number of 6 × 106 based on the chord length. Figure 9.14 shows the spanwise surface pressure profiles normalized by the local maximum values at three chordwise locations on the delta wing in comparison with the pressure tap data obtained at Mach 0.4 and Reynolds number based on the chord length of 6 × 106. Overall, the extracted surface pressure data are in reasonable agreement with the pressure tap data although the Reynolds numbers and Mach numbers in these cases are different. Furthermore, according to the discussions on the skin friction divergence ∇
τ in the previous studies (Chen et al. 2021a; Chen and Liu 2022, 2023), the attachment line with ∇
τ > 0 at the symmetrical line of the delta wing approximately corresponds to a high surface pressure region. The two separation lines with ∇
τ < 0 near the leading edges approximately correspond to lower surface pressure regions. Therefore, the extracted surface pressure field is consistent with the corresponding skin friction field.

220

9

Surface Pressure from Skin Friction

Fig. 9.14 Spanwise surface pressure profiles at three chordwise locations on a 65°-delta wing at AoA of 10° and comparison between the extracted result and the pressure tap data obtained at Mach 0.4 and Reynolds number based on the chord length of 6 × 106. (From Cai et al. (2022))

9.4.2

Square Junction Flow

GLOF measurements in a square junction flow were conducted by Kakuta et al. (2010) in the Tohoku-University Basic Aerodynamic Research Wind Tunnel with the test section of 300 mm width, 300 mm height, and 760 mm length. The test model was a square cylinder that had 40 mm × 40 mm cross-section and 100 mm height, which was mounted on the flat plate. The freestream velocity was set at 50 m/ s. The incident angle relative to the freestream was set at 0° for the square cylinder. The Reynolds number based on the equivalent diameter D of the cylinder was Re D = 1:3 × 105 . The local Reynolds number was Re x = 7:8 × 105 for the location of the front of the cylinder at 230 mm from the flat plate leading edge. It was confirmed by hot-wire measurement that the incoming boundary layer was laminar under these conditions. In GLOF measurements, perylene-mixed silicone oil was used. Figure 9.15 shows skin friction vectors and lines on the floor surface in the square junction flow obtained by using the GLOF method, respectively, where a typical GLOF image and the normalized skin friction field are shown as backgrounds. The

9.4

Applications

221

Fig. 9.15 Skin friction field obtained by using the GLOF method on the square junction flow: (a) skin friction vectors on a typical GLOF image and (b) skin friction lines on the normalized skin friction magnitude field. (From Cai et al. (2022))

Fig. 9.16 Normalized surface pressure field extracted from the skin friction field in the square junction flow: (a) surface pressure and (b) skin friction lines on the normalized surface pressure field. (From Cai et al. (2022))

skin friction topology is clearly delineated by skin friction lines in Fig. 9.15b. The primary separation line bifurcates at the front saddle, which is associated with the primary horseshoe vortex formed at the front of the square cylinder. Two attachment lines originate from the front corners of the square cylinder, which are also the on-wall footprints of the primary horseshoe vortex. There are two spiraling nodes behind the cylinder, which are the time-averaged footprint of the vortex shedding in the wake. There is a saddle between the two nodes near the back surface of the cylinder. Figure 9.16a shows the normalized surface pressure field extracted from the skin friction field when Φ is set as a negative constant. The correspondence between the surface pressure and skin friction lines is shown in Fig. 9.16b. The high-pressure

222

9 Surface Pressure from Skin Friction

region occurs at the front of the cylinder. The lowest surface pressure region occurs behind the cylinder, which is associated with the spiraling nodes in the wake. The primary separation line corresponds to the lower pressure bands. The two high surface pressure bands connected to the front corners of the square cylinder are associated to the attachment lines. These observations are consistent with the analysis on the skin friction divergence (Chen et al. 2021a; Chen and Liu 2022, 2023). An attachment line with ∇
τ > 0 and a separation line with ∇
τ < 0 approximately correspond to a higher surface pressure region and a lower surface pressure region, respectively. In addition, the two spiraling nodes behind the square cylinder are sinks with ∇
τ < 0 since skin friction lines spiral inwardly there. Therefore, it is expected that they correspond to the lower surface pressure region. From this topological perspective, the extracted surface pressure pattern is overall reasonable.

Chapter 10

Heat Flux from Surface Temperature Visualizations

This chapter describes extraction of the surface heat flux from a time sequence of surface temperatures obtained in time-resolved surface temperature measurements with temperature sensitive paint (TSP) in high-speed flows. First, the analytical inverse heat transfer solution of the 1D heat conduction equation for the heat flux is derived for a polymer coating layer on a finite base (a semi-infinite base as a reduced case). This analytical solution is expressed as a convolution-type integral of the surface temperature history. Next, the numerical inverse heat transfer solution is described for the temperature-dependent polymer and base materials, which is useful for TSP measurements in high-enthalpy hypersonic wind tunnels. Further, based on the analogy between the 3D and 1D solutions of the heat conduction equation, the image deconvolution method is developed for correcting the lateral heat conduction effect on 1D heat flux calculation. As examples, the inverse heat transfer methods are applied to cone and wedge models in hypersonic wind tunnels and obliquely impinging sonic jets.

10.1 10.1.1

Analytical Inverse Heat Transfer Solution Equations and Laplace Transform

The determination of the surface heat flux fields from a time history of surface temperature fields measured by using the global techniques (such as TSP) in aerothermodynamic testing is an inverse problem. In TSP measurements in hypersonic wind tunnels, a thin sensor layer with a relatively low thermal conductivity is applied on the surface of a body (typically a metallic model). As illustrated in Fig. 10.1, TSP is usually air brushed on the surface of a white base coating layer applied to a metal model to enhance the luminescent emission. Illumination lights with a suitable wavelength range (such as UV light) are used to excite TSP. Digital cameras with optical filters are used to detect the luminescent emission with a longer © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_10

223

224

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.1 Illustration of a generic experimental setup for TSP/TP measurement

Fig. 10.2 A thin polymer layer on a base with a finite thickness where a coordinate system is defined

wavelength from TSP. The working principles of TSP are described by Liu et al. (2021d). The analytical inverse heat transfer solutions for TSP measurements were discussed by Liu et al. (2010a, 2011b, 2018a, b, 2019b). A transient solution of the 1D time-dependent heat conduction equation is sought for a coating layer (TSP/ basecoat) on a base with a finite thickness. The geometry and coordinate system are illustrated in Fig. 10.2. The heat conduction equation for a polymer layer is 2

∂ ∂ - ap 2 θp ðt, yÞ = 0, ∂y ∂t

ð10:1Þ

where θp ðt, yÞ = T p - T in is a temperature change in the polymer layer from a constant initial temperature (T in ) and ap = k p =cp ρp is the thermal diffusivity of the polymer. Here, k p , cp , and ρp are the thermal conductivity, specific heat, and density of the polymer, respectively.

10.1

Analytical Inverse Heat Transfer Solution

225

Similarly, the heat conduction equation for the base is 2

∂ ∂ θb ðt, yÞ = 0, - ab ∂t ∂y2

ð10:2Þ

where θb ðt, yÞ = T b - T in is a temperature change of the base from a constant initial temperature, ab = k b =cb ρb is the thermal diffusivity of the base, and k b , cb , and ρb are the thermal conductivity, specific heat and density of the base, respectively. It is assumed that both the polymer layer and base have the same initial temperature. Here, the thermal properties are treated as constants for both the polymer layer and base. The boundary condition at the polymer surface is qs ð t Þ - k p

∂ θp t, Lp = 0, ∂y

ð10:3Þ

where qs ðt Þ is the heat flux into the polymer surface and Lp is the thickness of the polymer layer. The boundary condition at the bottom of the base is kb

∂ θb ðt, - Lb Þ = hc θb ðt, - Lb Þ, ∂y

ð10:4Þ

where Lb is the base thickness and hc is the convective heat transfer coefficient that is treated as an empirical constant. The matching conditions at the interface between the polymer and base are θp ðt, 0Þ = θb ðt, 0Þ,

ð10:5Þ

and kp

∂ ∂ θp ðt, 0Þ = k b θb ðt, 0Þ: ∂y ∂y

ð10:6Þ

After applying the Laplace transform 1

Θp ðs, yÞ =

θp ðt, yÞ expð- st Þdt,

ð10:7Þ

0

Equation (10.1) becomes d2 Θp s = Θp ðs, yÞ: 2 a p dy Similarly, the transformed form of Eq. (10.2) is

ð10:8Þ

226

10

Heat Flux from Surface Temperature Visualizations

d2 Θb s = Θb ðs, yÞ: 2 a b dy

ð10:9Þ

The solutions for Eqs. (10.8)–(10.9) are obtained, and the corresponding transformed boundary and matching conditions are applied to determine the unknown coefficients. Therefore, the transformed heat flux at the polymer surface is given by Q s ðsÞ = k p

s=ap Θps ðsÞ GðsÞ,

ð10:10Þ

where Θps ðsÞ = Θp s, Lp is the transformed temperature change at the polymer surface. The function GðsÞ is defined by GðsÞ =

1 þ mðsÞ exp - 2Lp

s=ap

1 - mðsÞ exp - 2Lp

s=ap

mðsÞ =

,

ε - mðsÞ , 1 - εmðsÞ

p s - hc exp - 2Lb mðsÞ = p s þ hc

ð10:11aÞ ð10:11bÞ

s=ab ,

ð10:11cÞ

where ε = ð1 - εÞ=ð1 þ εÞ, ε=

k p ρp cp =kb ρb cb ,

hc = h c =

k b ρb c b :

Now, the main problem is to evaluate the inverse Laplace transform of Eq. (10.10) for the heat flux into the polymer surface as a function of the polymer surface temperature change. Clearly, the effect of the finitely thick base is contained in mðsÞ. As Lb → 1, we know mðsÞ → 0 and mðsÞ → 0. Thus, the solution for a semi-infinite base can be recovered. In addition, as jsj → 1 (or t → 0), the solution for a semi-infinite base is approached as a limit.

10.1.2 Inverse Solution The inversion of Eq. (10.10) for a general case is sought. Equation (10.10) is rewritten as

10.1

Analytical Inverse Heat Transfer Solution

227

p Qs ðsÞ = k p = ap s Θps ðsÞK ðsÞ,

ð10:12Þ

1 1 þ mðsÞ exp - 2Lp s=ap : K ð sÞ = p s 1 - mðsÞ exp - 2Lp s=ap

ð10:13Þ

where

First, the inversion of K ðsÞ is k ðt Þ =

1 2π i

γþi1 γ - i1

expðs t ÞK ðsÞds,

ð10:14Þ

p where i = - 1. To calculate the inversion, following a standard procedure (Carslaw and Jaeger 1963; Smith 1966), we consider a contour integral I=

expðst ÞK ðsÞds:

ð10:15Þ

The function K ðsÞ has a branch point at s = 0. The integral, Eq. (10.15), along a closed contour C in Fig. 10.3 is zero. The integral I A ′ A over the segment A’A approaches the integral in Eq. (10.15) as the radius R of a large circle goes to infinity, where A and A’ are at γ þ iβ and γ - iβ, respectively. The segment AB is given by s = R expðiθÞ,

Fig. 10.3 Contour for inversion of the Laplace transform

228

10

Heat Flux from Surface Temperature Visualizations

where tan - 1 ðβ=γ Þ ≤ θ ≤ π =2 and jK ðsÞj < CR - 1=2 as R → 1. Therefore, I AB → 0 as R → 1. Similarly, I A ′ B ′ → 0, I BC → 0, and I C ′ B ′ → 0 as R → 1. The circular contour is given by s = r 0 expðiθÞ around s = 0, and the contour integral is I DD0 → 0 as r 0 → 0, where r 0 is the radius of the small circle around s = 0. The segments CD and D′C′ are given by, respectively, s = r exp½iðπ - δÞ and s = r exp½ið- π þ δÞ, where δ is a finite angle and r is the radial coordinate. The contour integral I CD is given by I CD =

expðst ÞK ðsÞds = CD

R r0

K 1 ðt, r Þ exp - t re - iδ - iδ dr ,

ð10:16Þ

where eiδ=2 1 - A2 þ 2A sinðβÞ i K 1 ðt, r Þ = p , r i 1 þ A2 - 2A cosðβÞ

ð10:17Þ

and other parameters are defined as A = jmjE, β = α þ ϕ, φ = argðmÞ, m=

ð10:18aÞ

p 2L ε-m η - hc exp - p b η , η = i e - iδ=2 r , ,m= 1 - εm ab η þ hc

2Lp 2Lp π-δ p π-δ p E = exp - p cos r , α = - p sin r, ap ap 2 2 ε = ð1 - εÞ=ð1 þ εÞ, ε =

k p ρp cp =kb ρb cb , hc = hc =

k b ρb c b :

ð10:18bÞ ð10:18cÞ ð10:18dÞ

It is found that I CD and I D ′ C ′ are conjugate. After taking the limits r 0 → 0 and R → 1, we have k ðt Þ =

-1 1 ðI þ I D ′ C ′ Þ = 2πi CD π

1

ReðΦÞ dr,

ð10:19Þ

0

where Re denotes the real value of a complex function defined as 1 1 - A2 þ 2A sinðβÞ i exp - t re - iδ - iδ=2 : Φðt, r Þ = p r 1 þ A2 - 2A cosðβÞ

ð10:20Þ

Therefore, by applying the convolution theorem to Eq. (10.12), an analytical solution for the heat flux at the polymer surface on a base with a finite thickness is given by

10.1

Analytical Inverse Heat Transfer Solution

kp qs ð t Þ = p ap

t

k ðt - τ Þ

0

229

dθps ðτÞ dτ , dτ

ð10:21Þ

where θps ðt Þ = T t, Lp - T in is the temperature change at the polymer surface from the initial temperature T in . Further, by using a transformation of ξ2 = t r, Eq. (10.20) can be expressed as a Gaussian-like integral for faster convergence in computations. Therefore, the analytical inverse solution for the heat flux at the polymer surface on a base with a finite thickness is expressed as kp qs ð t Þ = p π ap

t 0

W ðt - τÞ dθps ðτÞ p dτ , dτ t-τ

ð10:22Þ

where the function of W ðt Þ is defined as 2 W ðt Þ = p Re π

1 0

1 - A2 þ 2A sinðβÞ i exp - ξ2 e - iδ - iδ=2 dξ, 1 þ A2 - 2A cosðβÞ

ð10:23Þ

and δ is a finite positional angle for the segments CD and D′C′ of the contour for the inversion of the Laplace transform (see Fig. 10.3). The effects of the finite base thickness Lb and the heat transfer on the bottom are contained in the terms in Eqs. (10.18a–10.18d). For a polymer coating on a semiinfinite base, when Lb → 1 and δ → 0 (δ ≠ 0), Eq. (10.13) becomes 1 1 þ ε exp - 2Lp s=ap : K ð sÞ = p s 1 - ε exp - 2Lp s=ap

ð10:24Þ

Inversion of Eq. (10.24) by evaluating the contour integral is given (Liu et al. 2010a). In this case, since A = ε and β = α at δ = 0, Eq. (10.22) is reduced to the solution where Eq. (10.23) is reduced to W ðt Þ =

2 1 - ε2 p π

1 0

exp - ξ2 dξ : p 1 þ ε - 2ε cos 2Lp ξ= ap t 2

ð10:25Þ

In particular, for a homogenous semi-infinite base with ε = 1 and ε = 0 (due to kp ρp cp = kb ρb cb ), Eq. (10.22) recovers the classical solution on which the CookFelderman method is based on since Eq. (10.25) becomes a Gaussian integral, i.e., W ðt Þ = 1 (Carslaw and Jaeger 2000; Cook and Felderman 1970). This means that the function W ðt Þ contains the effects of the polymer layer, the finite thickness of the base, and the heat transfer on the backside of the base.

230

10

Heat Flux from Surface Temperature Visualizations

In the conventional inversion of the Laplace transform, the positional angle δ of the segments CD and D′C′ of the contour approaches zero (i.e., δ → 0). Mathematically, the contour integral should not depend on this control parameter δ. Thus, δ could be set to zero. However, for δ = 0, Eq. (10.22) describes a nonphysical case where A = jmjE exhibits an oscillation that does not decay to zero as Lb → 1, and thus the classical solution cannot be recovered. For this application, it is important that δ should be sufficiently small but finite to achieve good accuracy of the inverse solution. The discrete form of Eq. (10.22) used to calculate heat flux is kp qs ðt n Þ ffi p π ap

n

θps ðt i Þ - θps ðt i - 1 Þ p p × W ðt n - t i Þ þ W ðt n - t i - 1 Þ : tn - ti þ tn - ti - 1 i=1 ð10:26Þ

Equation (10.26) is a generalized form of the Cook-Felderman method, where the effects of a polymer coating and a finite base of any material are contained in the factor W ðt Þ. Like the Cook-Felderman method (Cook and Felderman 1970), the numerical scheme in Eq. (10.26) is easy to implement. A short account on the stability of the analytical inverse solution is given. The perturbations Q ′s and Θ ′ps on the base fields of the surface heat flux and temperature in the Laplace transform domain are considered. Since the inverse solution is given as a convolution, according to Eq. (10.12), the relation between Q ′s and Θ ′ps is written as Q ′s ðsÞ = T ðsÞ Θ ′ps ðsÞ, where the transfer function is p T ðsÞ = kp = ap s K ðsÞ: The use of the Cauchy-Schwarz inequality yields an estimate kQ ′s k = TΘ ′ps ≤ kT k Θ ′ps , where kk denotes the L2-norm in a domain s 2 ½0, 1 which corresponds to that in the time domain from t = 0 to t = 1. For kT k < 1, Θ ′ps < 1 leads to kQ ′s k < 1, that is, a finite perturbation of the surface temperature leads to a bounded heat flux perturbation. In this sense, the analytical inverse solution is at least marginally stable.

10.1

Analytical Inverse Heat Transfer Solution

231

Table 10.1 Properties of Mylar, aluminum, and aluminum alloy Properties Thermal conductivity k (W/m-K) Density ρ (kg/m3) Specific heat c (J/kg-K)

Mylar 0.15 1420 1090

Al 204 2700 904

Al 6016 Alloy 167 2700 896

Fig. 10.4 Simulation results for the step function of the heat flux: (a) temperature histories at the Mylar surface and Mylar-Al interface and (b) surface heat flux histories recovered by using the inverse solutions for the finite base and semi-infinite base. Note that the recovered solution for the finite base overlaps the given distribution. (From Liu et al. (2018a))

10.1.3

Solution Validation

To validate the analytical inverse solution, a Mylar layer of Lp = 20 μm on a finite aluminum base of Lb = 10 mm with hc = 0 at its bottom (the adiabatic wall condition) is considered (Liu et al. 2018a, b). Table 10.1 lists the properties of Mylar and aluminum. In simulations, three typical histories of the surface heat flux are specified, which are the step function, triangular function, and exponential transitional function. The time-dependent temperature fields of the polymer layer and the base are obtained by solving the governing equations with the boundary and matching conditions (Eqs. 10.1–10.6) using a forward time centered space finite difference scheme. Figure 10.4a shows the temperature histories at the Mylar surface and Mylar-Al interface after the step function of the heat flux is imposed on the Mylar surface. Simulations indicate that for a sufficiently small value of δ (e.g., δ ≤ π=64) the recovered heat flux by using the inverse solution for the finite base approaches to the true one. Therefore, the inverse solution given by Eq. (10.22) with δ = π=480 is used to calculate the heat flux for the finite base. Figure 10.4b shows the surface heat flux qs recovered by using the inverse solution based on the surface temperature history in Fig. 10.4a, which is in excellent agreement with the given step function

232

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.5 Simulation results for the triangular function of the heat flux, (a) temperature histories at the Mylar surface and Mylar-Al interface, and (b) surface heat flux histories recovered by using the inverse solutions for the finite base and semi-infinite base. Note that the recovered solution for the finite base overlaps the given distribution. (From Liu et al. (2018a))

distribution (the recovered curve overlaps the given one). For comparison, the inverse solution for the semi-infinite base is also applied, which gives the result that increasingly deviates from the given distribution as time increases. This indicates that the effect of the finite thickness becomes appreciable as time elapses. Similarly, the results for the triangular function of the heat flux are shown in Fig. 10.5, indicating that the heat flux history recovered by using the inverse solution for the finite base is accurate in contrast to that given by the solution for the semiinfinite base. To mimic the starting process of a short-duration hypersonic wind tunnel with a run time from milliseconds to seconds, the exponential transitional function of the heat flux is considered, and the recovered results by using both the inverse solutions for the finite base and the infinite base are shown in Fig. 10.6 for comparison. The accuracy of the recovered heat flux depends on the selection of δ, which is shown in Fig. 10.7. It is found that δ should be sufficiently small but finite as discussed in Sect. 10.1.3. For example, for δ = π=480, the recovered result is in excellent agreement with the true one.

10.2 10.2.1

Numerical Inverse Heat Transfer Solution Numerical Algorithm

Most analytical methods are developed based on the assumption that the thermal properties (thermal conductivity, specific heat, and density) of TSP (a polymer layer) are independent of temperature. In fact, the thermal properties are sensitive to temperature. When a change of surface temperature is significantly large in high-

10.2

Numerical Inverse Heat Transfer Solution

233

Fig. 10.6 Simulation results for the exponential transitional function of the heat flux, (a) temperature histories at the Mylar surface and Mylar-Al interface, and (b) surface heat flux histories recovered by using the inverse solutions for the finite base and semi-infinite base. Note that the recovered solution for the finite base overlaps the given distribution. (From Liu et al. (2018a)) Fig. 10.7 Effect of the parameter δ on the recovered heat flux by using the inverse solution for the finite base. (From Liu et al. (2018a))

enthalpy hypersonic tunnels, the assumption of constant thermal properties may produce a considerable error in heat flux calculation if the temperature effects on the thermal properties are not considered. In this case, since no exact analytical solution can be obtained, a numerical inverse method should be developed. Given a history of the polymer surface temperature obtained from TSP measurements, an iterative algorithm was proposed by Cai et al. (2011, 2018) to solve this two-region inverse heat conduction problem, which has the following procedures.

234

10.2.1.1

10

Heat Flux from Surface Temperature Visualizations

Finite Difference Method for Direct Problem

The direct (forward) heat conduction problem is to find the temperature fields θp ðt, yÞ and θb ðt, yÞ by solving the governing equations with the boundary and matching conditions, Eqs. 10.1–10.6, using the standard finite difference method for the general heat conduction equation (Morton and Mayers 1994; Fletcher 1988; Hoffman 1992).

10.2.1.2

Initial Determination

For initial estimation, a linear matrix equation relating the surface heat flux and surface temperature data is proposed, i.e., θps = Φ qs , where θps and qs denote a measured polymer surface temperature vector and a surface heat flux vector, respectively, and Φ is the relation matrix. If Φ is defined and nonsingular, we have qs = Φ - 1 θps :



However, the actual functional relation between the temperature data θps and heat flux qs is complicated, which is related to the thermal properties of the polymer and base layers. As an initial step of the iterative algorithm, we try to find the approximate relation matrix Φ0 based on the two assumptions. First, it is assumed that the inverse heat conduction problem to be linear, i.e., M

θps = j=1

Pj qjs ,

where the vector Pj is the surface temperature response to a unit step heat flux at time j and qjs is the heat flux magnitude at time j. Another assumption is that the effect of heat flux at time j on the surface temperature change is instantaneous. If the history of the surface temperature response to the unit step heat flux is T 1 , T 2 , ⋯, T M

T

,

then Pj = 0, ⋯, 0, T j , 0, ⋯, 0 , ðj = 1, 2, ⋯, M Þ T

and a diagonal matrix Φ0 has the nonzero elements T 1 , T 2 , ⋯, T M diagonal line.

only in the

10.2

Numerical Inverse Heat Transfer Solution

235

To obtain Φ0 , a unit step heat flux is applied to the system of equations and the surface temperature is determined by using the numerical method for the direct problem for a step function of heat flux. From the initial condition for the polymer, we know T 1 = 0, and hence the first row and first column of Φ0 are zero vectors. The first row and first column of Φ0 are therefore discarded. Because of discarding the data, the first heat flux estimate q1s cannot be determined. Fortunately, we can set q1s to be zero in most time. For the nonsingular matrix Φ0 , the history of heat flux can be estimated by q2s - M = Φ ′ 2--1M θ2ps- M : Since Φ02 - M is a diagonal matrix, it is easy to compute the inverse matrix. The initial (coarse) determination of heat flux is simple and fast. For some simple simulated histories of heat flux, the recovered results are good enough as the initial approximation for the subsequent optimization. Hereafter, we use Φ to denote the nonsingular matrix Φ02 - M by neglecting the influence of the initial condition. 10.2.1.3

Optimization

The determination of heat flux can be mathematically described as the following optimization problem, i.e., 2

J ðqs Þ = θps - θ0ps ðqs Þ

→ min,

ð10:27Þ

where kk is the L2-norm, qs is the surface heat flux to be found, and θ0ps is the estimated surface temperature responses to qs that is calculated by solving the system of equations for the positive problem. The difference between the estimated temperature and TSP-measured temperature is minimized, and an iterative method to solve the optimization problem is described in Algorithm 10.1. The error bound ε in calculations of heat flux is typically set at 10-5. Algorithm 10.1. Iterative Method Input: measured surface temperature θps (TSP data) Initialization: ◊ initialize error tolerance ε and construct matrix Φ ◊ compute coarse determination of heat flux qs1 using θps = Φqs ◊ from the direct heat conduction equations, compute θ0ps1 ◊ calculate the temperature error T err = θps - θ0ps1

2

◊k=1 While T err > ε, repeat Step 1. from the matrix form Φqerr = T err , compute qerr Step 2. qskþ1 = qsk þ qerr Step 3. compute surface temperature θ0pskþ1 using the direct heat conduction equations (continued)

236

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.8 Simulated heat flux with starting step changes followed by a sinusoidal change. (From Cai et al. (2011))

Step 4. update T err = θps - θ0pskþ1

2

Step 5. k = k þ 1 End Output: heat flux estimate qsk - 1

10.2.2

Solution Validation

To illustrate the effects of temperature dependencies of the thermal properties, as shown in Fig. 10.8, we consider the simulated heat flux with a starting step change followed by a sinusoidal change into a 0.01 mm PVC layer covered on an aluminum base and a Nylon-6 base. This history of the heat flux is used to simulate the transient starting process in the Boeing/AFOSR Mach-6 Quiet Tunnel at Purdue University (Liu et al. 2010a; Cai et al. 2011). The specific heat and thermal conductivity of PVC are fitted by the empirical formulas, respectively, i.e., cðT Þ = 1540 þ 4ðT - 293Þ kðT Þ = 0:16 - 0:0004ðT - 293Þ,

10.2

Numerical Inverse Heat Transfer Solution

237

Fig. 10.9 Temperature history at the polymer surface and interface between the 0.01 mm thick PVC sheet and a semi-infinite base responding to the simulated heat flux: (a) aluminum base and (b) Nylon-6 base. (From Cai et al. (2011))

where T is temperature in Kelvin, c is in J/kg-K, and k is in W/m-K. It is assumed that the density of PVC is independent of temperature, i.e., ρ = 1300 kg/m3. Therefore, the thermal diffusivity of PVC is að T Þ =

0:16 - 0:0004ðT - 293Þ k : = c ρ ½1540 þ 4ðT - 293Þ  1300

For simplicity, it is assumed that the thermal properties of the base (aluminum or Nylon-6) are constant. Figure 10.9 shows the temperature histories responding to the simulated heat flux change, where T surface and T interface are the temperatures at the polymer surface and interface between the polymer and base, respectively. Based on the surface temperature history simulated by using the temperaturedependent thermal properties, as shown in Fig. 10.10, the heat flux history calculated by using the numerical inverse method has good convergence to the original heat flux after six iterations for a 0.01 mm PVC layer on a semi-infinite aluminum base. In addition, Fig. 10.10 shows a comparison with the analytical inverse method that is valid only for the constant thermal properties. As expected, in this case, the numerical inverse heat transfer method can recover the correct heat flux history, while the analytical inverse method has a considerable error. The simulated surface temperature for a PVC layer on a semi-infinite Nylon-6 base is also considered. Figure 10.11 shows a comparison of the recovered results between the numerical inverse method and the analytical inverse method. In this case, the analytical inverse method can still achieve as good recovery of heat flux as the numerical inverse method. Interestingly, the temperature dependencies of the thermal properties do not significantly affect the calculation of the surface heat flux in the case of a low-conductive base. The insensitivity of heat flux calculation on the thermal properties for a low-conductive base like Nylon-6 has been found in an error analysis given by Liu et al. (2010a).

238

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.10 Comparison of the recovered heat flux by using the numerical inverse method and analytical method for a 0.01 mm PVC layer on a semi-infinite aluminum base, where the thermal properties of PVC are temperature-dependent. (From Cai et al. (2011))

Fig. 10.11 Comparison of the recovered heat flux by using the numerical inverse method and analytical method for a 0.01 mm PVC layer on a semi-infinite Nylon-6 base, where the thermal properties of PVC are temperature-dependent. (From Cai et al. (2011))

10.3

Correcting Lateral Heat Conduction Effect

10.3 10.3.1

239

Correcting Lateral Heat Conduction Effect Analogy Between 1D and 3D Direct Solutions

The analytical method described in Sect. 10.1 is the 1D method that ignores the lateral heat conduction effect on a 3D body that could be significant at certain locations where the spatial gradient in the heat flux is large. The large gradient in the heat flux could be found near shock waves, shock/shock interactions, shockwave/boundary-layer interactions, boundary-layer transition, flow separations, and strong vortex/wall interactions. The application of the 1D method to a complex 3D body in complex flows may have some errors in some regions with the high surface curvature. In general, to take the lateral heat conduction effect into account in heat flux calculation, an inverse solution for the 3D time-dependent heat conduction equation can be sought by using a finite difference method or finite-element method coupled with an optimization method when suitable boundary conditions on the given geometry are imposed. Estorf (2005) obtained a formal analytical solution of the 3D time-dependent heat conduction equation by using the Fourier transform for a semi-infinite body and then solved an inverse problem by using the regularization method for the transformed heat flux in the Fourier space. Further, the solution for heat flux in the physical space is given by inverting the Fourier transform. Liu et al. (2011b) proposed the image deconvolution method that is coupled with the 1D analytical inverse method to calculate more accurate heat flux fields by correcting the lateral heat conduction effect. The lateral heat conduction effect is corrected by solving a convolution-type integral equation with a Gaussian filter (kernel) that relates a heat flux field obtained by using the 1D inverse method on a surface to the true heat flux field. To understand the lateral heat conduction effect, we examine the formal relationship between the 1D and 3D solutions of the heat conduction equation. We consider an element of a polymer layer and a semi-infinite base, as illustrated in Fig. 10.12, Fig. 10.12 A thin polymer layer on a semi-infinite base where a coordinate system is defined

240

10

Heat Flux from Surface Temperature Visualizations

where y is the coordinate normal to the surface between the polymer layer and base and x and z are the coordinates on the surface. The 3D unsteady heat conduction equation for a polymer layer is 2

2

2

∂θp ∂ θp ∂ θ p ∂ θ p þ þ , = ap ∂t ∂ x2 ∂ y2 ∂z2

ð10:28Þ

where θp ðt, x, y, zÞ = T p - T in is a temperature change in the polymer layer from a constant initial temperature (T in ) and ap = k p =cp ρp is the thermal diffusivity of the polymer. Here, k p , cp , and ρp are the thermal conductivity, specific heat, and density of the polymer. The same equation is used for the temperature change θb ðt, x, y, zÞ in the base. By applying the Fourier transform to Eq. (10.28) θp =

1

1

-1

-1

θp exp½ - i ðux þ vzÞ dxdz,

ð10:29Þ

the transformed equation is 2

∂ θp ∂θp - u2 þ v 2 θ p , = ap ∂t ∂ y2

ð10:30Þ

where θp = θp ðt, y, u, vÞ. Further, applying the Laplace transform to Eq. (10.30) 1

Θp ðs, yÞ =

θp ðt, yÞ expð- st Þdt,

ð10:31Þ

0

Equation (10.30) becomes d 2 Θp = s=ap þ u2 þ v2 d y2

Θp :

ð10:32Þ

The boundary condition at the polymer surface is qs ðt, x, zÞ - kp ∂θp ðt, x, L, zÞ=∂ y = 0, where qs is the heat flux into the polymer surface and L is the polymer layer thickness. The boundary condition at infinity in the base is θb ðt, x, - 1, zÞ = 0:

10.3

Correcting Lateral Heat Conduction Effect

241

The corresponding solution for the base has the same form as Eq. (10.32) in the transformed domain. The matching conditions at the interface between the polymer and base are θp ðt, x, 0, zÞ = θb ðt, x, 0, zÞ, k p ∂θp ðt, x, 0, zÞ=∂y = k b ∂θb ðt, x, 0, zÞ=∂y: After using the boundary conditions and matching conditions at the interface, we have the transformed temperature change at the polymer surface: p Θps ðs, u, vÞ =

ap H ðs0 ÞQs ðs, u, vÞ, kp

ð10:33Þ

where Qs ðs, u, vÞ is the transformed heat flux at the polymer surface. The function H ðs0 Þ is defined by 1 1 - ε exp - 2L s0 =ap , H ðs0 Þ = p s0 1 þ ε exp - 2L s0 =ap

ð10:34Þ

where s 0 = s þ ap u2 þ v 2 , ε = ð1 - εÞ=ð1 þ εÞ, ε=

kp ρp cp =k b ρb cb :

The inverse Laplace transform of H ðs0 Þ is hðt, u, vÞ =

1 2π i

γþi1 γ - i1

expðst ÞH ðs0 Þds = exp - ap u2 þ v2 t

Gðt Þ : ð10:35Þ

where a Green function is defined as Gðt Þ = =

1 2π i

γ0þi1 γ ′ - i1

expðs0 t ÞH ðs0 Þds0 :

ð10:36Þ

Therefore, the inverse Laplace transform of Eq. (10.33) is p θps =

ap kp

t 0

Gðt - τÞ exp -

ap ð u2 þ v 2 Þ qs ðτ, u, vÞdτ : t-τ

ð10:37Þ

242

10

Heat Flux from Surface Temperature Visualizations

Further, the inverse Fourier transform of Eq. (10.37) leads to a relation between the surface temperature change and the heat flux into the surface of the polymer layer: p

ap kp

θpsð3DÞ ðt, x, zÞ =

t 0

Gðt - τÞ hqs ig ðτ, x, z; σ ðt - τÞÞdτ ,

ð10:38Þ

where the spatially averaged heat flux is h qs i g =

1

1

-1

-1

g½ x - x0 , z - z0 ; σ ðt - τÞ qs ðτ, x0 , z0 Þ dx0 dz0 :

ð10:39Þ

The filter function in Eq. (10.38) is the Gaussian distribution: gðx, z; σ Þ =

x2 þ z2 1 exp , 2 2 σ2 2π σ

ð10:40Þ

where σ ðt Þ = 2 ap t describes the extent of the lateral diffusion. The size of the filter functions characterizes the effect of the lateral heat conduction, and the standard deviation σ defines the affected region. Similarly, by using the Laplace transform, the direct solution for the 1D unsteady heat conduction equation is given by p θpsð1DÞ ðt Þ =

ap kp

t 0

Gðt - τÞ qsð1DÞ ðτÞdτ :

ð10:41Þ

The inverse solution of Eq. (10.41) for qsð1DÞ is actually given by the 1D analytical inverse solution (see Sect. 10.1). To explore the connection between the 1D and 3D solutions, qs ðτ, x0 , z0 Þ in Eq. (10.38) is expanded as a Taylor series at a fixed point ðx, zÞ on a surface. Thus, we have 2

hqs ig = qs ðτ, x, zÞ þ

1 2 ∂ qs g, ðx0 - xÞ 2 ∂x2

2

þ ðx,zÞ

1 2 ∂ qs g, ðz0 - zÞ 2 ∂z2

, ðx,zÞ

ð10:42Þ where the inner product is defined as hg, f i =

1

1

-1

-1

g½x - x0 , z - z0 : σ  f ðx0 , z0 Þ dx0 dz0 :

ð10:43Þ

The derivation of Eq. (10.42) utilizes the fact that the moments hg, x0 - xi, hg, z0 - zi, and hg, ðx0 - xÞðz0 - zÞi vanish at the fixed point ðx, zÞ because of the

10.3

Correcting Lateral Heat Conduction Effect

243

symmetry of the Gaussian function. Substitution of Eq. (10.42) into Eq. (10.38) yields p

ap kp

θpsð3DÞ ðt, x, zÞ = ð1 þ ΔλÞ

t

Gðt - τÞ qs ðτ, x, zÞdτ ,

ð10:44Þ

0

where Δλ is a relative surface temperature change due to the lateral heat conduction that is given by 2

Δλ =

1 ∂ qs β 2 xx ∂x2

2

ðx,zÞ

1 ∂ q þ βzz 2s 2 ∂z

,

ð10:45Þ

ðx,zÞ

and the coefficients in Eq. (10.45) are defined as t

βxx =

Gðt - τÞ g, ðx0 - xÞ2 dτ

0

t

, Gðt - τÞ qs ðτ, x, zÞdτ

0 t

βzz =

Gðt - τÞ g, ðz0 - zÞ2 dτ

0

t

:

Gðt - τÞ qs ðτ, x, zÞdτ

0

The analogy between Eqs. (10.41) and (10.44) gives the connection between the 3D solution and the 1D solution, which is simply written as θpsð3DÞ = ð1 þ ΔλÞθpsð1DÞ , since the integral in the RHS of Eq. (10.44) is actually the 1D solution applied to the fixed point ðx, zÞ on a surface. The factor Δλ mainly depends on the second derivatives of qs along the surface. The order estimates give 2

2

∂ qs =∂x2  jΔqs j=σ 2 , ∂ qs =∂z2  jΔqs j=σ 2 , βxx  σ 2 =jqs j, βzz  σ 2 =jqs j: Therefore, Δλ  jΔqs j=jqs j, and when the relative change of qs along a surface is small except in certain isolated region, Δλ ≪ 1.

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10.3.2 Convolution-Type Integral Equation When Δλ is treated as a time-independent factor (it may weakly depend on time), ð1 þ ΔλÞθpsð1DÞ = θpsð3DÞ gives t 0

Gðt - τÞ ð1 þ ΔλÞqsð1DÞ ðτ, x, zÞ - hqs ig ðτ, x, z; σ ðt - τÞÞ dτ = 0:

ð10:46Þ

Since t is arbitrary, the integrand in Eq. (10.46) must be zero. Further, for Gðt Þ > 0, we obtain a convolution-type integral equation, i.e., ð1 þ ΔλÞqsð1DÞ ðt, x, zÞ =

1

1

-1

-1

g½ x - x0 , z - z0 ; σ r ðt Þqs ðt, x0 , z0 Þ dx0 dz0 , ð10:47Þ

where σ r ðt Þ is a standard deviation that represents the extent of the lateral heat conduction effect, Δλ is a relative surface temperature change due to the lateral heat conduction, and t is the time elapsing from the start-up of a wind tunnel. The filter in Eq. (10.47) is the Gaussian distribution: gðx, z; σ Þ =

x2 þ z 2 1 exp 2 2 σ2 2π σ

ð10:48Þ

In Eq. (10.47), the heat flux field qsð1DÞ ðt, x, zÞ obtained by using the 1D analytical inverse method on a surface in a general 3D case is interpreted as a spatially averaged heat flux. In most applications where Δλ ≪ 1 except at certain isolated region, Δλ can be set at zero in the first-order approximation. In a sense, Eq. (10.47) is an approximate model since σ r ðt Þ should be modeled. The linear relation is proposed as a model, i.e., σ r ðt Þ = r 1 Ldiffu þ r 0 , where Ldiffu = 2 ap t is the thermal diffusion length scale and r1 and r0 are empirical coefficients to be determined. Furthermore, since the analysis of heat conduction is made for a flat surface, Eq. (10.47) is a reasonable model for a surface whose curvature radius is much larger than Ldiffu . Therefore, given qsð1DÞ , qs can be recovered by solving the convolution-type integral equation Eq. (10.47) as an inverse problem. Interestingly, this problem to solve Eq. (10.47) for qs is the exactly same as the classical image deconvolution problem or the image restoration problem (Helstram 1967; Banham and Katsaggelos 1997). This coincidence is not surprising since image blurring is generally modeled by a diffusion process. In image-based surface

10.3

Correcting Lateral Heat Conduction Effect

245

Fig. 10.13 Two-step processing: the 1D inverse method coupled with the image deconvolution method. (From Liu et al. (2011b))

temperature measurements, qsð1DÞ and qs are usually given in the image plane, and therefore they are conveniently treated as images in processing. Therefore, the existing image deconvolution algorithms can be directly applied to this problem. The total variation regularization method developed by Oliveira et al. (2009) is used. Based on the above discussions, the two-step method is proposed to calculate heat flux fields from a time sequence of surface temperature images, where the 1D inverse method is used in tandem with the image deconvolution method. Figure 10.13 is a flowchart of the two-step processing. The 1D analytical inverse method is applied to a time sequence of temperature images one pixel by one pixel (one point by one point on a surface) to obtain heat flux images at different times. Then, the resulting heat flux images are further processed one image by one image by using the image deconvolution method to reconstruct a time sequence of more accurate heat flux images. The key idea of the two-step method is that heat flux calculation on a surface is carried out without considering the lateral heat conduction effect as the first-order approximation, and then correcting the lateral heat conduction effect is treated separately as the image deconvolution problem.

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10.3.3 Solution Validation To facilitate image processing, Eq. (10.47) is projected onto the image plane. A special case is considered, where the image plane of a camera is parallel to the measured plane. In this case, the orthographical projection transformation between the surface coordinates ðx, zÞ in meters and the image coordinates xpix , zpix in pixels is particularly simple, i.e., ðx, zÞ = λ - 1 sx xpix , sz zpix , where λ is a nondimensional scaling constant and ðsx , sz Þ are the factors converting pixel to meter (meter/pixel) in the x- and z-coordinates. Therefore, Eq. (10.47) is written in the image coordinates as ð1 þ ΔλÞ qsð1DÞ t, xpix , zpix =

1

1

-1

-1

g xpix - x ′pix , z - z ′pix ; σ r,pix ðt Þ

ð10:49Þ

qs t, x ′pix , z ′pix dx ′pix dz ′pix , where the filter function is g xpix , zpix ; σ r,pix =

ðsx =sz Þ2 x2pix þ z2pix 1 exp : 2π σ 2r,pix 2 σ 2r,pix

ð10:50Þ

The standard deviation σ r,pix ðt Þ in pixels will be determined by optimization in simulations. A suitable model is σ r,pix ðt Þ = ðr 1 Ldiffu þ r 0 Þðλ=sz Þ, where Ldiffu = 2 ap t is the thermal diffusion length scale and r1 and r0 are empirical coefficients to be determined in simulations for different materials. Simulations are conducted to examine the capability of the two-step method for correcting the lateral heat conduction effect. A CCD camera/lens system with a principal distance c = 70 mm (approximately the focal length) is located at 990 mm away from the measured plane that is parallel to the image plane. Therefore, the scaling constant is λ = 0.0707. If the size of a CCD sensor is 10 mm × 10 mm and the numbers of both the horizontal and vertical pixels are 1000, the converting factors are sx = sz = 10 - 5 m/pixel, and the factor λ=sz is 0:707 × 104 pixel/m. A uniformly distributed heat flux source of 1000 W/m2 is suddenly applied onto a surface in a square region of 9.1 mm × 9.1 mm that corresponds to 64 pixels × 64 pixels in the image plane, while the heat flux source is zero outside this square region. The time history of the surface heat flux is given by

10.3

Correcting Lateral Heat Conduction Effect

247

qs ðt, x, zÞ = q0 ðx, zÞH ðt Þ, where H ðt Þ is a Heaviside function (H ðt Þ = 0, if t = 0 and H ðt Þ = 1, if t > 0), and q0 ðx, zÞ = 1000, if ðx, zÞ 2 D =D q0 ðx, zÞ = 0, if ðx, zÞ2 where D denotes the square region. As a typical case, we consider a 0.25 mm thick Mylar layer on a semi-infinite base of Nylon whose thermal properties are given by Liu et al. (2011b). A time sequence of temperature images at the polymer surface is calculated by solving the positive heat conduction problem over a time span of 3 s in a time step of 0.1 s after switching on the heat flux source. Figure 10.14 shows typical surface temperature distributions at t = 1.5, 2 and 2.5 s, which clearly indicates the effect of the lateral heat conduction near the edges of the square region. Then, a zero-mean white noise with a SNR of 10 dB is added on the surface temperature images.

Fig. 10.14 Surface temperature changes generated by a uniform heat flux of 1000 W/m2 suddenly imposed on a square region on a Mylar layer on Nylon at (a) 1.5 s, (b) 2 s, and (c) 2.5 s. (From Liu et al. (2011b))

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Fig. 10.15 Surface heat flux distributions on a Mylar layer on Nylon calculated by using the 1D inverse method from a time sequence of noisy temperature images at (a) 1.5 s, (b) 2 s, and (c) 2.5 s. (From Liu et al. (2011b))

The surface heat flux images are calculated pixel by pixel by using the 1D inverse method, Eq. (10.22), from a time sequence of the corresponding noisy surface temperature images. Figure 10.15 shows typical surface heat flux distributions at t = 1.5, 2, and 2.5 s obtained by using the 1D analytical inverse method. The random noise patterns in these images mimic those in typical image-based heat flux measurements. The resulting random noise in heat flux in Fig. 10.15 is contributed by the random error sources in both the time and space domains. The deconvoluted heat flux images from the blurred noisy images in Fig. 10.15 are obtained by using the algorithm developed by Oliveira et al. (2009) to solve Eq. (10.49). For a Mylar layer on Nylon, the optimal standard deviation (in pixels) is σ r,pix ðt Þ = ðr 1 Ldiffu þ r 0 Þðλ=sz Þ, where r 1 = 0:505 and r0 = 3:56 × 10 - 5 m that are determined by an optimization procedure. Figure 10.16 shows the deconvoluted surface heat flux distributions at t = 1.5, 2, and 2.5 s. These results indicate that the sharp edges in the heat flux distributions are recovered, and the random noise is removed by using the image deconvolution method.

10.3

Correcting Lateral Heat Conduction Effect

249

Fig. 10.16 Surface heat flux distributions on a Mylar layer on Nylon calculated by using the two-step method from a time sequence of noisy temperature images at (a) 1.5 s, (b) 2 s, and (c) 2.5 s. (From Liu et al. (2011b))

For direct comparison with the given heat flux distribution in simulations, Fig. 10.17 shows the surface heat flux distributions along the symmetrical axis across the square region calculated by using the 1D inverse method and the two-step method at t = 1.5, 2, and 2.5 s. Improvement by using the image deconvolution method is evident to remove the effect of the lateral heat conduction effect. For quantitative comparison, a relative error is defined as error =

qcal ði, jÞ - qgiven ði, jÞ qgiven ði, jÞ

,

where qcal and qgiven are the calculated and given heat flux images, respectively. Figure 10.18 shows the relative errors in the heat flux fields calculated by using the 1D inverse method and the two-step method. A significant decrease in the relative error is achieved by using the image deconvolution method.

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Heat Flux from Surface Temperature Visualizations

Fig. 10.17 Surface heat flux distributions along the symmetrical axis across the square region on a Mylar layer on Nylon calculated by using the 1D inverse method and the two-step method at (a) 1.5 s, (b) 2 s, and (c) 2.5 s. (From Liu et al. (2011b))

10.4 10.4.1

Applications 25°/45° Indented Cone at Mach 11

To examine the analytical and numerical inverse methods, raw TSP images on the sharp 25°/45° indented cone model at Mach 11 were processed by Liu et al. (2010a) using the analytical inverse method and Cai et al. (2011) using the numerical inverse method, which were originally obtained by Hubner et al. (2002) in the 48-inch Shock Tunnel at Calspan-University of Buffalo Research Center. The model was made of stainless steel. The detailed description of the experiments was given by Hubner et al. (2002), and here only data relevant to heat flux calculation using the analytical method are given. Over 60 platinum thin-film heat transfer gauges were installed along a ray of the model, providing heat flux data for comparison with TSP. The reported measurement accuracy of the gauges was ±5%, and the measurement resolution was 5 kW/m2.

10.4

Applications

251

Fig. 10.18 Relative errors in the heat flux fields calculated by using the 1D inverse method and the two-step method. (From Liu et al. (2011b))

TSP was Ru-phen in a non-oxygen-permeable polyurethane binder. The thickness of TSP was 10 μm, and the white polyurethane basecoat was 40 μm. Thus, the total thickness of the polymer layer was 50 μm. The measured thermal conductivity and diffusivity of the polyurethane layer were 0.48 W/K-m and 2.7 × 10-7 m2/s, respectively. In image processing, the raw flow-off and flow-on TSP images were corrected by subtracting the background intensity that was contributed by the ambient light and leakage of the optical filter before taking a ratio between the flow-off and flow-on images. Then, the ratio images were converted to the temperature images by using a priori calibration data. The analytical inverse method was used to obtain the heat flux fields for this case without considering the temperature dependencies of the thermal properties of the materials (Liu et al. 2010a). In the numerical inverse heat flux calculation, it was assumed that the polymer layer (TSP plus the base coating) had the same temperature dependencies as PVC in the thermal properties (Cai et al. 2011). The thermal properties of the stainless steel base were assumed to be constant. Figure 10.19a shows a typical surface temperature image selected from a total of 16 temperature images in the run and the averaged heat flux image in an interval of 4–6 ms in which the flow was stabilized. Figure 10.19b shows the time-averaged heat flux field calculated by using the numerical inverse method. The heat flux profiles along the centerline are shown in Fig. 10.20, where the numerical inverse method incorporates the temperature dependencies of the thermal properties in contrast to the analytical inverse method. The numerical inverse method gives the improved result that is

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Heat Flux from Surface Temperature Visualizations

Fig. 10.19 (a) Surface temperature image and (b) time-averaged heat flux image of the sharp 25°/ 45° indented cone model at Mach 11. Note that the original TSP images are from Hubner et al. (2002)

Fig. 10.20 Time-averaged heat flux profiles along the centerline on the sharp 25°/45° indented cone model at Mach 11. (From Cai et al. (2011))

10.4

Applications

253

closer to the data given by an array of thin-film heat flux gauges. The effect of the temperature dependencies of the thermal properties is pronounced in this case because the increase of the polymer temperature on the metal base is relatively large (about 20 K) in this tunnel.

10.4.2

7°-Half-Angle Cone at Mach-6

TSP measurements on a 7°-half-angle aluminum cone were performed in the Boeing/AFOSR Mach-6 Quiet Tunnel at Purdue University (Liu et al. 2010a, 2011b, 2013). The cone had a total length of 0.4 m and the radius of the stainless steel nose tip was about 1 mm. The interchangeable nose was 0.15-m-long. The whole cone was coated with LustreKote paint that was composed of white primer and glass MonoCote. The purpose of this coating is twofold. The white coating will enhance the luminescence emission of TSP detected by a camera for achieving a high SNR. Furthermore, for a metal model, an insulating coating on the surface will increase the change of surface temperature for the imposed heat flux to improve the accuracy in heat flux calculation. TSP, Ru(bpy) in Chromaclear auto paint, was coated on the top of the LustreKote paint. The mean paint thickness was about 250 μm. Figure 10.21 shows a generic setup of a CCD camera with a band-pass optical filter for detecting the luminescent emission from TSP and a blue LED array for illuminating TSP. A 16-bit CCD camera and a blue LED array whose emission peak is at 460 nm were used. A time sequence of 100 TSP images was acquired by the camera at 32 f/s for each run. Figure 10.22 shows an imaged region of TSP

Fig. 10.21 Experimental setup of a CCD camera and a UV LED array for illuminating TSP on a model at the Purdue Mach-6 tunnel. (From Liu et al. (2011b))

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Heat Flux from Surface Temperature Visualizations

Fig. 10.22 Imaged region of TSP on a 7°-half-angle circular cone. (From Liu et al. (2011b))

Fig. 10.23 Heat flux images on a 7°-half-angle circular cone at Mach-6 at t = 3 s: (a) image obtained by using the numerical inverse method and (b) image improved by using the image deconvolution method. (From Liu et al. (2011b))

measurements near the middle section of the cone. A square roughness element with a 1.27 mm × 1.27 mm cross section and a height of 0.28 mm was placed at 0.13 m from the nose tip to trigger the boundary-layer transition to turbulence. Surface temperature images were obtained from a sequence of TSP images, and then heat flux images were calculated at every pixel by using the numerical inverse method and analytical inverse method. It was assumed that the TSP had the same thermal properties as PVC. Figure 10.23a shows the heat flux image obtained by using the 1D numerical inverse method on the 7°-half-angle circular cone at Mach6 at t = 3 s from starting the tunnel. In this case, the effect of the temperature dependencies of the thermal properties of the polymer is negligible for a small change of temperature on the Nylon-6 base. The significant spike noise found in these images is typical in TSP measurements particularly when the measured temperature change is small (less than a few degrees). Figure 10.23b shows the corresponding heat flux image improved by applying the image deconvolution method to Fig. 10.23a to correct the lateral heat conduction effect and remove the random noise (see Sect. 10.3.3). These images clearly indicate the formation and development of a turbulent wedge triggered by the roughness element in the boundary layer, which is evidenced by the difference in heat flux between the turbulent and laminar flow regions. Figure 10.24 shows the heat flux profiles in the vertical direction in images across the cone at the streamwise locations x = 0.16, 0.18, 0.21, and 0.23 m (marked by a, b, c, and d in Fig. 10.22, respectively) at t = 3 s from starting the tunnel. In fact, these profiles are the projected ones of those on a circular arc of a local cross section of the cone. The local values of heat flux given by the laminar boundary layer solution on

10.4

Applications

255

Fig. 10.24 Heat flux profiles across the cone at (a) x = 0.16 m, (b) 0.18 m, (c) 0.21 m, and (d) 0.23 m at t = 3 s from starting the tunnel. The results are obtained by the numerical inverse method with the temperature-dependent thermal properties in comparison with the laminar boundary layer solution. Further, the results are improved by using the image deconvolution method. (From Liu et al. (2011b))

the cone are also plotted in Fig. 10.24 for comparisons. Then, the improved distributions of heat flux are obtained by applying the image deconvolution method to correct the lateral heat conduction effect and reduce the random spike noise. This improvement is particularly appealing for the result at x = 0.16 m where the random noise is very large relative to the signal. At x = 0.16 and 0.18 m where the turbulent wedge is narrow in its width and weak in its magnitude of heat flux, the TSP-measured values of heat flux outside of the wedge are consistent with the prediction by the laminar boundary layer solution on the cone. When the turbulent wedge considerably diffuses (expands) in the lateral direction as it develops downstream, the TSP-measured value of heat flux outside of the turbulent wedge deviates gradually from the prediction by the laminar boundary layer solution. This deviation is more evident at x = 0.23 and 0.26 m, indicating increasing influence of the turbulent wedge diffused along the lateral direction.

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Fig. 10.25 Experimental setup: (a) jet nozzle, pressure chamber and aluminum flat plate and (b) camera and LED light. (From Liu et al. (2019b))

10.4.3

Obliquely Impinging Sonic Jet

Using TSP, Liu et al. (2019b) measured the time-dependent surface temperature, heat flux and Nusselt number fields on a preheated, rectangular aluminum plate due to an obliquely impinging sonic jet at different total pressures. The analytical inverse method was used to determine the heat flux from the temperature measurements. The normalized ensemble-averaged heat transfer fields (surface temperature, heat flux, and Nusselt number) exhibit self-similarity (time-invariance) for different test conditions and therefore are found to exhibit a somewhat universal nature. Figure 10.25 shows the impinging jet setup and camera/light system. A converging nozzle with a 6.35 mm exit diameter was connected to a settling pressure chamber where the total pressure was set using a regulator and monitored using a pressure gauge. Compressed air was delivered through a 25.4-mm-long steel pipe which was sufficiently long such that the stagnation temperature of the compressed air was near the room temperature. The impingement surface was the upper surface of an aluminum 6016 alloy plate (304.8-mm-long,203.2-mm-wide, and 9.525-mmthick). Two thermistors were installed on the upper and bottom surfaces of the plate at the middle to monitor plate temperature. The jet was mounted onto a three-axis translation stage and the impingement distance could be controlled using two components of the stage. The impingement distance was fixed at six jet diameters, i.e., H=D = 6. The geometrical impingement angle was set at 15° using mounting blocks attached to the nozzle. Ru-based TSP was coated on a white base coating on the impingement plate, which had the typical absorption spectrum of 250–400 nm and emission spectrum of 550–650 nm. As shown in Fig. 10.25b, a LED light was used to illuminate TSP and a 12-bit CCD camera with a long-pass optical filter (>550 nm) was used to acquire TSP images. When the jet impinges on the surface of a preheated finite Al plate, the surface temperature change θps = T ps - T in is a function of time, where T in is an initial surface temperature of the plate. To demonstrate the self-similarity of surface

10.4

Applications

257

Fig. 10.26 Surface coordinate system on a typical temperature image where the black circle indicates the reference location. The color bar indicates surface temperature change in K. (From Liu et al. (2019b))

temperature fields in this flow, a typical under-expanded jet is considered, where the preheated base temperature was 45 °C, the total pressure was 29 psia, the jet exit Mach number was one, and the Reynolds number based on the nozzle diameter was 1.86 × 105. Figure 10.26 shows the surface coordinate system on a typical raw temperature image where the black circle indicates the reference location. The color bar indicates surface temperature change in K. Figure 10.27a shows three typical fields of the surface temperature change at t = 1, 3, and 5 s. The corresponding heat flux fields are obtained by applying the analytical inverse method to the time sequence of surface temperature images, as shown in Fig. 10.27b. Note that the heat flux is negative since heat enters to the flow from the preheated base. To describe the time-dependent behavior of heat transfer fields, the characteristic temperatures and heat fluxes are taken at the position of the maximum temperature change and a reference location at the downstream of the jet marked in Fig. 10.26. The characteristic temperature change and heat flux at these positions are shown in Figs. 10.28a, b as a function of time. Liu et al. (2019b) gave a theoretical argument indicating that the self-similarity of the heat transfer fields could be achieved in this flow after suitable normalization. The normalized surface temperature change is introduced, i.e., θps ðx=DÞ =

θps ðx, t Þ - θps ðxref , t Þ , max θps ðx, t Þ - θps ðxref , t Þ

ð10:51Þ

where xref =D = ð10:4, 0Þ is the reference location and θps ðxref , t Þ and max θps ðx, t Þ - θps ðxref , t Þ are the functions of time as shown in Fig. 10.28a. Figure 10.29 shows the normalized surface temperature profiles (θps -profiles) in the x-axis and y-axis across the origin at different times in the span of 1–7 s. Interestingly, all the data collapse and the profiles become largely time-invariant. This indicates that the θps -fields are approximately self-similar.

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Fig. 10.27 Snapshot fields: (a) surface temperature change (in K) and (b) heat flux (in W/m2), at t = 1, 3, and 5 s for the preheated base temperature of 45 °C. (From Liu et al. (2019b))

Similarly, the normalized heat flux field is defined as qs ðx=DÞ =

qs ðx, t Þ - qs ðxref , t Þ , maxj qs ðx, t Þ - qs ðxref , t Þ j

ð10:52Þ

where qs ðxref , t Þ and maxj qs ðx, t Þ - qs ðxref , t Þ j are the functions of time as shown in Fig. 10.28b. As shown in Fig. 10.30, the normalized heat flux profiles (qs-profiles) in the x-axis and y-axis at different times collapse as well. The qs -fields are approximately self-similar. Further, as shown in Fig. 10.31, the θps -field and hqs iT-field T

represent the more universal heat transfer distributions, where hiT is the timeaveraging operator. The Nusselt number is used for impinging jet heat transfer, which is defined as

10.4

Applications

259

Fig. 10.28 The characteristic quantities as a function of time at the location of the maximum value and the reference location for the preheated base temperature of 45 °C: (a) surface temperature, (b) heat flux, and (c) Nusselt number. (From Liu et al. (2019b))

Fig. 10.29 Normalized surface temperature profiles across the origin at different times: (a) profiles in the x-axis and (b) profiles in the y-axis for the preheated base temperature of 45 °C. (From Liu et al. (2019b))

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Fig. 10.30 Normalized surface heat flux profiles across the origin at different times: (a) profiles in the x-axis and (b) profiles in the y-axis for the preheated base temperature of 45 °C. (From Liu et al. (2019b))

Fig. 10.31 Normalized fields: (a) surface temperature, (b) heat flux, and (c) Nusselt number, where the circular mark indicates the reference location for the preheated base temperature of 45 °C. (From Liu et al. (2019b))

Nu =

qs ðx, t Þ hD D = , kf kf T ps ðx, t Þ - T aw

ð10:53Þ

where h is the convective heat transfer coefficient, k f is the thermal conductivity of fluid (e.g., air), D is the nozzle exit diameter, and T aw is the adiabatic wall temperature as a reference value for scaling. A key problem is how to determine

10.4

Applications

261

Fig. 10.32 Determination of the adiabatic wall temperature: (a) relation between the surface heat flux and temperature where the data are collected from five different locations at the initial base temperature of 45 °C and (b) the measured adiabatic wall temperature as a function of the preheated base temperature. (From Liu et al. (2019b))

Fig. 10.33 Normalized Nusselt number profiles across the origin at different times: (a) profiles in the x-axis and (b) profiles in the y-axis for the preheated base temperature of 45 °C. (From Liu et al. (2019b))

T aw that cannot be directly measured since the adiabatic wall does not exist in the real world. For complex flows such as impinging sonic jets, simplified estimation methods for T aw are not accurate. In fact, T aw can be experimentally determined from measured time histories of the surface temperature and heat flux at different locations. As shown in a qs-T ps plot in Fig. 10.32a, data over a time span of 1–7 s at five randomly selected locations collapse into a near-linear curve. Therefore, extrapolating these data via the linear regression gives a value of T ps at qs = 0, which is just an experimentally determined adiabatic wall temperature T aw = 317:9 K in this case. This value is approximately equal to the preheated base temperature T b = 318:12 K (45 °C) before a run. Figure 10.32b shows that T aw varies linearly with T b .

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The time history of Nu at the location of the maximum heat transfer is shown in Fig. 10.28c, indicating that maxðNuÞ changes with time much more gradually than its counterparts in the temperature and heat flux after a transient stage of the impinging jet. As shown in Fig. 10.33, the normalized Nu profiles by maxðNuÞ in the x-axis and y-axis at different times collapse. The normalized time-averaged Nusselt number field, hNuiT = maxðNuÞ, is shown in Fig. 10.31c, indicating the weaker but visible pattern associated with the impinging jet compared to the corresponding θps -field and hqs iT -field. In this sense, the Nu-fields exhibit the T

self-similarity.

10.4.4

In Situ Calibration for TSP on Finite Base

The analytical inverse heat transfer solution for a finite base was applied by Liu and Risius (2019) to in situ calibration in TSP measurements in high-enthalpy shortduration hypersonic wind tunnels. The thermal penetration parameter of TSP was determined by in situ calibration based on the data obtained using a reliable heat flux sensor at a reference location. The in situ calibration method was developed based on the analytical inverse heat transfer solution, Eq. (10.22), for a two-layer structure. The thermal penetration parameters in Eq. (10.22) are ηp = ρp cp kp and ηb = ρb cb kb for the polymer and base, respectively. For a metallic base where the temperature does not change much during a run, the parameter ηb for the base material is fixed and known. Then, the parameter ηp for the polymer is determined by in situ calibration based on the measured heat flux qs,m ðt Þ obtained by a reliable heat flux sensor at a selected reference location. To determine ηp , the following objective function is minimized, i.e., J ηp = qs ðt Þ - qs,m ðt Þ

2

→ min,

ð10:54Þ

where qs ðt Þ is the estimated heat flux by using Eq. (10.22) from the TSP-measured surface temperature change near the selected reference location and kk denotes the L2-norm in a suitable time interval after the transient stage of a run. When the heat flux approaches to a steady-state value after the transient stage, the time-averaged value of qs,m ðt Þ can be used in Eq. (10.54). To examine the in situ calibration method, Liu and Risius (2019) reprocessed a sequence of 50 TSP images on a 15° wedge obtained by Risius et al. (2017) during a run of 10 ms the German Space Center (DLR) High Enthalpy Shock Tunnel in Göttingen (HEG). The wedge model, consisting of a flat aluminum plate (80-mm-

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Applications

263

Fig. 10.34 In situ calibration based on the data obtained by using the heat flux sensor: (a) surface temperature history and (b) surface heat flux history, where the time-averaged value given by the heat flux sensor is marked in (b). (From Liu and Risius (2019))

long, 20-mm-wide, and 2.07-mm-thick) with a sharp leading edge, was placed at an angle of 15° to the flow direction. The plate surface was coated with TSP, the DLR OV322 paint (Eu-based luminophore and polyurethane), where the active layer, screen layer, and primer layer were 81, 57, and 27 μm, respectively. The total thickness of the composite polymer layer was 165 μm. A coaxial thermocouple was mounted through a small hole in the wedge for point-wise temperature and heat flux measurement, providing the data for in situ calibration. For in situ calibration, the heat flux histories are calculated by using the analytical inverse method at neighboring points near the heat flux sensor. In calculation, the properties of the aluminum base are fixed at ρb = 1420 kgm - 3 , cb = 904 J kg - 1 K - 1 , and kb = 204 Wm - 1 K - 1 . Thus, the thermal penetration parameter for the base is ηb = 4:97 × 108 kg2 K - 2 s - 5 . The aluminum base thickness is Lb = 2:07 mm. The polymer layer thickness is Lp = 165 μm. Since this polymer layer is a relatively thick, the characteristic time L2p =ap is in the order of seconds. Therefore, during a run of 10 ms, the heat transfer coefficient hc on the backside of the aluminum plate is set at zero. The thermal penetration parameter of the polymer layer ηp is adjusted as a parameter to minimize the objective function defined in Eq. (10.54) where the time-averaged value of qs,m ðt Þ after the transient stage is used. Figure 10.34 shows the time histories of the surface temperature and heat flux obtained by using TSP for ηp = 6 × 106 kg2 K - 2 s - 5 in comparison with the timeaveraged value from the heat flux sensor. This value of ηp determined by in situ calibration will be used in heat flux computation. Figure 10.35 shows images of the surface temperature change θps = T ps - T in and the heat flux qs on the wedge at t = 4, 6, 8, and 10 ms. The time-dependent surface heat flux images are calculated by using the analytical inverse method from a sequence of the surface temperature images, where the thermal penetration

264

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.35 (a) Surface temperature change (in K) and (b) heat flux (in MW/m2) at four instants. (From Liu and Risius (2019))

Fig. 10.36 Profiles of (a) surface temperature change (in K) and (b) surface heat flux (in MW/m2) in the x-coordinate along the centerline at four instants. (From Liu and Risius (2019))

10.4

Applications

265

Fig. 10.37 Determination of the adiabatic wall temperature based on the relationship between the surface heat flux and temperature, where the data are collected from four different locations. (From Liu and Risius (2019))

parameter determined by in situ calibration is ηp = 6 × 106 kg2 K - 2 s - 5 . Figure 10.36 shows the profiles of the surface temperature change and heat flux in the x-coordinate along the centerline at t = 4, 6, 8, and 10 ms. The surface temperature and heat flux on the wedge increase with time. In aerothermodynamics, the Stanton number is often used, which is defined as

St =

qs , ρ1 c1 u1 T ps - T aw

ð10:55Þ

where u1 , ρ1 , and c1 are the velocity, air density, and air specific heat in the freestream condition, respectively, T ps is the temperature at the polymer surface, and T aw is the adiabatic wall temperature. To determine T aw , the qs -T ps plots at four locations are shown in Fig. 10.37, in which the data collapse, exhibiting a near-linear behavior over a range of T ps . The linear regression of the data gives the extrapolated value of T ps at qs = 0, which is just the adiabatic wall temperature T aw = 296 K.

266

10

Heat Flux from Surface Temperature Visualizations

Fig. 10.38 Stanton number profiles in the x-coordinate along the centerline at four instants. (From Liu and Risius (2019))

Figure 10.38 shows the Stanton number profiles in the x-coordinate along the centerline at t = 4, 6, 8, and 10 ms. The Stanton number is nearly constant in the region, but it decreases as time increases, indicating the time-dependent development of the boundary layer on the wedge. Interestingly, compared to the heat flux fields, the Stanton number fields are much more homogeneous. In particular, comparison between Fig. 10.36b and Fig. 10.38 indicates that not only some small heat flux patterns are largely removed, but also the global Stanton number distribution becomes flat by normalization using the surface temperature field. This means that the proportional relation between the surface heat flux and temperature is approximately valid.

Chapter 11

Analysis of Physics-Based Optical Flow

This chapter describes the mathematical aspects of the variational optical flow solution to preserve discontinuities in a velocity field extracted from flow visualization images. The uniqueness of the solution and the convergence of a successive approximation sequence are proven. First, the bounded variation (BV) space is briefly reviewed, and an integral representation of the measure ∇
(gu) is given to analyze the physics-based optical flow equation. Then, the general variational formulation with the L1-norm is applied to the equation, which regularizes a velocity filed while preserving its discontinuities in a velocity field. In this case, the minimization problem admits a unique solution in the BV space. Since it is difficult to directly solve the nonlinear Euler-Lagrange equation associated with the functional with the L1-norm, a successive approximate method is proposed, and a convergence analysis of the successive approximation is given. Next, the numerical algorithm is developed and simulations are conducted to evaluate the accuracy of the algorithm.

11.1

Mathematical Analysis

A rigorous mathematical analysis of the optical flow in the brightness constraint equation in the space of functions of bounded variations (BV) was given by Aubert et al. (1999) and Aubert and Kornprobst (1999). However, a mathematical analysis for the physics-based optical flow method is lacking. Clearly, it is worthwhile to extend the methodology of Aubert et al. (1999) to the physics-based optical flow. Wang et al. (2015) presented an analysis of the solution of the physics-based optical flow equation, Eq. (2.71), via the variational formulation with a general constraint for preserving discontinuities in velocity fields. Since Eq. (2.71) has the term ∇
ðguÞ rather than u
∇g in the original Horn-Schunck brightness constraint equation proposed for computer vision, the associated Euler-Lagrange equations are different. Therefore, some mathematical treatments for Eq. (2.71) differ from those of Aubert

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8_11

267

268

11

Analysis of Physics-Based Optical Flow

et al. (1999) although some derivations are formally the same. The contents in this chapter are based on the work of Wang et al. (2015).

11.1.1

BV Space

Let Ω 2 R2 denote the 2D image bounded set and x 2 Ω be any point in it. A recorded image sequence is described by an intensity function g : Ω × ½0, T  → gðx, t Þ, and it is assumed that the data are Lipschitz: g 2 W 1,1 ðΩ × RÞ. The optical flow is denoted by uðxÞ = ðu1 ðxÞ, u2 ðxÞÞT : We denote space of vector-valued functions by bold characters. The physicsbased optical flow equation is ∇
ðguÞ þ gt = f , where a simple time-differential notation gt = ∂g=∂t is used. Liu and Shen (2008) solved this equation via a variational method based on the quadratic smoothness constraint, but they did not consider discontinuities of velocity fields. Here, we apply the method of Aubert et al. (1999) to this problem, which can be rigorously justified based on the space of functions of bounded variations (BV). We first briefly review the relevant notations and definitions following Aubert et al. (1999). The space of functions of bounded variations denoted by BVðΩÞ is defined by BVðΩÞ = v 2 L1 ðΩÞ :

Ω

jvj≔sup

Ω

v divðϕÞdx < 1, 8ϕ 2 C10 ðΩÞ2 , jϕj1 ≤ 1

If v 2 BVðΩÞ and Dv is the gradient in the sense of distributions, it is a vectorvalued Radon measure. The set of Radon measure is denoted by M ðΩÞ. The strong, weak, and weak* convergences in a space V ðΩÞ are denoted by !, !, and V ð ΩÞ

V ðΩÞ

!, respectively. The product topology of the strong topology of L1 ðΩÞ for v and

V ðΩÞ

the weak  topology of measures for Dv are called the weak  topology of BV and will be denoted by BV - w . Every bounded sequence in BVðΩÞ admits a subsequence converging in BV - w . The approximate upper and lower limits are defined by

11.1

Mathematical Analysis

269

fv > t g \ Bρ ðxÞ =0 ρ2 , fv < t g \ Bρ ðxÞ =0 v - ðxÞ = sup t 2 ½ - 1, þ1 : limþ 2 ρ ρ→0

vþ ðxÞ = inf t 2 ½ - 1, þ1 :

lim

ρ → 0þ

where Bρ ðxÞ is the ball of center x with radius ρ. We denote by Sv the jump set, i.e., the set of points x where vþ ðxÞ is different with v - ðxÞ: Sv = fx 2 Ω : v - ðxÞ < vþ ðxÞg: Choosing a normal nv ðxÞ (x 2 Sv) pointing toward the largest value of v, we have the following decompositions: Dv = ∇v
dx þ C v þ ðvþ - v - Þnv
H NjSv- 1 ,

ð11:1Þ

where ∇v is the density of the absolutely continuous part of Dv with respect to the Lebesgue measure dx, Cv is the Cantor part, and H N - 1 is the Hausdorff measure of dimension N - 1. Then, if v 2 BVðΩÞ, we define

Ω

φðDvÞ =

Ω

φð∇vÞdx þ φ1 ð1Þ

Ω∖Sv

j Cv j þφ1 ð1Þ

Ω

ðvþ - v - ÞdH N - 1 ,

ð11:2Þ

where φ1 is defined by φ1 ðzÞ = lim t → 0 φðtzÞ=t. This function is lower semicontinuous in the topology BV - w .

11.1.2

An Integral Representation of the Measure —
ðguÞ

To study the physics-based optical flow equation, we need an integral representation of the measure ∇
ðguÞ. To make a sense to the Radon measure in this case, we define it by duality: h∇
ðguÞ, ϕi≔ -

Ω

gu
∇ϕ dx, 8ϕ 2 C 10 ðΩ; RÞ:

ð11:3Þ

Chen and Frid (1999) analyzed the divergence-measure fields and presented the corresponding product rule. In the following part, we will establish a product rule for the measure ∇
ðguÞ in a similar way. The following definition is introduced and then a useful lemma is given.

270

11

Analysis of Physics-Based Optical Flow

Definition 11.1 Let Ω be an open set and F 2 BVðΩ; RN Þ; we define j div F j ðΩÞ≔sup

Ω

F
∇ϕ dx; ϕ 2 C10 Ω; RN , ϕðxÞj ≤ 1j x 2 Ω, :

By using this definition, the following lemma is presented without a proof since the details can be found in Theorem 1.17 given by Giusti (1984) and Theorem 1.2 given by Chen and Frid (1999). Lemma 11.1 Let F 2 BVðΩ; RN Þ, then there exists a sequence F j in C 1 ðΩ; RN Þ such that lim j→1 Ω

F j - F dx = 0, lim

j→1 Ω

divF j dx = j divF j ðΩÞ: The following theo-

rem gives a product rule for the measure. Theorem 11.1 (Product rule): Let F 2 BVðΩÞ and g is Lipschitz continuous. Then divðgF Þ = gdivðF Þ þ F
∇g:

ð11:4Þ

Proof Let F j be a sequence in Lemma 11.1. It is implied that divF j → divF weakly in MðΩÞ. Since g is Lipschitz continuous over all compact sets contained in Ω, then gdivF j þ F j
∇g → gdivF þ F
∇g, weakly in MðΩÞ. On the other hand, div gF j → divðgF Þ in the sense of distribution. Taking the limit in the identity div(gFj) = gdivFj + Fj
∇g in the sense of distributions and using the fact that C1 0 ðΩÞ is dense in C 0 ðΩÞ, we can give Eq. (11.4). Now, an integral representation of the measure ∇
ðguÞ is given in the following theorem. Theorem 11.2 If g 2 W 1,1 ðΩÞ and u 2 BVðΩÞ \ L1 ðΩÞ, then N Ω

∇
ðguÞ =

Ω

divðguÞdx þ i=1 N

þ i=1

Sσ i

gei


Ω∖Sσi

n ui u þ i

gei
C ui

- ui-

, dH

ð11:5Þ

N -1

where divðguÞ denotes the density of the absolutely continuous part of ∇
ðguÞ with respect to the Lebesgue measure. Proof If g 2 W 1,1 ðΩÞ and u 2 BVðΩÞ \ L1 ðΩÞ, then ∇
ðguÞ = g∇
u þ u
∇g by Theorem 11.3. We note that

11.1

Mathematical Analysis

271

g∇
u = ρ1
Du1 þ ρ2
Du2 , where ρi = gei 2 W 1,1 Ω, R2 ði = 1, 2Þ and ei is the canonical basis, i.e., the vector with 1 in the ith coordinate and 0 elsewhere. According to Anzellotti (1983), the pairing ρi
Dui is a Radon measure and absolutely continuous with respect to j Dui j. Thus, we have

Ω

ðρi
Dui Þ =

Ω

θi ðρi , Dui , xÞ jDui j ,

where θi ðρi , Dui , xÞ = ρi ðxÞ

dDui ðxÞ: d jDui j

Then, we obtain

Ω

ðρi
Dui Þ =

g Ω

þ

∂ui dx þ ∂xi

Sσ i

Ω∖Sσ i

ρi
C ui

dH N - 1 ρi
n u i uþ i - ui

:

ð11:6Þ

Finally, substituting ρi = gei into Eq. (11.6) and using the product rule, we deduce Eq. (11.5). Further, using Theorem 11.4, we can obtain the lower semicontinuity of the Radon measure j ∇
ðguÞ j. Theorem 11.3 If g 2 W 1,1 ðΩÞ and u 2 BVðΩÞ \ L1 ðΩÞ, then the Radon measure N Ω

j∇
ðguÞj =

Ω

jdivðguÞj dx þ i=1 N

þ i=1

Sui

Ω∖Sui

jgei
Cui j ð11:7Þ

jgei
nui uþ j dH N - 1 i - ui

is lower semicontinuous in the topology BV - w . Proof According to Rudin (1966) (Theorem 6.13), if v is a positive measure on M ðΩÞ, g 2 L1v ðΩÞ and λ are the measure defined by λðBÞ = we have

g dv, B

272

11

j λ j ðB Þ =

B

Analysis of Physics-Based Optical Flow

jgj dv:

ð11:8Þ

Thus, Eq. (11.7) can be deduced from Eq. (11.8). To prove the lower semicontinuity, let uk converge to u in the BV - w topology. Then, we have h∇
ðguÞ, ϕi = -

Ω

gu
∇ϕdx

= - lim

,

k→1 Ω

guk
∇ϕdx ≤ - lim inf j∇
guk j ðΩÞ k→1

and further j∇
ðguÞj ðΩÞ ≤ lim inf j∇
guk j ðΩÞ: k→1

11.2

Successive Approximation and Convergence

To solve the physics-based optical flow equation Eq. (2.71), we apply the general variational method proposed by Aubert et al. (1999), which regularizes a velocity field while preserving its discontinuities. The functional to be minimized is J ðuÞ =

Ω

j∇
ðguÞ þ gt - f j þαr

2 j=1

Ω

φ Duj þ αh

Ω

cðxÞkuk2 dx ,

ð11:9Þ

where αr and αh are positive constants and jj
jj is the usual Euclidian norm. We make the following hypotheses: (C1) φ: R → Rþ is an even, convex, and nondecreasing function on Rþ . (C2) There exist constants b1 > 0 and b2 ≥ 0 such that b1 x - b2 ≤ φðxÞ ≤ b1 x þ b2 for all x 2 Rþ . (C3) cðxÞ verifies c 2 C 1 ðΩÞ. (C4) There exist mc > 0 such that cðxÞ 2 ½mc , 1, 8x 2 Ω. Since u belongs to BVðΩÞ, the divergence ∇
ðguÞ is a Radon measure. Thus, the interpretation of ðj
jÞ is the total variation of the measure given by j∇
ðguÞ þ gt - f j ðΩÞ = Note that

Ω

j∇
ðguÞ þ gt - f j :

ð11:10Þ

11.2

Successive Approximation and Convergence

Ω

273

j∇
ðguÞj and φðDui Þ

are lower semicontinuous in the topology BV - w . Therefore, we can claim that the functional J defined on BVðΩÞ \ L1 ðΩÞ by N

J ðuÞ =

Ω

j∇
ðguÞ þ gt - f j þ i=1

Ω

φðDui Þ þ

Ω

ckuk2 dx

ð11:11Þ

is lower semicontinuous in the topology BV - w . Then, similar to Aubert et al. (1999), using the coerciveness and lower semicontinuity of the functional J ðuÞ, we have the following theorem. Theorem 11.4 The minimization problem

inf J ðuÞ admits a unique solution

u2BVðΩÞ

in BVðΩÞ. Since it is difficult to directly solve the nonlinear Euler-Lagrange equation associated with the functional J, an approximate algorithm and a convergence analysis of successive approximations are given by following the methodology of Aubert et al. (1999).

11.2.1 A Result of Γ-Convergence In order to construct an approximation of the functional J, the following notation is introduced first. Given a function f verifying the hypotheses (C1–C4), the function f ε is defined as

f ε ðt Þ =

f 0 ð εÞ 2 εf 0 ðεÞ t þ f ð εÞ 2ε 2 f ðt Þ εf 0 ð1=εÞ 2 f 0 ð1=εÞ t þ f ð1=εÞ 2 2ε

if t ≤ ε, if ε ≤ t ≤ 1=ε,

ð11:12Þ

if t ≥ 1=ε:

We have 8ε, f ε ≥ f and 8t, lim ε → 0 f ε ðt Þ = f ðt Þ. Under this definition, we denote the functions associated to jtj and φðt Þ by φ1,ε and φ2,ε , respectively. Now, a functional J ε is defined by J ε : BVðΩÞ → R,

274

11

J ε ðuÞ =

Ω

φ1,ε ðdivðguÞ þ gt - f Þdx þ þ

þ1

Ω

cðxÞkuk2 dx

Analysis of Physics-Based Optical Flow 2

j=1

Ω

φ2,ε

∇uj

dx

if u 2 W 1,2 ðΩÞ,

ð11:13Þ

otherwise:

It will be proved that the sequence J ε Γ-converges to J in the following theorem. Theorem 11.5 Under the hypotheses (C1–C4), the sequence J ε defined by Eq. (11.13) Γ-converges to J as ε approaches to 0 for the L2 -strong topology. Proof Define the functional J J : BVðΩÞ → R J ðuÞ =

J ðuÞ if u 2 W 1,2 ðΩÞ, þ1

otherwise:

It is known that J ε is a decreasing sequence converging pointwise to J. This means that J ε Γ-converges to the lower semicontinuous envelope of J in BVðΩÞ or to the relaxed functional R J . Since J is lower semicontinuous in the BV - w topology, it is sufficient to prove that R J = J. Theorem 11.6 Under the hypotheses (C1–C4), the minimization problem inf

J ε ðuÞ

u2W 1,2 ðΩÞ

admits a unique solution uε . Moreover, the sequence uε converges for the L2-strong topology to the unique minimizer of J in BVðΩÞ. Proof See Proof of Proposition 4.2 given by Aubert et al. (1999).

11.2.2

The Half-Quadratic Minimization

Following Aubert et al. (1999), we want to find an approximation solution to uε since it is difficult to directly solve the Euler-Lagrange equation associated with the functional J ðuÞ defined in Eq. (11.9). Here, the half-quadratic minimization is used. Let us introduce the functional given by J dε : W 1,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ → R

11.2

Successive Approximation and Convergence

J dε ðu, a, bÞ =

Ω

275

aðdivðguÞ þ gt - f Þ2 þ 2

þ j=1

Ω

bj ∇uj

2

1 dx a

þ ψ 2,ε bj

ð11:14Þ dx þ

Ω

cðxÞkuk2 dx

where a and b are dual variables and ψ 2,ε is a convex decreasing function. To minimize J dε with respect to all variables, we perform minimizations with respect to each variable alternatively. More precisely, given u0 , a0 , b0 2 W 2,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ, we use the following algorithm proposed by Aubert et al. (1999): unþ1 = arg min J dε ðu, an , bn Þ,

ð11:15Þ

u2W 2,2 ðΩÞ

anþ1 = bnþ1 =

a2L

2

arg min J dε unþ1 , a, bn , ðΩÞ, ε ≤ a ≤ 1=ε

arg min b2L ðΩÞ, εφ2,ε ′ ð1=εÞ ≤ b ≤ φ2,ε ′ ðεÞ=ε 2

J dε unþ1 , anþ1 , b :

ð11:16Þ ð11:17Þ

Now we want to show that the sequence unþ1 converges strongly to uε in L2 . Let us write the optimality conditions associated to Eqs. (11.15)–(11.17): 8 v = ðv1 , v2 ÞT 2 W 2,2 ðΩÞ, Ω

2

an ∇
gunþ1 þ gt - f ∇
ðgvÞ þ

j=1

bnj ∇unþ1
∇vj þ cðxÞunþ1
v dx = 0, j

ð11:18Þ anþ1 =

φ1,ε 0 ðj∇
ðgunþ1 Þ þ gt - f jÞ 1 =ε_ ^ 1=ε, j∇
ðgunþ1 Þ þ gt - f j 2 j∇
ðgunþ1 Þ þ gt - f j

bnþ1 j

=

φ2,ε 0

∇unj

2 ∇unj

where the map t → ε1 _ t ^ ε2 is defined as

,

ð11:19Þ

ð11:20Þ

276

11

ε1 _ t ^ ε 2 =

Analysis of Physics-Based Optical Flow

ε1

if t ≤ ε1 ,

t

if ε1 < t < ε2 ,

ε2

if t ≥ ε2 :

Following the general methodology of Aubert et al. (1999), we can obtain the convergence result using the boundness of kun kW 1,p0 ðΩ0 Þ and kun kL1 ðΩ0 Þ . The estimations presented here are quite different from the case considered by Aubert et al. (1999), which are given by the following two lemmas. Lemma 11.2 Let ðun , an , bn Þ 2 W 2,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ be a minimizing sequence of J dε . Let Ω0 be an open subset strictly included in Ω with the boundary ∂Ω 2 C. Then, there exists p0 > 2 such that 0

un 2 W 1,p ðΩ0 Þ

8n:

ð11:21Þ

Moreover, we can find a constant M independent of n such that kun kW 1,p0 ðΩ0 Þ ≤ M

8n

ð11:22Þ

kun kL1 ðΩ0 Þ ≤ M

8n:

ð11:23Þ 0

Proof We have W 1,2 ðΩ0 Þ ⊂ Lq ðΩ0 Þ, 8q 2 ½2, þ1Þ, and W 1,p ðΩ0 Þ ⊂ L1 ðΩ0 Þ with a continuous embedding. The Sobolev embedding theorem permits kun kLq ðΩ0 Þ ≤ C kun kW 1,2 ðΩ0 Þ ,

ð11:24Þ

kun kL1 ðΩ0 Þ ≤ C kun kW 1,p0 ðΩ0 Þ :

ð11:25Þ

Since it is easy to deduce Eq. (11.23) from Eq. (11.22) by using Eq. (11.25), we only focus on Eq. (11.22). It has been shown that the sequence fun g is a Cauchy sequence in W 1,2 ðΩÞ. Thus, we can find a constant M such that kun kW 1,2 ðΩ0 Þ ≤ M. Then, Eq. (11.24) implies kun kLq ðΩ0 Þ ≤ M

8q 2 ½2, þ1Þ,

ð11:26Þ

where M depends only on q. To estimate the gradient of un , the Euler-Lagrange equations are used, i.e.,

11.2

Successive Approximation and Convergence

277

∂ n ða ð∇
ðguÞ þ gt - f ÞÞ ∂x1 : ∂ n ða ð∇
ðguÞ þ gt - f ÞÞ = cðxÞu2 - g ∂x2

div bn1 ∇u1 = cðxÞu1 - g div bn2 ∇u2

ð11:27Þ

Fortunately, a result from Meyers (1963) in the scalar case can be adopted. There exists p0 > 2 and a constant C > 0 such that ∇unj

Lp0 ðΩ0 Þ

≤ C cðxÞuj - an - 1 g

∂ ð∇
ðgun Þ þ gt - f Þ ∂xj

L2 ðΩ0 Þ

:

Lemma 11.2 permits kukW 2,2 ðΩ0 Þ ≤ M. Since g, gt , f , and an belong to L2 ðΩÞ and u is uniformly bounded in W 2,2 ðΩ0 Þ, we can find a constant M such that ∇unj

Lp0 ðΩ0 Þ

≤ M:

ð11:28Þ

Combining Eq. (11.26) with Eq. (11.28), we deduce Eqs. (11.21) and (11.22). Lemma 11.3 Let ðun , an , bn Þ 2 W 2,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ be a minimizing sequence of J dε . Let Ω0 be an open subset strictly included in Ω with the boundary ∂Ω0 2 C. Then, we have kun kW 2,2 ðΩ0 Þ ≤ C kun kW 1,2 ðΩÞ :

ð11:29Þ

Proof Since un is a minimizing sequence, it is a weak solution of the associated Euler-Lagrange equation Eq. (11.27) that can be written in a divergence form: 2

→

→

div Ai,j ðxÞ∇uj þ b i
∇u þ c i
u = hi ,

-

i = 1, 2,

ð11:30Þ

j=1

where the coefficient matrices are A1,1 = A2,1 =

an g2 þ bn1

0

0

bn1

0

0

an g 2

0

→

, →

,

A1,2 = A2,2 =

0 an g2 0

,

0

bn2

0

0

an g2 þ bn2

,

∇u≔ð∇u1 , ∇u2 ÞT , b i , and c i are vector-valued functions and hi is a real-valued function, which are composed of g, gt , f , and an . Further, some notations are introduced. We denote

278

11

A1,2 : A2,2

A1,1 A2,1

A=

Analysis of Physics-Based Optical Flow

By simple computation, 8x 2 R4 , we have xT Ax ≥ min bn1 , bn2 jxj2 ≥ θjxj2

ð11:31Þ

where θ = εφ2,ε 0 ð1=εÞ. We denote difference quotient by vðx þ hek Þ - vðxÞ , h

Dhk vðxÞ =

ðh 2 R, h ≠ 0, k = 1, 2Þ,

where ek is kth standard coordinate vector. Since un is a weak solution of Eq. (11.30), for any v = ðv1 , v2 Þ 2 W 1,2 0 ðΩÞ, we have 2 j=1

Ω

Ai,j ðxÞ∇uj
∇vi dx =

→

Ω

f i vi dx

ði = 1, 2Þ,

ð11:32Þ

→

where f i = hi - b i
∇u - c i
u. Let us denote 2

T L≔ i, j = 1

Ω

Ai,j ∇uj
∇ Dk- h η2 Dhk ui dx,

2

T R≔ i=1

Ω

ðhi - bi
∇u þ ci
uÞvi dx:

According to the arbitrariness of v, choosing vi = - Dk- h η2 Dhk ui and substituting it into Eq. (11.32), we obtain T L = T R:

ð11:33Þ

On the one hand, the inequality

Ω

jvi j2 dx ≤ C

2

Ω

j∇uj2 þ η2 Dhk ∇u dx,

implies TR ≤

θ 4

2

Ω

η2 Dhk ∇u dx þ C

Ω

jhj2 þ juj2 þ j∇uj2 dx

where h = ðh1 , h2 ÞT . On the other hand, by using the formula

ð11:34Þ

11.2

Successive Approximation and Convergence

Ω

vDk- h wdx = -

T L can be expressed as

Ω

2 Ω

i, j = 1

η2 Ai,j Dhk ∇uj
Dhk ∇ui dx,

Ai,j Dhk ∇uj
2η∇ηDhk ui þ Dhk Ai,j ∇uj
2η∇ηDhk ui

2 i, j = 1

wDhk vdx and ∇ Dhk u = Dhk ð∇uÞ,

T L = T a þ T b , where

T a≔

T b≔

279

þ Dhk Ai,j ∇uj
η2 Dhk ∇ui dx

Ω

Applying Eq. (11.31) to T a yields Ta ≥ θ

2

Ω

η2 Dhk ∇u dx:

ð11:35Þ

Since η and Ai,j are smooth, we have jT b j ≤ C

Ω

η jDhk ∇u Dhk u jþηj Dhk ∇u ∇u jþηj Dhk u ∇u jdx:

Using Cauchy’s inequality, the above inequality can be expressed as jT b j ≤ ε

2

Ω

η2 Dhk ∇u dx þ

C ε

2

Dhk u þ j∇uj2 dx:

Ω

ð ε > 0Þ

Choosing ε = θ=2 and using the inequality 2

Ω

Dhk u dx ≤ C

Ω

j∇uj2 dx,

we obtain the following estimate jT b j ≤

θ 2

2

Ω

η2 Dhk ∇u dx þ C

Ω

j∇uj2 dx:

ð11:36Þ

The estimates Eqs. (11.35) and (11.36) imply TL ≥

θ 2

2

Ω

η2 Dhk ∇u dx - C

Finally, considering the relation

Ω

j∇uj2 dx

ð11:37Þ

280

11 2

Ω

0

Dhk ∇u dx ≤

Analysis of Physics-Based Optical Flow 2

Ω

η2 Dhk ∇u dx,

and combining Eqs. (11.33), (11.34), and (11.37), we deduce 2

Ω

0

Dhk ∇u dx ≤ C

jhj2 þ juj2 þ j∇uj2 dx:

Ω

Using the properties of difference quotient (Evans 1998), we have kukW 2,2 ðΩ0 Þ ≤ C kukW 1,2 ðΩÞ þ khkL2 ðΩÞ : Since h is smooth and its norm is a constant, we obtain Eq. (11.29). The convergence theorem is stated as follows. A proof is given for the completeness and convenience of reading, which is similar to that given by Aubert et al. (1999). Theorem 11.7 (Convergence theorem). Let u0 , a0 , b0 2 W 2,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ be given. Then the sequence n ðu , an , bn Þ is convergent. Moreover, un converges in L2 -strong topology to the unique minimizer of J ε . Proof We will complete the proof in three steps. Step 1: To show that ðun , an , bn Þ are Cauchy sequences, following Aubert et al. (1999) and using the optimality conditions Eqs. (11.18)–(11.20), we obtain U n  J dε ðun , an , bn Þ - J dε unþ1 , an , bn ≥ minðε, mc Þ un - unþ1

ð11:38Þ

W 1,2 ðΩÞ

V n  J dε unþ1 , an , bn - J dε unþ1 , anþ1 , bn ≥ ε3 an - anþ1

L2 ð Ω Þ

,

ð11:39Þ

W n  J dε unþ1 , anþ1 , bn - J dε unþ1 , anþ1 , bnþ1 ≥ cðεÞ bn - bnþ1

L2 ðΩÞ

:

ð11:40Þ

Denoting T n  J dε ðun , an , bn Þ, we have U n þ V n þ W n = T n - T nþ1 : Since T n is a positive and nonincreasing sequence, it is convergent. Further, since fU n g fV n g and fW n g are positive sequences, they approach to zeros separately.

11.2

Successive Approximation and Convergence

281

Moreover, since Ω is bounded and the sequence fan g and fbn g are also bounded, we deduce from the preceding inequalities: un - unþ1 an - anþ1

Lp

→ 0,

W 1,2 ðΩÞ

→ 0 8p

bn - bnþ1

ð11:41Þ

ðas n → þ 1Þ

Lp ðΩÞ

→ 0:

ð11:42Þ ð11:43Þ

Step 2: To show that the optimality conditions for approaches to zeros, we denote I nþ1 ðΩÞ = =

Ω

Ω

2

anþ1 ∇
gunþ1 þ gt - f ∇
ðgvÞ þ

j=1

bnþ1 ∇unþ1
∇vj þ cðxÞunþ1
v j j

anþ1 - an ∇
gunþ1 þ gt - f ∇
ðgvÞ þ

2 j=1

bnþ1 - bnj ∇unþ1
∇vj : j j

ð11:44Þ Introducing an open subset Ω0 strictly included in Ω, we have I nþ1 ðΩÞ = I nþ1 ðΩ=Ω0 Þ þ I nþ1 ðΩ0 Þ:

ð11:45Þ

For the first term on the RHS of Eq. (11.45), since anþ1 and bnþ1 are bounded independently of n, we can find a constant C such that jI nþ1 ðΩ=Ω0 Þj ≤ CkvkW 1,2 ðΩ=Ω0 Þ :

ð11:46Þ

For the second term, by considering the boundness in Lemma 11.2, for all p and p′ such that 1 1 1 þ þ = 1, p p0 2 the following inequality holds, i.e., jI nþ1 ðΩ0 Þj ≤ C anþ1 - an 2

þ j=1

bnþ1 - bnj j

Lp ðΩ0 Þ

Lp ðΩ0 Þ

∇unþ1 j

unþ1 0

Lp ð Ω 0 Þ

0

W 1,p ðΩ0 Þ

∇vj

kvkL2 ðΩ0 Þ þ k∇
vkL2 ðΩ0 Þ þ M

L2 ð Ω0 Þ

→ 0,

n → 1:

ð11:47Þ Equations (11.46) and (11.47) indicate

282

11

div b nj ∇u nj þ g

Analysis of Physics-Based Optical Flow

∂ n ða ð∇
ðgun Þ þ gt - f ÞÞ - cðxÞu nj ∂x j

S nj

U nj

T nj

! 0:

W 1,2 ðΩÞ ′

ð11:48Þ

Step 3: In the following, we consider the limit in each term. Since fun g is bounded in W ðΩÞ and compact in L2 ðΩÞ, we can extract a subsequence, still denoted as fun g, such that 1,2

un

! u, ∇u nj

L2 ðΩÞ

! ∇u j

L 2 ð ΩÞ

ðweaklyÞ,

and then un

! u:

W 1 ,2 ð Ω Þ ′

Therefore, U nj converges in W 1,2 ðΩÞ0 . Further, 8v 2 W 1,2 ðΩÞ, T n ð ΩÞ =

Ω

g∇ðan ð∇
ðgun Þ þ gt - f ÞÞ
v - g∇ðað∇
ðguÞ þ gt - f Þ
vdx:

ð11:49Þ

Using the property of the divergence, through simple calculations, we can rewrite the above equation in the following way: T n ð ΩÞ =

Ω

½ðan - aÞð∇
ðgun Þ þ gt - f Þ þ a∇
ðgðun - uÞÞ∇
ðgvÞdx:

If Ω0 is an open subset strictly included in Ω, we can decompose T n ðΩÞ into two parts: T n ðΩÞ = T n ðΩ0 Þ þ T n ðΩ=Ω0 Þ: The following inequalities hold, i.e., jI nþ1 ðΩ=Ω0 Þj ≤ C kvkW 1,2 ðΩ=Ω0 Þ , jT n ðΩ0 Þj ≤ C kan - akLp ðΩ0 Þ kun kW 1,p0 ðΩ0 Þ kvn kW 1,2 ðΩ0 Þ þCkun - ukW 1,2 ðΩ0 Þ kvn kW 1,2 ðΩ0 Þ

Similar to I n , T nj converges in W 1,2 ðΩÞ0 . Taking Eq. (11.17) into account, we have proved

:

11.2

Successive Approximation and Convergence

div

φ2 , ε

0

∇u nj

2 ∇u nj

283

∇u nj

!

W 1 ,2 ð Ω Þ ′

ð11:50Þ

0

∂ φ1, ε ðj∇
ðguÞ þ gt - f jÞ -g ð∇
ðguÞ þ gt - f Þ ∂x j 2 j ∇
ðguÞ þ gt - f j

þ cu j  S j :

Equation (11.21) in Lemma 11.2 can be rewritten in the form A u nj

! S j,

ð11:51Þ

W 1,2 ðΩÞ ′

where A : W 1,2 ðΩÞ → W 1,2 ðΩÞ0 AðϕÞ = div

φ2,ε 0

∇ϕ

2 ∇ϕ

∇ϕ :

Now, we show that A uj = Sj . Since A corresponds to the derivative of a convex functional and it is a monotone operator, we have A unj - AðϕÞ, unj - ϕ ≥ 0, 8ϕ 2 W 1,2 ðΩÞ0 : When n approaches to infinity, we have

AðϕÞ, unj

=

φ2,ε 0 Ω

∇ϕ

2 ∇ϕ

∇ϕ
∇unj → AðϕÞ, uj ,

A unj , unj → Sj , uj , and therefore Sj - AðϕÞ, uj - ϕ ≥ 0: Choosing ϕ = uj þ hv, 8h > 0, we obtain 0 8v 2 C 1 c ðΩ Þ,

Sj - A uj þ hv , v ≤ 0. Thus,

Sj , v ≤ A u j , v : We can deduce Sj = A uj , which completes the proof.

ð11:52Þ

284

11.3

11

Analysis of Physics-Based Optical Flow

Numerical Algorithm

In order to compute the physics-based optical flow u = ðu1 , u2 Þ, we give a discrete scheme to solve the Euler-Lagrange equation: ∂ ðað∇
ðguÞ þ gt - f ÞÞ ∂x1 ∂ ðað∇
ðguÞ þ gt - f ÞÞ divðb2 ∇u2 Þ = cðxÞu2 - g ∂x2 divðb1 ∇u1 Þ = cðxÞu1 - g

ð11:53Þ

where b = ðb1 , b2 Þ and a can be determined by minimizing the functional J dε introduced in Sect. 11.2. For a function h, let hx = ∂h=∂x. Using the following relations divðbi ui Þ = ∇bi ∇ui þ bi Δui ,

ði = 1, 2Þ

ð11:54Þ

∇
ðguÞ = g∇
u þ u
∇g,

ð11:55Þ

we have ∂ ðað∇
ðguÞÞÞ ∂x

= = =

∂ ½aðg∇
u þ u
∇gÞ ∂x ∂g ∂g ∂ ∂u1 ∂u2 þ þ a u1 þ a u2 ag ∂x ∂y ∂x ∂y ∂x ∂ agy ∂ðagx Þ ∂u ∂u u2 þ agx 1 þ agy 2 u1 þ ∂x ∂x ∂x ∂x 2 2 ∂ðagÞ ∂u1 ∂u2 ∂ u1 ∂ u2 þ þ þ þ ag ∂x ∂y ∂x2 ∂x∂y ∂x

and ∂ ðað∇
ðguÞÞÞ ∂y

= = =

∂ ½aðg∇
u þ u
∇gÞ ∂y ∂g ∂ ∂u1 ∂u2 ∂g þ þ a u1 þ a u2 ag ∂x ∂y ∂y ∂y ∂x ∂ agy ∂ðagx Þ ∂u ∂u u1 þ u2 þ agx 1 þ agy 2 ∂y ∂y ∂y ∂y þ

∂ðagÞ ∂u1 ∂u2 þ ∂y ∂x ∂y

2

þ ag

Substitution of the above relations into Eq. (11.53) yields

2

∂ u1 ∂ u2 þ ∂x∂y ∂2 y

11.3

Numerical Algorithm

285

∇b1 ∇u1 þ b1 Δu1 = cu1 - g

∂ agy ∂ðagx Þ ∂u ∂u u2 þ agx 1 þ agy 2 u1 þ ∂x ∂x ∂x ∂x

∂ðagÞ ∂u1 ∂u2 þ ∂x ∂y ∂x ∂ - g ½aðgt - f Þ ∂x

þ

2

þ ag

2

∂ u1 ∂ u2 þ ∂x2 ∂x∂y

ð11:56Þ

and ∇b2 ∇u2 þ b2 Δu2 = cu2 - g þ

∂ agy ∂ðagx Þ ∂u ∂u u2 þ agx 1 þ agy 2 u1 þ ∂y ∂y ∂y ∂y

∂ðagÞ ∂u1 ∂u2 þ ∂x ∂y ∂y

-g

2

þ ag

2

∂ u1 ∂ u2 þ ∂x∂y ∂2 y

ð11:57Þ

∂ ½aðgt - f Þ ∂y

Using the three to five-point discrete approximations, we define δx , δy , δxy , δ2x , δ2y , 2 2 and δ2 respectively, as discrete approximations to ∂=∂x, ∂=∂y, ∂ =∂x∂y, ∂ =∂x2 , 2 ∂ =∂y2 , and Δ2 in the following notations: ∂u1 ∂x

i,j

∂u1 ∂y

i,j

= ðδx u1 Þi,j = u1iþ1,j - u1i - 1,j =ð2hÞ, = δy u1

i,j

= u1i,jþ1 - u1i,j - 1 =ð2hÞ,

2

∂ u1 ∂x∂y

i,j

= δxy u1

i,j

= u1iþ1,jþ1 - u1iþ1,j - 1 - u1i - 1,jþ1 þ u1i - 1,j - 1 =ð4hÞ, u1 xi,j = u1iþ1,j þ u1i - 1,j ,

u1i,j = u1iþ1,jþ1 þ u1i - 1,jþ1 þ u1iþ1,j - 1 þ u1i - 1,j - 1 , 2

∂ u1 2 ∂ x

i,j

= δ2x u1

i,j

= u1iþ1,j þ u1i - 1,j - 2u1i,j = h2 = h - 2 u1 xi,j - 2h - 2 u1i,j :

Δu1 = u1iþ1,jþ1 þ u1i - 1,jþ1 þ u1iþ1,j - 1 þ u1i - 1,j - 1 - 4u1i,j = h2 = h - 2 u1i,j - 4h - 2 u1i,j : Similarly, we can define discrete approximations of

286

11

∂u2 ∂x

2

∂u2 ∂y

, i,j

, i,j

Analysis of Physics-Based Optical Flow

2

∂ u2 ∂x∂y

∂ u2 ∂y2

, i,j

i,j

, u2 yi,j , u2i,j and Δu2 :

Therefore, in the above notations, Eqs. (11.56) and (11.57) are written as b1x ðδx u1 Þi,j þ b1y δy u1

i,j

þ b1 h - 2 u1i,j - 4h - 2 u1i,j = cu1i,j - g½ðagx Þx u1i,j þ agy x u2i,j

þagx ðδx u1 Þi,j þ agy ðδx u2 Þi,j þ ðagÞx ðδx u1 Þi,j þ δy u2 þag h

-2

u1 xi,j

- 2h

-2

b2x ðδx u2 Þi,j þ b2y δy u2

u1i,j þ δxy u2 i,j

i,j

 - agðgtx - f x Þ,

i,j

þ b2 h - 2 u2i,j - 4h - 2 u2i,j = cu2i,j - g½ðagx Þy u1i,j

þ agy y u2i,j þ agx δy u1

i,j

þ agy δy u2

þag h - 2 u2 yi,j - 2h - 2 u2i,j þ δxy u1

i,j

i,j

þ ðagÞy ðδx u1 Þi,j þ δy u2

i,j

Þ - ag gty - f y :

Further, simplification of the above representations leads to - 4b1 h - 2 - c þ gðagx Þx - 2ag2 h - 2 u1i,j þ g agy x u2i,j = - b1x - gagx - gðagÞx ðδx u1 Þi,j - b1y δy u1 - gðagÞx δy u2

i,j

- ag2 δxy u2

i,j

- b2y þ gagy þ gðagÞy δy u2

i,j

y

- 2ag2 h - 2 u2i,j

- ag2 δxy u1

i,j

- gagy ðδx u2 Þi,j

- b1 h - 2 u1i,j - ag2 h - 2 u1 xi,j - agðgtx - f x Þ,

gðagx Þy u1i,j þ - 4b2 h - 2 - c þ g agy = - gðagÞy ðδx u1 Þi,j - gagx δy u1

i,j

i,j

- b2x ðδx u2 Þi,j

- b2 h - 2 u2i,j - ag2 h - 2 u2 yi,j - ag gty - f y :

The above linear system can be represented as Ai,j

u1i,j u2i,j

ð11:58Þ

= di,j ,

where the coefficient matrix is Ai,j =

gðagx Þx - 2ah - 2 g2 - 4b1 h - 2 - c

and the RHS vector is

gðagx Þy

g agy g agy

x

- 2ah - 2 g2 - 4b2 h - 2 - c y

11.3

Numerical Algorithm

d i,j =

287

gaðf x - gtx Þ - gagx þ gðagÞx þ b1x ðδx u1 Þi,j ga f y - gty - gðagÞy ðδx u1 Þi,j - gagx δy u1 -

gagy ðδx u2 Þi,j þ gðagÞx δy u2

i,j

gagy þ gðagÞy þ b2y δy u2

i,j

- h-2

b1 u1i,j þ ag

2

-

-

i,j

b1y δy u1

i,j

b2x ðδx u2 Þi,j

ag2 δxy u2

i,j

ag2 δxy u1

i,j

u1 xi,j

b2 u2i,j þ ag2 u2 yi,j

Hence, the optical flow u1i,j , u2i,j can be obtained by solving the linear system through u1i,j u2i,j

= Ai,j- 1 d i,j :

The blockwise Jacobi iterative method can be used to find the global physicsbased optical flow u = ðu1 , u2 Þ by minimizing J dε as described in Eq. (11.15). We have the following algorithm described in Algorithm 11.1. The initialization of the physics-based optical flow u0 can be obtained using the Horn-Schunck optical flow estimator. The convergence of unþ1 has been proved in the previous section. The variables a and b = ðb1 , b2 Þ are recomputed in each iteration, as described above, where the map t → ε1 _ t ^ ε2 is defined as

ε1 _ t ^ ε 2 =

ε1 t

if t ≤ ε1 , if ε1 < t < ε2 ,

ε2

if t ≥ ε2 :

and φ2,ε p is a function associated to φðt Þ. For simplicity, defining φðt Þ = 2 1 þ t 2 - 2, we have

φ2,ε ðt Þ =

φ 0 ð εÞ 2 εφ0 ðεÞ t þ φð ε Þ 2ε 2 φðt Þ φ0 ð1=εÞ εφ0 ð1=εÞ 2 t þ φð1=εÞ 2 2ε

if t ≤ ε if ε ≤ t ≤ 1=ε if t ≥ 1=ε

288

11

Analysis of Physics-Based Optical Flow

Algorithm 11.1 Input: Given u0 , a0 , b0 2 W 1,2 ðΩÞ × L2 ðΩÞ × L2 ðΩÞ Initialization: • initialize error tolerance ε • n=0 While T err > ε, repeat nþ1 Step 1. Obtain unþ1 = unþ1 from solving the following equations: 1 , u2

∂ n ða ð∇
ðguÞ þ gt - f ÞÞ ∂x1 ∂ n = cðxÞu2 - g ða ð∇
ðguÞ þ gt - f ÞÞ ∂x2

div bn1 ∇u1 = cðxÞu1 - g div bn2 ∇u2

Step 2. anþ1 = ε _ j∇
ðgunþ11Þþg Step 3. compute

bnþ1 j

=

φ2,ε 0

t

-fj

^ 1=ε,

∇unj

2 ∇unj

Step 4. update T err = unþ1 - un Step 5. n = n þ 1

2 2

End Output: the solution unþ1

11.4

Simulations

To examine the above algorithm, simulations were conducted on synthetic images where the displacement field is generated by using a given velocity field (Wang et al. 2015). Therefore, the accuracy of the proposed algorithm can be directly compared with the truth and the Liu-Shen method (Liu and Shen 2008; Liu 2017).

11.4.1

Flow over a Vortex Pair

First, grid images (480 × 640 pixels and 8 bits) are generated where the intensity profile across a grid line is Gaussian. The grid images simulate laser-inducedfluorescence-tagged grid images for velocity measurements (Koochesfahani and Nocera 2007). A synthetic velocity field, which is generated by superposing an Oseen vortex pair and a uniform flow, is imposed in the images. Two Oseen vortices

11.4

Simulations

289

Fig. 11.1 Grid image pair where the displacement field is generated by superposition of an Oseen vortex pair and a uniform flow: (a) t = 0 s and (b) t = 0.02 s. (From Wang et al. (2015))

are placed at ðm=3, n=2Þ and ð2m=3, n=2Þ in an image, respectively, where m (480) and n (640) are the numbers of rows and columns of the images. The circumferential velocity of an Oseen vortex is given by uθ = ðΓ=2πr Þ 1 - exp - r2 =r 20 , where the vortex strengths are Γ = ± 7000 (pixel)2/s and the vortex core radius is r0 = 15 pixels. The uniform flow velocity is 10 pixels/s. The grid image deformed by the velocity field after a time step Δt is generated using a discretized form of the physics-based optical flow equation for f = 0. Figure 11.1a shows a non-deformed 40 × 40 grid image. A deformed image in Fig. 11.1b is generated at Δt = 0:02 s, where the maximum displacement is about 1 pixel. Figure 11.2 shows velocity vectors and streamlines extracted from a pair of the grid images by using the present method. The initial Lagrange multiplier is 400, and then it is automatically adjusted by an iterative procedure. It is found that the result is not sensitive to the selection of the initial Lagrange multiplier. For comparison, the Liu-Shen method is also used (Liu 2017), where the Lagrange multipliers are 50 in the Horn-Schunck estimator for initial estimation and 400 for refined estimation, respectively. The extracted profiles of the x-component of velocity across the two vortex cores are shown in Fig. 11.3. The present method gives a better estimation of the maximum velocity just outside the vortex cores where the velocity experiences a large change. The mean optical flow error, kuk - uk,exa k = m - 1 n - 1

m

n

juk ði, jÞ - uk,exa ði, jÞj, i=1 j=1

is calculated in the whole image, where the subscript “exa” denotes the exact velocity distribution. The mean error in the whole image is 0.024 pixels/unit-time

290

11

Analysis of Physics-Based Optical Flow

Fig. 11.2 Extracted velocity field from a pair of the grid images by using the present method: (a) velocity vectors and (b) streamlines. (From Wang et al. (2015))

Fig. 11.3 Profiles of the x-component of velocity across the vortex cores extracted from a pair of the grid images by using the present method and the Liu-Shen method. (From Wang et al. (2015))

for the present method compared to 0.022 pixels/unit-time for the Liu-Shen method. In a large portion of the region, the present method is comparable to the Liu-Shen method based on the L2 norm in the regularization functional. The local error in both the methods depends on the image intensity gradient (Liu and Shen 2008).

11.4

Simulations

291

Fig. 11.4 Cloud image pair where the displacement field is generated by a simulated normal shock wave in the flow from the left to right: (a) t = 0 s and (b) t = 0.02 s. (From Wang et al. (2015))

11.4.2

Flow Across a Normal Shock

A synthetic flow across a normal shock wave is applied to a cloud image (a portion of Jupiter’s Great Red Spot image) to generate the second cloud image with a displacement field, as shown in Fig. 11.4. Then, the velocity field is extracted from the cloud image pair using the present method and the Liu-Shen method. Figure 11.5 shows the velocity magnitude maps. It is indicated that the present method gives a cleaner and sharper velocity jump across the shock than the Liu-Shen method. This is further illustrated in the spanwise-averaged velocity distribution across the shock wave in Fig. 11.6. The present method performs better to capture the sharp velocity change across the shock wave since it is designed to preserve discontinuous changes in a velocity field. The mean error in the whole image is 0.063 pixels/unit-time for the present method compared to 0.104 pixels/unit-time for the Liu-Shen method.

11.4.3

Quasi-2D Turbulence

Quasi-2D turbulence is simulated by randomly distributing 25 Oseen vortices with random strengths [Γ = - 10, 000 to 10, 000 (pixel)2/s] and sizes (r0 = 4 - 30 pixels) in a sand image. The sand image pair is shown in Fig. 11.7. Figure 11.8a shows velocity vectors on the velocity magnitude field extracted by using the present method from a pair of the sand images. Figure 11.8b shows streamlines on the vorticity field. The profiles of velocity across the vertical centerline are shown in Fig. 11.9. The mean error in the whole image is 0.028 pixels/unit-time for the present method compared to 0.029 pixels/unit-time for the Liu-Shen method. Figure 11.10

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Fig. 11.5 Extracted velocity magnitude fields of a simulated normal shock wave from a pair of the cloud images by using (a) the Liu-Shen method and (b) the present method. (From Wang et al. (2015)) Fig. 11.6 Profiles of the x-component of velocity across the normal shock wave obtained by using the present method and the Liu-Shen method. (From Wang et al. (2015))

shows the kinetic energy spectra of the simulated 2D turbulence extracted from a pair of the sand images, where the theoretical “-3” power-law spectrum for 2D turbulence is plotted for reference (Kraichnan 1967). Here, the units are (pixels/unittime)2 for the kinetic energy and (pixel)-1 for the wavenumber, which can be converted into the physical units when the converting factors are given. The energy spectra extracted by using the present method and the Liu-Shen method from are in good agreement with the truth.

11.4

Simulations

293

Fig. 11.7 Sand image pair where the displacement field is generated by the simulated 2D turbulence: (a) t = 0 s and (b) t = 0.02 s. (From Wang et al. (2015))

Fig. 11.8 Extracted velocity field of the simulated 2D turbulence from a pair of the sand images by using the present method: (a) velocity vectors on the magnitude field and (b) streamlines on the vorticity field. (From Wang et al. (2015))

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Fig. 11.9 Profiles of velocity across the vertical centerline in the simulated 2D turbulence extracted from a pair of the sand images: (a) the x-component of velocity and (b) the y-component of velocity. (From Wang et al. (2015))

Fig. 11.10 Kinetic energy spectra of the simulated 2D turbulence extracted from a pair of the sand images by using the present method and the Liu-Shen method, where the theoretical “-3” powerlaw spectrum for 2D turbulence is plotted for reference. (From Wang et al. (2015))

Appendix A: Useful Results in Differential Geometry

A.1 Decomposition of Gradient Operator on Surface Chen et al. (2019) introduced a useful lemma on a generalized gradient operator in a 3D Euclidean space, which is stated as follows. For a tensor field Ψ that is defined in a neighborhood of a surface denoted by ∂B, the decomposition of a gradient operator holds, i.e., ½∇∘Ψ∂B = ∇∂B ∘Ψ∂B þ n∘

∂Ψ ∂n

,

ðA:1Þ

∂B

where the subscript ∂B denotes a quantity at the surface, n is the unit normal vector of the surface, ∂Ψ=∂n is the wall-normal derivative of Ψ on the surface, and the notation ∘ denotes a reasonably defined product operator (such as the dot product , the cross product × , or the tensor product ). In particular, if Ψ is a scalar field, ∇Ψ is the gradient of the scalar field in a 3D Euclidean space, and ∇∂B Ψ∂B is the gradient of the scalar field on the surface. This lemma is proved below. For a surface coordinate system, a point on the surface is represented by x = ðx1 , x2 Þ in the conventional notations in differential geometry. We can construct a 3D spatial curvilinear coordinate systemðx, x3 Þ = ðx1 , x2 , x3 Þ, where x3 is the wallnormal coordinate. According to the mathematical properties of the coordinate system described in Sect. A.2, the gradient operator∇ in this coordinate system is expressed as ∇ = gi x, x3

∂ ∂ ∂ ∂ = g1 x, x3 þ g2 x, x3 þ nðxÞ 3 ∂x ∂xi ∂x1 ∂x2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8

ðA:2Þ

295

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where gi (i = 1, 2, 3) are the local contravariant base vectors in the 3D spatial curvilinear coordinate systemðx, x3 Þ. In addition, the surface gradient operator∇∂B in the surface coordinate systemðx1 , x2 Þ can be expressed as ∇∂B = gα ðxÞ∂=∂xα ðα = 1, 2Þ: Using Eq. (A.2), when the operators ∘ and ∂=∂xi are exchangeable, we have ∂ ½∇∘Ψ∂B = gi ðx, x3 Þ ∂x i ∘Ψ ∂B α 3 ∂Ψ ∂Ψ = g ðx, x Þ∘ ∂xα þ nðxÞ∘ ∂x 3

∂B

= ∇∂B ∘Ψ∂B þ n∘ ∂Ψ ∂n

ðA:3Þ ∂B

A.2 Geometric Properties of Surface As illustrated in Fig. A.1a, the positional vector of a point on the surface is given by rs = rs ðxÞ = rs ðx1 , x2 Þ, where x 2 Ds ⊂ ℝ2 are the curvilinear coordinates on the surface. For a fixed point on the surface, there exist two parameterized curves called the x1 -curve and x2 -curve passing through it. Along the x1 -curve, the coordinate x2 is fixed, while the coordinate x1 varies. Similarly, along the x1-curve, the coordinate x1 is fixed, while the coordinate x2 varies. The local covariant base vectors are defined as gα ðxÞ = ∂rs ðxÞ=∂xα , ðα = 1, 2Þ where g1 and g2 are tangent to the x1 curve and x2-curve, respectively. Then, the unit normal vector of the surface at the point x can be defined as n = g1 × g2 =jg1 × g2 j. The local contravariant base vectorsfgα g are defined in such a fashion that the

Fig. A.1 (a) Surface coordinate system and (b) 3D curvilinear coordinate system near the surface. (From Chen et al. (2019))

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297

condition gα  gβ = δαβ (α, β = 1, 2 ) is satisfied, where δβα is the Kronecker delta symbol (δβα = 1 if α = β and δβα = 0α ≠ β). Then, the metric tensor can be defined as I = gαβ gα

gβ ,

where gαβ = gα  gβ and gαβ = gα  gβ . It can be proved that I is symmetric, i.e., gαβ = gβα . In order to measure the curvature of the surface, the curvature tensor is defined as K = bαβ gα

gβ ,

where bαβ = n  ∂gβ =∂xα (α, β = 1, 2) are the coefficients (Chern et al. 1999; Huang 2003; Xie 2014). Here, K is a symmetric tensor with bβα = bαβ . The intrinsic equations of the base vectors in differential geometry are ∂gα = Γγαβ gγ þ bαβ n, ∂xβ ∂n = - bαγ gγ , ∂xα ∂gα = - Γαβγ gγ þ bαβ n, ∂xβ

ðA:4Þ

where bαβ = gαγ bγβ are the coefficients and Γγαβ = ∂gα =∂xβ  gγ (α, β, γ = 1, 2) is called the Christoffel symbol. In differential geometry, a 3D curvilinear coordinate system can be reconstructed based on the surface coordinate system. As shown in Fig. A.1b, a point in the neighborhood of the surface can be described as (Huang 2003) r = rs ðxÞ þ x3 nðxÞ, where ðx, x3 Þ 2 Ds × ð- δ, δÞ ⊂ ℝ3 and δ is a positive constant. This 3D coordinate system also has its own local covariant base vectors and local contravariant base vectors, which are defined as gi ðxÞ = ∂rðxÞ=∂xi and gi  gj = δij

ði, j = 1, 2, 3Þ:

Then, we have the following expressions: gi ðx, x3 Þ = δji - x3 bji gj ðxÞ g3 ðx, x3 Þ = nðxÞ

,

ði, j = 1, 2Þ

ðA:5Þ

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gi ðx, x3 Þ =

ð1 - 2x3 H mean Þδik þ x3 bik 1 - 2x3 H mean þ ðx3 Þ2 K Gauss

gk ð xÞ

,

ði, k = 1, 2Þ

ðA:6Þ

g ðx, x Þ = n 3

3

where H mean = ðκ1 þ κ2 Þ=2 is the mean curvature, K Gauss = κ 1 κ 2 is the Gaussian curvature, and κ 1 and κ2 are the two principal curvatures, respectively. When these base vectors are restricted on the surface, they are reduced to the local surface base vectors, i.e., gα ðx, x3 = 0Þ = gα ðxÞ g3 ðx, x3 = 0Þ = nðxÞ gα ðx, x3 = 0Þ = gα ðxÞ g3 ðx, x3 = 0Þ = nðxÞ

ðα = 1, 2Þ

ðA:7Þ

ðα = 1, 2Þ

ðA:8Þ

Therefore, we have the gradientoperators: ∇ = gi x, x3 ∇∂B = gα ðxÞ

∂ ∂ ∂ = gα x, x3 þ nðxÞ 3 ∂xi ∂xα ∂x

ðα = 1, 2Þ

∂ ∂ ∂ = g1 ðxÞ 1 þ g2 ðxÞ 2 : ðα = 1, 2Þ ∂xα ∂x ∂x

ðA:9Þ ðA:10Þ

Appendix B: Open-Source Programs

B.1 Open Optical Flow B.1.1 Basics The Matlab program package “OpenOpticalFlow” is made for extraction of highresolution velocity fields from various flow visualization images (Liu 2017), which is publicly available in the folder “OpenOpticalFlow_v1” in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics. This program is a useful tool for researchers to use OFM in various flow measurements. Here, the essence of OFM is briefly recapitulated for convenience of reading. Then, we describe the main program, relevant subroutines, and selection of the relevant parameters in optical flow computation. Examples are given to demonstrate the applications of OFM. The working principles of OFM are described in Chap. 2. The physics-based optical flow equation in the image coordinates is ∂g=∂t þ ∇  ðguÞ = f ,

ðB:1Þ

where u = ðu1 , u2 Þ is the optical flow; g is the normalized image intensity that is 2 proportional to the radiance received by a camera; ∇ = ∂=∂ xi and ∇2 = ∂ =∂xi ∂xi (i = 1, 2 ) are the gradient operator and the Laplace operator in the image plane, respectively; and the source termf is related to the boundary term and diffusion term. The corresponding Euler-Lagrange equation is g∇½∂g=∂t þ ∇  ðguÞ - f  þ α∇2 u = 0,

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8

ðB:2Þ

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Appendix B: Open-Source Programs

where α is a Lagrange multiplier. The standard finite difference method is used to solve Eq. (B.2) with the Neumann condition∂u=∂n = 0 on the image domain boundary ∂D for the optical flow (Liu and Shen 2008; Wang et al. 2015). For ∇  u = 0 and f = 0, Eq. (B.1) is reduced to the Horn-Schunck optical flow equation (Horn and Schunck 1981): ∂ g=∂ t þ u  ∇g = 0:

ðB:3Þ

The corresponding Euler-Lagrange equation is ð∂g=∂t þ u  ∇gÞ∇g - α∇2 u = 0:

ðB:4Þ

B.1.2 Implementation General Consideration The program files are contained in a folder named “OpenOpticalFlow.v1.” The main program is “Flow_Diagnostics_Run.” In the main program, a pair of images is loaded, the relevant parameters are set, images are preprocessed, optical flow computation with a coarse-to-fine iteration is conducted, and the results are plotted. The subroutine for solving Eq. (B.2) is “liu_shen_estimator.” In computations, the solution of the Horn-Schunck optical flow equation (“horn_schunck_estimator”) is used as an initial approximation for faster convergence. The combination of the subroutines “liu_shen_estimator” and “horn_schunck_estimator” constitutes the key elements of this optical flow program. The optical flow processing can be made in a selected rectangular domain of interest for regional flow diagnostics when the index “index_region” is set at 1. Otherwise, the processing is conducted in the whole image domain when the index “index_region” is set at 0. The input files and relevant parameters for optical flow computation are listed in Table B.1. The image files “Im1” and “Im2” could be in the conventional image format such as tif, bmp, or jpeg. The input parameters are discussed in the following subsections. The optical flow u = ðu1 , u2 Þ is calculated by solving Eq. (B.2). Table B.2 lists the outputs. The output data are “ux0” and “uy0” that are the coarse-grained velocity components in the image plane (x, y), which are extracted from the downsampled images in the coarse-to-fine process. The refined velocity components “ux_corr” and “uy_corr” are obtained by using the coarse-to-fine scheme. The unit of the velocity given by the program is pixels/unit-time, where the unit time is the time interval between two input images. The velocity can be converted to the physical unit of m/s after the relation between the image plane and the 3D object space is established through cameracalibration/orientation. The vorticity, strain rate, and second

Appendix B: Open-Source Programs

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Table B.1 Inputs Input files and parameters Image pair Lagrange multipliers

Notation Im1, Im2 lambda_1 for the HornSchunck estimator lambda_2 for the Liu-Shen estimator size_filter size_average

Unit – –

Note tif, bmp, jpeg, etc. Regularization parameter in variational solution

Pixel Pixel

Scale factor for downsampling of original images Number of iterations

scale_im

–

Removing random image noise Correction for local illumination intensity change Reduction of initial image size in coarse-to-fine scheme

no_iteration

–

Indicator for regional diagnostics (0 or 1)

index_region

–

Gaussian filter size Gaussian filter size

Iteration in coarse-to-fine scheme “0” for whole image; “1” for selected region

Table B.2 Outputs Output files Coarse-grained velocity Refined velocity

Notation ux0, uy0 ux_corr, uy_corr

Unit Pixels/ unit-time Pixels/ unit-time

Note Based on downsampled images Refined result with full spatial resolution in coarse-to-fine process

invariant can be calculated based on a high-resolution velocity field. Furthermore, this program can be adapted for processing of a sequence of image pairs such that the statistical quantities of the flow can be obtained.

Lagrange Multiplier The program “OpenOpticalFlow” uses the Horn-Schunck estimator (“horn_schunck_estimator”) for an initial solution and Liu-Shen estimator (“liu_shen_estimator”) for a refined solution of Eq. (B.2). In the main program, the Lagrange multipliers “lambda_1” and “lambda_2” are selected for the HornSchunck and Liu-Shen estimators, respectively. For example, “lambda_1” = 20 and “lambda_2” = 2000 for typical computations. There is no rigorous theory for determining the Lagrange multiplier in the variational formulation of the optical flow equation. The Lagrange multiplier acts like a diffusion coefficient in the corresponding Euler-Lagrange equations. Therefore, a larger Lagrange multiplier tends to smooth out finer flow structures. In general, the smallest Lagrange multiplier that still leads to a well-posed solution could be selected by a trial-and-error process. However, within a considerable range

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of the Lagrange multipliers, the solution is not significantly sensitive to its selection. Simulations based on a synthetic velocity field for a specific measurement could be carried out to determine the Lagrange multiplier by using an optimization scheme. Filtering Preprocessing of images is sometimes required to remove the random noise by using a Gaussian filter. The standard deviation (std) of a Gaussian filter is selected, depending on the noise level in a specific application (e.g., the std of a Gaussian filter is 4–6 pixels for images of 480 × 520 pixels). In the main program, the mask size of a Gaussian filter is given by the parameter called “size_filter,” and the std of the Gaussian filter is 0.6 of the mask size. Correction for Illumination Change The underlying assumption in optical flow computation is that the illumination light intensity keeps constant (or time-independent) in flow visualization, which is valid in the well-controlled laboratory conditions. In some situations, however, the illumination intensity field could change in a time interval between two successively acquired images. For example, when the Jupiter’s atmosphere structures were imaged by spacecraft, the illumination intensity field from the Sun could change in a relatively long time interval (hours) during image acquisition depending on the relative movement between the Sun, Jupiter, and spacecraft. In this case, correction for this illumination intensity change is required before applying the optical flow method to these images. In the first step in this subroutine, the overall illumination change in the whole image is corrected by normalizing both the images. In the second step, a simple scheme for correcting local illumination intensity changes is also based on the application of a Gaussian filter. The selection of the std (or size) of a Gaussian filter is important to determine the local averaged intensity value for correction. This size of the Gaussian filter is given by the parameter called “size_average” in the main program. When the size of a filter is too large, the local variation associated with the illumination intensity change in images cannot be corrected. On the other hand, when the filter size is too small, the apparent motion of patterns/features in the two images would be artificially reduced after the procedure is applied. The selection of the filter size is a trial-and-error process depending on the pattern of illumination change in a specific measurement, and simulations on a synthetic velocity field are used to determine the suitable size. Coarse-to-Fine Scheme OFM as a differential approach is more suitable for extraction of high-resolution velocity fields with small displacement vectors from images of continuous patterns (typically less than 5 pixels depending on the size of image patterns). When the displacements in the image plane are large (e.g., more than 10 pixels), the error in optical flow computation would be significant. In this case, a coarse-to-fine iterative scheme can be used, which is implemented in a loop in the main program.

Appendix B: Open-Source Programs

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First, the original images are suitably downsampled by a suitable scale factor (such as 0.5) so that the displacements in pixels are small enough (1–5 pixels). A scale factor is specified by the parameter called “scale_im” for downsampling. For example, “scale_im” = 0.5 means that the original images are reduced to 50% of the original size. A coarse-grained velocity field is obtained by applying the optical flow algorithm to the downsampled images. Then, the resulting coarse-grained velocity field is used to generate a synthetic shifted image with the same spatial resolution as the original image #1 (i.e., the first one in the two successive images) by using an image-shifting (or image-warping) algorithm in which a translation transformation is used for large displacements and the discretized optical flow equation is used for sub-pixel correction. Next, the velocity difference field between the synthetically shifted image and the original image #2 is determined by using the optical flow algorithm, and it is added on the initial velocity field for correction or improvement. Thus, a refined velocity field is successively recovered by iterations to achieve a better accuracy. The iteration number is given by “no_iteration.” Usually, one or two iteration is sufficient. When “no_iteration” is set at 0, the coarse-to-fine scheme is disabled.

B.1.3 Example: White Ovals on Jupiter The White Ovals are distinct storms in the Jupiter’s atmosphere. Figure B.1 shows two successive near-infrared continuum filtered (756 nm) images of the White Ovals. Three sets of images were taken on February 19, 1997 at a range of 1.1 million kilometers by the Solid State Imaging system aboard NASA’s Galileo spacecraft. Each is taken 1 hour apart. Unlike discrete particle images, these images have continuous patterns that are particularly suitable for the application of the optical flow method. It is noticed that the illumination from the Sun was considerably changed in a local and nonuniform fashion. Thus, the correction for this change is made by using the subroutine “correction_illumination.m” before applying the

Fig. B.1 A pair of images of the White Ovals on Jupiter: (a) Image #1 and (b) Image #2

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Appendix B: Open-Source Programs

Fig. B.2 Velocity vectors of the White Ovals on Jupiter: (a) coarse-grained field and (b) refined field

Fig. B.3 Streamlines of the White Ovals on Jupiter: (a) coarse-grained field and (b) refined field

optical flow method. Figure B.2a, b show the coarse-grained and refined velocity vector fields extracted from the White Ovals images, respectively. Figure B.3 shows the corresponding streamlines. Figure B.4 shows the refined vorticity field normalized by its maximum value. It is indicated the interactions between the three cyclonic vortices. The balloon-shaped vortex is seen between the two well-formed ovals. In optical flow computation in this case and the following cases, the Lagrange multipliers in the Horn-Schunck and Liu-Shen estimators are set at 20 and 2000, respectively.

B.1.4 Hybrid Optical-Flow-Cross-Correlation Method for PIV OFM is more suitable to images of continuous patterns in various flow visualizations. There are some intrinsic problems for applying OFM to PIV images. As a differential approach, OFM needs accurate computations of the time derivative and the spatial gradient of the image intensity field. In general, in OFM, the

Appendix B: Open-Source Programs

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Fig. B.4 Vorticity field of the White Ovals on Jupiter

displacements should be smaller than the characteristic length scale of tractable features in the image plane. In PIV images, the tractable features are particles that usually are 1–3 pixels, and thus the displacements should be smaller than about 3 pixels in optical flow computation. For a pair of successive PIV images, the temporal and spatial differentiations of the image intensity can be evaluated only when discrete particles in the first image are connected to the corresponding ones in the second image. Nevertheless, particle images are actually the distributions of spatially spiky noises in the image intensity, which are not ideal for temporal and spatial differentiations. In particular, when the displacements of particles are much larger than their sizes, the temporal and spatial differentiations of the image intensity cannot be correctly calculated. From this perspective, PIV images pose probably a challenging case for application of OFM. OFM is applicable to PIV images only for sufficiently small displacements. When particles in the first images are in neighborhoods of the corresponding ones in the second image even though they are disconnected, a coarse-to-fine scheme could be applied, which is also known as the Pyramid method in computer vision. First, an initial coarse-grained velocity is obtained from suitably downsampled and filtered PIV images by using OFM. Then, a shifted image is generated from the first image based the coarse-grained velocity filed. Further, the displacements between the shifted first image and the second image are calculated by using OFM for correction of the coarse-grained velocity field. Iterations can be made for asymptotic improvement. The cross-correlation method performs well for PIV images where the displacements are suitably large. In contrast, the optical flow method is applicable to PIV images with sufficiently small displacements. Naturally, a combination of the correlation method and the optical flow method could be promising to provide a robust tool of processing PIV images with large displacements to obtain high-resolution

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Appendix B: Open-Source Programs

Fig. B.5 Block diagram for the hybrid method for PIV

velocity fields. Liu et al. (2020) gave a systematical evaluation of a hybrid opticalflow-cross-correlation method (referred to as the hybrid method hereafter). This method overcomes the limitations of OFM applied to PIV images with large displacements. Liu and Salazar (2021) described an open-source Matlab program “OpenOpticalFlow_PIV_v1.” The program files are available in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics.

The coarse-to-fine scheme applied to OFM can be adapted for PIV images, as illustrated in Fig. B.5. The main procedures of the hybrid method are briefly described below. From a pair of successive PIV images, the correlation method rather than OFM (the Horn-Schunck estimator) is used for initial estimation of a coarse-grained displacement field since it is less sensitive to the displacement magnitude, random noise, and illumination change in PIV images. Then, a shifted image 1 is generated by using a shifting scheme based on the coarse-grained displacement field, where the spatial resolution of the original image 1 is restored by using the interpolation. Further, from the shifted image 1 and the image 2, the residual displacement field is extracted by using OFM since the residual displacement magnitude is small. The corrected displacement field is reconstructed by superposing the coarse-grained displacement field and the residual displacement field. Theoretically, the procedures can be iterated for further improvement. Usually, a single iteration is sufficient for the correction.

Appendix B: Open-Source Programs

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In this hybrid program, in order to make a stand-alone Matlab program of the hybrid method, the modified functions from the open-source Matlab software “PIVlab” is used (Thielicke and Stamhuis 2014). Optical flow computation is carried out using “OpenOpticalFlow.”

B.2 Open GLOF B.2.1 Basics “OpenSkinFrictionFromGLOF” is an open-source program in Matlab for extraction of high-resolution skin friction fields from GLOF images, which is publicly available the folder “OpenSkinFrictionFromGLOF_v1” in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics. The principles of the GLOF method are described in Chap. 3. In GLOF measurements, the thin-oil-film equation projected onto the image plane has the same form as the physics-based optical flow equation Eq. (B.1), i.e., ∂g=∂t þ ∇  ðgτ Þ = f ,

ðB:5Þ

where τ is an equivalent skin friction vector, ∇ = ∂=∂xi (i = 1, 2) is the gradient operator in the image plane, and the source termf represents the effects of the pressure gradient and gravity. The corresponding Euler-Lagrange equations is g∇½∂g=∂ t þ ∇  ðgτ Þ - f  þ α∇2 τ = 0,

ðB:6Þ

2

where ∇2 = ∂ =∂xi ∂ xi (i = 1, 2) is the Laplace operator in the image plane. The Neumann condition∂τ=∂n = 0 is imposed on an image domain boundary ∂D. Since Eq. (B.6) has the same form as the physics-based optical flow equation, the same numerical algorithm for the optical flow problem is used. From a pair of consecutive GLOF images, the numerical solution of Eq. (B.6) gives a snapshot solution for the equivalent skin friction τ, and thus the relative projected skin friction vector τ=g is obtained. The time-averaged skin friction is reconstructed by averaging a sequence of snapshot solutions.

B.2.2 Implementation The program files are contained in the folder “OpenSkinFrictionFromGLOF_v1.” The main program is “GLOF_Diagnostics_Run.” The central part is the optical flow computation using the subroutine “snapshot_solution_fun” that calls a subroutine

308

Appendix B: Open-Source Programs

“OpticalFlowPhysics_fun.” This optical flow subroutine is essentially the same as the open-source program “OpenOpticalFlow” described in Sect. B.1, which uses the Horn-Schunck estimator (“horn_schunck_estimator”) and the Liu-Shen estimator (“liu_shen_estimator”). In “GLOF_Diagnostics_Run,” a sequence of GLOF images is loaded, and the snapshot solutions of skin friction are obtained by using the optical flow method for selected pairs of GLOF images. Then, a relative (normalized) skin friction field is reconstructed by averaging the snapshot solutions. In addition, some preprocessing can be made such image filtering for removing random noise, image masking, and image downsampling. The input parameters are the same as those in “OpenOpticalFlow.” A pair of GLOF images is denoted by “Im1” and “Im2” that could be in the conventional image format such as tif, bmp, or jpeg. The time step between “Im1” and “Im2” should be suitably selected depending on the sequence of images at a frame rate. In general, for optical flow computation, a typical displacement between “Im1” and “Im2” is less than 5 pixels, which can be controlled by image downsampling. The time step between two snapshot solutions and the total number of snapshot solutions for averaging (superposition) should be selected to cover the evolution of the luminescent oil-film. “lambda_1” is the Lagrange multiplier for the Horn-Schunck estimator for initial estimation, and “lambda_2” is the Lagrange multiplier for the Liu-Shen estimator for refined estimation. “size_filter” is the Gaussian filter size for removing random noise in images in pixels. For operational simplicity, other parameters like “index_masking” are fixed. The optical flow processing can be made in a selected rectangular domain of interest for regional flow diagnostics when the index “index_region” is set at 1. Otherwise, the processing is conducted in the whole image domain when the index “index_region” is set at 0. The selection of the Lagrange multiplier and filtering are discussed in Sect. B.1.

B.2.3 Example: Square Junction Flow GLOF diagnostics was conducted on the flat plate around a square cylinder with a 51 mm × 51 mm cross section and a 121 mm height in the junction flow in the same flow conditions as those in the case of the circular cylinder. The ratio between the incoming boundary-layer thickness and the equivalent diameter was 0:03 and the Reynolds number based on the equivalent diameter was 7:3 × 104 , where the equivalent diameter is D = 4Ssq =π = 57:5 mm, and Ssq is the cross-section area of the square cylinder. Luminescent oil was a mixture of silicone oil (200 cs) with a small amount of oil-based UV dye. The resulting luminescent oil emitted the radiation at a wavelength of about 590 nm when it was illuminated by UV lamps. Before turning on the tunnel, the luminescent oil was brushed on the flat plate surface. After the flow was turned on, the luminescent oil evolution was recorded at 25 f/s using a CMOS camera with a 550-nm-long pass optical filter.

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Fig. B.6 A typical pair of GLOF images ((a) first image and (b) second image) in the square junction flow

Fig. B.7 Time-averaged skin friction field extracted by using the GLOF method in the square junction flow: (a) skin friction vectors and (b) skin friction lines

Figure B.6 shows a typical pair of GLOF images obtained in the square junction flow contained in the GitHub site. Figure B.7 shows the time-averaged skin friction field given by using the averaging of snapshot solutions extracted from the GLOF images in the square junction flow: skin friction vectors and skin friction lines. The spatial resolution of the extracted skin friction field is one vector per pixel. The separation and attachment lines associated with the horseshoe vortex around the square cylinder are clearly identified.

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B.3 Open Skin Friction from Surface Quantity B.3.1 Basics The Matlab program packages for extraction of high-resolution skin friction field from surface quantity image (pressure, temperature, and scalar concentration) are in the folders: “OpenSkinFrictionFromPressure_v1,” “OpenSkinFrictionFromTemperature_v1,” “OpenSkinFrictionFromScalar_v1,” which are available in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics. The programs in these folders for visualization images of other surface quantity (surface temperature or scalar concentration) are the same since the generic optical flow equation is the same for these surface quantities. As discussed in Chaps. 4, 5, and 6, for visualizations of surfacetemperature, scalar concentration, and pressure, the relations between skin friction and these surface quantities are written in a generic form, i.e., G þ τ  ∇g = 0,

ðB:7Þ

where g is a measurable quantity and G is a source term, which are defined differently depending on specific visualization technique used in experiments. For surface pressure visualization, G = - μf Ω and g = p∂B , where f Ω is the boundary enstrophy flux (BEF). For surface temperature visualization, G = - μf Q and g = T ∂B and for binary mass transfer visualization, G = - μf M and g = ϕ1 . The source termsf Ω , f Q and f M are defined in Chaps. 4, 5, and 6, respectively. Equation (B.7) has the same form as the Horn-Schunck optical flow equation where G = ∂g=∂t. The corresponding Euler-Lagrange equation is obtained, i.e., ðG þ τ  ∇gÞ∇g - α∇2 τ = 0,

ðB:8Þ 2

where α is a Lagrange multiplier and ∇ = ∂=∂ xi and ∇2 = ∂ =∂xi ∂xi (i = 1, 2) are the gradient operator and Laplace operator in the image plane, respectively. Given G and g, Eq. (B.8) can be solved numerically for τ = ðτ1 , τ2 Þ with the Neumann condition∂τ=∂n = 0 imposed on a domain boundary ∂D. Since Eq. (B.7) is valid instantaneously, unsteady skin friction fields can be extracted from unsteady surface temperature, scalar concentration, and pressure. Since the source termG is not known exactly, the approximate method is applied to the problem of extracting skin friction from the surface quantity. First, in the zeroth-order approximation, a known base flow (such as a boundary layer flow) is considered, which satisfies

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Gð0Þ þ τ ð0Þ  ∇gð0Þ = 0,

ðB:9Þ

where the superscript “0” denotes the base flow. A composite g-field (or a perturbed g-field) on a surface is given by gð1Þ = gð0Þ þ Δg,

ðB:10Þ

where Δg is the g-variation. The first-order τ -field denoted by τ ð1Þ can be described by the first-order approximate equation: Gð0Þ þ τ ð1Þ  ∇gð1Þ = 0,

ðB:11Þ

where Gð0Þ is the base-flow source term. Therefore, a τ ð1Þ -field can be obtained by solving the Euler-Lagrange equation, Eq. (B.8), with the known source term for the base flowG = Gð0Þ . The base-flow surface temperature and its gradient are given by gð0Þ = c0 þ ðc1 =2mÞðx - x0 Þ2m , ∂gð0Þ =∂x = c1 ðx - x0 Þ2m - 1 ,

ðB:12Þ

where c0 and c1 are proportional coefficients, x0 is the virtual origin of the boundary layer, and m is a power-law exponent. Accordingly, skin friction and the source term in the base flow are given in the power-law relations, i.e., τð0Þ = c2 ðx - x0 Þð3m - 1Þ=2 , Gð0Þ = c3 ðx - x0 Þð7m - 3Þ=2 ,

ðB:13Þ

where c2 and c3 are proportional coefficients. The power-law distributions of surface scalar concentration, skin friction, and source term serve as a local approximation in laminar and turbulent flows in applications. The parameters m and x0 can be determined by fitting surface scalar concentration data. In general, when these proportional coefficients are not given, a τ -field obtained by this approximate method is a relative field or normalized field.

B.3.2 Implementation As an example, the Matlab program package “OpenSkinFrictionFromPressure_v1” is considered for extraction of skin friction field from surface pressure field. The main program is “Skin_Friction_from_Pressure_Run.” The central part is the optical flow computation using the subroutine “Optical_Flow_generic” that is a solver for Eq. (B.8). The subroutine “p_bef_images_base_flow_power_law” is used to generate the images of surface pressure, source term (such as BEF), and skin friction with the power-law distributions in the main stream direction in the base flow. The

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subroutine “Superposition_base_variation_image” generates the composite surface pressure image (the gð1Þ-field) and the corresponding base-flowBEF image (the Gð0Þfield). In “Skin_Friction_from_Pressure_Run.m,” the composite surface pressure image (the gð1Þ-field) and the base-flowBEF image (the Gð0Þ-field) are loaded. The Lagrange multiplier “lambda” is given (e.g., 10-4). There is no rigorous theory for determining the Lagrange multiplier. In general, the smallest Lagrange multiplier that still leads to a well-posed solution could be selected by a trial-and-error process.

B.3.3 Example: Square Junction Flow PSP measurements were conducted in the Tohoku-University Basic Aerodynamic Research Wind Tunnel (Kakuta et al. 2010). This is a suction-type wind tunnel that has a solid wall test section of 300 mm width, 300 mm height, and 760 mm length. In junction flow measurements, the test model was a 3D square cylinder that has 40 × 40 mm cross section and 100 mm height. The test model was vertically mounted on the flat plate and could be rotated by a turntable. PSP measurements were conducted mainly on the floor around the model. The freestream velocity was set at 50 m/s in PSP measurement. The incident angle relative to the free-stream was set at 0 degrees for the square cylinder. The Reynolds number based on the model length was ReD = 1.3 × 105 for the square cylinder and 1.8 × 105 for the diamond cylinder. The local Reynolds number is Rex = 7.8 × 105 for the location of the front of the cylinder at 230 mm from the flat plate leading edge. It was confirmed by hot wire measurement that the incoming boundary layer was laminar state under these conditions. Figure B.8a shows a normalized surface pressure field obtained from PSP measurements in this junction flow. For comparison, GLOF skin friction diagnostics

Fig. B.8 (a) Normalized surface pressure image and (b) BEF image in the square junction flow

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Fig. B.9 Time-averaged skin friction lines in the square junction flow, which are extracted from the surface pressure and BEF images in Fig. B.8

were conducted at the same test conditions, where perylene-mixed silicone oil was used. From this extracted skin friction field and the surface pressure field obtained by using PSP, a BEF field is reconstructed by using the relation f Ω = μ - 1 τ  ∇p. Figure B.8b shows the BEF field reconstructed from PSP and GLOF measurements. From the surface pressure field and BEF field, a skin friction field is extracted by solving Eq. (B.8). Figure B.9 shows extracted skin friction lines, exhibiting interesting skin friction topology on the floor surface. A saddle is located at the upstream of the square cylinder, from which the primary necklaced separation line is originated. In addition, attachment lines are originated from the sides of the cylinder. The primary separation and attachment lines are associated with a single large horseshoe vortex forming in the front of the cylinder. Behind the cylinder, a combination of the saddle and the spiraling sink nodes (foci) are observed, which are the time-averaged on-wall footprints of the shedding wake structures.

B.4 Open Pressure from Velocity B.4.1 Basics The Matlab program package “OpenPressureFromVelocity_v1” is used for extraction of pressure filed, which is available in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics.

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As discussed in Chap. 8, for extraction of a pressure field from measured velocity fields, the relation between pressure and velocity is required. Based on the NS equations for an incompressible flow, the gradient of the nondimensional total pressure P is given by ∇P = Φ,

ðB:14Þ

where the RHS of Eq. (B.14) is defined as Φ = - St ∂u=∂t þ Re - 1 ∇2 u - ω × u,

ðB:15Þ

P, u, and ω are the normalized total pressure, velocity, and vorticity, respectively, ∇ and ∇2 are the normalized gradient and Laplace operators, respectively, Re is the Reynolds number, and St is the Strouhal number. The reason for using the total pressureP rather than pressure p in Eq. (B.14) is that well-known Bernoulli’s integral P = const: is automatically a reduced case of Eq. (B.14) for a steady, irrotational, and inviscid flow. The first, second, and third terms in Eq. (B.15) represent the local acceleration, viscous diffusion, and Lamb vector, respectively. For a 2D flow, taking the dot product between Eq. (B.14) and a nondimensional test vector field m = mx , my , we have the projected relation: m  ∇P = Φm ðx, yÞ:

ðB:16Þ

where Φm = m  Φ is the projection of Φ on m. For a steady, inviscid irrotational flow with Φm = 0, since m is arbitrarily selected, Eq. (B.16) gives a relation P = const: in the whole flow field, which is Bernoulli’s integral relating pressure to velocity. When m is a constant vector, applying the variational method with the smoothness constraint to Eq. (B.16), we have the Euler-Lagrange equation: 2

A

2

2

∂ ∂ ∂ þ C 2 P=F þ 2B ∂x2 ∂x∂y ∂y

ðB:17Þ

where A = m2x þ α, B = mx my C = m2y þ α, F = m  ∇Φm , and α is a Lagrange multiplier. The determinant of Eq. (B.17) for α > 0 is Δ = B2 - AC = - α m2x þ m2y þ α < 0: Therefore, Eq. (B.17) is an elliptic-type partial differential equation. The Neumann condition and Dirichlet condition are imposed on ∂D.

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315

B.4.2 Implementation The Matlab program package “OpenPressureFromVelocity_v1” is used for extraction of a pressure field from a velocity field. The main program is “Velocity2Pressure,” where the velocity component fields, ðu, vÞ, are read as inputs, and then they are downsampled to the suitable size W × H (e.g., 65 × 86) to accelerate the processing and to avoid out-of-memory issues. The pressure field p is obtained from ðu, vÞ by the numerical solution of Eq. (B.17). The subroutine “phi_vec_finite_diff_Hiemenz” is used to generate Φm in Eq. (B.16). The subroutines “CoeffMatrix” and “RHS_Vec” generate the coefficient matrix and the RHS vector of a linear system of equations developed from the discrete scheme of Eq. (B.17). Finally, the u, v, and p data will be stored in the folder “Results.”

B.4.3 Example: Hawkmoth Flight The high-resolution velocity fields around a freely flying hawkmoth (Manduca) were calculated and applied to the pressure extraction problem, as described in Chap. 8. In this example, to show a continuous development of the flow structures, a short sequence with four frames of Schlieren images is shown in Fig. B.10, where the rectangular tracking domains (regions of interest) are marked. The left boundary of the tracking domain moves by following the wing tip to capture the developing

Fig. B.10 Typical frames of Schlieren images of a freely flying hawkmoth and tracking domains (regions of interest (a)-(d)) at four consecutive phases for pressure calculation

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Fig. B.11 Pressure fields extracted from velocity fields in the regions of interest (a)-(d) near a freely flying hawkmoth at four consecutive phases

vortex structures. The velocity fields are calculated by OFM in the tracking domains. The normalized pressure fields extracted from the velocity fields are shown in Fig. B.11.

B.5 Open Surface Pressure from Skin Friction B.5.1 Basics The Matlab program package “OpenSurfacePressureFromSkinFriction_v1” is used for extraction of surface pressure from skin friction, which is available in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics. The relation between skin friction (τ) and surface pressure (p) is given, i.e., τ  ∇p = Φ,

ðB:18Þ

where Φ = μf Ω is a source term and f Ω is defined as a virtual source term in Chap. 4. Given τ and Φ, to solve Eq. (B.18) for p, the variational formulation is described in Chap. 9. The Euler-Lagrange equation LðpÞ = F, where

ðB:19Þ

Appendix B: Open-Source Programs 2

L=A

317 2

2

∂ ∂ ∂ ∂ ∂ þ 2B þC 2þD þE , ∂x ∂y ∂x2 ∂x∂y ∂y

A = τ2x þ α, B = τx τy , C = τ2y þ α, D = τx ∇  τ þ τ  ∇τx , E = τy ∇  τ þ τ  ∇τy , and F = Φ∇  τ þ τ  ∇Φ, α is a Lagrange multiplier. The determinant of Eq. (B.19) is Δ = B2 - AC = - α jτ j2 þ α : Thus, for α > 0, the determinant is Δ < 0, indicating that Eq. (B.19) is an elliptic partial differential equation. The Neumann condition ∂p=∂n = 0 is imposed on the boundary ∂D.

B.5.2 Implementation In “OpenSurfacePressureFromSkinFriction_v1,” there are two main programs “main_simulation” and “main_experiments” in the “Code” folder, which call a numerical solver for Eq. (B.19). The first one is for the simulation cases, while the second one is for the experiments. The program can be run directly if the skin friction data have been prepared in the “Data” folder. In “main_experiments,” skin friction data obtained from GLOF measurements (see Chap. 3) are loaded, and then are downsampled to accelerate the processing and to avoid out-of-memory issues. After that, the surface pressure field is calculated from the loaded skin friction data, and the results will be stored into the corresponding folder. In the approximate method described in Chap. 9, the source termΦ is set as a negative constant in the program. Computational tests indicate that the extracted surface pressure remains stable when the constant source termΦ and the Lagrange multiplierα keep in reasonable ranges. In “main_simulation.m,” in addition to the skin friction data, the corresponding ground truth of surface pressure is loaded too in a simulation case (such as the Falkner-Skan flow and a Delta wing flow in CFD). Hence, the extracted surface pressure field by using the algorithm from the skin friction data can be compared with the ground truth.

B.5.3 Example: 65°-Delta Wing GLOF skin friction measurements on a 65°-delta wing in a low-speed wind tunnel provide an example for the application of the algorithm, as described in Chap. 9. The obtained skin friction field on the upper surface of the delta wing using the GLOF method at AoA of 10° is shown in Fig. B.12a. The skin friction vectors are superposed on the a typical GLOF image. The normalized surface pressure field is

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Fig. B.12 Surface pressure field extracted from a skin friction field obtained by using the GLOF method on a 65°-delta wing at AoA of 10°: (a) skin friction vectors on a typical GLOF image and (b) normalized surface pressure field with skin friction lines

extracted by using the algorithm from the skin friction field, as shown in Fig. B.12b, where the source termΦ is set as a negative constant and the Lagrange multiplier is α = 0:1.

B.6 Open Heat Flux from Surface Temperature B.6.1 Basics The Matlab program package “OpenHeatFluxFromTSP_v1” is used for calculation of surface heat flux from a time sequence of surface temperature measured by TSP (or other temperature-sensitive coating techniques), which is available in the GitHub site: https://github.com/Tianshu-Liu/Open_Global_Flow_Diagnostics. As shown in Chap. 10, the analytical inverse solution for the heat flux at the polymer surface on a base with a finite thickness is expressed as kp qs ð t Þ = p πap

t 0

W ðt - τÞ dθps ðτÞ p dτ, dτ t-τ

ðB:20Þ

where θp ðt, yÞ = T p - T in is a temperature change in the polymer layer from a constant initial temperature (T in ) and the function W ðt Þ defined in Eq. (10.23) depends on the thermal properties of the polymer and base. For the polymer, ap = kp =cp ρp is the thermal diffusivity of the polymer, and k p , cp , and ρp are the

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thermal conductivity, specific heat, and density of the polymer, respectively. For the base, ab = kb =cb ρb is the thermal diffusivity of the base, and k b , cb , and ρb are the thermal conductivity, specific heat, and density of the base, respectively. The discrete form of Eq. (B.20) used to calculate heat flux is kp qs ð t n Þ ffi p πap

n i=1

θps ðt i Þ - θps ðt i - 1 Þ p p × W ðt n - t i Þ þ W ðt n - t i - 1 Þ : ðB:21Þ tn - ti þ tn - ti - 1

The numerical scheme in Eq. (B.21) is easy to implement. The function of W ðt Þ can be evaluated as a function of time, which is independent of the position.

B.6.2 Implementation The main program “Heat_Flux_Calc_Impingin_Jet_Finite_ConstDiffu_Images” in “OpenHeatFluxFromTSP_v1” is used to calculate the surface heat flux from a time sequence of surface temperature images obtained by TSP in a sonic impinging jet, which can be directly applied to other surface temperature measurements. A sequence of surface temperature images is loaded, and the thermal properties of the polymer and base are given, including the thermal diffusivity, the thermal conductivity, specific heat, and density. The function W ðt Þ is evaluated as a function time using the subroutine “W_fun_nathan.” A time sequence of the surface heat flux fields (images) is calculated by calling the subroutine “HF_Analy_FiniteBase_W1_images” that is based on the analytical solution Eq. (B.21). Another main program used to calculate the surface heat flux along a column or row is “Heat_Flux_Calc_Impingin_Jet_Finite_ConstDiffu_Scans.” The subroutines for the analytical inverse heat transfer method are contained in the folder “Analytical.” The subroutines for the numerical inverse heat transfer method are contained in the folder “Numerical.” Other useful subroutines are contained in the folder “Core funcs.”

B.6.3 Example: Obliquely Impinging Sonic Jet Some sample surface temperature images obtained in an obliquely impinging sonic jet are included in “OpenHeatFluxFromTSP_v1.” The main results are presented in Sect.10.4.3 in Chap. 10.

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Index

A Absorption, 16, 24, 25, 30, 32, 33, 256 Adiabatic wall temperature, 260, 261, 265 Airfoil flow, 153, 166–173 Angle of attack (AoA), 61, 71–74, 76–81, 119, 149–152, 168–173, 213–215, 217–220, 317, 318 Approximate method, 7, 10, 83, 89–93, 100, 102, 104, 111, 117–122, 130, 131, 135, 143, 145, 201, 208, 212, 267, 310, 311, 317 Attachment lines, 6, 73, 78, 80, 83, 86, 95, 103, 104, 106, 108, 122, 124–126, 158, 169, 204, 206, 213, 218, 219, 221, 222, 309, 313

B Background-oriented Schlieren (BOS), viii, 5, 13, 57–60 Base flow, 7, 90–96, 102, 117–121, 130, 143, 145, 310–312 Bernoulli’s equation, 93, 182, 189 Binary mass transport equation, 7, 135, 136 Boundary enstrophy flux (BEF), 7, 10, 83, 86, 87, 89–92, 94, 102, 118, 158, 163, 201, 206–208, 210, 213, 214, 310–313 BV space, 267–269

C Camera, 9, 13–17, 26, 28–30, 35, 37, 46, 49, 57, 64–66, 71, 72, 76, 77, 80, 100, 139, 144,

149, 166, 168, 173, 196, 217, 223, 246, 253, 256, 299, 300, 308 Camera calibration, 14, 80, 300 Christoffel symbol, 85, 116, 297 Cloud tracking, 13, 42–48 Collinearity equations, 14, 77, 80 Cone, 12, 100, 101, 223, 250–255 Continuity equation, 13, 18, 21, 23, 34, 86 Contravariant base vectors, 84, 113, 296, 297 Control surfaces, 4, 5, 16, 17, 19, 20, 22, 23, 26, 28–30, 32, 34–36 Convergence, 89, 161, 187, 229, 237, 267, 268, 272–283, 287, 300 Convolution-type integral equation, 11, 239, 244–245 Covariant base vectors, 85, 296, 297 Critical points, 6, 70, 71, 73, 78, 83, 93, 95, 98, 158 Curvilinear coordinate system, 113, 295–297

D Deconvolution, 11, 223, 239, 244, 245, 248, 249, 254, 255 Delta wing, 6, 7, 10, 11, 135, 139, 148–151, 201, 213–220, 317 Density-varying flow, 13, 20–24 Differential geometry, 83, 84, 112, 295–298 Disperse phase number equation, 24, 27, 29

E Energy equation, 7, 111, 112, 136

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 T. Liu, Z. Cai, Inverse Problems in Global Flow Diagnostics, https://doi.org/10.1007/978-3-031-42474-8

331

332 Enstrophy, 8, 84, 85, 95–99, 153, 157–160, 162–167 Error analysis, 5–7, 10, 39, 61, 68–69, 89, 161, 177, 182–183, 191, 206–207, 237 Euclidean space, 295 Euler-Lagrange equation, 3, 5–8, 10, 13, 38, 39, 53, 67–68, 87–91, 95, 102, 106, 117, 118, 142, 143, 160, 163, 166, 177, 179, 183–185, 192, 201, 203, 205, 208, 267, 273, 274, 277, 284, 299, 300, 310, 311, 314, 316 Evolution equations, 8, 153, 155–160 Extinction, 24, 25, 33, 34

F Falkner-Skan flow, 10, 92–95, 127, 129, 132, 201, 209–213, 317 Flow diagnostics, 300, 308 Flow visualization, 1, 2, 4–5, 13, 16, 35, 36, 64, 70, 267, 299, 302, 304 Fluctuation energy, 155, 157, 159, 166, 168–170, 174 Fourier transform, 239, 240, 242

G Gaussian curvature, 298 Global flow diagnostics, 1, 2, 9, 13, 38, 61, 70 Global luminescent oil-film (GLOF), 5–6, 10, 11, 61–81, 150, 151, 171–173, 201, 217, 218, 220, 221, 307–309, 312, 313, 317, 318 Gradient operator, 4, 17, 34–36, 84, 88, 93, 114, 117, 142, 154, 160, 295–296, 298, 299, 307, 310 Great Red Spot (GRS), 5, 43–46, 291

H Half-quadratic minimization, 274–283 Hawkmoth flight, 195–199, 315–316 Heat conduction equation, 11, 223–225, 234, 235, 239, 240, 242 Heat flux, 1, 3, 7, 11, 12, 111, 116, 127–131, 154, 223–266, 318–319 Hiemenz flow, ix, 10, 177, 188–190, 192, 193 Hypersonic wind tunnels, 11, 12, 223, 232, 262

Index I Ill-posed problem, 89, 161, 183 Image, 1, 87, 119, 135, 163, 177, 205, 239, 268, 310 Image coordinates, 13–15, 36, 41, 66, 87, 142, 160, 246, 299 Image intensity, 1, 2, 4, 15, 16, 18–24, 28, 30, 34–36, 39, 40, 53, 56, 65, 68, 77, 89, 137, 142, 165, 290, 299, 304, 305 Image plane, 2, 4, 6, 13, 14, 19, 20, 32, 36–38, 40, 41, 59, 61, 65–67, 77, 78, 87–89, 94–99, 117, 120, 121, 139, 140, 142, 160, 182, 202, 245, 246, 299, 300, 302, 305, 307, 310 Impinging jets, 5, 9, 130–133, 135, 144–149, 153, 172, 174–176, 256, 258, 262, 319 In situ calibration, 68, 103, 104, 208, 262–266 Inverse heat transfer, 3, 11, 12, 223–238, 262, 319 Inverse Laplace transform, viii, 226, 241 Inverse problems, 1–8, 10–13, 37, 64, 87, 177, 201, 223, 239, 244

J Junction flow, 6, 11, 71, 75–81, 201, 220, 221, 308, 309, 312, 313

K Kronecker delta, 85, 297

L Lagrange multiplier, 2, 5, 10, 12, 37, 40, 55, 67, 69, 88, 89, 102, 106, 117, 131, 142, 145, 147, 150, 160, 161, 163, 164, 166, 167, 184, 188, 193, 195, 202, 204–206, 210, 212–218, 289, 300–302, 304, 308, 310, 312, 314, 317, 318 Laplace operator, 18, 23, 36, 38, 67, 88, 117, 142, 160, 299, 307, 310, 314 Laplace transform, 3, 11, 223–227, 229, 230, 240, 242 Laser-induced fluorescence, 1, 16 Lateral heat conduction, 11, 223, 239–251, 254, 255 Lens distortion, 14 Light-path-averaged velocity, 4, 13, 17, 21, 27, 31, 35, 36, 198 Luminescence, 9, 68, 138, 156, 253

Index M Mass transfer, 34, 135, 141, 144, 145, 148, 156, 159, 310 Mass transfer flux, 136, 138, 139, 156 Mean curvature, 86, 114, 116, 298 Method of characteristics, 177, 179, 181–183, 188–193, 199 Metric tensor, 297 Minimization, 67, 184, 202, 267, 273–275

N Navier-Stokes (NS) equations, 7, 62, 83, 177 Neumann condition, 3, 5–8, 10, 13, 38, 67, 88, 89, 117, 142, 160, 177, 184, 185, 201–206, 300, 307, 310, 314, 317 Neutron beam, 32, 34, 35 Neutron radiography (NR), 1, 4, 5, 13, 31, 32, 49–52 Nodes, 45, 46, 70, 71, 73–75, 80, 95–99, 147, 148, 183, 188–192, 194, 195, 221, 222, 313 North polar vortex, 43 Nusselt number, 111, 127, 129–132, 159, 256, 258–262

O Object space, 4, 13–15, 17, 26, 30, 31, 35, 37, 59, 65, 66, 77, 78, 80, 300 Oil-film, 1, 2, 5–6, 99, 217, 308 Oil-film thickness, 5, 62, 64, 65, 69, 77 Optical coordinate, 25, 28, 29 Optical density, 34, 49 Optical depth, 26, 27, 29 Optical flow, 2, 4, 5, 11, 36, 44, 53, 55, 66–68, 87, 145, 147, 150, 177, 196, 267–294, 299–319 Optical flow equation, 2, 4, 8, 13, 36, 38, 40, 41, 50, 54, 61, 66, 67, 70, 88, 153, 160, 267–269, 272, 289, 299–301, 303, 307, 310

P Particle image velocimetry (PIV), 1, 4, 9, 13, 30, 31, 36, 39, 41, 42, 53–55, 57, 304–306 Particulate flows, 4, 13, 24–30, 35 Perspective center, 14 Perspective projection, 2, 13–15, 36, 144 Photogrammetry, 14

333 Poincare-Bendixson index formula, 46, 61, 70, 98 Poisson equation, 9, 23, 58, 59, 179–181, 205, 206 Power-law, 91–93, 100–102, 105, 118, 120, 121, 129, 130, 132, 133, 143, 183, 209, 212, 292, 294, 311 Prandtl number, 127, 128 Pressure, 1, 16, 62, 84, 112, 158, 177, 201, 256, 310 Pressure sensitive paint (PSP), 1, 6, 7, 9, 10, 64, 83, 87, 89, 100, 102–104, 106, 135, 137–139, 142, 144, 145, 147, 156, 312, 313 Principal distance, 14, 246 Principal point, 14 Projected motion equation, 4, 13, 16, 18, 19, 22, 23, 27, 31, 32, 34, 35 Projection, 4, 10, 13–16, 18, 62, 65–67, 79, 84, 87, 139, 160, 177, 178, 202, 246, 314

R Radiance, 13, 15, 16, 18, 25–31, 35, 141, 299 Reynolds analogy, 7, 111, 127–133 Reynolds number, 72, 76, 86, 91, 93, 100, 101, 119, 127, 129, 144, 149, 162, 165, 168, 173, 217, 219, 220, 257, 308, 312, 314

S Saddles, 46, 70–75, 80, 95–99, 148, 221, 313 Scalar, 2–4, 6–8, 10, 17, 35, 86, 87, 135–153, 156–160, 180, 277, 295, 310, 311 Schlieren image, 11, 13, 20–22, 196–198, 315 Schlieren technique, viii, 20, 57 Second invariant, 47, 48, 179–181, 198, 199, 300 Separated flows, 6, 7, 70–75, 83, 92–99, 105, 130, 168, 201 Separation bubbles, 7, 78, 80, 102–104, 108, 168, 169 Separation lines, 72, 73, 80, 106, 108, 122, 124–126, 150, 158, 204, 213, 218, 219, 221, 222, 313 Shadowgraph image, vii, 1, 13, 22–23 Sherwood number, 159 Shock wave, 7, 57–60, 83, 90, 91, 99–109, 239, 291, 292 Shock-wave/boundary-layer interaction (SWBLI), 7, 83, 91, 99, 120, 239 Skin friction, 1, 61, 83, 111, 135, 153, 201, 307

334 Skin friction divergence, 10, 93, 122–126, 158, 203–206, 208, 219, 222 Skin friction topology, 61, 70–75, 83, 91, 102, 104, 106, 107, 119, 122, 203–205, 220, 313 Snapshot solutions, 6, 61, 64, 69–70, 72, 150, 307–309 Solid angle, 15, 16, 25 Source term, 2, 4–8, 10, 18, 22, 36, 58, 59, 66, 67, 86, 91, 111, 116, 118, 127, 128, 130, 135, 136, 143, 155–160, 163, 166, 177–180, 189, 202, 205–208, 210, 217, 218, 299, 307, 310, 311, 316–318 Stanton number, 111, 265, 266 Strouhal number, 314 Sublimating coatings, 7, 135, 137, 140–142 Successive approximation, 90, 267, 272–283 Surface, 1, 13, 61, 83, 111, 135, 153, 188, 201, 223, 310 Surface coordinate system, 84, 257, 295–297 Surface curvature, 83, 85, 86, 136, 154, 239 Surface optical flow (SOF), 8–9, 153–176 Swept wing, 111, 119–125 Switch points, 70, 71, 73–75

T Taylor-series expansion solution, 62, 92, 93, 108, 124, 126, 203 Temperature, 1, 16, 111, 143, 153, 223, 310 Temperature sensitive paint (TSP), 1, 6–9, 11, 12, 64, 111, 118–120, 130, 153, 154, 166, 168, 172, 173, 223, 224, 232, 233, 235, 250–254, 256, 262–266, 318, 319

Index Thin-oil-film equation, 5, 6, 61–64, 307 Tikhonov functional, 37 Topology, 45, 103, 126, 130, 268, 269, 271–274, 280 Total pressure, 10, 93, 100, 119, 177, 178, 187, 189, 210, 213, 256, 257, 314 Transmittance image, 1, 4, 13, 23–28 Transonic flow, 111, 119–125 Transport equations, 2, 4, 13, 17, 35, 135 Turbulent channel flow, 9, 153, 162–167 Turbulent wedges, 7, 111, 119–126, 254, 255 Two-phase flow, 13, 31, 32, 34, 49–52

V Variational method, 35–42, 61, 67–70, 83, 87–92, 111, 117–119, 130, 135, 142–143, 160–161, 179, 183–188, 192–196, 201–204, 268, 272, 314 Velocity, 1–5, 8–11, 13, 16, 17, 22, 24, 27, 30, 31, 34–37, 40–56, 62, 72, 76, 84, 91–95, 108, 112, 124–127, 129, 130, 132, 144, 147, 148, 153, 155, 159, 163, 168, 173, 177–199, 202–205, 209, 210, 217, 220, 265, 267, 268, 272, 288–294, 299–305, 312, 314–316 Vorticity, 8, 46–48, 52, 55, 84, 86, 87, 157, 165, 179, 198, 204, 291, 293, 300, 304, 305, 314

W Wedges, 12, 92, 93, 122, 124–126, 209, 212, 223, 255, 262, 263, 265, 266