Non-Ideal Compressible Fluid Dynamics for Propulsion and Power: Selected Contributions from the 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion & Power, NICFD 2018, October 4-5, 2018, Bochum, Germany [1st ed.] 9783030496258, 9783030496265

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Lecture Notes in Mechanical Engineering

Francesca di Mare Andrea Spinelli Matteo Pini   Editors

Non-Ideal Compressible Fluid Dynamics for Propulsion and Power Selected Contributions from the 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion & Power, NICFD 2018, October 4–5, 2018, Bochum, Germany

Lecture Notes in Mechanical Engineering Series Editors Fakher Chaari, National School of Engineers, University of Sfax, Sfax, Tunisia Mohamed Haddar, National School of Engineers of Sfax (ENIS), Sfax, Tunisia Young W. Kwon, Department of Manufacturing Engineering and Aerospace Engineering, Graduate School of Engineering and Applied Science, Monterey, CA, USA Francesco Gherardini, Dipartimento Di Ingegneria, Università Di Modena E Reggio Emilia, Modena, Modena, Italy Vitalii Ivanov, Department of Manufacturing Engineering Machine and Tools, Sumy State University, Sumy, Ukraine

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Francesca di Mare Andrea Spinelli Matteo Pini •

•

Editors

Non-Ideal Compressible Fluid Dynamics for Propulsion and Power Selected Contributions from the 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion & Power, NICFD 2018, October 4–5, 2018, Bochum, Germany

123

Editors Francesca di Mare Ruhr University Bochum Bochum, Germany

Andrea Spinelli Politecnico di Milano Milan, Milano, Italy

Matteo Pini Delft University of Technology TU Delft, The Netherlands

ISSN 2195-4356 ISSN 2195-4364 (electronic) Lecture Notes in Mechanical Engineering ISBN 978-3-030-49625-8 ISBN 978-3-030-49626-5 (eBook) https://doi.org/10.1007/978-3-030-49626-5 © Springer International Publishing AG 2020 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, express 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

Preface

The progressive replacement of traditional, fossil fuel-based energy transformation processes through a worldwide increase in the usage of alternative, renewable primary energy sources has boosted interest for thermodynamic engines operating with fluids other than steam, air and flue gases. Especially, the availability of low-quality (i.e. low temperature) heat sources, as it is the case concentrated solar systems, geothermal and biomass plants, makes use of classic working fluids, such as water, no longer viable. Similarly, efforts to move to delocalized, small-capacity (about 0.1–10 MW) power plants and mobile powertrains still running on C-based fuels but with a virtually vanishing C-footprint as backup for volatile renewable sources have brought renewed attention to waste heat recovery applications based on unconventional technologies involving non-ideal fluid flows mostly centred on the concept of the organic Rankine cycle (ORC) and of supercritical CO2 (sCO2) technology. Such systems operate with working fluids (e.g. siloxanes, refrigerants, CO2) and in thermodynamic states exhibiting thermo-physical behaviour largely departing from that of an ideal gas, often close to the critical point, either in vapour or in two-phase conditions. Though technologies exploiting the properties of non-ideal flows find already a widespread application in space propulsion (rocket engines) and in the oil and gas industry, their relevance and popularity are also growing in residential applications especially due to the advent of the next generation of heat pumps. Whilst representing an attractive evolution of energy transformation processes, the migration to trans- or supercritical and generally unsteady operations poses new challenges for the design, optimization and maintenance of systems and their components. The predictability of the performance and life cycle of individual components requires in particular an accurate description of the thermodynamic and transport properties of the working fluid, on one side, and a faithful modelling of all relevant flow features on the other side, especially in combination with turbulence, compressibility and possibly phase change effects, as it is the case in turbomachinery (compressors and expanders).

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Preface

The NICFD conference series has been thought as a biennial forum to promote the exchange of scientific information, to encourage and consolidate the interaction between researchers and professionals in the field of non-ideal compressible fluid dynamics (NICFD), a sector of fluid mechanics dealing with flows of dense vapours, supercritical fluids and two-phase fluids, whose properties significantly depart from those of ideal gases. A Special Interest Group (SIG 49) was established in 2019 in the frame of ERCOFTAC, the European Research Community On Flow, Turbulence And Combustion. Key topics of the conference focus on: experiments, fundamentals, numerical methods, optimization and uncertainty quantification (UQ), critical and supercritical flows, turbulence and mixing; multi-component fluid flows, applications in organic Rankine cycle (ORC) power systems, applications in supercritical carbon dioxide (sCO2) power systems, steam turbines; and cryogenic flows, condensing flows in nozzles, cavitating flows, supercritical/transcritical fluids in space propulsion. More information about the conference is available on the website at www.rub.de/ nicfd2018. Four keynote lectures reviewed the state-of-the-art and illustrated future studies and applications. Professor Matthias Ihme, Stanford University, USA, spoke about the “Progress and challenges in the modeling of transcritical combustion: molecular structure, numerical methods, and applications”, Prof. Michael Pfitzner, Universität der Bundeswehr München, Germany, gave an interesting talk about “Large-Eddy Simulations of inert and reacting transcritical real gas flows”, Dr. Alexis Giauque, École Centrale de Lyon, France, reported on “Turbulent dense gas flow modelling using DNS” and Prof. Jeong Ik Lee, Korea Advanced Institute of Science and Technology, South Korea, discussed “Issues with non-ideal fluid properties for developing supercritical CO2 power cycle technology”. The submissions presented in this volume offer a focused and systematic selection of works spanning a broad spectrum of topics within the domain of non-ideal compressible flows, from the development of suitable numerical tools for high-fidelity flow modelling and simulation and fundamental research in the behaviour of turbulence in dense gases, to the construction of reduced models for turbomachines and chemical reaction, and to the challenge and potential of large-scale experiments in rocket engines. All of these submissions went through a rigorous peer review process. In all, 18 contributions were presented at the conference in 2018; eight were selected for the lecture series; each of them was reviewed by three members of the Scientific Committee and finally accepted. We would like to thank all the authors and participants of NICFD 2018 and the members of the Scientific Committee, for providing guidance during the organization of the conference and for managing the revision of the papers. We are also grateful to all reviewers for their invaluable help to attain the high scientific quality of the contributions collected here. Special thanks to Emre Karaefe, Pascal Post and

Preface

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Dr. Andreas Schramm for their great technical and editorial support at the conference and during the review process and also to the complete TTF team for their assistance. October 2018

Francesca di Mare Conference Chair Matteo Pini Andrea Spinelli Co-chairs

Organization

NICFD 2018 is organized by the Chair of Thermal Turbomachines and Aeroengines (TTF), Ruhr University Bochum, in cooperation with the research group Propulsion and Power (PP) of the TU Delft and the laboratory for Compressible-fluid dynamics for Renewable Energy (CREA) of Politecnico di Milano.

Executive Committee Conference Chair Francesca di Mare

Ruhr University Bochum, Germany

Co-chairs Matteo Pini Andrea Spinelli

Delft University of Technology, The Netherlands Politecnico di Milano, Italy

Scientific Committee Paola Cinnella Piero Colonna Rene Pecnik Matteo Pini Michael Oschwald Klaus Hannemann Joseph Oefelein Alberto Guardone

Arts et Métiers ParisTech (ENSAM), France Delft University of Technology, The Netherlands Delft University of Technology, The Netherlands Delft University of Technology, The Netherlands German Aerospace Center, Germany German Aerospace Center, Germany Georgia Tech, USA Politecnico di Milano, Italy

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Andrea Spinelli Francesca di Mare Michael Pfitzner

Organization

Politecnico di Milano, Italy Ruhr University Bochum, Germany Universität der Bundeswehr München, Germany

Contents

Experiments Experimental Investigations of Heat Transfer Processes in Cooling Channels for Cryogenic Hydrogen and Methane at Supercritical Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Haemisch, Dmitry Suslov, and Michael Oschwald

3

Cryogenic Flows Efficient Handling of Cryogenic Equation of State for the Simulation of Rocket Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Fechter, Tim Horchler, Sebastian Karl, Klaus Hannemann, Dmitry Suslov, Justin Hardi, and Michael Oschwald

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Numerical Methods Non-equilibrium Model for Weakly Compressible Multi-component Flows: The Hyperbolic Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Barbara Re and Rémi Abgrall Pressure-Based Solution Framework for Non-Ideal Flows at All Mach Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christoph Traxinger, Julian Zips, Matthias Banholzer, and Michael Pfitzner Towards Direct Numerical Simulations of Shock-Turbulence Interaction in Real Gas Flows on GPUs: Initial Validation . . . . . . . . . . Pascal Post and Francesca di Mare Direct Numerical Simulation of Turbulent Dense Gas Flows . . . . . . . . . Alexis Giauque, Christophe Corre, and Aurélien Vadrot

33

46

59 76

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Contents

Fundamentals Numerical Investigation of Supersonic Dense-Gas Boundary Layers . . . Luca Sciacovelli, Donatella Passiatore, Xavier Gloerfelt, Paola Cinnella, and Francesco Grasso

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Entropy Generation in Laminar Boundary Layers of Non-Ideal Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Matteo Pini and Carlo De Servi Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Experiments

Experimental Investigations of Heat Transfer Processes in Cooling Channels for Cryogenic Hydrogen and Methane at Supercritical Pressure Jan Haemisch1(B) , Dmitry Suslov1 , and Michael Oschwald1,2 1

2

Institute of Space Propulsion, DLR Lampoldshausen, Hardthausen am Kocher, Germany [email protected] Institut f¨ ur Strahlantriebe und Turbomaschinen, RWTH Aachen University, Aachen, Germany http://dlr.de/ra Abstract. A rising demand for efficient and reusable rocket engines leads to the development of a new generation of methane fueled rocket engines. The most crucial part is the optimal design of the cooling system, with minimal hydrodynamic losses. Therefore a precise knowledge of the heat transfer processes in the combustion chamber and primarily in the cooling channels is necessary. Cooling channels with a high aspect ratio (height-to-width-ratio) in a wall material with high thermal conductivity are known to improve cooling efficiency with only moderate increase in hydrodynamic losses. In this paper tests will be presented, that were performed with a cylindrical combustion chamber. This chamber is divided into 4 sections around the circumference, each containing cooling channels with different aspect ratios (1.7, 3.5, 9.2 and 30). Cryogenic hydrogen and liquid methane at temperatures as low as 60 K for hydrogen and 130 K for methane respectively were used as cooling fluids. Results show a distinct thermal stratification for both coolants and a very high influence of changing fluid properties close to the critical point for methane. Keywords: Liquid rocket engine · Methane Regenerative cooling · High aspect ratio

1

· Hydrogen · Reusability ·

Introduction

Liquid rocket engines, that burn a fuel and an oxidiser inside a combustion chamber are the main propulsion system to transport payload into earth orbit. The combination of liquid oxygen (LOX) and liquid hydrogen (lH2) has been investigated since the 1960s and is used in many first and upper stage engines (such as: Vulcain, Vinci, RD-0120, SSME, etc.). Nowadays methane is favored as a propellant for a new generation of rocket engines. It is expected to be cheaper and simpler to use compared to hydrogen. c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 3–16, 2020. https://doi.org/10.1007/978-3-030-49626-5_1

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The performance of regenerative cooling gets particularly important regarding reusable engines or expander cycle engines, where the regenerative cooling provides the power for the turbo-machinery. In both cases an efficient and well known cooling system is crucial. The geometry of cooling channels has a great influence in heat transfer. The advantage of cooling channels with a high aspect ratio in both heat transfer and hydrodynamic losses was early identified [2]. Manufacturing problems were believed to be the only limit for the aspect ratio until the occurrence of thermal stratification was proposed as a limit for very high aspect ratio cooling channels. Due to a limited mixing of cooling fluid, hot fluid remains at the bottom of the cooling channel and cold fluid remains at the top of the channel. Such a stratified flow limits the cooling performance. This effect has been intensively studied numerically [11] and experimentally for hydrogen [21] and nitrogen [20]. Experimental data with methane as a coolant are rare [9,19], [18] and to the authors knowledge, limited to demonstrator tests and electrical heated pipes. In the following sections, experimental tests at real conditions will be described for both hydrogen and methane as coolants. A comparison will point out the similarities and especially the differences in the case of regenerative cooling with high aspect ratio cooling channels. 1.1

Physical Properties of the Cooling Fluids

The physical properties of the fluid have a huge influence on the cooling capabilities. Table 1 lists the critical temperature Tc and pressure Pc for the two cooling fluids. Figure 1 shows the operational conditions of the present tests for hydrogen and methane with respect to the critical point. The diagram also shows the iso-lines of the specific heat at constant pressure for methane. Hydrogen enters the cooling channels at much higher temperatures and pressures than the critical values and can be described as an ideal fluid. In contrast to that, methane enters the cooling channels at supercritical pressure but subcritical temperature. It is then heated up by the combustion chamber and crosses the Widom line. The Widom line is the extension of the Coexistence line into the supercritical domain and is defined as the connection of points where the specific heat at constant pressure has its maximum value [1]. In this area, the physical properties of a fluid are strongly affected by small changes in pressure and temperature. This transcritical behavior is what most affects the cooling capabilities of methane and is therefore as a real gas application of special interest. Table 1. Critical points for hydrogen and methane [8] Tc [K]

kg Pc [bar] ρc [ m 3]

Hydrogen 32.94 12.84 Methane 190.56 45.99

31.36 162.66

Heat Transfer for Hydrogen and Methane

5

14 12

hydrogen

10 P/Pcr [-]

Widom-line 8 6 4 methane

2 0 0.5

1

1.5

2 2.5 T/Tcr [-]

3

3.5

4

Fig. 1. Operational conditions for hydrogen and methane on a reduced pressure reduced temperature diagram

2

Experimental Setup

To study the heat transfer phenomena in high and low aspect ratio cooling channels, tests were performed with a cylindrical research combustion chamber segment. The segment has a length of L = 200 mm and is divided into four sections around the circumference, each containing certain cooling channel geometry (rectangular cooling channels with different aspect ratios (height-to-width-ratio) (see Fig. 2 and Table 2)). The cooling channels are wire-cut and closed by electrodeposit copper and nickel. The channels were highly polished to ensure a fairly smooth surface of about ks ≈ 0.2 µm. Orifices are used in the setup to ensure similar mass flows for each sector to make the different geometries comparable. The so called HARCC-Segment (High Aspect Ratio Cooling Channels) is part of the combustion chamber “D”. This combustion chamber has an inner diameter of D = 80 mm and an injector head, that contains 42 coaxial injector elements [14,21]. The injector head is specifically designed to ensure an identical mass flow in every injector element. The large number of injectors establish an even heat flux distribution around the circumference. To exclude uncompleted burning processes and influences of the injector head on the heat flux in axial direction, the 200 mm long Standard-Segment is placed between injector head and HARCC-Segment. This ensures an axial distance of 2.75 · D to the injector head and therefore an even heat flux distribution on the analyzed cooling channels. Previous investigations show that the combustion processes are already completed after about 2.5 · D [15].

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Fig. 2. Cut through the HARCC-Segment that reveals the 4 channel geometries [21] Table 2. Geometry of the investigated cooling channels Sector

S1 S2

S3

S4

Height [mm] Width [mm] Distance to the hot gas [mm] Aspect ratio Hydraulic diameter [mm] Number of cooling channels (per 90◦ )

2 1.2 1.1 1.7 1.5 25

9 0.3 1.1 30 0.58 38

4.6 0.5 1.1 9.2 0.9 34

2.8 0.8 1.1 3.5 1.24 29

The coolant mass flow that feeds the Standard-Segment is used as propellant for the combustion. In contrast the coolant mass flow that feeds the HARCCSegment is dumped afterward and can therefore be controlled independently from the remaining combustion chamber. This offers great flexibility for a variety of different measurement points. Measurement Devices. The HARCC-Segment is extensively instrumented with measurement techniques. The mass flow is measured independently for every sector, using measurement turbines. Temperature and pressure sensors are located in every manifold to characterize the averaged heat flux, using calorimetric method, for each sector (see Sect. 2). The cooling channels are equipped with differential pressure sensors that enable to instantly measure the pressure loss. The thermal field is measured with 80 thermocouples, 20 per sector. These are integrated in the combustion chamber wall between two cooling channels at

Heat Transfer for Hydrogen and Methane

7

four axial positions and five distances to the combustion chamber wall (See Fig. 3 and Table 3). To ensure a defined measurement position, the thermocouples are thermally insulated and spring-pressed into their cylindrical holes. The springs ensure a constant force (F = 3N) so that the tip has constant thermal contact. This method significantly improves the measurement signal [16,17]. With an inverse method it is possible to reconstruct the thermal field and to characterize the heat transfer behavior inside the entire cooling channel [5,6,10].

Fig. 3. Position of the 20 thermocouples for structural temperature measurements (not to scale)

Table 3. Position of the thermocouples in the structure of the HARCC-Segment Axial position

P1 P1 P3

P4

Distance from the coolant inlet [mm] 52 85 119 152 Radial position

T1

T2

T3

T4

T5

Distance from the hot gas side [mm] 0.7 1.1 1.5 1.9 7.5

Test Procedure. A typical test procedure aims for stable combustion and constant properties inside the combustion chamber. Then the coolant mass flow of the HARCC-Segment is reduced stepwise until a maximum allowable temperature at the hot gas side surface is reached. Every point is held for at least 10 s until stationary conditions are reached. Regions that are not completely stable are not considered in the evaluation. Every measurement is averaged over 1 s. This time interval is the optimal compromise between measurement time and measurement accuracy [17]. The tests were performed at the European Research and Technology Test Facility P8 [4]. This test facility enables investigations at typical rocket engine operating conditions. In the frame of these tests it was achieved chamber pressures up to Pcc = 65bar, coolant pressure between P = 58 and P = 160bar and high heat fluxes up to q˙w = 26 MmW 2 . A picture of the combustion chamber inside the test cell during a hot run can be seen in Fig. 4.

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Fig. 4. Picture of the combustion chamber in the test cell during a hot run

Heat Flux Measurement. The calorimetric method is a well established approach to measure heat flux at high thermal load. The heat flux can be estimated by measuring the enthalpy increase of the cooling fluid giving a global averaged heat flux. The enthalpy h is a function of temperature and pressure, which is both measured in the individual inlet and outlet manifolds. The heat flux is then calculated as: q˙ =

m ˙ · (h(p, T )out − h(p, T )in ) AHG

(1)

where m ˙ is the coolant mass flow and AHG is the hot gas side surface.

3

Results

In this section two test cases, one for methane and one for hydrogen are compared. The test cases are chosen due to a similar combustion chamber pressure, P cc ≈ 50bar, and thrust. Even though the mass flow, heat flux and temperature are different. At the end of the section, all gathered test results are considered to develop a Nusselt corelation. The thrust depends on the mass flow and specific impulse F = m ˙ · ISP . ˙ CH4,tot · ISP,CH4 . For the same thrust this leads to m ˙ H2,tot · ISP,H2 = m ˙ CH4 and The total mass flow is the sum of fuel mass flow m ˙ H2 respectively m m ˙ O2 . oxidizer mass flow m ˙ O2 . Both are linked with the mixture ratio ROF = m˙ H2/CH4 That leads to a methane mass flow of: m ˙ CH4 =

ISP,H2 1 + ROFH2 m ˙ H2 ISP,CH4 1 + ROFCH4

(2)

Because the coolant mass flow is not equal to the mass flow that is fed into the combustion chamber, the ratio of coolant mass flow to injector mass flow HARCC has to be considered. In the analyzed test cases the ratio m˙ m is 0.82 for the ˙ IN J

Heat Transfer for Hydrogen and Methane

9

hydrogen test case and 0.94 for the methane test case. So the equivalent mass flow for the methane test case is relatively higher compared to the hydrogen test case (Table 4). Table 4. Test cases to directly compare hydrogen and methane

3.1

Hydrogen

Methane

Pcc [bar] ROF [−] F [kN ] ] m ˙ IN J [ kg s ] m ˙ HARCC [ kg s m ˙ HARCC [−] m ˙ IN J

49.1 3.9 17.0 0.79 0.65 0.82

50.2 2.0 17.8 1.85 1.74 0.94

Re [−] q˙w [ MmW 2 ] Tin [K]

160.000–870.000 46.000–470.000 22.0 13.6 62.6 138.6

Hot Gas Side Temperature

One main constraint for the design of regenerative cooling is the hot gas side surface temperature. Especially for the life-time, the hot gas surface temperature is a limiting factor. Figure 5 shows a comparison of hot gas side surface temperature for hydrogen and methane for different aspect ratios along the cooling channel length. Although the heat flux for methane is much smaller and the mass flow is relatively higher, the hot gas side surface temperature is much higher. Especially the low aspect ratio cooling channels exhibit much higher hot gas wall temperatures. For the high aspect ratio channel, S3 & S4, the wall temperatures are similar for both test cases both for the absolute temperature values and for the temperature increase along the length. The low aspect ratio channels, S1 & S2, have much higher temperature values and temperature gradients both for the hydrogen and the methane test case. Not only is the hot gas side temperature much higher, but also the structural temperature. To compare the decrease of temperature in radial direction the temperature is normed to the measurement position T1 , that is closest to the hot gas side. dT (i) = Ti − T1 , i = 1 . . . 5

(3)

Figure 6 shows the temperature difference for all cooling channels in radial direction r for measurement position 3. The temperature decrease is much steeper for hydrogen compared to methane, resulting in very low temperatures. Especially the cooling channel with the highest aspect ratio exhibits temperature values close to the inlet temperature of the coolant.

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(a) hydrogen

(b) methane

Fig. 5. Comparison of hot gas side wall temperature

To quantify the different decrease of temperature along the cooling channel circumference, the rib efficiency ηR was calculated. The rib efficiency is defined as the ratio between actual heat flux and the heat flux that could be obtained when the whole rib has the temperature at the root of the fin. Thus this value is an indicator how strong the decrease between root and top of the fin is. Figure 7 summarizes these values for all sectors and both coolants. For the low aspect ratio cooling channels the efficiencies are high and similar between hydrogen and methane. This indicates that the temperature decrease in radial direction is lower and similar in both test cases. For the high aspect ratio channels Fig. 7 reveals very low values and moreover even less efficiencies for the hydrogen test case. This result indicates a very strong thermal stratification in the structure of these cooling channels that is stronger for hydrogen compared with methane. Thermal Stratification. As it was already proposed by Kacynski [7] and indirectly numerically and experimentally confirmed for hydrogen [13,21], thermal stratification occurs in high aspect ratio cooling channels and lowers the heat transfer. A very high aspect ratio limits the fluid mixing and therefore hot cooling fluid remains at the bottom of the cooling channel whereas cold fluid remains at the top of the channel. Such a stratified flow has significantly lower heat transfer capabilities. Figure 8 shows the temperature for measurement point T5 that is 7.5 mm away from the hot gas side. The fluid temperatures at the inlet (Tin (H2/CH4) and outlet manifold (Tout (AR : 9.2/AR : 30)) are displayed and connected with a straight line. Thus this line roughly represents the mean fluid temperature. For cooling channel S3 (AR: 30) the temperature of the measurement point T5 is lower than the fluid temperature, for both coolants hydrogen and methane. For cooling channel S4 (AR: 9.2) the fluid temperature and the structural temperature are very close. This finding would not be possible for an even fluid temperature distribution and can only be explained with a strong thermal

Heat Transfer for Hydrogen and Methane

(a) hydrogen

11

(b) methane

Fig. 6. Decrease of temperature in radial direction for the different aspect ratios on measurement position 3. dT = Ti − T1 methane hydrogen

S

S

S

S

Fig. 7. Comparison of rib efficiency for all sectors

stratification and therefore a lower fluid temperature at the top part of the channel and a higher temperature at the lower part. Figure 8 also indicates a stronger stratification for hydrogen compared to methane. The reason for this behaviour is the higher heat transfer coefficient for hydrogen. A very high amount of heat is picked up in the lower part of the channel and therefore there is no much energy left to be transported along the fin to the upper part of the channel. For the low aspect ratio cooling channels, S1 (AR: 1.7) and S2 (AR: 3.5), the structural temperature is clearly above the fluid temperature and no indication of thermal stratification can be seen. Summarizing this section one can say that methane as fuel has a lower heat flux but a higher hot gas side wall temperature. The thermal stratification is less distinct and the temperature differences in the structure are less pronounced resulting in potentially lower thermal stresses.

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Q3 T5 Q4 T4 T3 T2 T1

Q2 Q1

Fig. 8. Temperatures at measurement point T5 and fluid temperature for high aspect ratio cooling channels. Dimension of cooling channels are up to scale

3.2

Nusselt Number Data Reduction

The results of the present experiments and those of a similar test campaign with the same combustion chamber that was realized by Alexander Woschnak in 2003 with hydrogen as coolant [21] are transferred to a Nusselt number. N uexp = α·dH Ac λ . Where dH is the hydraulic diameter (dH = 4 · Uc ) and λ is the thermal conductivity. The heat transfer coefficient α is assumed to be constant along the cooling q˙w channel circumference for all aspect ratios. It is approximated as: α = THG −Tb · sHG . q ˙ is the heat flux determined by the calorimetric method, T is the w HG schannel sHG hot gas wall temperature and Tb is the fluid bulk temperature. The ratio schannel refers to the ratio of hot gas side surface and cooling channel surface. H The Nusselt number can be related to the Reynolds (Re = G·d η ) - and η·cP ˙ per cooling channel Prandtl (P r = λ ) numbers. G is the mass flow rate, m, Area Ac and η is the viscosity. A slightly modified Dittus-Boelter equation (Eq. 4 [3]) was chosen to match the experimental values. N ucorr = a(AR) · Re0.8 P r0.4

(4)

To respect the different aspect ratios, the coefficient a was adapted individually for each sector. An equation then relates these coefficients to the aspect ratios. In that way correlations for a were found for hydrogen and methane that represent the experimental values for all aspect ratios.  e   The coefficient a follows the form a = b · ec·AR + d · TTHG . b The parameter b, c, d and e for hydrogen and methane respectively are listed in Table 5.

Heat Transfer for Hydrogen and Methane

13

e    for hydrogen and Table 5. Parameter for the Coefficient a = b · ec·AR + d · TTHG b methane b

c

d

e

Hydrogen 0.0055 −0.2015 0.0052 0 Methane 0.0286 −0.0238 −0.0046 −1

Note that incase of methane, in contrast to hydrogen, the wall-to-bulk teme was considered because it was found to have a strong perature ratio TTHG b impact. In the case of hydrogen the ratio has no remarkable influence. The uncertainty with these correlations are below ±25% for 97.2% of all data points for hydrogen and below ±25% for 84.3% of the methane test cases. Figure 9 shows the correlation for all data points for hydrogen and methane.

500

+25%

600

400

400

-25%

300 200

AR:1.7 AR:3.5 AR:9.2 AR:30

100 100

200

300

400

500

Nu exp

Nu exp

500

0 0

+25%

300

-25%

200 AR:1.7 AR:3.5 AR:9.2 AR:30

100 0 0

Nu corr

(a) hydrogen

100

200

300

400

Nu corr

(b) methane

Fig. 9. Nusselt correlation for hydrogen and methane compared to the experimental data. N ucorr = a · Re0.8 P r0.4

Keeping in mind, that the exponents for Reynolds and Prandtl number were set to the widely accepted values 0.8 & 0.4 and the Dittus-Boelter correlation was only slightly modified to consider different aspect ratios, the level of uncertainty is very good. For the estimation of heat transfer to a supercritical fluid in a rocket combustion chamber an uncertainty of ±25% is a very good result [12] Due to the rough environment, precise measurements are difficult to obtain. Therefore experimental uncertainties due to measurements sensors are higher than in a laboratory environment. Influencing factors apart from experimental scattering that can explain the deviation of the data points are: – rectangular shape of cooling channels – non-uniform heating of the cooling channels

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– roughness inside the cooling channels – in the case of methane: vicinity to the critical point The last point, the vicinity to the critical point, was surveyed in more detail. The deviation, F , of the calculated to the experimental determined Nusselt num(N ucorr −N uexp ) · 100. ber is given as: F = N uexp 40 measurement points Widom line

2.2

P/Pcr

2

20

1.8

10

1.6

0

1.4

-10

0.9

1

1.1

1.2

1.3

deviation, F [%]

30

-20

T/Tcr Fig. 10. Deviation of Nusselt correlation compared to experimental values as a function of reduced temperature and reduced pressure (methane)

Figure 10 shows the deviation of the Nusselt correlation as a function of reduced temperature and reduced pressure with respect to the critical values for methane. The blue dots indicate measurement points and the dashed red line represents the Widom line. It can be seen that the maximum deviation is close to the critical point. Moreover the slope of maximum deviation matches the progress of the Widom line. It is therefore evident, that what most influences the deviation in the case of methane is the rapid change in fluid properties that can not be considered by the correlation. Note that this trend is independent from the aspect ratio and solely depends on the vicinity to the Widom line.

4

Conclusion

A systematic experimental analysis of supercritical hydrogen and transcritical methane in cooling channels of rocket combustion chambers at real conditions was completed. The influence of the aspect ratio was evaluated with 4 different cooling channel geometries. Clear experimental evidence of thermal stratification

Heat Transfer for Hydrogen and Methane

15

in high aspect ratio cooling channels has been found for hydrogen and methane. The stratification is more distinct for hydrogen and results in a higher temperature decrease in the radial direction. A Nusselt correlations of the Dittus-Boelter type was found for both coolants that matches the experimental data for all aspect ratios. The accuracy of these correlations strongly depends on the distance to the critical point and is therefore a major issue for methane as coolant. The cooling capability of methane is affected by the vicinity to the critical point and therefore real gas effects have to be considered carefully.

References 1. Banuti, D.T.: Crossing the widom-line - supercritical pseudo-boiling. J. Supercrit. Fluids 98(03), 12–16 (2015) 2. Carlile, J., Quentmeyer, R.: An experimental investigation of high-aspect-ratio cooling passages. In: 28th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 6–8 July 1992, Nashville, TN, AIAA-92-3154 (1992) 3. Dittus, P., Boelter, L.: Heat transfer in automobile radiators of the tubular type. Univ. Calif. Publ. Eng. 2(13), 371 (1930) 4. Haberzettl, A., Gundel, D., Bahlmann, K., Oschwald, M., Thomas, J., Kretschmer, J., Vuillermoz, P.: European research and technology test bench p8 for high pressure liquid rocket propellants. In: 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit (2000) 5. Haemisch, J., Suslov, D., Oschwald, M.: Experimental analysis of heat transfer processes in cooling channels of a subscale combustion chamber at real thermal conditions for cryogenic hydrogen and methane. In: 6th Space Propulsion Conference, 14–18th May 2018, Seville (2018) 6. Haemisch, J., Suslov, D., Oschwald, M.: Experimental study of methane heat transfer deterioration in a subscale combustion chamber. J. Propul. Power 35 (2019) 7. Kacynski, K.J.: Thermal stratification potential in rocket engine coolant channels. Technical report, NASA, Lewis Research Center, Cleveland, Ohio, NASA-TM4378, E-6135, NAS 1.15:4378 (1992) 8. Linstrom, P., Mallard, E.W.G.: NIST Chemistry WebBook, NIST Standard Reference Database Number 69, National Institute of Standards and Technology, Gaithersburg MD, 20899. http://webbook.nist.gov. Accessed 7 Nov 2015. P.J. Linstrom and W.G. Mallard, Eds 9. Negishi, H., Daimon, Y., Kawashima, H.: Coupled combustion and heat transfer simulation for full-scale regenerative cooled thrust chambers. Space Propul. In: 4th Space Propulsion Conference, 19–22th May 2014, Cologne (2014) 10. Oschwald, M., Suslov, D., Haemisch, J., Haidn, O., Celano, M., Kirchberger, C., Rackemann, N., Preuss, A., Wiedmann, D.: Measurement of heat transfer in liquid rocket combustors. High Pressure Flows for Propulsion Applications, AIAA Progress in Astronautics and Aeronautics Series (2020) 11. Pizzarelli, M.: Modeling of Cooling Channel Flow in Liquid-Propellant Rocket Engines. Ph.D. thesis, Universita degli Studi Roma “La Sapienza” (2007) 12. Pizzarelli, M.: The status of the research on the heat transfer deterioration in supercritical fluids: a review. Int. Commun. Heat Mass Transf. 95, 132–138 (2018)

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13. Pizzarelli, M., Nasuti, F., Onofri, M.: Analysis on the effect of channel aspect ratio on rocket thermal behavior. In: 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 30 July–1 August 2012, Atlanta, Georgia (2012) 14. Sender, J., Suslov, D., Deeken, J., Gr¨ oning, S., Oschwald, M.: “l42” technology demonstrator: Operational experience. In: 5th Space Propulsion Conference, 02– 06th May 2016, Roma (2016) 15. Suslov, D., Arnold, R., Haidn, O.J.: Investigation of two dimensional thermal loads in the region near the injector head of a high pressure subscale combustion chamber. In: 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, 05–08th January 2009, Orlando, Florida (2009) 16. Suslov, D., Woschnak, A., Greuel, D., Oschwald, M.: Measurement techniques for investigation of heat transfer processes at European research and technology test facility p8. In: 1st EUCASS, 04–07th July 2005, Moscow (2005) 17. Suslov, D., Woschnak, A., Sender, J., Oschwald, M., Haidn, O.: Investigation of heat transfer processes in cooling channels of rocket engines at representative operating conditions. In: SPACE 2003, Moscow - Kaluga, 15–19 Sept 2003, Russian Federation (2005) 18. Torres, Y., Stefanini, L., Suslov, D.: Influence of curvature in regenerative cooling system of rocket engine. Prog. Propuls. Phys. 1 (2009) 19. Votta, R., Battista, F., Salvatore, V., Pizzarelli, M., Leccese, G., Nasuti, F., Meyer, S.: Experimental investigation of transcritical methane flow in rocket engine cooling channel. Appl. Thermal Eng. 101, 61–70 (2016) 20. Wennerberg, J., Anderson, W.E., Haberlen, P.A., Jung, H., Merkle, C.L.: Supercritical flows in high aspect ratio cooling channels. In: 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 10–13 July 2005, Tucson, Arizona (2005) 21. Woschnak, A., Suslov, D., Oschwald, M.: Experimental and numerical investigations of thermal stratification effects. In: AIAA 2003-4615 39th Joint Propulsion Conference, 20–23 July, Huntsville, Alabama (2003)

Cryogenic Flows

Efficient Handling of Cryogenic Equation of State for the Simulation of Rocket Combustion Chambers Stefan Fechter1(B) , Tim Horchler1 , Sebastian Karl1 , Klaus Hannemann1 , Dmitry Suslov2 , Justin Hardi2 , and Michael Oschwald2 1

German Aerospace Center, Institute of Aerodynamics and Flow Technology, Spacecraft Department, 37073 G¨ ottingen, Germany [email protected] 2 German Aerospace Center, Institute of Space Propulsion, Lampoldshausen, 74239 Hardthausen, Germany

Abstract. The simulation of cryogenic flows in rocket combustion chambers is challenging because we have to consider a reactive mixture over a wide temperature and density range. This necessitates the use of more advanced fluid models that impose additional computation overhead. In this study we compare two equation of state (EOS) mixture approximation approaches for cryogenic flows in rocket combustion chambers and present a computationally efficient implementation within a Reynolds Averaged Navier Stokes (RANS) context. The numerical study is validated with experimental results of a lab-scale rocket combustion chamber with optical access. Keywords: CFD

1

· Cryogenic fluids · EOS · Combustion chamber

Introduction

The accurate numerical simulation of cryogenic combustion and atomization processes in rocket combustion chambers is a key element for the design of future space transportation vehicles. One major challenge is to correctly predict the flow, mixing and combustion of cryogenic fuels and, ultimately, the thermal loads in combustion chambers of reusable rocket systems by means of validated numerical methods. The focus of this contribution is on the development and assessment of mixture equation of state (EOS) models of cryogenic fluids for application in Reynolds Averaged Navier Stokes (RANS) methods. The general EOS approximation methodology though is applicable to every kind of fluid solver. The challenges of cryogenic, reactive flows in rocket combustion chambers include the consistent approximation of thermodynamic effects as well as the resolution of the flame front. In detail this includes the following aspects: – Representation of an accurate mixture EOS ranging from cryogenic temperatures at the injector up to high temperature within the flame. c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 19–30, 2020. https://doi.org/10.1007/978-3-030-49626-5_2

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– Resolution of the phase front with a density variation from O(1000) kg/m3 in the liquid part at cryogenic temperatures (e.g. for oxygen) up to densities of O(0.1) kg/m3 in the flame. This is an additional discontinuity within the flow field that has to be treated numerically. – Different turbulent length scales in the liquid and vapor phases with strongly varying physical properties. The phase boundary acts as mixture barrier due to the density difference. – Chemical kinetic model for use with cryogenic fluids: For the LOx/H2 combustion the detailed chemistry is a fast option. For more complex kinetics parametrized chemical approaches, e.g. Flamelet methods for non-ideal gases have to be used. The extensions to non-ideal gas flamelets is described in Horchler et al. [10]. In Zips et al. [27] another approach is used to extend an ideal-gas flamelet approach for these conditions. Despite the well-known kinetics for the hydrogen-oxygen propellants (as considered here in this study), a lot of literature is focusing at the moment on (non-adiabatic) flamelet methods, see e.g. Ma et al. [15] or Zips et al. [27], with the focus towards the more complex Methane chemistry description for future launcher systems. A qualitative comparison of flamelet methods with application to a subscale rocket engine is published in Perakis et al. [18]. In open literature, in the context of numerical prediction of cryogenic injection processes it is an open question how cryogenic sub- and supercritical injection process, especially within rocket combustion chambers, can be modeled consistently (see e.g. the discussion in Oefelein [17], Bellan [3]) and especially which physical processes are important to characterize the breakup and the evaporation processes correctly. At these conditions, as found e.g. in rocket combustion chambers, all fluid properties have a strong sensitivity to changes related to the description of thermodynamic quantities. Detailed previous investigations of the thermodynamic structure of LOx/H2 diffusion flames have been performed by Banuti et al. [2]. They showed, based on higher-resolved Large-Eddy simulations, that non-linear mixture effects are limited to a small area at the phase interface. With steady-state RANS methods on a typical much coarser grid, it is not possible to resolve such fine segments at the phase interface. The objective of the paper is the following: we compare two numerical methodologies to efficiently handle cryogenic flows in combustion chambers using a RANS based simulation framework based on the German Aerospace Center (DLR) inhouse flow solver TAU. This includes the numerical framework for an efficient evaluation of cryogenic mixture states as well as state of the art mixture rules for viscosity and heat transfer coefficients. The first approach is based on a cubic EOS description of the EOS mixture states that is the state of the art modeling for cryogenic fluids due to its simplicity and its cheap evaluation. However, the density estimates without additional correction models are poor, especially for the liquid phase. In the second approach, we propose an adaptive high-order tabulation of the fluid properties based on accurate EOS (for example EOS [21,22] and many others that are expensive to evaluate during solver runtime) for one non-ideal component (based on the adaptive high-order

Efficient Handling of Cryogenic EOS

21

tabulation ideas of Dumbser et al. [6] for one fluid). The non-ideal component is mixed based on ideal-gas mixture rules with the other species under the assumption that no non-ideal mixing processes are present. This assumption is fulfilled for steady-state hydrogen-oxygen simulations as shown by Banuti et al. [1]. In this study we investigate the influence of the EOS description onto the solution of a subscale single-injector combustion chamber. It is still an open question, to which extend the non-linear mixing effects can be resolved within RANS methods and which influence these effects have on the flame structure in a rocket combustion chamber. In detailed investigations of Banuti et al. [2] only a small region of nonlinear mixture effects can be found within hydrogen-oxygen diffusion flames. Within coarse RANS resolutions it is not possible to resolve these small layer close to the flame and phase boundary. Thus, an open question is, whether this non-linearity has an influence onto the overall flame structure and wall heat transfer rates. The content of the paper is the following: In the first part the numerical and thermodynamic model used for the simulation of cryogenic rocket combustion chambers is discussed. Here we also detail the adaptive high-order tabulation method for non-ideal fluid properties. In the second part we analyze in detail the influence of the EOS description onto the solution of a sample single-injector combustion chamber at supercritical conditions.

2 2.1

Numerical and Physical Model Flow Solver

The baseline ideal gas flow solver is the DLR TAU Code for the compressible Navier-Stokes equations with various RANS turbulence models. Only a brief overview is given here, detailed descriptions can be found in Hannemann [8], and Karl [12]. TAU is a hybrid grid, finite volume, compressible flow solver. It has been verified for a variety of steady and unsteady flow cases, ranging from sub- to hypersonic Mach numbers. Only models relevant for the present work are discussed in the following. The mixture of chemically reacting flows governed by the compressible Navier-Stokes equations is solved using a Godunov type finite volume method. Second order spatial accuracy is obtained using MUSCL reconstruction at cell interfaces. The low Mach numbers and high density gradients, which are a challenge for a compressible flow solver, can be dealt with using a MAPS+ Riemann solver by Rossow [20]. For the modeling of the turbulent terms within the RANS approach we apply a Spalart-Alamaras turbulence model [22]. In the present work, TAU’s finite rate chemistry model is used in thermal equilibrium. Chemical source terms are calculated from Arrhenius’ equations, backward reaction rates are calculated using the forward rates and the equilibrium constants. The mechanism is Jachimowski’s [11] 6 species, 7 reaction model, as reported by Gerlinger [7]. The considered species are H2 , O2 , OH, H2 O, O, H.

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For the ideal gas components the species laminar viscosities are estimated using Blottner et al. [5] curve fits. The individual values are combined using Wilke’s mixing rule [25]. The thermal conductivity is computed using a modification of the Eucken correction by Hirschfelder et al. [9], the mixture heat conductivity is determined following Zipperer and Herning [26]. High pressure corrections for the viscosity and thermal conductivity are used that are valid for cryogenic liquids and gases (e.g. viscosity described in Lemmon and Jacobsen [13] for pure oxygen). These corrections are provided in a similar way to the thermodynamic corrections using adaptive tabulation methods described in the following section. In the following parts we focus on the numerical methodology for the consistent thermodynamic description for cryogenic mixtures. 2.2

Thermodynamic Model for Cryogenic Mixtures

The modeling of thermodynamic mixtures within cryogenic combustion chambers has been investigated by many researchers in the last decades. Due to the cryogenic injection conditions of the fuel and the oxidizers, an ideal gas description does not provide the correct densities (see e.g. in thermodynamic textbooks of Poling et al. [19]; Niedermeier et al. [16], Bellan [4]). However, the correct description of the injection conditions is crucial for the determination of the injection velocities at the injector head. This has a significant effect on the determination of combustion chamber pressure, one main characteristic for combustion chamber simulations. Thus, more advanced EOS models are needed to describe the fluid behavior at cryogenic conditions. Most EOS can be written in a generalized way decomposing the fluid behavior into an ideal gas part and a residual part q=

N 

Xi qiid (ρi , T ) +

i=1

N 

r Xi q0,i (ρ, T ) + Δq r (ρ, T, X)

(1)

i=1

for a general fluid quantity q based on the mole fraction of component i, Xi . r The EOS approximation is based on an ideal part qiid and a residual part q0,i that describes the derivation from the ideal gas state for each component i. An additional mixture departure function Δq r =

N  N 

r Xi Xj qij (ρ, T )

(2)

i=1 j=1 r for the components i and j. This additional with a binary departure function qij mixing term may be added depending on the species pairing. Several choices that model the residual part include (Fig. 1):

– Cubic mixture EOS models, e.g. Peng-Robinson or Soave-Redlich-Kwong EOS, that are based on a corresponding state principle to estimate the mixture state. This mixture model includes a non-linear mixture term and is

Efficient Handling of Cryogenic EOS A

B

aa , ba

A ab , b b

Mixture model

23

B

aa , b a

ab , b b

EOS

EOS

a ˆ, ˆb EOS = f (ˆ a, ˆb)

Mixture model

Mixture

Mixture

Non-linear mixture multiple non-ideal gases

Multi-Fluid mixing model one non-ideal gas

Fig. 1. EOS mixture modeling approaches for cryogenic fluids based on the description in Banuti et al. [1]. One sphere represents one fluid component of the mixture.

widely applied for the simulations of cryogenic flows because of its simplicity and its cheap application. The detailed description can be found in any textbook, e.g. see Poling et al. [19]. – Multi-Fluid mixture model, as introduced by Banuti et al. [1], that is a suitable description for mixtures with one non-ideal component (mostly the oxidizer). An important limitation is that all mixing processes are limited to the validity region of the ideal gas mixture rules. This requirement is fulfilled for attached hydrogen-oxygen diffusion flames in case the fluids are injected by coaxial injectors because the flame front prevents any non-ideal mixing processes of hydrogen into oxygen. Its main advantage is that its application is cheap (in case a tabulation approach for the residual part is used) and it allows the combination of high-accurate reference EOS for the description of the fluid behavior, e.g. EOS described in [14,21] that determines the residual term qires . The nonlinear mixture departure term Δq r is set to zero. In the next paragraph we detail the efficient tabulation approach for the residual part r . q0,i

2.3

Efficient Adaptive Tabulation and Evaluation of Fluid Properties

The efficient tabulation of the residual part is inspired by the ideas of Dumbser et al. [6] that applied a similar tabulation approach to cavitation problems of one single species. To obtain a minimal storage footprint we use a high-order approximation of the fluid properties together with a quadtree.

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In this study the adaptive high-order tabulation method acc. to [6] was chosen. This methodology combines an accurate storage of the EOS data within a data format that has a low memory footprint with a fast evaluation speed. For example, the file size of an equidistantly spaced tabular for oxygen is about 1.5 Gb while with the adaptive high-order approach only 200 Mb are needed for the same approximation quality. Building Step. In the tabulation process we discretize the state space Σ = [ρmin , ρmax ] × [Tmin , Tmax ] with an initially 2minlevel × 2minlevel equidistant quadtree mesh with a user-defined parameter minlevel. We use hierarchically l nested grids,  each  subdomain  lΣ is again decomposed into 2×2 cells of the  l where l 0 0 extent Δρ , ΔT = Δρ , ΔT /2 . Here the index l = 0, 1, . . . , L denotes the level of the approximation and L is the smallest refinement level. Each subdol l l = [ρli−1/2 , ρli+1/2 ]×[Ti−1/2 , Ti+1/2 ]. The indices main consists of the elements Eij i and j define the position of the element in its individual subgrid. Each element E is mapped onto the unit element E = [0, 1] × [0, 1] with the introduction of element local coordinates ξ and η as ρl = ρli−1/2 + ξΔρl ,

l T l = Ti−1/2 + ηΔT l .

(3)

Based on the approximation of the state space we introduce a polynomial representation of the algebraic relations for the quantity q as qhp (ξ, η) ≡ Iq =

p 

qˆrs ψrs (ξ, η) ,

qˆrs = q(ξr , ηs ),

(4)

r,s=0

using the nodal basis functions ψrs (ξ, η), the expansion coefficient qˆrs and the polynomial degree of the approximation p. In case the L∞ -error of the polynomial approximation in one grid cell i, j satisfies the criterion   qh (ρ, T ) − q(ρ, T ) L∞ = max > t , (5) q(ρ, T ) ∞ the corresponding grid cell is refined and an additional approximation level is introduced until the maximum level L is reached. Hereby, t denotes the target error of the approximation in the EOS table. The L∞ -error is sampled on a sufficiently high number of equidistantly spaced verification points nVP  p. As a point distribution in reference space we chose Legendre-Gauss points. Evaluation Step. In an EOS table the quadtree information, the refinement information as well as cell-errors are stored for all refinement levels. To reduce the memory footprint the loaded table only contains the tabulated polynomial data at the smallest refinement level. The quadtree data structure is only required for grid traversal. That level is controlled by a lower bound error l ≥ t . During the simulation the iteration of the quantity q (e. g. temperature) is replaced by an evaluation of the polynomial basis (4) in the cell containing the requested region. That cell is found by a grid traversal through the quadtree

Efficient Handling of Cryogenic EOS

25

from top, down to the cell the data point is located in. It is realized using an efficient integer-based bisection approach. In short, there are two main components for the EOS evaluation: The relatively fast grid traversal within the quadtree and the interpolation of the solution on the final data set, where the cost strongly depends on the polynomial degree employed. 2.4

Validation for Cryogenic Oxygen

In Fig. 2, a comparison of the density estimation of oxygen is shown. Except close to the critical point, the cubic EOS as well as the tabulated EOS behavior are able to reproduce the fluid behavior well. 8 SRK EOS 1,000

NIST

ideal gas

cp [kJ/(kg K)]

ρ [kg/m3 ]

SRK EOS

NIST

500

0

100

200

T [K]

300

400

6

4

2

0

100

200

T [K]

300

400

Fig. 2. EOS behavior of oxygen at different pressures (100 bar, 60 bar, 50 bar, 40 bar) including a comparison to reference data using the EOS of Lemmon & Jacobsen [13]. Note that the tabulated density in the Multi-Fluid resembles the reference solution for a pure fluid.

3

Comparison to BKC Combustion Chamber Experiments

The DLR subscale thrust chamber model “C” (designated BKC) at the DLR test facility P8 in Lampoldshausen has been chosen for validation of the numerical results. The experimental results and boundary conditions are described by Suslov et al. [23,24] where an extensive description of the experimental test facility and the measurement techniques applied can be found. The experiments were designed to provide extended data for high pressure LOx/GH2 combustion. Three different combustion chamber pressures from 40 to 60 bar were measured with three different O/F-ratios that cover the full range from sub- to supercritical combustion. We focus on the description of the numerical grid and boundary conditions and consider a supercritical load point for comparison.

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Table 1. Measured test parameters taken from Suslov et al. [23]. Load step (O/F)injector (O/F)cc Tinj, H2 Tinj, LOx vinj, H2 vinj, LOx

3.1

Table 2. Summary of the main experimental and numerical parameter O/Fcc (O/Finj ) p [bar]

3 [-] [-] [K] [K] [m/s] [m/s]

4 1.32 158 114 350 30

Experiment 1.1 (4) MultiFluid mixing Cubic EOS

60.9 61.0 61.1

Numerical Domain and Physical Model

In this investigation we consider a two-dimensional axisymmetric model for the flow in the single-injector combustion chamber. In Fig. 3 a sketch of the computational domain is shown. The tubes of the injector are not modeled here and replaced by a stub tube with the respective experimental inflow conditions. In this study we do not consider thermal radiation. The location of the boundary conditions is indicated in Fig. 3 with the injector boundary conditions specified in Table 1. The combustion chamber pressures are compared in Table 2. H2 window cooling inlet

Adiabatic wall Nozzle (cooled) Outflow

H2 inlet O2 inlet

Symmetry axis

Fig. 3. Sketch of the computational domain used in the numerical modeling (not to scale). The numerically chosen boundary conditions are indicated in the sketch.

3.2

Flame Structure

In Figs. 4 and 5 we compare the structure of the flame within the combustion chamber. Additionally, we superimpose the density of the dense oxygen core as it would appear on shadowgraph images based on the density field. An arrow indicates the experimentally measured length of the LOx core based on the half intensity criterion (see Suslov et al. [23] for more information). The overall flame shape and length is independent of the chosen EOS mixture model; the LOx core length is unaffected.

Efficient Handling of Cryogenic EOS

27

(a) Multi-Fluid mixing

(b) Cubic mixing LOx-core length

Fig. 4. Flame structure within the BKC combustor.

(a) Multi-Fluid mixing

(b) Cubic mixing

Fig. 5. Temperature contours within the BKC combustor.

3.3

Wall Temperature and Pressure Measurements

In the experimental measurement the pressure and temperature distribution at the combustor walls were measured using pressure and temperature transducers. Due to the fact that data points from different test runs are combined, that is needed to be able to change the window position to provide optical access of the whole combustion chamber, the scatter in the experimental data is quite large. In Fig. 6 the numerical and experimental wall pressures and temperatures are compared to the numerical results. The wall pressure distribution shown in Fig. 6(a) of the multi-fluid mixing and the cubic mixing approach fit withing less than 1 % of the average experimental wall pressure. The wall temperature distribution in Fig. 6(b) is well predicted by the numerical models and fits very well to the experimentally measured values. Both mixture approaches provide reasonable predictions for the wall temperature.

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p / pcc [-]

1.1

1

Experiment MultiFluid mixing Cubic EOS

0.9

0

50

100

150

200

250

300

350

400

450

400

450

Distance from injector head x [mm]

(a) Pressure 3 Experiment MultiFluid mixing Cubic EOS

T / Twall,0 [-]

2.5 2 1.5 1 0

50

100

150

200

250

300

350

Distance from injector head x [mm]

(b) Temperature

Fig. 6. Comparison of wall pressure and temperature measurements.

It should be noted that the wall temperature distribution is dominated by the injection of the hydrogen massflow that is needed for the window cooling.

4

Conclusion

In this numerical study the influence of non-linear mixing rules for the EOS description onto the flow and flame structure as well as the wall heat fluxes of a subscale rocket combustion chamber. Detailed investigations of the dense LOxcore length as well as on the wall temperature and pressure distribution were performed. The numerical model was compared to experimental measurements of the BKC combustion chamber at supercritical conditions. The simplified MultiFluid mixing model with tabulated EOS data is able to reproduce all features of the combustion chamber including LOx-core length and the wall predictions at a slight increase physical complexity compared to a standard ideal-gas flow solver. The main limitation of this mixing model is its assumption of ideal-gas mixing rules. Thereby, the challenges of the tabulation approach are to provide a consistent and accurate tabulation method. One suitable method is a quadtree based

Efficient Handling of Cryogenic EOS

29

high-order tabulation and evaluation method that combines the advantages of fast and accurate evaluation as well as low memory footprint. The numerical simulations showed a good agreement to the experimental data. Distinct differences between the models could only be investigated for the LOx-core intensity profile. The main advantage is the reduced computational time for cases with few thermodynamic non-ideal species for which non-linear mixing effects do not have a major impact, one example for that are attached cryogenic LOx/H2 diffusion flames. Acknowledgments. The present work was conducted in the framework of the German Aerospace Center (DLR) project TAUROS (TAU for Rocket Thrust Chamber Simulation) focusing on the qualification and advancement of the DLR flow solver TAU for liquid rocket thrust chamber applications. The financial support of the DLR Space Research Programmatic is highly appreciated.

References 1. Banuti, D.T., et al.: An efficient multi-fluid-mixing model for real gas reacting flows in liquid propellant rocket engines. Combust. Flame 168, 98–112 (2016) 2. Banuti, D.T., et al.: Thermodynamic structure of supercritical LOX–GH2 diffusion flames. Combust. Flame 196, 364–376 (2018) 3. Josette, B.: Supercritical (and subcritical) fluid behavior and modeling: drops, streams, shear and mixing layers, jets and sprays. Progress Energy Combust. Sci. 26(4), 329–366 (2000) 4. Bellan, J.: Theory, modeling and analysis of turbulent supercritical mixing. Combust. Sci. Technol. 178(1–3), 253–281 (2006) 5. Blottner, F.G., Johnson, M., Ellis, M.: Chemically reacting viscous flow program for multi-component gas mixtures. Technical report, Sandia Labs, Albuquerque, N. Mex. (1971) 6. Dumbser, M., Iben, U., Munz, C.-D.: Efficient implementation of high order unstructured WENO schemes for cavitating flows. Comput. Fluids 86, 141–168 (2013). ISSN 0045-7930 7. Peter, G., Helge, M., Dieter, B.: An implicit multigrid method for turbulent combustion. J. Comput. Phys. 167(2), 247–276 (2001) 8. Hannemann, V.: Numerische Simulation von Stoß-Stoß-Wechselwirkungen unter Ber¨ ucksichtigung von chemischen und thermischen Nichtgleichgewichtseffekten. Ph.D. thesis (1997) 9. Hirschfelder, J.O., et al.: Molecular Theory of Gases and Liquids, vol. 26. Wiley, New York (1954) 10. Horchler, T., et al.: Comparison of detailed chemistry and flamelet combustion modeling in a H2/LOx subscale combustion chamber. In: 17th International Conference on Numerical Combustion, May 2019 11. Jachimowski, C.J.: An analytical study of the hydrogen-air reaction mechanism with application to scramjet combustion (1988) 12. Karl, S.: Numerical investigation of a generic scramjet configuration. Ph.D. thesis. University of Dresden (2011) 13. Lemmon, E.W., Jacobsen, R.T.: Viscosity and thermal conductivity equations for nitrogen, oxygen, argon, and air. Int. J. Thermophys. 25(1), 21–69 (2004)

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14. Lemmon, E.W., McLinden, M.O., Friend, D.G.: Thermophysical properties of fluid systems. In: Linstrom, P.J., Mallard, W.G. (eds.) NIST Chemistry WebBook, vol. 69. National Institute of Standards and Technology, Gaithersburg 15. Ma, P.C., et al.: Numerical framework for transcritical real-fluid reacting flow simulations using the flamelet progress variable approach. In: 55th AIAA Aerospace Sciences Meeting, p. 0143 (2017) 16. Niedermeier, C.A., et al.: Large-eddy simulation of turbulent trans- and supercritical mixing. In: AIAA paper 2950, p. 2013 (2013) 17. Oefelein, J.C.: Thermophysical characteristics of shear-coaxial LOX–H2 flames at supercritical pressure. Proc. Combust. Inst. 30(2), 2929–2937 (2005) 18. Perakis, N., et al.: Qualitative and quantitative comparison of RANS simulation results for a 7-element GOX/GCH4 rocket combustor. In: 2018 Joint Propulsion Conference, p. 4556 (2018) 19. Poling, B.E., Prausnitz, J.M., O’Connel, J.P.: The Properties of Gases and Liquids, 5th edn. McGraw Hill Book Co., New York (2007) 20. Rossow, C.-C.: Extension of a compressible code toward the incompressible limit. AIAA J. 41(12), 2379–2386 (2003) 21. Setzmann, U., Wagner, W.: A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 Kat pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 20(6), 1061–1155 (1991) 22. Spalart, P.R., Allmaras, S.R.: A one equation turbulence model for aerodynamic flows. AIAA J. 94 (1992) 23. Suslov, D.I., Hardi, J.S., Oschwald, M.: Full-length visualisation of liquid oxygen disintegration in a single injector sub-scale rocket combustor. In: Aerospace Science and Technology (2019) 24. Suslov, D.I., et al.: Hot-fire testing of liquid oxygen/hydrogen single coaxial injector at high-pressure conditions with optical diagnostics. Progress Propul. Phys. 11, 391–406 (2019) 25. Wilke, C.R.: A viscosity equation for gas mixtures. J. Chem. Phys. 18(4), 517–519 (1950) 26. Zipperer, L., Herning, F.: Beitrag zur Berechnung der Z¨ ahigkeit technischer Gasgemische aus den Z¨ ahigkeiten der Einzelbestandteile. In: Das Gas-und Wasserfach, vol. 4, p. 49 (1936) 27. Zips, J., M¨ uller, H., Pfittzner, M.: Efficient thermo-chemistry tabulation for non-premixed combustion at high-pressure condition. Flow Turbulence Combust. 101(3), 821–850 (2018)

Numerical Methods

Non-equilibrium Model for Weakly Compressible Multi-component Flows: The Hyperbolic Operator Barbara Re(B) and R´emi Abgrall Institute of Mathematics, Universit¨ at Z¨ urich, 8057 Z¨ urich, Switzerland [email protected]

Abstract. We present a novel pressure-based method for weakly compressible multiphase flows, based on a non-equilibrium Baer and Nunziato-type model. Each component is described by its own thermodynamic model, thus the definition of a mixture speed of sound is not required. In this work, we describe the hyperbolic operator, without considering relaxation terms. The acoustic part of the governing equations is treated implicitly to avoid the severe restriction on the time step imposed by the CFL condition at low-Mach. Particular care is taken to discretize the non-conservative terms to avoid spurious oscillations across multimaterial interfaces. The absence of oscillations and the agreement with analytical or published solutions is demonstrated in simplified test cases, which confirm the validity of the proposed approach as a building block on which developing more accurate and comprehensive methods.

Keywords: Non-equilibrium multiphase flows Diffuse interface methods

1

· Low-mach scheme ·

Introduction

A particular instance of non-ideal compressible fluid dynamics (NICFD) concerns multiphase or multi-component flows, which are composed by two (or more) phases or immiscible fluids with different physical or chemical properties. Technical applications involving multiphase flows requires often to deal with weakly compressible, inviscid flows, e.g. in the transport of CO2 for carbon capture and storage (CCS) [1], or in incidental configurations of water nuclear power plants [2]. As an example, consider the former application, which has also motivated the present work. In normal-operating conditions, the transport pipelines are primarily designed to contain liquid or super-critical CO2 , but multiphase flows may occur due to fluctuating CO2 supply or during transient events [1]. The transported flow may contain also diverse impurities, whose state is not necessarily the same of the main component. In this regime, the velocity is considerably smaller than the speed of sound, i.e. the Mach number is very low. However, compressibility effects cannot be neglected as the time evolution of the flow field c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 33–45, 2020. https://doi.org/10.1007/978-3-030-49626-5_3

34

B. Re and R. Abgrall

strictly depends on the pressure waves and an accurate thermodynamic model is required to correctly estimate the propagation speed of these waves. Therefore, the goal of this work is to develop a computational fluid dynamics (CFD) tool well-suited for this kind of flow fields, which provides simultaneously an adequate level of model flexibility, in terms of multiphase topology and thermodynamic behavior, and a specific treatment for low-Mach regimes. From the numerical point of view, the standard schemes widely used for inviscid compressible flow fields containing shock waves, contact discontinuities, and strong rarefactions, are inadequate for weakly compressible flows, even singlephase ones. The causes of this failure, usually referred to as low-Mach limit, are multiple. First, the use of an explicit time integration scheme increases dramatically the computational time. Indeed, at low Mach, the problem is stiff and the acoustic effects result in a severe time step restriction imposed by the CFL condition [3]. A further difficulty concerns accuracy. Generally, approaching the limit M = 0, the numerical solution of the compressible Euler equations does not converge to solutions satisfying the equations for incompressible flows, except for a restricted class of special initial data (the so-called well-prepared case). In particular, when a flux difference splitting method is used, the discrete scheme cannot recover the correct scaling of the pressure with the Mach number and this error increases as the Mach number goes to zero [4]. The low-Mach limit has been investigated since long time in single-phase CFD and the literature proposes mainly two strategies for the development of accurate and efficient numerical methods for weakly compressible flows. The first one aims to improve the low-Mach behavior of compressible schemes by means of preconditioning or artificial viscosity [5], but it leads to problem-specific solutions, hard to be generalized, and to increased computational costs. Alternatively, compressibility can be added to incompressible schemes, by treating implicitly the acoustics and circumventing the issues related to the stiffness of the governing equations [6]. This latter research field was pioneered by Harlow and Amsden [7] and it has represented the core of several recent and outgoing research activities aimed at developing Mach-uniform schemes, e.g., [8–11]. In this work, we exploit some of these techniques to build a robust and flexible method for weakly compressible, non-equilibrium multiphase flows. While conceiving a CFD tool for multiphase flows, two of the main challenges to handle are the multiscale nature of the field and the presence of the dynamic interfaces that separate distinct fluids (or phases). These issues can be addressed in one of two ways: by interface-tracking methods or by interface capturing methods. The former ones, as for instance level-set, volume-of-fluid, front-tracking, and ad hoc arbitrary Lagrangian-Eulerian methods, fully resolve the interfaces and smooth out the fluid properties across them. The latter ones dynamically capture the interface as part of the numerical solution, by treating each fluid as a separate continuous. These methods are also known as multi-fluid or diffuse interface method [12]. This work focuses on this second class, which is more suitable to deal with the dynamic creation of interfaces. More specifically, we adopt a Baer and Nunziato (BN)-type model [13] (also known as 7-equation), which

Weakly Compressible Multi-component Flows: The Hyperbolic Operator

35

copes with the multitude of scales characterizing multiphase flows by means of averaging procedures. Assuming non-equilibrium of phasic pressures, velocities and internal energies, BN-type models are capable to describe different flow topologies, such as disperse flows as well as resolved interfaces separating two (nearly) pure fluids. Moreover, they describe each component through its own equations of state, and they do not require the definition of a mixture speed of sound, which may be discontinuous at the phase boundary. Some research activities have been already devoted to weakly compressible multiphase flows, but most of them focus on simplified BN-type models in which equilibrium in the pressure [14] and also in the velocity and the temperature [15] is imposed. On the contrary, Coquel et al. [2] adopt a full non-equilibrium BNtype model, but they make the simplifying assumption of isentropic flows. Apart from the broader generality, a key feature that motivates the use of the BN model is the hyberbolicity, which is not guaranteed by the 6-equation one-pressure model. In this work, we work with the symmetric variant proposed by Saurel and Abgrall [16], which includes pressure and velocity relaxation terms to model how the phasic equilibrium is reached at the interface. The goal of the paper is to present a preliminary 1D version of a novel numerical method for weakly compressible, non-equilibrium multiphase flows and to assess it for two-component flows. In Sect. 2, we present the model and the numerical discretization, devoting particular care to the discretization of the non-conservative terms involving the gradient of the volume fraction, which couple the evolution of the different phases. Then, in Sect. 3, we verify its validity in simplified test cases, aimed at recovering the single-phase behavior and verifying the correct discretization of non-conservative terms, i.e. the absence of spurious oscillations across multi-material interfaces. Finally, Sect. 4 draws the conclusions and outlines the future steps towards the long-term work.

2

Numerical Method

A sound strategy to attain an efficient numerical method for weakly compressible flows consists in approximating the sonic terms, which in the low-Mach limit are associated with an almost infinite sound propagation rate, in an implicit fashion. Typically, the methods that pursue this strategy for single-phase flows exploit a staggered description of the flow variables to avoid spurious pressure oscillations, and they solve the governing equations in a segregate way by updating the variables to the next time level through intermediate sub-steps. In low-Mach regimes, pressure-based methods are considered more suitable than density-based ones, because of the weak coupling between these two variables. Indeed, the density is approximately constant, while the pressure is not. If density is used as primary variable, small errors in density result in large errors in the pressure. In the followings, we illustrate how we have conceived a pressure-based, segregated strategy for non-equilibrium multiphase flows.

36

2.1

B. Re and R. Abgrall

The Continuous Model

The first key ingredient of the proposed model is a pressure decomposition, which is required to retrieve the correct order of pressure fluctuations and to converge to a correct approximation of the incompressible Euler equations [3]. We replicate here the choice of [8] and we define the dimensionless pressure as P =

P − Pr ρr u 2r

where

2r ρr u = Mr2 , Pr

(1)

where ρ is the density, u the velocity, and the subscript r and the accent · indicate reference and dimensional quantities, respectively. Mr is a reference Mach number expressing the overall compressibility of the flow field. The scaling (1) filters out the acoustics and ensures that pressure oscillations scale with the correct order in the low-Mach limit. Indeed, assuming an asymptotic expansion of the pressure as P = P (0) + Mr2 P (1) + O(Mr3 ), we can show that the pressure   oscillations that is PP(0) = PP(0)−−PPr , is of O(1) as Mr → 0 [3]. We can interpret the r scaling (1) also as a decomposition between the thermodynamic pressure (Pr ) and a component that satisfies the momentum equation [9]. We want to apply this scaling the symmetric BN-type model proposed by Saurel and Abgrall [16], which gives the possibility to use the same equations and the same numerical methods at all computational cells. In this preliminary work, we focus on the hyperbolic part, so we do not consider the relaxation terms. Considering only 2 phases (denoted with subscripts i and i∗ ), the dimensional form of the governing equations in 1D is ∂αi ∂αi +u I  ∂ x ∂t ∂(αi ρi ) ∂(αi m  i) + ∂ x ∂ t ∂(αi m  i ) ∂(αi m  iu i + αi Pi ) + ∂ x ∂ t     ∂ α ( E + P ) m  / ρ i ) i i i i i ∂(αi E +  ∂ x ∂t

=0

(2a)

=0

(2b)

= PI

∂αi ∂ x

∂αi = PI u I ∂ x

(2c) (2d)

where (2b)–(2d) are repeated for the phase i∗ . This system is written for the volume fraction α and for the conservative variables ρ, m, and E = e + m2 /(2ρ) of each phase. In addition to the standard Euler equations, the system includes transport terms that stem from the averaging process that underlies the BN-type model. To close the system, we have to define the interfacial pressure and veloc[16], we define ity (PI and uI ). Following   them as the weighted averages among the phases, i.e. PI = i αi Pi and uI = i αi mi /(αi ρi ), although different alternatives are available in the literature. The system (2) has to be complemented by a thermodynamic model for each fluid. In this preliminary work, we use the stiffened gas model, but we leave the formulation as general as possible, to be able to easily add more accurate equations of state in the future.

Weakly Compressible Multi-component Flows: The Hyperbolic Operator

37

Since our goal is the development of a pressure-based method, we need to formulate the governing equations in primitive variables instead of conservative ones. We are aware of the possible problems that may arise from working with a non-conservative formulation, but at the moment our targets are low-Mach flows, so not experiencing strong shock waves; then a local correction is sufficient to converge to the correct weak solution. Recently, this idea has been successfully applied to multiphase flows modeled according to the simplified 5equations model [17]. Therefore, we substitute Eq. (2d) with the following: αi

∂ Pi ∂ Pi ∂ ui ∂αi + αi ρi  = ρi  + α ui c2i c2i,I ( uI − u i )  ∂ x ∂ x ∂ x ∂t

(3)

where ci is the standard speed of sound of phase i and ci,I is a kind of interfacial 

i speed of sound, which we have defined as c2i,I = χi +κi PIρ+e , with χ = ∂P and ∂ρ i e  ∂P  κ = ∂e ρ . We highlight that this model, as all 7-equation BN-type ones, does not require to define a speed of sound for the mixture, which can be considered as an open-issue in multiphase research. The final step to formulate the target model is to make dimensionless the BN-type model given by (2a)–(2c), and (3) according to the scaling given by (1). The variables that do not involve pressure are scaled as usual, according to the reference density, velocity, and length (Lr ). On the other hand, the scaling of the thermodynamic variables, and in particular of the speed of sound, requires more care. Inserting Eq. (1) into the definition of  c2i , we have

 

r P P +e 2 Pr 2 = c2 + κ  c2 = χ + κ u r + κ u r ρ ρ ρr ρ

with

Pr =

Pr u 2r ρr

(4)

where c2 = χ+κ P ρ+e has the same expression of the dimensional speed of sound. A similar result is obtained also for  c2i,I . Omitting all the passages for brevity, the dimensionless formulation reads ∂αi ∂αi + uI =0 ∂t ∂x ∂(αi ρi ) ∂(αi mi ) + =0 ∂t ∂x ∂(αi mi ) ∂(αi mi ui + αi Pi ) ∂αi + = PI ∂t ∂x ∂x   ∂Pi ∂Pi ∂αi 2 2 ∂ui 2 + αi ui + αi ρi ci Mr αi = Mr ρi c2i,I (uI − ui ) ∂t ∂x ∂x ∂x  ∂(αi ui ) ∂αi − + κi u I . ∂x ∂x

(5a) (5b) (5c)

(5d)

We can observe that the first three equations are identical to their dimensional counterparts, while the pressure equation has a special expression, thanks to ∂(αi ui ) i = 0. This can be viewed as the which, as Mr → 0, we have ∂α ∂t + ∂x

38

B. Re and R. Abgrall

multiphase counterpart of the kinematic constraint for the incompressible single phase flows: with only one phase, α = 1 and ui = u, so the previous expression simplifies to ∂u ∂x = 0, which is the 1D version of ∇ · u = 0. 2.2

The Discretization

As common in low-Mach schemes for single phase flows, we exploit a staggered description of the flow variables to avoid spurious pressure oscillations. This means that the scalar, thermodynamic variables are stored at the cell centers, while the vectorial, velocity-related quantities are stored at the cell faces. Under this configuration, it is natural to solve the governing equations in a segregated way, as the computational cells for the momentum equation are different from the ones for the density and energy equation. The Semi-implicit Temporal Discretization. The solution at time level tn+1 is computed through intermediate sub-steps in which the convective velocity is approximated explicitly. Moreover, the computation of the momentum is split into two sub-steps: first an intermediate momentum m∗ is computed by approximating explicitly the pressure in (5c), then it is updated when the pressure P n+1 is known. More precisely, the following steps are performed for each phase: i) the density equation (5b) is solved by using uni ; ii) the volume fraction equation (5a) is solved by using unI (only for the phase i, then αi∗ = 1 − αi ); iii) the intermediate momentum (αm)∗ = (αρi )n+1 u∗i is computed solving  ∂ (αi mi )∗ uni + αin+1 Pin (αi mi )∗ − (αi mi )n ∂αn+1 + = PIn i ; (6) Δt ∂x ∂x iv) the pressure-equation (5d) is solved by treating implicitly the velocity divergence (which is mandatory to avoid too severe restrictions on the time step) and explicitly the terms related to the speed of sound, that is  n+1 n+1 n  − Pin ∂un+1 2 n+1 Pi ∗ ∂Pi Mr αi + ui + Mr2 ρi c2i + κi αin+1 i Δt ∂x ∂x n+1 n  2 2 ∂α (7) = Mr ρi ci,I + κi (uI − ui )∗ i ∂x v) the momentum is updated by solving the difference between (5c) and (6):   ∂ (αi mi )n+1 − (αi mi )∗ uni (αi mi )n+1 − (αi mi )∗ + Δt ∂x  ∂ αin+1 (Pin+1 − Pin ) ∂αn+1 + = (PIn+1 − PIn ) i . (8) ∂x ∂x

Weakly Compressible Multi-component Flows: The Hyperbolic Operator

39

The Staggered Spatial Discretization. We use a standard first-order, finitevolume discretization with Rusanov fluxes, so we highlight here only the peculiar features of the proposed scheme. Given a 1D domain, we divide it in Njs scalar cells of uniform width Δxj and Njv = Njs + 1 cells for the vectorial variables, whose width Δxj+ 12 is equal to Δxj for the internal cells, except for the first and the last cells which have the width Δxj /2. The equations for α, αρ, P are solved over the scalar (node-centered) cells Cj , while the equations for αm are solved over the vector cells Cj+ 12 . When a mapping from the vectorial to the scalar cells, or vice versa, is required, we use a weighted average according to the cell sizes. We denote the spatially discrete values by a further subscript j or j + 12 , e.g., (αi )nj is the volume fraction of phase i at cell j at time step n. The staggered configuration facilitates the computation of some integral terms: for instance, the velocity on the face between the cells Cj and Cj+1 is unequivocally uj+ 12 . Similarly, we can easily discretize the pressure gradient over the cell Cj+ 12 through a central difference between Pj+1 and Pj . Particular care is mandatory while discretizing the non-conservative terms, which involve the gradient of α. As known, a superficial discretization of these products may lead to the onset of oscillations across interfaces separating fluids with different material properties [18]. In compressible multiphase flows, a guideline is provided by the so-called Abgrall’s criterion (or pressure nondisturbance condition), which states that “a two-phase flow, uniform in pressure and velocity must remain uniform on the same variables during its temporal evolution” [19]. So, applying the derived discretization to a uniform flow, we can obtain an oscillation-free discretization of the non-conservative terms. This procedure clearly depends on the adopted discretization scheme, but it has been already applied to BN-type models in previous works of Saurel et al. [16,20]. According to the previous remarks, the discrete equations solved at each time step are i) = (αi ρi )nj − (αi ρi )n+1 j rus,ρ Fj+ 1 = 2

 Δt rus,ρ rus,ρ Fj+ 1 − Fj− 1 2 2 Δxj

with

(9)

1 1 n+1  n+1  (αi ρi )n+1 (αi ρi )n+1 (ui )n |(ui )n − | j+1 + (αi ρi )j j+1 − (αi ρi )j j+ 1 j+ 1 2 2 2 2

ii) (αi )n+1 = (αi )nj − j

Δt Hu (αin+1 , unI )j Δxj

(10)

where Hu is the discretization of the non-conservative term. It is obtained by combining this equation with the one at step i): starting from uniform density and velocity, we impose that the density remains constant, obtaining Hu (αin+1, un I )j =

1  n+1  n (αi )n+1 j+1 − (αi )j−1 (uI )j 2   n+1 n+1 −|(uI )n + (αi )n+1 j | (αi )j+1 − 2(αi )j j−1

40

B. Re and R. Abgrall

iii) (αi mi )∗j+ 1 = (αi mi )nj+ 1 − 2

2

Δt Δxj+ 1

 rus,m Fj+1 − Fjrus,m

  n+1 n + (αi )n+1 (Pi )nj − HP (αin+1 , PIn , u∗i )∗j+ 1 j+1 (Pi )j+1 − (αi )j 2

2

(11) where Fjrus,m is the standard Rusanov flux for the conservative variables (αi mi )∗ and the convective velocities (ui )n , while the approximation (HP )∗j+ 1 of the non-conservative term is obtained by imposing the Abgrall’s 2 criterion:   rus,ρ,¯u  n+1 n+1  vol ∗ + Kj+ (HP )∗j+ 1 = (PI )n − Fjrus,ρ,¯u 1 (ui )j+ 1 Fj+1 j+ 1 (αi )j+1 − (αi )j 2

2

2

2

 rus,ρ,¯ u where = [1−Δxj+ 12 Δxj ] and Fj+1 is defined as in (9) but between the mapped densities (αi ρi )j+ 12 and (αi ρi )j− 12 . The last term of HP results from the mapping of (αi ρj ) at the left-hand side,1 but it is non-null only at the vector boundary cells, where Δxj+ 12 = Δxj . vol Kj+ 1 2

iv)  Δx Mr2 (αi )n+1 (Pi )n+1 − (Pi )nj Δtj = j j   M 2 (αi )n+1  n+1 (Pi )n+1 (ui )∗j+ 1 − |(ui )∗j+ 1 | − r 2 j j+1 − (Pi )j 2 2   n+1 n+1 ∗ ∗ + (Pi )j − (Pi )j−1 (ui )j− 1 + |(ui )j− 1 | 2 2

  2 n+1 n+1 n n 2 n (ui )j+ 1 − (ui )n+1 − Mr (ρi )j ((ci )j ) + (κi )j (αi )j j− 12 2  2  ∗  n+1 n n 2 n ∗ + Mr (ρi )j ((ci,I )j ) + (κi )j Hu (uI − ui ), αi j (12) n+1

n+1

∗

∗

which has been derived from (7) re-writing u∗ ∂P∂x = ∂P ∂x u − P ∗ ∂u ∂x and using the Rusanov fluxes. For analogy with the density equation, the non-conservative term is discretized by the same operator Hu defined in step ii). − u∗i ) and v) we observe that if in Eq. (8) we define δαi mi = (αi ρi )n+1 (un+1 i n+1 n − P , it has the same shape of (6). Therefore we obtain the δP = P same discretization as in step iii). We highlight that the velocity in the second right-hand side term of (12) has to be discretized implicitly, but at step iv) the velocities (ui )n+1 are still

1

The discretization of HP is obtained by assuming uniform (ui )n and (Pi )n in (11) the discretization of and substituting into its left-hand side (αi mi )∗j+ 1 = (αi ρi )n+1 j+ 1

from (9). (αi ρi )n+1 j+ 1 2

2

2

Weakly Compressible Multi-component Flows: The Hyperbolic Operator

41

unknown. However, we can derive an approximation from the expression of step v), discharging the differences in the convective term. Thus, we use   n+1 ∗ Δt − (αi )n+1 (δPi )n+1 (ui )n+1 1 = (ui )j+ 1 + Δx j+1 (δPi )j+1 − (αi )j j j+ 2 2 j+ 1 2   n+1 n+1 +(δPI )n+1 (α ) − (α ) . (13) (αi ρi )n+1 1 i i j+1 j j+ j+ 1 2

2

However, the previous expression contains the interfacial velocity PIn+1 , which depends on the pressure at time tn+1 of both phases. Consequently, the use of (13) in (12), i.e. the implicit discretization of the acoustic part, makes the solution of the pressure equations coupled among all phases.

3

Results and Discussion

We have tested the proposed numerical method on some simplified problems to verify its correctness. In the presented tests, we use the stiffened gas equation of state, which describes molecular agitation and repulsive effects in a simplified way, but facilitates the analytical solution of some simple problems [22]. The tests have been run on a single Intel Xeon Processor E5-2640 @2.60 GHz, and they required less than 2 min each. Test 1. The first test is the advection of a volume fraction variation in a uniform pressure and velocity field, and its aim is the numerical verification of Abgrall’s criterion and the stability at CFL> 1. The fluids and the initial conditions are described in Table 1 (left) and Fig. 1. The final time is 1 ms, divided in 80 time steps. The resulting acoustic CFL conditions (i.e., considering u + c) are 10 for fluid 1 and 2.7 for fluid 2. The initial conditions are correctly preserved and no spurious oscillations are generated, as demonstrated in Fig. 1. Table 1. Thermodynamic parameters of the two fluids (water and air). cp is the isobaric specific heat capacity, γ is the ratio between the isobaric and the isochoric specific heat capacities, and P∞ is a parameter of the stiffened gas model. In principle, the ratio of specific heats γ for liquids should be close to 1. However, in the stiffened gas model, γ results from a tuning process, so a larger value may be required to match the values of density and speed of sound, in the interval of interest. Despite this fact and its simplicity, the stiffened-gas model provides a qualitatively correct agreement with more accurate model for water [21]. Finally, we remark that several sets of parameters (γ, P∞ ) have been proposed in the literature, and we have chosen the ones displayed above to be able to compare our results with the ones in the given references.

B. Re and R. Abgrall 1.1 1

-0.25

0

0.25

0.5

101 100 99 -0.5

-0.25

Volume fraction)

1

0.5

0 -0.5

-0.25

0

0

0.25

0.5

Position (m)

0.25

0.5

Position (m)

Density phase 1, (kg/m3)

Position (m)

1.3 1052 1.25 1051 1.2 1050 1.15 1049 -0.5

-0.25

0

0.25

1.1 0.5

Position (m)

Phase 1

Density phase 2, (kg/m3)

0.9 -0.5

Velocity, (m/s)

Pressure, (bar)

42

Phase 2

Fig. 1. Test 1: volume fraction transport in a uniform pressure and velocity field, solution at the final time tF = 1 ms. The domain is x = −0.5 m ≤ x ≤ 0.5 m, discretized with Njs = 500 scalar cells. Initial conditions are: u0 = 100 m/s, P 0 = 105 Pa, ρ01 = 1050 kg/m3 and ρ02 = 1.2 kg/m3 . The volume fraction profile is initially centered at x = 0, with α1 = 0.3 at left and α1 = 0.7 at right. 2

120

100

80 70 60

1.5 80 1

60 40

0.5 20

Velocity 2, (m/s)

Velocity 1, (m/s)

Pressure, (bar)

100 90

50 -0.5 -0.4 -0.3 -0.2 -0.1

0

0 -0.5 -0.4 -0.3 -0.2 -0.1

0.1 0.2 0.3 0.4 0.5

Position (m)

0

0 0.1 0.2 0.3 0.4 0.5

Position (m)

0.2

0.1

0 -0.5 -0.4 -0.3 -0.2 -0.1

0

0.1 0.2 0.3 0.4 0.5

90 80

1030

70 1025

60 50

1020 -0.5 -0.4 -0.3 -0.2 -0.1

Position (m)

0

0.1 0.2 0.3 0.4 0.5

Density phase 2, (kg/m3)

Mach number

0.3

Density phase 1, (kg/m3)

1035

Position (m) Phase 1, analytical Phase 1, numerical

Phase 2, analytical Phase 2, numerical

Fig. 2. Test 2: water/air shock-tube with no phase mixing, solution at the final time tF = 0.2 ms, after 100 time steps. The grid spacing is Δxj = 0.002 as in the other tests. The initial pressures are PL0 = 100 bar (left) and PR0 = 50 bar (right), while the temperature is T 0 = 308.15 K, uniform. The volume fraction is α = 0.5 everywhere. The numerical solution is compared to the analytical one for each phase.

Test 2. The second test is a water/air shock-tube with uniform volume fraction α1 = 0.5. Initially, the left and right chambers have different pressures. The difference in pressure is moderate, in order to avoid excessively high Mach numbers. Because of the absence of relaxation terms, each fluid evolves as a single phase, so it is possible to compute the analytical solution, given the fluid parameters

Weakly Compressible Multi-component Flows: The Hyperbolic Operator 2

Velocity 1, (m/s)

4 2 0 -0.6

-0.4

-0.2

0

0.2

0.4

1

400

0

200

-1 -0.6

0.6

-0.4

Position (m)

-0.2

0

0.2

0.4

Position (m)

1

5

0.8 0.6 0.4 0.2 0 -0.6

-0.4

-0.2

0

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0.6

Density phase 1, (kg/m3)

Volume fraction)

0 0.6

1060 4 1050 3 1040 2 1030 1 1020 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

Density phase 2, (kg/m3)

Pressure, (bar)

6

Velocity 2, (m/s)

600

10 8

43

Position (m)

Position (m)

Phase 1

Phase 2

Fig. 3. Test 3: “Smooth shock tube test case” as in [20], solution at the final time tF = 0.35 ms, after 700 time steps. The initial pressures are PL0 = 10 bar (left) and PR0 = 1 bar (right) and the velocity is 0 for both phases. The densities ρ01 = 1050 kg/m3 and ρ02 = 1.2 kg/m3 are uniform along the domain.

in Table 1 and initial conditions in Fig. 2. A good agreement is achieved, even if the rarefaction fans are smeared, but we can expect it since the discretization is only first-order accurate. We highlight that the pressure wave in the liquid (phase 1) moves at the speed of sound c1 ≈ 1525 m/s. Considering such a high propagation speed, we remark that the implicit treatment of the acoustics is crucial to use large time integration steps without introducing instabilities. Test 3. Finally, we reproduce a multiphase test for which the results without relaxation terms are available in [20]. The fluids are the same as in Test 1, but now a pressure difference between the left and right state is also imposed. In Fig. 3, we observe a good agreement with results in [20], expect for the pressure profile of phase 1. This difference can be explained by the fact that they use a simplified Riemann solver according to which no pressure wave is present in this phase. Furthermore, we notice a strange behavior of the numerical solution across the interface: in each phase, one point is not aligned with the neighboring ones. We believe that it could be generated by the mapping from the scalar to the vector grid and that it could be smooth down by adding the relaxation terms, however we will further investigate this issue.

4

Conclusions

In this work, we have presented a semi-implicit method for multiphase flows based on the Baer and Nunziato model, which avoids severe time step restrictions at low-Mach regimes. We have proposed a pressure-based method, which,

44

B. Re and R. Abgrall

thanks to a special scaling of the pressure, recovers in the limit Mr → 0 the incompressible formulation of the governing equations. Moreover, we have taken particular care in the discretization of the non-conservative terms, in order to avoid spurious oscillations across the multi-material interfaces. The preliminary tests have confirmed the implicit treatment of acoustics and the fulfillment of Abgrall’s criterion. These results indicate the applicability of the proposed approach and allow us to move on to the next step of the longterm work. Currently, we are working on the inclusion of the relaxation terms. Future steps are the implementation of more accurate thermodynamic models, of second- and high-order discretization. The flexibility of the BN-type model will allow also the investigation of flows containing more than 2 phases. Acknowledgments. This publication has been produced with support from the NCCS Centre, performed under the Norwegian research program Centres for Environment-friendly Energy Research (FME). The authors acknowledge the following partners for their contributions: Aker Solutions, Ansaldo Energia, CoorsTek Membrane Sciences, Emgs, Equinor, Gassco, Krohne, Larvik Shipping, Norcem, Norwegian Oil and Gas, Quad Geometrics, Shell, Total, V˚ ar Energi, and the Research Council of Norway (257579/E20).

References 1. Munkejord, S.T., Hammer, M.: Int. J. Greenhouse Gas Control 37, 398 (2015). https://doi.org/10.1016/j.ijggc.2015.03.029 2. Coquel, F., H´erard, J.M., Saleh, K.: J. Comput. Phys. 330, 401 (2017). https:// doi.org/10.1016/j.jcp.2016.11.017 3. Wenneker, I., Segal, A., Wesseling, P.: Int. J. Numer. Methods Fluids 40(9), 1209 (2002). https://doi.org/10.1002/fld.417 4. Guillard, H., Viozat, C.: Comput. Fluids 28(1), 63 (1999). https://doi.org/10. 1016/S0045-7930(98)00017-6 5. Guillard, H., Murrone, A.: Comput. Fluids 33(4), 655 (2004). https://doi.org/10. 1016/j.compfluid.2003.07.001 6. Xiao, F.: J. Comput. Phys. 195(2), 629 (2004). https://doi.org/10.1016/j.jcp.2003. 10.014 7. Harlow, F.H., Amsden, A.A.: J. Comput. Phys. 3(1), 80 (1968). https://doi.org/ 10.1016/0021-9991(68)90007-7 8. Bijl, H., Wesseling, P.: J. Comput. Phys. 141(2), 153 (1998). https://doi.org/10. 1006/JCPH.1998.5914 9. Munz, C.D., Roller, S., Klein, R., Geratz, K.J.: Comput. Fluids 32(2), 173 (2003). https://doi.org/10.1016/S0045-7930(02)00010-5 10. Kwatra, N., Su, J., Gr´etarsson, J.T., Fedkiw, R.: J. Comput. Phys. 228(11), 4146 (2009). https://doi.org/10.1016/J.JCP.2009.02.027 11. Ventosa-Molina, J., Chiva, J., Lehmkuhl, O., Muela, J., P´erez-Segarra, C.D., Oliva, A.: Int. J. Numer. Methods Fluids 84(6), 309 (2017). https://doi.org/10.1002/fld. 4350 12. Saurel, R., Pantano, C.: Ann. Rev. Fluid Mech. 50, 105 (2018). https://doi.org/ 10.1146/annurev-fluid-122316-050109 13. Baer, M.R., Nunziato, J.W.: Int. J. Multiphase Flow 6, 861 (1986)

Weakly Compressible Multi-component Flows: The Hyperbolic Operator

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14. Pandare, A.K., Luo, H.: J. Comput. Phys. 371, 67 (2018). https://doi.org/10. 1016/j.jcp.2018.05.018 15. Bernard, M., Dellacherie, S., Faccanoni, G., Grec, B., Penel, Y.: ESAIM Math. Model. Numer. Anal. 48(6), 1639 (2014). https://doi.org/10.1051/m2an/2014015 16. Saurel, R., Abgrall, R.: J. Comput. Phys. 150(2), 425 (1999). https://doi.org/10. 1006/JCPH.1999.6187 17. Abgrall, R., Bacigaluppi, P., Tokareva, S.: Comput. Fluids (2017). https://doi.org/ 10.1016/J.COMPFLUID.2017.08.019 18. Abgrall, R., Karni, S.: J. Comput. Phys. 229(8), 2759 (2010). https://doi.org/10. 1016/j.jcp.2009.12.015 19. Abgrall, R.: J. Comput. Phys. 125(1), 150 (1996). https://doi.org/10.1006/JCPH. 1996.0085 20. Saurel, R., Chinnayya, A., Carmouze, Q.: Phys. Fluids 29(6), 1 (2017). https:// doi.org/10.1063/1.4985289 21. Paill‘ere, H., Corre, C., Garc´ıa Cascales, J.: Comput. Fluids 32, 891 (2003). https://doi.org/10.1016/S0045-7930(02)00021-X 22. Haller, K.K., Ventikos, Y., Poulikakos, D.: J. Appl. Phys. 93(5), 3090 (2003). https://doi.org/10.1063/1.1543649

Pressure-Based Solution Framework for Non-Ideal Flows at All Mach Numbers Christoph Traxinger(B) , Julian Zips, Matthias Banholzer, and Michael Pfitzner Institute for Thermodynamics, Bundeswehr University Munich, 85577 Neubiberg, Germany [email protected]

Abstract. In this work, we present an all Mach number pressure-based solution framework suitable for the simulation of application-relevant high-pressure flow configurations. A cubic equation of state is applied for the accurate description of the thermodynamic state. Different formulations of the pressure equation for sub- and supersonic flows are discussed. The framework is employed to simulate one sub- and one supersonic test case, where experimental data are available. The comparison of the simulation results with experimental shadowgraphy and Schlieren images shows very good agreement.

Keywords: Pressure-based CFD thermodynamics

1

· All-speed · Real-gas

Introduction

Computational fluid dynamics (CFD) is an indispensable tool in the academic research and for a thorough investigation of fluid flow devices. Since computational power is steadily rising, the degree of detail and the widespread use of CFD is continuously increasing. This demands for robust and at the same time efficient numerical algorithms which at its best can be applied to a wide scope of problems ranging from low to high Reynolds numbers, Mach numbers, temperatures and pressures. In typical engineering applications, high Reynolds numbers are present and therefore, the selection of the numerical algorithm is usually done with respect to the Mach number whereby the flow regimes can be roughly divided into the incompressible and the compressible regime. In general, two types of solvers are widely used for the numerical simulation of fluid flow [1]: density-based and pressure-based. Density-based solvers are historically applied in compressible flow problems (high Mach number) and consider the density as a primary variable. Therefore, the pressure is determined from the thermodynamic equation of state. These solvers have disadvantages when the Mach number approaches zero as density variations vanish in the incompressible limit and the problem becomes ill-conditioned. Many different approaches have been developed to overcome these deficiencies which include for instance c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 46–58, 2020. https://doi.org/10.1007/978-3-030-49626-5_4

Pressure-Based Solution Framework for Non-Ideal Flows

47

=

1

pr = p/pc [-]

Z

2

0.90 Z=

pre-conditioning, see, e.g., Weiss and Smith [2], or the introduction of an artificial compressibility, see, e.g., Choi and Merkle [3]. Pressure-based solvers use the pressure as primary variable and can therefore be directly applied to incompressible cases (low Mach number). In contrast to density-based solvers, where the conservation equations are typically solved in a coupled manner, pressurebased solvers usually employ a sequential approach. Furthermore, the pressure is solved in a Poisson-like equation which can be derived from the momentum and continuity equations. Therefore, the pressure is used to force mass conservation by manipulating the velocity field which makes it suitable for the incompressible regime. By explicitly considering the pressure-density coupling, this pressurebased solver can be applied to compressible (high Mach number) flows. In the present work, it is our objective to present a pressure-based CFD framework which is able to simulate fluid flows at all Mach numbers. In contrast to other investigations, see, e.g., Moukalled and Darwish [4] or Xiao et al. [5], the thermodynamic closure is not achieved by assuming an ideal gas state but by applying a framework consider8 ing real-gas effects, i.e., accounting for the nonideality of the fluid. This type of closure 6 makes the solver suitable for Z = 1. the investigation of a vast vari05 4 ety of application-relevant highpressure fluid flow devices like DE liquid-propellant rocket engines 2 (LRE), diesel engines (DE) LRE GE and gas engines (GE). The necessity of considering real1 1.5 2 2.5 3 gas effects based on typical injection and operating condiTr = T /Tc [-] tions is highlighted in Fig. 1. Fig. 1. General reduced pressure-temperature Here, a pressure-temperature diagram showing typical injection conditions: The diagram together with iso-lines solid black line denotes the vapor-pressure curve for the compressibility factor determined at the critical point (pr = Tr = 1). Z = pv (RT )−1 are shown. The dashed line corresponds to the Widom-line. Only at the iso-line Z = 1, the The iso-lines highlight regions of constant com- ideal gas assumption is strictly pressibility factors Z = pv (RT )−1 . valid.

Governing Equations

The equations governing the flow of multicomponent, compressible fluids are the mass and momentum conservation equations together with the transport ˙ Yk , jk equations for the energy and the different species. Let ρ, t, u, σ, ht , q, represent the density, time, velocity vector, stress tensor, total enthalpy, heat flux, mass fraction and mass flux of species k, respectively. The equations read:

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∂ρ + ∇ · (ρu) = 0, ∂t ∂ (ρu) + ∇ · (ρuu) = ∇ · σ, ∂t ∂ρht ∂p + ∇ · (ρht u) = + ∇ · (τ · u) − ∇ · q, ˙ ∂t ∂t ∂ (ρYk ) + ∇ · (ρYk u) = −∇ · jk . ∂t For Newtonian fluids, the stress tensor reads     2 T σ = − p + μ∇ · u I + μ ∇u + (∇u) = −pI + τ, 3

(1) (2) (3) (4)

(5)

where, τ and μ are the viscous stress tensor and the dynamic viscosity, respectively. In addition, the total enthalpy in Eq. (3) denotes the sum of the static 2 enthalpy h and the kinetic energy 12 u . In this work, the heat flux q˙ and the mass flux jk are deduced under the unitary Lewis-number assumption  −1 Le = κ (ρcp D) = 1  κ κ ∂h  q˙ = − ∇h + ∇p, (6)  cp cp ∂p  T,Y

jk = −ρD∇Yk ,

(7)

whereby the second term on the right-hand side of Eq. (6) can be neglected for low Mach number flows. In Eqs. (6) and (7) κ, cp and D denote the thermal conductivity, the specific heat at constant pressure and the diffusion coefficient, respectively. Soret, Dufour and radiation effects are neglected. For the turbulence closure in the Large-Eddy Simulations (LESs) presented in Sect. 6, the Smagorinsky model [6] with Cs = 0.17 is applied.

3

Real-Gas Thermodynamics

For closing the system of governing equations (1)–(4), relations for the thermal and caloric properties are required. For the coupling of density and pressure, we apply the cubic equation of state (EoS) of Peng and Robinson [7] which takes into account attractive and repulsive forces by means of parameters a and b, respectively. The pressure-explicit form of this EoS reads [7]: p=

a RT − 2 . v − b v + 2vb − b2

(8)

Here, R is the universal gas constant and v is the molar volume. For the consideration of multicomponent mixtures, the concept of a one-fluid mixture in combination with mixing rules is applied [8] a=

Nc Nc i=1 j=1

zi zj aij

and

b=

Nc i=1

z i bi ,

(9)

Pressure-Based Solution Framework for Non-Ideal Flows

49

where z = {z1 , ..., zNc } is the vector of mole fractions consisting of Nc different species. Pseudo-critical combination rules [9] are employed in this study to determine the off-diagonal elements (i = j) of aij . In addition to the thermal properties, the caloric properties are derived by applying the departure function formalism, see the work of Poling et al. [8] for further details. In this concept, the respective property, e.g., the enthalpy h, is divided into an ideal gas (ig) and a real gas (rg) part, i.e.: ⎞ ⎛  ρ  ∂Z  dρ + Z − 1⎠ . (10) −T h = hig (T, z) + Δhrg (T, p, z) = hig + RT ⎝  ∂T  ρ 0 ρ

where R is the specific gas constant. The real-gas contribution Δhrg can be straight-forwardly derived from the applied cubic EoS. The ideal gas part is determined using the seven-coefficient NASA polynomials [10]. The empirical correlation of Chung et al. [11] is employed to calculate the viscosity μ and the thermal conductivity κ, respectively.

4

The Pressure Equation

In order to arrive at a system of linear algebraic equations the finite volume method (FVM) is applied which yields a weak formulation of the governing equation (1)–(4) [1]. For the introduction of this method, the general convectiondiffusion equation for an intensive property ζ is considered: ∂ρζ + ∇ · (ρζu) = ∇ · (ρDζ ∇ζ) + Sζ . ∂t

(11)

Here, Dζ denotes the diffusion coefficient and Sζ is used for any explicit source term. Applying the first order Euler scheme, the Gauss theorem and the midpoint integration to the different terms of Eq. (11) yields: ρn ζ n − ρn−1 ζ n−1 ∂ρζ dΩ ≈ Ω, (12) Δt Ω ∂t #» ∇ · (ρζu) dΩ ≈ (uρ)f ζf · S f , (13) Ω

f

Ω

∇ · (ρDζ ∇ζ) dΩ ≈ Ω

f

#» (ρDζ ∇ζ)f · S f ,

Sζ dΩ ≈ Sζ Ω.

(14)

(15)

#» Here, Ω and S f are the cell volume and the surface normal vector, respectively. The superscript in Eq. (12) denotes the time level and the subscript f in Eqs. (13) and (14) indicates the interpolation from cell to face centers which results from the application of the Gauss theorem.

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By applying the FVM to the momentum conservation Eq. (2) and recasting the convection term according to #» ∇ · (ρuu) = (ρu)f (u)f · S f = φf (u)f = aP uP + aN uN (16) f

f

N

a semi-discrete form of the momentum equation can be derived [12]: aN uN − ∇p = HP − ∇p. aP uP = SP −

(17)

N

Here, P is the index of the respective cell and N denotes the adjacent cells. Furthermore, aP are the diagonal coefficients of the matrix and HP contains all off-diagonal contributions (aN ) and source-terms (SP ). Substituting Eq. (17) into the mass conservation equation (1) yields the final form of the semi-discrete pressure equation:      HP ρ ∂ρ +∇· ρ ∇p = 0 . (18) −∇· ∂t aP aP This pressure equation is elliptical in nature and can also be derived in a continuous form by assuming constant density and taking the divergence of the momentum Eq. (2)   ∂ρu 2 ∇ p = Δp = ∇ · − − ∇ · (ρuu) + ∇ · τ . (19) ∂t In the real-gas community, see, e.g., Ries et al. [13] and Lapenna et al. [14], Eq. (18) is referred to as low Mach approach, as it does not include any modification for the pressure dependency of the density. In our experience, this approach allows the simulation of moderate density stratifications in the subsonic regime. In addition, it is important to note, that the pressure equation (18) is different to the pressure equations applied in density-based solvers for stabilization purpose, see, e.g., Terashima and Koshi [15] or Lacaze et al. [16]. In real-gas flows density-based frameworks suffer spurious pressure oscillations as the pressure is evaluated from the equation of state based on the filtered density and internal energy. To overcome this issue, some groups replace the energy equation with a pressure transport equation, which should not be confused with Eq. (18). 4.1

All-Mach Extension

In general, the governing equations of high speed and compressible flows have a hyperbolic character which is the main reason why density-based solvers are so well suited for simulating this kind of flows [1]. To utilize the pressure-based approach for this kind of flows, the elliptical pressure equation (18) has to be adjusted to account for the variation of density with respect to pressure. This can be done in a straight-forward manner by replacing the density in Eq. (18) with ρ = Ψ p. (20)

Pressure-Based Solution Framework for Non-Ideal Flows

51

Independent of the fluid model, Ψ can be expressed as Ψ=

1 . ZRT

(21)

For an ideal-gas (Z = 1), Ψ corresponds to the isothermal compressibility −1 ∂ρ/∂pT = (RT ) . In the real-gas case, Z is unequal to unity, see Fig. 1. The compressibility factor Z can be directly obtained from the applied cubic EoS. This evaluation is a crucial part in a real-gas, pressure-based framework, as at sub-critical conditions up to three real solutions of the cubic EoS exist out of which the thermodynamically and physically most meaningful solution has to be selected. Usually, the root with the smallest Gibbs energy is chosen. Applying again the FVM and the first order Euler time-marching scheme together with Eq. (20) yields a hyperbolic pressure equation: Ω

Ψ n pn − Ψ n−1 pn−1 Δt

+

⎡ ⎤ ⎡ ⎤    HP ρ  #» n n #» ⎣ ⎣ Ψp f · S f ⎦ + ∇p · S f ⎦ = 0. aP f aP f f f



(22)

Generally, this equation can be used to simulated trans- and supersonic flows. However, based on our experience, this approach can only be used for simulations with moderate density stratifications. For high-speed flows, like they occur during the fuel injection, where underexpanded jets with strong shock and expansion structures form, modifications have to be made. In our solver, we apply a hybrid approach where the advective fluxes include both contributions due to the bulk velocity and due to propagation of waves. The solver is based on the work of Kraposhin et al. [17], where the surface interpolation approach of Kurganov and Tadmor (KT) is applied. Using this hybrid approach, we are able to perform simulations with Mach numbers up to 5.5, i.e., up to the hypersonic flow regime, see Sect. 6.2. For a more detailed discussion and introduction of the solver see Traxinger et al. [18]. 4.2

Real-Gas Treatment in Subsonic Flows

Under real-gas conditions, thermodynamic properties depend on both temperature and pressure. In high-speed flows featuring real-gas effects the approaches described in the previous section can still be applied as the main source for density variations is the pressure. In subsonic flow, however, Eq. (18) has to be modified as the density variations are mainly induced by temperature changes. Large density stratifications occur for instance during the transcritical injection at supercritical pressure. Following typical pressure correction approaches in the literature, see, e.g., Ferziger and Peric [19] or Moukalled and Darwish [4], the density variation in the time derivative of Eq. (18) is approximated based on the total differential of the density     Nc Nc ∂ρ  ∂ρ  ∂ρ  ∂ρ  dYi = Ψh dp+  dh+ ρi dYi . dρ =   dp+  dh+ ∂p  ∂h  ∂Yi  ∂h  i=1 i=1 h,Y

p,Y

h,p,Yj =Yi

p,Y

(23)

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Recasting this total differential into a Taylor series truncated after the first term and neglecting all variations except for the pressure gives   (24) ρ = ρn−1 + Ψhn−1 p − pn−1 . Here, the superscript n − 1 refers to the last time or iteration step. Substituting Eq. (24) into Eq. (18) yields        ∂ ρn−1 − Ψhn−1 pn−1 ∂Ψhn−1 p HP ρ + +∇· ρ ∇p = 0 . (25) −∇· ∂t ∂t aP aP Note, that the density in the third term on the left-hand side of Eq. (25) is not replaced by Eq. (24). With this pressure equation flows with larger density stratification can be simulated compared to the low Mach equation (18).

Origin Initialization Momentumpredictor Species equation

t = t + Δt

Pressure equation Correct u

j ≤ jmax ?

5 i=i+1

PISO

Evaluate thermodynamics

j =j+1

Energy equation

yes

SIMPLE

no i ≤ imax ?

yes

no Evaluate turbulence/ sub-grid model

yes

t < tSimulation ? no End

Fig. 2. Flowchart of the numerical algorithm.

Solution Algorithm

The pressure-based approach described above demands for a segregated solution algorithm which is realized in the opensource toolbox OpenFOAM 4.1 [20]. In our framework we apply a PIMPLE approach which is a combination of the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) [21] and the PressureImplicit with Splitting of Operators (PISO) [22] method. In Fig. 2 the basic structure of the numerical framework is illustrated. In the combined approach, the SIMPLE method forms the outer loop motivated by the predictor-corrector procedure of the velocity field u. The result of this predictor step is achieved with the old pressure value from the last time/iteration step. The mass

Pressure-Based Solution Framework for Non-Ideal Flows

53

conservation is fulfilled by correcting the velocity field according to the solution of the pressure equation, which enforces continuity. The inner loop is based on the PISO method. In contrast to ideal gas flows, the species and energy equations are coupled to the pressure equation and solved inside the PISO loop as the thermodynamic properties in real-gas flows depend on both temperature and pressure. The evaluation of the turbulence/sub-grid model is carried out at the end of the PIMPLE approach before advancing to the next time level. For the determination of the time-step, we use the convective Courant-Friedrich-Lewy (CFL) number criterion.

6

Numerical Results

In the following, two different test cases are presented in order to demonstrate the capability of the pressure-based CFD framework. For both cases, experimental results in terms of shadowgraphy and Schlieren images are available for validation. The first case corresponds to a subsonic injection case. The second test case is a highly supersonic test case. 6.1

Subsonic Injection

In the first test case, n-hexane is injected into a quiescent nitrogen atmosphere at elevated pressure. The reference experiments were conducted at the University of Stuttgart [23] and were the focus of a recent study [24] on multicomponent phase separation. For the injection of n-hexane a commercial single-hole Diesel injector with a diameter of 0.236 mm is used. Due to mechanical modifications of the Diesel injector, the injection is almost isobaric and can therefore be considered a low Mach number test case. For the present work, we select the case where n-hexane is injected at a total temperature of 600 K, for more details see Traxinger et al. [24]. Due to the high injection temperature no multicomponent phase separation occurs but real-gas effects are still present. The test chamber is pressurized at 50 bar using nitrogen as chamber fluid and the temperature corresponds to ambient conditions which is 293 K. Large-Eddy Simulations were conducted for this test case using Eq. (25) as pressure equation. Turbulent inlet boundary conditions were generated in a separate incompressible LES of the injector pipe and have been extracted over time and interpolated onto the inlet of the LES grid. In the LES, the injector is resolved with 30 cells in radial direction corresponding to a smallest cell size of 7.87 µm. The CFL number was set to 0.4 and second order schemes for the time and spatial integration have been used. To stabilize the numerical simulation, total variation diminishing (TVD) limiter have been applied to the convective terms of the enthalpy and the species equations. In Fig. 3, the experimental and numerical results of the fully developed jet are shown up to a distance of x/D ≤ 45. A highly turbulent jet is visible which gradually dissolves into the environment. In the experiment, almost no dark areas are present for x/D  30 which can be similarly observed in the LES. The

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r/D [-]

10

experimental opening angle: ≈ eight degrees

Experiment

0 experminetal length of the dark core: x/D ≈ 22

−10

r/D [-]

10

experimental opening angle: ≈ eight degrees

Simulation

0 |∇ · ∇ρ|

−10

[kg/m5 ]

2 · 106

0

2 · 107

experminetal length of the dark core: x/D ≈ 22

15

30

45

x/D [-]

Fig. 3. Fully developed jet structure of a n-hexane jet injected into a quiescent nitrogen atmosphere at p = 50 bar. Top: Experimental results by means of shadowgraphy; Bottom: Instantaneous, line-of-sight integrated LES result. Reprinted figure with permission from Traxinger et al. [24]. Copyright (2020) by the American Physical Society.

dark core length extracted from the experimental data by means of a binarized image compares very well to the result of the numerical simulation. From a fluid dynamics point of view, both jets feature a similar opening angle of approximately eight degrees highlighted by the dashed line. This line was fitted to the shadowgraphy of the experiment and has been superimposed onto the LES results. Looking deeper into the jet topology, finger-like structures can be seen on the outer surface for x/D  15. These phenomena are typical in supercritical jets and underline the single-phase character of the jet mixing process and hence the absence of droplets or any vaporization process. 6.2

Supersonic Injection

Jets being injected from a reservoir at a high pressure pt into an environment with low ambient pressure p∞ form shock and expansion patterns depending on the nozzle pressure ratio N P R = pt /p∞ . For NPRs between 2.08 and 3.85, the shock and expansion structure form a diamond-like pattern of reflected oblique shocks which is usually referred to as moderately underexpanded jet [25]. When the N P R is larger than 3.85, a highly underexpanded jet results which shows one

Pressure-Based Solution Framework for Non-Ideal Flows

55

Fig. 4. Comparison of the experimental Schlieren image (left) and the numerical result by means of an instantaneous snapshot of the line-of-sight integrated axial density gradient magnitude (right). The Schlieren image has been taken from Xiao et al. [27]. Reprinted by permission from Springer Nature Customer Service Centre c (2019). GmbH: Springer Nature, Flow, Turbulence and Combustion, Xiao et al. [27], 

strong normal shock followed by weak oblique shocks. The first shock is usually named Mach disk and its size and distance depends on the pressure ratio. As test case for high Mach number flows, an experimental investigation recently conducted by Fond et al. [26] has been used. In this experiment, Argon is injected through a single-hole injector (D = 1.54 mm) into itself at a N P R of 120 and the ambient pressure is set to 1 bar. The total temperature in the high pressure reservoir corresponds to 274 K and the temperature in the chamber is 285 K. Schlieren images are available and can be found in the work of Xiao et al. [27], see left-hand side of Fig. 4. A Large-Eddy Simulation was conducted based on pressure equation (22) together with the KT flux formulation. The mesh was set up according to the fine grid reported by Xiao et al. [27]. For the temporal and spatial integration second order accurate schemes have been employed and the interpolation from cell to face values was done using the van Albada TVD-limiter. In contrast to the work of Xiao et al. [27] who applied the cubic EoS for the determination of the thermal properties only, we employ the fully consistent thermodynamic framework discussed in Sect. 3. Due to the moderate temperatures T < 300 K and the low back-pressure of 1 bar as well as the strong underexpansion, a region of two-phase flow upstream of the Mach disk appears. The presence of this separation process was also concluded by Fond et al. [26] based on bright regions in their Mie scattering images. Similar findings have been reported by Baab et al. [28] for underexpanded n-hexane jets injected into nitrogen. Based on first a-priori analysis, we found that the expected thermodynamic states are located in both the metastable as well as in the spinodal region. Therefore, the simulation with a fully consistent thermodynamic framework is only stable if phase separation is taken into account which was done in this study. In order to employ our recently implemented and validated multicomponent phase separa-

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tion framework, see Traxinger et al. [24], argon was diluted with 1% nitrogen. This does not significantly alter the vapor-liquid equilibrium compared to the pure fluid and hence enables us to conduct stable simulations of this test case. In Fig. 4, the experimental and the numerical results of the near-injector field x ≤ 16 mm are shown by means of Schlieren images and the computed axial density gradient magnitude, respectively. Both images show the formation of a highly underexpanded jet and the presence of a Mach disk at x ≈ 12.5 mm. Due to the reflecting shocks and the strong expansion fans (Prandtl-Mayer expansion) downstream of the inlet, a characteristic shock barrel forms. Both the axial as well as the radial extension of the shock structure of the numerical simulation are in good agreement with the experimental Schlieren image.

7

Conclusion

Industrial fluid flow devices like liquid-propellant rocket engines, Diesel engines and gas engines cover a wide range of operating conditions in terms of Mach number, temperature and pressure. For the application of numerical tools during the design process, computational algorithms incorporating two main features are required: robustness in all Mach number regimes and accurate description of the fluid flow thermodynamic state. In this work, an all Mach number pressure-based real-gas CFD framework is presented. The thermodynamic closure is achieved by applying a cubic equation of state. Different pressure equations for sub- and supersonic flows are introduced and discussed. The framework is employed to simulate one sub- and one supersonic test case, where experimental data are available for comparison. The comparison of the simulation results with the experimental shadowgraphy and Schlieren images shows overall a very good agreement. Acknowledgments. Financial support has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft – DFG) in the framework of the Sonderforschungsbereich Transregio 40 and Munich Aerospace (www.munich-aerospace. de).

References 1. Hirsch, C.: Numerical Computation of Internal and External Flows. ButterworthHeinemann, Oxford (1997) 2. Weiss, J.M., Smith, W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33(11), 2050–2057 (1995) 3. Choi, D., Merkle, C.L.: Application of time-iterative schemes to incompressible flow. AIAA J. 23(10), 1518–1524 (1985) 4. Moukalled, F., Darwish, M.: A high-resolution pressure-based algorithm for fluid flow at all speeds. J. Comput. Phys. 168(1), 101–130 (2001) 5. Xiao, C., Denner, F., van Wachem, B.G.M.: Fully-coupled pressure-based finitevolume framework for the simulation of fluid flows at all speeds in complex geometries. J. Comput. Phys. 346, 91–130 (2017)

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6. Smagorinsky, J.: general circulation experiments with the primitive equations: i. The basic experiment. Mon. Weather Rev. 91(3), 99–164 (1963) 7. Peng, D., Robinson, D.B.: A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15(1), 59–64 (1976) 8. Poling, B.E., Prausnitz, J.M., O’Connell, J.P.: The Properties of Gases and Liquids. McGraw-Hill, New York (2001) 9. Reid, R.C., Prausnitz, J.M., Poling, B.E.: The Properties of Gases and Liquids. McGraw-Hill, New York (1987) 10. Goos, E., Burcat, A., Ruscic, B.: Report ANL 05/20 TAE 960. Technical report (2005). http://burcat.technion.ac.il/dir 11. Chung, T., Ajlan, M., Lee, L., Starling, K.E.: Generalized multiparameter correlation for nonpolar and polar fluid transport properties. Ind. Eng. Chem. Res. 27(4), 671–679 (1988) 12. Jasak, H.: Error analysis and estimation for the finite volume method with applications to fluid flows. Ph.D. thesis, University of London (1996) 13. Ries, F., Obando, P., Shevchuck, I., Janicka, J., Sadiki, A.: Numerical analysis of turbulent flow dynamics and heat transport in a round jet at supercritical conditions. Int. J. Heat Fluid Flow 66, 172–184 (2017) 14. Lapenna, P.E., Ciottoli, P.P., Creta, F.: Unsteady non-premixed methane/oxygen flame structures at supercritical pressures. Combust. Sci. Technol. 189(12), 2056– 2082 (2017) 15. Terashima, H., Koshi, M.: Approach for simulating gas-liquid-like flows under supercritical pressures using a high-order central differencing scheme. J. Comput. Phys. 231(20), 6907–6923 (2012) 16. Lacaze, G., Schmitt, T., Ruiz, A., Oefelein, J.C.: Comparison of energy-, pressureand enthalpy-based approaches for modeling supercritical flows. Comput. Fluids 181, 35–56 (2019) 17. Kraposhin, M., Bovtrikova, A., Strijhak, S.: Adaptation of Kurganov-Tadmor numerical scheme for applying in combination with the PISO method in numerical simulation of flows in a wide range of mach numbers. Procedia Comput. Sci. 66, 43–52 (2015) 18. Traxinger, C., Zips, J., Banholzer, M., Pfitzner, M.: A pressure-based solution framework for sub- and supersonic flows considering real-gas effects and phase separation under engine-relevant conditions. Comput. Fluids 202, 104452 (2020) 19. Ferziger, J.H., Peric, M.: Computational Methods for Fluid Dynamics. Springer, Berlin (2002) 20. OpenFOAM 4.1. https://openfoam.org/ 21. Patankar, S.V., Spalding, D.B.: A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. In: Numerical Prediction of Flow, Heat Transfer, Turbulence and Combustion, pp. 54–73. Elsevier (1983) 22. Issa, R.I., Gosman, A.D., Watkins, A.P.: The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J. Comput. Phys. 62(1), 66–82 (1986) 23. Baab, S., F¨ orster, F.J., Lamanna, G., Weigand, B.: Combined elastic light scattering and two-scale shadowgraphy of near critical fuel jets. In: 26th Annual Conference on Liquid Atomization and Spray Systems (2014) 24. Traxinger, C., Pfitzner, M., Baab, S., Lamanna, G., Weigand, B.: Experimental and numerical investigation of phase separation due to multi-component mixing at high-pressure conditions. Phys. Rev. Fluids 4(7), 074303 (2019) 25. Donaldson, C., Snedeker, R.S.: A study of free jet impingement. Part 1. Mean properties of free and impinging jets. J. Fluid Mech. 45(2), 281–319 (1971)

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26. Fond, B., Xiao, C., T’Joen, C., Henkes, R., Veenstra, P., van Wachem, B.G.M., Beyrau, F.: Investigation of a highly underexpanded jet with real gas effects confined in a channel: flow field measurements. Exp. Fluids 59(10), 160 (2018) 27. Xiao, C., Fond, B., Beyrau, F., T’Joen, C., Henkes, R., Veenstra, P., van Wachem, B.G.M.: Numerical investigation and experimental comparison of the gas dynamics in a highly underexpanded confined real gas jet. Flow Turbulence Combust. 1–33 (2019) 28. Baab, S., Lamanna, G., Weigand, B.: Two-phase disintegration of high-pressure retrograde fluid jets at near-critical injection temperature discharged into a subcritical pressure atmosphere. Int. J. Multiphase Flow 107, 116–130 (2018)

Towards Direct Numerical Simulations of Shock-Turbulence Interaction in Real Gas Flows on GPUs: Initial Validation Pascal Post(B)

and Francesca di Mare

Department of Mechanical Engineering, Chair of Thermal Turbomachines and Aeroengines, Ruhr-Universit¨ at Bochum, 44801 Bochum, Germany [email protected]

Abstract. A better understanding of turbulent flows in presence of strong compressible and real gas effects is in high demand for future improvements in engineering applications, such as turbomachines for regenerative power production. Therefore, a solver for future investigation of shock-turbulence interaction in presence of strong real gas effects by means of direct numerical simulation with multi-parameter equations of state utilizing the computational power of GPUs is carefully designed and initially validated using the example of carbon dioxide. It relies on a hybrid energy-consistent WENO scheme based on a shock sensor. The implementation is validated with a set of test cases, comprising the Shu–Osher problem, the inviscid Taylor-Green vortex and the compressible decay of homogeneous isotropic turbulence with eddy shocklets. In absence of appropriate and well documented experimental results, the test cases are adapted to real gas in regions where the gas behavior can be deemed near to the ideal limit and results are compared with the validated ideal gas configurations. Keywords: Real gas DNS · GPU

1

· Validation · Shock-turbulence interaction ·

Introduction

Flows in which shock waves and turbulence interact dynamically still represent a research field of utmost importance in science and engineering due to their occurrence in a wide range of applications such as process engineering, regenerative energy conversion cycles or aerospace propulsion [1]. Operating pressures in modern propulsion and energy systems are continuously increasing to attain higher efficiencies such that the working fluid reaches conditions close to and above the critical state, where strong real gas effects arise, as e.g. in supercritical carbon dioxide turbines. Steam and Organic-Rankine-Cycle turbines or wet air effects in the intake of a jet engine represent other examples of high relevance, where real gas effects and shock-turbulence interaction (STI) coexist. As well-documented, detailed experiments and measurements at these conditions are demanding and c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 59–75, 2020. https://doi.org/10.1007/978-3-030-49626-5_5

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extremely rare, the investigation of STI in real gas flows, especially close to the critical point, remains largely unexplored. However, numerical experiments by means of direct numerical simulation (DNS) utilizing the computational power of graphics processing units (GPUs) represent a suitable way of addressing this need and gain a better understanding of turbulent flows in presence of strong compressible and real gas effects. Nowadays, the design process of technical systems (e.g. turbomachines) is primarily based on a Reynolds-averaged Navier-Stokes (RANS) equations approach, often coupled to an ideal gas thermodynamic formulation. However, the accuracy of the involved models in real gas flows has not been properly assessed up to now due to the lack of experimental or numerical reference data [1]. Future design of technical systems will be based increasingly on predictive computational fluid dynamics (CFD) with scale resolving simulations like large-eddy simulations (LES) playing a central role and where all relevant real gas dynamics must be accurately described. Here again, subgrid scale (SGS) models for LES have not been assessed for their application in real gas flows [2] which is crucial for the predictive nature of such kind of computations. Even assuming an idealized gas behavior, SGS modeling in presence of shocks is still not conclusively solved [3]. Very little attention has also been paid to the development of a truly compressible wall model, as it is generally assumed that Morkovin’s hypothesis holds [4]. This is no longer true in presence of shock front oscillations [5] and at least questionable for flows with strong real gas effects, as e.g. close to the critical point. Transition in presence of such effects has also not yet been exhaustively investigated. In conclusion, a wide range of reliable DNS results are needed for the detailed understanding of the mentioned phenomena, to assess existing RANS and LES models in real gas applications and to potentially provide a database for the development and calibration of improved models [1], as done, e.g., in [6] under ideal gas assumptions. Research on DNS of compressible real gas flows has drawn attention only lately, with a primary focus on the decay of homogeneous turbulence in the presence of strong dense-gas effects in Bethe-Zel’dovich-Thompson (BZT) fluids relevant to ORC applications, for which expansion shocks may occur at particular thermodynamic conditions in the transonic and supersonic flow regions [1,2,7,8], suggesting that dense-gas effects influence the turbulent dynamics when turbulent Mach numbers are sufficiently high. Wall bounded flows with conditions close to the critical point have been in focus at low Mach numbers only [9– 11]. For detailed numerical experiments it is essential to minimize all sources of errors, involving the equations of state (EoS) and constitutive relations for viscosity and thermal conductivity whenever real gas effects do matter, especially in the proximity of the critical point. Yet, cubic or virial EoS like the MartinHou [12] EoS are primarily used in the conducted DNS, which are not very accurate close to the critical point, because multi-parameter EoS, very accurate even in the closest proximity of the critical point [13], are highly computationally demanding.

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In addition to these physical issues, the proper numerical treatment of STI is still subject to ongoing research [3,14]. Its accurate numerical treatment is challenging due to the contradictory requirements to minimize any numerical dissipation for a precise representation of the turbulent spectrum (especially at the small scales) whilst providing stability in the presence of shock waves typically accomplished by introducing numerical dissipation [3]. Even in absence of shocks, the straightforward discretization of the compressible Navier-Stokes equations in divergence form works only as long as nonlinearities are weak. In high-Reynolds number fluid turbulence, numerical instabilities are encountered necessitating the use of upwinding, filtering, energy-consistent or entropy-consistent schemes [15], especially in under-resolved compressible turbulence [3]. Energy-consistent schemes attempt to replicate the energy-preservation properties of the governing equations in the discrete sense [15] by splitting the convective derivatives in the advective fluxes. Different split forms have been proposed [16–18] and equivalent local discrete conservation forms have been developed [19,20]. When shock waves enter the solution, all methods for smooth turbulent flows suffer from spurious Gibbs oscillations near shock jumps, which may lead to nonlinear instabilities [15]. The onset of oscillations in the proximity of shocks can be avoided (or at least limited) by either shock-fitting or shock-capturing approaches, whereas the latter is feasible for general application [15]. Shock waves are extremely thin regions of the order of the mean-free path length, suggesting that the continuum hypothesis in the derivation of the governing equations is not appropriate to resolve their physical structure [14]. For accurate and stable resolutions on a computational grid, traditional shock-capturing methods rely on a conservative form of the equation to ensure convergence to the correct weak solution and additional numerical dissipation in the vicinity of the shock, resulting in smearing the shock over a few grid points [14]. Such traditional shockcapturing methods are not appropriate for STI problems due to the inevitable numerical dissipation in smooth turbulence regions and their high computational demand. To remedy this, an obvious idea is to combine a baseline spectral-like scheme with shock-capturing capabilities through the local replacement with a classical shock-capturing approach forming a hybrid scheme or through a nonlinear filter for the controlled addition of shock-capturing dissipation [15]. A key role in this class of schemes is played by shock sensors that must be defined in such a way that numerical dissipation is effectively confined in shocked regions, so that it does not pollute smooth parts of the flow field [15]. Hybrid methods thus offer the potential for shock-turbulence calculations with substantially reduced dissipation and computational runtime [3]. A detailed comparison to other numerical methods capable of simultaneously handling turbulence and shock waves including WENO, artificial diffusivity, adaptive characteristic-based filter and shock fitting is presented in [14]. Their results indicate that the hybrid method is minimally dissipative and leads to sharp shock resolution and well-resolved broadband turbulence, provided the shock sensor is defined appropriately.

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The challenges posed by high-fidelity simulations of strongly non-ideal flows must be answered by tackling both computational efficiency and parallelization strategies. Graphical processing units (GPUs) are nowadays a widely adopted choice to perform scientific and engineering simulations and a great variety of algorithms have already been implemented and optimized for this highly parallel architecture, including solution methods for the Lattice Boltzmann equations, particle transport and molecular dynamics [21]. The solution of the unsteady, tree-dimensional Navier-Stokes in CFD represents even on modern supercomputers a challenging task due to the high computational demand [21]. Especially the DNS of turbulent flows, the most demanding type of CFD simulation, is ideally suited for the parallization for GPU architectures but still a demanding task, as the algorithms have to be adapted for the GPU’s type of data-parallelism. The first detailed attempt of porting a DNS code for the compressible Navier-Stokes equations based on energy-consistent schemes resulted in speedup of approximately a factor of 22 [22]. Recently, energy- and entropy-consistent schemes [23] and particle-laden turbulent flow simulations [24] on GPUs have been presented. A speedup larger than a factor of 100 for the solution of hypersonic flows with detailed state-to-state air kinetics is reported in [25], another DNS code for the solution of decaying compressible turbulence based on the gas kinetic method is presented in [26]. In this work we present an implementation of a hybrid energy-consistent WENO scheme aimed for the detailed numerical investigation of STI in real gas flows and its validation for a selected set of test cases. We test the direct use of a multi-parameter EoS using the Span-Wagner EoS [27] for carbon dioxide, which is the most accurate EoS so far known. This has been possible due to the use of the modern highly parallel GPU architecture. In the future, adequate tabulations thereof with sufficient high accuracy for DNS [28] will also be implemented. Another challenge addressed here is the rigorous initial validation of a real gas solution method, which requires the adaption of conventional numerical treatment based on ideal gas assumptions. In absence of appropriate real gas test cases, the initial and boundary conditions pertinent to the available ideal gas test cases are adapted in regions where a near ideal gas behavior holds to allow a direct comparison with verified solution methods for ideal gas. The article is organized as follows. First, the mathematical model of a gas flow with generalized expressions for thermodynamic state equations and constitutive relations is introduced in Sect. 2. Section 3 describes the implemented hybrid solution method followed by its evaluation over a set of test cases in Sect. 4. Finally, the article ends with some concluding remarks and an outlook for future work.

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Mathematical Model

In this work the three-dimensional Navier-Stokes equations in conservative form are considered and solved. These read   (1) ∂t w + div F a − F d = 0,

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closed by generalized expressions for thermodynamic state equations and transport properties. The vector of unknowns comprises the conservative variables density, specific momentum and specific total internal energy density   (2) w (x, t) := , u, E with the advective and diffusive fluxes being   F a (w) := u, u ⊗ u + p (, e) I, u (E + p (, e) /)   F d (w) := 0, τ, τ · u − q ,

(3) (4)

where I denotes the identity tensor and the relation of specific total and static internal energy (5) E := e + u · u/2. The thermodynamic pressure p (, e) is a generalized function of density and internal energy, computed directly form the conservative variables  (w) = , e (w) = (E − u · u / (2)) /,

(6)

and therefore the natural choice as the independent thermodynamic state variables. The same holds for temperature, molecular viscosity and thermal conductivity coefficient appearing in viscous shear stress tensor and heat flux vector    τ := μ (, e) gradu + (gradu) − 2/3 divu I (7) q := −λ (, e) grad T (, e) .

(8)

Appropriate constitutive and state equations must be provided for the considered working fluid. In this work, the example of carbon dioxide is used with its most accurate available EoS from Span and Wagner [27]. This EoS is formulated in terms of the state variables density and temperature necessitating the iteration T = T (, e). For validation purposes, results based on the ideal gas assumption are also shown, for which the pressure is given by p (, e) = (γ − 1) e, γ = const..

3

(9)

Solution Method

As customary in computational gas dynamics [15], the method-of-line approach is adopted, where the resulting ODE in time is integrated by a standard 4th -order Runge-Kutta scheme. For better accuracy and robustness [15,29], the diffusive terms are discretized in non-conservative form by 6th -order explicit central differences. In order to avoid odd-even decoupling, a dedicated second-derivative

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discretization is used for the appearing second order terms. To ensure conservation, the advective fluxes are discretized in local discrete conservative form [19]. For the resulting numerical fluxes, approximating the advective fluxes at the uniformly spaced grid interfaces, a hybrid treating is adopted. The basic idea is to distinguish regions of “smooth” turbulence from regions dominated by shock waves based on a shock sensor and applying appropriate numerical fluxes to the respective region [3,14]. The numerical treatment reflects thus the diverse physical nature of turbulence and shocks by a linear combination of a non-dissipative central flux and a shock-capturing WENO flux, which also guarantees optimal performance as the computationally elaborate WENO flux is used only when necessary. The quality of such a hybrid discretization is largely dependent on the formulation of the shock sensor, which is difficult to design for universal application [14]. Here, we employ a Ducros et al. [30] type dilatation-vorticity-sensor as presented in [3,14], the fluxes at the interface are switched checking all neighboring points that are used in the central flux stencil to ensure that central differencing is not done across any shocks. This formulation works well for traditional STI, but does not find contact discontinuities [3], fails entirely whenever vorticity is not present (degenerating in this case to a sensor of flow compression regions [14]) and behaves inferior on very coarse grids, where the vorticity is poorly captured [14]. To remedy, [14] suggest to average the denominator in some sense (locally, or in homogeneous directions) in order to prevent unnecessary switching in random regions of small vorticity. A further analysis of shock sensors is also given in [14]. It must be noted that for the one-dimensional Shu-Osher problem the simple shock sensor formulation of Visbal and Gaitonde [31] is used. The most obvious choice to discretize the governing equations in smooth flow regions is the application of high-order finite central difference approximations to the divergence of the advective flux [15]. These approximations can be explicit or implicit (compact) [32] with coefficients optimized for maximal formal accuracy (order) or behavior in wave number-frequency space (dispersion-relationpreserving schemes [33,34]). However, their direct application does only work for wave propagation problems in which nonlinearities are weak, typically leading to numerical instability in high-Reynolds number fluid turbulence, which requires the use of upwinding, filtering, energy-consistent or entropy-consistent schemes [15]. In the present work, we rely on explicit 6th -order accurate central flux formulations in split form (often improperly referred to as skew-symmetric form). These represent energy-consistent nonlinearly stable numerical schemes, replicating the energy-preservation properties of the governing equations in the discrete sense [15]. The use of explicit formulations does not only allow a straightforward parallel implementation on GPU architectures, but also guarantees local discrete conservation contrary to compact approximations [15]. Different split forms have been proposed [16–18] and formulated in terms of corresponding fluxes for a local discrete conservative form [19,20]. All three split forms are implemented, the Kennedy and Gruber split form is used by default due to its improved robustness.

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In discontinuous shock regions a 5th -order accurate conservative finite difference WENO (standard WENO-JS [35]) scheme of characteristic-wise Roe-type with entropy (algorithm 2.3 in [36] with a local Lax-Friedrichs flux splitting) and carbuncle (H-correction of [37], implemented as described in [38]) fixes for minimal numerical dissipation is used by default, as described in detail in [39]. Our implementation follows the derivation in [40].

4

Test Problems

For the validation of the hybrid solver implementation we use test cases as described in [14]. This allows a direct comparison of the results and is efficient due to the coarse grids used. First, results of the one-dimensional Shu-Osher problem containing a well-defined shock wave are presented, followed by an adaption for carbon dioxide. Next, the inviscid Taylor-Green vortex test case is considered, a problem with purely smooth broadband turbulence features. Finally, the compressible decay of homogeneous isotropic turbulence (DHIT) with broadband spectra and eddy shocklets is presented for ideal gas and for real gas. 4.1

Shu-Osher Problem

The Shu-Osher problem [36] represents a one-dimensional STI idealization, in which a normal shock front propagates with M = 3 into an artificially perturbed density field. The goal of this test case is to test the capability of the solution method to accurately capture a shock wave, its interaction with an unsteady density field, and the waves propagating downstream of the shock [14]. The Euler-equations are solved for an ideal gas equation of state with γ = 1.4 within a domain x ∈ [−5, 5] with a mesh of Δx = 0.05 initialized by  (3.857143, 2.629369, 10.33333) if x < −4 (, u, p) = . (10) (1 + 0.2 sin (5x) , 0, 1) otherwise Figure 1 shows an evaluation of density, velocity and entropy Δs/cv = ln(p/γ ) after Δt = 1.8 for the WENO implementations with global LaxFriedrichs (LF) and local Lax-Friedrichs (LLF) flux-splitting and the WENORoe implementation with entropy and carbuncle fixes. The hybrid solver combines the Kennedy and Gruber split form with the WENO-Roe implementation. Because this test case is one-dimensional, the default shock sensor is replaced by the shock sensor of Visbal and Gaitonde [31]. The reference solution is computed on a mesh containing 1600 cells and validated against the results given in the original paper [36]. The interaction between shock and entropy disturbance (resulting from the disturbed density field) generates acoustic and entropy waves downstream of the moving shock front, complicating the interpretation of the density plot typically shown [14]. Therefore, entropy and velocity are shown separately to isolate each family of waves [14]. All results yield the correct shock location xs ≈ 2.39 and

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Fig. 1. Evaluation of the Shu-Osher problem after Δt = 1.8

capture the shock reasonably well. The location of the contact discontinuity at the leading entropy wave is xc ≈ 0.69, the location of the leading acoustic wave is xa ≈ −2.75 [14]. The most prominent difference between the schemes lies in the entropy waves. Downstream of the shock, the amplitudes of the entropy waves decrease as they propagate downstream due to the numerical dissipation. WENO-LF is more dissipative than WENO-LLF, which nearly matches WENORoe. The hybrid method is much less dissipative, even better results are reported in [14] using a more elaborate shock sensor. Already this simple one-dimensional test case illustrates the fact that, while an accurate treatment of the shock is important, the properties of the numerical scheme away from the shock also matter in order to achieve accurate results in the entire domain [14].

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Fig. 2. Evaluation of the real gas adaption for carbon dioxide of the Shu-Osher problem after Δt = 6.23 × 10−3

To validate the real gas implementation of WENO and hybrid schemes for real gas, the Shu-Osher problem is adapted. The initial thermodynamic state upstream of the M = 3 shock wave is chosen such that near ideal gas behavior is to be expected Z ≈ 1 and real gas and ideal gas results are to be almost identical. The Rankine-Hugoniot condition for ideal gas with the local γ is applied to determine the upstream state. The initial conditions are given by  (5.223 586, 814.9174, 982 531.2) if x < −4 (, u, p) = , (11) (1.0594 + 0.2 sin (5x) , 0, 100 000) otherwise and the results in Fig. 2 are evaluated after Δt = 6.23 × 10−3 . To have a better resolution of the wave downstream of the moving shock, a mesh resolution of 400 cells is chosen, the ideal gas computation is conducted with γ = 1.2 and R = 190 J/kg/K. The real gas computations for carbon dioxide using the Span-Wagner equation of state [27] are about a factor of 100 computationally more expensive than corresponding ideal gas computations. Evaluations of density and velocity in Fig. 2 show, that both, the real gas implementations of the hybrid and the WENO-Roe schemes match the ideal gas reference results as expected. 4.2

Inviscid Taylor-Green Vortex

To assess the effect of dissipation of the different schemes in the absence of shock waves, the inviscid Taylor-Green vortex test case [41] is considered. From a wellresolved initial condition, the inviscid vortex begins stretching and producing ever smaller scales, constituting a non-regularized problem with no lower bound on the length scale [3,14] (see also Fig. 3). It is solved without any regularization other than the numerical method, testing their preservation of kinetic energy, enstrophy growth and stability in presence of under-resolved motions [14].

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Fig. 3. Isosurfaces of p = 99.8 colored by Mach number for hybrid computation of the inviscid Taylor-Green vortex test case 30

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Fig. 4. Temporal evolution of mean quantities for the inviscid Taylor-Green vortex on a 643 grid

The Euler-equations are solved for an ideal gas with γ = 5/3 and the periodic 3 domain [0, 2π) is discretized by a uniform 643 grid. The initial condition is given by =1   u = sin x cos y cos z, − cos x sin y cos z, 0 (12) (cos 2z + 2) (cos 2x + cos 2y) − 2 , p = 100 + 16 with a sufficiently high mean pressure to make the problem essentially incompressible and to allow a comparison to the semi-analytical results for the

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enstrophy growth of Brachet et al. [42]. Due to the incompressibility, the kinetic energy should remain constant, while the enstropy grows rapidly [14]. Isosurfaces of pressure colored by the Mach number are shown in Fig. 3 for times t = (0, 5, 10) for the hybrid computation. These show the general regimes through which the flow evolves in time. From the smooth initial conditions at t = 0, the structures are beginning to interact. At t = 5 the flow is still laminarlike and dominated by large-scale structures. By t = 10, the flow has transitioned to turbulence-like, with smaller scale structures forming because of nonlinear interactions producing a range of length scales in the flow. Figure 4 shows the temporal evolution of mean kinetic energy and enstrophy, averaged over all space allowing to assess the influence of the dissipation on this problem. The dilatation-based shock sensor of the hybrid scheme never activates the WENO flux for this problem   when the grid is chosen carefully (if e.g. a grid point is located in x = 0, 0, 0 the vorticity in this point is 0 and the shock sensor fails; therefore the grid is equally shifted by Δx/2; enhancements of the shock sensor as proposed in [14] are also viable), allowing the non-dissipative central scheme to preserve the kinetic energy. Also for the enstrophy, hybrid and central scheme show an identical rapid growth very similar to the results Brachet et al. [42]. Furthermore, hybrid and central schemes match the spectral (F-SF23N) results of Shu et al. [43] on the same 643 grid. Also their WENO5 results are similar to those of the WENO-LLF and WENO-Roe implementations. The WENO schemes are certainly stable, but at the expense of a large dissipation that causes under-prediction of both the kinetic energy and the enstrophy for t  2. At this point, the WENO scheme begins adapting its stencils leading to drastically increased numerical dissipation [14]. Again, WENO-LF is more dissipative than WENO-LLF, which nearly matches the dissipation of the WENO-Roe implementation. All WENO implementations agree reasonable with the semi-analytical results for the enstrophy growth, the influence of the numerical dissipation is evident at the latest from t  3. 4.3

Decay of Homogeneous Isotropic Turbulence with Shocklets

As the final test case, the compressible DHIT setup of [14] is considered. Due to the sufficiently high turbulent Mach number  (13) Mt = u · u/ a , eddy shocklets [44,45] (weak shock waves) develop spontaneously from the turbulent motions, such that the test case represents a combination of the previously considered tests with broadband motions in the presence of randomly distributed shocks. 3 The periodic domain [0, 2π) is discretized by a uniform 643 grid. The NavierStokes equations are solved for an ideal gas with γ = 1.4 and Pr = 0.71. The viscosity is assumed to follow a power-law 3/4

μ/μref = (T /Tref )

,

(14)

70

P. Post and F. di Mare 1 0.8

 ω · ω / u2rms,0 /λ20

2 u · u / 3urms,0



reference hybrid WENO

0.6 0.4 0.2

0

1

2 t/τ

3

30 20 10 0

4

0



3

4

3

4

1.5

0.2 0.15 0.1 0.05 0

2 t/τ

(b) enstrophy

  2 θ / u2rms,0 /λ20

  2 2 δT 2 / (γ − 1) T0 Mt,0

(a) kinetic energy

1

0

1

2 t/τ

3

(c) temperature variance

4

1

0.5

0

0

1

2 t/τ

(d) dilatation

Fig. 5. Temporal evolution of mean quantities for the compressible DHIT test case on a 643 grid, the reference are DNS results of [14]

Fig. 6. Isosurfaces of Q-criteria with 1 × 105 colored by Mach number for the compressible DHIT test case

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where the reference state is specified to match Mt,0 = 0.6 and Reλ,0 = 100, with the Taylor-scale Reynolds number based on RMS velocity and Taylor microscale Reλ =  urms λ/ μ  urms = u · u /3  

2 λ2 = u2x / ( ∂ux /∂x ) .

(15)

The initial velocity field is purely solenoidal and randomized with the algorithm detailed in the appendix of [14] to match the energy spectrum   2 (16) E(k) ∼ k 4 exp −2 (k/k0 ) , with the most energetic wave number chosen as k0 = 4 yielding λ0 = 2/k0 . As initial density and pressure fields are specified constant, the initial conditions are not in acoustic equilibrium and a field of background acoustic waves develops and persists throughout the simulation [14]. Similarly, there are initial entropy modes [14]. Large acoustic and entropy modes are desirable for testing the schemes. The reference in Fig. 5 is the DNS result on a 2563 grid presented in [14]. Temporal evolution of mean kinetic energy, enstrophy, temperature variance and dilatation is evaluated over eddy turn-over time τ = λ0 /urms,0 .

(17)

The minimally dissipative hybrid method agrees reasonably well with the reference solution for all quantities. Slight deviations are found in temperature variance and dilatation and may be a result of deviating initial velocity fields due to the random nature of the generation process. The WENO implementation underpredicts all quantities compared to the reference, revealing its dissipative nature for broadband motions. Still, the results are reasonable. It particularly underpredicts vorticity and dilatation, which is consistent with the fact that the WENO procedure damps the small-scale motions [14]. This effect is also evident from Fig. 6 showing Q-criteria isosurfaces colored by the Mach number of the initial field, as well as hybrid and WENO scheme results after t/τ = 4 in comparison, where the smallest scales are only present in the hybrid results. To validate the real gas implementation of WENO and hybrid schemes for the complete Navier-Stokes equations, the DHIT test case described in Sect. 4.3 is adapted for carbon dioxide. The initial thermodynamic state is chosen as p = 1 bar and  = 1 kg/m3 , for which Z ≈ 1, allowing the direct comparison to the validated ideal gas results with γ = 1.224 and R = 188.82 J/kg/K. To match Reλ,0 = 100 the viscosity is scaled as in [1] and computed by the power-law with Tref = 529.61 K and μref = 0.606 kg/m/s. The temporal evolution of mean kinetic energy, enstrophy, temperature variance and dilatation over eddy turnover time are shown in Fig. 7. The real gas results of both, WENO and hybrid schemes, match the ideal gas reference in all presented quantities, validating the solver for the real gas application.

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 ω · ω / u2rms,0 /λ20

u · u / 3u2rms,0



reference hybrid WENO

0.6 0.4 0.2

0

1

2 t/τ

3

30 20 10 0

4

0

3

4

3

4

1.5  2  θ / u2rms,0 /λ20

  2 2 δT 2 / (γ − 1) T0 Mt,0 

2 t/τ

(b) enstrophy

(a) kinetic energy 0.2 0.15 0.1 0.05 0

1

0

1

2 t/τ

3

(c) temperature variance

4

1

0.5

0

0

1

2 t/τ

(d) dilatation

Fig. 7. Temporal evolution of mean quantities for the compressible DHIT test case adapted to carbon dioxide on a 643 grid, reference are the ideal gas results

5

Conclusions

This work presents the development and rigorous initial validation near the ideal gas limit of a solver for future investigation of shock-turbulence interaction in real gas flows by means of direct numerical simulation utilizing the computational power of graphics processing units. Its application is shown for the highly accurate but demanding multi-parameter equation of state for carbon dioxide by Span and Wagner [27]. The development is needed to contribute to a better understanding of turbulent flows in presence of strong compressible and real gas effects, as is in high demand for future improvements in engineering applications, like modern turbomachines. The solution method is of hybrid nature, switching between a baseline energy-consistent split scheme of 6th -order in smooth turbulence regions and a 5th -order accurate conservative finite difference WENO scheme of characteristic-wise Roe-type with entropy and carbuncle fixes for minimal numerical dissipation in shocked regions. Time integration is conducted applying the classical 4th -order Runge-Kutta method. The implementation is

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validated showing results for a set of test cases, comprising the Shu-Osher problem with a well-defined shock wave, the inviscid Taylor-Green vortex test case with purely smooth turbulence and a combination thereof, the compressible decay of homogeneous isotropic turbulence with broadband turbulence and eddy shocklets. The real gas implementation is validated in regions of near ideal gas behavior, allowing the direct comparison of real gas and ideal gas results. An obvious next step is further validation in regions with strong real gas effects, e.g., by comparison to exact so0lutions of the classical Riemann problem close to the critical point. On this basis, detailed studies of shock-turbulence interaction in real gas flows will be conducted. The DNS results will also be used to assess existing RANS and LES models in real gas applications and to provide a database for the development and calibration of improved models. Further developments will include the extension of the solution method to nonuniform grids, the computation of combustion processes and its use on multiple GPUs. An essential improvement will be the use of tabulation techniques e.g. for the equation of state with an accuracy sufficient for its application in numerical experiments exploiting all features of the GPU architecture for maximal speedup.

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Direct Numerical Simulation of Turbulent Dense Gas Flows Alexis Giauque(B) , Christophe Corre, and Aur´elien Vadrot LMFA - Laboratoire de M´ecanique des Fluides et d’Acoustique, Ecole Centrale de Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France [email protected]

Abstract. In order to assess the specific characteristics of turbulence in dense gas flows with respect to ideal gas flows, Direct Numerical Simulations are performed for both FC-70 described using Martin-Hou Equation of State (EoS) and a reference ideal gas, in the case of a forced Homogeneous Isotropic Turbulence (HIT) configuration and of a temporal compressible mixing layer configuration. The forced HIT shows that the statistically stationary turbulent kinetic energy (TKE) spectrum follows quite closely the one obtained in the incompressible case even when the turbulent Mach number is large. It is shown that the weakening of compressible dissipation of the TKE can be related to the decoupling of the density fluctuations from the velocity as well as to the strong weakening of compression shocklets in the dense gas case. The mixing layer case at a convective Mach number Mc = 1.1 shows that the wellknown compressibility-related reduction of the mixing layer momentum thickness growth rate is not significantly influenced by the EoS. Yet the unstable growth leading to the self-similarity phase is enhanced by the dense gas thermodynamics. Keywords: Turbulence · Dense gas · Direct Numerical Simulation · Forced Homogeneous Isotropic Turbulence · Compressible mixing layer

1

Introduction

The numerical simulation of compressible turbulent dense gas (DG) flows in the field of Organic Rankine Cycle (ORC) turbine design currently relies on RANS or LES turbulence closure models which have all been developed and calibrated in the context of ideal gas turbulent flows (see for instance Dura Galiana et al. [1]). The tool of choice to be used in order to assess the potential specificities of turbulence in a dense gas flow is Direct Numerical Simulation (DNS), which enables the resolution of every turbulent scales. The present study provides a comparative analysis of compressible turbulence characteristics for an ideal and a dense gas, based on the DNS of a forced Homogeneous Isotropic Turbulence (HIT) configuration and the DNS of a compressible mixing layer. While HIT can be seen as representative of a small flow region in the inter-blade space of an c Springer International Publishing AG 2020  F. di Mare et al. (Eds.): NICFD 2018, LNME, pp. 76–87, 2020. https://doi.org/10.1007/978-3-030-49626-5_6

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ORC turbine, the mixing layer would be representative of the blade wake region. DNS of decaying HIT in a dense gas has been previously studied (see Sciacovelli et al. [2], Giauque et al. [3]) but the rapid decrease of the turbulent Mach number in this case precludes the analysis of turbulence in a quasi-stationary context, which motivates to turn to forced HIT. The paper is organized as follows: Sect. 2 reviews the general methodology used in the present study to perform DNS of forced HIT and mixing layer. Section 3 is devoted to the comparative analysis of dense and ideal gas DNS results for both flow configurations.

2

Direct Numerical Simulation Methodology

Numerical Solver DNS are performed using the explicit and unstructured numerical solver AVBP (see Schonfeld et al. [4]). The code solves the full compressible NavierStokes equations using a 2-step time-explicit Taylor-Galerkin scheme (TTGC or TTG4A) for the hyperbolic terms based on a cell-vertex formulation (see Colin and Rudgyard [5]). The schemes provide high spectral resolution as well as low numerical dissipation and dispersion. They also ensure third-order space accuracy and third-order (TTGC) or fourth-order (TTG4A) time accuracy. These schemes have been extensively validated in the context of Large Eddy Simulation (see Giauque et al. [6]) but their accuracy can be considered as the low limit for performing Direct Numerical Simulations (DNS). In order to make sure the results are reliable, a grid convergence study is systematically performed for each case of the DNS database. Equation of State (EoS) and Transport Coefficients The DG considered in this study is FC-70 (see Guardone and Argrow [7] for the fluid properties). Its thermodynamic behavior close to the critical point is accurately described using the EoS developed par Martin and Hou [8], hereafter denoted MH EoS. Using MH EoS provides in particular an accurate description of the dense gas region where the fundamental derivative Γ becomes less than 1, with Γ defined as:    c4 ∂ 2 v  ρ ∂c  v 3 ∂ 2 p  Γ = 2 2 = 3 2 = 1 +  2c ∂v  2v ∂p  c ∂ρ  s s s  where v is the specific volume, ρ the density, c = ∂p/∂ρ|s the speed of sound, p the pressure and s the entropy. In the case of a BZT gas, the inversion zone where Γ < 0 is also accurately described using MH EoS. Transport properties are modeled using the model developed by Chung et al. [9]. When needed and for comparison purposes, the perfect gas (PG) EoS is used in combination with Sutherland model to describe the molecular viscosity temperature dependence and a heat conductivity derived from a constant Prandtl number fixed to 0.71.

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Shock Capturing Methodology In highly compressible turbulent flows, very localized shockwaves (shocklets) are expected to occur. To preserve the positivity of the solution in regions where very strong gradients exist, a hyperviscosity β is introduced in the viscous stress tensor τij , using the approach of Cook and Cabot [10]. Hyperviscosity may be viewed as an additional pressure term thickening the shock front. It acts on the very sharp velocity gradients characterizing shocks but goes back to zero in zones where the velocity evolves smoothly. More details on the model and its implementation can be found in Dauptain et al. [11]. Turbulence Forcing Methodology A turbulence forcing methodology is used to reach statistical stationarity in the HIT configuration. The numerical tool used for the study being a temporal DNS solver, the linear forcing model proposed by Lundgren [12] and studied by Rosales and Meneveau [13] is retained. Initially introduced for incompressible turbulence, it has been since analyzed in the compressible context by Petersen and Livescu [14] and is similar to the natural Reynolds shear stress production mechanism in the turbulent kinetic energy equation. Once the source term is computed following this classical procedure, a large scale filtering is applied to add kinetic energy only to the four largest wavelengths in the domain. In addition a sink term is added to the energy equation, removing the overall heat resulting from the dissipation of the TKE at the Kolmogorov scale in the domain.

3 3.1

Analysis of the DNS Results Forced HIT Case

Five computations have been performed as reported in Table 1. The initial Taylor Reynolds number is set to 100 for all simulations, initialized using an incompressible homogeneous isotropic turbulent velocity field following the Passot-Pouquet of maximum initial kinetic energy defined by spectrum, with ke the wavelength  the relationship Reλ = 15 ∗ 2πurms /(νke ). The initial thermodynamic state is selected in the inversion zone of FC-70: pinit /pc = 0.9866, vinit /vc = 1.5733 with pc , vc respectively the critical pressure and critical specific volume. Given the low speed of sound in the dense gas, a 20 m/s turbulent velocity translates into a turbulent Mach number equal to 0.8. In the case of Perfect Gas computations, a 63 m/s turbulent velocity has to be imposed in order to reach the same turbulent Mach number. Comparison of TKE Spectra The study of largely compressible turbulence by Kritsuk et al. [15] shows that Kolmogorov’s cascade phenomenology, which leads to the well known −5/3 decay of the TKE spectrum with the wavenumber, must be corrected to take into account compressibility. When the flow is significantly compressible, the velocity power spectrum defined as /2 follows rather a −2 slope while the TKE  √ 2 spectrum defined as < ρui > /2 follows a −3/2 slope. It can be inferred 

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Table 1. Characteristics of the DNS test cases (Forced HIT) Case name EoS

Mesh size (number of cells)

Turbulent velocity (m/s)

Turbulent Mach Number

Initial Taylor Re number

MH1

Martin & Hou 5403

20

0.8

100

MH2

Martin & Hou 6753

20

0.8

100

MH3

Martin & Hou 8003

20

0.8

100

PG1

Perfect gas

5403

63

0.8

100

PG2

Perfect gas

6753

63

0.8

100

that, as the turbulent Mach number increases, the difference between velocity power spectrum and TKE spectrum expresses the influence of compressibility on turbulence. The velocity power spectrum and the TKE spectrum obtained using the PG EoS (PG2) and the MH EoS (MH2) are compared at t = 15τ in Fig. 1. All spectra are normalized by their value for k = 4k0 so as to remove from the comparison any forcing-related effect (k0 = 2π/L being the minimum wavenumber, with L the computational domain size). The inertial range is restricted to a narrow region between k = 4k0 and k = 20k0 in the spectral space. This is likely due to the moderate Taylor Reynolds number reached in the present study. Figure 1 therefore also features a close-up view of this region, where the velocity power spectrum departs only slightly from the TKE spectrum when using MH EoS. Contrary to the PG case, compressibility effects have a limited influence on DG flow turbulence. Additionally, the slope of both spectra is close to −5/3 in the inertial range, which corresponds to the incompressible scaling (for a turbulent Mach number of 0.8). As far as PG is concerned, the spectra followed by the velocity power and TKE are more significantly separated from each other as expected. The velocity power spectrum slope is close to the expected −2 slope, while the TKE spectrum follows more closely a −3/2 slope. At this stage, an important observation can be made: for the studied Taylor Reynolds number, compressibility effects in the DG flow are almost suppressed in the inertial range, even at a fairly large value of the turbulent Mach number. The dissipation region close to the Kolmogorov wavenumber (beyond k = 100k0 ) reveals another interesting difference between DG and PG flows: the slope for both the velocity power and the TKE spectra is much steeper in the PG case. This difference is believed to be due at least in part to a different behavior of the shocklets between the two types of flows, calling for the in-depth analysis of shocklets performed in the following paragraph. Shocklets Analysis The shocklets analysis for the compressible forced HIT configuration is first focused on the comparison of the relationships between pressure and density jumps across shocklets for the PG and DG cases. The entropy jump through the shocklets is also studied because one of the interesting features of DG for turbomachinery applications is that much weaker compression shocks are expected,

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Fig. 1. Comparison (t = 15τ ) of TKE and velocity power spectra as a function of EoS.

leading to reduced entropy losses. Secondly, since the dense gas DNS using MH EoS is performed with an initial mean thermodynamic state located in the inversion region where the fundamental derivative Γ is negative, the occurrence of expansion shocklets for the dense gas simulation is also investigated. The marching cube algorithm proposed by Samtaney et al. [16] is implemented to analyze both PG and DG compression shocklets and possibly DG expansion shocklets. The iso-surfaces of zero density Laplacian (Δρ = 0) are detected in the flowfield. In order to retain only iso-surfaces corresponding to actual shocklets, the generalized Rankine-Hugoniot condition across the shocklet:   1 1 + (1) 2(h2 − h1 ) = (p2 − p1 ) ρ1 ρ2 is verified between left and right states distant from 4h (h is the cell characteristic length) in the direction normal to the discontinuity surface since it is known from previous studies (see Giauque et al. [3]) that the computed shock is numerically spread over four cells. Since Eq. (1) is strictly valid in the inviscid context only, jumps satisfying the relation within 10% of the mean enthalpy are eventually retained. Note the generalized Rankine-Hugoniot relation remains valid regardless of the shocklet velocity itself, which can be complex to compute. Finally, to ensure the detected flow region is indeed a shocklet, it is checked the selected candidate iso-surface is associated with a local velocity divergence large enough with respect to the rms value of the turbulent flow velocity divergence. This threshold value is set to −3 for compression shocklets and to 3 for expansion

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shocklets (DG only). When using the PG EoS, Eq. (1) can be simplified (since γ p h = γ−1 ρ ) into: ρ2 (γ + 1)p2 /p1 + γ − 1 = ρ1 (γ − 1)p2 /p1 + γ + 1

(2)

When FC-70 is described using the PG EoS, density and pressure jumps are expected to be almost the same as the constant value of γ in that case is very close to unity: γ = 1.0084. Using γ = 1 +  with  = 8.4 10−3