Aerodynamics of Tandem Wing Aircraft: From Dinosaurs to UAVs and Supersonic Planes 3031237765, 9783031237768

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Illia Kryvokhatko

Aerodynamics of Tandem Wing Aircraft From Dinosaurs to UAVs and Supersonic Planes

Aerodynamics of Tandem Wing Aircraft

Illia Kryvokhatko

Aerodynamics of Tandem Wing Aircraft From Dinosaurs to UAVs and Supersonic Planes

Illia Kryvokhatko E&D Product Development; Mechanical & Aerospace SAMI Advanced Electronics Riyadh, Saudi Arabia

ISBN 978-3-031-23776-8 ISBN 978-3-031-23777-5 https://doi.org/10.1007/978-3-031-23777-5

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Katia, Alisa, and Vilia who was waiting for this book for so long.

Preface

The author’s objective in this book was to present information on tandem wings’ aerodynamics in a comprehensive, intelligible, and relatively interesting manner to students of aerospace specialties, aircraft engineers, aircraft model designers, and researchers. Also, it may be attractive (except Chap. 2, as there are too many formulas) to a wide range of readers who are curious about aircraft history and development. The author believes the given work will be helpful not only for designers intending to develop a tandem wing aircraft but for aerodynamicists in general. There are a lot of features (such as interference of lifting surfaces resulting in additional longitudinal and lateral moments and altering the effectiveness of control surfaces, interaction of free vortices, stability, and trimming tuning) that may be useful in adjacent fields of science. All of it makes the aerodynamics of tandem wing aircraft more complicated than conventional layout but more exciting to study. The book is based on the author’s PhD thesis (2015, in Ukrainian) with major improvements and additions from different studies between 2015 and 2022. In this work, many remarks of aerodynamic experts were taken into account; the aircraft review was significantly supplemented with recent examples; here are new research sections about the connection between geometric parameters and aerodynamic characteristics (influence of airfoils and setting angles, dihedral angles of the wings, control surface effectiveness, installation of winglets, features of foldable configuration), recommendations regarding general development of tandem wing aircraft, etc. Riyadh, Saudi Arabia

Illia Kryvokhatko

vii

Acknowledgments

The author is grateful to professor Vitalii V. Sukhov for suggesting the topic of research and encouragement to write a book after the PhD thesis was done; to Oleksandr M. Masko for providing 3D model for CFD research (Sect. 3.7) as well as technology and equipment for the wind tunnel model manufacturing; to Oleksandr V. Pulava for his help in the manufacturing of the abovementioned model; to professor Yevhenii O. Shkvar for CFD consulting; to Mihail D. Melnikov for consulting regarding the experiment; to Volodymyr I. Andrus for consulting regarding aircraft static stability; to Konstantin O. Predachenko for critics of the method and for illustrations of the aircraft in Chap. 1; to Nataliia P. Tsentylo for invaluable help with arrangement of the wind tunnel test; to Yurii V. Yakovlev for collaboration and sharing valuable experimental data. The author does not claim to be a designer of any aircraft mentioned in the book.

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Contents

1

Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

Effect of Geometric Parameters on Aerodynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

3 4

Recommendations Regarding Aerodynamic Design of Tandem Wing Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

xi

About the Author

Illia Kryvokhatko graduated with honors from Faculty of Aerospace Systems (currently Institute of Aerospace Technologies) of National Technical University of Ukraine “Igor Sikorsky KPI” in 2010 (Master of Aerospace). From 2009 to 2015 – Aerodynamic engineer in wind tunnel lab at Antonov Company (Kyiv, Ukraine). In 2015 he defended PhD thesis in specialty Aerodynamics and Gas Dynamics of Aircraft. The topic was Method of Aerodynamic Characteristics Determination for Tandem-Scheme Aircraft. Professor Vitalii V. Sukhov was a scientific advisor. From 2015 to 2020 – Head of Aircraft Aerodynamic Design sector at Antonov Company. From 2012 to 2020 – part-time Lecturer/Associate Professor at National technical university of Ukraine “Igor Sikorsky KPI” Lecturing of disciplines Aerodynamics and Hydraulics, Aircraft Aerodynamics, Computational fluid dynamics, Modern design of aircraft. Lecturing and Lab works in Experimental Aerodynamics (wind tunnel tests). Scientific advisor of bachelors (19 people) and masters (19 people). Since 2020 the author has been working as Lead Aerospace Systems Analyst in SAMI Advanced Electronics (Riyadh, Saudi Arabia).

xiii

Symbols and Abbreviations

AC FWL GEV HT MAC UAV VT index 1 index 2 index eq index I index II index fus a = ∂CL/∂α a0 = ∂Cl/∂α AR = b2/S AReff b c Сd C
d2 CD isol CD int СD СD0 CD min СD і Cf Cl

Aerodynamic Characteristics Fuselage Waterline Ground-Effect Vehicle Horizontal Tail Mean Aerodynamic Chord Unmanned Aerial Vehicle Vertical Tail Forward wing (in other sources: fore, front, leading) Rear wing (in other sources: hind, aft, back, trailing) Equivalent wing Normal scheme Tandem wing configuration Fuselage Lift slope (derivative of lift coefficient versus angle of attack) Airfoil lift slope Aspect ratio of the wings Effective aspect ratio of the wings Wingspan; average span of forward and rear wings Chord; exceedance (difference in rear and forward wingspans) Airfoil drag coefficient (2D) Airfoil drag coefficient of rear wing at high turbulence Drag coefficient of isolated wings (no interference) Drag coefficient of the wings under interference Drag coefficient (3D) Drag coefficient at zero lift (CL = 0) Minimum drag coefficient Induced drag coefficient Skin-friction coefficient Lift coefficient of an airfoil (2D); rolling moment coefficient

xv

xvi

Symbols and Abbreviations

C βl

Rolling moment coefficient derivative versus sideslip angle; static rolling stability degree Lift coefficient (3D) Maximum lift coefficient Pitching moment coefficient (positive is nose-down) Pitching moment coefficient for isolated wings (no interference) Pitching moment coefficient at zero lift Pitching moment coefficient derivative versus angle of attack Pitching moment coefficient derivative versus lift coefficient; static longitudinal stability degree; longitudinal static margin Yaw moment coefficient

СL CL max Cm Cm iso Cm0 C αm C cmL Cn = C βn

Mz qSb

Cy dw D e I kint kd kV = q2/q1 l liso lx0, h0 lx, h L L/D (L/D)max M My nelev q = ρV2/2 Re r r0 S Suf SM SHL t t = t=c TR

Yawing moment coefficient derivative versus sideslip angle; static directional stability degree Lateral/side force coefficient Diameter of a fuselage at a conjunction with a wing Drag Oswald coefficient Turbulence intensity Coefficient of wing-fuselage interference Correction factor (for a distance between wing free vortices) Coefficient of airflow deceleration after forward wings Distance between wing-tip vortices Distance between tip vortices of isolated wings Distances between 25% of MAC of forward and rear wings along and perpendicular to a fuselage waterline, respectively Distances between 25% of MAC of forward and rear wings along and perpendicular to a free stream, respectively Lift Lift-drag ratio Maximum lift-drag ratio Mach number Pitching moment Effectiveness factor of elevators at subsonic speeds Dynamic pressure Reynolds number Distance to the axis of the vortex Radius of the viscous core Planform wing area (top view) Planform wing area occupied by the fuselage Cross-section area of the fuselage (midship) Area of wing high-lift devices Airfoil thickness Dimensionless airfoil thickness Wing taper ratio (a tip chord length divided by a root chord)

Symbols and Abbreviations

Vt V1 хac xac = xac =ceq xp xcg xт Y α α0 αS αsh β ε ε21 ε22 εfoil εfree ε0 εα φ Δφ = φ2 – φ1 Λ ν ρ σ θ

xvii

Tail volume ratio Free-stream velocity (velocity of an undisturbed flow) Longitudinal coordinate of the aerodynamic center Dimensionless longitudinal coordinate of the aerodynamic center Dimensionless coordinate of the center of pressure Dimensionless coordinate of the center of gravity Dimensionless coordinate of boundary layer transition point Lateral/side force Angle of attack, AoA Angle of attack at zero lift (CL = 0) Stalling angle of attack Angle of aerodynamic shading (rear wings/tail blanket) Sideslip angle Wash (angle): ε < 0 is downwash, ε > 0 is upwash Downwash angle on rear wings induced by forward wings’ vortices (mutual induction) Downwash angle on rear wings induced by rear wings’ vortices (self-induction) Downwash from a wing lifting (attached) vortex Downwash from wing free vortices Downwash after forward wings at α1 = α01 Derivative of downwash versus angle of attack Setting angle of a wing Decalage (difference in setting angles of rear and forward wings) Wing sweep angle (positive means backswept wings) Induced-thrust coefficient (Butler’s correction) Air density Prandtl coefficient (interference factor) Wing dihedral angle (positive is tips up; negative aka anhedral)

Chapter 1

Historical Review

1.1

Introduction

While the so-called “normal” or conventional aerodynamic scheme of aircraft has been studied in detail, modern aviation materials and numerical methods of aerodynamic optimization give a chance to rehabilitate unconventional configurations, in particular, with a tandem arrangement of the wings [1, p.19]. This aerodynamic scheme is defined by a presence of forward wings and rear wings of comparable areas. How are tandem wings related to other aerodynamic configurations? If area of the forward wings becomes much higher than one of the rear wings, we come to a conventional or normal scheme that means two wings and horizontal tail. If the area of the forward wings becomes much less, then we switch to a canard. If we add sweepback to the forward wings and sweepforward to the rear wings and connect them directly (or with additional vertical surface), the aircraft will be called a joined wing (or a box wing, respectively [2]). When is it reasonable to consider tandem wing design? For a small aircraft (a manned one or a drone) designed for a maximum flight time (not distance), tandem scheme can be the choice, as in 1936 it set a world record for endurance [3, p.146]. And nowadays there are UAVs of this kind with empty weight of 14 kg that are able to fly for up to 25 h which is an outstanding performance. On the other hand, tandem wing configuration is reasonable from structural point of view for UAVs with folded wings and limited dimensions (e.g., tube-launched) as it allows significantly decrease wingspan (approximately by 40%). Reducing the wingspan is also crucial for large aircraft with low-wing loading (weight-to-wing area ratio), such as solar-powered aircraft and UAVs, flying in sparse stratospheric air. Obviously, a wingspan of 60 m makes the airfield base more convenient and safer than a hundred-meter wings.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. Kryvokhatko, Aerodynamics of Tandem Wing Aircraft, https://doi.org/10.1007/978-3-031-23777-5_1

1

2

1

Historical Review

Tandem configuration is a typical solution for ground-effect vehicles (GEV) as it allows to ensure longitudinal stability, which is often problematic for these aircraft, and take full advantage of ground effect for high-efficiency operations [4, p.17]. For a convertiplane, it is easier to provide stability and controllability during takeoff and landing with not two but four propellers, and then it is reasonable to place them on four wings of comparable areas [5]. The basis and approximately half of the book’s content are the author’s PhD thesis [6].

1.2

Review of Tandem Wing Aircraft

Long before man began to build airplanes and even before man appeared, the aerodynamic configuration with tandem wings was implemented by nature. Dinosaur Microraptor used feathered front and rear limbs to glide in Cretaceous period (Fig. 1.1). Modern researches in wind tunnel show that anatomically probable changes in the position of the limbs do not lead to significant changes in aerodynamic characteristics. It should be noted that according to the experimental results, the Microraptor had a low lift-drag ratio (about 4.7, while modern birds have 10–12) and was stable only at high lift coefficients [7]. The reason for this is not only the “choice” of the aerodynamic scheme but also the smaller wingspan of the rear wings; the small-aspect ratio of the wings, especially the rear; as well as the large area of the tail. However, according to paleontologists, this did not prevent Microraptor from hunting birds of that time. At the very beginning of aircraft history, people did not know about adverse wing-wing interference, and projects with many wings (polyplanes) were very popular, including those with two pairs of wings spaced along the horizontal line of the fuselage. In monograph by Bowers [1, p.19], there is a comprehensive review of tandem wing aircraft created before the 1980s (most of them are ridiculous and

+lift +drag

down

biplane

force sensor adjustable sting

5 cm sprawled

tent

open jet wind tunnel

angle of attack +side force

air flow

sprawled

tent

38 cm

without leg or tail feathers without leg or tail feathers

Fig. 1.1 Plan view and wind tunnel model of Microraptor [8]

+drag

+p

itch

angle of attack

1.2

Review of Tandem Wing Aircraft

3

Fig. 1.2 Quickie aircraft [10]

Fig. 1.3 А-8 aircraft

will not be considered in the given book; reasons of the failures are presented in [4, p.25]). Bowers concluded that usage of tandem wings results in gain of airframe mass, and along with lifting capacity increasing, the aircraft drag also goes up. Most of the projects date back to the first half of the twentieth century, after which the scheme was not used for a long time. The idea was revived in 1977 in Quickie homebuilt aircraft (developed by B. Rutan, T. Jewett, G. Sheehan). This single seater has a takeoff weight of 217 kg, total wing area of 4.98 m2, and wingspan of 5.2 m (Fig. 1.2). Engine power of only 18 hp. allows to achieve a top speed of 203 km/h. Success of development is provided by the use of the newest constructional materials (fiberglass, polyfoam) allowing to provide high quality of the surfaces of the aircraft and to apply high-quality laminar airfoils [1, p.34]. With approximately the same span and area of forward and rear wings, this aircraft can be considered a tandem, although its designers called it “canard” as the elevators are on the forward wings and ailerons are on the rear wings. Based on this model, two-seater modifications Q2 and Q200 were created with 64- and 105-hp engines. In 1985, a maiden flight of A-8 was performed. This aircraft produced by Aeroprakt Ltd. is very similar to Quickie (Fig. 1.3). Forward wing area is 2.47 m2, rear wing area 2.44 m2, takeoff weight 223 kg, empty weight 143 kg, maximum liftdrag ratio 12, top speed 220 km/h, never exceed speed 300 km/h, maximum g-load 6, takeoff run 150 m, landing run 150 m, engine power 35 hp., and rate of climb (vertical speed) 5 m/s [9]. The aircraft is made entirely of plastic, forward wing airfoil is RAF-32, and rear wing airfoil is FX 60-126. Vought Company (USA) conducted studies of tandem wing aircraft models (Fig. 1.4). Elevons (elevators + ailerons) occupied half of a span and a quarter of a chord on each wing, and deflecting these controls on an angle of ~9.8° provides balancing in a range of lift coefficient from 0 to 1.4. The model showed good stall characteristics with a tendency to intense diving, buffeting was much weaker, and

4

1

Historical Review

Fig. 1.4 Model tested by Vought Company

Fig. 1.5 Design №1 by Vought Company

induced drag was much less than for a monoplane [11]. Directional stability was close to one of the conventional schemes. From experiment, it was concluded that the advantage of the tandem wing over the monoplane in induced drag is even greater than expected theoretically. However, this model is unsuitable for implementation due to its aeroelasticity problems, as the rear wing with vertical surfaces acts as a large T-tail unit and creates an asymmetric system in terms of rigidity. Therefore, another configuration of the tandem-scheme aircraft was developed (Fig. 1.5). In this design, the forward wings have about zero dihedral angle, but the rear wings have untypically high dihedral angles of more than 30°. Thus, a significant vertical gap between the wingtips provides low induced drag (in Sect. 3.5, modern CFD methods prove that it is reasonable to have big dihedral angles for tandem wings). The rear wingtips have significant negative dihedral angles and airfoils of winglets. These wingtip devices provide course stability, and their effectiveness as a vertical tail increases due to their location away from the nacelles. The increase in the wing area due to the addition of the winglets is compensated by their positive effect on the induced drag. The model is stable about all three axes. There was also a design with propellers instead of jets (Fig. 1.6). In 1987, АТТТ aircraft prototype by Scale Composites performed its maiden flight [12]. It was conceived as a short-range military transport with up to

1.2

Review of Tandem Wing Aircraft

5

Fig. 1.6 Design №2 by Vought Company

Fig. 1.7 A prototype of ATTT aircraft

14 paratroopers. Outstanding characteristics of its short takeoff and landing are provided by aerodynamic configuration with three lifting surfaces: two pairs of tandem wings and a T-tail (Fig. 1.7). The wings are placed almost at the same level with small dihedral angle of the forward wings (~5°) and small anhedral angle (negative dihedral angle) of the rear wings (~5°). The wings are connected by engine nacelles that allow to improve their stiffness, decrease mass, and solve a problem of necessary fuel volume. Each wing is equipped with double-section flap and aileron, and there are four different airfoils in the wing design. This layout provides high takeoff lift-to-drag ratio of about 20 and maximum lift coefficient of 3.35. During cruising, all wings and horizontal tail generate positive lift. 70% of the airframe is made of composites: length 22 m, forward wingspan 18.5 m, rear wingspan 23.4 m, maximum takeoff weight 25,500 kg, and payload 5670 kg. At an altitude of 900 m, its cruise speed reaches 600 km/h.

6

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Historical Review

Fig. 1.8 Project 52 (a superheavy cargo aircraft)

Project 52 was under development at Myasishchev Experimental MachineBuilding Plant in the late 1980s [13, p.2]. It would have been a transport aircraft with a record weight of payload 200–500 tons. The tandem scheme was considered that allowed to maximize lift and include the payload in the airframe structure (Fig. 1.8). Optimal airfoil design and choice of decalage angle would minimize negative interference of the wings and reduce the aerodynamic losses. Surfaces of vertical stabilizer were installed on the rear wings. The development of the aircraft was stopped at the design stage. In 1998, Burt Rutan’s experimental aircraft called Proteus made its first flight, and then it set a few world records (Fig. 1.9). Aerodynamic configuration between tandem and canard with high-aspect ratio of the wings ensures small induced and total drag. The forward wings are situated higher, so rear ones never get into the aerodynamic trace (wake). Main wing area is 27.9 m2; canard (forward wing) area is 16.6 m2. Main wing span is 28 m; forward wing span is 19.7 m. The gross weight is 5670 kg, fuel capacity 2720 kg, empty weight 2676 kg, airspeed 352 km/h at 20,000 ft. and 518 km/h at 40,000 ft., and cruise Mach number 0.42. This plane was able to fly for 18 h at an altitude of 20 km [14]. In the beginning of the twenty-first century, Bell and Boeing began to develop a convertiplane Quad TiltRotor (QTR) with a payload of 26 tons [15]. Its takeoff takes place with the help of four propellers located at the wingtips, with vertically oriented axes of the engines. In cruise, the axes of the engines are oriented along the fuselage, and the aircraft is basically a typical tandem scheme (Fig. 1.10). To provide clearance between propellers and ground, it has both forward and rear high wings with slightly lower forward ones. Draganfly Tango UAV was introduced in 2006 (Fig. 1.11). Its patented wings provided high flight performance including smooth stall. The UAV has a length of

1.2

Review of Tandem Wing Aircraft

7

Fig. 1.9 World recordholder Proteus

Fig. 1.10 Quad TiltRotor aircraft

Fig. 1.11 Draganfly Tango UAV

1.2 m, wingspan of 1.5 m, takeoff weight of 2.8 kg, payload weight of 1.14 kg, cruising speed of 40 km/h (stalling 24 km/h, maximum 100 km/h), maximum altitude of 640 m, and flight time of 50 min. The forward wing is located slightly above the rear. Ailerons were placed on the rear wings. Note the use of winglets on both forward and rear wings to increase lift-to-drag ratio. The drone has V-shaped tail (the winglets also participate in directional stability) and a tractor (pulling) propeller [16]. Switchblade UAV was designed for a tube launch, and its wings deploy after start (Fig. 1.12). Original design (2011) with weight of 2.5 kg was rebranded as Switchblade 300 to not be mixed up with Switchblade 600 (2020) with maximum weight of 54 kg (Fig. 1.13). It is operator-controlled and sends streaming video from an electro-optical sensor; after finding a target and the end of the data transfer about

8

1

Historical Review

Fig. 1.12 Switchblade 300 UAV

Fig. 1.13 Switchblade 600 UAV

dislocation of the enemy, it folds the wings and turns into a kamikaze drone with a small warhead [17]. The wings are almost coplanar (at one level), and thin plate was used in the role of the airfoil which can be justified for the drone with a lifecycle of up to 10 min. So this UAV illustrates another implementation of tandem scheme, but it is by no means an example of high aerodynamic performance. The UAVs were intensively used against Russian invaders in Ukraine. In 2008, a patent application was submitted for an invention, the scientific significance of which is to identify positive interference of a two-wing system [18]. The rear wing is located above the forward; it has an airfoil and aspect ratio to make lift slope greater than that of the forward wing (with the same airfoils, it is enough to have a higher aspect-ratio of the rear wing), which ensures longitudinal stability of the system (Fig. 1.14). The distance between the trailing edge of the forward wing and the leading edge of the rear wing is 0.5–1.5 of the length of the MAC of the rear wing. The rear wing area is about 1.5 times smaller than the forward one. The described principle of operation is that the rear wing reduces the suction over the trailing edge of the forward wing, so the separation area of flow on the forward wing is smaller than for the wing without interference. The stalling angle of attack of the two-wing system has been increased to values greater than 30°. Nevertheless, this tandem wing configuration was not implemented in the modern aircraft. Among drawbacks, there are the following:

1.2

Review of Tandem Wing Aircraft

9

Fig. 1.14 Wing system with “favorable interference”

Flow

A

A

A-A

– Low-aspect ratio of wings (~3 of forward and ~4.5 for rear) results not only in high stalling angle of attack but also in low maximum lift-drag ratio. – Drop of lifting capacity of the rear wings happens for small angles of attack (α < 5°) because of the flow acceleration in the gap between the wings and on the lower surface of the rear wing (suction on any lower surface means negative lift); so we have not only positive interference for the forward wings (compared to the isolated wings) but also inevitable negative interference for the rear wings. – At medium angles of attack (~10–15° for the presented geometry), the rear wings get into aerodynamic shadow of the forward wings. The low airflow velocity results in further decreasing of lift; in total, this layout is the opposite to the configuration with flap that is placed lower than the forward lifting surface. – At higher angles of attack, flow separation occurs on the rear wing first (the forward is expected to be protected by the interference), so it loses its lift, and two-wing system gets uncompensated nose-up moment and falls into spin. It is exactly the opposite to how one should design tandem-scheme aircraft. Basically, for any aerodynamic scheme with more than one lifting surface (conventional, tandem, canard, joined-wing), separation must begin on its forward surface (wing, canard, or stabilizer). In 2009, tube-launched radio-controlled Coyote UAV was introduced (Fig. 1.15). It is not exactly a kamikaze drone, but it was designed with reusability factor up to five times. The device with a wingspan of 1.47 m and a fuselage length of 0.79 m is intended for operations at heights of 150–365 m (although its practical ceiling is 6100 m) with a maximum takeoff weight of 6.4 kg, cruising speed of 110–140 km/h, and endurance of 1.5 h. The design of the UAV is foldable with the forward high wing and the rear low wing [19] that is aerodynamically better than Switchblade’s configuration. A vertical gap between wings is ~110 mm, and a stagger is ~480 mm. The same foldable design with forward wings on the upper surface and rear wings on the lower surface of the fuselage, two-fin tale, and rectangular cross section of the fuselage was implemented in Piranha UAV (Fig. 1.16). Relative stagger is less than for Coyote UAV. The gross weight is about 2.3 kg of which payload is 0.7 kg, cruise

10

1

Historical Review

Fig. 1.15 Coyote UAV

Fig. 1.16 Piranha UAV

speed around 76 km/h, endurance 10–20 min, wingspan 750 mm, and fuselage length 476 mm [20]. A larger class of drones is presented by Talisman (Fig. 1.17). This Italian UAV has a wingspan of 3.6 m and an MTOW of 50 kg, speed range of 53–155 km/h (optimal is 94 km/h), and flight time up to 25 h [21]. The main characteristics are: – Very stable and safe behavior under all flight conditions. – Very high MTOW if compared to the empty weight (it can carry on board 2.4 times its empty weight in payload and fuel). – The design is inherently protected against stall and spin [22, 23]. After this UAV, projects appeared to develop its descendants with maximum takeoff weight of 150, 300, and 750 kg. In 2011 at Dubai exhibition, a much bigger aircraft named United 40 MALE UAV was introduced (Fig. 1.18). Its wingspan is 17.5 m and a fuselage length is 11 m. Claimed lift-drag ratio equals 43 that is a unique value for motorized aircraft. Flight time of the UAV is not less than 25 h [24]. Tandem UAVs are being designed in Ukraine. Kharkiv Aviation University has created Poisk-2 system (Search-2) with takeoff weight of 60 kg (payload 15 kg), wingspan of 2.6 m, fuselage length of 2.1 m, cruising speed of 120 km/h, maximum speed of 180 km/h, and flight duration of 5 h [25]. The launch is performed from a catapult or a moving vehicle, the landing on a guided parachute. The UAV is

1.2

Review of Tandem Wing Aircraft

11

Fig. 1.17 Talisman UAV

Fig. 1.18 United 40 MALE UAV

dynamically stable in the turbulent layer of low-altitude atmosphere and has high maneuverability and high lift-drag ratio in a wide range of angles of attack and the center of gravity positions. At the same university, two UAVs named Inspector-1 and Pchiolka (Bee) of the similar configuration were designed [26]. The first one is intended for certification and control of pipelines, has a takeoff weight of 250 kg, a fuselage length of 3.1 m, a wingspan of 4.8 m, a cruising speed of 150 km/h, and a flight duration of up to 10 h (Fig. 1.19). It has a catapult launch or takes off from a moving vehicle. Transforming unmanned aerial system Pchiolka is designed for remote monitoring of objects and territories (Fig. 1.20). Weight equals 35–75 kg, cruising speed 50–150 km/h, and lift-drag ratio to 15. The specifics of all three lastly mentioned UAVs are the high and relatively short fuselages and the fins on the middle of the rear wings’ span.

12

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Historical Review

Fig. 1.19 Inspector-1 UAV

Fig. 1.20 Pchiolka UAV

In 2016, Ukroboronprom (Ukrainian Defense Industry) presented Dragonfly 1603 UAV designed for reconnaissance (Fig. 1.21). Maximum takeoff weight is less than 4 kg, hand-launched and parachute-landed, wingspan 1.9 m, length 1.25 m, cruise speed 50 km/h, maximum speed 120 km/h, endurance 2 h, and flight distance 140 km. The aerodynamic configuration is characterized by a tractor propeller, highaspect ratio wings, smooth winglets, and one-keel vertical tail [27, 28]. Pilum UAV is a kamikaze drone similar to Switchblade and is able to start from a container or from a carrier UAV (Fig. 1.22). The cruise speed is 90 km/h, and maximum one is 110 km/h, which with endurance of 30 min gives a range of 50 km to which 2 kg of payload (warhead) can be delivered. Maximum total weight is 10 kg and maximum altitude 2 km [29]. Flight tests were performed successfully [30].

1.2

Review of Tandem Wing Aircraft

13

Fig. 1.21 Dragonfly 1603 UAV

Fig. 1.22 Pilum kamikaze drone

A tandem wing micro-UAV with a fast-assembled airframe was developed in 2010 in Australia [31]. The aerodynamic configuration follows the most of abovementioned UAVs with a forward high wing and rear low wing, a significant stagger, and a vertical gap (Fig. 1.23). To increase the wing-aspect ratio, the wings are made detachable and aerodynamically identical to telescopic. In 2014, famous American company Lockheed Martin introduced Vector Hawk UAV (Fig. 1.24) designed for both air and underwater traveling [32]. The UAV options include both fixed-wing and folding structure, possible tube launch, and vertical takeoff. Its maximum weight is 4 pounds (1.75 kg) of which payload is 0.34 kg. The fixed-wing version has a cruising speed of 30 knots (56 km/h) and a

14

1

Historical Review

Fig. 1.23 Airframes of micro-UAV

Fig. 1.24 Vector Hawk UAV

Fig. 1.25 Trident UAV

maximum of 70 knots (130 km/h), with flight duration of up to 2.5 h. The winglets on the rear wing point downward and act as a vertical tail. The propeller is pulling, and the controls are located on both forward and rear wings and occupy about 70% of the span [33]. Trident UAV of folding configuration is aimed for payload delivery and atmospheric research (Fig. 1.25). It has a wingspan and a fuselage length of 3 ft. (0.91 m); a maximum takeoff weight of 5 pounds (2.2 kg), of which the payload is 2 pounds (0.87 kg); a flight duration of 25 min; a stall speed of 40 knots (74 km/h), and a cruising speed of 50 knots (93 km/h). The propeller is pushing; the tail is X-shaped [34]. In 2017, maiden flight of Tango2 UAV (modification of Tango UAV of 2006) was performed (Fig. 1.26). A vertical gap between the wings has been increased, and

1.2

Review of Tandem Wing Aircraft

15

Fig. 1.26 Tango2 UAV

Fig. 1.27 Spectre UAV

the propeller has become a pushing impeller, with an engine nacelle controlled and used as a rudder. The structure is made of composite materials, weight is 5.9 kg, payload is 1 kg, and cruising speed is 12–14 m/s (43–50 km/h). In different sources, its flight time is specified between 1.5 and 3 h [35, 36] and even higher with additional batteries and solar panels. In 2015, a convertiplane UAV named ERA-101 manufactured by Aeroxo Ltd. flew for the first time. With a maximum takeoff weight of 24 kg (of which 7 kg is a payload), the UAV has a wingspan of 1.3 m, maximum speed of 200 km/h, and flight range up to 500 km. The main purpose is monitoring of remote infrastructure, search and rescue missions, environmental monitoring, and cargo delivery. The declared price is $40,000–$50,000 [37]. Spectre is a tilt-wing, electric VTOL combat air system capable of quickly transitioning to forward flight mode unveiled in 2018 in the United Kingdom (Fig. 1.27). Designed with an integrated modular payload bay Spectre will weigh about 100 kg including 25 kg of payload (that may consist of several missiles). Spectre will have a cruise speed of 180 km/h, cruising altitude of less than 100 m, and combat range of 10 km with flight endurance of more than 60 min. The wingspan equals 2 m on which eight paired propeller units are placed (two on each wing).

16

1

Historical Review

Fig. 1.28 Original design of Eraole aircraft

The Spectre system can “find and fix” beyond line-of-sight threats or use “watch and wait” mode with a top attack capability. Other mission module options include resupply payloads, improved sensors, or electronic warfare payloads. Spectre UAS can be used as a single system or as a scalable cooperative swarming capability [38, 39]. Eraole is a modern French single-seat eco-friendly aircraft with solar panels on both forward and rear wings (Fig. 1.28). It is aimed for transatlantic flight. Nevertheless, the solar panels cover only 25% of energy demand, while a biofuel combustion engine provides 55%, and soaring in upstreams adds the rest 20%. Wingspan of the aircraft is 14 m, length 7.5 m, weight 750 kg, and maximum speed 140 km/h, but the most efficient energy consumption is achieved at 65 km/h [40, 41]. It should be admitted that during the known flight test, Eraole became not a purely tandem wing aircraft due to installation of a small horizontal stabilizer [42]. Vahana project by European corporation Airbus was designed as a demonstrator in order to create city airlines with the help of a certified electric automatic manned aircraft of vertical takeoff and landing (Fig. 1.29). Its expected speed is 175 km/h at a flight range of 80 km. Operation as air taxi, cargo carrier, ambulance, and rescue vehicle is considered [5, 43]. Similar to the mentioned above is Lilium Jet, the first electrical vertical takeoff and landing jet that is technically a canard, not a tandem [44]. As compared to Vahana design, Lilium Jet changes the thrust direction by rotating parts of the wings and canard surfaces, not the whole wings. This ducted electric vectored thrust (DEVT) is the core technology. The interior has four-seat or six-seat configuration. Between 2015 and 2022, five generations of technology demonstrators have been built. The aim is to get certification with both EASA and FAA. An American electric single-seat amphibious aircraft with vertical takeoff and landing Opener BlackFly flew for the first time in 2014 (Fig. 1.30). The airframe is manufactured of carbon fiber polymer. The tractor propellers on the forward and rear wings (eight in total) make it similar to Vahana, but the wings are not rotatable. Each engine weighs 2 kg and creates 59 kg of thrust. Behind each engine in the wing, there

1.2

Review of Tandem Wing Aircraft

17

Fig. 1.29 Vahana air taxi by Airbus

Fig. 1.30 Opener BlackFly aircraft

are two batteries to power it. The aircraft is controlled by elevons located on all the wings. Parachute landing is possible. The empty weight is 158 kg, and the allowable weight of the pilot is 104 kg. The wingspan and fuselage length are about equal (4.1 m). Declared cruising speed is 96 km/h, and range is 32 km (with 20% reserve), while during supercharging 80% charge is provided in 25 min. The aircraft was designed to meet the airworthiness standards of ultralight aircraft FAR 103 with low noise and no emissions. It is easy to operate to such a degree that in the United States it does not require a pilot license (only writing exam and training on the aircraft itself) [45, 46]. Emotion aircraft is similar to the previous examples – with propellers on the leading edges of all wings (Fig. 1.31). The project is currently at the stage of flight tests. After thorough CFD optimization, Emotion is aimed to have a vertical takeoff and landing when all 16 electric engines are oriented up. Span equals 5.5 m for the forward wings and 6.06 for the rear wings, and the length of the fuselage is 4.2 m. In

18

1

Historical Review

Fig. 1.31 Emotion aircraft

Fig. 1.32 Nuuva V300 UAV

horizontal flight, a range of 600 km should be provided at a cruise speed of 250 km/h [47] with lift-drag ratio of 16 [48]. While air taxi development still faces some problems because of extreme high safety requirements [49], it is slightly easier to certify a cargo UAV. Hybrid VTOL UAV Nuuva V300 by Pipistrel (Slovenia) will have a maximal takeoff weight of 1700 kg with a typical fuel weight of 65 kg and a maximum payload of 460 kg in a 3 m3 cargo compartment (Fig. 1.32). The drone has eight electric engines for takeoff and landing and one IC engine for horizontal flight. Wingspan equals 13.2 m and wing area 23 m2. Its fast cruise speed is supposed to be 220 km/h, economy cruise speed 165 km/h, maximum cruise altitude 6000 m, and endurance up to 12 h. It is expected to have a typical range of 300 km but maximum of 2500 km [50]. An order for 15 Nuuva V300 was placed in April 2022 [51]. A group of researchers from the Chinese Academy of Sciences is developing a passenger plane that can travel at a speed of 6000–8000 km/h with a payload of 5 tons or 50 passengers [52]. The aircraft model is designed with two pairs of wings of comprehensive planform and was successfully tested in wind tunnel at Mach number 5–7. The principle of using positive interference at supersonic speeds is that a shock wave from the forward low wing gets to the lower surface of the rear high wing at the point of maximum airfoil thickness (Fig. 1.33). As the pressure increases

1.2

Review of Tandem Wing Aircraft

19

'R

'L

'D M>1 shock wave

Fig. 1.33 Conceptual design of I Plane and Supersonic tandem wing shock waves

after the shock, the lifting force of the rear wing increases, and the drag decreases that is equivalent to additional thrust [53, p.453]. There are also numerical investigations of this layout called Busemann biplane for Mach number up to 2.5 [54]. In the following chapters, transonic and supersonic aerodynamics of tandem wing aircraft is not considered. Implementation of tandem wings may be reasonable for atmospheric satellites (also called pseudo-satellites). This type of UAVs is solar-powered, so wing area is big to place enough solar panels. Also, they require high lift-drag ratio that results in high-aspect ratio, and wingspan increases dramatically. It becomes much easier to exploit if the lifting surface is split into two. HAPS (high-altitude pseudo-satellite) ApusDuo15 by UAVOS has a wingspan of 15 m (Fig. 1.34), payload of 2 kg, maximal takeoff weight of 43 kg, and lift-drag ratio over 30. At a latitude of 20°, the UAV can fly infinitely long above 12 km. Takeoff can be performed from a grass runway surface with electric winch or from a paved runway under its own power using landing gears. Three persons are needed to run with the UAV during the takeoff. The design with no high lift devices reduces weight and costs and increases reliability. Control of pitch and roll is provided by changing of the angle of attack in particular sections of the wings. Flight test in atmospheric conditions was successfully performed in October 2020 when the UAV reached an altitude of 19 km despite the specified service ceiling being 18 km [55–57]. There are also many modern studies of flapping tandem wings – usually for insect-scale Reynolds numbers [58–60]. This transient aerodynamics was left outside the scope of the presented book.

20

1

Historical Review

Fig. 1.34 HAPS Apus Duo15

1.3

Conclusions

Thus, the aerodynamic layout called “tandem wings” in the early twenty-first century is increasingly exploited for both unmanned and piloted aircraft. The number of respectable aircraft building companies, adopted this configuration for different use cases, proves that it is not just a coincidence, but a well-grounded decisions. Tandem wings have shown a number of advantages (compactness, high lift-drag ratio and, as a result, long flight endurance, anti-spin protection, solving stability and controllability issues for convertiplanes, and ground-effect vehicles) and are prospective for further research to optimize parameters and achieve high flight characteristics for wider implementation among different types of aircraft.

References 1. Bowers P (1984) Unconventional aircraft. TAB Books, Blue Ridge Summit. 2. Kroo I (2005) Nonplanar wing concepts for increased aircraft efficiency. VKI lecture series on innovative configurations and advanced concepts for future civil aircraft, 6–10 June 2005. https://lf5422.com/wp-content/uploads/2014/08/vki_nonplanar_kroo-1.pdf 3. Sobolev DA (1989) Samolioty osobyh shem (Aircraft of special layouts). Mashinostroenie, Moscow 4. Minardo A (2014) The tandem wing: theory, experiments, and practical realisations. Dissertation, Politecnico Di Milano, Milano 5. Vahana. The next technological breakthrough in urban air mobility (2019) The Index Project. https://theindexproject.org/award/nominees/2645. Accessed 2 Oct 2022 6. Kryvokhatko IS (2015). Metod vyznachennya aerodynamichnyh kharakterystyk litalnogo aparata skhemy tandem (Method for aerodynamic characteristic determination of tandem wing aircraft). Dissertation, National Aviation University, Kyiv.

References

21

7. Dyke G, de Kat R, Palmer C et al (2013) Aerodynamic performance of the feathered dinosaur microraptor and the evolution of feathered flight. Nat Commun 2489(4):1–9. https://doi.org/10. 1038/ncomms3489 8. Evangelista D, Cardona G, Guenther-Gleason E, Huynh T, Kwong A, Marks D et al (2014) Aerodynamic characteristics of a feathered dinosaur measured using physical models. Effects of form on static stability and control effectiveness. PLoS One 9(1):e85203. https://doi.org/10. 1371/journal.pone.0085203 9. Letatelnye apparaty (Flying Vehicles) (2007). http://aeroclub.com.ua/?module=articles&c= La&b=3&a=2. Accessed 2 Oct 2022 10. Gideon E (2003) File:Rutan quickie q2.jpg. https://commons.wikimedia.org/wiki/File:Rutan_ quickie_q2.jpg. Accessed 2 Oct 2022 11. Wolkovitch J (1979) Subsonic VSTOL aircraft configurations with tandem wings. J Aircr 16(9):605–611. https://doi.org/10.2514/3.58574 12. Scott W (1988) Scaled composites tests low-altitude, long-range capability of ATTT aircraft. Aviation Week Space Technol 18:26 13. Pogodin V (2004) Tandem – novoe slovo v aviatsii? (Is Tandem a new word in aviation?). http://dlib.eastview.com/browse/doc/6439130. Accessed 2 Oct 2022 14. Gibbs Y (2017) NASA Armstrong fact sheet: proteus high-altitude aircraft. https://www.nasa. gov/centers/armstrong/news/FactSheets/FS-069-DFRC.html. Accessed 2 Oct 2022 15. Sklar M (2007) Integrated defense systems. Diversity in design. https://www.boeing.com/news/ frontiers/archive/2006/december/i_ids03.pdf. Accessed 2 Oct 2022 16. DraganFly Tango Manuals (2007). https://www.manualslib.com/manual/641088/DraganflyTango.html. Accessed 2 Oct 2022 17. Switchblade 300 (2022). Army recognition. https://www.armyrecognition.com/us_american_ unmanned_aerial_ground_vehicle_uk/switchblade_300_miniature_loitering_munition_sui cide_drone_data_fact_sheet.html. Accessed 2 Oct 2022 18. Biryukov II (2008) Dvuhkrylyevaya sistema s polozhitelnoy interferentsiyey (Two-wing system with favorable interference) RU Patent 2381142, 7 Aug 2008 19. Coyote (2017) NavalDrones. http://www.navaldrones.com/Coyote.html. Accessed 2 Oct 2022 20. 2011 SCOAR Spring Meeting CIRPAS Piranha Handout (2011) Manualzz. The universal manuals library. https://manualzz.com/doc/8939294/2011-scoar-spring-meeting-cirpas-pira nha-handout. Accessed 2 Oct 2022 21. Talisman S (2019) Elytron Aeronautica. http://www.elytron-aeronautica.com/en/platforms/ talisman/. Accessed 2 Oct 2022. 22. UAV Navigation Carries Out A New Adaptation In The Shortest Period Of Time (2022) UAV Navigation. https://www.uavnavigation.com/company/news-and-events/uav-navigationcarries-out-new-adaptation-shortest-period-time. Accessed 2 Oct 2022 23. Talisman Unmanned Aerial Vehicle (2022) Verdict Media Limited. https://www.aerospacetechnology.com/projects/talisman-unmanned-aerial-vehicle/. Accessed 2 Oct 2022 24. Osborne T (2011) Dubai Airshow 2011: Adcom unveils United 40 MALE UAV. Shephard Press Limited. https://www.shephardmedia.com/news/uv-online/dubai-airshow-2011-adcomunveils-united-40-male-ua/. Accessed 2 Oct 2022 25. Bespilotnyy letatelnyy apparat Sapsan (Sapsan UAV) (2014). Ekonomicheskie novosti (Economic News). https://enovosty.com/armiya/full/375-bespilotnyj-letatelnyj-apparat-sapsan. Accessed 2 Oct 2022 26. Grebenikov AG, Myalitsa AK et al (2009) Problemy sozdaniya bespilotnyh aviatsionnyh kompleksov v Ukraine (Problems of unmanned aerial system development in Ukraine). KhAI Kharkiv 42:111–119 27. Military Uncrewed Systems Handbook. Issue 30 (2022) Shepard Media. https://handbooks. shephardmedia.com/view/553589908/174/#zoom=true. Accessed 3 Oct 2022 28. Products. Dragonfly-1603 (2022) Arkeik Space Technologies. http://www.arkeik.com/ products/. Accessed 3 Oct 2022 29. Pilum (2022) ADrones https://adrones.com.ua/drones/pilum/. Accessed 3 Oct 2022

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Historical Review

30. Kamikaze drone PILUM (2019). A.Drones. https://youtu.be/OtK7tvu0ht4. Accessed 3 Oct 2022 31. Treble M (2013) UAV body. https://web.archive.org/web/20131107170648/http://www. marctreble.com/portfolio/glare/. Accessed 2 Oct 2022 32. Vector Hawk Small Unmanned Aircraft System (sUAS) (2016) Naval technology. https://www. naval-technology.com/projects/vector-hawk-small-unmanned-aircraft-system-suas/. Accessed 3 Oct 2022 33. Lockheed Martin unveils the Vector Hawk (2014) Defense update. http://defense-update. com/20140514_lockheed-martin-unveils-the-vector-hawk.html. Accessed 3 Oct 2022 34. Trident (2015) Unmanned integrated systems. http://uis.sg/author/uisadmin/. Accessed 3 Oct 2022 35. Draganfly Announces New Fixed-Wing Aircraft The Tango2 (2017) UAS Weekly. https:// uasweekly.com/2017/05/10/draganfly-announces-new-fixed-wing-aircraft-tango2/. Accessed 3 Oct 2022 36. Draganflyer Tango2 (2017) Draganfly Inc. https://www.draganfly.com/aircraft-panel.html. Accessed 3 Oct 2022 37. ERA-101 (2017) Avia Pro. https://avia.pro/blog/era-101-tehnicheskie-harakteristiki-foto. Accessed 3 Oct 2022 38. MBDA unveils Spectre Combat UAV Concept (2018) UAS Vision. https://www.uasvision. com/2018/09/21/mbda-unveils-spectre-combat-uav-concept/. Accessed 3 Oct 2022 39. The MBDA Spectre (2018) Military factory. https://www.militaryfactory.com/aircraft/detail. asp?aircraft_id=2025. Accessed 3 Oct 2022 40. The Eraole Challenge (2020) Eraole. https://eraole.com/en/the-eraole-challenge/. Accessed 3 Oct 2022 41. Eraole, l’avion du future (2013) Technologies de Pointe. https://up-magazine.info/technologiesa-la-pointe/technologies/1946-eraole-l-avion-du-futur/. Accessed 3 Oct 2022 42. Sigler D (2018) Eraole in flight – further and higher. Sustainable Aviation Foundation. http:// sustainableskies.org/eraole-flight-higher/. Accessed 3 Oct 2022 43. 5 Best Personal Aircraft - Passenger Drones (Flying Taxis) and Flying Cars (2017) TerkRecoms – Tech TV. https://www.youtube.com/watch?v=_dNPjqLyxSI. Accessed 3 Oct 2022 44. Introducing the first electric vertical take-off and landing jet (2022) Lilium. https://lilium.com/ jet. Accessed 3 Oct 2022 45. Opener Reveals Ultralight eVTOL (2019) Aviation Publishing Group. https://www.avweb. com/recent-updates/business-military/opener-reveals-ultralight-evtol/. Accessed 3 Oct 2022 46. BlackFly (2018) Opener. https://opener.aero/. Accessed 3 Oct 2022 47. Unparalleled performances (2021) Emotion Aircraft SL. https://www.emotion-aircraft.com/ technicalspecs. Accessed 4 Oct 2022 48. Designed for superior aerodynamics (2021) Emotion Aircraft SL. https://www.emotion-aircraft. com/aerodynamics. Accessed 4 Oct 2022 49. Flying taxis are taking off to whisk people around cities (2019) The Economist. https://www. economist.com/science-and-technology/2019/09/12/flying-taxis-are-taking-off-to-whisk-peo ple-around-cities. Accessed 4 Oct 2022 50. NUUVA V300 (2022) Pipistrel by Textron eAviation. https://www.pipistrel-aircraft.com/ aircraft/nuuva-v300/#tab-id-3. Accessed 4 Oct 2022 51. Pipistrel and Lobo Leasing sign partnership (2022) Pipistrel by Textron eAviation. https://www. pipistrel-aircraft.com/pipistrel-and-lobo-leasing-sign-partnership-and-place-order-for-15nuuva-v300-hvtol-aircraft/. Accessed 4 Oct 2022 52. Illmer A (2018) Examining China’s hypersonic transport plans. BBC News, Singapore. https:// www.bbc.com/news/business-43151175 53. Votyakov VD (1972) Aerodinamika letatelnyh apparatov i gidravlika ih sistem (Aerodynamics of aircraft and hydraulics of their systems) VVIA named after prof. NE Zhukovsky, Moscow

References

23

54. Patidar VK, Yadav R, Joshi S (2016) Numerical investigation of the effect of stagger on the aerodynamic characteristics of a Busemann biplane. Aerosp Sci Technol 55:252–263. https:// doi.org/10.1016/j.ast.2016.06.007 55. New UAVOS HAPS ApusDuo variant completes test flight (2019) Military+Aerospace Electronics. Indeavor Business Media. https://www.intelligent-aerospace.com/unmanned/arti cle/14071325/uavos-haps-apusduov-test. Accessed 4 Oct 2022 56. Host P (2020) Update: UAVOS flight tests HAPS ApusDuo in unstable atmospheric conditions. Janes. www.janes.com/defence-news/news-detail/update-uavos-flight-tests-haps-apusduo-inunstable-atmospheric-conditions. Accessed 4 Oct 2022 57. High-Altitude Pseudo-Satellite ApusDuo Aircraft (2020) UAVOS Inc. https://www.uavos.com/ products/fixed-wing-uavs/apusduo-atmospheric-satellite. Accessed 4 Oct 2022 58. Arranz G, Flores O, García-Villalba M (2020) Three-dimensional effects on the aerodynamic performance of flapping wings in tandem configuration. J Fluids Struct. https://doi.org/10.1016/ j.jfluidstructs.2020.102893 59. Peng L, Zheng M, Pan T, Su G, Li Q (2021) Tandem-wing interactions on aerodynamic performance inspired by dragonfly hovering. R Soc Open Sci. https://doi.org/10.1098/rsos. 202275 60. Bie D, Li D (2022) Numerical analysis of the wing–wake interaction of tandem flapping wings in forward flight. Aerosp Sci Technol 121. https://doi.org/10.1016/j.ast.2022.107389

Chapter 2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Methods for determination of tandem wing aircraft aerodynamic characteristics are divided into the same groups as for conventional configuration (with two wings, a fuselage, horizontal and vertical tail surfaces): • Analytical methods – based on theoretical aerodynamics and generalized dependencies from previous experimental data • CFD (computational fluid dynamics) methods – the most powerful of which solve Reynolds-averaged Navier-Stokes (RANS) equations with usage of different turbulence models, e.g., Menter’s (K-ω SST) • Experimental methods – wind tunnel tests of aircraft models • Flight tests In this chapter, we will see that for the tandem scheme, both analytical and numerical methods are less developed than for the conventional one. The most reliable source of aerodynamic characteristics is tests in a wind tunnel or in flight.

2.1

Physics of the Airflow Around a Tandem Wing Aircraft

Specifics of the tandem scheme, which must be taken into account by analytical or numerical methods, are interference of the wings (forward) and represented in three aspects on the rear wing: 1. Flow retardation (deceleration) 2. Flow turbulization 3. Downwash and upwash (less significant on forward wing from rear wing) The first two effects are significant then and only then if the rear wings are placed directly in the aerodynamic trace (shadow) of the forward wings. In this case, the rear wings generate smaller lift (because of low initial flow speed) and bigger drag (turbulent boundary layer from its leading edge). This configuration would be poor © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 I. Kryvokhatko, Aerodynamics of Tandem Wing Aircraft, https://doi.org/10.1007/978-3-031-23777-5_2

25

26

2 z

Determination of Tandem Wing Aircraft Aerodynamic Characteristics y

z

G1

V

O

dw

l’

G1

y

G1

V

A

b1

x

T1

b1

l’ G1

G1

y

x

T1 G1

Fig. 2.1 A simplified scheme to determine velocities induced by attached and tip vortices Fig. 2.2 Downwash (-) and upwash (+) zones from a tip vortex of the forward wing

also from longitudinal stability considerations: during getting in and out of the trace, the lift would change dramatically, and that results in pitch oscillations. The only exception (or rather a mitigating factor) is the vehicle with the rear wings in a propeller slipstream (see Figs. 1.29, 1.30, and 1.31). Then the flow retardation from the forward wing is less significant, but the lift-drag ratio remains low due to high friction drag in a completely turbulent boundary layer. The third aspect of the wing-wing interference (nonuniform downwash/upwash) is essential for all real configurations of the aircraft, as the effect of tip vortices shows itself on a larger scale, and true angles of attack will be different across the rear wing span. This side of interference can be reduced by moving the forward wings’ tip vortices away from the plane of the rear wings, but neglecting it completely would be a very rough approximation. A primitive representation of the wing vortex system is given in Fig. 2.1. According to Joukowski theorem [1, p.99], wings can be replaced by attached vortex (also called bound vortex) that creates horseshoe vortex together with two tip vortices (or free vortices). It is more accurate to describe the shape of the tip vortices taking into account their contraction to the symmetry plane (Fig. 2.2). But for two pairs of wings (forward and rear), the real vortex picture becomes even more comprehensive because of interaction of the vortices, i.e., mainly the repulsion of unidirectional vortices on the same side from the symmetry plane (Fig. 2.3). A

2.1

Physics of the Airflow Around a Tandem Wing Aircraft

V12

1

2

1

27

V12

2

R12 1

2

R12 1

2

Fig. 2.3 Repulsion of unidirectional vortices and attraction of oppositely directed vortices: V12, velocity induced by vortex 1; R12, force applied to vortex 2 by vortex 1

Fig. 2.4 Trajectories of tip vortices of a tandem wing UAV model (photomontage, top view, bottom-up position)

distance to oppositely directed vortices on the other side of the symmetry plane is an order of magnitude greater, so the interaction with them can be neglected. Real trajectories of the tip vortices in a tandem-scheme UAV were determined experimentally and are presented in Figs. 2.4 and 2.5 [2]. It should be noted that for this wind tunnel model due to structural deformations under the action of the counterweight, a decalage (difference in setting angles of the wings) was positive (Δφ = φ2 - φ1 = +4°), whereas in reality it is negative (Δφ = -1° to -4°). Therefore, in the experiment, the forward wings created less lift than they should, and the forward tip vortex was weaker and deviated more than for a real tandem wing aircraft. The position of the vortex of the forward wing relative to the rear wing determines distribution of circulation (lift) over the rear wing, i.e., its induced drag coefficient (Fig. 2.6). Reaching high angles of attack, the flow separation should begin on the forward wings earlier than on the rear ones (Fig. 2.7). Then the loss of lift of the forward wings provides longitudinal stability as the aircraft lowers its nose and returns to the flight range of angles of attack.

28

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Fig. 2.5 Trajectories of tip vortices of a tandem wing UAV model (photomontage, side view, bottom-up position) elliptic distribution of circulation (minimum drag) free (tip) vortex

factual circulation distribution for a rectangular wing with no twist

elliptic distribution

circulation on a rear wing with wing-wing interference

circulation for a rectangular wing with no twist (no wing-wing interference)

Forward wing Rear wing

Fig. 2.6 Spanwise circulation distribution on the forward (left) and rear (right) wings. Position of the tip vortex symbolically coincides with a tip cross section of the forward wing

Fig. 2.7 Flow separation only on the forward wings of the UAV (α = 16°; data from Ref. [3])

2.2

Analytical Methods

In Chap. 1, we saw that tandem scheme is mostly used for low subsonic Mach numbers, with rectangular or tapered wings and small or zero sweep angles. These cases are of the main interest for presented analytical methods.

2.2

Analytical Methods

29

This section is devoted to a direct analytical method, which is based on the calculation of definite integrals by simple numerical methods (at the level of Mathcad, it allows you to calculate the flow downwash/upwash in seconds and more accurately than classical methods). Much of the formulas mentioned below were derived about a century ago, but classical methods for determining the flow wash on the rear wing and the coefficients of lift and drag are based on significant simplifications of the physical picture and mathematical operations and are described in detail in [4]. Note only that the classical calculation is based on Prandtl theory, which includes the hypothesis of a horseshoe vortex and the flow wash approximate integration in the general form. So determination of the flow wash on the forward and rear wings can be done by expression: C L2 c2 pffiffiffiffiffi C c kV , ε21 = k 21 pL1ffiffiffiffiffi 1 , ð2:1Þ 4π b1 b 4π kV 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2x þ b2 þ h2 þ lx lx l2x þ b2 þ h2 - l2x þ c2 þ h2 , - 2 де k12 = ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 l þ h 2 x lx þ c þ h þ lx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l x 2 þ b2 þ h2 þ l x l k21 = ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l x 2 þ b2 þ h2 - l x 2 þ c 2 þ h2 , þ 2 x 2 lx 2 þ c2 þ h2 þ lx lx þ h ε12 = k 12

b is average wingspan (b = (b1 + b2)/2). с is wing exceedance (c = (b1 - b2)/2). lx and h are gaps between the wings in flow direction and perpendicularly to it, respectively; they depend on the angle of attack. Total induced drag can be calculated as: 1 Di = πq

 2   2  L1 L L1 L2 þ 2σ þ 22 , b1 b2 b21 b2

where σ is Prandtl coefficient (or interference factor) that equals: σ=

1 b 2 þ h2 : ln 8 c 2 þ h2

ð2:2Þ

Refinement of the classical theory is given in [5, p.410] where a solution of the differential-integral equation by a Fourier series expansion is performed taking into account the 39th order of terms of series, not the first (classical Prandtl theory). However, the method still has a singularity at zero vertical gap between wings. In addition, it is based on the theory of a horseshoe vortex which is not exactly accurate as evidenced by the visual tests.

30

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Analytical determination of pitching moment was performed in [6, 7]. The only problem is the exact definition of the flow downwash/upwash. This work presents a more accurate analytical definition of the flow wash which avoids the approximate calculation of integrals in general and takes into account the contraction (shrinking) of tip vortices to the symmetry plane.

2.2.1

Flow Deceleration

Determination of flow deceleration is reasonable using experimental data [8, p.119] (Fig. 2.8). In the graph for M = 0.25, it can be seen that if the rear wings are located below the forward, the flow deceleration (as well as reducing the Reynolds number) on the rear wings can be neglected. This aircraft configuration is the most appropriate and simple to calculate, if we do not take into account other factors, such as ground effect. The biggest flow retardation (minimum coefficient of airflow deceleration kV) is demonstrated at z/c1 ≈ 0.3; then, it equals kV = q2/q1 ≈ 0.93. Given the fact that as a rule the wingspan of the rear wing is larger and only part of it gets into the aerodynamic trace of the forward wings, we can approximately write: kV = k V1

Fig. 2.8 Flow retardation (1-kV) as a decimal fraction for various Mach numbers at lift coefficients up to 0.35

l1 l -l l þ 1 2 1 = 1 - 1 ð1- kV1 Þ: l2 l2 l2

2.2

Analytical Methods

2.2.2

31

Flow Turbulization

The effect of flow turbulence on rear wings is difficult to accurately calculate analytically. In general, many studies have been conducted on the effect of the turbulence intensity on airfoils’ aerodynamic characteristics [9], but even with a given geometry, the turbulence intensity behind the forward wings is uneven and depends on angle of attack. However, with rationally selected parameters, i.e., with a vertical spacing of the wings (e.g., high wing + low wing), the rear wings are not in the area of increased turbulence, so this factor can be neglected. If the rear wings in a certain flight regime appear in the aerodynamic trace of the forward, then the aerodynamic characteristics of the rear airfoil should be calculated as with a fully turbulent boundary layer. The corresponding increase in the drag coefficient (if the span of the forward wings is smaller) will be:   ΔC d2 = Cd2 - C d2 b1 =b2 or regarding the total area of the forward and rear wings:     ΔCd2 = C d2 - Cd2 b1 =b2  S2 =ðS1 þ S2 Þ = Cd2 - C d2 b1 c2 =ðS1 þ S2 Þ, where from empirical formulas [10, p.132]: Cd2 = 1:85 C f ηc ;   - 0:183 - 0:09xт ; Cf = ð0:0046 - 0:00267xт Þ 10 - 6 Re ηc = 1:1 þ ð0:0286 - 0:0238xт Þt; During the calculation of C d2 in both formulas, xт = 0, so the whole boundary layer is turbulent. The influence of the aerodynamic trace on the lift coefficient depends on the shape of a particular airfoil but still shows itself to a lesser extent than the effect on the drag coefficient. Boundaries of the aerodynamic trace from the forward wings can be estimated on the same graph as the flow retardation (see Fig. 2.8). Numerical methods were used to assess the effect of increased flow turbulence on the aerodynamic characteristics of the forward and rear wings for two-dimensional or the simplest three-dimensional (rectangular wing compartments) cases [11]. Different effect of the initial turbulence intensity on the aerodynamic characteristics of the forward and rear airfoils was revealed. The use of different turbulence models leads to different quantitative results. In general cases, k-ω turbulence models should be preferred, as they are focused on solving small-scale turbulence and have been successful in modeling near-wall flows at significant pressure gradients [12, p.21]. Although later significant contradictions were found between numerical and

32

2 Determination of Tandem Wing Aircraft Aerodynamic Characteristics

experimental methods in determining wing interference, Ansys correctly identifies some patterns. Thus, the turbulence intensity slightly affects the lift and pitch moment in the area of their linear dependence on angle of attack which is consistent with the previous experimental data [9].

2.2.3

Downwash and Upwash

The main aspect of wing-wing interference is a change in actual angles of attack of the rear wing, and for any real tandem-scheme aircraft, it must always be considered. Hereafter, we determine only the vertical component of the flow wash Vz as the most important, neglecting longitudinal Vx and lateral Vy component. To more accurately determine the flow wash and induced drag of configuration, it is necessary to rely on experimental data under the conditions of similarity of vortex systems [13]. Due to the fact that tip vortices of forward and rear wings are repelled, their position is slightly different from the analytical determination for an isolated wing. As with turbulence, where the vertical spacing of the wings is recommended, for a layout with rationally chosen geometric parameters, the vortices do not approach each other, and their interaction has little effect on the integrated aerodynamic characteristics. Analytically downwash/upwash can be determined as following: (a) According to approximate Horner formula [14, p.43], an average true angle of attack of rear wings is less than geometrical AoA by a value: ε2 = ε21 þ ε22 = - 1:6

CL1 C L2 , πe1 AReff1 πe2 AReff2

ð2:3Þ

where the first term represents downwash from the forward wings (mutual induction) and the second term stands for self-induction of the rear wings. The first summand of this expression is essentially the averaging of Munk empirical formulas [4, p.55] for the downwash on the tail of monoplane and biplane, respectively: ε = - 1:5

CL CL and ε = - 1:8 πeAReff πeAReff

Note that the expression does not take into account even the ratio of the wingspans or the intervals between the wings along ox and oz that do affect the value of the downwash. (b) A more accurate calculation is performed by determining the velocities from the attached and tip vortices of the forward wings at all sections of the rear wings according to Biot-Savart formula. CFD methods (Fig. 2.9) and, what is more

2.2

Analytical Methods

33

Fig. 2.9 Tip vortex of a rectangular wing visualized in XFLR5 software

important, wind tunnel tests [15, p.51] showed that the tip vortices of the wings are oriented in the vertical plane along the external flow velocity. Meanwhile, in the horizontal plane, these vortices are tightened at a certain distance to the symmetry plane. In fact, vortices in the vertical plane can deviate from a straight path significantly. Due to the repulsion of vortices, there is a curvature of the forward vortices (see Fig. 2.5), which due to the smaller setting angle of the forward wings during the experiments was much weaker than the rear (usually in the tandem-scheme rear vortex is weaker and must bend more). Position of Tip Vortices In general case, the distance between tip vortices is [8, p.102]: 1 liso = Г0

Z0:5b Г ðyÞ dy: - 0:5b

For a trapezoidal isolated wing with a taper ratio (TR), the distance between the vortices in the Trefftz plane can be presented approximately [8, p.103]: liso = bð0:64 þ 0:25TRÞ: That for a rectangular wing gives the value of liso/b = 0.89 independent of aspect ratio. From the theoretical standpoint, the aspect ratio has some effect on the position of free vortices (Fig. 2.10) [16]. From integration of this graph, we obtain the ratios of 0.866, 0.890, 0.905, and 0.912 for AR = 6.28, 9.42, 12.57, and 15.71, respectively. So we have pretty good agreement between data from two independent sources as the first one considers general aviation aircraft with aspect ratios around 8–10. Only for much higher ARs (typical for gliders and solar planes) it is reasonable to consider its effect on free vortex position. Taking into account fuselage effect [17, p.412], we obtain:

34

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Fig. 2.10 Distribution of circulation along straight wings of different aspect ratios

Table 2.1 Correction factor kd dw/b1 kd

0 1.0

0.1 0.980

0.2 0.970

0.3 0.968

0.4 0.968



0

l = b1

0.5 0.970

0.6 0.972

0.7 0.980

0.8 0.985

 0:25 k þ dw , 0:64 þ TR1 d

0.9 0.995

1.0 1.0

ð2:4Þ

where dw is a fuselage diameter at a wing-fuselage conjunction and kd is a correction factor determined according to Table 2.1: Velocities Induced by Tip Vortices In an arbitrary point with coordinates (x, y, z) regarding the center of the attached vortex (the middle of the forward wings – see Fig. 2.1) for low Mach number, the induced velocity is [4, p.44]: 1

0 V Z ð yÞ = -

-

-



0

Г1 ð0:5l - yÞ 0

2

4π ð0:5l - yÞ þ

z2

x C

B @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 x2 þ z2 þ ð0:5l0 - yÞ 0 1

Г1 ð0:5l0 þ yÞ 0

2

4π ð0:5l þ yÞ þ 0

z2

x C

B @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 0 2 2 x þ z þ ð0:5l þ yÞ 0

1

0

Г1 x 0:5l þ y 0:5l - y B C @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 4πðx2 þ z2 Þ 2 2 x2 þ z2 þ ð0:5l0 - yÞ x2 þ z2 þ ð0:5l0 þ yÞ

where Г1 = 0:5C L1 V 1 bS11 is forward wings’ vortex circulation for rectangular wings [1, p.277] and for tapered wings Г1 = C L1 bS11 V 1 1:285þ10:5TR1 [8, p.103].

2.2

Analytical Methods

35

Here, the first two terms correspond to the free vortices and the last to the attached vortex of the forward wings. Substituting the intensity of the vortex Г1 and given the fact that the flow wash is always measured in a few degrees, i.e.: ε21 = arctan

VZ VZ V ≈ = pffiffiffiffiffiZ V2 V2 kV V 1

(where V2 is an average flow velocity on the rear wings), after simple transformations, we obtain the final formula: 2 ε21 ðyÞ = -

þ

þ

0

1

0

0:5l - y CL1 S1 6 x B C pffiffiffiffiffi 4 @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 0 2 2 8π kV b1 ð0:5l - yÞ þ z 0 2 2 x þ z þ ð0:5l - yÞ 1 0

x 0:5l0 þ y C B × @1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 0 2 2 ð0:5l þ yÞ þ z x2 þ z2 þ ð0:5l0 þ yÞ 0 0

x2

13

0

0:5l þ y 0:5l - y x B C7 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5: 2 þz 2 2 0 0 2 2 2 2 x þ z þ ð0:5l - yÞ x þ z þ ð0:5l þ yÞ ð2:5Þ

The analysis of these formulas in general is quite complicated, so mainly only special cases are considered: points that are very distant from the wings or those that lay in a symmetry plane of the aircraft [17]. For a conventional aircraft scheme, it is practically sufficient to determine the flow downwash in the symmetry plane of the horizontal tail as a shift to the left leads to an increase in the downwash from the left vortex and a decrease in the downwash from the right, so the total downwash varies slightly. But for the rear wings in the tandem scheme, it is necessary to determine the wash at each point on the wingspan, as the wash even varies from downwash in root sections to upwash at wingtips (see Fig. 2.2). To determine the lift coefficient, it is enough to determine the average wash on the wing. For the calculation of the induced drag coefficient (and lateral moments at sideslip), it is necessary to take into account the change in the spanwise lift distribution on the rear wings. If sweepback and dihedral angles of the wings are zero, taking into account the flow deceleration behind the forward wings, we obtain the average values of the induced velocity and the flow wash on the rear wings due to interference with the forward:

36

VZ

2

1 avg = b2

Zb2 =2 - b2 =2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

V Z avg V Z avg 1 V Z ðyÞdy, ε21 = arctan ≈ = V2 V2 b2

Zb2 =2 ε21 ðyÞdy: - b2 =2

In general, the indefinite integral can be calculated only approximately, on which the classical methods are based. But the calculation of a definite integral using publicly available software is performed in a fraction of a second with a given accuracy of E-5. The calculation results in Mathcad with Romberg method and adaptive method differ only in the fourth decimal place, but Excel is also good enough. Viscous Core of the Vortex The drawback of Biot-Savart formula is that it has a singularity point at the vortex axis where the induced velocity will be infinite. In real gas, any vortex has a viscous core in which the velocity increases with distance from the vortex axis [1, p.67]. If the wing and the axis of the vortex are closer to each other than the radius of the vortex, then the above approximation will not be valid. To get around this singularity point, there are two approaches: either to assume that the radius is small compared to the characteristic dimensions (height of the box) [18] or, more accurately, to represent the viscous core of radius r0 with linear dependence of the tangential velocity – the so-called Rankine vortex model (Fig. 2.11).

Fig. 2.11 Tangential velocities around a vortex. (Based on data from Ref. [19])

2.2

Analytical Methods

37

According to different studies, including wind tunnel experiment [20] and flight test with C-130 and Boeing-757 [21], the radius of the tip vortex is below 1.25–1.4% of the span. In the current experimental studies (Sect. 2.3) with indirect method for wing lift coefficients from 0.38 to 0.93, the radius was determined between 0.7% and 1.3%. So, if in exploitation range of AoA the distance between the vortices and rear wings is higher than r0 = 0.014b1 (and this condition is recommended for aerodynamic design of a tandem wing aircraft), it is possible to use all the formulas mentioned above. Otherwise, maximum-induced velocity should be determined (e.g., at y = 0.5l′ + r0, z = 0) and then the induced velocity to be interpolated between zero and this maximum in a range of y = 0.5l′ to y = 0.5l′+r0. Maximum velocity can be also calculated from the expression: V max =

Г1 , 2πr01

for different lift and circulation of the forward wings as we already know the radius of the viscous core. On the other hand, any real wing is not flat, and the vortex core size has an order of magnitude as the wing thickness (e.g., in the experiment from Sect. 2.4: 15 mm and 13 mm, respectively). The real vortex does not go through the wing; it will extend along on the upper or lower surface of the wing. So using Rankine vortex model may not provide more accurate results. Meanwhile, in this experimental study, the vortex cores’ sizes do not show any noticeable change with the distance from the wing. It agrees with a PIV research performed in another wind tunnel [22]. So in the scales of an aircraft, there is no need to apply more complicated models such as Lamb-Oseen vortex that decays due to viscosity. Geometrical Considerations The picture of the flow is complicated if the wings have dihedral angles (V-shape) which is really used in practice. By the classical method, dihedral angles can be considered only as a change in the height of the wing box Δh and the wingspan. However, this is not entirely correct, as intensities of the tip vortices with and without wing dihedral angles are different, and the displacement of the attached vortex is not taken into account. In the presence of the dihedral angle of the forward wings θ1, free vortices rise up by Δh = 0.5 l′ sin θ1 and the center of every attached vortex by Δz = 0.25 l′ sin θ1 (Fig. 2.12). In the presence of the dihedral angle of the rear wings θ2, vertical gap h0 changes, and when integrating the downwash across the rear wings’ span, we should keep in mind that not only the y-coordinate but also z-coordinate changes (Fig. 2.13). The case of the dihedral angles of different signs for forward and rear wings is presented in Fig. 2.14. To determine the average wash at zero angle of attack, the following should be inserted in Eq. 2.5:

38

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Fig. 2.12 Geometrical effect of forward wings’ dihedral angle

Fig. 2.13 Geometrical effect of rear wings’ dihedral angle

Fig. 2.14 Geometrical effect of both forward and rear wings’ dihedral angles

x = lx0 , z = h0 þ jyj tan θ2 - 0:5l0 tan θ1 where lx0 and h0 are distances between quarter-chords of the forward and rear wings along and perpendicular to the fuselage waterline (in body-fixed frame), and integrate in a range from -0.5 b2 cos θ2 to 0.5 b2 cos θ2:

2.2

Analytical Methods

39

ε21 ≈

1 b2 cos θ2

0:5bZ2 cos θ2

ε21 ðyÞdy

ð2:6Þ

- 0:5b2 cos θ2

Here, we consider b2 a wingspan at θ2 = 0. For a nonzero angle of attack from geometric considerations, the stagger and the height between the wings in the wind frame will be written as: x = lx = lx0 cos α þ ðz0 þ jyj tan θ2 - 0:5l0 tan θ1 Þ sin α, z = - lx0 sin α þ ðz0 þ jyj tan θ2 - 0:5l0 tan θ1 Þ cos α, where y0 is a vertical gap between the root sections of the wings at α = 0 (if the rear wings are higher then y0 > 0). ρV 2 ρV 2s In the case of sweepback angles mg = CL cruise cruise 2 S and mg = C L max 2 S (in the future, we consider them small) and wings’ bending under aerodynamic loads: x = ðlx0- 0:5l0 tan Λ1 þ jyj tan Λ2 Þ cos α þ ðz0 þ jyj tan θ2 - 0:5l0 tan θ1 - h1 þ h2 Þ sin α,

ð2:7Þ

z = - ðlx0- 0:5l0 tan Λ1 þ jyj tan Λ2 Þ sin α þ ðz0 þ jyj tan θ2 - 0:5l0 tan θ1 - h1 þ h2 Þ cos α, where h1, h2 are bending values of the forward and rear wings, respectively, in body-fixed frame; they are also functions of lift and coordinate y. Knowing only maximal deflection of the wing at one angle of attack, it is good enough to interpolate the bending from root to tip sections as follows: h2 ð yÞ =

y - D=2 Δh max 2 : l1 =2 - D=2

So, using Eqs. 2.4, 2.5, 2.6, and 2.7, we obtain the average wash on the rear wings from the forward wings at arbitrary angle of attack. In the next subsections, the method will be extended in case of sideslip angle. Often, to determine the aerodynamic characteristics of an aircraft, the wash angle of the flow is represented as: ε21 = ε0 þ εα α: Approximately, we can take a derivative of Horner formula (Eq. 2.3) and obtain a1 εα = - 1:6 πe1 AR . It is more accurate to take the derivative of Eq. 2.6 and get eff1

40

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Table 2.2 Values of (-εα) at lx0/b1 = 0.3 h0/b1 -0.01 -0.03 -0.05 -0.07 -0.10

b2/b1 0.9 0.300 0.274 0.256 0.239 0.217

1.0 0.274 0.257 0.242 0.227 0.205

1.05 0.270 0.249 0.232 0.218 0.198

1.1 0.248 0.234 0.220 0.207 0.189

1.2 0.209 0.203 0.194 0.186 0.172

1.3 0.182 0.178 0.172 0.165 0.155

1.5 0.145 0.143 0.139 0.135 0.127

1.1 0.186 0.172 0.162 0.153 0.142

1.2 0.161 0.153 0.147 0.140 0.131

1.3 0.138 0.134 0.129 0.125 0.118

1.5 0.107 0.105 0.103 0.101 0.097

Table 2.3 Values of (-εα) at lx0/b1 = 0.6 h0/b1 -0.01 -0.03 -0.05 -0.07 -0.10

b2/b1 0.9 0.210 0.193 0.182 0.173 0.161

1.0 0.173 0.174 0.169 0.162 0.152

1.05 0.190 0.175 0.166 0.158 0.147

εα = dε21 =dα and ε0 = - εα α01 :

ð2:8Þ

Quantitative Evaluation of Downwash To evaluate the order of these parameters, it is convenient to use tabular data. Consider the case of θ1 = θ2 = Λ1 = Λ2 = 0°, CL 1 = 0.5 (α0 = - 3°, α = 2.5°), and TR1 = 1, d = c1 = 0.1b1, and write down the value of the coefficient εα versus dimensionless wing stagger (lx0/b1), vertical gap (h0/b1), and wingspan ratio (b2/b1) (Tables 2.2 and 2.3). At intermediate values of the parameters, the flow downwash can be interpolated. Note that when the angle of attack changes, the vertical interval and the values of εα also change slightly (decrease when the wings are moved from each other in the vertical plane). Nevertheless, according to Horner formula, all the values given in the tables would be the same (εα ≈ - 0.33), which is not true. For example, increasing the wingspan of the rear wings from 0.9b1 to 1.5b1 results in a twice smaller downwash (mainly because of the upwash in the rear wings’ tip sections). Comparisons of the calculation of the average flow wash on the rear wings for a typical geometry of tandem wings, considered in Sect. 2.4, according to the proposed method, as well as the classical method implemented by Betz (Eq. 2.1) [4 p.64], and Horner formula (Eq. 2.3) are shown in Table 2.4. Obviously, for this configuration, Horner formula exaggerates the flow wash in absolute values relative to other two methods, for example, at the angle of attack α = 5.8° half as much. The flow wash calculated by the classical method (Betz) coincides with the results of the proposed method with high accuracy (±0.06°). That is, to determine the flow wash on the rear wings, it is sufficient to use the classical

2.2 Analytical Methods

41

Table 2.4 The flow wash values on the rear wings (θ1 = θ2 = 0°) α, ° 0 2 3.9 5.8 7.8 9.8 11.7 13.7 15.7

ε21 (direct method), ° 0.41 -0.05 -0.45 -0.81 -1.14 -1.42 -1.63 -1.81 -1.91

CL1 -0.145 0.019 0.176 0.331 0.491 0.645 0.780 0.906 1.000

ε21 (Betz), ° 0.40 -0.05 -0.46 -0.83 -1.18 -1.47 -1.69 -1.86 -1.95

ε21 (Horner), ° 0.5 -0.1 -0.6 -1.2 -1.8 -2.3 -2.8 -3.3 -3.6

approximate method. However, as will be shown below, the direct method also allows to take into account the moderate values of the wings’ dihedral angles and sweepback angles, as well as sideslip angles of the aircraft to determine the lateral aerodynamic characteristics (rolling moment and yawing moment).

2.2.4

Lift Coefficient

As a tandem-scheme aircraft has no stabilizer, then lift is generated only by wings and fuselage: C L = CL1

S1 S þ C L2 2 þ C L S S

fus

SМ , S

ð2:9Þ

where S = S1 + S2 is a total planform area of the wingsCL fus is lift coefficient of the fuselageSМ is a cross-section area of the fuselage To build the graph of lift coefficient versus angle of attack CL(α) for wings with practically sufficient accuracy, it is enough to determine zero-lift AoA t = t=c, lift slope a (derivative of lift coefficient with AoA), stalling AoA αs, and maximum lift coefficient CL max. For forward wings of rectangular planform and with one airfoil across the span without twist and dihedral angles, it can be written [1, p.290; 17, p.237]: CL

1

= a01

α - α01 þ φ1 α - α01 þ φ1 = a01 01 1 þ π e1aAR 1 þ ARa01eff 1 0:375 eff 1

where φ1 is wings’ setting angle (often we count the angles of attack from the MAC of the forward wings, so φ1 = 0) α01 is zero-lift angle of attack for the forward wings (can be taken equal to this value for the isolated airfoil)

42

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

a01is lift slope for the airfoil of the forward wings (always higher than for the whole wings) AReff 1 =

=

AR 1þSuf =S1

b21 =S1 1þSuf =S1

b21 S1 þSuf

=

is effective aspect ratio of the forward wings.

If the airfoil aerodynamic characteristics are set as a table: CL

1

= Cl

1

α - α01 = Cl α - α01 þ π e CARl1eff1

1

α - α01 Cl 1 α - α01 þ AR 0:375 eff 1

In the presence of small dihedral angles and/or small sweep angles, the formulas take the form: CL CL

1

1

= a01

= Cl

1

α - α01 þ φ1 a01 1 cos θ1 cos Λ1 þ π e1 AReff 1

ð2:10Þ

α - α01 l1 þ π e1CAR eff1

ð2:11Þ

α - α01 cos θ1 cos Λ1

On the linear part of the graph CL2(α) for the rear wings with no dihedral or sweep angles:  CL2 = a2 ðα - α02 þ ε þ ΔφÞ = a2

 CL2 α - α02 þ ε21 þ Δφ , πe2 AReff2

where α02 is zero-lift angle of attack for rear wings (isolated) and its airfoil. Δφ is the angle between the chord of the rear wings and the direction relative to which the angles of attack are measured; the difference between the angles of the rear and forward wings (decalage Δφ = φ2 - φ1), if the angles are measured from the chord of the forward wings. Given the retardation of the flow, the dihedral angles and the sweep angles of the wings, we write:  C L2 = kV a2 α - α02 -

 C L2 þ ε21 þ Δφ cos θ2 cos Λ2 , πe2 AReff2

where θ2 is the dihedral angle of the rear wings Λ2 is their sweepback angle (from the 25% chord line), negative for sweepforward wings Solving the last equation in regard to CL2, CL

2

=

a02 a02 1 kV cos θ2 cos Λ2 þ π e2 AReff 2

ðα - α02 þ ε21 þ ΔφÞ:

ð2:12Þ

In the case of a tabular setting of the lift coefficient of the wing airfoil Cl 2, taking into account that a02 = Cl 2/(α - α02), we can obtain:

2.2

Analytical Methods

43

CL

2

= C l2

α - α02 þ ε21 þ Δφ α - α02 Cl2 kV cos θ2 cos Λ2 þ π e2 AReff 2

ð2:13Þ

Here, all angles must be represented in radians (!). In Eqs. 2.12 and 2.13, we can substitute the previously defined coefficients εα and ε0 from Eq. 2.8 and obtain: C L2 =

a02 a02 1 kV cos θ2 cos Λ2 þ πe2 AReff2 C L2 = Cl2

ðð1 þ εα Þα - α02 - εα α01 þ ΔφÞ

ð1 þ εα Þα - α02 - εα α01 þ Δφ α - α02 C l2 k V cos θ2 cos Λ2 þ πe2 AReff2

ð2:14Þ ð2:15Þ

From Eqs. 2.10 and 2.14, it is easy to find the derivatives a1 and a2: a1 =

1 1 a01 cos θ1 cos Λ1

þ

1 πe1 AReff1

and a2 =

1 þ εα 1 þ πe2 AR eff2

1 kV a02 cos θ2 cos Λ2

ð2:16Þ

For a system of wings and for a whole aircraft, we can get: S1 S þ a2 2 , S S S S S a = a1 1 þ a2 2 þ afus М S S S a = a1

ð2:17Þ ð2:18Þ

If the wings are located close and the influence of the rear wings on the forward cannot be neglected, then the lift coefficients are determined from a system of equations: S2 S = C L1 iso þ a1 k12 C L2 2 ; S1 S1 S1 S CL2 = C L2 iso þ a2 ε21 = C L2 iso þ a2 k21 C L1 1 , S2 S2 CL1 = C L1 iso þ a1 ε12

where k12 and k21 are constants, as in all models the downwash is proportional to the lift coefficient of the corresponding wings. In the first equation, S1 is a reference area that is why we have S2/S1 next to the rear wings’ lift coefficient. If we neglect the values of the second order of smallness, then: k12 =

∂ε12 ∂ε21 ; k21 = : ∂C L2 ∂C L1

where ε21 can be found from Eqs. 2.5 and 2.6, and ε12 in the same way:

44

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

2 ε12 ¼ -

þ

-

0

1

0

0:5l2 - y C L2 S2 6 x B C 4 @1 - qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 8πb2 ð0:5l2 0 - yÞ2 þ z2 2 0 2 2 x þ z þ ð0:5l2 - yÞ 0 1 x 0:5l2 0 þ y B C × × @1 - qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 0 2 2 0 ð0:5l2 þ yÞ þ z 2 2 x þ z þ ð0:5l2 þ yÞ 0 0

0

13

0:5l2 þ y 0:5l2 - y x B C @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5, x2 þ z 2 2 2 0 0 2 2 2 2 x þ z þ ð0:5l2 - yÞ x þ z þ ð0:5l2 þ yÞ 1 ε12 ≈ b1 cos θ1

0:5bZ1 cos θ1

ε12 ðyÞdy - 0:5b1 cos θ1

Then: C L1 =

cL1 iso þ a1 k12 C L2 iso S2 =S1 C þ a2 k21 C L1 iso S1 =S2 , CL2 = L2 iso 1 - a1 a2 k12 k21 1 - a1 a2 k12 k21

It is also possible to determine a zero-lift angle of attack for rear wings in the presence of forward wings α02 . If the wings have no aerodynamic and geometric twist, then for the isolated rear wings, the zero-lift angle of attack is the same as for its airfoil (α02 ≈ - 2f 2 rad, wheref 2 is a dimensionless camber of the airfoil). If the wings have aerodynamic or geometric twist, then this angle is determined about mean aerodynamic chord. Zero-lift angle of attack of rear wings α02 under the wash from forward wings equals: α02 - α02 þ ε21 þ Δφ = 0: With the roughly approximated ε21 from Eq. 2.1, after simple transformations, we get: α02 =

ðπe1 AReff1 þ a01 Þðα02 - ΔφÞ þ 1:6a01 α01 : πe1 AReff1 - 0:6a01

It is more precise to find α02 graphically, building the dependence of CL2(α). If the wing consists of several sections (like a telescopic wing), then evaluation of α0 is possible, assuming that lift of each section is proportional to its area (by formula L2 = a2(α - α02 + ε21 + Δφ)q2S2, while factors a2 and q2 do not vary significantly for different sections, and downwash angle ε22 is neglecting, as for the whole rear wings the lift equals zero:

2.2

Analytical Methods

α02 = α02

45

O S2O =S2

þ

n X

ðα02 i - Δφi ÞS2i =S2 ,

i=1

where Δφi is a setting angle for i section about the reference section. The proposed method of determining the lift coefficients of both wings is developed for the area of linear dependence on the angle of attack and gives an error between the stalling angle of attack of the airfoil and the stalling angle of attack of the wing, for which the airfoil’s coefficient reaches its maximum and the wings’ lift continues to grow. In this area, it is necessary to estimate the stalling angle of attack of the wing on the basis of previous experience and interpolate the calculated and estimated values. For example, for practical purpose, we can assume that for a rectangular wing with aspect ratio of AR = 6 [4, p.39]: CL

2.2.5

max

= Cl

max =1:07:

Drag Coefficient

The main difficulty in determining drag coefficient of a tandem wing aircraft is the calculation of the induced component taking into account the mutual influence of the forward and rear wings. The induced drag of true tandem wings (not canard or biplane) is dominated by the downwash effect generated by the forward wings on the rear wings [23]. It is not correct for canard case when the forward lifting surface is few times smaller than the rear one. Some experiments show that for proper configurations, the aerodynamic interference between the wing and canard increases the CL max of the canard and does not degrade the CL max of the wing [24]. The direct method of calculation is to extend the formula derived for the monoplane [1, p.284] and to find circulation in every section of the rear wings: 0 B ГðyÞ ¼ 0:5aðyÞcðyÞV 0 @αgeom ðyÞ þ 1

þ

1 4πV 0

Zb=2 - b=2

dГðy1 Þ=dy1 dy1 y1 - y

V z attached ðyÞ þ V z tip ðyÞ A pffiffiffiffiffi kV V 0

where the last summand represents flow wash from the forward wings. Then, lift and drag, respectively, are

46

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Fig. 2.15 Theoretical induced drag for tandem wings (equal span) and monoplane with btail/ b = 0.4, h/b = 0.2. (Based on data from Ref. [25])

Zb=2 L = ρV 0 Zb=2 Di = - ρ - b=2

2

ГðyÞdy - b=2

61 Г ð yÞ 4 4π

Zb=2 - b=2

3 dГðy1 Þ=dy1 7 dy15 dy y1 - y

The exact solution of these equations is difficult even for a monoplane (i.e., isolated pair of wings without a flow wash from the forward wings), so, as a rule, an approximate solution is obtained by decomposing into a Fourier series. Mathematical transformations of such expressions are well known [4], although they are quite lengthy. A practical method for determining the induced drag is based on the classical Prandtl theory, according to which [25]: Di =

1 πq

 2   2  L2 L1 L1 L2 þ þ 2σ : 2 b1 b2 b1 b22

ð2:19Þ

Benefits of tandem wings over conventional layout are shown in Fig. 2.15. Total drag of a tandem wing aircraft consists of drags of the forward and rear wings (taking into account the interference between them and with the fuselage), the fuselage, and the vertical tail. The drag of the wing consists of profile component (sum of friction and form drags) and induced component. Induced drag is

2.2

Analytical Methods

47

determined theoretically and profile by methods of numerical aerodynamics (calculation code XFOIL has made a good showing). We assume that the wingspans are of the same order and the rear wings are NOT directly in the aerodynamic trail (slipstream) of the forward, i.e., the rear are NOT above the forward in a range of 20–45% of MAC of the forward wing (see Fig. 2.8) [8, p.119]. Induced drag consists of self-induction and mutual induction [4, p.68], which is convenient to extract from the wing drag. Then the drag coefficient of a tandemscheme aircraft can be written as: CD = CD

1

S1 S þ CD 2 2 þ ΔC D S S

ind21

S þ CD fus М þ C D S

VT

SVT S



ð2:20Þ

where ΔCD ind21 represents mutually induced drag shown below:   CD fus SSМ þ CD VT SSVT is drag coefficient of the fuselage and the vertical tail, referred to area of the equivalent wings (total area of forward and rear wings). The drag coefficient of the forward wing CD

will be written in the form:

C2L 1 , πe1 AReff1   Suf 1 , = C 1k 01 d 1 int 1 S1

CD CD

1

1

= CD

01

þ

ð2:21Þ ð2:22Þ

where Suf 1 is a planform area of the forward wings occupied by the fuselage. For high wings, the interference factor is about kint ≈ 1.0, for low wings (that rear wings more often are) kint = 0.25...0.6 [26, p.376]. So, for the rear wings: CD CD

2

= CD

 02 = k V C d

02

þ

C2L 2 , πe2 AReff2

 2 1- k int

 Suf 2 : 2 S2

ð2:23Þ ð2:24Þ

where Cd2 is rear wings’ airfoil drag (taking into account higher turbulence intensity after the forward wings, if needed). Coefficient of mutually induced drag is: ΔC D

ind 21

= 2σ

C L1 C L2 S1 S2 πðS1 þ S2 Þb1 b2

ð2:25Þ

where σ is Prandtl coefficient from Eq. 2.2: h = y by Eq. 2.5. As shown below, the calculation of the polars by Eqs. 2.20, 2.21, 2.22, 2.23, 2.24, and 2.25 leads to an overestimation of the drag compared to experimental data. In this case, Eqs. 2.20, 2.21, 2.22, 2.23, and 2.24 are not in doubt. The very concept of determining the induced drag through the Prandtl coefficient is also confirmed by

48

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Fig. 2.16 Induced thrust coefficient. (Butler correction to Prandtl coefficient)

H21

CL1,CL 2

V

CD ind

H’21, H’12

H’21, H’12

H21 V

CL1,CL2 CD ind

Fig. 2.17 Block diagram of the classical (left) and the proposed approaches to determine mutual induced drag in the tandem scheme

modern mathematical methods [5], but for typical aircraft configurations, the errors are 7–16% compared to the experiment [4, p.260]. The task is to define the value of Prandtl coefficient more precisely, as the magnitude of the flow wash differs according to the new method and to the classical theory. Mathematical refinement of Prandtl coefficient taking into account the 20 terms of the Fourier series was performed by G.F. Butler [5] and came to the reduction of the first term in Eq. 2.19, which must be multiplied by (1-ν). Here, ν was called an induced thrust coefficient and with an accuracy of 5% can be taken equal to the optimal value νopt (Fig. 2.16 For a canard aircraft, if b1 = 0.5b2, the coefficient reaches the largest values (νopt ≈ 0.3) and if z/b2 = 0.05, L1 = 0.25, and L2 = 0.75, the correction will be 6% of the total induced drag calculated according to Prandtl theory. For a typical tandem configuration b1 = 0.9b2, CL1 = 0.6, CL2 = 0.4, and even at a small vertical interval, the induced thrust coefficient is low: νopt ≈ 0.05. Thus, for a typical tandem configuration, Butler correction to the Prandtl coefficient is about 2% of the total induced drag. The correction is less than 1% of the total vehicle drag (including profile wing drag, fuselage, and vertical tail drag) and is within the accuracy of measurements in wind tunnel and in flight tests. A new inverse approach to the classical one is proposed. In the 1930s and 1940s, dimensionless coefficients ε′12 and ε′21 were determined for the wings’ geometrical parameters in tandem configuration [4, p.65, 83]; from that coefficients of the flow wash ε21 were calculated and then lift coefficient and Prandtl coefficient σ for drag. Since now we can more accurately determine the wash ε21, then we can calculate the coefficients ε′12 and ε′21 for this value and then Prandtl coefficient (Fig. 2.17).

2.2

Analytical Methods

49

Flow downwash on the rear wings from the forward wings can be found as: ε21 = ε021

C L1 c1 pffiffiffiffiffi 4π k V b2

) ε021 =

4πε21 b2 pffiffiffiffiffi kV C L1 c1

At the same time, Prandtl coefficient was derived as: σ=

ε012 þ ε021 8

If we neglect the influence of the rear wing on the forward, then ε012 = 0 and σ=

π ε21  b2 pffiffiffiffiffi kV 2 CL1  c1

ð2:26Þ

If the wing stagger is relatively small (lx0 < 3c1) and we cannot neglect the influence of the rear wings on the forward, then:   ε012 þ ε021 π ε12  b1 ε21  b2 pffiffiffiffiffi k þ σ= = V 8 2 C L2  c2 C L1  c1

ð2:27Þ

For a typical tandem aircraft, such as those mentioned in Chap. 1, the impact of the rear wings on the forward is small. The tip vortices of the rear wings reduce the true angle of attack on the forward wings, but the attached vortex of the rear wing, on the contrary, increases that angle (Fig. 2.18). Therefore, they largely compensate for each other’s influence, and within the accuracy of the analytical method, this feedback in the wing-wing interference can be neglected.

Fig. 2.18 Upwash and downwash from rear wings

50

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

Regarding drag coefficient, flexibility of the airframe of tandem configuration should be mentioned, as it affects the aerodynamic characteristics more significantly than for conventional scheme. For example, bending of the wings leads to a change of the vertical gap between the wings and so of the flow wash and Prandtl coefficient. As wind tunnel tests showed, even with a rigid forward wing and a flexible rear wing, increasing the angle of attack changes the height of the wing box. That is, for the upper location of the forward wings and the lower location of the rear, with increasing angle of attack, Prandtl coefficient decreases as the vortices are being removed. But if the upper wing is harder than the lower, then with increasing angle of attack, Prandtl coefficient decreases more slowly, and the mutually induced drag will be greater. Thus, the method was developed to determine the aerodynamic coefficients of lift and drag for the tandem scheme, taking into account three types of interference between the wings. The method is based on the hypothesis of horseshoe (U-shaped) vortices taking into account their contraction to the symmetry plane similar to a monoplane, with Biot-Savart formula for calculation of the induced velocities and limitation for the vortex core (if needed). It is necessary to take into account such features of the tandem scheme as free vortices of the forward wings, passing along the upper or lower surface of the rear wing. Real vortices in a viscous gas have a low pressure inside. Therefore, their passage minimally above/below the rear wing may cause a significant increase/ decrease in the lift of the rear wing, a change in the mutually induced drag, as well as a shift of the aircraft aerodynamic center back/forward. Thus, a slight difference in geometric parameters can lead to a significant difference in aerodynamic characteristics, in particular in the maximum lift drag of the aircraft, but more importantly such design can face problems with dynamic pitch stability. Therefore, the largest errors of the method will occur when the free vortices of the forward and rear wings are close and when the front vortex passes close to the surface of the rear wing. In the first case, the vortices repel and change shape, which to some extent changes the magnitude of the average flow wash on the rear wing and the distribution of the wash on the rear wing span (and, so, induced drag). In the second case, the low pressure inside the vortex increases the lift of the rear wing if the vortex passes close to its upper surface and decreases if close to the lower surface. Of course, modern CFD methods are more precise in determining the flow wash and are applicable for complex wing shapes (e.g., with a few sections of different dihedral angles and sweepback angles). On the other hand, they are more timeconsuming and also not free from arbitrary assumptions, e.g., in most numerical studies, the vortex core radii are much higher than were determined experimentally [21].

2.2

Analytical Methods

2.2.6

51

Pitching Moment Coefficient

Determination of longitudinal static stability is based on determining the lift coefficients on the linear dependence versus AoA according to Eqs. 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, and 2.15. An issue occurs for swept wings as it is problematic to determine the longitudinal moment arm for the lift distributed spanwise. In this section, the sweep angles are considered small, so you can use the average values of the flow downwash/upwash on the rear wings. Aircraft pitching moment about the center of gravity is (Fig. 2.19):   M = M 0 þ L xac- xcg , where M = Cm ρV2 ceq ðS1 þ S2 Þ is pitching moment. 2 M 0 = C m0 ρV2 ceq ðS1 þ S2 Þ is zero-lift pitching moment (at CL = 0). 2 2 2 L = C L1 ρV2 S1 þ C L2 ρV2 S2 = ðC L1 S1 þ CL2 S2 Þ ρV2 is lift. At that, wing-wing interference, dihedral angles, and sweep angles are taken into account in the lift coefficients CL1 and CL2. In common case, dimensionless pitching moment coefficient is: 2

  C m = C m0 - xac - xcg C L = Cm0 þ C CmL C L :

ð2:28Þ

Note that for usual decalage angle (negative), when aircraft’s CL = 0, wings’ lift coefficients are nonzero: CL1 > 0, CL2 < 0. C L1 S1 = - C L2 S2 : Then: CL

  - εα α01 - α02 þ Δφ S2 : = a ð α α ÞS = a α þ 1 1 01 1 2 1 þ εα

From here, we find the AoA of zero lift for the system of wings: cL1

Fig. 2.19 Pitching moment scheme

cm01

cL2 x1 G

cm02 x2

52

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics α

α0 =

02 - Δφ a1 α01 S1 þ a2 ε α01 þα S2 1þεα , a1 S1 þ a2 S2

where a1 and a2 are determined from Eq. 2.16. At that angle of attack, pitching moment coefficient equals: x S c 1 S1 c S þ a1 ðα0 - α01 Þ 1 1 þ C m02 2 2 ceq S ceq S ceq S   εα α01 þ α02 - Δφ x2 S2 - a2 α0 , ceq S 1 þ εα

C m0 = C m01

where x1 and x2 are distances from the center of gravity to quarter-chord lines of the forward and rear wings (or to the quarters of their mean aerodynamic chords if the wings are swept) as shown in Fig. 2.19; aerodynamic center of the wing can be within a narrow range of 20–25% of the chord depending on the airfoil [8, p.44]; so, it is possible to find the position of the AC more accurately by simple numerical methods (as well as values of Cm01, Cm02). Then for any angle of attack on the linear part of CL(α), a system of forward and rear wings has moment coefficient: c 1 S1 c S x S þ a1 ðα - α01 Þ 1 1 þ C m02 2 2 ceq S ceq S ceq S   α ε α01 þ α02 - Δφ x2 S2 - a2 α , ceq S 1 þ εα

C m = C m01

C αm = a1

x 1 S1 x S - a2 2 2 ceq S ceq S

ð2:29Þ ð2:30Þ

Dividing the Eq. 2.30 by Eq. 2.17, we obtain: CCmL =

x1 S1 x2 ∂Cm Cm a1 ceq S - a2 ceq = = a ∂C L a1 SS1 þ a2 SS2

S2 S

ð2:31Þ

If we measure the position of the center of gravity from a quarter of the forward wings’ MAC (xcg = x1 =ceq ), after simple transformations, we will receive: xac = xcg - C CmL = -

lx0 a2 S 2 : a1 S1 þ a2 S2 ceq

ð2:32Þ

Without taking into account the influence of the power plant, the aircraft will have angle-of-attack static stability on conditions that:

2.2

Analytical Methods

53

a1 xac1 S1 < a2 xac2 S2 :

ð2:33Þ

where we measure aerodynamic center coordinates from the center of gravity. The effect of a fuselage on the longitudinal moment is relatively insignificant, as in the tandem scheme the arms of the forward and rear wings are much larger than the arm of the fuselage. In addition, the moment created by the engine thrust is taken into account in the same way as for the conventional scheme (thrust multiplied by its arm, i.e., a distance from the thrust vector to the center of gravity). Knowing the aerodynamic center coordinate, we can choose the aircraft center of gravity position with a given coefficient of longitudinal static stability C CmL . Depending on the controls’ effectiveness, it can be in the range from -0.03 (extreme aft CG) to -0.30 (extreme forward CG) and even further when the elevators occupy the entire span of all wings (but balancing losses will be high). Nowadays, it is reasonable to use simple CFD method such as XFLR5 for determining longitudinal moment at low angles of attack (before flow separation occurs). It allows varying the dihedral angles and sweeping angles providing more accurate results than analytical method. For high angles of attack (when flow separation appears on some wing), simple CFD methods (vortex lattice or panel) are not applicable, and more sophisticated one, with Navier-Stokes equation solving, should be used. Nevertheless, to provide the longitudinal stability at high AoA, all we need is to get separation occurred on the forward wings before on the rear wings. The simplest way for the same airfoil on both wings is to have small negative decalage (1–4°). Then on the rear wing, the true angle of attack in average would be less by this decalage angle and the angle of downwash from the forward wings, and the rear wings will be protected from stalling.

2.2.7

Rolling Moment Coefficient

From previous studies, it is known that determination of the lateral static stability of the tandem wing or canard aircraft has a specificity compared to conventional scheme: for a nonzero sideslip angle, vortex zone from the forward lifting surface (a canard or a wing) creates additional rolling moment [27, p.59]. It should be noted that in open sources no methods have been found to assess the lateral stability of such aircraft taking into account interference of the lifting surfaces. Consider the forces that create the rolling moment of the tandem-scheme aircraft at arbitrary sideslip angle (Fig. 2.20). In general, there are forces along y-axis on zarms and forces along z-axis on y-arms. The rolling moment from a fuselage in most cases can be neglected, as this force is applied to the small arm (although, for S-shaped fuselage and no wings’ dihedral angles and no vertical tail, it is possible to take the effect into account; then it is being done in the same way as for conventional configuration [17, p.485]).

54

2

Determination of Tandem Wing Aircraft Aerodynamic Characteristics

'cL1 'cL2

'cyfin

y1

zfin c.g.

y2

z1 z2

'cy1

cl 0 → cl1  0

'ε 21right 0 → Cm1  0

'ε 21