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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Aerospace Series (PEP)
- Sprache: Englisch
Unsteady Aerodynamics A comprehensive overview of unsteady aerodynamics and its applications The study of unsteady aerodynamics goes back a century and has only become more significant as aircraft become increasingly sophisticated, fly faster, and their structures are lighter and more flexible. Progress in the understanding of flow physics, computing power and techniques, and modelling technologies has led to corresponding progress in unsteady aerodynamics, with a wide range of methods currently used to predict the performance of engineering structures under unsteady conditions. Unsteady Aerodynamics offers a comprehensive and systematic overview of the application of potential and vortex methods to the subject. Beginning with an introduction to the fundamentals of unsteady flow, it then discusses the modelling of attached and separated, incompressible and compressible flows around two-dimensional and three-dimensional bodies. The result is an essential resource for design and simulation in aerospace engineering. Unsteady Aerodynamics readers will also find: * MATLAB examples and exercises throughout, with codes and solutions on an accompanying website * Detailed discussion of most classes of unsteady phenomena, including flapping flight, transonic flow, dynamic stall, flow around bluff bodies and more * Validation of theoretical and numerical predictions using comparisons to experimental data from the literature Unsteady Aerodynamics is ideal for researchers, engineers, and advanced students in aerospace engineering.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
About the Companion Website
1 Introduction
1.1 Why Potential and Vortex Methods?
1.2 Outline of This Book
References
2 Unsteady Flow Fundamentals
2.1 Introduction
2.2 From Navier–Stokes to Unsteady Incompressible Potential Flow
2.3 Incompressible Potential Flow Solutions
2.4 From Navier–Stokes to Unsteady Compressible Potential Flow
2.5 Subsonic Linearised Potential Flow Solutions
2.6 Supersonic Linearised Potential Flow Solutions
2.7 Vorticity and Circulation
2.8 Concluding Remarks
References
Notes
3 Analytical Incompressible 2D Models
3.1 Introduction
3.2 Steady Thin Airfoil Theory
3.3 Fundamentals of Wagner and Theodorsen Theory
3.4 Wagner Theory
3.5 Theodorsen Theory
3.6 Finite State Theory
3.7 Concluding Remarks
3.8 Exercises
References
Notes
4 Numerical Incompressible 2D Models
4.1 Introduction
4.2 Lumped Vortex Method
4.3 Gust Encounters
4.4 Frequency Domain Formulation of the Lumped Vortex Method
4.5 Source and Vortex Panel Method
4.6 Theodorsen's Function and Wake Shape
4.7 Steady and Unsteady Kutta Conditions
4.8 Concluding Remarks
4.9 Exercises
References
Note
5 Finite Wings
5.1 Introduction
5.2 Finite Wings in Steady Flow
5.3 The Impulsively Started Elliptical Wing
5.4 The Unsteady Vortex Lattice Method
5.5 Rigid Harmonic Motion
5.6 The 3D Source and Doublet Panel Method
5.7 Flexible Motion
5.8 Concluding Remarks
5.9 Exercises
References
Note
6 Unsteady Compressible Flow
6.1 Introduction
6.2 Steady Subsonic Potential Flow
6.3 Unsteady Subsonic Potential Flow
6.4 Unsteady Supersonic Potential Flow
6.5 Transonic Flow
6.6 Concluding Remarks
6.7 Exercises
References
Notes
7 Viscous Flow
7.1 Introduction
7.2 Impulsively Started Flow around a 2D Flat Plate at High Angles of Attack
7.3 Flow Around a 2D Circular Cylinder
7.4 Flow Past 2D Rectangular Cylinders
7.5 Concluding Remarks
7.6 Exercises
References
Note
A Fundamental Solutions of Laplace's Equation
A.1 The 2D Point Source
A.2 The 2D Point Vortex
A.3 The Source Line Panel
A.4 The Vortex Line Panel
A.5 The Horseshoe Vortex
A.6 The Vortex Line Segment
A.7 The Vortex Ring
A.8 The 3D Point Source
A.9 The 3D Point Doublet
A.10 The Source Surface Panel
A.11 The Doublet Surface Panel
References
Note
B Fundamental Solutions of the Linearized Small Disturbance Equation
B.1 The Subsonic Doublet Surface Panel
B.2 The Acoustic Source Surface Panel
B.3 The Acoustic Doublet Surface Panel
B.4 The Supersonic Source Surface Panel
References
C Wagner's Derivation of the Kutta Condition
Reference
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Aerodynamic results for NACA 2412 airfoil at by the vortex pane...
List of Illustrations
Chapter 2
Figure 2.1 A body translating and rotating in a still fluid.
Figure 2.2 A body translating and rotating in a steady free stream.
Figure 2.3 A static body in an unsteady free stream.
Figure 2.4 Flow induced by a 3D point (a) source and (b) sink.
Figure 2.5 Flow induced by a 3D point doublet...
Figure 2.6 Potential flow on and around an impermeable sphere immersed in a ...
Figure 2.7 Potential and pressure distribution on the surface of the imperme...
Figure 2.8 Definition of control volume...
Figure 2.9 Control volume definition in a flow with potential...
Figure 2.10 Discretisation of a surface into quadrilateral panels.
Figure 2.11 Potential distribution in flowfield with potential...
Figure 2.12 Flow domain around a wing.
Figure 2.13 Potential distribution in flowfield around an impermeable sphere...
Figure 2.14 Comparison of numerical and exact solutions for the potential ar...
Figure 2.15 Potential variation with...
Figure 2.16 Variation of...
Figure 2.17 Definition of potential and normal velocity either side of the s...
Figure 2.18 Propagation of acoustic disturbances in subsonic flow.
Figure 2.19 2D section of flowfield induced by a compressible point source w...
Figure 2.20 2D section of flowfield induced by a compressible point source w...
Figure 2.21 Domain of validity of a steady supersonic source for...
Figure 2.22 Propagation of acoustic disturbances in supersonic flow.
Figure 2.23 The 2D section of the velocity field induced by a steady superso...
Figure 2.24 A surface ...
Figure 2.25 Lamb–Oseen vortex. (a) Velocity in...
Figure 2.26 Comparison of velocities induced by the Lamb–Oseen and Vatistas‐...
Figure 2.27 Velocity induced by segment of vortex filament on point ...
Figure 2.28 Four snapshots of Prandtl's visualisation of the flow around and...
Figure 2.29 Flow domain around a wing and its wake.
Chapter 3
Figure 3.1 Thickness distribution and camber line of a NACA 3412 airfoil.
Figure 3.2 Thin airfoil theory representation of an impermeable camber line....
Figure 3.3 Total aerodynamic load and its components.
Figure 3.4 Thin airfoil theory lift and moment predictions for different NAC...
Figure 3.5 Upwash components due to motion of 2D flat plate. (a) Upwash due ...
Figure 3.6 Conformal transformation from circle to flat plate. (a) Circle pl...
Figure 3.7 Transformation of the wake from the circle plane to the flat plat...
Figure 3.8 Singularity placement in circle plane.
Figure 3.9 Potential (left) and stream function (right) induced by a constan...
Figure 3.10 Velocity vector field induced by a constant source distribution ...
Figure 3.11 Streamlines and velocity vectors of flow originating outside and...
Figure 3.12 Potential (left) and stream function (right) induced by a consta...
Figure 3.13 Velocity vector field induced by a constant vortex distribution ...
Figure 3.14 Streamlines and velocity vectors of flow originating outside and...
Figure 3.15 Flow around a flat plate at constant angle of attack in a consta...
Figure 3.16 Wake history after impulsive start of a flat plate airfoil.
Figure 3.17 Wake vorticity distribution in the wake and lift time response a...
Figure 3.18 Flow around a flat plate in an impulsively started flow. (a) a...
Figure 3.19 Wagner's function.
Figure 3.20 Wagner's function estimates. (a) Bessel functions and (b) approx...
Figure 3.21 Steady pressure contours around the flat plate of Example 3.8.
Figure 3.22 Parabolic velocity field.
Figure 3.23 Drag variation with time acting on an impulsively started flat p...
Figure 3.24 Modelling a continuous time signal by discrete steps.
Figure 3.25 Aerodynamic states and loads of sinusoidally oscillating 2D pitc...
Figure 3.26 Amplitude variation of circulatory and non‐circulatory loads wit...
Figure 3.27 Ratio of unsteady to quasi‐steady load amplitudes.
Figure 3.28 Wake oscillation in time and space.
Figure 3.29 Variation of magnitude and phase of Theodorsen's function with r...
Figure 3.30 Aerodynamic load response predictions for sinusoidally pitching ...
Figure 3.31 Predictions of aerodynamic load variation with instantaneous pit...
Figure 3.32 Typical aeroelastic section with pitch and plunge degrees of fre...
Figure 3.33 Magnitude and phase of frequency response of typical aeroelastic...
Figure 3.34 Time response of typical aeroelastic section at two airspeeds. (...
Figure 3.35 Elliptic coordinate system.
Figure 3.36 Polynomial functions of Eq. (3.231). (a) ...
Figure 3.37 Damped sinusoidal functions of Eq. (3.231). (a) ...
Figure 3.38 Functions ...
Figure 3.39 Sums of all symmetric and antisymmetric functions ...
Figure 3.40 Accuracy of Eq. (3.276) for four values of...
Figure 3.41 Lift acting on an impulsively started airfoil and Wagner functio...
Figure 3.42 Aerodynamic states and loads of sinusoidally oscillating 2D pitc...
Figure 3.43 Theodorsen's function estimate from finite state theory. (a) Amp...
Chapter 4
Figure 4.1 Lumped vortex representation of unsteady flow around 2D airfoil....
Figure 4.2 Single lumped vortex representation of steady flow around 2D flat...
Figure 4.3 Flow vectors around 2D flat plate modelled by a single lumped vor...
Figure 4.4 Discretisation of camber line by lumped vortex panels.
Figure 4.5 Solution of flow around static NACA 24XX airfoil using the lumped...
Figure 4.6 Lift and pitching moment coefficient variation with angle of atta...
Figure 4.7 Shedding procedure for wake lumped vortices. (a)...
Figure 4.8 Prescribed wake shape and aerodynamic loads for an impulsively st...
Figure 4.9 Free wake shape and aerodynamic loads for an impulsively started ...
Figure 4.10 Detailed wake shape after impulsive start for...
Figure 4.11 Detailed wake shape after impulsive start using Vatistas et al. ...
Figure 4.12 Aerodynamic load dependence on vortex model for an impulsively s...
Figure 4.13 Two simple transverse gust models. (a) Sharp‐edged gust and (b) ...
Figure 4.14 Free wake shape and aerodynamic loads for a sharp‐edge gust enco...
Figure 4.15 Motion‐induced flow velocities on flat plate at an angle of atta...
Figure 4.16 Types of wake modelling for a sinusoidally pitching and plunging...
Figure 4.17 Snapshots of the wake behind a plunging airfoil. (a) Experiment...
Figure 4.18 Free wake shape and aerodynamic loads for a plunging NACA 0015 a...
Figure 4.19 Wake jet effect.
Figure 4.20 Snapshots of the wake behind a pitching airfoil. (a) Experiment...
Figure 4.21 Free wake shape and aerodynamic loads for a pitching NACA 0015 a...
Figure 4.22 Mean drag/thrust produced by sinusoidally pitching wing.
Figure 4.23 Wake behind sinusoidally pitching airfoil at two values of the r...
Figure 4.24 Amplitude and phase of lift, drag and pitching moment coefficien...
Figure 4.25 Panel discretisation of a 2D airfoil.
Figure 4.26 Velocities induced by source panel...
Figure 4.27 Original and modified geometries of the NACA 2412 airfoil. (a) T...
Figure 4.28 Pressure distribution around static NACA2412 airfoil calculated ...
Figure 4.29 Aerodynamic loads acting on static NACA2412 airfoil using the so...
Figure 4.30 Convergence of SVPM predictions for lift and pitching moment wit...
Figure 4.31 Points upstream of the trailing edge, along which the potential ...
Figure 4.32 Calculation of perturbation potential on airfoil's surface. (a) ...
Figure 4.33 The unsteady source and vortex panel scheme of Basu and Hancock ...
Figure 4.34 Wake panel placement at the trailing edge of NACA four‐digit air...
Figure 4.35 Effect of using the two different roots of Eq. (4.165) on the st...
Figure 4.36 Time response of lift and drag coefficients acting on a NACA 001...
Figure 4.37 Effect of airfoil thickness on impulsive lift and drag responses...
Figure 4.38 Mean thrust coefficient, mean power coefficient and propulsive e...
Figure 4.39 Circulatory and non‐circulatory lift acting on a pitching and pl...
Figure 4.40 Ratio of unsteady to quasi‐steady circulatory lift amplitude for...
Figure 4.41 Discretisation of a 2D airfoil by vortex panels with linearly va...
Figure 4.42 Pressure distribution around a NACA 2412 airfoil at...
Figure 4.43 Flow around the trailing edge of a NACA 2412 airfoil at...
Figure 4.44 Convergence of VPM predictions for steady lift and pitching mome...
Figure 4.45 Discretisation of a 2D airfoil by vortex panels with linearly va...
Figure 4.46 Mean thrust coefficient, mean power coefficient and propulsive e...
Figure 4.47 Flow around the trailing edge of a pitching and plunging airfoil...
Chapter 5
Figure 5.1 Swept and tapered trapezoidal wing geometry. (a) Top view, (b) si...
Figure 5.2 Flow behind a lifting finite wing. (a) Visualisation of flow behi...
Figure 5.3 Finite wing modelling using horseshoe vortices.
Figure 5.4 Spanwise induced and effective angles of attack.
Figure 5.5 Elliptical wing planform.
Figure 5.6 Sectional modelling of wing. (a) Circle plane and (b) Flat plate ...
Figure 5.7 Single vortex ring placed on wing.
Figure 5.8 Jones' vortex ring model. (a) Superposition of vortex rings at...
Figure 5.9 Vortex ring of span...
Figure 5.10 Downwash induced at midspan of bound vortex by a row of vortex r...
Figure 5.11 Lift coefficient response of impulsively started elliptical wing...
Figure 5.12 Drag coefficient response of impulsively started elliptical wing...
Figure 5.13 Lift coefficient response of impulsively started elliptical wing...
Figure 5.14 Camber surface of a half‐wing.
Figure 5.15 Vortex lattice panelling scheme. (a) Complete wing and (b) Singl...
Figure 5.16 Two different panel numbering schemes for the VLM. (a) Double in...
Figure 5.17 Vortex ring characteristics. (a) Vortex ring and geometric panel...
Figure 5.18 Forces acting on VLM grid.
Figure 5.19 Wing tested by Kolbe and Boltz ([1951]) and its VLM model. (a) A...
Figure 5.20 Vortex strength at the trailing edge for the steady VLM.
Figure 5.21 Lift and drag curves computed by the VLM. Experimental data by K...
Figure 5.22 Spanwise normal force distribution for three angles of attack. E...
Figure 5.23 Lift‐curve slopes and induced drag factors of straight rectangul...
Figure 5.24 Vortex strength at the trailing edge for the unsteady VLM.
Figure 5.25 Wake shape behind impulsively started elliptical wing.
Figure 5.26 Aerodynamic load coefficient variation with time after impulsive...
Figure 5.27 Aerodynamic load coefficient variation with time after impulsive...
Figure 5.28 Effect of aspect ratio on the load responses after an impulsive ...
Figure 5.29 Effect of taper ratio on the load responses after an impulsive s...
Figure 5.30 Effect of sweep on the load responses after an impulsive start f...
Figure 5.31 Effect of twist on the load responses after an impulsive start f...
Figure 5.32 Types of wake modelling for a sinusoidally pitching and plunging...
Figure 5.33 Mean thrust coefficient variation with reduced frequency for a p...
Figure 5.34 Lift and drag responses of the pitching and plunging wing of Fig...
Figure 5.35 Free wake shape behind plunging rectangular wing with...
Figure 5.36 Mean thrust coefficient and effective angle of attack variation ...
Figure 5.37 Body‐fixed and inertial axes systems. (a) Body‐fixed and (b) ine...
Figure 5.38 Simulation of rectangular wing with...
Figure 5.39 Aerodynamic loads acting on the leading edge spanwise segments o...
Figure 5.40 Wing model and kinematics tested by Queijo et al. ([1956]). (a) ...
Figure 5.41 Sideforce and roll aerodynamic stability derivatives predicted b...
Figure 5.42 Wing and wake surfaces and unit normal vector definitions.
Figure 5.43 Wing and wake discretized into rectangular panels.
Figure 5.44 Wing tested by Applin and Gentry Jr. ([1990])
Figure 5.45 Two different panel numbering schemes for the SDPM. (a) Double i...
Figure 5.46 Wing and wake discretized into quadrilateral panels for steady f...
Figure 5.47 Kutta condition for the SDPM.
Figure 5.48 Steady pressure distribution around rectangular wing predicted b...
Figure 5.49 Total lift and drag coefficient predicted by the SDPM. Experimen...
Figure 5.50 Out‐of‐plane mode shapes of a cantilevered flat plate wing. (a) ...
Figure 5.51 Tip bending amplitude and phase of plunging rectangular half‐win...
Figure 5.52 Two snapshots of the deformed shape of the very flexible wing fo...
Figure 5.53 Mean thrust, mean propulsive power and propulsive efficiency var...
Figure 5.54 Wing and mode shape used by Lessing et al. ([1960]). (a) Half‐wi...
Figure 5.55 Mean pressure distribution predicted by the SDPM at...
Figure 5.56 Real and imaginary parts of oscillatory pressure distribution pr...
Chapter 6
Figure 6.1 Half‐wing tested by Lockman and Seegmiller ([1983]), showing the ...
Figure 6.2
Pressure distribution predicted by the SDPM at three spanwise mea
...
Figure 6.3 Discretization scheme for the Doublet Lattice Method. (a) Complet...
Figure 6.4 LANN half‐wing with pressure taps.
Figure 6.5 Calculation of camber line derivatives of the LANN wing. (a) Inte...
Figure 6.6
Amplitude and phase of mean pressure jump predicted by the DLM ac
...
Figure 6.7 Amplitude and phase of oscillating pressure jump predicted by the...
Figure 6.8 Mean pressure distribution predicted by the SDPM around LANN wing...
Figure 6.9 Real and imaginary parts of oscillatory pressure distribution pre...
Figure 6.10 Comparison of mean pressure jump across the LANN wing predicted ...
Figure 6.11 Comparison of oscillating pressure jump across the LANN wing pre...
Figure 6.12 Comparison of pressure jump distribution on the LANN wing obtain...
Figure 6.13 NLR 7301 supercritical airfoil with oscillating flap tested by Z...
Figure 6.14 Mean and oscillating pressure distribution predicted by the SDPM...
Figure 6.15 Subsonic (left) and supersonic (right) leading edges of the F‐5 ...
Figure 6.16 F‐5 half‐wing model tested by Tijdeman et al. (1979b). (a) Half‐...
Figure 6.17 Mach box grid for F5 wing for...
Figure 6.18 Oscillating pressure jump across the F‐5 wing predicted by the M...
Figure 6.19 Wing and diaphragm panels for the F‐5 wing at two Mach numbers. ...
Figure 6.20 Mean pressure jump across the F‐5 wing predicted by the Mach pan...
Figure 6.21 Oscillating pressure jump across the F‐5 wing predicted by the M...
Figure 6.22 Mach contours and pressure distribution around NASA SC(2)‐0714 a...
Figure 6.23 Steady pressure distribution predicted by the transonic SDPM aro...
Figure 6.24 Experimental and interpolated mean pressure jump distributions a...
Figure 6.25 Corrected DLM mean pressure jump across LANN wing for...
Figure 6.26 Corrected DLM oscillatory pressure jump across LANN wing for...
Figure 6.27 Corrected SDPM mean pressure jump across LANN wing for...
Figure 6.28 Corrected SDPM oscillatory pressure jump across LANN wing for...
Chapter 7
Figure 7.1 Conceptual view of attached and separating boundary layers. (a) F...
Figure 7.2 Three steady stall mechanisms. (a) Trailing edge stall, (b) leadi...
Figure 7.3 Lift curves and drag polars of three different airfoils at...
Figure 7.4 Lift and drag curves of the NLR‐7301 airfoil at...
Figure 7.5 Aerodynamic load response predictions for sinusoidally pitching 2...
Figure 7.6 Aerodynamic load response predictions for sinusoidally pitching 2...
Figure 7.7 Aerodynamic load response predictions for sinusoidally pitching 2...
Figure 7.8 Aerodynamic load response predictions for sinusoidally pitching 2...
Figure 7.9 Modified lumped vortex scheme for separated flows.
Figure 7.10 Vorticity around flat plate at...
Figure 7.11 Time response of circulation in the separated shear layer...
Figure 7.12 Time response of normal and chordwise force coefficient...
Figure 7.13 Aerodynamic load response predictions for sinusoidally pitching ...
Figure 7.14 Snapshots of the vorticity field in the flow around a circular c...
Figure 7.15 Variation of Strouhal number and mean drag coefficient with Reyn...
Figure 7.16 Approximate modelling of boundary layer.
Figure 7.17 Vortex shedding strategy for circular cylinder.
Figure 7.18 Snapshots of the discrete vortex positions at four time instance...
Figure 7.19 Time and frequency response of aerodynamic loads acting on circu...
Figure 7.20 Mean pressure distribution around circular cylinder at...
Figure 7.21 Rectangular cylinder of side ratio...
Figure 7.22 Definition of panel vertices on rectangular cylinder.
Figure 7.23 Strouhal number and mean drag coefficient of flow around rectang...
A
Figure A.1 Real potential and stream function induced by a source of strengt...
Figure A.2 Velocity field induced by a source and a sink...
Figure A.3 Real potential and stream function induced by a vortex of strengt...
Figure A.4 Velocity field induced by vortices of opposite strength...
Figure A.5 The 1D source panel.
Figure A.6 Flow induced by a source panel with constant unit strength.
Figure A.7 The 1D vortex panel.
Figure A.8 Flow induced by a vortex panel with constant unit strength.
Figure A.9 Horseshoe vortex.
Figure A.10 Downwash induced by a horseshoe vortex.
Figure A.11 Vortex line segment and the velocity it induces at point P.
Figure A.12 Velocities induces at point...
Figure A.13 Velocity field induced by vortex rings of constant unit strength...
Figure A.14 Velocity field induced by source surface panels of constant unit...
Figure A.15 Quadrilateral panel lying on...
Figure A.16 Velocity field induced by doublet surface panels of constant uni...
B
Figure B.1 Quadrilateral panel lying on...
Figure B.2 Quadrilateral supersonic source panel, control panel, Mach lines ...
C
Figure C.1 Wagner's flow model.
Guide
Cover
Table of Contents
Series Page
Title Page
Copyright
Preface
About the Companion Website
Begin Reading
A Fundamental Solutions of Laplace's Equation
B Fundamental Solutions of the Linearized Small Disturbance Equation
C Wagner's Derivation of the Kutta Condition
Index
End User License Agreement
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Aerospace Series
David Allerton. Principles of Flight Simulation
Dora Musielak. Scramjet Propulsion: A Practical Introduction
Allan Seabridge, Mohammad Radaei. Aircraft Systems Classifications: A Handbook of Characteristics and Design Guidelines
Douglas M. Marshall. UAS Integration into Civil Airspace: Policy, Regulations and Strategy
Paul G. Fahlstrom, Thomas J. Gleason, Mohammad H. Sadraey. Introduction to UAV Systems, 5th Edition
James W. Gregory, Tianshu Liu. Introduction to Flight Testing
Ashish Tewari. Foundations of Space Dynamics
Egbert Torenbeek. Essentials of Supersonic Commercial Aircraft Conceptual Design
Mohammad H. Sadraey. Design of Unmanned Aerial Systems
Saeed Farokhi. Future Propulsion Systems and Energy Sources in Sustainable Aviation
Rama K. Yedavalli. Flight Dynamics and Control of Aero and Space Vehicles
Allan Seabridge, Ian Moir. Design and Development of Aircraft Systems, 3rd Edition
Gareth D. Padfield. Helicopter Flight Dynamics: Including a Treatment of Tiltrotor Aircraft, 3rd Edition
Craig A. Kluever. Space Flight Dynamics, 2nd Edition
Trevor M. Young. Performance of the Jet Transport Airplane: Analysis Methods, Flight Operations, and Regulations
Andrew J. Keane, Andros Sobester, James P. Scanlan. Small Unmanned Fixed‐wing Aircraft Design: A Practical Approach
Pascual Marques, Andrea Da Ronch. Advanced UAV Aerodynamics, Flight Stability and Control: Novel Concepts, Theory and Applications
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Unsteady Aerodynamics
Potential and Vortex Methods
Grigorios Dimitriadis
University of LiègeBelgium
This edition first published 2024.
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Names: Dimitriadis, Grigorios, 1972‐ author.
Title: Unsteady aerodynamics : potential and vortex methods / Grigorios
Dimitriadis.
Description: Hoboken, NJ : Wiley, 2024. | Series: Aerospace series |
Includes index.
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Preface
The term ‘unsteady aerodynamics’ is used to denote fluid flow problems whereby either a body moves in a fluid in a time‐varying fashion or the flow is time‐varying in itself. The reader should also take note of the subtitle: potential and vortex methods. Although physical phenomena related to unsteady fluid motion are discussed in detail, the focus of this book is on modelling methods. Furthermore, the subtitle makes it clear that there will be no discussion in this book of what is commonly referred to as Computational Fluid Dynamics. Even though many numerical approaches will be presented, none of them will rely on discretising the entire flowfield around the body. All numerical solutions will be obtained by discretising the surface of the body and shedding vorticity in its wake, propagating the latter using a Lagrangian approach.
The book is called ‘Unsteady Aerodynamics’ and not Unsteady Fluid Dynamics because the main focus is on flow over wings. Nevertheless, many of the methodologies presented here could be applied to hydrodynamic problems, and the validation data used for some of the theories were obtained in water tunnels. Even though unsteadiness is the main subject area, several steady aerodynamic problems are presented and discussed in detail. However, the book is not an introduction to all aerodynamics or fluid dynamics in general. This means that the reader should have good background knowledge of basic aerodynamics.
The emphasis of this book is on application so that all theories are accompanied by practical examples solved by means of Matlab and C codes. These codes are available to the reader on the Wiley website; they have been tested on Matlab version 2020a but could also be compatible with other versions. The C codes are written as Matlab mex functions and should be compiled appropriately. It is the reader's responsibility to do so, neither Wiley nor the author will provide technical support. The Mathworks website includes a list of compatible compilers for different architectures here: https://www.mathworks.com/support/requirements/supported-compilers.html. The reader should note that the purpose of the codes is to illustrate the examples and the underlying theories. They solve the particular problems for which they were written, but they should not be seen as general unsteady aerodynamic analysis codes that can be directly applied to different problems.
Chapter 1 is a brief introduction to steady and unsteady aerodynamics and provides a more detailed outline of the book. Chapter 2 presents the fundamentals of unsteady flow, focusing on the concepts and equations that will be of use throughout the book. Classical 2D unsteady potential theory is presented in Chapter 3, with a focus on analytical solutions. Chapter 4 discusses numerical 2D potential flow solutions and concentrates on interesting physical phenomena such as thrust production and propulsive efficiency. Chapter 5 introduces the aerodynamic analysis of finite wings in incompressible flows, initially by means of analytical methods and then using numerical techniques. Compressible unsteady flows are treated in Chapter 6, which addresses subsonic, supersonic and transonic aerodynamic problems. Finally, Chapter 7 introduces viscous unsteady flows featuring significant flow separation.
I would like to take this opportunity to thank my colleagues Thomas Andrianne, Adrien Crovato, Thierry Magin and Ludovic Noels who took over my teaching during my sabbatical year. A particularly warm thank you goes to Kyros Iakinthos and Pericles Panagiotou, who welcomed me into their research group at the Aristotle University of Thessaloniki during that year. I would also like to thank Adrien Crovato for spotting a negative steady drag issue in the source and doublet panel code I developed for this book, Thomas Lambert for our exchanges on the Vatistas model, Mariano Sánchez Martínez for carrying out Euler simulations on the LANN wing, Johan Boutet for our collaboration on unsteady lifting line methods and Vincent Terrapon and Boyan Mihaylov for a late‐night Zoom session discussing compound rotations (among other things) during the lockdown.
August 2023
Grigorios Dimitriadis
Liège and Thessaloniki
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/dimitriadis/unsteady_aerodynamics
The website includes sample programs with MATLAB codes and solutions.
1Introduction
Unsteady aerodynamics refers to flow of air over bodies whose velocity field is changing in time. The causes of unsteadiness can be
Translational and rotational acceleration of the body relative to the fluid. The vast majority of this book is devoted to this type of unsteadiness.
Upstream or free stream unsteadiness. In the atmosphere, this phenomenon is referred to as atmospheric turbulence and is caused by a variety of meteorological and geographical phenomena. In the laboratory, we encounter wind tunnel turbulence. There will be no discussion of this type of unsteadiness in this book.
Turbulence in boundary layers, which is a ubiquitous source of unsteadiness in most practical flows. A short discussion of laminar and turbulent boundary layers is included in
Chapter 7
.
Flow that separates from the surface of the body, either instantaneously or permanently. This type of flow is inherently unsteady, even when the motion of the body is steady. We will discuss this type of unsteadiness in
Chapter 7
.
Other sources of unsteadiness, such as acoustics, jet impingement, wake interaction, and thermal effects, lie beyond the scope of this book.
The vast majority of practical airflows will feature some degree of turbulence upstream in the boundary layer and in the wake of the body. As a consequence, nearly all aerodynamics is unsteady. However, aircraft, rotors, wind turbines and other engineering structures are generally designed to operate under attached flow conditions, so that the turbulence is confined to a thin layer of fluid in contact with the surface. Under such conditions, the effect of turbulence is averaged and therefore the flow can be treated as steady. Then, the major source of unsteadiness becomes the motion of the body itself. Conversely, civil engineering structures are mostly aerodynamically bluff bodies; even though they seldom move, they are subjected to significant unsteadiness due to separated flow and upstream turbulence.
This book deals mostly with wings and therefore the source of unsteadiness it will address most of the time is body motion. Our ancestral prototype of flight is bird flight, which involves flapping wings. Yet the first man‐made flying objects were kites which in their simplest form do not flap or deform in any way. The first gliders and aircraft also had fixed wings, and flapping blades were introduced in helicopter rotors much later. From a practical point of view, it is clearly easier to work with steady aerodynamics. This is also the case from a mathematical point of view; the flow equations are simpler and easier to solve. From the experimental point of view too, setting up and measuring a steady flow is more straightforward.
Even fixed‐wing aircraft undergo unsteady motion, both rigid and flexible. Rigid aircraft motion is the field of study of flight dynamics; aircraft have both oscillatory and non‐oscillatory rigid body eigenmodes that cannot be predicted adequately using purely steady aerodynamic analysis. We will give an example of the calculation of aerodynamic stability derivatives in Chapter 5. Furthermore, aircraft structures are flexible and are becoming increasingly so. The study of vibrating structures in an airflow is the subject area of aeroelasticity. Again, a steady or quasi‐steady aerodynamic analysis is insufficient to predict aeroelastic phenomena. Chapter 3 includes one example of a direct application of unsteady aerodynamics to flutter prediction. Nevertheless, all of the methods presented in this book can be used for flight dynamic, aeroelastic or combined aeroservoelastic analysis.
1.1 Why Potential and Vortex Methods?
The equations of fluid flow are notorious for being unsolvable. The Millennium Prize (Clay Mathematics Institute, [2000]) for proving the existence and smoothness of solutions to the 3D Navier–Stokes equations was still unclaimed at the time of writing of this book and the original US$1 million prize money had already depreciated to US$575,000 due to inflation. Numerical solutions of these equations are possible, but turbulence renders them impractical. In order to capture all the spatial scales of turbulence at a Reynolds number encountered in aeronautical practice, the computational requirements of a direct numerical simulation of the Navier–Stokes equations exceed the capabilities of even the fastest and biggest modern computers. Therefore, in order to model practical problems, we resort to solving easier equations. These can be averaged or filtered versions of the original Navier–Stokes relations or simpler equations that are developed after making assumptions about the physics of the flow.
The fastest solutions are obtained for potential flow equations, whereby the flow is assumed to be inviscid, irrotational and isentropic, if not incompressible. Even though a significant amount of the physics of fluid flow is discarded in order to obtain such solutions, their range of validity can include many aeronautical applications under nominal operating conditions. For example, potential flow methods are the industrial standard for aircraft aeroelastic calculations. As long as the flow remains attached to the surface, its Reynolds number is high and there are no strong shock waves, potential methods can provide fast and reliable solutions to practical engineering problems. Their main advantage is that they do not require the calculation of the solution in the entire flowfield; calculations on the surface of the body and in its wake are sufficient, and the computational cost of such solutions is very low. Even separated flows can be approximated in this manner, by shedding vortices from the separated flow region of the body into the wake.
Potential flow approaches for steady aerodynamics are presented in detail in many textbooks, notably Katz and Plotkin ([2001]). Gülçat ([2016]) discusses many potential flow methods for unsteady aerodynamics for various flow conditions, from incompressible to hypersonic. Potential flow techniques for unsteady transonic flows are also presented in Landahl ([1961]) and Nixon ([1989]). The present book focuses on application; each method is presented in detail and applied to practical, usually experimental test cases. Furthermore, the book is accompanied by computer codes in the Matlab programming environment that can be used to solve these test cases. The text and computer codes should be studied in parallel. It is hoped that this application‐based approach will help the reader to develop a deeper understanding of the various methodologies.
The focus on potential and vortex methods means that the reader will not find any information in this book on what is commonly referred to as Computational Fluid Dynamics (CFD). The latter requires the numerical solution of the flow in a very wide region around the body, usually by means of finite volume, finite element or finite difference discretization. Even though most of the methods discussed in this book are numerical, they only require the discretization of the surface of the body; the wake is treated in a Lagrangian manner so that its vorticity propagates at the local flow velocity. Readers interested in unsteady CFD can consult alternative texts, such as Tucker ([2014]).
1.2 Outline of This Book
Chapter 2 constitutes an introduction to the mathematics of unsteady flow. The full flow equations are presented, and their simplifications using specific flow assumptions are derived. Both compressible and incompressible flow equations are presented, and their boundary conditions are discussed. Solutions to these equations are developed, and their implementation by means of Green's theorem is described in detail. The chapter finishes with a discussion of vorticity and the viscous flow equations.
Chapter 3 introduces the classical unsteady aerodynamic theories for 2D incompressible inviscid flow. The modelling of a flat plate airfoil oscillating in a flow is presented, and analytical equations for the resulting aerodynamic loads are derived. The example of impulsive airfoil motion is used in order to introduce the Wagner function while oscillating motion is used for the definition of Theodorsen's function. It is shown how general small amplitude motion can be represented using these theories and the generation of thrust or drag due to unsteady phenomena is explained. Finally, finite state theory is derived in detail.
Chapter 4 introduces numerical methods that can be used to model 2D inviscid unsteady flow with higher fidelity, for example by modelling more accurately the wake behind an oscillating airfoil or by representing the geometry of airfoils with non‐negligible thickness. Three such methods are presented, all with their advantages and disadvantages. They are used in order to demonstrate the physics of the wake behind oscillating airfoils and the mechanisms of thrust generation. Comparison of the predicted aerodynamic loads to experimental results demonstrates how the higher fidelity of numerical methods can represent more of the physics of the real phenomenon. Furthermore, it is shown that all numerical panel methods can be linearized and transformed to the frequency domain in order to obtain faster aerodynamic load predictions for harmonically oscillating wings.
Chapter 5 presents unsteady aerodynamic theories for 3D finite wings. It starts with a description of finite wing geometry. Then, analytical solutions are developed, starting with an impulsively started elliptical wing. Unsteady lifting line theories are discussed before detailing two numerical panel methods, the Vortex Lattice Method and the Source and Doublet Panel Method. These approaches are applied to several practical problems, such as the calculation of aerodynamic stability derivatives and prediction of the unsteady pressure distribution on a flexible wing.
Chapter 6 treats 3D compressible unsteady flows. Subsonic flow is treated first, with the presentation of the Doublet Lattice Method and the subsonic Source and Doublet Panel Method. Supersonic flow is modelled by means of the Mach box and Mach panel techniques. Then, unsteady transonic flows are discussed, and their modelling by means of field panel techniques is outlined. Finally, steady and unsteady corrections that allow subsonic approaches to model transonic flows are presented.
Chapter 7, which is the last chapter of this book, addresses viscous flows. It starts with a brief presentation of the boundary layer and its separation and then proceeds to discuss leading edge separation and dynamic stall. Finally, the Discrete Vortex Method is used to model highly separated flows around bluff bodies.
References
Clay Mathematics Institute (2000). Millennium problems.
https://www.claymath.org/millennium-problems
(accessed 14 March 2023).
Gülçat, U. (2016).
Fundamentals of Modern Unsteady Aerodynamics
, 2e. Springer.
Katz, J. and Plotkin, A. (2001).
Low Speed Aerodynamics
. Cambridge University Press.
Landahl, M.T. (1961).
Unsteady Transonic Flow
. Dover Publications, Inc.
Nixon, D. (ed.) (1989).
Unsteady Transonic Aerodynamics
,
Progress in Astronautics and Aeronautics
, vol. 120. AIAA.
Tucker, P.G. (2014).
Unsteady Computational Fluid Dynamics in Aeronautics
,
Fluid Mechanics and Its Applications
, vol. 104. Springer.
2Unsteady Flow Fundamentals
2.1 Introduction
This chapter introduces the concepts and equations that will be used throughout the rest of the book. It is not intended as an introduction to all fluid dynamics and many results will be taken for granted. We will not introduce the continuum assumption or constitutive fluid models and we will not derive the flow equations; there are several good textbooks that do. The main focus lies in deriving the compressible and incompressible potential flow equations, discussing their boundary conditions and developing fundamental solutions for these equations.
2.2 From Navier–Stokes to Unsteady Incompressible Potential Flow
For a Newtonian fluid, the flow equations (see for example Anderson Jr. ([1985]) or Kuethe and Chow ([1986])) are given by the continuity equation
and the momentum equations, also known as the Navier–Stokes equations,
where is the fluid density, , , the flow velocities in the , and directions, the pressure and the dynamic viscosity. The Navier–Stokes equations reduce to the Euler equations for inviscid flow by setting , such that
We can write the Euler equations in a different form that does not contain derivatives of the density. We first multiply the continuity equation by to obtain
and then we write Eq. (2.5) as
Subtracting Eq. (2.8) from this latest expression yields
Carrying out similar operations to Eqs. (2.6) and (2.7), we obtain
Despite the lack of density derivatives, Eqs. (2.9)–(2.11) can still describe compressible flow if is not constant; the only simplification we have imposed is to ignore viscosity.
2.2.1 Irrotational Flow
We can apply a further simplification by assuming that the flow is irrotational, that is
where and
from which we obtain the three irrotationality relationships:
Equation (2.9) can be rewritten as
and re‐arranged in the form
Substituting from the irrotationality relationships of Eq. (2.13), the momentum equation simplifies to
Carrying out similar operations to Eqs. (2.10) and (2.11), we obtain the irrotational Euler equations
Furthermore, we define the velocity potential function such that
Substituting these definitions into Eqs. (2.14)–(2.16) leads to
2.2.2 Laplace's and Bernoulli's Equations
We now apply the final simplification by assuming that the flow is incompressible so that the density is constant everywhere in the flowfield and at all times. The continuity Eq. (2.1) becomes
Substituting from the definition of the potential in expressions (2.17) results in
which is known as Laplace's equation and can also be expressed as
where .
The inviscid, irrotational and incompressible assumptions have simplified the continuity equation from the form (2.1) to the form (2.22), which is a linear partial differential equation with a single unknown, the potential . The irrotational Euler equations can also be simplified using the incompressible assumption; we can multiply Eqs. (2.18)–(2.20) by their respective spatial differentials to obtain
Since is constant, exchanging the order of the time and space derivatives in the first terms of each of the equations and integrating in space results in
which is known as the unsteady Bernoulli equation. Note that all three momentum equations lead to the same Bernoulli expression. The constant of integration can be written as without loss of generality, where is a reference pressure. Then the unsteady Bernoulli equation becomes
Using the definition of the potential, it can also be written as
where . Bernoulli's equation is valid everywhere in an incompressible, inviscid and irrotational flowfield but the reference pressure needs to be specified. As it stands, Eq. (2.27) relates conditions at any point to a point in space where the pressure is equal to , the total flow speed is zero and the time derivative of the potential is also equal to zero. Bernoulli's equation can be written more intuitively by relating conditions at point to a faraway point where the pressure is , the total flow speed is and the potential is constant or zero, that is
The quantity is known as the dynamic pressure and is the far‐field dynamic pressure. The pressure coefficient is defined as
and, using Eq. (2.29)
Equations (2.23) and (2.31) will be used to solve all incompressible flow problems presented in this book, for which viscous phenomena are not important. Laplace's equation is a second‐order linear partial differential equation with a single unknown, the potential . Once it is solved, the potential can be substituted into Bernoulli's equation in order to calculate the pressure anywhere in the flow. Since Laplace's equation is of second order, it requires two boundary conditions.
2.2.3 Motion in an Incompressible, Inviscid, Irrotational Fluid
Consider a body in unsteady motion in still air. Points on the body's surface are denoted by vector with respect to a static origin; they are defined as the solutions of the equation . The body is rotating, translating and deforming so that the velocity of the points on the surface is denoted by . Consequently, if the equation of the surface is at time , at time , it becomes
Expanding as a Taylor series around gives
where is the vector . Therefore, recalling that ,
where the notation signifies ‘evaluate at and ’. Now, as the surface of the body is moving with velocity , we can approximate as
Substituting back into Eq. (2.32) yields
or dividing throughout by
Finally, taking the limit of this latest expression as ,
Equation (2.33) describes how the surface of the body deforms with respect to its own velocity . However, we are also interested in how the fluid deforms due to the motion of the body's surface.
We denote the coordinates of any point in the fluid by . Assuming that the air is still, far from the body its velocity will be zero. This is known as the far‐field boundary condition, which states that the flow disturbance caused by the body decays to zero as , where
As the flow velocity is given by , the far‐field condition can be formulated mathematically as
Far from the body, the potential must be constant so that its derivatives in all directions are equal to zero. This constant value of the potential may be chosen to be equal to zero.
Close to the body, the flow will be disturbed by the body's motion and therefore the flow velocity and potential will be non‐zero. The objective of unsteady potential flow modelling is to calculate the flow velocities , , on the surface of the body. From these, the pressure around the body, , can be evaluated from Eq. (2.29); the total aerodynamic force acting on the body is then given by
where is a unit vector normal to the surface at point and time , while denotes an integral over the entire surface of the body and is an infinitesimal element of this surface. The fundamental flow equation to be solved is Laplace's equation (2.22), which requires two boundary conditions. One of them is the far‐field condition but we still need to define the second. Assuming that the surface of the body is impermeable, the layer of fluid in contact with the surface will also obey and, hence, Eq. (2.32). The velocity of the fluid on the surface is given by so that for the fluid,
Substituting in Eq. (2.32), dividing by and taking the limit as leads to
Furthermore, dividing throughout by gives
The quantity is in fact the unit vector normal to the surface , so that
Finally, we solve Eq. (2.33) for and substitute the result into Eq. (2.36) to obtain
Equation (2.37) is known as the impermeability boundary condition, or zero normal flow condition. It states that the relative velocity between the fluid and the surface in a direction normal to the surface must be equal to zero so that no flow can cross the solid boundary. The far‐field equation (2.34) and the impermeability equation (2.37) are the two boundary conditions that are required to obtain solutions of Laplace's equation.
Example 2.1 Determine the impermeability boundary condition for a sphere that translates with speed and whose radius is a function of time.
We will select the centre of the sphere as the body datum. Since we are measuring all distances from the centre of the sphere, the equation of the surface of the sphere is
The solutions of this equation are the points on the surface such that . The quantity evaluated at becomes
and its magnitude is
so that the normal vector on the surface is given by
The deformation velocity of the surface of the sphere has magnitude and its direction is radial, that is parallel to . Recalling that the sphere also translates with speed , the total velocity of the surface in the normal direction is given by
Substituting this latest result in the boundary condition of Eq. (2.37) gives
or after substituting for from expression (2.38)
As expected, this equation states that the flow velocity in a direction normal to the surface of the sphere must be equal to the normal component of the motion of the surface, if the flow is not to penetrate inside the sphere.
As the flow is inviscid, the fluid velocity tangent to the surface is non‐zero and contributes directly to the surface pressure. If is a unit vector tangent to the surface and parallel to the local flow direction, then the tangential flow velocity is given by and the pressure on the surface is obtained from Eq. (2.28) as
where is the relative velocity between the fluid and the body's surface in the tangential direction.
Now, let us consider the special case where the body does not deform, it only translates with velocity and rotates rigidly around a point with rotational velocity . The surface velocity is given by
The situation is depicted in Figure 2.1; is the instantaneous position of the rotation centre with respect to the origin and is the instantaneous position of a point on the surface, again with respect to . Both and translate with velocity , while also rotates with velocity around . The figure also draws the unit vectors normal and tangent to the surface at point , and , respectively. For this rigid case, the far‐field boundary condition is still given by Eq. (2.34). The impermeability boundary condition of Eq. (2.37) becomes
where is the velocity of point on the surface of the body.
Figure 2.1 A body translating and rotating in a still fluid.
Next, we assume that the translation velocity is composed of a steady component, which we will call and an unsteady component . In such cases, it is customary to set the fluid in motion with velocity , known as the free stream velocity, and to only attribute the unsteady component to the body, as seen in Figure 2.2. The free stream has components ; usually, they are chosen such that and . The angle is the free stream angle of attack , while the angle is the free stream sideslip angle . If is the magnitude of the free stream velocity and , then
The relative velocity between the fluid and the body does not change, but as the steady velocity component is attributed to the flow, the flow's potential will increase. The total flow potential, , is the sum of the potential due to the free stream and the perturbation potential due to the shape of the body and its unsteady motion, . It can be written as
where is the free stream potential, defined as . Consequently, the boundary conditions must be adapted to
Far‐field. The total potential far from the body is equal to while the perturbation velocity tends to zero, that is
This means that the total potential is not a perturbation quantity.
Impermeability. The velocity anywhere in the flow is now , so that Eq. (
2.40
) becomes
Figure 2.2
