The objective of this book is to provide university students with a solid foundation for understanding the numerical methods employed in today's CFD and to familiarise them with modern CFD codes by hands-on experience. It is also intended for engineers and scientists starting to work in the field of CFD or for those who apply CFD codes. Due to the detailed index, the text can serve as a reference handbook too.
Each chapter includes an extensive bibliography, which provides an excellent basis for further studies.
The accompanying CD-ROM contains the sources of 1-D and 2-D Euler and Navier-Stokes flow solvers (structured and unstructured) as well as of grid generators. Provided are also tools for Von Neumann stability analysis of 1-D model equations. Finally, the CD-ROM includes the source code of a dedicated visualisation software with graphical user interface.
Computational Fluid Dynamics (CFD) is an important design tool in engineering and also a substantial research tool in various physical sciences as well as in biology. The objective of this book is to provide university students with a solid foundation for understanding the numerical methods employed in today's CFD and to familiarise them with modern CFD codes by hands-on experience. It is also intended for engineers and scientists starting to work in the field of CFD or for those who apply CFD codes. Due to the detailed index, the text can serve as a reference handbook too. Each chapter includes an extensive bibliography, which provides an excellent basis for further studies.
Front Cover 1
Dedication 3
Computation of Supersonic Flow over Flying Configurations 4
Copyright Page 5
Contents 6
About the Author 14
Preface 16
Acknowledgments 18
Chapter 1 Zonal, Spectral Solutions for the Three-Dimensional, Compressible Navier–Stokes Layer 20
1.1 Introduction 20
1.2 Three-dimensional, partial-differential equations of compressible Navier–Stokes layer (NSL) 21
1.3 The spectral variable and the spectral forms of the velocity's components and of the physical entities 23
1.4 The first and second derivatives of the velocity's components 24
1.5 The implicit and explicit forms of the boundary conditions at the NSL's edge 27
1.6 The dependence of the density function R versus the spectral velocity, inside the NSL 29
1.7 Dependence of absolute temperature T versus the spectral velocity, inside the NSL 30
1.8 The scalar forms of the NSL's impulse's partial-differential equations and their equivalent quadratical algebraic equations 31
1.9 Determination of spectral coefficients of the velocity's components by solving an equivalent quadratical algebraic system, via the collocation method 34
1.10 An original iterative method to solve a quadratical algebraic system 35
1.11 Conclusions 37
References 38
Chapter 2 Hyperbolical Potential Boundary Value Problems of the Axial Disturbance Velocities of Outer Flow, at NSL's Edge 39
2.1 Introduction 39
2.2 Basic equations 40
2.3 Full-linearized partial-differential equations of the flow over flattened, flying configurations 45
2.4 The characteristic hypersurfaces of the partial-differential equations of second order 47
2.4.1 The classification of quasi-linear partial-differential equations of second order 47
2.4.2 The characteristic's condition and the characteristic hypersurface 49
2.5 The linearized pressure coefficient C[sub(p)] on flying configurations 52
2.6 The linearized boundary value problems for flying configurations, at moderate angles of attack & #945
2.7 Definitions and properties of the thin and thick-symmetrical components of the thick, lifting flying configurations 54
2.8 The disturbance regions produced by a moving point in subsonic and supersonic flow 57
2.9 Disturbance regions and characteristic surfaces produced by triangular wings, in supersonic flow 59
2.10 Disturbance regions and characteristic surfaces produced by trapezoidal wings, in supersonic flow 68
2.11 Disturbance regions and characteristic surfaces produced by rectangular wings, in supersonic flow 71
2.12 The boundary value problems for the axial disturbance velocities on thin and thick-symmetrical wedged triangular wing components, in supersonic flow 72
2.13 Conclusions 75
References 76
Chapter 3 Computation of Axial Disturbance Velocities on Wedged Wings, in Supersonic Flow, at NSL's Edge 77
3.1 General considerations 77
3.2 The conical flow of first order 79
3.2.1 Definition of the conical flow 79
3.2.2 The Germain's complex plane 80
3.2.3 The Germain's compatibility conditions for the conical flow 82
3.2.4 The Carafoli's hydrodynamic analogy for the conical flow 82
3.2.5 The principle of the minimal singularities for the wedged triangular wings 84
3.3 The boundary conditions for the wedged triangular wings, in the Germain's plane 87
3.3.1 Introduction 87
3.3.2 The boundary conditions of the fictitious, complex potentials U and U* on the real axis of the Germain's complex plane 87
3.3.3 The wedged triangular wings with one subsonic and one supersonic leading edge 91
3.3.4 The wedged triangular wings with two supersonic leading edges 92
3.4 The solutions of direct boundary value problems for U and U* on wedged triangular wing components 97
3.4.1 The auxiliary plane & #967
3.4.2 The affine transformed wing and the transformed complex plane x 97
3.4.3 The contribution of a subsonic leading edge on the thin wedged triangular wing 99
3.4.4 The contributions of ridges of the thin and thick-symmetrical wedged triangular wings 101
3.4.5 The contribution of the supersonic leading edge on the thin wedged triangular wing 103
3.4.6 The contributions of the leading edges on the thick-symmetrical wedged triangular wings 104
3.5 The complex axial disturbance velocities U and U* on the wedged triangular wing components 104
3.5.1 Introduction 104
3.5.2 The complex axial disturbance velocity U on the thin wedged triangular wing 105
3.5.3 The complex axial disturbance velocity U* on the thick-symmetrical wedged triangular wing 107
3.6 The axial disturbance velocities u and u* on the wedged delta wing components 110
3.7 The axial disturbance velocities u and u* on the wedged trapezoidal wing components 114
3.8 The axial disturbance velocities u and u* on the wedged rectangular wing components 120
3.9 Conclusions 121
References 122
Chapter 4 Computation of Axial Disturbance Velocities on Flying Configurations with Arbitrary Shapes, in Supersonic Flow, at NSL's Edge 125
4.1 General considerations 125
4.2 The theory of high conical flow of nth order 126
4.2.1 Definition of the high conical flow of the nth order and the homogeneity conditions 126
4.2.2 The Germain's compatibility conditions for the high conical flow of nth order 130
4.2.3 The Carafoli's hydrodynamic analogy for the high conical flow of nth order 131
4.2.4 The boundary conditions of the fictitious, complex potentials F[sub(f)] and Fast*[sub(f)], on the real axis of the Germain's complex plane 133
4.3 The principle of minimal singularities for the high conical flow of nth order 138
4.4 The solutions of boundary value problems of fictitious complex potentials F[sub(f)] and F*[sub(f)], on triangular wings 140
4.5 The axial disturbance velocities on the thin and thick-symmetrical triangular wings with arbitrary shapes 148
4.6 The axial disturbance velocities on delta wings with arbitrary shapes 154
4.7 The axial disturbance velocities on trapezoidal wings with arbitrary shapes 156
4.8 The axial disturbance velocities on rectangular wings with arbitrary shapes 159
4.9 The axial disturbance velocities on non-integrated or integrated delta wing-fuselage configurations 161
4.10 The axial disturbance velocities on non-integrated or integrated delta wing-fuselage configurations with movable leading edge flaps 166
4.11 Determination of the constants of axial disturbance velocities on flying configurations 170
4.12 Conclusions 171
References 172
Chapter 5 The Aerodynamical Characteristics of Flying Configurations with Arbitrary Shapes, in Supersonic Flow 175
5.1 General considerations 175
5.2 The computation of the aerodynamical characteristics of the delta wings 177
5.3 The computation of the aerodynamical characteristics of delta wing-fuselage configurations 184
5.4 The computation of the aerodynamical characteristics of delta wing-fuselage configurations, fitted with leading edge flaps, in open positions 191
5.5 The computation of the lift, pitching moment and drag coefficients of the rectangular wings 199
5.6 Conclusions 204
References 204
Chapter 6 The Visualizations of the Surfaces of Pressure Coefficients and Aerodynamical Characteristics of Wedged Delta and Wedged Rectangular Wings, in Supersonic Flow 207
6.1 Introduction 207
6.2 The three-dimensional visualizations of the C[sub(p)]-surfaces on the LAF's wedged delta wing, in supersonic flow 208
6.3 Visualizations of the behaviors of the C[sub(p)]-surfaces on a wedged delta wing, by crossing of sonic lines 218
6.4 Visualizations of the surfaces of lift and pitching moment coefficients of LAF's wedged delta wing and of their asymptotical behaviors, by crossing of sonic lines 220
6.5 The visualization of the inviscid drag coefficient's surface of the LAF's wedged delta wing and of its asymptotical behavior, by crossing of sonic lines 221
6.6 The polar surface of the LAF's wedged delta wing and its asymptotical behavior, by crossing of sonic lines 223
6.7 The visualizations of the C[sub(p)]-surfaces on the LAF's wedged rectangular wing 226
6.8 The behaviors of the C[sub(p)]-surfaces by changing of the LAF's wedged rectangular wing from long to short, at & #965
6.9 The three-dimensional visualizations of surfaces of aerodynamical characteristics of LAF's wedged rectangular wing 234
6.10 The polar surface of the LAF's wedged rectangular wing, in supersonic flow 238
6.11 Conclusions 239
References 241
Chapter 7 Qualitative Analysis of the NSL's Asymptotical Behaviors in the Vicinity of its Critical Zones 243
7.1 Introduction 243
7.2 Reduction of quadratical, elliptical and hyperbolical algebraic equations to their canonical forms 245
7.3 The asymptotical behaviors of quadratical algebraic equations with variable free term 247
7.3.1 General considerations 247
7.3.2 The qualitative analysis of the behaviors of quadratical, elliptical, algebraic equations in the vicinity of their black points 248
7.3.3 The qualitative analysis of the behaviors of quadratical, hyperbolical, algebraic equations in the vicinity of their saddle points 256
7.4 The qualitative analysis of elliptical and hyperbolical, quadratical, algebraic equations with variable coefficients of free and linear terms 266
7.4.1 General considerations 266
7.4.2 The collapse of the elliptical QAEs along their critical parabola 267
7.4.3 The degeneration of the hyperbolical QAEs along their critical parabola 269
7.5 The Jacobi determinant and the Jacobi hypersurface 270
7.6 The aerodynamical applications of the qualitative analysis of the QAEs 271
7.7 Conclusions 272
References 273
Chapter 8 Computation of the Friction Drag Coefficients of the Flying Configurations 275
8.1 Introduction 275
8.2 Computation of the inviscid lateral velocity & #965
8.3 The coupling between the NSL's slopes and the velocity field 282
8.4 Computation of friction and total drag coefficients of the delta wings 283
8.5 Conclusions 285
References 286
Chapter 9 Inviscid and Viscous Aerodynamical Global Optimal Design 288
9.1 Introduction 288
9.2 The optimum–optimorum theory 290
9.3 Inviscid aerodynamical global optimal design, via optimum–optimorum theory 292
9.4 Inviscid aerodynamic global optimal design of delta wing model ADELA, via optimum–optimorum theory 296
9.5 Inviscid aerodynamic global optimal design of fully-integrated wing/fuselage models FADET I and FADET II 298
9.6 The iterative optimum–optimorum theory and the viscous aerodynamical optimal design 302
9.7 Proposal for a fully-optimized and fully-integrated Catamaran STA 304
9.8 Conclusions 306
References 307
Chapter 10 Comparison of the Theoretical Aerodynamical Characteristics of Wing Models with Experimental-Determined Results 311
10.1 Introduction 311
10.2 The aims of the experimental program 312
10.3 Determination of experimental-correlated values of aerodynamical characteristics and of interpolated values of pressure coefficient 316
10.4 Comparison of theoretical aerodynamical characteristics of LAF's wedged delta wing model with experimental results 318
10.4.1 The description of LAF's wedged delta wing model 318
10.4.2 The computation of axial disturbance velocities on the upper side of wedged delta wings 318
10.4.3 The comparison of the theoretical and experimental-correlated values of C[sub(l)] and C[sub(m)] 323
10.5 Comparison of theoretical aerodynamical characteristics of LAF's double wedged delta wing model with experimental results 330
10.5.1 The description of LAF's double wedged delta wing model 330
10.5.2 Computation of downwashes and of axial disturbance velocities on double wedged delta wing 333
10.5.3 Comparison of theoretical and experimental-correlated C[sub(l)] and C[sub(m)] of LAF's double wedged delta wing 335
10.6 Comparison of theoretical aerodynamical characteristics of LAF's wedged delta wing model, fitted with a conical fuselage, with experimental results 338
10.6.1 Description of LAF's delta wing-fuselage model 338
10.6.2 The computation of downwashes and of axial disturbance velocities on the wedged delta wing model, fitted with conical fuselage 339
10.6.3 Comparison of the theoretical and experimental-correlated values C[sub(l)] and C[sub(m)] of LAF's wedged delta wing model, fitted with a conical fuselage 343
10.7 Comparison of theoretical aerodynamical characteristics of LAF's fully-optimized delta wing model ADELA with experimental results 346
10.7.1 Description of LAF's fully-optimized delta wing model ADELA 346
10.7.2 The computation of downwashes and of axial disturbance velocities on the fully-optimized delta wing model ADELA 349
10.7.3 Comparison of theoretical and experimental-correlated values of C[sub(l)] and C[sub(m)] of LAF's fully-optimized delta wing model ADELA 351
10.8 Comparison of theoretical aerodynamical characteristics of LAF's wedged rectangular wing model with experimental results 355
10.8.1 Description of LAF's wedged rectangular wing model 355
10.8.2 The computation of axial disturbance velocities on wedged rectangular wing model 358
10.8.3 The comparison of theoretical and experimental-correlated values of C[sub(l)] and C[sub(m)] of LAF's wedged rectangular wing 359
10.9 Comparison of theoretical aerodynamic characteristics of LAF's cambered rectangular wing model with experimental results 362
10.9.1 Description of LAF's cambered rectangular wing model 362
10.9.2 Computation of the axial disturbance velocities on LAF's cambered rectangular wing model 363
10.9.3 The comparison of theoretical and experimental-correlated values of C[sub(l)] and C[sub(m)] of LAF's cambered rectangular wing 366
10.10 Conclusions 368
References 371
Final Remarks 373
Outlook 375
Author Index 376
A 376
B 376
C 376
D 376
E 376
F 376
G 376
H 376
J 377
K 377
L 377
M 377
N 377
O 377
P 377
R 378
S 378
T 378
V 378
W 378
Y 378
Z 378
Subject Index 380
A 380
B 381
C 382
D 385
E 387
F 388
G 390
H 391
I 391
J 392
K 393
L 393
M 394
N 396
O 396
P 397
Q 399
R 399
S 400
T 403
U 405
V 405
W 407
X 408
Z 408
Plate Section 410
Governing Equations
2.1 The Flow and its Mathematical Description
Before we begin with the derivation of the basic equations describing the behaviour of the fluid, it may be convenient to clarify what the term ‘fluid dynamics’ stands for. It is, in fact, the investigation of the interactive motion of a large number of individual particles. These are in our case molecules or atoms. That means, we assume the density of the fluid is high enough, so that it can be approximated as a continuum. It implies that even an infinitesimally small (in the sense of differential calculus) element of the fluid still contains a sufficient number of particles, for which we can specify mean velocity and mean kinetic energy. In this way, we are able to define velocity, pressure, temperature, density and other important quantities at each point of the fluid.
The derivation of the principal equations of fluid dynamics is based on the fact that the dynamical behaviour of a fluid is determined by the following conservation laws, namely:
1. the conservation of mass,
2. the conservation of momentum, and
3. the conservation of energy.
The conservation of a certain flow quantity means that its total variation inside an arbitrary volume can be expressed as the net effect of the amount of the quantity being transported across the boundary, of any internal forces and sources, and of external forces acting on the volume. The amount of the quantity crossing the boundary is called flux. The flux can be in general decomposed into two different parts: one due to the convective transport and the other one due to the molecular motion present in the fluid at rest. This second contribution is of a diffusive nature it is proportional to the gradient of the quantity considered, and hence it will vanish for a homogeneous distribution.
The discussion of the conservation laws leads us quite naturally to the idea of dividing the flow field into a number of volumes and to concentrate on the modelling of the behaviour of the fluid in one such finite region. For this purpose, we define the so-called finite control volume and try to develop a mathematical description of its physical properties.
Finite control volume
Consider a general flow field as represented by streamlines in Fig. 2.1. An arbitrary finite region of the flow, bounded by the closed surface ∂Ω and fixed in space, defines the control volume Ω. We also introduce a surface element dS and its associated, outward pointing unit normal vector
Figure 2.1 Definition of a finite control volume (fixed in space).
The conservation law applied to an exemplary scalar quantity per unit volume U says that its variation in time within Ω, i.e.,
is equal to the sum of the contributions due to the convective flux – amount of the quantity U entering the control volume through the boundary with the velocity
further due to the diffusive flux – expressed by the generalised Fick’s gradient law
where κ is the thermal diffusivity coefficient, and finally due to the volume as well as surface sources, Qv, , i.e.,
After summing up the above contributions, we obtain the following general form of the conservation law for the scalar quantity U
(2.1)
where U* denotes the quantity U per unit mass, i.e., U/ρ.
It is important to note that if the conserved quantity would be a vector instead of a scalar, the above Equation (2.1) would be formally still valid. But in difference, the convective and the diffusive flux would become tensors instead of vectors the convective flux tensor and the diffusive flux tensor. The volume sources would be a vector , and the surface sources would change into a tensor . We can therefore write the conservation law for a general vector quantity as
(2.2)
The integral formulation of the conservation law, as given by the Equations (2.1) or (2.2), has two very important and desirable properties:
1. if there are no volume sources present, the variation of U depends solely on the flux across the boundary ∂Ω and not on any flux inside the control volume Ω;
2. this particular form remain valid in the presence of discontinuities in the flow field like shocks or contact discontinuities [1].
Because of its generality and its desirable properties, it is not surprising that the majority of the CFD codes today is based on the integral form of the governing equations.
In the following section, we shall utilise the above integral form in order to derive the corresponding expressions for the three conservation laws of the fluid dynamics.
2.2 Conservation Laws
2.2.1 The Continuity Equation
If we restrict our attention to single-phase fluids, the law of mass conservation expresses the fact that mass cannot be created in such a fluid system, nor it can disappear. There is also no diffusive flux contribution to the continuity equation, since for a fluid at rest, any variation of mass would imply a displacement of the fluid particles.
In order to derive the continuity equation, consider the model of a finite control volume fixed in space, as sketched in Fig. 2.1. At a point on the control surface, the flow velocity is , the unit normal vector is , and dS denotes an elemental surface area. The conserved quantity in this case is the density ρ. For the time rate of change of the total mass inside the finite volume Ω we have
The mass flow of a fluid through some surface fixed in space equals to the product of (density) × (surface area) × (velocity component perpendicular to the surface). Therefore, the contribution from the convective flux across each surface element dS becomes
Since by convection always points out of the control volume, we speak of inflow if the product is negative, and of outflow if it is positive and hence the mass leaves the control volume.
As stated above, there are no volume or surface sources present. Thus, by taking into account the general formulation of Eq. (2.1), we can write
(2.3)
This represents the integral form of the continuity equation – the conservation law of mass.
2.2.2 The Momentum Equation
We may start the derivation of the momentum equation by recalling the particular form of Newton’s second law which states that the variation of momentum is caused by the net force acting on an mass element. For the momentum of an infinitesimally small portion of the control volume Ω, (see Fig. 2.1) we have
The variation in time of momentum within the control volume equals
Hence, the conserved quantity is here the product of the density and the velocity,. i.e
The convective flux tensor, which describes the transfer of momentum across the boundary of the control volume, consists in the Cartesian coordinate system of the following three components
The contribution of the convective flux tensor to the conservation of momentum is then given by
The diffusive flux is zero since there is no diffusion of momentum possible for a fluid at rest. Thus, the remaining question is now, what are the forces the fluid element is exposed to? We can identify two kinds of forces acting on the control volume:
1. External volume or body forces, which act directly on the mass of the volume. These are for example gravitational, buoyancy, Coriolis or centrifugal forces. In some cases, there can be electromagnetic forces present as well.
2. Surface forces, which act directly on the surface of the control volume. They result from only two sources:
(a) the pressure distribution, imposed by the outside fluid surrounding the volume,
(b) the shear and normal stresses, resulting from the friction between the fluid and the surface of the volume.
From the above, we can see that the body force per unit volume, further denoted as , corresponds to the volume sources in Eq. (2.2). Thus, the contribution of the body (external) force to the momentum conservation is
The surface sources consist then of two parts – of an isotropic pressure component and of a viscous stress tensor , i.e.,
(2.4)
with being the unit tensor (for tensors see, e.g., [2]). The effect of the surface sources on the control volume is sketched in Fig. 2.2. In Section 2.3, we shall elaborate the form of the stress tensor in more detail, and in particular show how the normal and the shear stresses...
| Erscheint lt. Verlag | 20.12.2006 |
|---|---|
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik | |
| Technik ► Bauwesen | |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-08-052967-4 / 0080529674 |
| ISBN-13 | 978-0-08-052967-7 / 9780080529677 |
| Haben Sie eine Frage zum Produkt? |
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