* Computational work and experimental results show the real world application of computational results
* Easy computation and visualization of inviscid and viscous aerodynamis characteristics of flying configerations
* Includes a fully optimized and integrated design for a proposed supersonic transport aircraft
Computation of Supersonic Flow over Flying Configurations is a high-level aerospace reference book that will be useful for undergraduate and graduate students of engineering, applied mathematics and physics. The author provides solutions for three-dimensional compressible Navier-Stokes layer subsonic and supersonic flows. Computational work and experimental results show the real-world application of computational results Easy computation and visualization of inviscid and viscous aerodynamic characteristics of flying configurations Includes a fully optimized and integrated design for a proposed supersonic transport aircraft
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
Zonal, Spectral Solutions for the Three-Dimensional, Compressible Navier–Stokes Layer
Contents
1.2 Three-dimensional, partial-differential equations of compressible Navier-Stokes layer (NSL)
1.4 The first and second derivatives of the velocity’s components
1.5 The implicit and explicit forms of the boundary conditions at the NSL’s edge
1.6 The dependence of the density function R versus the spectral velocity, inside the NSL
1.7 Dependence of absolute temperature T versus the spectral velocity, inside the NSL
1.10 An original iterative method to solve a quadratical algebraic system
1.1. Introduction
The start equations considered in this book are the partial-differential equations (PDE) of the Navier-Stokes layer (NSL), deduced in [1] and [2], without any simplifications.
In this chapter are proposed own, zonal, spectral solutions for the compressible stationary NSL over flattened flying configurations (FC), as in [3]–[15]. These developed NSL’s solutions present the following new characteristics, as in [8]–[12]:
• the velocity’s components uδ, vδ, wδ are written as products between the potential velocities at the edge and polynoms in the spectral variable η with free spectral coefficients, namely ui, vi, wi;
• the absolute temperature T and the here-introduced density function R = ln ρ are supposed to be written in the form of sums between their wall values and terms, which are also products of their edge values (obtained from the outer potential flow, at the edge) with polynoms in η also with free spectral coefficients ri and ti;
• all these spectral coefficients are used to fulfill the NSL’s PDE as exactly as possible, in an arbitrary chosen number of points;
• the seven boundary conditions for the velocity at the NSL’s edge, which are usually written as implicit relations, are here explicitly written. Seven spectral coefficients are no longer free, because the boundary conditions at the edge are now automatically fulfilled;
• in the NSL’s PDE a change of variable is proposed. The here-called density function R = ln ρ is used instead of the density ρ. This change of variable is introduced in the continuity’s PDE. A linear PDE, only versus the first partial derivatives of the density function R is obtained. If a spectral form for R is also used, its spectral coefficients ri are obtained only as functions of the velocity’s coefficients, by solving a linear algebraic system;
• these spectral forms also allow us to write the other physical entities only as functions of the velocity’s spectral coefficients;
• at the NSL’s edge, reinforced potential solutions over the modified FC, obtained after the solidification of the NSL, are used instead of the outer parallel flow, taken by Prandtl at the edge of his boundary layer. The author uses the original three-dimensional, hyperbolic, potential solutions, presented in the Chaps 3 and 4, at the NSL’s edge. These reinforced potential solutions are deduced by using the hydrodynamic analogy of E. Carafoli. The NSL’s solutions presented here fulfill the correct asymptotical behaviors along the leading edges and the ridges and satisfy the last behavior, and, for supersonic flow, the boundary condition on the characteristic surface of the FC is satisfied, due to the outer potential flow. The NSL’s solutions presented here are also reinforced zonal, spectral, NSL’s solutions.
Some applications of the zonal, spectral solutions for the three-dimensional compressible NSL’s partial differential equations are presented in Chaps 7 to 9 of this book.
1.2. Three-dimensional, partial-differential equations of compressible Navier-Stokes layer (NSL)
The partial differential equations of the three-dimensional compressible, stationary NSL on the FC are, after [1] and [2], the following:
• the continuity equation
(1.1)
• the impulse equations (1.2a)
(1.2a)
(1.2b)
(1.2c)
• the equations of the absolute temperature T and of the internal energy E are:
(1.3)
(1.4)
where , p(x1, x2, η), ρ and μ are the local velocity, pressure, density and viscosity, λ is the coefficient of the thermal conductivity of gas, Cp and Cv are the coefficients of specific heat by constant pressure and by constant volume, respectively, and φd is the following dissipation function:
(1.5)
By using the physical gas equation (i.e. here the equation of perfect gas), the pressure p, inside the NSL, can be expressed only as a function of the absolute temperature T and of the density function R, i.e.:
(1.6)
The viscosity μ depends only on the temperature T. If an exponential law is accepted, it results in:
(1.7)
where Rg is the universal gas constant, μ∞ and T∞ are the values of viscosity and absolute temperature of the undisturbed flow, E is the internal energy, a∞ is the sound’s propagation velocity in the undisturbed flow, κ is the ratio of the coefficients Cp and Cv and n1 is the exponent of the exponential law (n1 = 0.76 for air).
1.3. The spectral variable and the spectral forms of the velocity’s components and of the physical entities
Let us now denote δ+(x1, x2) and δ−(x1, x2) the thicknesses of the NSL on the upper and lower surfaces Z+(x1, x2) and Z−(x1, x2), respectively, of an arbitrary flattened FC. Further, only the NSL on the upper surface is considered, i.e. Z+(x1, x2) = Z(x1, x2) and δ+(x1, x2) = δ(x1, x2). Further a new coordinate η is proposed here, inside the NSL:
(1.8)
The first and second derivatives of these coordinates, used further, are the following:
(1.9a–c)
(1.10a–f)
Remarks
• The range of the dimensionless spectral coordinate η, inside the NSL, is 0 ≤ η ≤ 1, as wished.
• The coordinate η depends on the equation of the surface Z(x1, x2) and of the NSL’s thickness distribution δ. For the computation problems the equation of the surface is given.
The dimensionsless axial, lateral and vertical velocities uδ, vδ and wδ, inside the upper NSL (which is considered here only) are supposed to be expressed in the following spectral forms, as in [3]–[18], i.e.:
(1.11a–c)
Further the following spectral forms for the density function R = ln ρ and the absolute temperature T are used:
(1.12)
(1.13)
where Rw and Tw are the values of R and T at the wall and ue, ve, we, Re, Te are the edge values of uδ, vδ, wδ, R, T, which can be easily obtained from the outer inviscid...
| Erscheint lt. Verlag | 7.7.2010 |
|---|---|
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Naturwissenschaften ► Physik / Astronomie ► Mechanik | |
| Technik ► Bauwesen | |
| Technik ► Fahrzeugbau / Schiffbau | |
| Technik ► Luft- / Raumfahrttechnik | |
| Technik ► Maschinenbau | |
| Wirtschaft | |
| ISBN-10 | 0-08-055699-X / 008055699X |
| ISBN-13 | 978-0-08-055699-4 / 9780080556994 |
| Haben Sie eine Frage zum Produkt? |
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