Asymptotic Theory of Supersonic Viscous Gas Flows (eBook)

Front Cover 1
Asymptotic Theory of Supersonic Viscous Gas Flows 4
Copyright Page 5
Contents 8
Preface 16
Chapter 1 Flow in the Regions of Free Interaction Between a Supersonic Flow and a Boundary Layer 28
1.1 Derivation of the equations and boundary conditions 29
1.1.1 Estimates of the scales and characteristic values of the functions in disturbed flow regions 29
1.1.2 Asymptotic representations, equations, and boundary conditions 31
1.2 Flow near the separation point of the laminar boundary layer in a supersonic flow 36
1.2.1 Formulation of the problem and similarity law 36
1.2.2 Asymptotic behavior of the solution at minus infinity and results of the numerical solution of the problem 38
1.2.3 Results of calculations and comparison with experimental data 38
1.2.4 Note on the nature of upstream disturbance propagation in the interaction between the boundary layer and the outer flow 42
1.3 Separation far from the leading edge 45
1.4 Separation from a leading edge 49
Chapter 2 Other Types of Flows Described by Free Interaction Theory 52
2.1 Laminar boundary layer separation in a supersonic flow under conditions of low skin friction 52
2.1.1 Formulation of the problem. Estimation of the scales and characteristic values of the flow functions in the wall region 52
2.1.2 Equations and boundary conditions 54
2.1.3 Solution of the linear boundary value problem 57
2.2 Expansion flow 60
2.2.1 Asymptotic behavior of the solution, as & #958
2.2.2 Results of calculations 62
2.3 Other types of flows described by free interaction equations 64
2.3.1 Equations and boundary conditions for the case of a curvilinear body contour 66
2.3.2 Flow inside a corner somewhat smaller than & #960
2.3.3 Formulation of other problems for flows with free interaction 69
2.3.4 Integration of the equations 71
2.4 Elimination of boundary layer separation by means of slot suction 72
2.4.1 Formulation of the problem 73
2.4.2 Derivation of the equations and boundary conditions for regions 1 and 2 75
2.4.3 Solutions for nonlinear inviscid flow regions for & #916
2.4.4 Solutions for finite-length flaps and bodies with a bend in the contour for 1 & #8811
2.4.5 Flow past a flap deflected by an angle & #952
2.4.6 Flow patterns in the laminar boundary layer for finite flap deflection angles 84
Chapter 3 Viscous Gas Flows in Regions with Developed Locally Inviscid Zones and High Local Pressure Gradients 88
3.1 Formulation of the problem of the expansion flow near a corner point on a body in supersonic flow 88
3.1.1 Asymptotic expansions 90
3.1.2 Upstream disturbance decay 93
3.1.3 Boundary conditions for the viscous sublayer 32 95
3.1.4 Bringing the equations for region 33 into the standard form 97
3.1.5 Solution of the problem in the region of locally inviscid flow 22 98
3.2 Flow ahead of the base section of a body 102
3.2.1 Formulation of the problem and characteristic flow regions 102
3.2.2 Solution of the problem and comparison with experimental data 103
3.3 Reattachment of a supersonic flow to the body surface 104
3.3.1 Formulation of the problem and main flow regions 106
3.3.2 Nature of the locally inviscid flow in region 22 107
3.3.3 Solution for the problem of the locally inviscid flow in region 22 109
3.3.4 Viscous flow regions 113
3.3.5 Solution for the region with maximum friction and heat flux values 120
3.3.6 Discussion of the Chapman–Korst criterion 122
3.4 Problems with discontinuous boundary conditions describing laminar high-Reynolds-number flows 128
3.4.1 Structure of disturbed flow regions 129
3.4.2 Analysis of the regimes described by free interaction theory 134
3.4.3 Boundary value problem for the case & #949
3.4.4 Numerical solution of the problem 138
3.4.5 Analysis of nonlinear time-dependent flow patterns 140
3.4.6 Examples of numerical solutions of nonlinear time-dependent problems 143
3.5 Structure of chemically nonequilibrium flows at jumpwise variation of the temperature and catalytic properties of the surface 147
3.5.1 Formulation of the problem 148
3.5.2 Parameter scales, equations, and boundary conditions 150
3.5.3 Analysis of the flow in region IV near the point of jumpwise variation of the temperature and catalytic properties of the surface 156
3.5.4 Results of numerical calculations 159
Chapter 4 Flows Under Conditions of the Interaction Between the Boundary Layer and the Outer Flow Along the Entire Body Length 164
4.1 Regime of weak interaction with the outer flow 164
4.2 Moderate and strong interactions in a hypersonic flow 169
4.2.1 Flow nature in the locations of rapid variation of the boundary conditions 169
4.2.2 Equations and boundary conditions for the flat-plate flows in the presence of moderate and strong interactions 171
4.2.3 Study of the nature of the nonuniqueness of the boundary value problem 174
4.2.4 Results of calculations and comparison of the similarity law with the experimental data 179
4.3 Theory of hypersonic flow/boundary layer interaction for two-dimensional separated flows 184
4.3.1 Formulation of the problem, equations, and boundary conditions 184
4.3.2 Similarity criteria 189
4.4 Propagation of disturbances at strong distributed gas injection through the body surface to a supersonic flow 190
4.4.1 Formulation of the problem and derivation of the equations 190
4.4.2 Analysis of the solutions for region 1 192
4.4.3 Flow near the base section 194
4.4.4 Concluding remarks 197
4.4.5 Integration of Eqs. (4.36) 199
4.5 Detachment of a laminar boundary-layer 200
4.5.1 Formulation of the problem, equations, and boundary conditions 201
4.5.2 Results of the solution 204
4.6 Gas injection into a hypersonic flow 206
4.6.1 Formulation of the problem 206
4.6.2 Equations and boundary conditions 206
4.6.3 Self-similar solutions 209
4.6.4 Analysis of the N = O(1) regime 213
4.6.5 Dependence of the solution on the base pressure difference 215
4.6.6 Base pressure difference effect on the flow past an impermeable surface 217
4.7 Gas injection into a hypersonic flow (moderate injection) 217
4.7.1 Formulation of the problem and boundary conditions 219
4.7.2 Results of the solution 221
Chapter 5 Three-Dimensional Hypersonic Viscous Flows 226
5.1 Viscous flow over a low-aspect-ratio wing in the weak interaction regime (longitudinal-transverse interaction) 226
5.1.1 Special features of the formulation of the boundary value problem 226
5.1.2 Original relations and estimates 227
5.1.3 Equations and boundary conditions 228
5.1.4 Eigenvalue problem 230
5.1.5 Approximate calculation of the flow past a wing in the self-similar case 232
5.1.6 Finite-difference method for solving the problem 234
5.1.7 Numerical results 240
5.2 Formation of secondary flows on thin semi-infinite wings 242
5.2.1 Estimation of secondary flow parameters in boundary layers on thin wings 242
5.2.2 Thin semi-infinite wing at zero incidence 246
5.2.3 Plane cross-section law 250
5.3 Thin power-law wings in weak viscous-inviscid interaction 251
5.3.1 Formulation of the boundary value problem 252
5.3.2 On the nature of the pressure distribution 255
5.3.3 Certain features of the solution of boundary value problems 255
5.3.4 Characteristics of the self-similar solution 259
5.3.5 Approximate solution of the problem for delta wings 261
5.4 Strong viscous interaction regime on delta and swept wings 263
5.4.1 Formulation of the problem 263
5.4.2 Equations and boundary conditions 264
5.4.3 Strong viscous interaction on a delta wing 266
5.4.4 Solution in the vicinity of the leading edge 267
5.4.5 Strong viscous interaction on a swept plate 268
5.4.6 Propagation of disturbances from the trailing edge of a swept plate 269
5.4.7 Delta wing 271
5.5 Distinctive features of the symmetric flow over a thin triangular plate in the strong interaction regime 273
5.5.1 Equations and boundary conditions 273
5.6 Finite-length wings in the strong viscous interaction regime 279
5.6.1 Mathematical formulation of the problem 279
5.6.2 Aerodynamic characteristics of finite-length wings at zero incidence 282
5.6.3 Wings of finite length at an angle of attack 286
5.7 Wings of finite length in the moderate viscous interaction regime 291
5.7.1 Mathematical formulation of the problem 292
5.7.2 Aerodynamic characteristics of a wing at zero incidence 294
5.7.3 Angle-of-attack effect of the aerodynamic characteristics 297
Chapter 6 Supercritical and Transcritical Interaction Regimes: Two-Dimensional Flows 304
6.1 Distinctive features of boundary layer separation on a cold body and its interaction with a hypersonic flow 305
6.1.1 Formulation of the problem 305
6.1.2 Starting estimates 306
6.1.3 Solution for the hypersonic regime of weak viscous interaction 308
6.1.4 Discussion of the results 310
6.1.5 Supercritical regime of incipient separation 313
6.2 Distinctive features of interaction and separation of a transcritical boundary layer 316
6.2.1 Equations and boundary conditions 316
6.2.2 Flow in region 3 320
6.2.3 Classification of flow regimes 320
6.2.4 Properties of transcritical flows corresponding to curve AB 325
6.2.5 Properties of integral curves 327
6.3 Study of time-dependent processes of transcritical interaction between the laminar boundary layer and a hypersonic flow 331
6.3.1 Estimates of the scales 331
6.3.2 Formulation and solution of the boundary value problem 333
6.3.3 Conclusion 340
6.4 Analysis of the boundary layer flow near the trailing edge of a flat plate and in its wake in the strong hypersonic interaction regime 340
6.4.1 Formulation of the problem 340
6.4.2 Investigation of the plate wake flow in the vicinity of the point of subcritical-to-supercritical transition 342
6.4.3 Investigation of the flow in the vicinity of the transition point for a near-supercritical regime 344
6.4.4 Analysis of the flow in the vicinity of the trailing edge of a flat plate in the subcritical and supercritical regimes 347
6.4.5 Analysis of the flow in the vicinity of the trailing edge of a flat plate in the transcritical interaction regime 349
6.5 Global solution for the hypersonic flow over a finite-length plate with account for the wake flow 355
6.5.1 Formulation of the problem 355
6.5.2 Transformation of variables 357
6.5.3 Results of calculations 357
6.6 Strong interaction of the boundary layer with a hypersonic flow under local disturbances of boundary conditions 359
6.6.1 Formulation of the problem 359
6.6.2 Estimates of the orders of the flow parameters 360
6.6.3 Flow regime with finite pressure disturbances 363
6.6.4 Flow patterns with small pressure differences 368
6.6.5 Concluding remarks 370
Chapter 7 Three-Dimensional Hypersonic Viscous Flows with Supercritical and Subcritical Regions 372
7.1 Strong interaction between a hypersonic flow and the boundary layer on a cold delta wing 372
7.1.1 Equations and boundary conditions 372
7.1.2 Solution near the leading edge 374
7.1.3 Flow regimes 376
7.1.4 Analysis of the solution in the vicinity of the critical section 377
7.1.5 Aerodynamic characteristics of delta wings 384
7.1.6 Characteristics for supercritical boundary layers and wakes for an arbitrary wing planform 386
7.2 Propagation of disturbances in three-dimensional time-dependent boundary layers 390
7.2.1 Formulation of the problem 390
7.2.2 Determining subcharacteristic surfaces in time-dependent three-dimensional flows 392
7.2.3 Results of the numerical analysis 395
7.2.4 Two-dimensional flows 397
7.2.5 Three-dimensional boundary layer 399
7.3 Supercritical regimes of hypersonic flow over a yawed planar delta wing 399
7.3.1 Equations and boundary conditions 400
7.3.2 Results of the calculations 402
7.4 Existence of self-similar solutions in the supercritical region on a nonplanar delta wing in hypersonic flow 405
7.4.1 Equations and boundary conditions 405
7.4.2 Self-similar solutions 407
7.4.3 Results of calculations 409
7.5 Effect of strong cooling of the surface on the hypersonic viscous flow over a nonplanar delta wing 413
7.5.1 Equations and boundary conditions 414
7.5.2 Results of calculations 416
7.6 Self-similar flows with gas injection from the triangular plate surface into a hypersonic flow 420
7.6.1 Equations and boundary conditions 421
7.6.2 Reduction to self-similar form 424
7.6.3 Results of calculations 425
7.7 Mass transfer on a planar delta wing in the presence of a supercritical flow region in the boundary layer 430
7.7.1 Equations and boundary conditions 431
7.7.2 Results of calculations 431
7.8 Mass transfer on a nonplanar delta wing 436
7.8.1 Equations and boundary conditions 436
7.8.2 Results of calculations 438
7.9 Using the Newtonian passage to limit for studying the flow over a delta wing 444
7.9.1 Estimates of the flow parameters 445
7.9.2 Self-similar variables 451
7.9.3 Conditions of supercritical-to-m-subcritical flow regime transition 454
Chapter 8 Boundary Layer Flow Over Roughnesses at Body Surfaces 460
8.1 Flow over two-dimensional roughnesses 460
8.1.1 General formulation of the problem and classification of flow regimes 460
8.1.2 Flow over "short" roughnesses embedded in the wall region of the undisturbed boundary layer 462
8.1.3 Flow over "short" roughnesses with the formation of locally inviscid disturbed flow regions 466
8.1.4 Flow over roughnesses with a characteristic length equal in the order to the boundary layer thickness 473
8.1.5 Flow over "long" roughnesses whose length is greater than the boundary layer thickness 478
8.1.6 Classification diagram of the regimes of the flow over small two-dimensional roughnesses 483
8.1.7 Examples of solutions for the flow over two-dimensional roughnesses 484
8.1.8 Classification of the regimes of flow over roughness on a cold surface 492
8.2 Regimes of the flow over three-dimensional roughnesses 502
8.2.1 Flow over fairly narrow roughness of the type of a hole or a hill 502
8.2.2 Flow over streamwise-elongated narrow roughnesses 508
8.2.3 Compensation regime of the flow over roughnesses 513
8.3 Numerical investigation of the three-dimensional flow over roughnesses in the compensation interaction regime 524
8.3.1 Formulation of the problem and estimates for the scales 525
8.3.2 Boundary value problem 528
8.3.3 Numerical solution 530
Bibliography 536
Index 554
A 554
B 554
C 556
D 556
E 556
F 557
G 557
H 558
I 559
L 559
M 559
N 559
O 560
P 560
R 560
S 560
T 562
U 562
V 562
W 563
Y 563
Z 563