A numerical study of the viscous supersonic flow past a flat plate is presented. The objective is to investigate the supersonic flow at high angles of incidence where large flow gradients occur. The effect of the angle of incidence and the Reynolds number (Re) in the flow structure especially in the formation of the separation region is investigated. The study is based on the solution of the full Navier–Stokes equations by high resolution schemes, and it focuses on the supersonic flow over the plate at Re≤105. Results on fine computational grids are presented for flow angles up to 20°. The calculations reveal that the flow remain attached for angles of incidence less than a=5°. For a=5° and Re=105, separation of the flow at the trailing edge appeared. Increasing the flow angle (a≳5°) moves the separation point upstream while a reverse flow region forms for the entire range of the Reynolds numbers used in this study. The results reveal that for large angles of incidence, the variation of the Reynolds number has significant effects on the variation of the flow variables. The flow behind the trailing edge is also affected from the flow angle as well as from the Reynolds number. Comparisons are also presented between viscous and inviscid solutions. The comparisons show that the viscous effects are dominant on the upper surface of the plate as well as behind the trailing edge. These effects become stronger when the flow angle is a=20°.
REFERENCES
1.
H. W. Liepmann and A. Roshko, Elements of Gasdynamics (Wiley, New York, 1957).
2.
F.
Durst
and K.
Brieden
, “Die Überschallströmung der angestellten, ebenen Platte
,” Z. Angew. Math. Phys.
21
, 947
(1970
).3.
S.
Goldstein
, “On laminar boundary layer flow near a point of separation
,” Q. J. Mech. Appl. Math.
1
, 43
(1948
).4.
L.
Lees
and B. L.
Reeves
, “Supersonic separated and reattaching laminar flows
,” AIAA J.
2
, 1907
(1944
).5.
L.
Crocco
, “Considerations on the shock-boundary layer interaction
,” Proc. Conf. High Speed Aero.
1
, 75
(1955
).6.
K.
Stewartson
and P. G.
Williams
, “Self-induced separation
,” Proc. R. Soc. London Ser. A
312
, 181
(1969
).7.
V. Y.
Neiland
, “Towards a theory of separation of the laminar boundary layer in a supersonic stream
,” Izv. Akad. Nauk. SSSR, Mekh, Zhidk. Gaza
4
, 33
(1969
).8.
K.
Stewartson
, “On the flow near the trailing edge of a flat plate
,” Proc. R. Soc. London Ser. A
306
, 275
(1968
).9.
A. F.
Messiter
, “Boundary-layer flow near the trailing edge of a flat plate
,” SIAM J. Appl. Math.
18
, 241
(1970
).10.
V.
YA. Sychev
, “Concerning laminar separation
,” Izv. Akad. Nauk. SSSR, Mekh, Zhidk. Gaza
3
, 47
(1952
).11.
F. T.
Smith
, “The laminar separation of an incompressible fluid streaming past a smooth surface
,” Proc. R. Soc. London Ser. A
356
, 433
(1977
).12.
K.
Stewartson
, “On the flow near a trailing edge of a flat plate II
,” Mathematica
16
, 106
(1969
).13.
14.
A. F.
Messiter
, “Boundary-layer interaction theory
,” J. Appl. Mech.
50
, 1104
(1983
).15.
F. T.
Smith
, “On the high Reynolds number theory of laminar flows
,” IMA J. Appl. Math.
28
, 207
(1982
).16.
S. N.
Brown
and K.
Stewartson
, “Trailing-edge stall
,” J. Fluid Mech.
42
, 561
(1970
).17.
N.
Riley
and K.
Stewartson
, “Trailing edge flows
,” J. Fluid Mech.
39
, 193
(1969
).18.
F. T.
Smith
and K.
Stewartson
, “Plate-injection into a separated supersonic boundary layer
,” J. Fluid Mech.
58
, 143
(1973
).19.
P. G.
Daniels
, “Laminar boundary-layer reattachment in supersonic flow
,” J. Fluid Mech.
90
, 289
(1979
).20.
P. G.
Daniels
, “Numerical and asymptotic solutions for the supersonic flow near the trailing edge of a flat plate at incidence
,” J. Fluid Mech.
63
, 641
(1974
).21.
P. G.
Rizzeta
, O. R.
Burggraf
, and R.
Jenson
, “Triple-deck solutions for viscous supersonic and hypersonic flow past corners
,” J. Fluid Mech.
89
, 535
(1978
).22.
A.
Kluwick
, P.
Gittler
, and R. J.
Bodonyi
, “Viscous-inviscid interactions on axisymmetric bodies of revolutions in supersonic flow
,” J. Fluid Mech.
140
, 281
(1984
).23.
V. N.
Vatsa
and M. J.
Werle
, “Quasi-three-dimensional laminar boundary-layer separations in supersonic flow
,” Trans. ASME I: J. Fluids Eng.
99
, 634
(1977
).24.
C. E.
Jobe
and O. R.
Burggraf
, “The numerical solution of the asymptotic equations of trailing edge flow
,” Proc. R. Soc. London Ser. A
340
, 91
(1974
).25.
R. Chow and R. E. Melnik, “Numerical solutions of the triple-deck equations for laminar trailing edge stall,” Proceedings of the 5th International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics (Springer-Verlag, Berlin, 1976), Vol. 59.
26.
M.
Napolitano
, M. J.
Werle
, and R. T.
Davis
, “A numerical technique for the triple-deck problem
,” AIAA J.
17
, 699
(1979
).27.
M. J.
Werle
and V. N.
Vatsa
, “A new method for supersonic boundary layer separation
,” AIAA J.
12
, 1491
(1974
).28.
T. S.
Lewis
and L.
Sirovich
, “Approximate and exact numerical computation of supersonic flow over an airfoil
,” J. Fluid Mech.
112
, 265
(1981
).29.
J. L.
Steger
and R. F.
Warming
, “Flux vector splitting of the inviscid gas dynamic equations with applications to finite-difference methods
,” J. Comput. Phys.
40
, 263
(1981
).30.
B. van Leer, “Flux-vector splitting for the Euler equations,” Lecture Notes in Physics (Springer-Verlag, Berlin, 1982), Vol. 170, p. 507.
31.
P. L.
Roe
, “Approximate Riemann-solvers, parameters vectors, and difference schemes
,” J. Comput. Phys.
43
, 357
(1981
).32.
P.
Colella
and H. M.
Glaz
, “Efficient solution algorithms for the Riemann problem for real gases
,” J. Comput. Phys.
59
, 264
(1985
).33.
S.
Osher
and F.
Solomon
, “Upwind schemes for hyperbolic systems of conservations laws
,” Math. Comput.
38
, 339
(1981
).34.
A.
Eberle
, “Characteristic flux averaging approach to the solution of Euler’s equations
,” VKI Lecture Series, Comp. Fluid Dyn.
1987–04
, 1
(1987
).35.
A.
Harten
, “ENO schemes with subcell resolution
,” J. Comput. Phys.
83
, 148
(1989
).36.
H. C.
Yee
and A.
Harten
, “Implicit TVD schemes for hyperbolic conservation laws in curvilinear coordinates
,” AIAA J.
25
, 265
(1987
).37.
S. R. Chakravarthy, K. Y. Shema, and U. C. Goldberg, “Application of a new class of high accuracy TVD schemes to the Navier-Stokes equations,” AIAA Paper No. 85–0165 (1985).
38.
H. C. Yee, “Upwind and symmetric shock capturing schemes,” NASA Tech. Memo. TM-89464 (1987).
39.
D.
Drikakis
and S.
Tsangaris
, “On the solution of Navier-Stokes equations using improved flux vector splitting methods
,” Appl. Math. Modeling
17
, 282
(1993
).40.
D. Drikakis and S. Tsangaris, “Laminar and turbulent viscous compressible flows using improved flux vector splittings,” Proceedings of the Ningth GAMM-Conference on Numerical Methods in Fluid Mechanics, Lausanne, Notes on Numerical Fluid Mechanics, edited by J. B. Vos, A. Rizzi, and I. L. Ryhming (Vieweg, Braunschweig, 1992), Vol. 35, pp. 407–416.
41.
D.
Drikakis
and S.
Tsangaris
, “Local solution acceleration method for the Euler and Navier-Stokes equations
,” AIAA J.
30
, 340
(1992
).42.
D. Drikakis and F. Durst, “Numerical study of transonic turbulent separated flows,” Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, 16–18 August 1993, Vol. 2, p. 263–1.
43.
M. A. Schmatz, A. Brenneis, and A. Eberle, “Verification of an implicit relaxation method for steady and unsteady viscous and inviscid flow problems,” AGARD Report No. AGARD CP 437, 15–1, 1988.
44.
D. Drikakis and E. Schreck, “Development of parallel implicit Navier-Stokes solvers on MIMD multi-processor systems,” AIAA Paper No. 93–0062, 31st Aerospace Sciences Meeting and Exhibit, Reno, Nevada 11–14 January, 1993.
45.
D. Drikakis and E. Schreck, “Parallel multi-level calculations for viscous compressible flows,” ASME Conference, Washington, DC, June 1993, CFD Algorithms and Applications for Parallel Processors, edited by O. Baysal and V. Saxena (ASME, New York, 1993), Vol. 156.
46.
R. L. Sorenson, “A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson’s equations,” NASA Tech. Memo. TM-81198 (1980).
47.
A. Jameson, W. Schmidt and E. Turkel, “Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes,” AIAA Paper No. 81–1259, 1981.
48.
H. Schlichting, Boundary-Layer Theory (McGraw Hill, New York, 1968).
49.
Y. H.
Kuo
, “Viscous flow along a flat plate moving at high supersonic speeds
,” J. Aeronaut. Sci.
1
, 125
(1956
).50.
J. D. Anderson, Jr., Modern Compressible Flows: With Historical Perspective (McGraw-Hill, New York, 1982).
51.
N. F. Krasnov, Methods of Aerodynamic Calculations (Mir, Moscow, 1985).
52.
S. R.
Chakravarthy
, “High resolution upwind formulations for the Navier-Stokes equations
,” VKI Lecture Series, Comput. Fluid Dyn.
1988–05
, 1
(1988
).
This content is only available via PDF.
© 1994 American Institute of Physics.
1994
American Institute of Physics
You do not currently have access to this content.
