Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T14:50:41.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Compressible corner flow

Published online by Cambridge University Press:  29 March 2006

Bernard C. Weinberg  and
Stanley G. Rubin
Bernard C. Weinberg
Affiliation:
Polytechnic Institute of Brooklyn, Preston R. Bassett Research Laboratory, Farmingdale, New York Present address: Department of Fluid and Heat Transfer, Negev University, Beer-Sheva, Israel.
Stanley G. Rubin
Affiliation:
Polytechnic Institute of Brooklyn, Preston R. Bassett Research Laboratory, Farmingdale, New York
Get access Rights & Permissions [Opens in a new window]

Abstract

The viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated. In the absence of viscous-inviscid interactions and imbedded shock waves, a theory is developed that is valid throughout the subsonic and supersonic Mach number range. Within this limitation and the additional assumptions of unit Prandtl number and a linear viscosity-temperature law, a consistent set of governing equations and boundary conditions is derived. The method of matched asymptotic expansions is applied in order to distinguish the relevant regions in the flow field.

In the corner region the Crocco integral is shown to apply, even for a three-dimensional flow field. The equations governing the flow in the corner layer consist of four coupled nonlinear elliptic partial differential equations of the Poisson variety. Since they do not lend themselves to analytic solution, numerical methods are employed. Two such methods used here are the Gauss-Seidel explicit technique and the alternating direction implicit method. The merits of both techniques are discussed with regard to convergence rate, accuracy and stability. The calculations show that in cases where the Gauss-Seidel method fails to give converged solutions, owing to instability, the alternating direction implicit method does provide converged solutions. However, in cases where both methods are convergent, there is no appreciable difference in convergence rates. The numerical calculations were done on a CDC 6600 computer.

Results of calculations are presented for representative compressible-flow conditions. The extent of the corner disturbance is controlled by the Mach number and wall temperature ratio in a manner analogous to the two-dimensional boundary layer. A swirling motion is noted in the corner layer which is influenced to a great extent by the asymptotic cross-flow profiles. The skin-friction coefficient is shown to increase monotonically from zero at the corner point to its asymptotic two-dimensional value. For cold wall cases, this value is approached more rapidly. The asymptotic analysis indicates that for even colder wall cases, not considered here, an overshoot is possible.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 56 , Issue 4 , 28 December 1972 , pp. 753 - 774
Copyright
© 1972 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bloom, M. H. 1966 Remark on compressibility effects in the boundary-layer cross-flow near a corner. Polytechnic Institute of Brooklyn, PIBAL Rep. no. 969.
Bloom, M. H. & Rubin, S. G. 1961 High-speed viscous corner flow. J. Aero. Sci. 28, 145–157.Google Scholar
Bloom, M. H., Rubin, S. G. & Cresci, R. J. 1970 Three-dimensional viscous interactions. Proc. Symp. Viscous Interaction Phenomena in Supersonic and Hypersonic Flows, pp. 493–511. Dayton University Press.
Libby, P. A. 1966 Secondary flows associated with a supersonic corner region. A.I.A.A. J. 4, 1130–1132.Google Scholar
Libby, P. A. & Fox, H. 1963 Some perturbation solutions in laminar boundary-layer theory. Part 1. Momentum equation. J. Fluid Mech. 17, 433–439.Google Scholar
Pal, A. & Rubin, S. G. 1971 Asymptotic features of viscous flow along a corner. Quart. Appl. Math. 29, 91–108.Google Scholar
Peaceman, D. W. & Rachford, H. H. 1955 The numerical solution of parabolic and elliptic differential equations. J. Soc. Indust. Appl. Math. 3, 28–41.Google Scholar
Rubin, S. G. 1966 Incompressible flow along a corner. J. Fluid Mech. 26, 97–110.Google Scholar
Rubin, S. G. & Grossman, B. 1971 Viscous flow along a corner. Part 2. Numerical solution of corner-layer equations. Quart. Appl. Math. 29, 169–186.Google Scholar
Rubin, S. G. & Lin, T. C. 1972 Numerical methods for three-dimensional viscous flow problems. Application to the hypersonic leading edge. J. Comp. Phys. 9, 339–364.
Smith, G. D. 1965 Numerical Solutions of Partial Differential Equations. Oxford University Press.
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Oxford University Press.
Weinberg, B. C. 1972 Compressible corner flow. Ph.D. thesis, Polytechnic Institute of Brooklyn.