Abstract

This paper describes an accurate, robust and efficient methodology for solving two-dimensional steady transonic turbomachinery flows. The Euler fluxes are discretized in space using a hybrid multidimensional upwind method, which, according to the local flow conditions, uses the most suitable fluctuation splitting (FS) scheme at each cell of the computational domain. The viscous terms are discretized using a standard Galerkin finite element scheme. The eddy viscosity is evaluated by means of the Spalart-Allmaras turbulence transport equation, which is discretized in space by means of a mixed FS-Galerkin approach. The equations are discretized in time using an implicit Euler scheme, the Jacobian being evaluated by two-point backward differences. The resulting large sparse linear systems are solved sequentially using a preconditioned GMRES strategy. The proposed methodology is employed to compute subsonic and transonic turbulent flows inside a high-turning turbine-rotor cascade.

Issue Section:
Technical Papers
Keywords:
computational fluid dynamics, subsonic flow, transonic flow, flow instability, turbulence, Galerkin method, fluctuations, viscosity, turbines, rotors
Topics:
Flow (Dynamics), Mach number, Navier-Stokes equations, Numerical analysis, Rotors, Turbines, Turbomachinery, Turbulence, Viscosity, Cascades (Fluid dynamics), Transonic flow, Scalars, Jacobian matrices, Flux (Metallurgy)
1.
Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., 1997, Computational Fluid Mechanics and Heat Transfer, Second Ed., Taylor and Francis, London.
2.
Jameson, A., Schmidt, W., and Turkel, E., 1981, “Numerical Simulation of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes,” AIAA Paper 81-1259.
3.
Roe
,
P. L.
,
1981
, “
Approximate Riemann Solvers, Parameter Vectors and Difference Schemes
,”
J. Comput. Phys.
,
43
(
2
), pp.
357
–
372
.
4.
van der Weide, E., and Deconinck, H., 1996, “Positive Matrix Distribution Schemes for Hyperbolic Systems, With Applications to the Euler Equations,” 3rd ECCOMAS CFD Conference, Paris.
5.
Catalano, L. A., De Palma, P., Pascazio, G., and Napolitano, M., 1997, “Matrix Fluctuation Splitting Schemes for Accurate Solutions to Transonic Flows,” Lecture Notes in Physics, 490, Springer-Verlag, Berlin, pp. 328–333.
6.
De Palma, P., Pascazio, G., and Napolitano, M., 1999, “A Hybrid Multidimensional Upwind Scheme for Compressible Steady Flows,” AIAA Paper 99-3513.
7.
De Palma
,
P.
,
Pascazio
,
G.
, and
Napolitano
,
M.
,
2001
, “
A Multidimensional Upwind Solver for Steady Compressible Turbulent Flows
,”
CFD Journal
, Special No., pp.
287
–
295
.
8.
Bonfiglioli, A., 1998, “Multidimensional Residual Distribution Schemes for the Pseudo-Compressible Euler and Navier-Stokes Equations on Unstructured Meshes,” Lecture Notes in Physics, 515, Springer-Verlag, Berlin, pp. 254–259.
9.
Spalart
,
P. R.
, and
Allmaras
,
S. R.
,
1994
, “
A One-Equation Turbulence Model for Aerodynamical Flows
,”
Rech. Aerosp.
,
1
, pp.
5
–
21
.
10.
Struijs, R., Deconinck, H., and Roe, P. L., 1991, “Fluctuation Splitting Schemes for the 2D Euler Equations,” VKI LS 1991-01, Computational Fluid Dynamics, von Karman Institute, Belgium.
11.
Paille`re, H., 1995, “Multidimensional Upwind Residual Distribution Schemes for the Euler and Navier-Stokes Equations on Unstructured Grids,” Ph.D. thesis, Universite´ Libre de Bruxelles, Belgium, June.
12.
Mizukami
,
A.
, and
Hughes
,
T. J. R.
,
1985
, “
A Petrov-Galerkin Finite Element Method for Convection-Dominated Flows: An Accurate Upwinding Technique for Satisfying the Maximum Principle
,”
Comput. Methods Appl. Mech. Eng.
,
50
, pp.
181
–
193
.
13.
Roe, P. L., 1990, “Optimum Upwind Advection on a Triangular Mesh,” ICASE Report No. 90-75.
14.
van Leer, B., Lee, W. T., and Roe, P. L., 1991, “Characteristic Time-Stepping or Local Preconditioning of the Euler Equations,” AIAA Paper 91-1552.
15.
De Palma, P., Pascazio, G., and Napolitano, M., 1998, “A Hybrid Fluctuation Splitting Scheme for Transonic Inviscid Flows,” 4th ECCOMAS CFD Conference, Athens.
16.
De Palma, P., Pascazio, G., and Napolitano, M., 2002, “A Hybrid Fluctuation Splitting Scheme for Two-Dimensional Compressible Steady Flows,” Innovative Methods for Numerical Solution of Partial Differential Equations, M. M. Hafez and J. J. Chattot, eds., World Scientific Publishing, Singapore, pp. 303–333.
17.
Balay, S., Gropp, W. D., Curfman McInnes, L., and Smith, B. F., 1998, PETSc home page http://www.mcs.anl.gov/petsc.
18.
Sieverding, C. H., 1973, “Experimental Data on Two Transonic Turbine Blade Sections and Comparisons With Various Theoretical Methods,” VKI Report No. LS59, von Karman Institute, Belgium.
19.
Kiock
,
R.
,
Lethaus
,
F.
,
Baines
,
N. C.
, and
Sieverding
,
C. H.
,
1986
, “
The Transonic Flow Through a Plane Turbine Cascade as Measured in Four European Wind Tunnels
,”
ASME J. Eng. Gas Turbines Power
,
108
, pp.
277
–
285
.
20.
Arnone
,
A.
, and
Swanson
,
R. C.
,
1993
, “
Navier-Stokes Solver for Turbomachinery Applications
,”
ASME J. Turbomach.
,
115
, pp.
305
–
313
.
Copyright © 2005
 by ASME
You do not currently have access to this content.