Cell-centered finite volume solution of the two-dimensional Navier-Stokes equations

Cell-centered finite volume method with multistage time-stepping is successfully applied to two-dimensional mass-weighted, time-averaged Navier-Stokes equations for the computation of viscous flows. In the cell-centered scheme, flow quantities are associated with the center of a cell. Convective fluxes at the cell faces are evaluated by means of upwind Roe Flux Differencing Scheme (Roe FDS) with Monotone Upwind Schemes for Scalar Conservation Laws (MUSCL) approach. Green’s theorem is employed for evaluation of gradients in computation of viscous fluxes. Five stage hybrid time-stepping scheme is implemented for integration to steady state. Convergence is accelerated by utilizing local time stepping and residual smoothing. The accuracy of the present Navier-Stokes solver is verified by comparing flat-plate laminar boundary-layer solutions with theoretical solutions of Blasius and by comparing laminar airfoil solutions with those available in literature. Convergence down to machine zero attained in the computations indicates a good sign for the efficiency of the present solver. Turbulence closure for Reynolds stresses is obtained using two-layer algebraic eddy viscosity model of Baldwin and Lomax. Computed results for turbulent flows are validated with available experimental results.

PDF

___

1.Uygun, M., Tuncer, I.H., 2003, "A Computational Study of Subsonic Flows over A Medium Range Cargo Aircraft," AIAA paper 2003-3661.

2.Uygun, M., Tuncer, I.H., 2004, "Viscous Flow Solutions over CN-235 Cargo Aircraft," AIAA Journal of Aircraft, Vol. 41, No. 4, pp. 940-944.

3.Uygun, M., Kırkköprü, K., "Numerical Solution of the Euler Equations by Finite Volume Methods: Central versus Upwind Schemes," Journal of Aeronautics and Space Technologies, Vol. 2, No. 1, pp.47-55, January 2005.

4.Roe, P.L., 1981, "Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes," Journal of Computational Physics, 43:357-372.

5.Leer, B. V., 1977, "Towards the Ultimate Conservation Difference Scheme IV; A New Approach to Numerical Convection," Journal of Computational Physics, 23: 276-299.

6.Albada; G.D., Leer, B. V., and Roberts, W.W., 1982, "A Comparative Study of Computational Methods in Cosmic Gas Dynamics," Astron. Astrophysics, 108:76-84. 7.Baldwin, B.S., and Lomax, H., 1978, "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows," AIAA paper 1978-257.

8.Cebeci, T., and Smith, A.M.O., Analysis of Turbulent Boundary Layers, Academic Press, New York, 1974.

9.White, F.M., Viscous Fluid Flow, McGraw Hill Inc., New York, 1974.

10.Proceedings of the GAMM-Workshop on Numerical Simulation of Compressible Navier-Stokes Flows, INRIA, Sophia-Antipolis; Notes on Numerical Fluid Mechanics, Vol.18, Vieweg-Verlag, 1986.

11.Chakrabartty, S.K., 1989, "Numerical Solution of Navier-Stokes Equations for Two-Dimensional Viscous Compressible Flows," AIAA Journal, Vol. 27, No. 7, pp. 843-844.

12.Chakrabartty, S.K., 1990, "Vertex Based Finite-Volume Solution of the Two-dimensional Navier-Stokes Equations," AIAA Journal, Vol. 28, No. 9, pp. 1829-1831.

13.Swanson, R.C., Turkel, E., 1987, "Artificial Dissipation and Central Difference Schemes for The Euler and Navier-Stokes Equations," AIAA paper 1987-1107.

14.Thibert, J.J., Granjacques, M., and Ohman, L. H., 1979, "NACA 0012 Airfoil," Experimental Data Base for Computer Program Assessment, AGARD-AR-138.

15.Carlson, J.R., 1996, "High Reynolds Number Analysis of Flat-Plate and Separated Afterbody Flow Using Non-linear Turbulence Models," AIAA paper 1996-2544.

16.Roe, P.L.; Pike, J., "Efficient Construction and Utilization of Approximate Riemann Solutions," Computing Methods in Applied Sciences and Engineering, R. Glowinski, J.L. Lions (eds.), North Holland Publishing, The Netherlands, 1984.

17.Harten, A., Lax, P.D., Van Leer, B., 1983, "On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws," Soc. Indust. and Appl. Math. Rev., 25, No. 1.

18.Venkatakrishnan, V., 1993, "On the Accuracy of Limiters and Convergence to Steady State Solutions," AIAA Paper 1993-0880.

19.Jameson, A., Baker, T.J., 1983, "Solution of the Euler Equations for Complex Configurations," AIAA paper 1983-1929.