A fully implicit Navier-Stokes algorithm for unstructured grids incorporating a two-equation turbulence model

Transcription

1 Copyright 1996, American Institute of Aeronautics and Astronautics, Inc. AIAA Meeting Papers on Disc, January 1996 A , AIAA Paper A fully implicit Navier-Stokes algorithm for unstructured grids incorporating a two-equation turbulence model Y. Kergaravat Rutgers Univ., Piscataway, NJ D. Knight Rutgers Univ., Piscataway, NJ AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996 An implicit algorithm is developed for the 2D compressible Favre-averaged Navier-Stokes equations. It incorporates the standard k-epsilon turbulence model of Launder and Spalding (1974) and the low-reynolds-number correction of Chien (1982). The equations are solved using an unstructured grid of triangles with the flow variables stored at the centroids of the cells. The inviscid fluxes are obtained from Roe's flux difference split method. Linear reconstruction of the flow variables to the cell faces provides second-order spatial accuracy. Turbulent and viscous stresses as well as heat transfer are obtained from a discrete representation of Gauss's theorem. Interpolation of the flow variables to the nodes is achieved using a second-order-accurate method. Temporal discretization employs Euler, trapezoidal, or three-point backward differencing. An incomplete LU factorization of the Jacobian matrix is implemented as a preconditioning method. Results are presented for a supersonic turbulent mixing layer, a supersonic laminar compression corner, and a supersonic turbulent compression corner. (Author) Page 1

2 AIAA Paper No A Fully Implicit Navier-Stokes Algorithm for Unstructured Grids Incorporating a Two-Equation Turbulence Model Yan Kergaravat* and Doyle Knight* Department

3 Stokes algorithm, incorporating a turbulence model

4 where i,-j and Tij are respectively the components of the viscous stress tensor and the turbulent stress tensor,

5 The Chien model equates the dissipation e with the solenoidal dissipation e a although (21)

6

7 where

8 adjacent cells and may be written 6 (49) where Aj are the absolute values of the eigenvalues A 2 A 3 A 4 A 5 A 6 & ft -& (50) The Roe-averaged variables are _ + x/p^ _ + ^/fr a 2 = +

9 obtained for df/dy. The values of df/dx and df/dy are evaluated at midpoint p of face Jb (Fig. 2). The molecular viscosity is evaluated at the midpoint of face k using the formula V-

10 where Jacobian

11 4 Boundary Conditions

12 5.1 Supersonic Turbulent Mixing Layer The purpose of this computation is to assess the precision of the code for turbulent free shear flows and to test the computational efficiency of the preconditioning algorithm. A spatially developing supersonic turbulent mixing layer (as shown in Fig. 3) is computed. The same computation is performed with and without preconditioning. Figure 3: Supersonic turbulent mixing layer

13 OQ20 GL015 aoio Similarity X= O.OOS aooo acts aoio -0.01J -ao» aoio k Figure 6: Turbulence kinetic energy for supersonic turbulent mixing layer 0030 O01S Similarity X = y/x aooo aoio ,X Figure

14 different.

15 Unpreconditioned Preconditioned CPU time

16 The computed

17 References [1] Batina, J., "A Fast Implicit Upwind Solution Algorithm for Three-Dimensional Unstructured Dynamic Meshes", AIAA Paper, Paper No , [2] Buelow,

18 [21] White, F., Viscous Fluid Flow, Me Graw-Hill, Inc., [22] Whittaker, D., Slack, R., and Walters, R., "Solution Algorithms for the Two-Dimensional Euler Equations on Unstructured Meshes", AIAA Paper, Paper No ,1990. [23] Wilcox,

19 Copyright 1996, American Institute of Aeronautics and Astronautics, Inc. Fig. 16