Implementation of a Reynolds-Averaged Navier-Stokes Flow Solver

Implementation of a Reynolds-Averaged Navier-Stokes Flow Solver

Weiyang Lin, Lafayette K. Taylor, Kidambi Sreenivas

One of the essences in CFD is its capability of accurately predicting complex flows. Since many flows of engineering interest are turbulent, an suitable treatment of turbulent flows is therefore critical. While the flow field is sufficiently described by the Navier-Stokes equations, the accurate solution to them is usually not practical. Reynolds-averaged Navier-Stokes (RANS) models have been proven successful in the last few decades.

The goal of this report is to describe the details of the implementation of two well-known turbulence models in couple with the mean flow in three dimensional grids. One model is the Spalart-Allmaras standard (SA) model, where the turbulent molecular viscosity is modeled in one equation. The other is the Menter Shear Stress Transport (SST) model, in which the kinetic energy (per unit mass) of the turbulent fluctuations and the dissipation per unit turbulence kinetic energy are modeled in two equations. Boussinesq eddy-viscosity approximation is employed to couple the mean flow with the solutions to the turbulence models.

The following work is done on the infrastructure of 3D unstructured flow solver, with the adding of viscous terms and additional turbulence model(s).

Laminar flow over a flat plate is typically the first benchmark problem to solve in order to validate the code. The case is ran on a flat plate with length of 1.0. It has infinitely long width and infinitely small thickness (due to the symmetric boundary condition). The grid being used has 50 points on the x-direction (parallel to the flow) of the plate and 2 points on the z-direction; the upstream and downstream directions are extended towards the x-coordinates of -6.0 and 6.0 with 11 grid points, and all of the points are extruded towards the y-coordinates of 5.0 for 31 points in total with the initial spacing of 1E-3 to resolve the laminar boundary layers. A side look of the grid is shown in Fig. 1. In this case, the free-stream flow has the Reynolds number of 1E4 and a low Mach number of 0.2, this results in the general contour of u-velocity as is plotted in Fig. 2. The normalized velocity profile and skin friction are compared with the Blasius solutions below in Figs. 3 and 4.

Figure 1. A side look of the grid being used to simulate the laminar flow over a flat plate. Spacing off the solid wall is 1E-3

Figure 2. The u-velocity contour from the solution to the laminar flow over a flat plate. Flow parameters are: Re = 1E4, Mach = 0.2

Figure 3. Normalized u-velocity against eta plot in comparison with the Blasius solution

Figure 4. Skin friction Cf against Rex plot in comparison with the Blasius solution

The following cases are ran on the same flat plate as the laminar case. The initial spacing of 1E-5, 5E-6, 1E-6 and 5E-7 are chosen to resolve the turbulent boundary layers. The free-stream flow has the Reynolds number of 5E6 and a low Mach number of 0.2. The normalized velocity profile is compared with the Spalding solution below in Figs. 5-6. Both models are ran in 4 grids with different grid spacing off the wall. 

Figure 5. Velocity profile plot (solutions from SA model) in comparison with the Spalding solution

Figure 6. Velocity profile plot (solutions from SST model) in comparison with the Spalding solution

The RANS solver is ran on NACA0012 airfoil out of interest. Results from zero angle-of-attack with both SA and SST models are presented in Figs. 7 and 8, and the SST results are compared with the corresponding ones obtained from Tenasi (Fig. 9). Results from 5o angle-of-attack with SST model is presented in Fig. 10.

Figure 7. Eddy viscosity of turbulent flow over NACA0012, solution obtained from SA model

Figure 8. Eddy viscosity of turbulent flow over NACA0012, solution obtained from SST model

Figure 9. Eddy viscosity of turbulent flow over NACA0012, solution obtained from SST model by Tenasi

Figure 10. Results of turbulent flow over NACA0012 at 5o angle-of-attack, solution obtained from SST model

For implementation details and more results, please refer to the attached document [PDF] and/or contact the author.