The formulation, implementation and usage of a numerical solution verification code is described. This code uses the Richardson extrapolation procedure to estimate the order of accuracy and error of a computational program solution. It evaluates multiple solutions performed in numerical grid convergence studies to verify a numerical algorithm implementation. Analyses are performed on both structured and unstructured grid codes. Finite volume and finite element discretization programs are examined. Two and three-dimensional solutions are evaluated. Steady state and transient solution analysis capabilities are present in the verification code. Multiple input data bases are accepted. Benchmark options are included to allow for minimal solution validation capability as well as verification.
The Detached Eddy Simulation (DES) and steadystate Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling approaches are examined for the incompressible flow over a square cross-section cylinder at a Reynolds number of 21,400. A compressible flow code is used which employes a second-order Roe upwind spatial discretization. Efforts are made to assess the numerical accuracy of the DES predictions with regards to statistical convergence, iterative convergence, and temporal and spatial discretization error. Three-dimensional DES simulations compared well with two-dimensional DES simulations, suggesting that the dominant vortex shedding mechanism is effectively two-dimensional. The two-dimensional simulations are validated via comparison to experimental data for mean and RMS velocities as well as Reynolds stress in the cylinder wake. The steady-state RANS models significantly overpredict the size of the recirculation zone, thus underpredicting the drag coefficient relative to the experimental value. The DES model is found to give good agreement with the experimental velocity data in the wake, drag coefficient, and recirculation zone length.
The development of a unified one-equation model for the prediction of transitional and turbulent flows is described herein. An eddy-viscosity-transport equation for nonturbulent fluctuation growth based on that proposed by Warren and Hassan (Warren, E. S, and Hassan, H. A., "Transition Closure Model for Predicting Transition Onset" Journal of Aircraft, Vol. 35, No. 5, 1998, pp. 769-775) is combined with the Spalart-Allmaras one-equation model for turbulent fluctuation growth. Blending of the two equations is accomplished through a multidimensional intermittency function based on the work of Dhawan and Narasimha (Dhawan, S., and Narashima, R., "Some Properties of Boundary Layer Flow During Transition from Laminar to Turbulent Motion, " Journal of Fluid Mechanics, Vol. 3, No. 4, 1958, pp. 418-436). The model predicts both the onset and extent of transition. Low-speed test eases include transitional flow over a flat plate, a single-element airfoil, a multielement airfoil in landing config uration, and a circular cylinder. Simulations of Mach 3.5 transitional flow over a 5-deg cone are also presented. Results are compared with experimental data, and the spatial accuracy of selected predictions is analyzed.
This paper reports on the development of a unified one-equation model for the prediction of transitional and turbulent flows. An eddy viscosity--transport equation for nonturbulent fluctuation growth based on that proposed by Warren and Hassan is combined with the Spalart-Allmaras one-equation model for turbulent fluctuation growth. Blending of the two equations is accomplished through a multidimensional intermittency function based on the work of Dhawan and Narasimha. The model predicts both the onset and extent of transition. Low-speed test cases include transitional flow over a flat plate, a single element airfoil, and a multi-element airfoil in landing configuration. High-speed test cases include transitional Mach 3.5 flow over a 5{degree} cone and Mach 6 flow over a flared-cone configuration. Results are compared with experimental data, and the grid-dependence of selected predictions is analyzed.
Many Navier-Stokes codes require that the governing equations be written in conservation form with a source term. The Spalart-Allmaras one-equation model was originally developed in substantial derivative form and when rewritten in conservation form, a density gradient term appears in the source term. This density gradient term causes numerical problems and has a small influence on the numerical predictions. Further work has been performed to understand and to justify the neglect of this term. The transition trip term has been included in the one-equation eddy viscosity model of Spalart-Allmaras. Several problems with this model have been discovered when applied to high-speed flows. For the Mach 8 flat plate boundary layer flow with the standard transition method, the Baldwin-Barth and both k-{omega} models gave transition at the specified location. The Spalart-Allmaras and low Reynolds number k-{var_epsilon} models required an increase in the freestream turbulence levels in order to give transition at the desired location. All models predicted the correct skin friction levels in both the laminar and turbulent flow regions. For Mach 8 flat plate case, the transition location could not be controlled with the trip terms as given in the Spalart-Allmaras model. Several other approaches have been investigated to allow the specification of the transition location. The approach that appears most appropriate is to vary the coefficient that multiplies the turbulent production term in the governing partial differential equation for the eddy viscosity (Method 2). When this coefficient is zero, the flow remains laminar. The coefficient is increased to its normal value over a specified distance to crudely model the transition region and obtain fully turbulent flow. While this approach provides a reasonable interim solution, a separate effort should be initiated to address the proper transition procedure associated with the turbulent production term. Also, the transition process might be better modeled with the Spalart-Allmaras turbulence model with modification of the damping function f{sub v1}. The damping function could be set to zero in the laminar flow region and then turned on through the transition flow region.