Algebraic turbulence modeling for adaptive unstructured grids

Abstract

Algebraic turbulence models are quite common in flow numerical simulations. Their simplicity and low computation cost, as well as their applicability to specific classes of flows, have made them attractive compared to the more expensive multiple-equation models. Adaptive, unstructured grid algorithms have become quite popular over the last years, due to their flexibility in treating complex geometries, as well their high-resolution capability for local flow phenomena. However, application of algebraic turbulence models with unstructured grids proves to be nontrivial. The difficulties associated with implementation of those models with such grids have generally impeded use of unstructured grids for turbulent flow simulations. This paper presents a novel method of implementing an algebraic turbulence model with embedded, unstructured grids. The BaldwinLomax model was chosen in order to test the implementation. Modifications to the application of the original model on structured grids are suggested as well. Comparisons with both experiments and other numerical simulations are employed in order to evaluate accuracy of the implementation method. ONSIDERABLE progress has been made in the development of numerical methods for the solution of the Navier-Stokes equations. However, most of those methods are impractical for complicated flows in a design environment. The primary reason is poor efficiency, which makes it difficult to obtain accurate results. Very fine resolution is needed, which results in long computation times even with the use of available supercomputers. In general, the selection of the equations to be solved, the scheme, and the grid all are determined a priori by the user before starting the solution procedure, and quite often some and even all of these factors must be modified by the user in order to improve the results. The robustness of current numerical schemes as well as present computer capabilities has recently allowed a dramatic change in this philosophy. General algorithms have been developed that are flexible enough to adaptively adjust those parameters during the solution procedure without intervention by the user. One of the most efficient adaptive methods consists of local refinement by dividing grid cells. An initially coarse grid is embedded in regions with large flow gradients (e.g., boundary layers, shocks, wakes, etc.). The algorithm senses high gradient regions and automatically divides grid cells in such regions. This approach has been used for the resolution of shocks in flowfields described by Euler equations.1'2 The embedded scheme was used as part of an adaptive algorithm for turbulent flows.36 Algebraic turbulence models are quite common in aerodynamics numerical simulations. Their simplicity and low computation cost, as well as their accuracy for specific classes of flows, have made them attractive compared to the more expensive multiple-equation models. However, such models require nonlocal information that characterizes an entire shearlayer profile. The adaptive algorithm results in an unstructured mesh, which is a major issue in the implementation of an algebraic turbulence model. Grid interfaces frequently interrupt normal-to-wall meshlines, and, therefore, models that employ information from an entire shear-layer profile cannot be applied. Similarly, in the case of triangular meshes, such normal grid lines generally do not exist. A new method of