Extension of Baldwin-Lomax turbulence model to general three-dimensional flows

The Baldwin-Lomax turbulence model is extended to three-dimensional flows by replacing the directed normal distance with the curvilinear radial coordinate in general curvilinear coordinates. The modification, which makes the B-L model compatible with the Navier-Stokes solution associated with curvilinear meshes over complex geometries, offers physically realistic viscous coordinates and velocity profiles in case of highly curvy grid lines. It results in moderately higher eddy viscosity and thus skin friction values than the existing model in the range of angle of attack considered. For a given flow condition, the modified model takes slightly less computation time than the existing model. Any grid that works well with the existing model will also work well or better with the modified model. Introduction The Baldwin-Lomax turbulence model is used widely tooughout the computational fluid dynamics (CFD) community. It is a conventional, two-layer type algebraic model. The model is used in both two and three dimensions with very little modification. Generally, it is considered a good model for the prediction of attached and mildly separated flows as assessed by Rumsey and Vatsa. For flows having large separated regions, including the secondary separation, Degani and Schliff found difficulty hi proper evaluation of the scale length for determining the eddy viscosity in the outer layer and therefore, proposed a "first maximum" modification to the model so that the secondary separation can be detected. Although simple, the model seems to be the best turbulence model for vortical flows as recognized by the CFD community. For almost two decades since its introduction, the model has been applied to three-dimensional flow computations without being aware of the compatibility of the model with the Navier-Stokes solution in general curvilinear coordinates. There are two parameters in the model that are subject to compatibility problem: the directed normal distance and the magnitude of the vorticity. The former is highly incompatible * Senior Research Scientist, Marine and Aviation Department, Carderock Division. Associate Fellow AIAA. while the latter imposes an indirect effect. For instance, in the three-dimensional Navier-Stokes code such as the popular CFL3D code, the viscous distance is oriented "normal" to the wall as in the two-dimensional or axisymmetric case using the boundary-layer coordinates, although the code is mainly for three-dimensional applications. This practice is inconsistent with the solution of the threedimensional Navier-Stokes equations written in general curvilinear coordinates. For general three-dimensional flow over a complex geometry, the computational grid (structured grid) is usually generated by an elliptic solver and the resulting grid lines are mostly curvilinear away from the surface. Unlike in the case of the thin boundary layer, the shear layer in the Navier-Stokes solution extends along the radial coordinate line farther away from the wall; keeping the viscous coordinate distance perpendicular to the wall is highly inappropriate. In the present paper, the Baldwin-Lomax model is extended to general three-dimensional flows by properly replacing the straight normal distance with the curvilinear radial coordinate. The algebraic equations of die model are recast in general three-dimensional curvilinear coordinates so that the model is in concert with the Navier-Stokes solution. Numerical experiments were conducted to compare the results based on the existing and modified models. Baldwin-Lomax Turbulence Model The Baldwin-Lomax turbulence model is a two-layer type algebraic model. The inner layer is governed by the Prandd mixing length with van Driest damping, and the outer layer follows the Clauser approximation. Computed vorticity is used in determining the eddies in the inner layer, and in defining the reference mixing length required for the outer layer. For convenience, the original equations for the BaldwinLomax turbulence model are written here in conventional three-dimensional body-oriented coordinates x, y, z, for viscous calculations, where the coordinate x coincides with the body surface, i.e., the wall, hi the longitudinal direction, y is also on the body surface but in the transverse direction, and z is measured from the surface along a straight line normal to die tangent plane, see Fig. 1. Appearing in both the inner and outer layer equations, the magnitude of vorticity Q can be conveniently replaced by its three-dimensional counterpart; it is the normal distance z that has been overlooked. In case of a complex geometry with a structured grid, the grid is usually generated by an elliptic solver and the resulting grid lines are generally curvilinear away from (he wall. Maintenance of perpendicularity of the viscous distance to the wall can lead to erroneous results. Modified Baldwin-Lomax Model for Three-Dimensional Flows In applying the Baldwin-Lomax model to general threedimensional flows, the above equations can be transformed from three-dimensional body-oriented boundary-layer coordinates to general curvilinear coordinates where Fig. 1 Viscous flow in body-oriented coordinates The eddy viscosity in the inner layer is determined by h,, = p(0.4Dz)n (1) where p is fluid density, O. is the magnitude of the vorticity, and D is (he Van Driest damping factor that can be written in its original boundary-layer form: D = 1 exp (-z/26)