1. each specific application. made to improve the Smagorinsky model, and evaluated at centerline it has been applied successfully, both in its origii' nal and modified forms, to a number of simulations that include decaying isotropic turbulence,2 turbulent planar channel flow~,~-~ compressible turbulent fl~ws,~-' and turbulent mixing 1ayers.i These computations and others have identified the following major deficiencies of the Smagorinsky model: (1) the eddy viscosity constant is flow dependent, (2) limiting behavior near solid boundaries and in laminar regimes is incorrect, (3) the model does not account for the backscatter of energy from small to largescales, i1 4 the model is overly dissipative, and ( ) (5) the model does not account for compressibility effects.g is dictated by stability constraints of the algorithm rather than by the frequency content of the largescale structures. This can be a severe constraint, particularly for low-Mach number and wall-bounded flows. In addition, the situation is exacerbated since computations must be carried out for extended periods of time in order to collect statistical information. The implicit technique, though incurring increased computational expense per step, may afford a desirable alternative by allowing larger time increments.
2. The dynamic SGS model was first proposed by German0 et a1.12for incompressible flows, and extended by Moin et al.I3 for compressible applications. Its most general formulation is identical to that of Smagorinsky and Yoshizawa given by Eqs. 22-28. In this description, however, the model onstants' C and CI are computed as a function of time and space from the energy content of the resolved large-scale structures. This is, accomplished by introducing a test filter function G, with a filter width that is wider than the computational mesh, where its application is represented as for the model constants. Details of the derivation may be found in Refs. 12-14 which result in CA2 = ((Gj - +Lkkdij) Mij)
3. The aforementioned features of the numerical algorithm are embodied in an existing fullyvectorized time-accurate three-dimensional computer code FDL3D12', which has proven to be reliable for steady and unsteady fluid flow problems, including the simulation of flows over delta wings with leading-edge vortices,2212g-31 vortex breakdown,2g-31 and the direct numerical simulation of transitional wall jets32 and synthetic jet actuators.33 While attention of this investigation is concentrated on the implicit time-advancement method, FDLSDI also provides the optional fourthorder Runge-Kutta explicit integration scheme, implemented in low-storage form.34 Spatial derivatives which must be evaluated for the computation of SM were approximated by fourth-order explicit stencils, and the dynamic SGS model onstants' C, Cr, and Prt were numerically restricted to be non negative.