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tom solid wall is changed to a free-slip condition. By this changing of boundary conditions, the mechanism of turbulence generation is eliminated and the flow begins to decay. This problem is of interest to understand how the free-surface turbulence evolves far from the region of turbulence generation, e.g., in the wake of a vessel. The direct simulation by Pan and Banerjee showed that, near the free surface, the structure of turbulence is quasi two-dimensional, with the quasi-two-dimensional-region increasing in thickness in the decaying process. Thus, this test case is particularly severe for SGS models, which have a three-dimensional structure. The point at issue in this work is whether large eddy simulation, using SGS dynamic mixed models and, in particular, the dynamic two-parameter model, may reproduce the nearly two-dimensional nature of the turbulence near the free surface.

Mathematical formulation and numerical method

We now briefly summarize the mathematical formulation of the dynamic-mixed models used in the present simulations. More details can be found in (2) and (3).

For the incompressible and constant density flow considered here, the basic governing equations are the grid-filtered Navier-Stokes and continuity equations (omitted here for sake of brevity), in which the effect of the unresolved subgrid scales is represented by the SGS stress tensor:


The overbar denotes a grid-filtered variable.

In the dynamic two-parameter model, the SGS stress tensor is modeled as follows (3):


where: is the resolved strain rate tensor, and is the “modified Leonard stress” (5). It represents the resolved part of the SGS stress. The grid filter width is defined as:

where is the grid spacing in i direction. The two unknown coefficients C and K in (2) are computed dynamically in the way described below.

Following the procedure of Germano et al. (1), a test-filter, denoted by a hat, is applied to the governing equations; the sub-test scale stress tensor is then obtained:


It is modeled in the same manner as the SGS stress tensor:


where: and is the modified Leonard term for the test-scale filter (2) (3). The width of the test filter is assumed to be larger than that of the grid filter. The ratio between the width of the test and grid filters is denoted by α (=2 here as in most applications in the literature).

Using the Germano identity (1) and equations (2) and (4), we obtain:


where: and . Using a least square approach as suggested by Lilly (6), the unknown coefficients are evaluated by setting ∂Q/∂C=0 and ∂Q/∂K=0, where Q is the square of the error in (5).

The expression of the SGS stress tensor in the DMM is given by Eq. (2) in which K is set equal to unity (2). Following the same dynamic procedure as previously, the model parameter C can be obtained by setting ∂Q/∂C=0, Q being the square of the error in (5) with K=1.

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