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Space discretization of the production term in two-equation models

One of the source terms in turbulence model equations is the production rate of turbulent energy given by (6.9). The discretization of this term is carried out at center (i,j,k) with central differences and bilinear interpolations in which the fewest number of neighbouring nodal points are taken. Since we use as unknowns, the covariant derivative of the contravariant velocity components must be expressed in terms of flux components. We have

 

The partial derivative of the flux component can be approximated by central differences. The same interpolation rules as for the momentum equations are applied. All geometrical quantities are evaluated at the centre of a scalar cell. Closest to a boundary, some derivatives also contain virtual fluxes. These virtual quantities are expressed in internal fluxes by using linear extrapolation. In two-dimensional case, for example, at lower boundary we get:

The above discretization is not well-suited when a non-smooth grid is employed. An approach to discretize the production term in case of non-smooth grids is to integrate the Cartesian expression of over a finite volume so that no Christoffel symbols or metric tensors occur in the formulation. At this moment we restrict ourselves to the Boussinesq hypothesis for the modeling of the production of turbulent energy. From (6.24) it follows that the Cartesian expression for is given by

The remain task is to discretize the partial derivatives of the Cartesian velocity components with respect to at point (i,j,k). This can be done with the integration-path method. Here a "quick" approach is given. To approximate the -derivative of this derivative has to be expressed in terms of the derivative with respect to . Using the chain rule, one gets:

 

The approximation of (6.33) at point (i,j,k) leads to

Here represents the difference in , across the cell enclosing point (i,j,k), in direction. The differences are evaluated as:

The final expression of the approximation for

becomes

The velocity components on cell faces are to be obtained with

 

Since is only given at center of cell faces, discontinuous geometric quantities and fluxes have to be replaced by suitable definitions such that (6.38) is exact for constant on arbitrary grids. An example:

with



Tatiana Tijanova
Wed Mar 26 10:36:42 MET 1997