Dynamic Modeling of Turbulence

This paper presents a new systematic and generalized approach to model turbulence dynamically. The suggested approach is based on the variational technique to solve a system of equations where the number of unknowns is larger than the number of equations. Turbulence closure problem results when averaging the Navier-Stokes (N-S) equations. Averaging transforms the N-S equations from a determinate set of equations describing turbulent flow field to an indeterminate set of equations that need additional information. Unknown terms, Reynolds stresses, appear as a results of averaging; and the solution of the averaged N-S equations depends on the proper selection of Reynolds stresses. In the dynamic modeling formulation of turbulence, the Reynolds stresses are selected to produce a best solution of the averaged N-S equations. The Reynolds stresses are computed via optimizing a performance index ‘I’. In the optimization process the averaged N-S equations are considered as constraints. The performance index ‘I’ is defined as a measure of the quality of solution. Averaging can be considered as a process by which we lose some information about the flow field. The lost information appears partially in the unknown terms “Reynolds stresses”. Hence, the performance index should include some measure of information losses which occur as the result of averaging. Classical approach does not rely on the N-S equations, itself as a complete description of turbulence, to derive a suitable turbulence models. The new concept will use the N-S equations, combined with the physics of turbulence, for an optimal selection of turbulence model through ‘I’. In this approach the model is not specified in advance, but it will be developed dynamically with the solution.