The Use of Differential Constraints for Analyzing Turbulence Models

Abstract

In this article we introduce a concept based on the differential constraints method to examine the closure procedure in Turbulence Models. We show how this concept may be applied to study the problem of interaction and mixing between two semi-infinite homogeneous turbulent flow fields of different scales.

Invariant Sets and Explicit Solutions to a Third-Order Model for the Shearless Stratified Turbulent Flow

“…The results of this article continue our earlier studies of the problem of interaction and mixing between two semi-infinite turbulent flow fields of different scales given in [4,5] wherein it was proposed a concept based on the method of differential constraints [23,19] for examining the closure procedure for momentum equations in Parametric Turbulent Models. The key idea of this approach can be formulated shortly as follows: the algebraic expressions for the n-order moments of statistical characteristics of turbulent flow are determined as the equations of invariant sets (manifolds) of the corresponding differential equations.…”

“…This implies existence of a solution to (4.4), (4.5). In [4] we proven this result directly for a problem of type (4.4), (4.5) and noted that this solution is essentially different from the well-known Barenblatt's solution [2].…”

“…In [4,5] we studied the existence of selfsimilar solutions and presented in detail the qualitative properties of the solution obtained. A direct calculation shows that for stratified flows there are no selfsimilar solutions similar to a , w 2 a , w 3 a .…”

Invariant Manifolds in Parametric turbulent Models

“…it is assumed that the fifth-order cumulants are equal to zero. Therefore (see, [1], [24]), the following third-order mathematical model can be used Model 2:…”

“…A concept of algebraic expressions for the n-order moments of statistical characteristics of turbulent flow as the equations of invariant manifolds of the corresponding transport equations enables us to examine the closure procedure for momentum equations. The invariant manifolds were used in [1], [2] for investigating the problem of the development of a shearless mixing layer in the third-order closure model wherein we showed that this approach may be used to obtain new reductions of the models under consideration and to construct a class of explicit solutions. Due to the differential constraints derived, it was established that the equation of the invariant manifold (differential constraints of the model) coincides with the classical tensor-invariant Hanjalic-Launder algebraic model of the triple correlations for an unstratified flow and with the Zeman-Lumley model for a stratified flow.…”