Generalization of the RANS Equations Using Mean Modal Decomposition of the Navier Stokes Equations

A generalization in the Reynolds decomposition and averaging are proposed in this paper. The method is directly applied to the Navier Stokes (N-S) equations to construction of a generalized Reynolds Averaged Navier Stokes (RANS) equations. The formulation which is presented for the fields realized in a suitable ensemble, is based on a two part decomposition. One part is an approximate unique representation of the field and when reconstruction of the field, will repeat in all ensemble elements. The other part represents deviation of the real field from the approximate part and therefore is different in any mode and each ensemble element. The decomposition is applied in both spatial and temporal fashions. In the temporal decomposition, a system of Partial Differential Equations (PDEs) is obtained that is nonclosed, coupled and second order in space and its zeroth mode is the classical Reynolds averaged values of the field. In the spatial decomposition whereas, a first order system of nonclosed PDEs is obtained which could be seen as an alternative version of the Proper Orthogonal Decomposition (POD) or the Coherent Vortex Simulation (CVS) methods. In both fashions however, there are some terms that must be modeled just like as the classical closure problem in the RANS method. The method is applied on a one dimensional mixed random-nonrandom field and successfully extracted the coherent part of the field.