In the model, the depth-averaged partial differential k̃– and w̃ -equations are solved with the mean flow equations together in order to determine the distribution of turbulence eddy viscosity and diffusivity as well as other physical variables.
An axisymmetric, unstructured grid was used for solving the Eulerian, mean flow equations and the vertices were used to store mean statistics for solving the Lagrangian, fluid particle equations.
(4) There is no advantage to numerically coupling the two-equation model system to the mean flow equation system, in regard to convergence or accuracy.
While the mean-flow equations and the turbulence model equations are advanced in time in a loosely-coupled manner, their multigrid cycling is strongly coupled (FC-MG).
The novel time-marching approach relies on decoupled, implicit time integration, that is, the five mean-flow equations are solved separately from the seven Reynolds-stress closure equations.
Thanks to the scheme characteristics, a spatially second-order discretization method for the Reynolds stress model equations (exactly as applied to the mean-flow equations) can be used, free of stability difficulties within the fixed point iterations.
The method is applied with success to the mean curvature flow equation, the Kuramoto Sivashinsky equation, and to the Rayleigh Taylor instability in a Hele-Shaw cell, including the effect of surface tension.
Once obtained the interior (C^{1, alpha }) estimate of the solution uniform in (epsilon ), we write the mean curvature flow equation in non divergence form: begin{aligned} partial _t u_epsilon - sum _{i,j=1}^n a^epsilon _{ij} x,t) X_i^epsilon X_j^epsilon u_epsilon =0.
A variational method is used to deduce the acoustic wave equation, satisfied by the potential, for quasi-one-dimensional propagation, in a duct of varying cross-section, containing a low Mach number mean flow; both wave equations, for the acoustic potential and velocity, are reduced to a 'Schrödinger' form, by using the ray approximation as a factor, in the exact solution.
Based on the latter, an adjoint law of the wall that bridges the gap between the solid wall and the first grid node off the wall is proposed and used during the solution of the system of adjoint (to both the mean flow and turbulence) equations.
As extinction is approached the equations for mean flow and turbulence quantities reduce to the corresponding equations for an inert counterflow.
