A Strongly Coupled Time-Marching Method for Solving the Navier-Stokes and k-;gv; Turbulence Model Equations with Multigrid

to perhaps the fact that the turbulence model equations are solved only to obtain the eddy viscosity and also the Many researchers use a time-lagged or loosely coupled approach in solving the Navier–Stokes equations and two-equation turbuconvenience of simply adding separate routines to an exlence model equations in a time-marching method. The Navier– isting Navier–Stokes code. Consequently, many codes use Stokes equations and the turbulence model equations are solved a time-lagged or loosely-coupled approach in solving the separately and often with different methods. In this paper a strongly Navier–Stokes and two-equation turbulence model equacoupled method is presented for such calculations. The Navier– tions [7, 8]. Stokes equations and two-equation turbulence model equations, in particular, the k-g equations, are considered as one single set of In a typical iteration of a loosely coupled approach, the strongly coupled equations and solved with the same explicit timeNavier–Stokes equations are first solved with fixed eddy marching algorithm without time-lagging. A multigrid method, toviscosity and then the k-« or k-g equations are solved with gether with other acceleration techniques such as local time steps the newly updated flow field. Different solution methods and implicit residual smoothing, is applied to both the Navier– Stokes and the turbulence model equations. Time step limits due are often used for the Navier–Stokes and turbulence model to the source terms in the k-g equations are relieved by treating the equations. To some extent, the model equations look simappropriate source terms implicitly. The equations are also strongly pler than the Navier–Stokes equations, particularly after coupled in space through the use of staggered control volumes. the convection velocities are frozen in a loosely coupled The method is applied to the calculation of flows through cascades algorithm. However, this does not appear to yield an easier as well as over isolated airfoils. Convergence rate is greatly improved by the use of the multigrid method with the strongly coupled task for numerical solution. On the contrary, results seem time-marching scheme. Q 1996 Academic Press, Inc. to show slow convergence or incomplete convergence due to possible reasons such as numerical stiffness for some models, not well-defined boundary conditions, trouble