Direct Numerical Simulation of Spatial Transition to Turbulence Using Fourth-Order Vertical Velocity Second-Order Vertical Vorticity Formulation

A highly accurate algorithm has been developed to study the process of spatial transition to turbulence. The algorithmic details of the direct numerical simulation (DNS) of transition to turbulence in a boundary layer based on a formulation in terms of vertical velocity and vertical vorticity are presented. Issues concerning the boundary conditions are discussed. The linear viscous terms are discretized using an implicit Crank–Nicholson scheme, and a low-storage Runge–Kutta method is used for the nonlinear terms. For the spatial discretization, fourth-order compact finite differences have been used, as these have been found to have better resolution compared to explicit differencing schemes of comparable order. The number of grid points that are needed per wavelength is close to the theoretical optimum for any numerical scheme. The resulting time-discretized fourth-order equations are split up into two second-order equations, resulting in Helmholtz- and Poisson-type equations. The boundary conditions for the Laplacian of the vertical velocity are determined using an influence matrix method. A robust multigrid algorithm has been developed to solve the resulting anisotropic elliptical equations. For the outflow boundary, a buffer domain method, which smoothly reduces the disturbances to zero, in conjunction with parabolization of the Navier–Stokes equations has been used. The validation of the results for the DNS solver is made both for linear and weakly nonlinear cases.