Optimum algorithm for channel flow analysis in direct numerical simulation method

The objective of this work is to perform a direct numerical simulation of turbulent channel flow where all essential scales of motion are resolved due to variable time-stepping algorithm in various time advancement method and different discritized form of convection term. A pseudo spectral method (Fourier series in stream-wise and span-wise directions and Chebychev polynomial expansion in normal direction) is employed for the spatial derivatives. The time advancement is carried out by different semiimplicit and splitting schemes. Also Alternating and Linearized forms are added to four commonly used forms of the convective term, referred to as divergence, Convection, skew-symmetric, and rotational. Spectral method based on the primitive variable formulation is used in Cartesian coordinates with two periodic and one non-periodic boundary condition in three dimensional directions Ω=[0,4π]×[-1,1]×[0,2π]. The friction Reynolds number for channel flow is set to be Reτ=175 and the computational grids of 128×65×128 are used in the x, y and z directions, respectively. The comparison is made between turbulent quantities such as the turbulent statistics, wall shear velocity, standard deviation of u and total normalized energy of instantaneous velocities in different time-discretization methods and different forms of nonlinear term. The present results show that third-order timediscretizations forward Euler for explicit terms and backward Euler for implicit terms can minimize the computational cost of integration by maximizing the time step, while keeping the CFL number near a threshold in time-discretization schemes. Also, the de-aliased results of turbulence statistics do indicate that different expressions of nonlinear terms have minor discrepancy in pseudo spectral method. The results show that the most desirable approach is a combination of variable time stepping third order backward difference algorithm and rotational form, which provides reduced cost and further accuracy improvements.