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 semi-implicit 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 Omega=[0.4 pi] x [-1,1] x [0,2 pi]. The friction Reynolds number for channel flow is set to be Re-tau = 175 and the computational grids of 128x 65x 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 time-discretizations 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.