Direct Numerical Simulation of Transitions Toward Turbulence in Complex Channel Flows

Abstract

A numerical tool for the direct numerical simulation (DNS) of instability and transition to turbulence is presented and applied to problems of secondary instability of complex channel flows. The Navier-Stokes equations for incompressible flow are solved in generalized curvilinear coordinates so that channel flows may be investigated in which the walls of the channel are both curved and wavy. The channel geometry and the flow solution are assumed to be periodic in the streamwise and spanwise directions. A spectral collocation method is employed, in which the periodic directions are discretized using the Fourier collocation method, and the transverse direction is discretized using the Chebyshev collocation method. The time integration is performed with implicit coupling of velocity and pressure at each time step. Both fully- and semi-implicit second-order integration schemes were developed in this study. For the fully-implicit method, Newton’s method is directly applied to the solution of the nonlinear system of equations. The large linear algebra system obtained from the linearization of the spatial discretization and coupled velocity and pressure is solved using a preconditioned iteration scheme based on the Generalized Minimal Residual (GMRES) method. Preconditioning is performed through an approximate factorization of the linearized Navier-Stokes operator which decouples the solutions of the velocity and pressure updates during the iterative algorithm. The velocity and pressure sub-iterations are both solved using preconditioned GMRES as well. The velocity system is preconditioned by a block Jacobi (line-implicit) approximation. The pressure system is preconditioned by left and right Fourier transform operators followed by a block Jacobi approximation.