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.
This numerical technique was applied to several problems of instability and transition in curved channel flows and in curved channel flows with wall waviness. The numerical methodology was validated by carefully comparing the present results with those of Finlay, Keller and Ferziger (JFM, vol. 194, 1988) and Ligrani et al. (Phys. Of Fluids A, vol. 4, no. 4, 1992) for two- and three-dimensional Dean vortex flows in a curved channel. Also, new results were obtained for curved channel flows with two-dimensional small amplitude wall waviness. The waviness significantly altered the evolution of both Dean vortex and Tollmien-Schlichting wave instabilities. The traveling wave twisting Dean vortex solution of Ligrani et al. for Reynolds number 409 was repeated with wall waviness, and resulted in a highly oscillatory state. Waviness also modified the secondary instability of Tollmien-Schlichting waves at Reynolds number 5000 by forcing asymmetry in the three-dimensional A-vortex structures near the upper and lower walls of the channel. Finally, highly unsteady and complex results were obtained for saturated Tollmien-Schlichting waves in two-dimensional channel flow with large amplitude wall waviness at Reynolds number 5000. These cases were used to demonstrate the capabilities of the computational tool for DNS of instability and transition in complex channel flows.
This work is supported in part by AFOSR Grant No. F49620-93-1-0393, NSF Grant No. CTS-9512450 and Ohio Supercomputer Center Grant No. PES070-5.
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Duncan, B.D., Ghia, K.N. (1999). Direct Numerical Simulation of Transitions Toward Turbulence in Complex Channel Flows. In: Knight, D., Sakell, L. (eds) Recent Advances in DNS and LES. Fluid Mechanics and its Applications, vol 54. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4513-8_12
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DOI: https://doi.org/10.1007/978-94-011-4513-8_12
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