Development of an Incompressible Navier-Stokes Solver Involving Symplectic and Nonsymplectic Time Integrators

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This article develops a heat/flow solver for properly simulating incompressible viscous flow at high Reynolds numbers and natural-convection flow at high Rayleigh numbers. Our strategy of simulating these problems is to retain some rich geometric properties embedded in the invisid Euler equations. Thanks to the underlying theory of Clebsch velocity decomposition, which divides the velocity vector as the sum of three velocity components due to the potential, rotational, and viscous contributions, the mixed-type Navier-Stokes equations cast in primitive variables can be fractionally split into three corresponding equations which are coupled to each other. In the pure advection step, the symplectic Runge-Kutta integrator is employed to approximate the temporal derivative term so as to numerically retain the Hamiltonians embedded in the lossless Euler differential equations. In addition, an upwinding scheme with minimized phase error is applied to discretize the first-order spatial derivative terms. In the diffusion step, we approximate the time derivative term shown in the time-dependent parabolic equation using a time-stepping scheme which does not need to be symplectic and is thus different from that used in the first step for the calculation of Euler solutions. We then solve the velocity vector from the projection step, subject to the divergence-free constraint condition. The proposed method is validated through benchmark tests. The predicted results of the incompressible Navier-Stokes and natural-convection equations are also justified for the problems investigated at high Reynolds and high Rayleigh numbers, respectively.