Computational Fluid Dynamics Development

On the Earth, 2/3 of the matter is fluid such as the air in the atmosphere, the water in the sea, or even the blood in our human bodies. Most of the fluids flow in the state of turbulence, which means that the flow parameters such as velocity, pressure, density, etc are always randomly fluctuating with time. The turbulence will increase the friction drag of an airplane, the heat transfer of air conditioning, and enahnce the mixing between two fluids such as coffee and cream. For high speed aerodynamic flows, there is also a phenomenon called shock wave, which is an extremely thin surface. Across a shock wave, the flow parameters such as pressure, temperature, velocity jump. Such a jump is a discontinuity in the mathematics sense. The other example of discontinuity is the contact surface between air and water, across which the density jumps about 1000 times. Even though the density has a jump, the velocity and pressure at the contact surface are the same such as the ocean wave driven by the wind of hurricane.

The governing equations of fluid motion are the Navier-Stokes equations, which are based on mass conservation, Newton's 2nd law, and energy conservation. The Navier-Stokes equations are hence composed of partial differential derivatives in time and space. We all know that a derivative does not exist across a discontinuity. How to calculate the turbulent flow fields with shock waves or contact discontinuities is a very challenging and unsolved problem. One of our research interests is to simulate these turbulent flow fields using large eddy simulation (LES) and detached eddy simulation (DES). We have developed high order (5th and 7th order) finite difference weighted essentially non-oscillatory (WENO) schemes to capture the shock waves and contact discontinuities. We also developed a fully conservative 4th and 6th order central differencing schemes for the viscous terms in Navier-Stokes equations. In mathematics term, a numerical scheme that can resolve the discontinuities is called Riemann solver. In our research, a E-CUSP upwind scheme is developed and it can capture crisp shock profiles and exact contact surfaces. The E-CUSP scheme is used with those high order WENO schemes for turbulence simulation and fluid-structural interactions. An implicit Gauss-Seidel line relaxation with 2nd order temporal accuracy is used to avoid the approximate factorization errors (such as those in Beam-Warming scheme, Jameson's LU-SGS scheme) and achieve high convergence rate.

We have also developed the Non-Reflective boundary conditions for 3D unsteady Navier-Stokes equations to avoid wave reflections at the computational boundaries, which can contaminate the flow solutions and distort the simulated results. A general sub-domain boundary mapping procedure is created in our research for parallel computation of the flow fields. The parallel computation is to partition a large domain to multiple smaller sub-domains and conduct the calculation simultaneously on multiple processors to save wall clock time. We have built a Beowulf Linux cluster system in our CFD lab for parallel computing.

The following is a LES result animation of the vorticity field in a 3D turbulent flow passing a cylinder at Reynolds number of 3900 and incoming Mach number of 0.2.

Recent Publications: