An effective multigrid method for high‐speed flows

Abstract

We consider the use of a multigrid method with central differencing to solve the Navier-Stokes equations for high-speed flows. The time-dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for two-dimensional lamipar flow over a circular cylinder and a 15 degree compression ramp. iI IntroductionDuring the 1980's a wide variety of numerical schemes were investigated for solving the Euler and Navier-Stokes equations. Multistage time-stepping schemes with central differencing and multigrid acceleration [1,2,3] were demonstrated to be quite effective in computing subsonic and transonic flows 'ver aerodynamic components and configurations. With the recent resurgence of interest in high-speed flight vehicles, we now need to construct versatile algorithms for hypersonic flow. One must keep in mind that hypersonic flows represent a formidable challenge for any flow solver. I1 particular, strong shock and expansion waves can occur in the flow field, and the3 can interact with each other and with shear layers (i.e., boundary layers, jets, wakes). Such strong nonlinear behavior and interactions can easily cause divergence of any numerical integration procedure. This is especially true during the initial phase of a calculation with a time-dependent method. So even the most successful algorithms of the last decade may require significant modifications to be effective for hypersonic flows.An initial effort [4] to apply a central-difference multigrid algorithm to high-speed flows resulted in numerical difficulties that prevented the calculation of two-dimensional flows (i.e., blunt body and wedge type) with a Mach number higher than about 7. In order to compute such flows a low Courant-Friedrichs-Lewy (CFL) number was required. Thus four and five stage schemes were not practical, since there is substantial deterioration in the high frequency damping of the scheme due to the large reduction in the CFL number. The CFL restriction reduced the potential of the scheme as a viscous flow solver. More recently an algorithm utilizing a semicoarsening technique, a symmetric TVD formulation, and a three stage Runge-Kutta scheme [5] was proposed and used to compute high Reynolds number (laminar) Mach 10 flow over an airfoil at 10 degrees angle of attack. A good resolution of the bow shock wave and a reasonable convergence rate were obtained. The method of semicoarsening considered required a much more complicated cycle strategy than that employed with standard multigrid methods. In addition, it appears to be somewhat cumbersome to implement in three dimensions.It is our contention that standard multigrid techniques can be u:.,d in conjunction with central differencing to compute hypersonic flows effectively. To achieve such success with these techniques ...

The Aerodynamic and Dynamic Analysis ofThree Common 4.5mm Caliber Pellets in a Transonic Flow

“…The calculation can be regarded as converged when R error < 0:01% [10,14,15]. The time spent for solving the Navier-Stokes equations for a speci c Mach number is about ve minutes using an Intel ® Core TM 2 Duo CPU 2.53 GHz.…”

Multigrid Convergence Acceleration for Turbulent Supersonic Flows

“…While first employed for elliptic subsonic flows, this technique may also be applied to supersonic and hypersonic flows. [7][8][9][10][11][12] However, possible convergence accelerations for high-speed flows seem to be smaller than those achieved for low-speed cases. Using central difference schemes, the kind of artificial viscosity which is responsible for good shock capturing is also a crucial factor for the efficient use of multigrid techniques.…”

Multigrid for hypersonic viscous two- and three-dimensional flows

“…Both a convection limit as well as a diffusion limit must be taken into account in general. As shown in [15] the actual time step (Lt,,ct), based on a sufficien~t condition for stability, is determined as follows:…”