Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow

Abstract

We consider the solution of the reactive and non-reactive Euler equations on twodimensional domains that evolve in time. The domains are discretized using moving overlapping grids. In a typical grid construction, boundary-fitted grids are used to represent moving boundaries, and these grids overlap with stationary background Cartesian grids. Block-structured adaptive mesh refinement (AMR) is used to resolve fine-scale features in the flow such as shocks and detonations. Refinement grids are added to base-level grids according to an estimate of the error, and these refinement grids move with their corresponding base-level grids. The numerical approximation of the governing equations takes place in the parameter space of each component grid which is defined by a mapping from (fixed) parameter space to (moving) physical space. The mapped equations are solved numerically using a second-order extension of Godunov's method. The stiff source term in the reactive case is handled using a Runge-Kutta error-control scheme. We consider cases when the boundaries move according to a prescribed function of time and when the boundaries of embedded bodies move according to the surface stress exerted by the fluid. In the latter case, the Newton-Euler equations describe the motion of the center of mass of the each body and the rotation about it, and these equations are integrated numerically using a second-order predictor-corrector scheme. Numerical boundary conditions at slip walls are described, and numerical results are presented for both reactive and non-reactive flows in order to demonstrate the use and accuracy of the numerical approach.

Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement

“…The second special case involves a more difficult set of equations given by the reactive Euler equations. This set of equations was considered in our previous papers [2] and [3] for two-dimensional flow in stationary and moving domains. Here, our focus is on reactive and non-reactive flow in three dimensions for which we consider the nonlinear conservation equations given by…”

“…Once the general framework is established, we proceed in the next section to describe the extension of the numerical approach for parallel computations. It is worth noting that while the present discussion focuses on solving the IBVP given in (1), and the equations in (2) and (3) in particular, the numerical approach outlined here is more general and may be used to handle a wide range of time-dependent problems, such as those for the second-order Maxwell equations [37] and problems involving moving boundaries [3].…”

“…The first is a linear advection-diffusion equation, and the second is the (nonlinear) reactive Euler equations of gas dynamics. The first equation provides a useful test case to study the behavior of the numerical approach and to verify its accuracy quantitatively, while the second builds on our earlier work in [2,3] and illustrates the numerical approach for a more difficult set of equations. While a brief description of the discretization of these two PDEs is given, the emphasis of the discussion is on the extension of the numerical approach for parallel computations in three-dimensional geometries.…”

“…Other pioneering work includes that of Starius [6][7][8], Kreiss [9], and Steger et al [10] who referred to the approach as Chimera grids. Since this early work, the overlapping grid technique has been used successfully to solve a wide variety of problems in high-speed reactive flow [2,3,11], reactive and non-reactive multi-material flow [12,13], combustion [14], aerodynamics [15][16][17][18][19][20][21], blood flow [22], flows around ships [23], visco-elastic flows [24], and flows with deforming boundaries [25][26][27], among others. The use of AMR in combination with overlapping grids was considered by Brislawn et al [28], Boden and Toro [29], Meakin [21], Henshaw and Schwendeman [2,3], and Banks et al [12,13].…”

“…The software used to generate the results presented in this paper is freely available 3 . The parallel-AMR components that support regridding, interpolation, error estimation, etc.…”