On Validating an Astrophysical Simulation Code

We present a case study of validating an astrophysical simulation code. Our study focuses on validating FLASH, a parallel, adaptive-mesh hydrodynamics code for studying the compressible, reactive flows found in many astrophysical environments. We describe the astrophysics problems of interest and the challenges associated with simulating these problems. We describe methodology and discuss solutions to difficulties encountered in verification and validation. We describe verification tests regularly administered to the code, present the results of new verification tests, and outline a method for testing general equations of state. We present the results of two validation tests in which we compared simulations to experimental data. The first is of a laser-driven shock propagating through a multi-layer target, a configuration subject to both Rayleigh-Taylor and Richtmyer-Meshkov instabilities. The second test is a classic Rayleigh-Taylor instability, where a heavy fluid is supported against the force of gravity by a light fluid. Our simulations of the multi-layer target experiments showed good agreement with the experimental results, but our simulations of the Rayleigh-Taylor instability did not agree well with the experimental results. We discuss our findings and present results of additional simulations undertaken to further investigate the Rayleigh-Taylor instability.Comment: 76 pages, 26 figures (3 color), Accepted for publication in the ApJ

MFC: An open-source high-order multi-component, multi-phase, and multi-scale compressible flow solver

MFC is an open-source tool for solving multi-component, multi-phase, and bubbly compressible flows. It is capable of efficiently solving a wide range of flows, including droplet atomization, shock–bubble interaction, and bubble dynamics. We present the 5- and 6-equation thermodynamically-consistent diffuse-interface models we use to handle such flows, which are coupled to high-order interface-capturing methods, HLL-type Riemann solvers, and TVD time-integration schemes that are capable of simulating unsteady flows with strong shocks. The numerical methods are implemented in a flexible, modular framework that is amenable to future development. The methods we employ are validated via comparisons to experimental results for shock–bubble, shock–droplet, and shock–water-cylinder interaction problems and verified to be free of spurious oscillations for material-interface advection and gas–liquid Riemann problems. For smooth solutions, such as the advection of an isentropic vortex, the methods are verified to be high-order accurate. Illustrative examples involving shock–bubble-vessel-wall and acoustic–bubble-net interactions are used to demonstrate the full capabilities of MFC

A verification of steady state discontinuous solutions using the method of manufactured solutions for finite volume computational fluid dynamic codes

When applying the method of manufactured solutions (MMS) on computational fluid dynamic (CFD) software, it is traditionally a requirement that all solutions be continuous on the computational domain. This stipulation is limiting for the verification and validation of CFD solutions where discontinuities are frequent. This work details the development of a discontinuous MMS method for finite volume codes. The CFD code used throughout this research is a cell centered, finite volume, 1st order, Eulerian scheme within the software AVUS (Air Vehicles Unstructured Solver) which is combined with uniform structured grids. This code is used as a representative testing platform with the convenience of accessible source code. A piecewise technique is used for defining manufactured solutions which simulate discontinuities. Since source terms which allow arbitrary solutions in continuous MMS do not exist within Riemann solvers, conditions at the shock boundary are physically constrained by the Rankine-Hugoniot jump conditions. Upwind manufactured solutions are first initialized and a regression technique is then used to solve for solutions downwind of the discontinuity. It is shown that a change in regression error of four order of magnitude has no significant effect on an order of convergence test. When applying MMS on finite volume CFD codes, determining the exact solutions and source terms when the stored value is the integrated average over the control volume is a non-trivial and frequently ignored problem. MMS with discontinuities further complicates the problem of determining these values. To obtain low error and high convergence rates, linearly and quadratically exact transformations are proposed for cells split by discontinuities. These transformations are combined with a nine point Gauss quadrature method to achieve 4th order accuracy for fully general solutions and shock shapes. To begin testing, continuous MMS is first performed to ensure a verified code. AVUS is verified for 1st order solutions but retains lower order boundary conditions when solving 2nd order. The error is verified using a second academic CFD solver but is left unchanged for shock solutions which are inherently 1st order. Constant primitive, oblique shock solutions are then used to demonstrate a solution\u27s error dependence on grid alignment. Grid alignment is shown to play a vital role in the error surrounding a shock. Constant oblique solutions with a grid aligned shock result in no discretization error while a shock that passed through the interior of cells yields error upwards of 4% for the u-component velocity. A semi one-dimensional problem combined with a grid aligned shock is then used to demonstrate the error magnitude (\u3c 1%) due to the cell averaging on both sides of the discontinuity. Fully generic primitives and discontinuities are then introduced and grid convergence studies yielding 1st order results typically associated with shocks are used to verify the correctness of the code. Despite high errors near the region of the shock, similar patterns and orders of convergence are shown for both physical and mathematical shock shapes. Sub-linear convergences near 0.9, especially in the u-component velocity, indicate the presence of linearly degenerate characteristics which are typical of shock capturing schemes. The fully general solutions are used to show that errors in the Riemann solver can be identified with discontinuous MMS. Three coding errors are introduced which are not identified by continuous MMS. In two cases the discontinuous procedure indicates a coding error in the convergence test and the final error is classified as a efficiency mistake and is missed by all methods. Lastly, the method is replicated on an academic CFD code for a final validation procedure. Identical behavior and near identical errors suggest the robustness of the developed method despite fundamental differences in the shock capturing schemes of the two codes

Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility