A comparative study of accuracy of shock capturing schemes for simulation of shock/acoustic wave interactions (original) (raw)
Related papers
Numerical investigation of the interaction of acoustic disturbances with a shock wave
The interaction of acoustic waves with a shock wave is simulated numerically. Both sound reflec- tion and refraction are investigated. Numerical simulations are performed with resolving the interior viscous structure of shock transition. A high-order compact-difference scheme is used to solve the compressible Navier-Stokes equations. It is shown that in the limiting case when the wavelength of incident disturbances is much greater than the shock thickness, the results of simulations within a wide range of Mach numbers and angles of incidence of disturbances correspond very accurately to the predictions of the classical linear inviscid theory. The results obtained enable an unambiguous conclusion on the incorrectness of the recently proposed alternative theory of interaction of shock waves with small disturbances to be made.
Robustness versus accuracy in shock-wave computations
International Journal for Numerical Methods in Fluids, 2000
Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported (Quirk JJ. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18: 555-574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd -even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd -even decoupling. This analysis is applied to Roe's scheme (Roe PL, Approximate Riemann solvers, parameters vectors, and difference schemes, J. GRESSIER AND J.-M. MOSCHETTA 314 computations where shocks propagate along a wall. Before analyzing the way these failings appear, some results are given on classical upwind schemes such as Roe's scheme or Osher's scheme and more recent schemes such as the AUSM scheme [10] and EFMO . These schemes have been selected because of their shock-capturing capabilities which make them well-suited for the computation of high-speed flows.
Computational Considerations for the Simulation of Shock-Induced Sound
SIAM Journal on Scientific Computing, 1998
The numerical study of aeroacoustic problems places stringent demands on the choice of a computational algorithm, because it requires the ability to propagate disturbances of small amplitude and short wavelength. The demands are particularly high when shock waves are involved, because the chosen algorithm must also resolve discontinuities in the solution.
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, 2009
High-order methods that can resolve interactions of flow-disturbances with shock waves are critical for reliable numerical simulation of strong-shock and turbulence interaction problems. Such problems are not well understood due to limitations of numerical methods. Most of the popular shock-capturing methods are inherently dissipative and may incur numerical oscillations near the shock. Present paper is continuation of our previous work [1] on development and implementation of new algorithms based on shock-fitting and fronttracking methodology which can solve the flow with high-order accuracy near as well as away from the shocks. The shock-fitting algorithm avoids dissipation and possible numerical oscillations incurred in shock-capturing methods by treating shocks sharply. We explore two ways for shock-fitting: conventional moving grid setup and a new fixed grid setup with front tracking. In conventional shock-fitting method, a moving grid is fitted to the shock while in the newly developed fixed grid setup shock is tracked using Lagrangian points and is free to move across underlying fixed grid. High order front-tracking method gives satisfactory results for 2-D shock disturbance interaction problem. We also studied various methods to find shock velocity. It was found that conventional method of advancing the shock using characteristics based shock-acceleration relation gives more accurate results than other methods considered. Shock-turbulence interaction problems may produce nonlinearities like secondary shocks in the post-shock flow. Hence, we have combined our shock-fitting methodology with shock-capturing methods. Convergence analysis was carried out for combined shock-fitting and WENO methods for an unsteady shock disturbance problem and expected fifth order accuracy was achieved. It was found that, where applicable, shock-fitting algorithm provides superior results to those obtained from purely shock-capturing schemes.
Comparative study of high-resolution shock-capturing schemes for a real gas
AIAA Journal, 1989
The recently developed second-order explicit shock-capturing methods of the van Leer, Harten, and Yee types, in conjunction with the generalized flux-vector splittings of Vinokur and Montagnk, and a generalized Roe's approximate Riemann solver of Vinokur for a real gas are studied. The comparisons are made on different one-dimensional Riemann (shock-tube) problems for equilibrium air with various ranges of Mach numbers, densities, and pressures. Six different Riemann problems are considered. These tests provide a check on the validity of the generalized formulas, since theoretical prediction of their properties appears t o be difficult because of the non-analytic form of the state equation. The numerical results in the supersonic and low-hypersonic regimes indicate that these approaches produce good shock-capturing capability and that the shock resolution is only slightly affected by the state equation of equilibrium air. The difference in shock resolution between the various methods varies slightly from one Riemann problem to another, but the overall accuracy is very similar. For the onedimensional case, the relative efficiency in terms of operation-count for the different methods is within 30%. The main difference between the methods lies in their versatility in being extended to multidimensional problems with efficient implicit solution procedures. tAn abbreviated version will appear in the proceedings of the 7th GAMM Conference on Numerical Methods in Fluid Mechanics,
A hybrid semi-primitive shock capturing scheme for conservation laws
Electronic Journal of Differential Equations
A hybrid semi-primitive shock capturing scheme is presented for hyperbolic problems. Upwind based construction is done using explicit information on the wave propagation direction associated with the problem. This scheme captures the shock waves at right location but shows unphysical sonic expansion shock. This phenomena of unphysical expansion shock in the presence of expensive sonic point is not surprising and it is common for the wave based upwind schemes. A hybrid scheme approach using an iteration criteria based on sonic entropy fix is proposed to avoid such expansion shocks. Numerical results for scalar test problems are presented which show that proposed scheme captures the shock accurately.
An investigation of the internal structure of shock profiles for shock capturing schemes
Journal of Computational and Applied Mathematics, 2007
The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.
International Journal for Numerical Methods in Fluids, 2000
The present study addresses the capability of a large set of shock-capturing schemes to recover the basic interactions between acoustic, vorticity and entropy in a direct numerical simulation (DNS) framework. The basic dispersive and dissipative errors are first evaluated by considering the advection of a Taylor vortex in a uniform flow. Two transonic cases are also considered. The first one consists of the interaction between a temperature spot and a weak shock. This test emphasizes the capability of the schemes to recover the production of vorticity through the baroclinic process. The second one consists of the interaction of a Taylor vortex with a weak shock, corresponding to the framework of the linear theory of Ribner. The main process in play here is the production of an acoustic wave. The results obtained by using essentially non-oscillatory (ENO), total variation diminishing (TVD), compact-TVD and MUSCL schemes are compared with those obtained by means of a sixth-order accurate Hermitian scheme, considered as reference. The results are as follows; the ENO schemes agree pretty well with the reference scheme. The second-order accurate Upwind-TVD scheme exhibits a strong numerical diffusion, while the MUSCL scheme behavior is very sensitive to the value on the parameter i in the limiter function minmod. The compact-TVD schemes do not yield improvement over the standard TVD schemes.