Shock capturing methods have been widely used in the computation of inviscid flows with shock and contact discontinuities and they cancan be classified into two main categories, modern and classical. The classical methods including the McCormack method (72),72 the Lax-Wendrofff method (73), 73 and Beam-Warming method (74),74 use symmetric or central discretization schemes, while modern methods employ upwind-type differencing schemes. Modern methods include higher order schemes such as the Total Variation diminishing (TVD) scheme first proposed by Harten(75),75 the Flux-Corrected Transport (FTC) scheme introduced by Boris and Book (76),76 Monotonic Upstream-centered Schemes for Conservation Laws (MUSCL) based on the Godunov approach (77),77 and introduced by Van Leer (78),78 various Essentially Non-Oscillatory schemes (ENO) proposed by Harten et al.(79),79 Piecewise Parabolic Method (PPM) proposed by Woodward and Colella(80).80 Another important class of high-resolution schemes belongs to the approximate Riemann solvers proposed by Roe and Osher(81)(82).81,82
In general, when shock discontinuities are present in the flow, it is almost impossible to achieve a stable solution, free of unphysical numerical oscillations and nonlinear instabilities without introducing some numerical dissipation.
Modern shock capturing methods employ non-linear numerical dissipation, in a way that the amount of dissipation in any cell is adjusted according to gradients in neighboring cells which makes this scheme stable and accurate.
High-resolution schemes provide 2nd order or higher accuracy in numerical solutions in the vicinity of strong gradients such shocks and contact discontinuities. To avoid the generation of numerical oscillations associated with high order spatial discretization schemes, flux limiters are used. They only come into operation when sharp wave fronts are present and have the effect of limiting the spatial derivatives near shocks and discontinuities to physically realistic values which makes the solution free from spurious numerical oscillations. In smooth regions of the solution, flux limiters do not operate and spatial derivatives are represented by higher order spatial accuracy (75) (83)(84).75,83,84 The numerical method is said to be TVD, total variation diminishing when no new local extrema can be created within the solution spatial domain and the value of a local minimum is non-decreasing, and the value of a local maximum is non-increasing and therefore monotonicity is preserved. Godunov has shown that first order schemes preserve monotonicity and are therefore TVD. On the other hand, higher order schemes are not TVD and introduce spurious oscillations. These drawbacks are overcome with flux limiters which make the numerical scheme TVD(85).85 TVD limiters are designed to switch the spatial discretization scheme down to a first order accurate method in the vicinity of strong gradients. Away from the shocks and contact discontinuities, TVD simply do not operate and higher order discretization schemes are used in the majority of the flow while still capturing shock waves and strong gradients without obvious wiggles (75).75
Previous studies which modeled the complex transient flow structures in shock tubes and captured the shock and contact discontinuities by solving the Euler equations have used different numerical approaches.6,36,37,64-71 (64) (65) (66) (67) (68) (36) (69) (6) (37) (70)(71)
Argow64 (64)studied the evolution of non-classical flow fields in a conventional shock tube by solving the one-dimensional Euler equations. The equations were discredited by using the TVD-MacCormack (TVDM) predictor-corrector scheme, a finite volume variant of the MacCormack scheme which is second order accurate in space and time. A Minmod limiter was used and the courant condition was set to 0.6.
Loh and Liou71 (71)showed that complicated shock and contact discontinuities can be accurately resolved when the streamwise marching Langrangian method is adopted. The steady Euler equations were discretized using the finite difference first order scheme which was upgraded to the high resolution TVD scheme by using the Minmod flux limiter. The scheme is a variation of Van Leer's MUSCL scheme78(78) followed that given in Liou86 (86) and Liou and Hsu87(87).
Cocchi et al. 66 (66) proposed a correction to Godunov Type schemes that yields a perfect discontinuity. This method is based on a prediction step which makes use of any Euler scheme and a correction step based on a lagrangian approach. Two discretization methods were used; the first order Godunov scheme and the second order VanLeer scheme. The numerical diffusion was corrected for by the correction step which consists of interpolating the values at node points on both sides of the interface as function of values at neighboring points. Instabilities and oscillations were handled with the Total Variation Diminishing concept (TVD) and the Minmod limiter.
Petrie-Repar and Jacobs37 (37) employed a cell-centered finite volume code U2DE. The generalized MUSCL interpolation scheme was used to construct the left and right flow states and the Minmod limiter was applied to limit the oscillations in the flow domain. In addition, the Equilibrium flux Method (EFM) was used to calculate the flux array from the left and right edge flow states. EFM solves the Euler equations with added pseudo dissipation and in the hypersonic limit, becomes an upwind scheme. Grid adaption was implemented with the density and pressure gradient adaption performed every five time steps. Advancement in time was achieved by using the predictor-corrector explicit time scheme. The CFL condition was set to 0.5.
Burtschell and Zeitoun49 (49) investigated the interaction of two oblique axi-symmetrical shock waves in a supersonic flow by solving the Euler equations according to a cell-centered finite volume method on a two-dimensional structured grid. A second order accurate algorithm in space and time was used. The dissipative fluxes were replaced with central differences and the convective fluxes were computed by solving the Riemann problem replaced by the AUSM-M in the case of strong shocks. Instabilities were handled with both the Minmod and the Superbee-type limiters. The unsteady formulation of the discretized equation used a predictor corrector explicit time scheme with the CFL condition set to 0.8.
Jiang et al.70 (70) investigated the three-D propagation of the transmitted shock wave in a square cross section chamber numerically by solving the Euler equation. Discretization in space was accomplished using the dispersion-controlled scheme and the flux vector was split according to the Steger and Warming method with the help of the Minmod limiter. The time marching integration was performed using a 2nd order accurate Range-Kutta algorithm with the courant number set to 0.5. The dispersion controlled scheme requires that shock capturing schemes must have leading or lagging phase errors to avoid non-physical oscillations near the shock and contact discontinuities which can be achieved without resorting to additional artificial viscosity.
Cocchi et al.68 (68) proposed a hybrid formulation of conservative and non-conservative forms to solve the Euler equations in order to correctly estimate the temperature across shock and strong rarefaction waves in two-dimensional flows. A finite volume formulation of a MacCormack72 (72) scheme with second order artificial viscosity was applied. Numerical results were compared to results from classical schemes such as the first order Godunov scheme77 (77) and the second order Godunov-MUSCL Hancock scheme78(78) with an exact Riemann solver.
Takayama and Sun88 (88) performed numerical studies of shock diffraction phenomena in a two dimensional shock tube model by solving the Euler equations following a finite volume approach. The equations were discretized by the means of two 2nd order in space and time schemes; a centered scheme based on the predictor-corrector Lax- Wendroff scheme with added nonlinear artificial viscosity and an upwind MUSL-Hancock scheme (84).84 A Minmod limiter was used to flatten slopes of primitive variables and the fluxes through interfaces were determined by solving the HLLC approximate Riemann problem.
Chang and Kim67 (67) investigated the dynamics of inviscid shock waves in an expansion tube. The simulation was performed on an axi-symmetric unstructured triangular mesh using the Finite volume Galerkin algorithm .The FCT (Flux-Corrected Transport) discretization scheme was adopted by blending the low order fluxes with the higher order ones under the monotonicity constraint. Excessive Anti-diffusion from the low order scheme was corrected for by blending the artificial dissipation and the area weighted differencing of higher order increment. A limiter function that computes the minimum between the density and the energy was chosen to prevent overshoots in the solution. Solution was marched in time following the 3-stage Runge-Kutta time integration. The H refinement adaptive method was used for grid adaption every 5 or 7 time steps.