Shock-capturing Algorithm Research Articles

An efficient algorithm for the multicomponent compressible Navier–Stokes equations in low- and high-Mach number regimes

The goal of this study is to develop an efficient numerical algorithm applicable to a wide range of compressible multicomponent flows, including nearly incompressible low-Mach number flows, flows with strong shocks, multicomponent flows with high density ratios and interfacial physics, inviscid and viscous flows, as well as flows featuring combinations of these phenomena and various interactions between them. Although many highly efficient algorithms have been proposed for simulating each type of the flows mentioned above, the construction of a universal solver is known to be challenging. Extreme cases, such as incompressible and highly compressible flows, or inviscid and highly viscous flows, require different numerical treatments in order to maintain the efficiency, stability, and accuracy of the method.Linearized block implicit (LBI) factored schemes (see e.g. Briley and McDonald, 1980 [1], Beam and Warming, 1978 [2]) are known to provide an efficient way of solving the compressible Navier–Stokes equations (compressible NSEs) implicitly, allowing us to avoid stability restrictions at low Mach number and high viscosity. However, the methods’ splitting error has been shown to grow and dominate physical fluxes as the Mach number approaches zero (see Choi and Merkle, 1985 [3]). In this paper, a splitting error reduction technique is proposed to solve the issue. A shock-capturing algorithm from [4] is reformulated in terms of finite differences, extended to the stiffened gas equation of state (SG EOS) and combined with the LBI factored scheme to stabilize the method around flow discontinuities at high Mach numbers. A novel stabilization term is proposed for low-Mach number applications. The resulting algorithm is shown to be efficient in both low- and high-Mach number regimes. Next, the algorithm is extended to a multicomponent case using an interface capturing strategy (see e.g. Saurel and Abgrall, 1999 [5]) with surface tension as a continuous surface force (see Perigaud and Saurel, 2005 [6]). Special care is taken to avoid spurious oscillations of pressure and generation of artificial acoustic waves in the numerical mixture layer. Numerical tests are presented to verify the performance and stability properties for a wide range of flows.

Flood Inundation Modelling of Flash Floods in Steep River Basins and Catchments

The potential flood inundation extent can be estimated with flood inundation models, which can differ in the level of physical and numerical modelling complexity included in the solution procedure. In recent years, several studies have highlighted the benefits of shock-capturing flood inundation models, particularly when modelling a high Froude number or supercritical flows, or in areas prone to the occurrence of rapidly varying flood events, such as flash floods. Nonetheless, decision makers are often reluctant to implement more complex modelling tools into practical flood inundation modelling studies, unless evidence is provided to establish when such refined modelling tools should be used. The main objective of this study was to determine a general threshold value of the bottom slope that could be used by decision makers as an orientation guide to ascertain when to use a specific type of flood inundation model. The results obtained suggest that in torrential river basins or catchments (i.e., river basins and catchments with a bed slope generally greater than 1%), the flood inundation modelling should be conducted by using a flood inundation model that include shock-capturing algorithms in the model solution procedure.

A numerical study of nonlinear infrasound propagation in a windy atmosphere.

Direct numerical simulations of the two-dimensional unsteady compressible Navier-Stokes equations are performed to study the acoustic field generated by an infrasonic source in a realistic atmosphere. Some of the main phenomena affecting the propagation of infrasonic waves at large distances from the source are investigated. The effects of thermal and wind-related refraction on the signals recorded at ground level are highlighted, with particular emphasis on the phase shift induced by the presence of caustics in the acoustic field. Nonlinear waveform steepening associated with harmonic generation, and period lengthening, both of which are typical of large source amplitudes, are illustrated, and the importance of thermoviscous absorption in the upper atmosphere is clearly demonstrated. The role of diffraction in the shadow zone, around caustics and at stratospheric altitudes is also pointed out. The Navier-Stokes equations are solved using high-order finite-differences and a Runge-Kutta time integration method both originally developed for aeroacoustic applications, along with an adaptive shock-capturing algorithm which allows high-intensity acoustic fields to be examined. An improvement to the shock detection procedure is also proposed in order to meet the specificities of nonlinear propagation at long range. The modeling as well as the numerical results are reported in detail and discussed.

Appropriate model use for predicting elevations and inundation extent for extreme flood events

Flood risk assessment is generally studied using flood simulation models; however, flood risk managers often simplify the computational process; this is called a “simplification strategy”. This study investigates the appropriateness of the “simplification strategy” when used as a flood risk assessment tool for areas prone to flash flooding. The 2004 Boscastle, UK, flash flood was selected as a case study. Three different model structures were considered in this study, including: (1) a shock-capturing model, (2) a regular ADI-type flood model and (3) a diffusion wave model, i.e. a zero-inertia approach. The key findings from this paper strongly suggest that applying the “simplification strategy” is only appropriate for flood simulations with a mild slope and over relatively smooth terrains, whereas in areas susceptible to flash flooding (i.e. steep catchments), following this strategy can lead to significantly erroneous predictions of the main parameters—particularly the peak water levels and the inundation extent. For flood risk assessment of urban areas, where the emergence of flash flooding is possible, it is shown to be necessary to incorporate shock-capturing algorithms in the solution procedure, since these algorithms prevent the formation of spurious oscillations and provide a more realistic simulation of the flood levels.

The application of a Godunov-type shock capturing scheme for the simulation of waves from deep water up to the swash zone

A two-dimensional vertical (2DV) non-hydrostatic boundary fitted model based on a Godunov-type shock-capturing scheme is introduced and applied to the simulation of waves from deep water up to the swash zone. The effects of shoaling, breaking, surf zone dissipation and swash motions are considered. The application of a Godunov-type shock-capturing algorithm together with an implicit solver on a standard staggered grid is proposed as a new approach in the 2DV simulation of large gradient problems such as wave breaking and hydraulic jumps. The complete form of conservative Reynolds averaged Navier–Stokes (RANS) equations are solved using an implicit finite volume method with a pressure correction technique. The horizontal advection of the horizontal velocity is solved by an explicit predictor–corrector method. Fluxes are predicted by an exact Riemann solver and corrected by a downwind scheme. A simple total variation diminishing (TVD) method with a monotonic upstream-centered scheme for conservation laws (MUSCL) limiter function is employed to eliminate undesirable oscillations across discontinuities. Validation of the model is carried out by comparing the results of the simulations with several experimental test cases of wave breaking and run-up and the analytical solution to linear short waves in deep water. Promising performance of the model has been observed.

Cosmological shock waves: clues to the formation history of haloes

Shock waves developed during the formation and evolution of cosmic structures encode crucial information on the hierarchical formation of the Universe. We analyze an Eulerian AMR hydro + N-body simulation in a $\Lambda$CDM cosmology focused on the study of cosmological shock waves. The combination of a shock-capturing algorithm together with the use of a halo finder allows us to study the morphological structures of the shock patterns, the statistical properties of shocked cells, and the correlations between the cosmological shock waves appearing at different scales and the properties of the haloes harbouring them. The shocks in the simulation can be split into two broad classes: internal weak shocks related with evolutionary events within haloes, and external strong shocks associated with large-scale events. The shock distribution function contains information on the abundances and strength of the different shocks, and it can be fitted by a double power law with a break in the slope around a Mach number of 20. We introduce a generalised scaling relation that correlates the average Mach numbers within the virial radius of haloes and their virial masses. In this plane, haloes occupy different areas according to their early evolutionary histories: those with a quiet evolution have an almost constant Mach number independently of their masses, whereas haloes undergoing significant merger events very early in their evolution show a linear dependence with their masses. At high redshift, the halo distribution in this scaling relation forms a L-like pattern that changes due to the evolution of the haloes. The analysis of the propagation speed and size of the shock waves around haloes could give some hints on the formation time and main features of the haloes. (Abridged)

A note on compressive limiting for two-material flows

In this short note we describe a simple extension to the multi-material shock-capturing algorithm presented in (J. Comput. Phys. 2007; 223:262–297) that can be used to maintain sharp material interfaces. The method takes the form of an artificial compression which is designed so that the material indicator jumps across only a few cells but which does not excite physical instabilities in the flow. The advantages of the approach include its simplicity and flexibility in that it provides a parameter that effectively determines the captured interface thickness. Copyright © 2009 John Wiley & Sons, Ltd.

A characteristic-based shock-capturing scheme for hyperbolic problems

In order to suppress numerical oscillations of linear compact schemes around discontinuities, a characteristic-based flux splitting limited method is introduced instead of ENO/WENO or other shock-capturing algorithms. This method begins with upwind schemes and flux vector splittings. The upwind schemes are projected along characteristic directions in a different way, and their amplitudes are carefully controlled by a special limiter in order to meet entropy condition and to prevent non-physical oscillations. A fifth-order linear compact upwind scheme is modified by this method for solving problems involving discontinuities. The properties of the numerical algorithm are checked on some benchmark problems in one, two and three space dimensions. Numerical results show that it is high-order accurate with high resolution and oscillation-free.

A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation/erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum–minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well.

A kinetic approach to tunnelling at Schottky contacts

In this paper, we present a semiclassical kinetic approach to tunnelling through potential barriers, which can be applied to the simulation of planar semiconductor devices. The proposed model includes thermionic emission currents at the metal–semiconductor interface as well as tunnelling currents together with the effect of barrier lowering. The considered scattering mechanisms are electron–phonon and electron–impurity interactions. The numerical scheme used is a combination of multicell methods with high-order shock-capturing algorithms. To demonstrate the applicability of the developed method, we present the characteristics of several silicon based Schottky barrier diodes.

Numerical simulation of detonations in mixtures of gases and solid particles

This article examines the structure and stability of detonations in mixtures of gases and solid particles via direct numerical simulation. Cases with both reactive and inert particles are considered. First, the two-phase flow model is presented and the assumptions that it is based upon are discussed. Steady-wave structures admitted by the model are subsequently analysed. Next, the algorithm employed for the numerical simulations is described. It is a recently developed high-order shock-capturing algorithm for compressible two-phase flows. The accuracy of the algorithm has been verified through a series of code validation and numerical convergence tests, some of which are included in this article. Subsequently, numerical results for both one-dimensional and two-dimensional detonations are presented and discussed. These results show that the mass, momentum and energy transfers between the two phases result in detonation structures that are substantially different from those observed in the corresponding purely gaseous flows. The effect of certain important parameters, such as particle reactivity, particle volume fraction, and particle diameter, are examined in detail. The numerical results predict that increased particle reactivity suppresses the flow instabilities and increases the detonation velocities. It is further predicted that sufficiently high particle volume fractions can cause detonation quenching regardless of particle reactivity.

A high-wavenumber viscosity for high-resolution numerical methods

A spectral-like viscosity is proposed for centered differencing schemes to help stabilize numerical solutions and reduce oscillations near discontinuities. Errors introduced by the added dissipation can be made arbitrarily small by adjusting the power of the derivative in the viscosity term. The high-wavenumber viscosity is combined with a 10th-order compact scheme to produce an accurate and efficient shock-capturing method. The new scheme compares favorably with other shock-capturing algorithms.

An adaptive shock-capturing algorithm for solving unsteady reactive flows

A new double flux model is associated with the recent MUSCL-AUSM +-triad approach. This coupling allows to do rapid, accurate and stable numerical computations. 1-D and 2-D inert and reactive flows are presented.

A New Spherically Symmetric General Relativistic Hydrodynamical Code

In this paper we present a full general relativistic one-dimensional hydro-code which incorporates a modern high-resolution shock-capturing algorithm, with an approximate Riemann solver, for the correct modelling of formation and propagation of strong shocks. The efficiency of this code in treating strong shocks is demonstrated by some numerical experiments. The interest of this technique in several astrophysical scenarios is discussed.

Computational Aerodynamics for Aircraft Design

This article outlines some of the principal issues in the development of numerical methods for the prediction of flows over aircraft and their use in the design process. These include the choice of an appropriate mathematical model, the design of shock-capturing algorithms, the treatment of complex geometric configurations, and shape modifications to optimize the aerodynamic performance.

Modification of the Osher upwind scheme for use in three dimensions

The Osher upwind scheme, a shock-capturing algorithm based on an approximate Riemann solver, which has been developed for the solution of hyperbolic conservation laws, is briefly described, and the need for modification of the scheme for use in three dimensions is demonstrated. Two forms of a scheme modification for use on a three-dimensional arbitrary grid are then presented.