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The purpose of the present study is to compare the well-posedness criteria of the free-pressure two-fluid model, single-pressure two-fluid model, and two-pressure two-fluid model in a vertical pipe. Two-fluid models were solved using the Conservative Shock Capturing Method. A water faucet case is used to compare two-fluid models. The free pressure two-fluid model can accurately predict discontinuities in the solution field if the problem's initial condition satisfies the Kelvin Helmholtz instability conditions. The single-pressure two-fluid model can accurately predict the behavior of flows in which the two phases are poorly coupled. The two-pressure two-fluid model is an unconditionally well-posed one; if in the free-pressure two-fluid model and single-pressure two-fluid model, the range of velocity difference of two phases exceeds certain limits, the models will be ill-posed. The two-pressure two-fluid model produces more numerical diffusion than the free-pressure two-fluid and single-pressure two-fluid models in the solution field. High numerical diffusion of two-pressure two-fluid models leads to failure to better comply with the problem's analytical solution. Results show that a single-pressure model is a powerful model for numerical modeling of gas-liquid two-fluid flows in the vertical pipe due to a broader range of well-posed than free-pressure models and less numerical diffusion than the two-pressure model.

In this paper, a numerical study is performed in order to investigate the effect of the virtual viscosity on simulation of separated two-phase flow of gas-liquid. The governing equations solved by shock capturing method which can provide predicting the interface without the flow field solving. In this work, in order to calculate the numerical flux term, first-order centred scheme (Force scheme) was applied cause of its accuracy and appropriate validation. Analysis approves that the obtained stability range of this research is consistent with the classic Kelvin-Helmholtz instability equation only for the long wavelength with small amplitude. Results reveal that when the wavelengths are reduced, the specified range is not consistent and wavelength affects on instability range and it is over predicted. An algorithm for water faucet problem was developed in Fortran language. Short wavelength perturbations induce unbounded growth rates and make it impossible to achieve converging solutions. The approach taken in this article has been to adding virtual viscosity as a CFD technique, is used to remedy this deficiency.

A recently developed high-order implicit shock tracking method is novelly applied to a benchmark problem in two-dimensional compressible reactive flow, and results of remarkably high accuracy are a...

A computational approach is developed to simulate high-enthalpy shock propagation under strong nonequilibrium conditions, such as reentry experiments at NASA’s Electric Arc Shock Tube (EAST) facili...

Mathematical models of the dynamics of elastoplastic, granular, and porous media are reduced to variational inequalities for hyperbolic differential operators that are thermodynamically consistent in the sense of Godunov. On this basis, the concept of weak solutions with dissipative shock waves is introduced and a priori estimates of smooth solutions in characteristic conoids of operators are constructed, which suggest the well-posedness of the formulation of the Cauchy problem and boundary value problems with dissipative boundary conditions. Additionally, efficient shock-capturing methods adapted to solution discontinuities are designed.

As discussed in Chapter 4, generally the motion of fluids encountered in engineering applications is described by the Navier-Stokes equations. Considering today’s computational fluid dynamics capabilities, it is possible to numerically solve the Navier- Stokes equations for laminar flows (no turbulent fluctuations), transitional flows (using appropriate intermittency models), and turbulent flow (utilizing appropriate turbulence models). Given today’s computational capabilities, one may argue at this juncture that there is no need to artificially subdivide the flow regime into different categories such as incompressible, compressible, viscid or inviscid ones. However, based on the degree of complexity of the flow under investigation, a computational simulation may take up to several days, weeks, and even months for direct Navier- Stokes simulations (DNS). The difficulties associated with solving the Navier-Stokes equations are caused by the existence of the viscosity terms in the Navier-Stokes equations.

Time-accurate coupled fluid and structure solutions for the unsteady Euler equations are too expensive for routine aeroelastic computation. The small-disturbance Euler (SD-Euler) method is shown to...

High order accurate shock-capturing ENO schemes are mainly based on high order ENO reconstructions. Generally, classical ENO reconstruction methods show some relevant drawbacks. In this paper, we introduce a new ENO reconstruction procedure, the TVENO methods, to design high order accurate shock capturing methods for hyperbolic conservation laws, in order to overcome the drawbacks of classical ENO procedure. The main features of these new ENO method is that it substantially reduced smearing near discontinuities and gives a good resolution of corners and local extrema. Computational results of several one-dimensional numerical experiments for scalar conservation laws, including linear advection, Burger’s equation are given in support of our claim.

The links between coherent structures in the shear layer of an isothermal Mach 2 jet and steepened Mach waves just outside the jet are investigated using large-eddy simulation. Conditional averages...

This study is concerned with predicting surface heat transfer in viscous hypersonic flows using high-order discontinuous Galerkin (DG) methods. Currently, finite-volume (FV) schemes are most commonly employed for computing flows in which surface heat transfer is a target quantity; however, these schemes suffer from large sensitivities to a variety of factors, such as the inviscid flux function and the computational mesh. High-order DG methods offer advantages that can mitigate these sensitivities. As such, a simple and robust shock capturing method is developed for DG schemes. The method combines intraelement variations for shock detection with smooth artificial viscosity (AV) for shock stabilization. A parametric study is performed to evaluate the effects of AV on the solution. The shock capturing method is employed to accurately compute double Mach reflection and viscous hypersonic flows over a circular half-cylinder and a double cone, the latter of which involves a complex flow topology with multiple shock interactions and flow separation. Results show this methodology to be significantly less sensitive than FV schemes to mesh topology and inviscid flux function. Furthermore, quantitative comparisons with state-of-the-art FV calculations from an error vs. cost perspective are provided.

Purpose This paper aims to propose an adaptive method for the numerical solution of the shallow water equations (SWEs). The authors provide an arbitrary high-order method using high-order spline wavelets. Furthermore, they use a non-linear shock capturing (SC) diffusion which removes the necessity of post-processing. Design/methodology/approach The authors use a space-time weak formulation of SWEs which exploits continuous Galerkin (cG) in space and discontinuous Galerkin (dG) in time allowing time stepping, also known as cGdG. Such formulations along with SC term have recently been proved to ensure the stability of fully discrete schemes without scarifying the accuracy. However, the resulting scheme is expensive in terms of number of degrees of freedom (DoFs). By using natural adaptivity of wavelet expansions, the authors devise an adaptive algorithm to reduce the number of DoFs. Findings The proposed algorithm uses DoFs in a dynamic way to capture the shocks in all time steps while keeping the representation of approximate solution sparse. The performance of the proposed scheme is shown through some numerical examples. Originality/value An incorporation of wavelets for adaptivity in space-time weak formulations applied for SWEs is proposed.

In general, B-spline quasi-interpolation (BSQI)-based numerical schemes for hyperbolic conservation laws are unstable in nature. In the present work, we have developed the stable modified v...

A new adaptive weighted essentially non-oscillatory WENO- θ scheme in the context of finite difference is proposed. Depending on the smoothness of the large stencil used in the reconstruction of the numerical flux, a parameter θ is set adaptively to switch the scheme between a 5th-order upwind and 6th-order central discretization. A new indicator τ θ measuring the smoothness of the large stencil is chosen among two candidates which are devised based on the possible highest-order variations of the reconstruction polynomials in L 2 sense. In addition, a new set of smoothness indicators β k of the substencils is introduced. These are constructed in a central sense with respect to the Taylor expansions around the point x j . Numerical results show that the new scheme outperforms other comparing 6th-order WENO schemes in terms of improving the resolution at critical regions of nonsmooth problems as well as maintaining symmetry in the solutions.

It is well known that low diffusion Riemann solvers with minimal smearing on contact and shear waves are vulnerable to shock instability problems, including the carbuncle phenomenon. In the present study, we concentrate on exploring where the instability grows out and how the dissipation inherent in Riemann solvers affects the unstable behaviors. With the help of numerical experiments and a linearized analysis method, it has been found that the shock instability is strongly related to the unstable modes of intermediate states inside the shock structure. The consistency of mass flux across the normal shock is needed for a Riemann solver to capture strong shocks stably. The famous carbuncle phenomenon is interpreted as the consequence of the inconsistency of mass flux across the normal shock for a low diffusion Riemann solver. Based on the results of numerical experiments and the linearized analysis, a robust Godunov-type scheme with a simple cure for the shock instability is suggested. With only the dissipation corresponding to shear waves introduced in the vicinity of strong shocks, the instability problem is circumvented. Numerical results of several carefully chosen strong shock wave problems are investigated to demonstrate the robustness of the proposed scheme.

Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer. For super/hypersonic boundary layers, the external disturbances first interact with the shock ahead of the flying vehicles before entering the boundary layer. Since direct numerical simulation (DNS) is the only available tool for its comprehensive and detailed investigation, an important problem arises whether the numerical scheme, especially the shock-capturing method, can faithfully reproduce the interaction of the external disturbances with the shock, which is so far unknown. This paper is aimed to provide the answer. The interaction of weak disturbances with an oblique shock is investigated, which has a known theoretical solution. Numerical simulation using the shock-capturing method is conducted, and results are compared with those given by theoretical analysis, which shows that the adopted numerical method can faithfully reproduce the interaction of weak external disturbances with the shock.

This paper presents a numerical study using two-fluid model in order to compare the effect of hydrodynamic and hydrostatic models for pressure correction term in two-fluid model in gas-liquid two-phase flow modeling to provide a more accurate model. Two-fluid model is solved by Conservative Shock Capturing Method. The two-fluid model is applied using both hydrodynamic pressure correction term and hydrostatic pressure correction term for four test cases including Water Faucet Case, Water-Air Separation Case, Toumi's Shock Tube Case, and Large Relative Velocity Shock Tube Case. Hydrostatic pressure correction term is neglected for vertical geometry, therefore, in this geometry; two-fluid model cannot be hyperbolic. Thus, hydrostatic pressure correction term is not a stabilizing term. Also, in horizontal pipe and for atmospheric conditions, hydrostatic pressure correction term presents better results than hydrodynamic pressure correction term. But, in non-atmospheric conditions, hydrodynamic pressure correction term provides better results. Therefore, we should consider geometry (vertical or horizontal) and flow conditions (atmospheric or under-pressure) in order to select a proper pressure correction term for two-fluid model. Also, hydrodynamic pressure correction term in two-fluid equations system is hyperbolic in a boarder range than hydrostatic pressure correction term.

The recent discovery of shear shock wave generation and propagation in the porcine brain suggests that this new shock phenomenology may be responsible for a broad range of traumatic injuries. Blast-induced head movement can indirectly lead to shear wave generation in the brain, which could be a primary mechanism for injury. Shear shock waves amplify the local acceleration deep in the brain by up to a factor of 8.5, which may tear and damage neurons. Currently, there are numerical methods that can model compressional shock waves, such as comparatively well-studied blast waves, but there are no numerical full-wave solvers that can simulate nonlinear shear shock waves in soft solids. Unlike simplified representations, e.g., retarded time, full-wave representations describe fundamental physical behavior such as reflection and heterogeneities. Here we present a piecewise parabolic method-based solver for one-dimensional linearly polarized nonlinear shear wave in a homogeneous medium and with empirical frequency-dependent attenuation. This method has the advantage of being higher order and more directly extendable to multiple dimensions and heterogeneous media. The proposed numerical scheme is validated analytically and experimentally and compared to other shock capturing methods. A Riemann step-shock problem is used to characterize the numerical dissipation. This dissipation is then tuned to be negligible with respect to the physical attenuation by choosing an appropriate grid spacing. The numerical results are compared to ultrasound-based experiments that measure planar polarized shear shock wave propagation in a tissue-mimicking gelatin phantom. Good agreement is found between numerical results and experiment across a 40 mm propagation distance. We anticipate that the proposed method will be a starting point for the development of a two- and three-dimensional full-wave code for the propagation of nonlinear shear waves in heterogeneous media.

In the present study, a particle method of characteristics is put forward to simulate the evolution of one-dimensional shock waves in barotropic gaseous, closed-conduit, open-channel, and two-phase flows. All these flow phenomena can be described with the same set of governing equations. The proposed scheme is established based on the characteristic equations and formulated by assigning the computational particles to move along the characteristic curves. Both the right- and left-running characteristics are traced and represented by their associated computational particles. It inherits the computational merits from the conventional method of characteristics (MOC) and moving particle method, but without their individual deficiencies. In addition, special particles with dual states deduced to the enforcement of the Rankine–Hugoniot relation are deliberately imposed to emulate the shock structure. Numerical tests are carried out by solving some benchmark problems, and the computational results are compared with available analytical solutions. From the derivation procedure and obtained computational results, it is concluded that the proposed PMOC will be a useful tool to replicate one-dimensional shock waves.

The two-dimensional Riemann problem with polytropic gas is considered. By a restriction on the constant states of each quadrant of the computational domain such that there is only one planar centered wave connecting two adjacent quadrants, there are nineteen genuinely different initial configurations of the problem. The configurations are numerically simulated on a fine grid and compared by the 5th-order WENO-Z5, 6th-order WENO-źź6, and 7th-order WENO-Z7 schemes. The solutions are very well approximated with high resolution of waves interactions phenomena and different types of Mach shock reflections. Kelvin-Helmholtz instability-like secondary-scaled vortices along contact continuities are well resolved and visualized. Numerical solutions show that WENO-źź6 outperforms the comparing WENO-Z5 and WENO-Z7 in terms of shock capturing and small-scaled vortices resolution. A catalog of the numerical solutions of all nineteen configurations obtained from the WENO-źź6 scheme is listed. Thanks to their excellent resolution and sharp shock capturing, the numerical solutions presented in this work can be served as reference solutions for both future numerical and theoretical analyses of the 2D Riemann problem.

We introduce a new relativistic astrophysics code, SpECTRE, that combines a discontinuous Galerkin method with a task-based parallelism model. SpECTRE's goal is to achieve more accurate solutions for challenging relativistic astrophysics problems such as core-collapse supernovae and binary neutron star mergers. The robustness of the discontinuous Galerkin method allows for the use of high-resolution shock capturing methods in regions where (relativistic) shocks are found, while exploiting high-order accuracy in smooth regions. A task-based parallelism model allows efficient use of the largest supercomputers for problems with a heterogeneous workload over disparate spatial and temporal scales. We argue that the locality and algorithmic structure of discontinuous Galerkin methods will exhibit good scalability within a task-based parallelism framework. We demonstrate the code on a wide variety of challenging benchmark problems in (non)-relativistic (magneto)-hydrodynamics. We demonstrate the code's scalability including its strong scaling on the NCSA Blue Waters supercomputer up to the machine's full capacity of 22,380 nodes using 671,400 threads., 41 pages, 13 figures, and 7 tables. Ancillary data contains simulation input files

In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consists of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers' equation, and the 1-D Euler equations. For the latter, we examine the Shu-Osher model problem for turbulence-shockwave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity., Comment: 10 pages, 11 figures. Under review for TCFD

In river hydraulic, the classical shallow water equations are governed by the Saint-Venant system, are widely used to model water flow in rivers, lakes, reservoirs, coastal areas. As their solutions are typically non-smooth and even discontinuous, among all techniques numerical, the finite volume schemes are well suited to problems conservative, hyperbolic and nonlinear such as the model of Saint-Venant and also among the numerical schemes are among the most popular tools. In this study, we promote a finite volume method, on an unstructured grid, with shock-capturing capabilities by an automatic algorithm to adapt grids near higher gradient. The performance of the numerical method has been investigated by applying it on a problem of urban flow including an oblique hydraulic jump.

In this paper, one- dimensional cylindrically converging shock problem is investigated. The governing equations are Eulerian and Equation of State (EOS) of Mie- Gruneisen type is analyzed. It has been shown that the Lax- Friedrich’s scheme used is appropriate for the problem under investigation. In the present work, Dusty gas and Royce EOS with suitable material constants are worked out in detail to investigate the behavior of magnetic pressure on flow variables. The role of artificial viscosity in shock capturing methods is investigated using Lax- Friedrich’s scheme. The convergence and stability of the scheme is presented. It has been observed that with the increase in magnetic field strength, the flow variables tend to rise behind the shock front . The obtained results are in good agreement with the profiles obtained by some of the researchers.

A numerical implementation of a large number of Reynolds-averaged Navier–Stokes (RANS) models based on two-, three-, four-equation, and Reynolds stress turbulence models (using either the turbulent kinetic energy dissipation rate or the turbulent lengthscale) in an Eulerian, finite-difference shock-capturing code is described. The code uses third-order weighted essentially nonoscillatory (WENO) reconstruction of the advective fluxes, and second- or fourth-order central difference derivatives for the computation of spatial gradients. A third-order TVD Runge–Kutta time-evolution scheme is used to evolve the fields in time. Improved closures for the turbulence production terms, compressibility corrections, mixture transport coefficients, and a consistent initialization methodology for the turbulent fields are briefly summarized. The code framework allows for systematic comparisons of detailed predictions from a variety of turbulence models of increasing complexity. Applications of the code with selected K–ε based models are illustrated for each of the three instabilities. Simulations of Rayleigh–Taylor unstable flows for Atwood numbers 0.1–0.9 are shown to be consistent with previous implicit LES (ILES) results and with the expectation of increased asymmetry in the mixing layer characteristics with increasing stratification. Simulations of reshocked Richtmyer–Meshkov turbulent mixing corresponding to experiments with light-to-heavy transition in air/sulfur hexafluoride and incident shock Mach number Mas = 1.50, and heavy-to-light transition in sulfur hexafluoride/air with Mas = 1.45 are shown to be in generally good agreement with both pre- and post-reshock mixing layer widths. Finally, simulations of the seven Brown–Roshko Kelvin–Helmholtz experiments with various velocity and density ratios using nitrogen, helium, and air are shown to give mixing layer predictions in good agreement with data. The results indicate that the numerical algorithms and turbulence models are suitable for simulating these classes of inhomogeneous turbulent flows.

A two-dimensional wave model coupled with ice dynamics is developed to evaluate ice effects on shallow water wave propagation on a beach and in a channel. The nonlinear Boussinesq equations with ice effects are derived and solved by the hybrid technique of the Godunov-type finite volume method and finite difference method with the third-order Runge&ndash, Kutta method for time integration. The shock capturing method enables the model to simulate complex flows over irregular topography. The model is capable of simulating wave propagations accurately, including non-hydrostatic water pressure and wave dispersions. The ice dynamic module utilizes a Lagrangian discrete parcel method, based on smoothed particle hydrodynamics. The Boussinesq wave model is validated with an analytical solution of water surface oscillation in a parabolic container, an analytical solitary wave propagation in a flat channel, and experimental data on tsunami wave propagations. The validated model is then applied to investigate the interaction between ice and tsunami wave propagation, in terms of ice attenuation on tsunami wave propagations over a beach, ice deposition on the beach driven by the tsunami wave, and ice jam formation and release in a coastal channel with the intrusion of the tsunami wave. The simulated results demonstrated the interactions between tsunami waves and surface ice, including the maximum run up, ice movement along the beach, and ice jamming in a channel.

Summary The shock instability phenomenon is a well-known problem for hypersonic flow computation by the shock-capturing Roe scheme. The pressure checkerboard is another well-known problem for low-Mach-number flow computation. The momentum interpolation method (MIM) is necessary for low-Mach-number flows to suppress the pressure checkerboard problem, and the pressure-difference-driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high-Mach-number flows or low-Mach-number cells in numerical shock. Therefore, MIM should be retained for low-Mach-number flows and be completely removed for high-Mach-number flows and low-Mach-number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.

General conservation laws are considered, and the concept of a weak solution is extended to the case of an equation involving three space variables and time. Four-dimensional vector calculus is used to develop general jump conditions at a shock wave in the material. To illustrate the use of this result, jump conditions at a shock in unsteady three-dimensional compressible gas flow are presented. It is then proved rigorously that these reduce to the commonly assumed conditions in coordinates normal and tangential to the shock face. A similar calculation is also outlined for an unsteady three-dimensional shock in magnetohydrodynamics, and in a chemically reactive fluid. The technique is available for determining shock–jump conditions in quite general continuous media.

Geometrical shock dynamics (GSD) theory is an appealing method to predict the shock motion in the sense that it is more computationally efficient than solving the traditional Euler equations, especially for converging shock waves. However, to solve and optimize large scale configurations, the main bottleneck is the computational cost. Among the existing numerical GSD schemes, there is only one that has been implemented on parallel computers, with the purpose to analyze detonation waves. To extend the computational advantage of the GSD theory to more general applications such as converging shock waves, a numerical implementation using a spatial decomposition method has been coupled with a front tracking approach on parallel computers. In addition, an efficient tridiagonal system solver for massively parallel computers has been applied to resolve the most expensive function in this implementation, resulting in an efficiency of 0.93 while using 32 HPCC cores. Moreover, symmetric boundary conditions have been developed to further reduce the computational cost, achieving a speedup of 19.26 for a 12-sided polygonal converging shock.

This manuscript is concerned with the detection of shock discontinuities in the solution of conservation laws for high-order discontinuous Galerkin methods. A shock detector based on the entropy residual is proposed to distinguish smooth and non-smooth parts of the solution. The numerical analysis shows that the proposed entropy residual converges if the true solution is smooth and sufficiently regularized in space and time. To precisely localize discontinuities of different natures, an approach is developed that dynamically sets the threshold on the detection function, such that the detection criterion retains its sensitivity to the characteristics of the local solution. The implementation is conducted in an entropy-bounded discontinuous Galerkin framework, and numerical tests confirm the convergence property of the entropy-residual formulation and the effectiveness of the thresholding procedure. This shock detector is combined with an artificial viscosity scheme for shock stabilization. Comparison with other detectors is performed to demonstrate the excellent performance of the entropy-residual based shock detector for a wide range of problems on regular and triangular grids.

The generalized rheological method is used to construct a mathematical model of small deformations of a porous media with open pores. Changes in the resistance of the material to external mechanical impact at the moment of collapse of the pores is described using the von Mises–Schleicher strength condition. The irreversible deformation is accounted for with the help of the classic versions of the von Mises–Tresca–Saint-Venant yield condition and the condition that simulates the plastic loss of stability of the porous skeleton. Within the framework of the constructed model, this paper describes the analysis of the propagation of plane longitudinal compression waves in a homogeneous medium accompanied with plastic strain of the skeleton and densification of the material. A parallel computational algorithm is developed for the study of the elastoplastic deformation of the porous medium under external dynamics loads. The algorithm and the program are tested by calculating the propagation of plane longitudinal compression shock waves and the extension of the cylindrical cavity in an infinite porous medium. The calculation results are compared with exact solutions, and it is shown that they are in good agreement.

For problems with complex geometry, a numerical method is proposed for solving the three-dimensional nonstationary Euler equations on Cartesian grids with the use of hybrid computing systems. The baseline numerical scheme, a method for implementing internal boundary conditions on body-unfitted grids, and an iterative matrix-free LU-SGS method for solving the discretized equations are described. An efficient software implementation of the numerical algorithm on a multiprocessor hybrid CPU/GPU computing system is considered. Results of test computations are presented.

The Piecewise Polynomial Harmonic Method (PPHM) is a local third order accurate shock capturing method for hyperbolic conservation laws. In this paper, theoretical stability properties are presented. Using these properties, the convergence of the scheme for scalar conservation laws is obtained. A direct adaptation of these results should be used to derive the convergence of the PHM (Marquina, SIAM J. Sci. Comput. 15(4), 892---915, 1994). Finally, the method is tested in several classical problems in order to explore its numerical behavior.

This paper explores theoretical solutions to the three-dimensional (3D) shock/shock interaction induced by 3D asymmetrical intersecting compression wedges in supersonic inviscid flows. For Mach interactions, an analytical method known as spatial-dimension reduction, which transforms the problem of 3D steady shock/shock interaction into a two-dimensional (2D) pseudo-steady problem on cross sections, is used to obtain the solutions in the vicinity of the Mach stem. The theoretical solutions include the pressure, temperature, density, Mach number behind the Mach stem, and total pressure recovery coefficient. Numerical simulations are performed to validate the theoretical results. Here, the NND scheme is employed by solving 3D inviscid Euler equations, and good agreements are obtained. The asymmetry of the flow characteristics induced by the wedge angle and sweep angle are thoroughly discussed.

The interference of an impinging plane oblique shock wave with the viscous shock layer on a cylinder has been studied numerically for rarefied-gas flow regimes at the Reynolds numbers 15.5 ≤ ReR,0 ≤ 124 and Knudsen numbers 0.1 ≥ KnR,∞ ≥ 0.012. The calculations have been performed within the framework of the direct simulation Monte-Carlo technique [1] and the Navier-Stokes equations for a perfect gas using the shock capturing method [2]. The principal properties of flow parameters have been studied for five different types of interference at low Reynolds numbers. The differences with respect to the previously investigated interference regimes for high Reynolds numbers have been examined. The comparison analysis of numerical results and experimental data has been provided. It has been found that the local pressure and heat transfer coefficients on the surface of a cylinder may considerably (by a factor of 3.5) exceed the values on the edge stagnation line observed in the absence of interference. The type IV interference pattern observed in continuum flows has not been found in this study.

A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e.g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks., Comment: Accepted in Applied Mathematics and Computation

Broken wave characteristics in front of a vertical seawall were modeled and studied using a shock-capturing Boussinesq wave model FUNWAVE-TVD. Validation with the experimental data confirmed the capability of FUNWAVE-TVD in predicting the wave characteristics via the shock-capturing method. Compared to the results obtained from the Boussinesq model coupled with an empirical breaking model, the advantage of the present shock-capturing model for the broken waves near a vertical seawall was clearly revealed. A preliminary investigation of the effects of the key parameters, such as the incident wave height, water level at the seawall, and seabed slope, on the wave kinematics (i.e., the root mean square of the surface fluctuations and depth-averaged horizontal velocity) near the seawall was then conducted through a series of numerical experiments. The numerical results indicate the incident wave height and the water depth at the seawall are the important parameters in determining the magnitude of the wave kinematics, while the effect of the seabed slope seems to be insignificant. The role of the breaking point locations is also highlighted in this study, in which case further breaking can reduce the wave kinematics significantly for the coastal structures predominately subjected to broken waves.

We present a new high-order front-tracking method for hyperbolic systems of conservation laws for two gases separated by a tracked contact discontinuity, using a combination of a high-order Godunov algorithm and level set methods. Our approach discretizes the moving front and gas domains on a Cartesian grid, with control volumes determined by the intersection of the grid with the front. In cut cells, a combination of conservative and nonconservative finite volume quadratures provide small-cell stability. Global conservation is maintained using redistribution. We demonstrate second-order convergence in smooth flow and first-order convergence in the presence of shocks.

The efficiency of computational fluid dynamics simulations can be greatly enhanced by employing higher-order accurate numerical schemes which provide superior accuracy for a given cost. For unsteady turbulent flow simulations involving shocks, contacts, and/or material discontinuities, various higher-order shock capturing schemes are available in the literature. The desired numerical scheme should be free of spurious numerical oscillations across discontinuities and it should obtain higher-order accuracy in smooth flow regions in an efficient manner. Sufficient robustness is necessary for effectively utilizing these numerical methods in engineering and science applications. Three classes of higher-order shock capturing schemes are compared in this paper: (1) central finite-difference schemes with explicit artificial dissipation, (2) a compact centered finite-difference scheme with localized artificial diffusivity and (3) weighted essentially non-oscillatory schemes in both explicit and compact finite difference forms. Multiple variations of these methods were implemented and tested using a block-structured Cartesian mesh solver. The current paper assesses shock capturing capabilities as well as effects on the accuracy in smooth flow regions using a variety of test cases that range from canonical shock problems to homogeneous isotropic turbulence at a turbulent Mach number of 0.5 where shocklets are formed. Finally, a computational cost breakdown for each scheme is provided and the overall computational efficiency of the different schemes are compared to each other.

A multitemperature code intended for the numerical solution of the multicomponent gas dynamics equations in problems with a high energy density in matter is described. The velocities of all components with nonzero masses are assumed to be identical. Together with transport of gases with a tabulated equation of state, the code can include electron heat conduction, radiative transfer, energy exchange between the components, and chemical reactions. The gasdynamic part is based on Godunov’s scheme and an efficient Riemann solver with an approximate local equation of state.

The non-linear non-equilibrium nature of shock waves in gas dynamics is investigated for the planar case. Along each streamline, the Euler equations with non-equilibrium pressure are reduced to a set of ordinary differential equations defining a slow–fast system, and geometric singular perturbation theory is applied. The proposed theory shows that an orbit on the slow manifold corresponds to the smooth part of the solution to the Euler equation, where non-equilibrium effects are negligible; and a relaxation motion from the unsteady to the steady branch of the slow manifold corresponds to a shock wave, where the flow relaxes from non-equilibrium to equilibrium. Recognizing the shock wave as a fast motion is found to be especially useful for shock wave detection when post-processing experimental measured or numerical calculated flow fields. Various existing shock detection methods can be derived from the proposed theory in a rigorous mathematical manner. The proposed theory provides a new shock detection method based on its non-linear non-equilibrium nature, and may also serve as the theoretical foundation for many popular shock wave detection techniques.

When applying the method of manufactured solutions on computational fluid dynamic software, it is a requirement that all solutions be continuous on the computational domain. This stipulation is limiting for the verification and validation of numerical solutions where discontinuities are frequent. In an effort to adapt the standard method of manufactured solutions procedure, we propose a piecewise approach for modeling solutions with discontinuities. Linearly and quadratically exact transformations are used for determining the exact solutions and source terms for cells split by discontinuities. Upwind manufactured solutions are initialized and a least squares fit is used to solve for solutions downwind of the discontinuity such that the Rankine–Hugoniot conditions are satisfied. The codes used throughout this research are finite volume, 1st and 2nd order, inviscid schemes combined with uniform structured grids. This study shows that a modified method of manufactured solutions procedure can be performed wit...

The $$YZ\beta $$YZβ shock-capturing technique, which is residual-based, was introduced in conjunction with the Streamline-Upwind/Petrov---Galerkin (SUPG) formulation of compressible flows in conservation variables. It was later also combined with the variable subgrid scale (V-SGS) formulation of compressible flows in conservation variables and successfully tested on 2D and 3D computation of inviscid flows with shocks. In this paper we extend that combined method to inviscid flow computations with particle tracking and particle---shock interaction. Particles are tracked individually, assuming one-way dependence between the particle dynamics and the flow. We present two steady-state test computations with particle---shock interaction, one in 2D and one in 3D, and show that the overall method is effective in particle tracking and particle---shock interaction analysis in compressible flows.