Interaction of weak free-stream disturbance with an oblique shock： validation of the shock-capturing method

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.

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes （2nd-order and 3rd-order） and three compact upwind schemes （3rd-order, 5th-order and 7th-order） are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.

Simple and robust h-adaptive shock-capturing method for flux reconstruction framework

In this paper,a simple and robust shock-capturing method is developed for the Flux Reconstruction(FR)framework by combining the Adaptive Mesh Refinement(AMR)technique with the positivity-preserving property.The adaptive technique avoids the use of redundant meshes in smooth regions,while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning.The compatibility of these two significant features relies on a novel limiter designed for mesh refinements.It ensures the positivity of solutions on all newly created cells.Therefore,the proposed method is completely positivity-preserving and thus highly robust.It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance.The performance of the proposed method is examined in various numerical experiments.When solving Euler equations,the technique of Local Artificial Diffusivity(LAD)is additionally coupled to damp oscillations.More importantly,when solving Navier-Stokes equations,the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly.The implementation of the method becomes rather simple.

A new three-dimensional finite-volume non-hydrostatic shock-capturing model for free surface flow

In this paper a new finite-volume non-hydrostatic and shock-capturing three-dimensional model for the simulation of wave-structure interaction and hydrodynamic phenomena（wave refraction, diffraction, shoaling and breaking） is proposed. The model is based on an integral formulation of the Navier-Stokes equations which are solved on a time dependent coordinate system： a coordinate transformation maps the varying coordinates in the physical domain to a uniform transformed space. The equations of motion are discretized by means of a finite-volume shock-capturing numerical procedure based on high order WENO reconstructions. The solution procedure for the equations of motion uses a third order accurate Runge-Kutta（SSPRK） fractional-step method and applies a pressure corrector formulation in order to obtain a divergence-free velocity field at each stage. The proposed model is validated against several benchmark test cases.

THE THREE-POINT FIFTH-ORDER ACCURATE GENERALIZED COMPACT SCHEME AND ITS APPLICATIONS

A three-point fifth-order accurate generalized compact scheme (GC scheme) with a spectral-like resolution is constructed in a general way. The scheme satisfies the principle of stability and the principle about suppression of the oscillations, therefore numerical errors can decay automatically and no spurious oscillations are generated around shocks. The third-order TVD type Runge-Kutta method is employed for the time integration, thus making the GC scheme best suited for unsteady problems. Numerical results show that the GC scheme is shock-capturing. The time-dependent boundary conditions proposed by Thompson are well employed when the algorithm is applied to the Euler equations of gas dynamics.

A Hybrid Finite-Volume and Finite Difference Scheme for Depth- Integrated Non-Hydrostatic Model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.

On the Solution Accuracy Downstream of Shocks When Using Godunov-Type Schemes.I.Sources of Errors in One-Dimensional Problems

The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity approach.The proposed paper describes the numerical methods used in the study,and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems.The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.

Analysis of Discontinuity Detectors and Hybrid WCNS Schemes Based on Waveform Recognition

In this paper,we present a hybrid form of weighted compact nonlinear scheme(WCNS)for hyperbolic conservation laws by applying linear and nonlinear methods for smooth and discontinuous zones individually.To fulfill this algorithm,it is inseparable from the recognition ability of the discontinuity detector adopted.In specific,a troubled-cell indicator is utilized to recognize unsmooth areas such as shock waves and contact discontinuities,while avoiding misjudgments of smooth structures.Some classical detectors are classified into three basic types:derivative combination,smoothness indicators and characteristic decomposition.Meanwhile,a new improved detector is proposed for comparison.Then they are analyzed through identifying a series of waveforms firstly.After that,hybrid schemes using such indicators,as well as different detection variables,are examined with Euler equations,so as to investigate their ability to distinguish practical discontinuities on various levels.Simulation results demonstrate that the proposed algorithm has similar performances to pure WCNS,while it generally saves 50 percent of CPU time for 1D cases and about 40 percent for 2D Euler equations.Current research is in the hope of providing some reference and establishing some standards for judging existing discontinuity detectors and developing novel ones.

Efficient WENOCU4 scheme with three different adaptive switches

Although classical WENOCU schemes can achieve high-order accuracy by introducing a moderate constant parameter C to increase the contribution of optimal weights,they exhibit distinct numerical dissipation in smooth regions.This study presents an extension of our previous research which confirmed that adaptively adjusting parameter C can indeed overcome the inadequacy of the usage of a constant small value.Cmin is applied near a discontinuity while Cmax is used elsewhere and they are switched according to the variation of the local flow-field property.This study provides the reference values of the adaptive parameter C of WENOCU4 and systematically evaluates the comprehensive performance of three different switches(labeled as the binary,continuous,and hyperbolic tangent switches,respectively)based on an optimized efficient WENOCU4 scheme(labeled as EWENOCU4).Varieties of 1D scalar equations,empirical dispersion relation analysis,and multi-dimensional benchmark cases of Euler equations are analyzed.Generally,the dissipation and dispersion properties of these three switches are similar.Especially,employing the binary switch,EWENOCU4 achieves the best comprehensive properties.Specifically,the binary switch can efficiently filter more misidentifications in smooth regions than others do,particularly for the cases of 1 D scalar equations and Euler equations.Also,the computational efficiency of the binary switch is superior to that of the hyperbolic tangent switch.Moreover,the optimized scheme exhibits high-resolution spectral properties in the wavenumber space.Therefore,employing the binary switch is a more cost-effective improvement for schemes and is particularly suitable for the simulation of complex shock/turbulence interaction.This study provides useful guidance for the reference values of parameter C and the evaluation of adaptive switches.

TWO-STAGE FOURTH-ORDER ACCURATE TIME DISCRETIZATIONS FOR 1D AND 2D SPECIAL RELATIVISTIC HYDRODYNAMICS

This paper studies the two-stage fourth-order accurate time discretization[J.Q.Li and Z.F.Du,SIAM J.Sci.Comput.,38(2016)]and its application to the special relativistic hydrodynamical equations.Our analysis reveals that the new two-stage fourth-order accurate time discretizations can be proposed.With the aid of the direct Eulerian GRP(generalized Riemann problem)methods and the analytical resolution of the local“quasi 1D”GRP,the two-stage fourth-order accurate time discretizations are successfully implemented for the 1D and 2D special relativistic hydrodynamical equations.Several numerical experiments demonstrate the performance and accuracy as well as robustness of our schemes.

A high-resolution,hybrid compact-WENO scheme with minimized dispersion and controllable dissipation

In this paper, a high-resolution, hybrid compact-WENO scheme is developed based on the minimized dispersion and controllable dissipation reconstruction technique. Firstly, a sufficient condition for a family oftri-diagonal compact schemes to have independent dispersion and dissipation is derived. Then, a specific 4th order compact scheme with low dispersion and adjustable dissipation is constructed and analyzed. Finally, the optimized compact scheme is blended with the WENO scheme to form the hybrid scheme. Moreover, the approximation dispersion relation approach is employed to optimize the spectral properties of the nonlinear scheme to yield the true wave propagation behavior of the finite difference scheme. Several test cases are carried out to verify the high- resolution as well as the robust shock-capturing capabilities of the proposed scheme.

ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS ON THE EXPLICIT COMPACT SCHEMES II: EXTENSION OF THE STCE/CE METHOD ON NONSTAGGERED GRIDS

This paper continues to construct and study the explicit compact (EC) schemes for conservation laws. First, we axtend STCE/SE method on non-staggered grid, which has same well resolution as one in [1], and just requires half of the computational works. Then, we consider some constructions of the EC schemes for two-dimensional conservation laws, and some 1D and 2D numerical experiments are also given.

A Parameter–Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

A parameter-free limiting technique is developed for high-order unstructured-grid methods to capture discontinuities when solving hyperbolic conservation laws.The technique is based on a“troubled-cell”approach,in which cells requiring limiting are first marked,and then a limiter is applied to these marked cells.A parameter-free accuracy-preserving TVD marker based on the cell-averaged solutions and solution derivatives in a local stencil is compared to several other markers in the literature in identifying“troubled cells”.This marker is shown to be reliable and efficient to consistently mark the discontinuities.Then a compact highorder hierarchical moment limiter is developed for arbitrary unstructured grids.The limiter preserves a degree p polynomial on an arbitrary mesh.As a result,the solution accuracy near smooth local extrema is preserved.Numerical results for the high-order spectral difference methods are provided to illustrate the accuracy,effectiveness,and robustness of the present limiting technique.

ASecond-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary.The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls.Although the motion of the boundary could be coupled with the fluid,all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow.The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domainΩ_(R)⊃Ω.Ωe identify inner and ghost points.The inner points are the grid points located insideΩ,while the ghost points are the grid points that are outsideΩbut have at least one neighbor insideΩ.The evolution equations for inner points data are obtained from the discretization of the governing equation,while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables(density,velocities and pressure).Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains.Several numerical experiments are conducted to illustrate the validity of themethod.Ωe demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix.The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws,while retaining the attractive features of the original solver:the method is entropy-satisfying,differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field,in particular to the intermediate ones,since the full eigenstructure of the underlying hyperbolic system is used.To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws:Euler equations of compressible gasdynamics with ideal gas and real gas equation of state,classical and relativistic MHD equations as well as the equations of nonlinear elasticity.To the knowledge of the authors,apart from the Euler equations with ideal gas,an Osher-type scheme has never been devised before for any of these complicated PDE systems.Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in[9].

A HIGHLY ACCURATE NUMERICAL METHOD FOR FLOW PROBLEMS WITH INTERACTIONS OF DISCONTINUITIES

Focuses on a study wherein a type of shock fitting method has been used to solve dimensional flow problems with interactions of various discontinuities. Discretization of the transformed equations; Numerical results; Interactions of discontinuities.