Shock Discontinuities Research Articles

Validation of radiative transfer computation with Monte Carlo method for ultra-relativistic background flow

We developed a three-dimensional radiative transfer code for an ultra-relativistic background flow-field by using the Monte Carlo (MC) method in the context of gamma-ray burst (GRB) emission. For obtaining reliable simulation results in the coupled computation of MC radiation transport with relativistic hydrodynamics which can reproduce GRB emission, we validated radiative transfer computation in the ultra-relativistic regime and assessed the appropriate simulation conditions. The radiative transfer code was validated through two test calculations: (1) computing in different inertial frames and (2) computing in flow-fields with discontinuous and smeared shock fronts. The simulation results of the angular distribution and spectrum were compared among three different inertial frames and in good agreement with each other. If the time duration for updating the flow-field was sufficiently small to resolve a mean free path of a photon into ten steps, the results were thoroughly converged. The spectrum computed in the flow-field with a discontinuous shock front obeyed a power-law in frequency whose index was positive in the range from 1 to 10 MeV. The number of photons in the high-energy side decreased with the smeared shock front because the photons were less scattered immediately behind the shock wave due to the small electron number density. The large optical depth near the shock front was needed for obtaining high-energy photons through bulk Compton scattering. Even one-dimensional structure of the shock wave could affect the results of radiation transport computation. Although we examined the effect of the shock structure on the emitted spectrum with a large number of cells, it is hard to employ so many computational cells per dimension in multi-dimensional simulations. Therefore, a further investigation with a smaller number of cells is required for obtaining realistic high-energy photons with multi-dimensional computations.

High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows

Simulations of single- and multi-species compressible flows with shock waves and discontinuities are conducted using a weighted compact nonlinear scheme (WCNS) with a newly developed sixth order localized dissipative interpolation. In smooth regions, the scheme applies the central nonlinear interpolation with minimum dissipation to resolve fluctuating flow features while in regions containing discontinuities and high wavenumber features, the scheme suppresses spurious numerical oscillations by hybridizing the central interpolation with the more dissipative upwind-biased nonlinear interpolation. In capturing material interfaces between species of different densities, a quasi-conservative five equation model that can conserve mass of each species is used to prevent pressure oscillations across the interfaces. Compared to upwind-biased interpolations with classical nonlinear weights [1,2] and improved weights [3], and the interpolation with adaptive central-upwind weights for scale-separation [4], it is shown that WCNS with the proposed localized dissipative interpolation has better performance to simultaneously capture discontinuities and resolve smooth features.

1st International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion & Power

1st International Seminar on Non-Ideal Compressible-Fluid Dynamics for Propulsion & Power

Generalised Riemann problem for Euler system

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves (shock, rarefaction wave and contact discontinuity) is also introduced.

A new comment on the computation of non-conservative products using Roe-type path conservative schemes

We are interested in the numerical approximation of the discontinuous solutions of non-conservative hyperbolic systems. We more precisely consider a non-conservative formulation of the usual gas dynamics equations and show how to slightly modify the so-called Roe-type path-conservative schemes to properly capture the underlying shock discontinuities. Numerical evidences are proposed. The present note follows a first comment on the computation of non-conservative products in [1].

Molecular dynamics simulation of the Hugoniot states of boron suboxide

It has been previously demonstrated in a number of studies that the strength of boron carbide can be significantly reduced via introduction of a site vacancy. The site vacancies result in a discontinuous shock response at high pressure that is driven by bond formation between atoms which neighbor the vacancies. In this work, molecular dynamics simulations of boron suboxide (B6O) have been conducted to determine whether an analogous bond formation occurs at high pressure producing discontinuities in the B6O Hugoniot. Hugoniot data for ideal boron suboxide, a defect structure of boron suboxide, and a carbon doped structure are provided and the structural response of the oxygen atoms within the B6O crystal with increasing shock pressure is presented.

Research of influence of reduced-order boundary on accuracy and solution of interior points

The flow field with a high order scheme is usually calculated so as to solve complex flow problems and describe the flow structure accurately. However, there are two problems, i.e., the reduced-order boundary is inevitable and the order of the scheme at the discontinuous shock wave contained in the flow field as the supersonic flow field is low. It is questionable whether the reduced-order boundary and the low-order scheme at the shock wave have an effect on the numerical solution and accuracy of the flow field inside. In this paper, according to the actual situation of the direct numerical simulation of the flow field, two model equations with the exact solutions are solved, which are steady and unsteady, respectively, to study the question with a high order scheme at the interior of the domain and the reduced-order method at the boundary and center of the domain. Comparing with the exact solutions, it is found that the effect of reduced-order exists and cannot be ignored. In addition, the other two model equations with the exact solutions, which are often used in fluid mechanics, are also studied with the same process for the reduced-order problem.

Shock evolution in non-radiative supernova remnants

We present a new analytical approach to derive approximate solutions describing the shock evolution in non-radiative supernova remnants (SNRs). We focus on the study of the forward shock and contact discontinuity while application to the reverse shock is only discussed briefly. The spherical shock evolution of a SNR in both the interstellar medium with a constant density profile and a circumstellar medium with a wind density profile is investigated. We compared our new analytical solution with numerical simulations and found that a few percent accuracy is achieved. For the evolution of the forward shock, we also compared our new solution to previous analytical models. In a uniform ambient medium, the accuracy of our analytical approximation is comparable to that in Truelove&McKee (1999). In a wind density profile medium, our solution performs better than that in Micelotta et al. (2016), especially when the ejecta envelope has a steep density profile. The new solution is significantly simplified compared to previous analytical models, as it only depends on the asymptotic behaviors of the remnant during its evolution.

On the ions acceleration via collisionless magnetic reconnection in laboratory plasmas

This work presents an analysis of the ion outflow from magnetic reconnection throughout fully kinetic simulations with typical laboratory plasma values. A symmetric initial configuration for the density and magnetic field is considered across the current sheet. After analyzing the behavior of a set of nine simulations with a reduced mass ratio and with a permuted value of three initial electron temperatures and magnetic field intensity, the best ion acceleration scenario is further studied with a realistic mass ratio in terms of the ion dynamics and energy budget. Interestingly, a series of shock wave structures are observed in the outflow, resembling the shock discontinuities found in recent magnetohydrodynamic simulations. An analysis of the ion outflow at several distances from the reconnection point is presented, in light of possible laboratory applications. The analysis suggests that magnetic reconnection could be used as a tool for plasma acceleration, with applications ranging from electric propulsion to production of ion thermal beams.

Investigation of hypersonic inlet pulse-starting characteristics at high Mach number

An unsteady viscous numerical simulation is performed to study the hypersonic inlet pulse-starting in a shock tunnel under the condition of inflow Mach number higher than the design value with a simplified nozzle startup wave system. A local unstart of hypersonic inlet is obtained with the initial backpressure and temperature of test section equal to the hypersonic freestream static pressure and static temperature, respectively. During the hypersonic inlet pulse-starting process, the nozzle startup wave system (including the primary shock, contact discontinuity and secondary shock) propagates downstream and interacts with the hypersonic inlet, the secondary shock induces boundary layer separation on the compression ramp. A low velocity flow induced by the separation bubble enters into the internal duct with the passage of time. It thickens the boundary layer and then the cowl shock induces a large separation bubble on the compression side. The shock induced by the large separation bubble impinges on the cowl and induces a small separation bubble, which moves upstream and induces the reflection of the external compression shock at the cowl transiting from regular reflection to Mach reflection. Mach reflection results in the shock propagating forward to cause a shock detachment at the cowl lip. The hypersonic inlet is not fully started and this phenomenon is referred to as “local unstart of inlet”. Moreover, it is found that the hypersonic inlet starts with lower initial backpressure or higher initial temperature of test section, and the hypersonic inlet unstarts with higher initial backpressure. The effect mechanism is analyzed with the help of numerical simulations and unsteady one-dimensional shock theory. As there are three operation modes of hypersonic inlet at high Mach number: start, local unstart and unstart. The results of the hypersonic inlet pulse-starting in a shock tunnel should be carefully treated.

An efficient shock-capturing scheme for simulating compressible homogeneous mixture flow

This work is devoted to the development of a procedure for the numerical solution of Navier-Stokes equations for cavitating flows with and without ventilation based on a compressible, multiphase, homogeneous mixture model. The governing equations are discretized on a general structured grid using a high-resolution shock-capturing scheme in conjunction with appropriate limiters to prevent the generation of spurious solutions near shock waves or discontinuities. Two well-known limiters are examined, and a new limiter is proposed to enhance the accuracy and stability of the numerical scheme. A sensitivity analysis is first conducted to determine the relative influences of various model parameters on the solution. These parameters are adopted for the computation of water flows over a hemispherical body, conical body and a divergent/convergent nozzle. Finally, numerical calculations of ventilated supercavitating flows over a hemispherical cylinder body with a hot propulsive jet are conducted to verify the capabilities of the numerical scheme.

An Essentially Non-Oscillatory Spectral Deferred Correction Method for Conservation Laws

We present a computational method based on the Spectral Deferred Corrections (SDC) time integration technique and the Essentially Non-Oscillatory (ENO) finite volume method for the conservation laws (one-dimensional Euler equations). The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (Piece-wise Parabolic Method (PPM)) for solving the conservation laws is first carried out by Layton et al. in [Layton, A. T. and Minion, M. L. [2004] “Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics,” J. Comput. Phys. 194(2), 697–714]. Issues about this approach have been addressed and some improvements have been added to it in [Kadioglu et al. [2012] “A gas dynamics method based on the spectral deferred corrections (SDC) time integration technique and the piecewise parabolic method (PPM),” Am. J. Comput. Math. 1–4, 303–317]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM methods). We have tested the new SDC-ENO technique by solving several test problems involving moderate to strong shock waves and smooth/complex flow structures. Our numerical results show that we have numerically achieved the formally fourth-order convergence of the new method for smooth problems. Our numerical results also indicate that the newly proposed technique performs very well providing highly resolved shock discontinuities and fairly good contact solutions. More importantly, the discontinuities in the flow test problems are captured with essentially no-oscillations. We have numerically compared the fourth-order SDC-ENO scheme to the fourth-order SDC-PPM method for the same test problems. The results are similar for most of the test problems except in some cases the SDC-PPM method suffers from minor oscillations compared to SDC-ENO scheme being completely oscillation free.

An entropy-residual shock detector for solving conservation laws using high-order discontinuous Galerkin methods

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.

Slow shock and rotational discontinuity in MHD and Hall MHD models with anisotropic pressure

AbstractPressure anisotropy may modify the characteristics of magnetohydrodynamic (MHD) waves, in particular, the slow mode wave and the corresponding shocks and discontinuities. In this study the formation of slow shocks (SSs) in anisotropic plasmas is examined by solving the gyrotropic MHD and Hall MHD equations numerically for one‐dimensional Riemann problem. The MHD shocks and discontinuities are generated by imposing a finite normal magnetic field on the Harris type current sheet with a guide magnetic By component. It is shown that anomalous SSs moving faster than the intermediate wave or with positive density‐magnetic field correlation may be generated in gyrotropic MHD and Hall MHD models. Moreover, for some parameter values SSs may exhibit upstream wave trains with right‐handed polarization in contrast with the earlier prediction that SSs shall possess downstream left‐hand polarized wave trains based on the isotropic Hall MHD theory. For the cases of By ≠ 0, SSs with increased density and decreased magnetic field followed by noncoplanar intermediate mode or rotational discontinuity (RD)‐like structures similar to the compound SS‐RD structures observed in space plasma environments may possibly form in symmetric and asymmetric current layers. The Walén relation of these anomalous RDs without the correction of pressure anisotropy may significantly be violated.

Study of high-order-accurate limiters for time-dependent contact discontinuity and shock capturing

ABSTRACTIn most applications of computational fluid dynamics (CFD), it is very important to correctly project the interaction phenomena when we compute the typical discontinuous structure. Numerical methods for the conservation law are usually divided into two kinds: One is the “single stepping method”; another is the “semidiscrete scheme”. When the semidiscrete scheme handles the issues of high accuracy and operational process without oscillations, it is a very successful method. In this article, we consider the one-dimensional Euler equations for the Riemann problem of a typical shock tube without external active force to improve the nonphysical numerical oscillation problem near the contact discontinuity interface and shock wave. We try to improve the computational accuracy of numerical resolution of the Riemann problem and reduce nonphysical numerical oscillations through the AUSMDV numerical flux scheme with implementing distinct flux limiters and time-stepping methods. Using the conditions of single gas/dual gases, we carry out a series of comparisons for the accuracy of shock capturing as well as the capability of improving numerical oscillations.

Evaluation of rotated upwind schemes for contact discontinuity and strong shock

Rotated upwind schemes have been evaluated as carbuncle-free shock-capturing methods. In order to reveal the relation between the rotated upwind-differencing direction and the performance of rotated upwind schemes, theoretical and numerical investigations are made. Three commonly used upwind schemes, including Roe scheme, AUSM+ scheme and Van Leer scheme are investigated. According to the theoretical analysis, the dissipative terms of each upwind scheme are changed by the rotation procedure. Especially, the dissipative density term in the numerical flux function of rotated upwind schemes introduces extra dissipation. The numerical cases include simulations of laminar boundary layer, steady twin vortices and strong shock. The numerical results show that the carbuncle phenomenon can be eliminated by rotated upwind schemes, and the contact resolving capability of rotated upwind schemes is deteriorated by the rotation procedure. Monotonicity of the dissipation with rotation angle is also shown in the numerical results. Therefore, a novel rotation strategy that defines the rotation angle by a pressure weight function is introduced for tuning the dissipative effects adaptively. This strategy shows not only carbuncle-free result in hypersonic inviscid flow simulations, but also accurate results in viscous flow simulations.

A high-order flux reconstruction method with adaptive mesh refinement and artificial diffusivity on unstructured moving/deforming mesh for shock capturing

This article presents new progress of shock capturing on unstructured dynamic grids by using adaptive mesh refinement (AMR) method in conjunction with artificial diffusivity (AD) under the framework of flux reconstruction (FR)/correction procedure via reconstruction (CPR) method. The proposed AMR algorithm is applicable even when the grid undergoes dynamic motion. It features an innovative data structure that is capable of direct addressing in managing the system of cells on multiple levels of refinement. A conservative mortar method is used to handle non-conforming interfaces resulted from cell subdivision. The AD is added in regions of strong dilatation to counter the Gibbs oscillation that is disruptive to high-order solutions when shock discontinuities exist. Several test cases are used to verify that this FR-AMR-AD framework can achieve high-order accuracy for smooth flows, and obtain stable solution with the presence of shock discontinuity. By further limiting the shock within subdivided adjacent cells, a sharp shock solution can be obtained with reduced amount of AD addition, hence errors from adding dissipation, on dynamically refined grids.

A smoothness indicator for numerical solutions to the Ripa model

The Ripa model is the system of shallow water equations taking the water temperature fluctuations into account. For one-dimensional case, the Ripa model consists of three partial differential equations relating to three primitive variables, namely water depth, velocity, and temperature. The Ripa model is hyperbolic, and its solution can be discontinuous. When the Ripa model is solved using a conservative numerical method, the solution is usually diffusive around discontinuities. The diffusion at rough regions (around discontinuities, such as contact and shock discontinuities) makes the solution inaccurate. In practice, we want to know the places where the solution is accurate, and where it is inaccurate. That is, we want to know where the solution is smooth, and where it is rough. In this paper we propose the numerical entropy production to detect the smoothness of numerical solutions to the Ripa model. Numerical results show that the numerical entropy production is a robust smoothness indicator for numerical solutions to the Ripa model.

Shock-wave studies of anomalous compressibility of glassy carbon

The physico-mechanical properties of amorphous glassy carbon are investigated under shock compression up to 10 GPa. Experiments are carried out on the continuous recording of the mass velocity of compression pulses propagating in glassy carbon samples with initial densities of 1.502(5) g/cm3 and 1.55(2) g/cm3. It is shown that, in both cases, a compression wave in glassy carbon contains a leading precursor with amplitude of 0.135(5) GPa. It is established that, in the range of pressures up to 2 GPa, a shock discontinuity in glassy carbon is transformed into a broadened compression wave, and shock waves are formed in the release wave, which generally means the anomalous compressibility of the material in both the compression and release waves. It is shown that, at pressure higher than 3 GPa, anomalous behavior turns into normal behavior, accompanied by the formation of a shock compression wave. In the investigated area of pressure, possible structural changes in glassy carbon under shock compression have a reversible character. A physico-mechanical model of glassy carbon is proposed that involves the equation of state and a constitutive relation for Poisson’s ratio and allows the numerical simulation of physico-mechanical and thermophysical properties of glassy carbon of different densities in the region of its anomalous compressibility.

Direct numerical simulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns

The reflection on a wavy wall of a planar shock propagating at Mach number 1.5 in air is simulated in a two-dimensional geometry by solving the fully compressible Navier–Stokes equations. A high-order spectral difference numerical discretization is used over an unstructured mesh composed of quadrilateral elements. The shock discontinuity is handled numerically through a specific treatment, which is limited in space to a small portion of the computational cell through which the shock is travelling. In the conditions under investigation, the reflection on the wavy wall leads to a weak and smooth deformation of the shock front without singularities just after reflection. Long-living triple points (Mach stems) are spontaneously formed on the reflected shock at a finite distance from the wavy wall. They then propagate on the front in both directions and collide regularly, forming a periodic cellular pattern quite similar to that of a cellular detonation. Transverse waves, issued from the triple points, are generated in the shocked gas. As a result of their mutual interaction, a complex and strongly unsteady flow is produced in the shocked gas. The topology of the instantaneous streamline patterns is characterized by short-lived critical points that appear intermittently. Due to the compressible character of the unsteady two-dimensional flow, the topology of critical points which can be observed is more diverse than would be expected for incompressible two-dimensional flows. Some of them take the form of short-lived sources or sinks. The mechanism of formation and the dynamics of the triple points, as well as the instantaneous streamline patterns, are analysed in the present paper. The results are useful for deciphering the cellular structure of unstable detonations.