Riemann Shock Research Articles

Formation, Propagation and Reflection of 1D Normal Shocks in Riemann Shock Tube

The purpose of this paper is to consider 1D Riemann shock tube to investigate the formation and propagation of compression waves leading to formation, propagation and reflection of 1D normal shocks using simplified mathematical models commonly used in the published work as well as using complete mathematical models based on Conservation and Balance Laws (CBL) of classical continuum mechanics and constitutive theories for compressible viscous medium derived using entropy inequality and representation theorem. This work is aimed at resolving compression waves, the shock structure, shock formation, propagation and reflection of fully formed shocks. Evolutions obtained from the mathematical models always satisfy differentiability requirements in space and time dictated by the highest order of the derivatives of the dependent variables in the mathematical models investigated. All solutions reported in this paper including boundary conditions and initial conditions are always analytic. Solutions of the mathematical models are obtained using the space-time finite element method in which the space-time integral forms are space-time variationally consistent ensuring unconditionally stable computations during the entire evolution. Solution for a space-time strip or slab is calculated and is time marched upon convergence to obtain complete evolution for the desired space-time domain, thus ensuring time accurate evolutions. The space-time local approximation over a space-time element of a space-time strip or slab is p-version hierarchical with higher-order global differentiability in space and time, i.e., we consider scalar product approximation spaces in which k = (k1, k2) are the order of the space in space and time and p = (p1, p2) are p-levels of the approximations in space and time. Model problem studies are presented for different mathematical models and are compared with solutions obtained from the complete mathematical model based on CBL and constitutive theories for viscous compressible medium to illustrate the deficiencies and shortcomings of the simplified and approximate models in simulating correct physics of normal shocks.

Large-Time Asymptotic Stability of Riemann Shocks of Scalar Balance Laws

We prove the large-time asymptotic orbital stability of strictly entropic Riemann shock solutions of first-order scalar hyperbolic balance laws under piecewise regular perturbations provided that the source term is dissipative about endstates of the shock. Moreover, the convergence toward a shifted reference state is exponential with a rate predicted by the linearized equations about constant endstates.

Modelling supernova-driven turbulence

ABSTRACTHigh Mach number shocks are ubiquitous in interstellar turbulence. The Pencil Code is particularly well suited to the study of magnetohydrodynamics in weakly compressible turbulence and the numerical investigation of dynamos because of its high-order advection and time evolution algorithms. However, the high-order algorithms and lack of Riemann solver to follow shocks make it less well suited to handling high Mach number shocks, such as those produced by supernovae (SNe). Here, we outline methods required to enable the code to efficiently and accurately model SNe, using parameters that allow stable simulation of SN-driven turbulence, in order to construct a physically realistic galactic dynamo model. These include the resolution of shocks with artificial viscosity, thermal conductivity and mass diffusion; the correction of the mass diffusion terms and a novel generalisation of the Courant condition to include all source terms in the momentum and energy equations. We test our methods with the numerical solution of the one-dimensional (1D) Riemann shock tube, also extended to a 1D adiabatic shock with parameters and Mach number relevant to SN shock evolution, including shocks with radiative losses. We extend our test with the three-dimensional (3D) numerical simulation of individual SN remnant evolution for a range of ambient gas densities typical of the interstellar medium and compare these to the analytical solutions of Sedov–Taylor (adiabatic) and the snowplough and Cioffi et al. results incorporating cooling and heating processes. We show that our new timestep algorithm leads to linear rather than quadratic resolution dependence as the strength of the artificial viscosity varies, because of the corresponding change in the strength of interzone gradients.

Simulating cosmic ray physics on a moving mesh

We discuss new methods to integrate the cosmic ray (CR) evolution equations coupled to magneto-hydrodynamics (MHD) on an unstructured moving mesh, as realised in the massively parallel AREPO code for cosmological simulations. We account for diffusive shock acceleration of CRs at resolved shocks and at supernova remnants in the interstellar medium (ISM), and follow the advective CR transport within the magnetised plasma, as well as anisotropic diffusive transport of CRs along the local magnetic field. CR losses are included in terms of Coulomb and hadronic interactions with the thermal plasma. We demonstrate the accuracy of our formalism for CR acceleration at shocks through simulations of plane-parallel shock tubes that are compared to newly derived exact solutions of the Riemann shock tube problem with CR acceleration. We find that the increased compressibility of the post-shock plasma due to the produced CRs decreases the shock speed. However, CR acceleration at spherically expanding blast waves does not significantly break the self-similarity of the Sedov-Taylor solution; the resulting modifications can be approximated by a suitably adjusted, but constant adiabatic index. In first applications of the new CR formalism to simulations of isolated galaxies and cosmic structure formation, we find that CRs add an important pressure component to the ISM that increases the vertical scale height of disk galaxies, and thus reduces the star formation rate. Strong external structure formation shocks inject CRs into the gas, but the relative pressure of this component decreases towards halo centres as adiabatic compression favours the thermal over the CR pressure.

Riemann shock tube: 1D normal shocks in air, simulations and experiments

This paper presents numerical simulation of the evolution of one-dimensional normal shocks, their propagation, reflection and interaction in air using a single diaphragm Riemann shock tube and validate them using experimental results. Mathematical model is derived for one-dimensional compressible flow of viscous and conducting medium. Dimensionless form of the mathematical model is used to construct space–time finite element processes based on minimization of the space–time residual functional. The space–time local approximation functions for space–time p-version hierarchical finite elements are considered in higher order spaces that permit desired order of global differentiability of local approximations in space and time. The resulting algebraic systems from this approach yield unconditionally positive-definite coefficient matrices, hence ensure unique numerical solution. The evolution is computed for a space–time strip corresponding to a time increment Δt and then time march to obtain the evolution up to any desired value of time. Numerical studies are designed using recently invented hand-driven shock tube (Reddy tube) parameters, high/low side density and pressure values, high- and low-pressure side shock tube lengths, so that numerically computed results can be compared with actual experimental measurements.

Evolution, Propagation, Reflection, and Interactions of 1D-normal Shocks in Air and FC70 Using hpk Finite Element Computational Framework

In this paper, the mathematical models based on Navier-Stokes equations incorporating physical mechanism of viscosity and conductivity are utilized in h, p, k finite element computational framework to investigate 1-D normal shocks in air and FC70. In the case of air, shock evolution, propagation, repeated reflection and interactions are simulated using ideal gas law with constant viscosity and thermal conductivity. For FC70, numerical studies are presented for initial conditions inside the BZT zone as well as for those away from the BZT zone to investigate the possibility of compression as well as rarefaction shocks. The Van der Waals equation of state with constant transport properties is used for FC70. In all cases, “the rate of production of entropy per unit volume” is used to establish: (i) evolution of shock; (ii) sustained propagation; (iii) repeated reflections and subsequent sustained propagation; and (iv) interactions and sustained propagation of reflected shocks. In all numerical studies care is taken in choosing h, p, k computational parameters to ensure that the computational processes are free of measurable numerical dispersion and that the non-discretized form of the governing differential equations are satisfied accurately in the point wise sense and hence, ensuring time accuracy of the evolutions. 1-D Riemann shock tube is used as a model problem.

Higher-order quadrature-based moment methods for kinetic equations

Kinetic equations containing terms for spatial transport, body forces, and particle–particle collisions occur in many applications (e.g., rarefied gases, dilute granular gases, fluid-particle flows). The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows sufficiently far away from the Maxwellian limit. In previous work, a quadrature-based third-order moment closure was derived for approximating solutions to the kinetic equation for arbitrary Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the non-negative weights and velocity abscissas. Here, a robust inversion procedure is proposed for three-component velocity moments up to ninth order. By reconstructing the velocity distribution function, the spatial fluxes in the moment equations are treated using a kinetic-based finite-volume solver. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the kinetic equation, the mass, momentum and energy are conserved for arbitrary Knudsen and Mach numbers. The computational algorithm is tested for the Riemann shock problem and, for increasing Knudsen numbers (i.e. larger deviations from the Maxwellian limit), the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes.

A quadrature-based third-order moment method for dilute gas-particle flows

Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag, and particle–particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows away from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at any Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the Boltzmann equation, the mass, momentum and energy are conserved for arbitrary Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to any application that can be modeled by a kinetic equation (e.g., thermal and non-isothermal flows currently treated using lattice Boltzmann methods), and examples are given from the literature.

Strong and Weak Form of the Governing Differential Equations in Least Squares Finite Element Processes in h,p,k Framework

This paper presents an investigation of performance of the least squares finite element process in hpk mathematical framework utilizing: (i) governing differential equations (GDEs) containing highest order derivatives of the dependent variables (strong form of GDEs); (ii) GDEs containing only first order derivatives of the dependent variables and hence constituting a system of first order equations (weak form of GDEs) derived either directly from conservation laws or derived using auxiliary variables or auxiliary equations. It is shown that while the weak form of the GDEs may appear perfectly legitimate, the auxiliary equations in such forms cause irrecoverable inconsistencies in the resulting computational processes due to the fact that local approximations for the dependent variables and the auxiliary variables always remain inconsistent regardless of the choices p-level and the orders of the approximation spaces. The inconsistency of local approximations in the auxiliary equations may introduce spuriousness in the numerically computed solution that may progressively grow if the theoretical solution gradients are severe and isolated and may even cause failure of the computational process. It is shown that strong forms of the GDEs have no such problems and are highly meritorious over the weak forms in all aspects and hence are always the preferred choice but require the use of higher order local approximation spaces. One dimensional steady state convection diffusion equation (BVP) and one dimensional Riemann shock tube (IVP) are used as model problems to illustrate various concepts and to present numerical studies.

K‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

AbstractIn this paper, we consider and examine alternate finite element computational strategies for time‐dependent Navier–Stokes equations describing high‐speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non‐discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space–time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non‐degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space–time differential operators.We consider one‐dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. (2) Various choices of variables. The paper establishes and considers density, velocity and temperature as variables of choice. (3) Details of the space–time least squares (LS) integral forms (meritorious over all others in all aspects) are presented and choice of approximation spaces are discussed. (4) In all numerical studies we consider a viscous and conducting medium with ideal gas law, however results are also presented for non‐conducting medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space–time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non‐discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Comparisons are made with published results as well as theoretical solutions (when possible).It is established that space–time least squares processes are the only processes that yield variationally consistent space–time integral forms, and hence unconditionally non‐degenerate space–time computational processes, which when considered in higher‐order scalar product spaces provide the desired mathematical framework in which progressively higher‐order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the IVP in all aspects. Copyright © 2006 John Wiley & Sons, Ltd.

Detecting shock waves in cosmological smoothed particle hydrodynamics simulations

We develop a formalism for the identification and accurate estimation of the strength of structure formation shocks during cosmological smoothed particle hydrodynamics simulations. Shocks play a decisive role not only for the thermalization of gas in virializing structures but also for the acceleration of relativistic cosmic rays (CRs) through diffusive shock acceleration. Our formalism is applicable both to ordinary non-relativistic thermal gas, and to plasmas composed of CRs and thermal gas. To this end, we derive an analytic solution to the one-dimensional Riemann shock tube problem for a composite plasma of CRs and thermal gas. We apply our methods to study the properties of structure formation shocks in high-resolution hydrodynamic simulations of the Lambda cold dark matter (� CDM) model. We find that most of the energy is dissipated in weak internal shocks with Mach numbers M ∼ 2 which are predominantly central flow shocks or merger shock waves traversing halo centres. Collapsed cosmological structures are surrounded by external shocks with much higher Mach numbers up to M ∼ 1000, but they play only a minor role in the energy balance of thermalization. This is because of the higher pre-shock gas densities within non-linear structures, and the significant increase of the mean shock speed as the characteristic halo mass grows with cosmic time. We show that after the epoch of cosmic reionization the Mach number distribution is significantly modified by an efficient suppression of strong external shock waves due to the associated increase of the sound speed of the diffuse gas. Invoking a model for CR acceleration in shock waves, we find that the average strength of shock waves responsible for CR energy injection is higher than that for shocks that dominate the thermalization of the gas. This implies that the dynamical importance of shock-injected CRs is comparatively large in the low-density, peripheral halo infalling regions, but is less important for the weaker flow shocks occurring in central highdensity regions of haloes. When combined with radiative dissipation and star formation, our formalism can also be used to study CR injection by supernova shocks, or to construct models for shock-induced star formation in the interstellar medium.

Breakdown of waves described by exact solutions of the Thomas-Fermi model

The dynamics of large localized repulsive clouds is examined by means of exact non-stationary solutions of the one-dimensional Thomas-Fermi model. The nonlinear flattening of the cloud peak, the wave breakdown at the cloud peripheries, and the condensate velocity distributions are thus described. Our solutions, which can contain an arbitrary amount of free parameters, show the nonlinear evolution of an arbitrary initial wave form. A unique procedure for analyzing these solutions is presented. The difference between our breakdown matter wave solutions and the well known Riemann shock waves is stressed.

A 1-D gas dynamics code for subsonic and supersonic flows applied to predict EGR levels in a heavy-duty diesel engine

The development of a 1-D gas dynamics code for unsteady flow in internal combustion (IC) engines as well as its validation and application for predicting residual gas fraction are introduced in this paper. Some new approaches are presented for modelling flows in diverging ducts and for treating boundary conditions. These include the use of flow resistance correlation to describe separated flows and flows in bends. Excellent agreement with analytical solutions and test results has been obtained when the code was validated with fundamental gas dynamic problems, including converging-diverging nozzle flows with and without shocks; Fanno and Rayleigh flows; the Riemann shock tube problem; and engine rig experiments for modelling flow with different property gases. The code has been applied satisfactorily to predict the gas exchange process of a spark ignition (SI) engine following exhaust blow-down and exhaust gas recirculation (EGR) levels in a heavy-duty diesel engine.

Binary and ternary oscillations in a cubic numerical scheme

We study a central difference semi-discretization of the cubic scalar conservation law. Both spatial period-2 (binary) and period-3 (ternary) oscillations are stationary solutions of this scheme, and we find where each type is linearly stable. We observe numerically the formation of ternary oscillations, to the left of Riemann shock initial data with u r = 0, while binary oscillations form to the right of Riemann rarefaction data having u l = 0. We derive modulation equations for both wave patterns, using them to show that binary oscillations generated by the scheme are numerical artifacts, while computing an explicit solution for the ternary modulation equations. For positive initial data, the ternary modulation equations remain hyperbolic, while the solutions enter an elliptic region for data of both signs. Conditions under which solutions of the ternary modulation equations can be inverted to yield period-3 oscillations are also discussed.

Adaptive Smoothed Particle Hydrodynamics: Methodology. II.

This paper presents an alternative formulation of the ASPH algorithm for evolving anisotropic smoothing kernels, in which the geometric approach of Shapiro et al. (1996; Paper I) is replaced by an approach involving a local transformation of coordinates to those in which the underlying anisotropic volume changes appear to be isotropic. The ASPH method is presented in 2D and 3D, including a number of details not previously included in Paper I, some of which represent either advances or different choices with respect to Paper I. Among the advances included here are an asynchronous time-integration scheme with different time steps for different particles and the generalization of the ASPH method to 3D. The shock-tracking algorithm described in Paper I for locally adapting the artificial viscosity to restrict viscous heating just to particles encountering shocks, is not included here. Instead, we adopt a different interpolation kernel for use with the artificial viscosity, which has the effect of spatially localizing effects of the artificial viscosity. This version of the ASPH method in 2D and 3D is then applied to a series of 1D, 2D, and 3D test problems, and the results are compared to those of standard SPH applied to the same problems. These include the problem of cosmological pancake collapse, the Riemann shock tube, cylindrical and spherical Sedov blast waves, the collision of two strong shocks, and problems involving shearing disks intended to test the angular momentum conservation properties of the method. These results further support the idea that ASPH has significantly better resolving power than standard SPH for a wide range of problems, including that of cosmological structure formation. (Abridged)

Hybrid Formulations of smoothed particle hydrodynamics

In this paper the concept of hybrid methods or flux-limiter methods from the finite difference approach is applied to Smoothed Particle Hydrodynamics (SPH) to obtain improved results in problems which involve shocks. Several variations of the resulting hybrid SPH formulation are discussed and tested against the Riemann shock tube problem. This new technique allows the SPH continuity equation to be used successfully for this test problem. The work described here shows considerable promise and the methods should be further developed.

A fully implicit Navier-Stokes algorithm using an unstructured grid and flux difference splitting

An implicit algorithm is developed for the two-dimensional, compressible, laminar Navier-Stokes equations using an unstructured grid of triangles. A sell-centered data structure is employed with the flow variables stored at the centroids of the triangles. The algorithm is based on Roe's flux difference split method for the inviscid fluxes, and a discrete representation of the viscous fluxes and heat transfer using Gauss' Theorem. Linear reconstruction of the flow variables to the cell faces, employed for the inviscid terms, provides second-order spatial accuracy. Interpolation of the flow variables to the nodes is achieved using a second-order accurate method. Temporal discretization employs Euler, trapezoidal or 3-point backward differencing. The complete, exact Jacobian of the inviscid and viscous terms is derived. The algorithm is applied to the Riemann Shock tube problem, a supersonic laminar boundary layer on a flat plate, and subsonic viscous flow past an NACA0012 airfoil. Results are in excellent agreement with theory and previous computations.

A finite element method in space and time for relativistic spherical collapse

The finite element method is applied in space and time to spherically symmetric collapse in general relativity. The fluid equations are discretized with a weighted residual method in which isoparametric, bilinear approximations on quadrilaterals are used. The metric equations are constraint equations, and are approximated with cubic and linear functions on triangles. The time slices and nodal movement can be chosen arbitrarily, and a number of choices are investigated. These include a maximal slicing condition which requires an additional solution of a differential equation. The code is tested with the Riemann shock tube, and then compared with a mixed finite element/finite difference code and with a smoothed particle hydrodynamics code. Three particular simulations are used for comparisons: static, freefall collapse and collapse-and-bounce. The finite element method agrees well with the other codes, and will produce accurate results with extremely “bent” time slices.

Godunov computational fluid dynamics method for extreme flow velocities and any equation of state

An inviscid computational fluid dynamics code that can handle multiple component species, simple chemical reactions, a completely general equation of state, and velocities up to hundreds of kilometers per second is described. Radiation effects are not included. The code uses a sliding grid technique to handle multiple moving zones containing different media. Third-order extrapolations and interpolations are used to determine cell boundary values of the primitive variables. Limiting techniques then are applied to these values to maintain code accuracy and stability. First- and second-order Godunov procedures are used to calculate the cell boundary fluxes. The cell center values are updated in time using an explicit MacCormack predictor-corrector technique that is second-order accurate in time. The code was proofed with benchmark test cases, including Riemann's shock tube problem at pressure ratios up to 104, impacts of ideal-gas and dense media zones at closing velocities up to 220 km/s, detonation waves in stoichiometric hydrogen-oxyge n mixtures, ideal-gas wedge and cone flows, and a wedge flow of iron at a speed of 110 km/s. The results of these proof tests are discussed. The agreement between the CFD results and the exact solutions is found to be excellent.

A new finite element method for spherically symmetric relativistic collapse

In a previous paper a mixed finite element-finite difference numerical method was used to model relativistic spherical collapse. The method was unsatisfactory in some cases, and a new method is described in this paper. Einstein's equations are written in a standard form, and a weighted residual, moving finite element method is applied to derive a discretization. Tests are made on static and pressureless collapse, in both Newtonian and. relativistic situations. Shocks are smoothed using an artificial viscosity, and the results are compared to finite difference codes for the Riemann shock tube problem. Some collapse and bounce models are made, and the resulting shock is investigated. The code seems to be fast and accurate, with reasonable shock descriptions.