Shock Fitting Research Articles

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

Extrapolated Shock Fitting for Two-Dimensional Flows on Structured Grids

Over the years the development of structured-grid shock-fitting techniques faced two main problems: the handling of a moving discontinuity on a fixed background grid and the capability of simulating complex flow configurations. In the proposed work, the authors present a new shock-fitting technique for structured-grid solvers that is capable of overcoming the limitations that affected the different approaches originally developed. The technique presented here removes the tight link between grid topology and shock topology, which characterizes previous shock fitting as well as front tracking methods. This significantly simplifies their implementation and more importantly reduces the computational overhead related to these geometrical manipulations. Interacting discontinuities and shocks interacting with a solid boundary are discussed and analyzed. Finally, a quantitative investigation of the error reduction obtained with the approach proposed via a global grid convergence analysis is presented.

Dispersive shock waves for the Boussinesq Benjamin–Ono equation

AbstractIn this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin–Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin–Ono dispersion, is derived. This DSW solution is derived using two methods, DSW fitting and from a simple wave solution of the Whitham modulation equations for the BBO equation. The first of these yields the two edges of the DSW, while the second yields the complete DSW solution. As the Whitham modulation equations could not be set in Riemann invariant form, the ordinary differential equations governing the simple wave are solved using a hybrid numerical method coupled to the dispersive shock fitting which provides a suitable boundary condition. The full DSW solution is then determined, which is found to be in excellent agreement with numerical solutions of the BBO equation. This hybrid method is a suitable and relatively simple method to fully determine the DSW solution of a nonlinear dispersive wave equation for which the (hyperbolic) Whitham modulation equations are known, but their Riemann invariant form is not.

A novel stabilization method for high-order shock fitting with finite element methods

A moving-grid, shock-fitting, finite element method has been implemented that can achieve high-order accuracy for flow simulations with shocks. In this approach, element edges in the computational mesh are fitted to the shock front and moved with the shock throughout the simulation. The Euler or Navier-Stokes equations are solved on the moving mesh in an arbitrary Lagrangian-Eulerian framework. The method is implemented in two-dimensions in the context of a streamwise upwind Petrov-Galerkin finite element discretization with unstructured triangular meshes and mesh adaptation. It is shown that the shock interface motion equation has a wave nature, and disturbances can propagate along the shock interface. A SUPG stabilization term is introduced to the interface motion equation that is critical for ensuring that interface disturbances do not lead to non-convergent solution behavior. The formal order of accuracy of the scheme is verified, and the performance of the proposed scheme is assessed for both inviscid and viscous problems. It was found that the present scheme predicts smooth and noise-free surface heating for hypersonic flow over a cylinder with purely irregular triangular elements.

High-order shock-fitted detonation propagation in high explosives

A highly accurate numerical shock and material interface fitting scheme composed of fifth-order spatial and third- or fifth-order temporal discretizations is applied to the two-dimensional reactive Euler equations in both slab and axisymmetric geometries. High rates of convergence are not typically possible with shock-capturing methods as the Taylor series analysis breaks down in the vicinity of discontinuities. Furthermore, for typical high explosive (HE) simulations, the effects of material interfaces at the charge boundary can also cause significant computational errors. Fitting a computational boundary to both the shock front and material interface (i.e. streamline) alleviates the computational errors associated with captured shocks and thus opens up the possibility of high rates of convergence for multi-dimensional shock and detonation flows. Several verification tests, including a Sedov blast wave, a Zel'dovich–von Neumann–Döring (ZND) detonation wave, and Taylor–Maccoll supersonic flow over a cone, are utilized to demonstrate high rates of convergence to nontrivial shock and reaction flows. Comparisons to previously published shock-capturing multi-dimensional detonations in a polytropic fluid with a constant adiabatic exponent (PF-CAE) are made, demonstrating significantly lower computational error for the present shock and material interface fitting method. For an error on the order of 10 m/s, which is similar to that observed in experiments, shock-fitting offers a computational savings on the order of 1000. In addition, the behavior of the detonation phase speed is examined for several slab widths to evaluate the detonation performance of PBX 9501 while utilizing the Wescott–Stewart–Davis (WSD) model, which is commonly used in HE modeling. It is found that the thickness effect curve resulting from this equation of state and reaction model using published values is dramatically more steep than observed in recent experiments. Utilizing the present fitting strategy, in conjunction with a nonlinear optimizer, a new set of reaction rate parameters improves the correlation of the model to experimental results. Finally, this new model is tested against two dimensional slabs as a validation test.

Shock-Fitting Versus Shock-Capturing Modeling of Strong Shocks in Nonequilibrium Plasmas

In this paper, a supersonic flow of an argon plasma around a cylinder has been investigated comparing shock fitting and shock capturing techniques. Shock-capturing codes are algorithmically simple, but are plagued by a number of numerical troubles, particularly evident when the shocks are strong and the grids unstructured. On the other hand, shock-fitting algorithms allow to accurately compute solutions on coarse meshes, but tend to be algorithmically complex. The kinetic scheme adopted includes the argon metastable state as an independent species and takes into account for electron-atom and atom-atom processes. Electron density distributions have been reported.

A two-dimensional well-balanced numerical model for shallow water equations

Modeling flow discontinuities, due to a numerical approach, often pose severe challenges. In this way, a number of techniques like artificial viscosity (particularly for finite difference methods), shock fitting and etc. have been proposed. These techniques usually require ad-hoc terms which need to be adjusted through calibration. In this study, an efficient numerical model based on a shallow water equation is developed. The model uses a first-order centered (FORCE) scheme, in combination with the Surface Gradient Method, (SGM) for spatial discretization, and the third order Runge–Kutta algorithm for time integration. At first, it is demonstrated that the model is well-balanced, then, through several classical examples, such as a 1D dam-break on both dry and wet beds, transcritical flow over a bump, with and without shock, sub-critical flow over a bump, circular dam-break, small perturbation propagation, dam-break on a dry bed channel with varying widths and right-angled channel junctions, it is shown that the model is capable of capturing flow discontinuities. Furthermore, the model can simulate dry bed conditions, and also presents smooth symmetric results.

The Role of Mesh Generation, Adaptation, and Refinement on the Computation of Flows Featuring Strong Shocks

Within a continuum framework, flows featuring shock waves can be modelled by means of either shock capturing or shock fitting. Shock-capturing codes are algorithmically simple, but are plagued by a number of numerical troubles, particularly evident when shocks are strong and the grids unstructured. On the other hand, shock-fitting algorithms on structured grids allow to accurately compute solutions on coarse meshes, but tend to be algorithmically complex. We show how recent advances in computational mesh generation allow to relieve some of the difficulties encountered by shock capturing and contribute towards making shock fitting on unstructured meshes a versatile technique.

Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics

Progress in computer technology now allows the application of numerical methods to some practical multidimensional steady and unsteady gas flows. While selecting a numerical method, preference should be given to higher-order schemes, since they allow the study of the fine features of the solution and the reduction of the computation time. In general, however, solutions of non-stationary nonlinear equations of gas dynamics are not smooth and may include strong discontinuities, such as shock waves or contact discontinuities. This fact is important for choosing an appropriate numerical scheme. If the strong discontinuities are extracted from the simulation domain and traced (shock fitting), while the rest of the flow is smooth, it is possible to use numerical schemes with approximation close to the second order [1-3]. However, if the number of discontinuities increases with time leading to a complicated pattern, the application of the shock-fitting approach becomes difficult. In this case it is preferable to apply shock-capturing schemes [4-6], which have a lower order of approximation but allow the integration of the governing equations through the discontinuities without extracting their surfaces. Shock-capturing schemes of a higher order of approximation are currently under development [7-9]. In this paper, certain aspects of constructing a numerical scheme applicable to discontinuous flows of gas dynamics are discussed using the idea of minimizing the derivatives of the solution. A numerical scheme that is second-order accurate in space and first-order accurate in time is developed for the model equation ut+ux=0. The properties of this scheme are analyzed and the conditions for its stability and monotonicity are formulated. These results are then used in devising a similar scheme for the non-stationary one-dimensional equations of gas dynamics. The performance of the new scheme is compared to that of other schemes, using the break-up of an initial discontinuity (Riemann problem) as an example.

Deriving Features of Reacting Hypersonic Flow from Gradients at a Curved Shock

The shock jump relations for gradients of flow variables are presented in the case of a uniform steady freestream with a curved shock that may trigger an endothermic or exothermic chemical reaction. The rich information that can be obtained from these is used in order to derive field information for flow variables as well as details about the sonic line. This presentation collects, reviews, and extends work that has been published previously in various places. Comparisons with experiment and with numerical computation are given in some cases. The gradient-jump relations are likely to be useful for higher-order numerical schemes that use shock fitting.

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

A Shock Fitting Procedure Based on Monte Carlo Calculations: Application to Slow Shocks

Recent derived procedure of fitting the Rankine‐Hugoniot shock jump relations for fast MHD shocks (Lin et al., 2006) is here applied to slow‐mode shocks. The present paper is a continuation of Lin et al. (2006). We modify the method to be applicable to slow shocks. We have tested the new method via a simulated slow shock. The result shows that the method can find a best fit solution that is close to the “true answer” as long as the systematic errors in the synthetic data are not too large. We also apply the method to one interplanetary slow shock and two slow shocks observed in the magnetotail. The results have been compared to the previous studies. The new method of shock fitting is demonstrated successfully for identifying slow shocks.

A new shock fitting procedure for the MHD Rankine‐Hugoniot relations for the case of small He2+ slippage

To study MHD shocks in space, it is important to find the shock frame of reference from the observed plasma and magnetic field parameters. These shock parameters have to satisfy the Rankine‐Hugoniot relations. In this study we present a novel procedure for shock fitting of the one‐fluid anisotropic Rankine‐Hugoniot relations and of the time difference between two spacecraft observations in the case of small He2+ slippage. Here, a Monte‐Carlo calculation and a minimization technique are used. The observed variables including the upstream and downstream magnetic fields, plasma densities, plasma betas, plasma anisotropies, W (the difference between the downstream and upstream velocities, W ≡ V2 − V1), and Δt (the time difference between two spacecraft observations) are used in our procedure where V is defined as the center of mass velocity of plasmas. A loss function based on a difference between the calculated and the observed values is defined, and the best fit solution is found by searching for the minimum loss function value. For shocks that cannot be fitted well, we introduce two new parameters in the modified RH relations, one in the normal momentum flux and the other in the energy flux equations. These two parameters are interpreted as the equivalent “normal momentum” and “heat” fluxes needed in the RH relations. They provide two degrees of freedom in the system, and their amounts can be estimated from our procedure. Several synthetic shocks are given to verify our procedure. We also apply this procedure to two interplanetary shocks observed by both the WIND and Geotail spacecraft. The results demonstrate that our method works for both the synthetic and the real shocks. We have shown that our method can provide accurate shock normal estimations for perpendicular and parallel shocks as well. Given that our model is based on the RH relations that do not include the effect of alpha particle (He2+) slippage, it can only be applied to the cases with an ignorable slippage pressure tensor. We have investigated the pressure tensor due to alpha particle slippage using the WIND spacecraft data. It is found that in general the slippage pressure is small in comparison with the thermal pressure of the system and can be ignored. Thus our model can be applied to most interplanetary shocks observed near the ecliptic plane. However, when the slippage pressure is large, the magnetic coplanarity theorem is not valid any more. A more general model that involves slippage pressure tensor is a major and important development that is beyond the scope of the present study.

Interaction of a shock with a density disturbance via shock fitting

An accurate solution to the problem of a normal shock moving into still fluid with a density variation is presented. The solution is obtained using a shock fitted approach and Runge–Kutta time integration. Uniform third order accuracy of the scheme us demonstrated. Comparisons with shock captured solutions show that the fitted solution presented here is more accurate.

Standardized pseudospectral formulation of the inviscid supsersonic blunt body problem

A highly accurate pseudospectral numerical approximation to the generalized coordinate, nonconservative form of the Euler equations is implemented for supersonic flow over an axisymmetric blunt body geometry; shock fitting is employed to maintain global accuracy and minimize the corrupting influence of numerical viscosity. The variables in the Euler equations as well as the physical grid coordinates are collocated via Lagrange interpolating polynomials and the problem is then cast in the standard form of a large system of ordinary differential equations, d x / dτ= q ( x ) , which can be solved using standard solution techniques that do not require an explicit criteria for the minimum time step. Code verification is performed by demonstrating through a series of grid refinement tests that the error in the approximation to a Taylor–Maccoll solution converges to 10 −12. Grid refinement tests for flow over a blunt body show convergence of the numerical error also to 10 −12. The code is validated for supersonic flow over a blunt body by comparison with the modified Newtonian approximation for the surface pressure distribution and empirical predictions for the shock shape. The ability of the method to capture unsteady flow phenomena is demonstrated on the problem of a planar acoustic wave interacting with an attached shock.

Thirty-six years of shock fitting

Thirty-six years of shock fitting

Shock-fitted multidomain solution of supersonic flows

Shock fitting is presented as a way to compute shocks with spectral methods. With fitting, spectral approximation is performed on smooth portions of the flow and a precise description of a discontinuous shock is possible. Coupled with spectral domain decomposition, one has the ability to resolve efficiently and without stretching transformations, boundary layers and other regions in the flow where increased local resolution is needed. A supersonic blunt body problem is solved to provide an example of the use of the approach. Comparisons between computations and experimental measurements are made for body surface quantities, shock shape, and body normal direction profiles.

Shock Fitting and Numerical Modeling of Detonation Waves

Numerical modelling of detonation using shock fitting is described in detail. A complete set of jump conditions that hold across the detonation front is presented in a simple form. Validity and accuracy of the model has been established by comparison with published results and results of another model utilizing the method of integral relations. A brief description of the later model is given to highlight its validity and limitations.

A Conservative Shock Fitting Method on Unstructured Grids

A dynamic discontinuity tracking method to achieve shock fitting is presented. The method is based on the properties of Roe's scheme which becomes an exact Riemann solver when grid lines are aligned with the discontinuity. To make full use of this property, a technique that moves and adapts the grid in order to align and position element interfaces on the discontinuities is proposed. This algorithm is implemented with a conventional two-dimensional finite volume solver on unstructured triangular meshes. The method's ability to retrieve the exact solution is investigated on simple problems having analytical solutions.

Sensitivity algorithms for an inverse design problem involving a shock wave

A model inverse design problem is used to investigate the effect of flow discontinuities on the optimization process. The optimization involves finding the cross-sectional area distribution of a duct that produces velocities which closely match a targeted velocity distribution. Quasi-one-dimensional flow theory is used, and the target is chosen to have a shock wave in its distribution. The objective function which quantifies the difference between the targeted and calculated velocity distributions may become non-smooth due to the presence of the shock in the discretized flow field. This paper offers two techniques to resolve the resulting problems for the optimization algorithms. The first, shock fitting, involves careful integration of the objective function through the shock wave. The second, coordinate straining with shock penalty, uses a coordinate transformation to align the calculated shock with the target and then adds a penalty proportional to the square of the distance between the shocks. These techniques are evaluated and tested using several sensitivity methods, including finite-differences, direct and adjoint methods.