Shock-fitting Method Research Articles

Mesh-free simulation of two-phase fluid flow in porous media based on the shock-fitting method

The physical interactions between two immiscible fluids in porous media may lead to the formation of a shock front at the interface between the two fluid phases. Numerical methods for modeling fluid flow with shocks can be classified into two categories: the shock-capturing and the shock-fitting methods. The shock-fitting method can in general provide more accurate solution of the shock front than the shock-capturing method, but suffers from the extra complexity in accommodating the moving shock front inside the background mesh. In this study, we explore the possibility of integrating the mesh-free methods for solving partial-differential equations with the shock-fitting method. In this integrated method, which we call “mesh-free shock-fitting (MFSF)”, the nodes needed for mesh-free calculations of spatial derivatives are generated on the fly to adapt to the moving shock front. We demonstrate the implementation of MFSF using numerical experiments in one and two spatial dimensions. Preliminary results show that MFSF has the potential to provide more accurate solutions with lower computational cost than conventional shock-capturing methods and can simplify many of the operations in shock-fitting methods with unstructured meshes.

Hypersonic attachment-line instabilities with large sweep Mach numbers

Abstract

Convergence acceleration for high-order shock-fitting methods in hypersonic flow applications with efficient implicit time-stepping schemes

The research on hypersonic boundary layer transition is a recent focal point in fluid dynamics. One of the effective numerical methods for transition study is the high-order shock-fitting method, which is especially suitable for dealing with shock-related interactions and removing post-shock oscillations as compared with shock-capturing methods. Currently, explicit time-stepping schemes are mainly used for shock-fitting methods. In comparison, implicit schemes have received less attention due to the extra complexity arising from dynamic shock and moving mesh, even though they can accelerate the convergence speeds for steady flows by up to an order of magnitude. In this paper, two implicit time-stepping methods, GMRES (generalized minimum residual) and LR (line-relaxation), are realized for the efficient calculation of steady flow fields using the 5th-order shock-fitting method. The gas models include both calorically perfect gas and 5-species thermal-chemical non-equilibrium gas required for high-speed and high-enthalpy flows. The results of test cases showed that the convergence rates of GMRES+LR and LR are 4 to 20 times faster than the explicit/semi-implicit Runge-Kutta schemes. Therefore, the usage of implicit GMRES and LR as time-stepping schemes raises the computational efficiency for obtaining steady boundary layer flows, which helps the important high-order shock-fitting methods play a greater role in the study of hypersonic boundary layer stability and transition.

Extrapolated Shock Tracking: Bridging shock-fitting and embedded boundary methods

We propose a novel approach to approximate numerically shock waves. The method combines the unstructured shock-fitting approach developed in the last decade by some of the authors, with ideas coming from embedded boundary techniques. The numerical method obtained allows avoiding the re-meshing phase required by the unstructured fitting method, while guaranteeing accuracy properties very close to those of the fitting approach. This new method has many similarities with front tracking approaches, and paves the way to shock-tracking techniques truly independent on the data and mesh structure used by the flow solver. The approach is tested on several problems showing accuracy properties very close to those of more expensive fitting methods, with a considerable gain in flexibility and generality.

Numerical investigation of filling transients in small-scale pipelines with submerged outlet

An elastic model that combines the dead-end and the orifice at the downstream end is proposed in order to study the hydraulic transients during the rapid filling in small-scale pipelines with submerged outlet. In this model, the unsteady water flow is solved by the method of characteristics, the water-air interface is traced by a shock fitting method, and the air phase is controlled by the polytropic law. The model is validated by a filling test in a pipeline with orifice. Then the proposed model is applied to the filling transient analysis in a small-scale pipeline with submerged outlet. Finally, the effects of the submerged depth of the downstream reservoir on the filling transients are analyzed. This model provides a reasonable approximation of the submerged outlet boundary condition in small-scale pipelines.

A minimal hyperbolic system for unstable shock waves

We present a computational analysis of a 2×2 hyperbolic system of balance laws whose solutions exhibit complex nonlinear behavior. Traveling-wave solutions of the system are shown to undergo a series of bifurcations as a parameter in the model is varied. Linear and nonlinear stability properties of the traveling waves are computed numerically using accurate shock-fitting methods. The model may be considered as a minimal hyperbolic system with chaotic solutions and can also serve as a stringent numerical test problem for systems of hyperbolic balance laws.

Dispersive shock waves governed by the Whitham equation and their stability

Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, governed by the non-local Whitham equation are studied in order to investigate short wavelength effects that lead to peaked and cusped waves within the DSW. This is done by combining the weak nonlinearity of the Korteweg–de Vries equation with full linear dispersion relations. The dispersion relations considered are those for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120° peaked Stokes wave of highest amplitude. A dispersive shock fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges of the DSW. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude which is just below the DSW of maximum amplitude.

Higher-order jump conditions for conservation laws

The hyperbolic conservation laws admit discontinuous solutions where the solution variables can have finite jumps in space and time. The jump conditions for conservation laws are expressed in terms of the speed of the discontinuity and the state variables on both sides. An example from the Gas Dynamics is the Rankine–Hugoniot conditions for the shock speed. Here, we provide an expression for the acceleration of the discontinuity in terms of the state variables and their spatial derivatives on both sides. We derive a jump condition for the shock acceleration. Using this general expression, we show how to obtain explicit shock acceleration formulas for nonlinear hyperbolic conservation laws. We start with the Burgers’ equation and check the derived formula with an analytical solution. We next derive formulas for the Shallow Water Equations and the Euler Equations of Gas Dynamics. We will verify our formulas for the Euler Equations using an exact solution for the spherically symmetric blast wave problem. In addition, we discuss the potential use of these formulas for the implementation of shock fitting methods.

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.

Steady-State Analysis of Rotating Detonation Engine Flowfields with the Method of Characteristics

A method for modeling the internal flowfield in a rotating detonation engine is developed using shock-expansion theory combined with the steady two-dimensional isentropic method of characteristics. An analytical model using the oblique shock relations, the Prandtl–Meyer function, and the detonation jump conditions is used to determine the basic shock structure. Once the structure is known, a shock-fitted method of characteristics solution is marched out to generate the rest of the flowfield. Reactant injection is handled analytically by solving the conservation equations for a flow undergoing a sudden expansion along with the method of characteristics compatibility relations to provide a new boundary condition. A new solution is then initialized using information from the previous solution to calculate the new shock structure. This process is repeated until the solutions converge. The converged solution is the ideal steady-state solution of a rotating detonation engine in the wave-fixed reference frame. The goal of this model is to provide ideal performance estimates one step up from a basic thermodynamic model that are still fast enough for large parametric studies.

Flow Field Analysis on the Stagnation Streamline of a Blunt Body

The hypersonic flow on the stagnation streamline of a blunt body is analyzed with quasi one-dimensional (1-D) Navier-Stokes equations approximated by adopting the local similarity to the two-dimensional (2-D)/axisymmetric Navier-Stokes equations. The governing equations are solved using the implicit finite volume method. The computational domain is confined from the stagnation point to the shock wave, and the shock fitting method is used to find the shock position. We propose a boundary condition at the shock, which employs the shock wave angle in the vicinity of the stagnation streamline using the shock shape correlation. As a result of numerical computation conducted for the hypersonic flow over a sphere, the proposed boundary condition is shown to improve the accuracy of the prediction of the shock standoff distance. The quasi 1-D Navier-Stokes code is efficient in computing time and is reliable for the flow analysis along the stagnation streamline and the prediction of heat flux at the stagnation point in the hypersonic blunt body flow.

Simulation of Second-Mode Instability in a Real-Gas Hypersonic Flow with Graphite Ablation

A new high-order shock-fitting method with thermochemical nonequilibrium and finite-rate chemistry boundary conditions for graphite ablation is presented. The method is suitable for direct numerical simulation of boundary-layer transition in a hypersonic real-gas flow with graphite ablation. The new method is validated by comparison with three computational datasets and one set of experimental data. Direct numerical simulations were run for a 7 deg half-angle blunt cone at Mach 15.99 to find how graphite ablation and thermochemical nonequilibrium affect boundary-layer receptivity and instability. The real-gas simulation is compared with ideal-gas simulations that set their wall temperature and wall blowing from the real-gas simulation. Weak planar fast-acoustic waves in the freestream are used to perturb the steady base flow. A 525 kHz second-mode wave was found to be significantly unstable for the real-gas simulation, whereas in the ideal-gas simulations, no significant flow instability was seen. For the specific flow conditions tested, it was found that real-gas effects significantly destabilize second-mode waves.

Application of a shock-fitted spectral collocation method for computing transient high-speed inviscid flows over a blunt nose

Interaction of freestream disturbances with high-speed inviscid flow over a blunt nose is simulated utilizing a shock-fitted spectral collocation method. The unsteady flow computations are made through solving the 2D Euler equations by virtue of such a dissipation-free numerical algorithm for precise unsteady flow simulations. A shock-fitting technique is employed to accurately compute the unsteady shock motions and its interaction with monochromatic freestream disturbances of different conditions. A symmetry condition is proposed to accurately model the both steady and unsteady characters of the symmetry boundary, which allows the use of halved geometries and avoids the extra computational cost. The computational results for a cylinder at Mach 8.03 are presented and verified through comparisons with other numerical and promising analytical solutions. The stagnation line where the most energetic interactions take place is inspected carefully. The study shows the significant influence of the shock on its subsequent flow field in unsteady simulations and notable ability of the shock-fitted spectral collocation method implemented for the study of such problems.

Direct Numerical Simulation on the Receptivity, Instability, and Transition of Hypersonic Boundary Layers

The prediction of the laminar-turbulent transition of boundary layers is critically important to the development of hypersonic vehicles because the transition has a first-order impact on aerodynamic heating, drag, and vehicle operation. The success of transition prediction relies on a fundamental understanding of the relevant physical mechanisms. In the 20 years since the review by Kleiser & Zang (1991) on the direct numerical simulation (DNS) of the boundary-layer transition, significant progress has been made on DNS in the hypersonic flow regime and in the spatial DNS approach. Many high-order shock-capturing and shock-fitting finite-difference methods have been developed and extensively applied to numerical simulations of the hypersonic boundary-layer transition. DNS has become a powerful research tool and has led to discoveries of new transition mechanisms. This article reviews the recent progress of DNS on hypersonic boundary-layer receptivity, instability, and transition. The current status and future directions are also presented.

Numerical simulation of axisymmetric supersonic viscous flow over a blunt cone with a diagonal fourth-order finite difference method

The numerical solution of steady viscous supersonic axisymmetric flowfield modelled by thin-layer Navier–Stokes equations is computed over a blunt cone with the shock-fitting method and the diagonal fourth-order central difference scheme implemented. Owing to the presence of high-order terms of the Taylor series in the discretization of derivatives, this method has high accuracy and low numerical error (dispersion error) compared with low-order methods. The boundary-closure scheme plays an important role in the numerical stability of this method. Using a coarse grid in this method, the results of numerical solution are found to be very close to those obtained with a fine grid employing the implicit second-order (Beam–Warming) method. Higher accuracy of this method is identified relative to the second-order method when the grid is being refined. The convergence rate of this method is also higher than the second-order method. Furthermore, the convergence of the method can be adjusted to accommodate the computational hardware capabilities.

High-order shock-fitting methods for direct numerical simulation of hypersonic flow with chemical and thermal nonequilibrium

In recent years, much progress has been made in the direct numerical simulation of laminar-turbulent transition of hypersonic boundary layer flow. However, most of the efforts at the direct numerical simulation of transition previously have been focused on the idealized perfect gas flow or “cold” hypersonic flows. For practical problems in hypersonic flows, high-temperature effects of thermal and chemical nonequilibrium are important and cannot be modeled by a perfect gas model. Therefore, it is necessary to include the real gas models in the numerical simulation of hypersonic boundary layer transition in order to accurately predict flow field parameters. Currently most numerical methods for hypersonic flow with thermo-chemical nonequilibrium are based on shock-capturing approach at relatively low order of accuracy. Shock capturing schemes reduce to first-order accuracy near the shock and have been shown to produce spurious oscillations behind curved strong shocks. There is a need to develop new methods capable of simulating nonequilibrium hypersonic flow fields with uniformly high-order accuracy and avoid spurious oscillations near the shock. This paper presents a fifth-order shock-fitting method for numerical simulation of thermal and chemical nonequilibrium in hypersonic flows. The method is developed based on the state-of-the-art real gas models for thermo-chemical nonequilibrium and transport phenomena. Shock-fitting approach is used because it has the advantage of capturing the entire flow field with high-order accuracy and without any oscillations near the shock. The new method has been tested and validated for a number of test cases over a wide span of free stream conditions. The developed method is applied for the study of receptivity of free stream acoustic waves over a blunt cone for hypervelocity flow. Some preliminary results of the computations of the high order shock fitting method for the above mentioned study have also been presented.

Acoustic shock wave propagation in a heterogeneous medium: A numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.

Response of a Hypersonic Boundary Layer to Wall Blowing-Suction

In the laminar-turbulent transition of boundary-layer flows, the spatial development of boundary-layer waves, especially that of unstable waves, is important and indispensable. In this paper, the response of a Mach 8 flow over a 5.3° half-angle sharp wedge to wall blowing―suction is studied by numerical simulations. Steady base flow is obtained by solving the compressible Navier―Stokes equations with a combination of a fifth-order shock-fitting method and a second-order total-variation-diminishing scheme. In stability simulations, wall blowing―suction is introduced through an actuator on the wedge surface. The unsteady flow simulation is carried out using the shock-fitting method. The results show that mode F, mode S, acoustic waves, and entropy/vorticity waves are simultaneously excited by wall blowing―suction and coexist in the boundary layer just downstream of the actuator. For blowing―suction at a specific frequency, the results also show that mode S is strongly excited when the actuator is located upstream of the corresponding synchronization point. There is no significant amplification of pressure perturbation when the actuator is downstream of the synchronization point. This result represents a mixture of the receptivity and the downstream growth of mode S. The exact cause and mechanism of this result are not clear. However, such a result is obtained for wall blowing―suction at all frequencies considered in the current study. To excite a strong mode S at a specific frequency, the result indicates that it is necessary to place the blowing―suction actuator upstream of the corresponding synchronization point. Further theoretical analysis is needed to reveal the mechanism behind the numerical results.

Structure and kinetic properties of slow-mode shocks associated with magnetic reconnection in the near-Earth magnetotail

We study the structure and kinetic properties of slow-mode shocks near the plasma sheet boundary layer (PSBL) associated with magnetic reconnection by Cluster observation. The presence of slow-mode shocks is confirmed by traditional Rankine–Hugoniot (RH) analysis and Monte-Carlo shock fitting method. The Walén analysis, applied to the tailward flow associated with slow-mode shocks, also supports that plasma was accelerated across a Petschek-type slow-mode shock connected to the diffusion region. Back-streaming ions were observed on the shock layer, and cold ions were accelerated and heated by slow-mode shocks. In addition, whistler and electrostatic solitary waves were observed around the slow-mode shocks. These waves might be excited by the observed field-aligned electron beams near the shocks.

A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness

Laminar-turbulent transition of hypersonic boundary layers can be affected significantly by the existence of surface roughness. Currently many important mechanisms of roughness-induced transition are not well understood. In recent years, direct numerical simulation (DNS) has been extensively applied for investigating instability and transition mechanisms of hypersonic boundary layers. Most of the past DNS studies, however, have been based on body-fitted grids for smooth surfaces without roughness. Due to complex geometry of embedded roughness, the use of body-fitted grids can be very difficult for flow with arbitrary surface roughness. In this paper, we present a new high-order cut-cell method to overcome the natural complexities in grid generation around arbitrary surface of roughness. The new method combines a non-uniform-grid finite-difference method for discrete grid points near the solid boundary and a shock-fitting method for the treatment of the bow shock. The non-uniform-grid finite-difference formulas are expressed in a general explicit form so that they can be applied to different multi-dimensional problems without any modification. The computational accuracy of new algorithms of up to O( h 4) are tested on several one- and two-dimensional elliptic equations in irregular domains. In addition, the new method is applied to the simulation of the receptivity process of a Mach 5.92 flow over a flat plate under the combined effect of an isolated surface roughness element and surface blow and suction. A good agreement is found between the unsteady flow results and those obtained by a Linear Stability Theory (LST).