Shock Fitting Research Articles

Comparison of real gas simulations using different numerical methods

Two different numerical simulation codes (the SPLIT/NE code and the PoliTo code) are examined in this paper by comparing results for non-equilibrium flow cases using two different geometries, a hemisphere-cylinder-flare (axisymmetric) and a hyperbola-flare-cylinder configuration (axisymmetric and 2-D plane). The solution methods implemented in the codes are essentially different (finite difference, bow shock fitting, matrix splitting and Runge-Kutta high accuracy schemes for SPLIT/NE and finite volume, shock capturing, flux difference splitting and E.N.O. high accuracy schemes for PoliTo). The flow simulations are done employing physical models of different complexity (inviscid and viscous flows, chemically reacting flows in vibrational equilibrium and non-equilibrium, steady and unsteady flows). Comparing the solutions for the different codes, a very good agreement is obtained for both the hemisphere-cylinder-flare flow and the 2-D plane hyperbola-flare-cylinder flow. For relatively small flare angles, this is also true for the axisymmetric hyperbola-flare-cylinder flow; however, for larger flare angles, differences in the solutions can be detected which can be addressed to the formation of a strong recirculation bubble behind the detached flare shock.

Parallel, scalable parabolized Navier-Stokes solver for large-scale simulations

A massively parallel formulation for the solution of the parabolized Navier-Stokes equations has been developed for multiple instruction, multiple data (MIMD) computers. All functionality of the serial version of the code has been preserved in the parallel implementation, including grid generation, linear system solvers, and shock fitting. The computational domain is automatically decomposed and load balanced on individual processing elements. Performance timings are carried out for various MIMD architectures. Numerical tests indicate high parallel efficiency, with fixed solution time as the computational work is scaled with the number of processors. 13 refs.

Three-dimensional flow computations with shock fitting

The present paper provides a shock-fitting technique for solving inviscid transonic three-dimensional (3-D) flows. The continuous flow field is computed by means of an implicit fast Euler solver, which separately integrates compatibility conditions, written in terms of generalized Riemann variables along appropriate bicharacteristic lines. The continuous 3-D flow problem is thus reduced to a sequence of simple quasi 1-D problems. The shock wave is computed by means of a shock-fitting technique, which enforces the proper shock jumps by an explicit use of the Rankine-Hugoniot equations. The computed shock thus develops into a discontinuity as it is in reality. The merits of the present approach are demonstrated by means of a few simple applications and by comparison with corresponding results computed using a flux-difference splitting methodology.

Hypersonic viscous flow simulations

Computational work in continuum hypersonics and, in particular, the difficulties encountered during the development of the methods at the DLR Institute for Theoretical Fluid Mechanics are presented. Finite-difference schemes with implicit-explicit central discretisation and shock fitting and with upwind-TVD formulation and shock capturing are discussed, e.g., for strong shock-shock interactions. 3D flows of perfect gas at large angle of attack are considered, including those with radiation-adiabatic surface boundary conditions. 2D or axisymmetric as well as 3D flows in thermochemical equilibrium are simulated using either the Tannehill curve-fitting approach for air or the minimization procedure for Gibbs' free energy which is more general at the expense of larger cost. Nonequilibrium flow simulations are discussed for axisymmetric nitrogen and air model flows. With the help of an appropriate model equation for finite-rate chemistry it is shown that the usual Harten-Yee approach does not allow to propagate shock waves in a time-accurate fashion if large source terms are present.

Multiple Shock Compaction of Simple Type Powders by Punch Impact

Recently, we have elucidated some mechanical behaviors of powders during the compaction. The elucidation involves the constitutive relations of a powder medium under the multishock compaction, the qualitative behavior such as the similarities of the compaction processes, the die wall friction effect, and the uniformity of the final density distribution of the compact with a high density, and the quantitative behavior analyzed by the pseudo-viscosity method and the shock fitting. This review describes this behavior systematically.

Shock Fitting of Aggradational Profiles Due to Backwater

Depositional deltas form in the headwaters of most reservoirs. The deltas are characterized by a developing foreset slope that eventually approaches the submerged angle of repose. Deltas extend both upstream and downstream; downstream growth seriously depletes reservoir storage, while upstream evolution raises local ground‐water levels and increases flood frequency. Flow separates as it passes downstream over the delta lip into the deeper part of the reservoir. Because of this flow separation and attendant recirculation, traditional finite difference modeling approaches are invalid near the steep foreset slope and cannot model delta growth accurately. A method of numerically fitting a vertical shock face to the evolving delta is developed and illustrated. Conditions upstream of the shock are described with the traditional St. Venant equations; downstream conditions are constant. A one‐dimensional mobile‐bed computer model is developed and compared to a simulated reservoir in a laboratory flume. The simulated delta closely matches the growth and propagation rate of the observed delta.

Fresh look at floating shock fitting

A fast implicit upwind procedure for the two-dimensional Euler equations is described that allows accurate computations of shocked flows on nonadapted meshes. Away from shocks, the second-order accurate upwinding is based on the split-coefficient-matrix (SCM) method. In the presence of shocks, the difference stencils are modified using a floating shock fitting technique. Rapid convergence to steady-state solutions is attained with a diagonalized approximate factorization (AF) algorithm. Results are presented for Riemann's problem, for a regular shock reflection at an inviscid wall, for supersonic flow past a cylinder, and for a transonic airfoil. All computed shocks are ideally sharp and in excellent agreement with other numerical results or 'exact' solutions. Most importantly, this has been accomplished on unusually crude meshes without any attempt to align grid lines with shock fronts or to cluster grid lines around shocks.

Extension of Transonic Flow Computational Concepts in the Analysis of Cavitated Bearings

In this paper, an analogy between the mathematical modeling of transonic potential flow and the flow in a cavitating bearing is described. Based on the similarities, characteristics of the cavitated region and jump conditions across the film reformation and rupture fronts are developed using the method of weak solutions. The mathematical analogy is extended by utilizing a few computational concepts of transonic flow to numerically model the cavitating bearing. Methods of shock fitting and shock capturing are discussed. Various procedures used in transonic flow computations are adapted to bearing cavitation applications, e.g., type differencing, grid transformation, an approximate factorization technique, and Newton’s iteration method. These concepts have proved to be successful and have vastly improved the efficiency of numerical modeling of cavitated bearings.

Comparison of inviscid and viscous separated flows

INVISCID compressible flow separation on simple two-dimensional configurations has been investigated by solving the Euler equations numerically, using different computational schemes. The results were compared with numerical solutions of the Navier-Stokes equations. The computational schemes used in this study included a lambda scheme with shock fitting, a flux-vector splitting scheme for the Euler equations, and a flux-vector splitting scheme for the Navier-Stockes equations

A mixed‐variable continuously deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase‐change in porous media

AbstractThe conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed‐variable finite element computational scheme for a single evolution equation was presented in Part I.1 In this paper (Part II), we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two‐dimensional transport problems governed by a system of coupled non‐linear partial differential evolution equations. In particular, we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts, and abrupt, discontinuous, moving phase‐change interfaces. In this method, the potentials, such as the temperature, pressure and species concentration, and the corresponding fluxes, are permitted to jump in value across the phase‐change interfaces. The equations, and the jump conditions, governing the physical phenomena, which were specialized from a general multiphase, multiconstituent mixture theory, provided the basis for the development and implementation of a two‐dimensional numerical simulator. This simulator can effectively resolve steep continuous fronts (i.e. shock capturing) without oscillations or numerical dispersion, and can accurately represent and track discontinuous fronts (i.e. shock fitting) through adaptive grid deformation and redistribution. The numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in Part III2 of this paper.

Calculation of unsteady flows in turbomachinery using the linearizedEuler equations

A method for calculating unsteady flows in cascades is presented. The model, which is based on the linearized unsteady Euler equations, accounts for blade loading shock motion, wake motion, and blade geometry. The mean flow through the cascade is determined by solving the full nonlinear Euler equations. Assuming the unsteadiness in the flow is small, then the Euler equations are linearized about the mean flow to obtain a set of linear variable coefficient equations which describe the small amplitude, harmonic motion of the flow. These equations are discretized on a computational grid via a finite volume operator and solved directly subject to an appropriate set of linearized boundary conditions. The steady flow, which is calculated prior to the unsteady flow, is found via a Newton iteration procedure. An important feature of the analysis is the use of shock fitting to model steady and unsteady shocks. Use of the Euler equations with the unsteady Rankine-Hugoniot shock jump conditions correctly models the generation of steady and unsteady entropy and vorticity at shocks. In particular, the low frequency shock displacement is correctly predicted. Results of this method are presented for a variety of test cases. Predicted unsteady transonic flows in channels are compared to full nonlinear Euler solutions obtained using time-accurate, time-marching methods. The agreement between the two methods is excellent for small to moderate levels of flow unsteadiness. The method is also used to predict unsteady flows in cascades due to blade motion (flutter problem) and incoming disturbances (gust response problem).

A finite volume scheme with shock fitting for the steady Euler equations

Analysis of two alternative finite volume formulations, in respect of accuracy on non-uniform meshes and number of spurious modes, leads to a preference for the more compact cell vertex scheme over the cell centre scheme. The resulting equations are solved iteratively by using a Lax-Wendroff procedure as a smoother for a multigrid algorithm: then application of boundary conditions in a natural way leads rapidly to all individual residuals being driven close to zero—except at shocks. At shocks the residuals should not be zero and a shock-fitting procedure is introduced to avoid this inconsistency. Sharp, accurate solutions on a relatively coarse mesh are obtained for a channel (low problem in which the Zierep singularity is displayed, and for the NACA 0012 aerofoil.

Shock Embedding Discontinuous Solution of Elliptic Equation for Inverse Problem of Transonic S2 Flow

In the case of the inverse problem of S2 surface flow in a transonic turbomachine, if the meridional component of the relative velocity is subsonic, the equation governing the flow is elliptic. Through the use of a proper conservative form of the stream-function principal equation and embedding the shock relations into the principal equation, the transonic flow over the whole S2 surface containing the discontinuity of the passage shock can be calculated. A computer code employing this method has been programmed. The algorithm, which is only a little different from the subsonic code, is simple, accurate, and reliable. This program is particularly useful in the solution of the three-dimensional transonic flow in a fan or compressor through iterative computation of transonic S1 and S2 flows, when the former is carried out by the recently developed method of “separate-region calculation with shock fitting.” The new computer code is used to calculate the S2 flow in a typical transonic rotor and the difference between the result obtained by this method and that by the commonly used one, in which the shock discontinuity is not taken into account, can be clearly seen.

Efficient Euler solver with many applications

A computational technique for two-dimensional unsteady Euler equations, based on a A-formulation and explicit shock fitting, is presented. Any number of shocks, of any shape and type, and their interactions can be treated by this technique, which does not require complicated logic and does not perform redundant calculations. The code is fast and the results are very accurate. Examples (transonic airfoils, shocks in ducts, intake flows, multiple Mach reflections) are presented and discussed.

Kinetic energy distribution of ions generated by laser ionization sources

The time of flight (TOF) method has been utilized to measure kinetic energy distributions of ions generated by laser ionization sources. Measurements were carried out on carbon foil targets with pulses from a Q-switched ruby laser in the power density range from 10 6 to 10 10 W cm −2. the onset of plasma formation and appearance of positive and negative energy ions were observed. Interference of neighbouring lines could be resolved by decreasing the light intensity down to the plasma formation threshold. The energy spectrum consisted of a positive energy peak and negative energy tail centered around the ideal symmetrical peak. The appearnace of these characteristics depended on the laser power density applied. The positive energy part of the energy spectrum was related to the shock wave due to laser heating of the foil while the negative energy part could be explained by post-ionization of neutrals in the accelerating region. Shock fitting according to Godunov served as a rough estimate for plasma hydrodynamics, while an optical and charge transfer model as a possible description of the negative part of the energy spectrum were discussed. Encouraging correlations between measured and calculatd data were found.

The importance of being earnest about shock fitting

To identify observed plasma and magnetic field discontinuities in interplanetary space as magnetohydrodynamic shocks requires fitting the plasma and magnetic field data to Rankine‐Hugoniot conditions. By statistics and specific samples we show that unless the fluctuations in the observed parameters in the preshock and postshock regions are seriously taken into consideration, careful fitting may still lead to erroneous identification and description of a shock. An approach to the unbiased choice of the preshock and postshock values is to systematically consider all possible combinations of preshock and postshock parameters averaged over different time intervals measured away from the shock.

Three distinct stages of the unsteady flow behind the shock wave formed by the normal reflection of a planar strong point blast wave from a wall

The method of characteristics with shock fitting and a family of continuous C-characteristic lines are used for the analysis of the normal reflection of a planar strong point blast wave from a wall. Several techniques are devised in order to increase the accuracy of calculation for the problem including two complex singularities. New data for the flow field near the wall till very late time are obtained. Descriptions are given for three stages, each possessing distinct gasdynamic features such as the orientation of characteristics, paths of fluid particles, distribution of entropy, and so on.

The Convergence of Upstream Collocation in the Buckley-Leverett Problem

Abstract Upstream collocation is a fast and accurate scheme for simulating multiphase flows in oil reservoirs. In contrast to standard orthogonal collocation, upstream collocation yields numerical solutions to the Buckley-Leverett problem that converge to correct solutions physically. The failure of standard orthogonal collocation is not surprising, since the Buckley-Leverett problem as commonly stated is posed incompletely. The equal-area rule of Buckley and Leverett and the Welge tangent construction both specify additional constraints needed to close the problem properly. An error analysis of upstream collocation shows that this method forces convergence through an artificial dissipative term analogous to the "vanishing viscosity" used in shock fitting. This constraint is mathematically equivalent to the more familiar constructions and should prove beneficial in stimulating EOR schemes based on frontal displacement.

Transonic Cascade Flow Solved by Separate Supersonic and Subsonic Computations With Shock Fitting

A new method to solve the transonic flow past a plane cascade with inlet bow wave and a passage shock extending from the leading edge all the way across the flow channel is presented. In this method, different algorithms are used for supersonic and subsonic flow regions, respectively, and the aerothermodynamic relations of a connecting passage shock are satisfied. Within the supersonic region, the method of characteristics is used. For calculation of the subsonic flow, the stream function equation expressed with respect to nonorthogonal curvilinear coordinates and nonorthogonal velocity components is solved with a matrix method. By the use of Rankine-Hugoniot relations, the passage shock can be accurately determined and a definite jump of the aerothermodynamic quantities across the passage shock is obtained. All calculations involved are programmed and incorporated into a single code. The computational procedure is straightforward and simple, and the computational time is short. Some results of typical applications of this method are given in this paper. Agreement of the numerical results with the experimental data, including the location and shape of the passage shock wave, is very good.

Discussion: “Transonic Cascade Flow Solved by Separate Supersonic and Subsonic Computations With Shock Fitting” (Wu, Wenquan, Wu, Chung-Hua, and Yu, Dabang, 1985, ASME J. Eng. Gas Turbines Power, 107, pp. 329–335)

Discussion: “Transonic Cascade Flow Solved by Separate Supersonic and Subsonic Computations With Shock Fitting” (Wu, Wenquan, Wu, Chung-Hua, and Yu, Dabang, 1985, ASME J. Eng. Gas Turbines Power, 107, pp. 329–335)