Diffraction of normal shock wave by yawed wedges

The diffraction of normal shock with yawed wedges of small angles have been considered in this paper. Vorticity distribution of the fluid particle over the diffracted shock has been determined for several Mach numbers of the shock wave. The Mach reflection effects have also been investigated when the bend is concave to the flow.     

反射型激波风洞中激波与边界层的相互作用

本文研究了反射型激波风洞中由于非完全反射对激波与壁面边界层相互作用的影响,给出了在反射激波坐标系中计算边界层速度分布、温度分布和马赫数分布的计算方法.算例表明,在计及氮气的平衡真实气体效应的情形下,随着入射激波马赫数M_s的增大,边界层的最小马赫数从壁面处移到边界层内;随着喷管喉道面积的增大,边界层的最小马赫数、反射激波的分叉角α和分叉区后的射流速度均随之减小.计算结果与实验值相比是一致的.     

菱形口射流与超音主流的相互作用

基于在不同射流角(10°, 27.5°, 45°, 90°)和射流总压(0.1 MPa, 0.46 MPa)下,对音速次膨胀射流通过菱形口喷射到马赫5横穿主流的实验及圆形射流器的对比实验,研究了次膨胀射流与超音横穿主流相互作用流场, 实验包括横截面流场的Pitot和锥静压力, 获得横截面马赫数、 压力分布．结果表明近壁面低马赫数半圆区为尾区,尾区附近边界层减薄．脱体激波高度向自由流扩展,激波形状更弯曲, 低马赫数区较大．高射流压力和射流角增加羽流涡度,激波位置较高．90°菱形和圆形喷射器有更强的羽流涡度,但圆形喷射器的低马赫数区较小．前沿渐细的变壁面的斜面物增加羽流涡度,反之,双变壁面的斜面物减弱羽流涡度．通过表面激波形状、中心平面激波及横截面激波模化三维激波形状,激波总压损失用正激波关系式通过马赫数法向分量估计．激波总压损失随射流角和动压比的减小而减小,最大损失发生在90°圆形和菱形喷射器．     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

The structure of strong shock waves using the Fokker-Planck model

The structure of strong shock waves in monatomic gases is studied using the Fokker-Planck model to represent the particle collisions and the Mott-Smith distribution to describe the distribution function within the shock front. The differential equation governing the variation of the density within the shock is derived by using the variational principle. The thickness of the shock front is evaluated numerically for various monatomic gases for Mach numbers varying from 2 to 20, and besides, the variation of the shock thickness with viscosity is also studied for different gases. Several parameters of physical interest within the shock, such as density, temperature and mean velocity of flow are evaluated numerically and detailed curves showing their variation within the shock are presented for different Mach numbers. It is found that the temperature rises very steeply, reaches a maximum within a distance less than half the thickness of the shock and then diminishes slowly to attain its asymptotic downstream values. The variation of the mean velocity is slow for weak shocks, but for higher Mach numbers, the mean velocity diminishes steeply and reaches the downstream values within half the thickness of the shock.     

Shock Wave Structure in a Rarefied Polyatomic Gas Based on Extended Thermodynamics

A theory of the shock wave structure in a rarefied polyatomic gas is developed on the basis of the recent new approach to extended thermodynamics. We summarize the following points (i) and (ii) based on the previous study on this subject and also show the new point (iii): (i) The theory can explain the change of types of the shock wave structure observed experimentally with the increase of the Mach number from unity; the nearly symmetric shock wave structure (Type A, small Mach number), the asymmetric structure (Type B, moderate Mach number), and the structure composed of thin and thick layers (Type C, large Mach number). (ii) The theoretical prediction of the mass density profile agrees well with experimental data. (iii) The points (i) and (ii) are not strongly affected by the details of the temperature dependence of the bulk viscosity.     

Stability of a Mach configuration

We prove the stability of a Mach configuration, which occurs in shock reflection off an obstacle or shock interaction in compressible flow. The compressible flow is described by a full, steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic regions, and other two are subsonic regions. We prove that if the constant states in the supersonic regions are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic regions, are also slightly perturbed. Such a conclusion asserts the existence and stability of the general Mach configuration in shock theory. In order to prove the result, we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundaries, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined, we have to deal with an elliptic‐hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equations and elliptic equations, we establish the required estimates, which are crucial in the proof of the existence of a solution to the free boundary value problem. © 2005 Wiley Periodicals, Inc.     

Dissociation effects in hypersonic flow past a circular cone at an angle of attack

The analytical solution to the steady, compressible, non-viscous, inviscid hypersonic flow past a circular cone at an angle of incidence, with an attached Shockwave, in the presence of dissociation of air in the shock layer, has been obtained here under the assumption of thermal equilibrium. Expression for the velocity, pressure, temperature, density, velocity of air, Mach number, pressure, drag and lift coefficients have been obtained both in the shocklayer outside the vortical layer and on the surface of the cone inside the vortical layer.     

Mach configuration in pseudo-stationary compressible flow

This paper is devoted to studying the local structure of Mach reflection, which occurs in the problem of the shock front hitting a ramp. The compressible flow is described by the full unsteady Euler system of gas dynamics. Because of the special geometry, the motion of the fluid can be described by self-similar coordinates, so that the unsteady flow becomes a pseudo-stationary flow in this coordinate system. When the slope of the ramp is less than a critical value, the Mach reflection occurs. The wave configuration in Mach reflection is composed of three shock fronts and a slip line bearing contact discontinuity. The local existence of a flow field with such a configuration under some assumptions is proved in this paper. Our result confirms the reasonableness of the corresponding physical observations and numerical computations in Mach reflection.

In order to prove the result, we formulate the problem to a free boundary value problem of a pseudo-stationary Euler system. In this problem two unknown shock fronts are the free boundary, and the slip line is also an unknown curve inside the flow field. The proof contains some crucial ingredients. The slip line will be transformed to a fixed straight line by a generalized Lagrange transformation. The whole free boundary value problem will be decomposed to a fixed boundary value problem of the Euler system and a problem to updating the location of the shock front. The Euler system in the subsonic region is an elliptic-hyperbolic composite system, which will be decoupled to the elliptic part and the hyperbolic part at the level of principal parts. Then some sophisticated estimates and a suitable iterative scheme are established. The proof leads to the existence and stability of the local structure of Mach reflection.

     

Control of the transition between regular and mach reflection of shock waves

A control problem was considered that makes it possible to switch the flow between stationary Mach and regular reflection of shock waves within the dual solution domain. The sensitivity of the flow was computed by solving adjoint equations. A control disturbance was sought by applying gradient optimization methods. According to the computational results, the transition from regular to Mach reflection can be executed by raising the temperature. The transition from Mach to regular reflection can be achieved by lowering the temperature at moderate Mach numbers and is impossible at large numbers. The reliability of the numerical results was confirmed by verifying them with the help of a posteriori analysis.     

激波在异种气体中传播及诱导的剪切混合研究

利用二阶迎风TVD格式求解多组分,层流全N－S方程,针对直通道和突扩直通道,研究了马赫数为2和4的激波在H2和空气界面上的传播及诱导的燃料剪切混合,计算结果表明：（1）直通道中,剪切层中的激波阵面要发生畸变,存在对混合起主要作用的卷吸涡,激波马赫数不同,卷吸涡结构和横向混合的尺寸也不同,激波马赫数低,剪切混合效果好,（2）在突扩直通道中,马赫数为2和4的激波在H2中产生不同强度激波,在剪切层中都能产生顺时针,尺度较大的卷吸涡,后台阶增强了剪切层的混合。     

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with $h$-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.     

Global existence and stability of a stationary Mach reflection

We prove the global existence and stability of a wave structure containing a stationary Mach configuration,which occurs when an incident shock front hits a wall with a large incident angle.Our result shows that the data of the upstream flow and the pressure at downstream part jointly determine the whole flow,as well as the wave structure.Particularly,we show that the height of the Mach stem depends not only on the data of upstream flow,but also on the pressure at downstream flow.The flow with the assigned wave structure is governed by a free boundary value problem for the Euler system.In the problem the location of the triple point,the shock fronts and the contact discontinuity are all unknown,they are finally determined together with the solution.     

Computation of a shock wave structure in monatomic gas with accuracy control

The structure of a shock wave in a monatomic one-component gas was computed by solving the Boltzmann kinetic equation with accuracy controlled with respect to computational parameters. The hard-sphere molecular model and molecules with the Lennard-Jones potential were considered. The computations were performed in a wide range of Mach numbers with the accuracy no less than 3% for the shock front width and 1% for local values of density and temperature. The shock wave structure was studied in terms of macroscopic gas characteristics and in terms of the molecular velocity distribution function.     

Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

Verification of Complex Transonic Phenomena with CFD Codes

Classical analytic models for the theoretical behavior of transonic flows have guided the development of numerical simulation of practically relevant flows. But while operational successfully for the usual applications (conventional configuration high speed aerodynamics), CFD codes fail frequently once detailed information regarding even just inviscid flow structure for freestream Mach number close to unity is needed. In this contribution shock configurations are computed verifying analytical results for both the far field behavior of detached bow waves and the local structure of attaching bow waves. The first example is challenging the ability of a CFD code (DLR‐τ) to adapt its unstructured grid to a weak bow wave detached from the airfoil. Known analytical results for the asymptotic bow wave behavior at decreased Mach number → 1 serve for judging numerical results where practical case studies neither from CFD nor from experiment are available. In the second example the behavior of a bow wave attaching to a wedge for increased Mach number is studied, with grid adaption allowing for a verification of both the singular analytical model when attachment takes place and also the transition to plain supersonic wedge flow.     

Potential theory for regular and mach reflection of a shock at a wedge

If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 2θ_w through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter β = c₁θ²_w(γ + 1)b ranges from 0 to ∞. Here γ is the polytropic constant and C₁ is the sound speed behind the incident shock. For β > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For β < 1/2 Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case β = 0 can be solved. For 1/2 < β < 2 or even larger β the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens. The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for β sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point. © 1994 John Wiley & Sons, Inc.     

Collision of a solar wind shock wave with the Earth’s bow shock. Wave flow pattern

The wave pattern of the flow developed when a solar wind shock wave propagates along the surface of the Earth’s bow shock is studied. The investigation is carried out in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model in which the medium is assumed to be inviscid and non-heat-conducting and to have the infinite conductivity. The global three-dimensional pattern of the interaction which is a function of the latitude and longitude of elements on the surface of the bow shock is constructed as a mosaic of solutions to the problem of breakdown of a discontinuity developed between the states behind the impinging and bow shocks on the moving curve of intersection of their fronts. The investigation is carried out for typical solar wind parameters and interplanetary magnetic field strength in the Earth’s orbit and for several Mach numbers of the interplanetary shock wave, which makes it possible to trace the evolution of the flow developed as a function of the intensity of the shock perturbation of the solar wind. The solution obtained is necessary for interpreting measurements carried out by spacecraft located in the neighborhood of the Lagrange point and the Earth’s magnetosphere.     

Shock structure and multiple sub-shocks in binary mixtures of Eulerian fluids

The problem of sub-shock formation within a shock structure solution of hyperbolic systems of balance laws is investigated for a binary mixture of multi-temperature Eulerian fluids. The main purpose of this work is the analysis of the ranges of Mach numbers characterizing shock-structure solutions with different features, continuous or not, and to show the existence of ranges, below the maximum unperturbed characteristic velocity, for which each constituent of the mixture may develop a sub-shock within a smooth shock structure profile. The theoretical results are supported by numerical calculations.