COMPUTATION OF NONEQUILIBRIUM HYPERSONIC FLOW OVER CONCAVE CORNERS

1. IntroductionWe consider steady inviscid hyPersonic flow of air about a concave corner including chendcalreaction effects. Computation of nonequllibrium flow is duncult because of the steep gradielltsbehind the shock and an entroPy layer near the body For many cases the time scale of theChemical reactions is 1arger than, or close to, the time scale of the origina1 meChanical problemand the equllibrium chendstry mode1 is not realistic. Therefore, it is necessary to eva1uate thekinetics of c…     

The Second-Order Fluid in Cell (FLIC) Method for the One-Dimensional Unsteady Compressible Flow Problems

In this paper we suggest a second-order fluid in cell (FLIC) method for the one-dimensional unsteady compressible flow problems. The numerical result obtained by the present method is compared with the one obtained with the original FLIC method and the exact solution for a shock tube problem.     

Numerical solutions to the self‐similar transonic two‐dimensional nonlinear wave system

We present numerical results on a two‐dimensional Riemann problem governed by the self‐similar nonlinear wave system that gives rise to a transonic shock. We consider a configuration for a vertical incident shock moving to the right above a rectangular object. The incident shock then interacts with a sonic circle soon after it moves beyond the object, and creates a transonic region. We implement Lax–Liu positive schemes and Strang splitting, and obtain several numerical solutions for the model system. With the numerical results that we have obtained, we present several analyses of the transonic shock strengths and the positions of the transonic shocks with various Riemann data. Moreover, due to the presence of the corner of the object, numerical oscillations are apparent. We discuss regularity results for the solution near the corner of the object. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence results for coupled nonlinear systems approximating the rotating MHD flow over a rotating sphere near the equator

We study a coupled nonlinear system of differential equation approximating the rotating MHD flow over a rotating sphere near the equator. In particular, using the Schauder fixed point theorem, we are able to establish existence of solutions. Other results on similar systems show that the question of existence in not obvious and, hence, that the present results are useful. Indeed, the work of McLeod in the 1970s shows some nonexistence results for similar problems. From here, we are also able to discuss some of the features of the obtained solutions. The observed behaviors of the solutions agree well with the numerical simulations present in the literature.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

Approximate formulas for heat flows towards an ideally catalytic surface near a plane of symmetry

The three-dimensional flow of a chemically unstable viscous gas near a plane of symmetry of blunt bodies streamlined at the angle of attack, is considered. The investigation is carried out using a model of a thin, viscous shock layer. To a first approximation of the method of successive approximations for a uniform gas simple formulas are obtained for the distribution of the heat flux over the surface, referred to its value at the stagnation point. It is shown that for a chemically unstable gas the distribution of the heat flux along an ideally catalytic surface depends only slightly on the conditions prevailing within the incident flow, is determined mainly by the geometrical characteristics of the body, and is described quite satisfactorily by the formulas obtained. The accuracy of these formulas is determined by comparison with numerical computations carried out for bodies of various shapes, moving at different angles of attack along a planing trajectory of re-entry into the Earth's atmosphere, and during re-entry into the atmosphere at a constant velocity.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Effect of bulk viscosity in supersonic flow past spacecraft

In this paper, we consider the effect of bulk viscosity in various hydrodynamic problems. We numerically study this effect on the front structure of the one-dimensional stationary shock wave and on the flow past blunt body. We estimate the effect of the bulk viscosity coefficient (BVC) on the heat transfer and drag of a sphere in a supersonic flow, apparently for the first time, by the numerical solution of parabolized Navier–Stokes equations. The solution is obtained by an original fast convergent method of global iterations of the longitudinal pressure gradient. The directions of further investigations of bulk viscosity are suggested.     

粉尘气体中激波绕射的数值研究*

本文针对稀相气固两相体系,选取双流体耦合模型,综合运用算子分裂技术和高精度高分辨率数值方法,研究了激波在粉尘气体中沿90°拐角的绕射特性,揭示了固相颗粒及其物性改变对激波绕射特征和波后流场结构的影响.     

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100 km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.     

Fluid–structure interaction solver for compressible flows with applications to blast loading on thin elastic structures

In this study, we report the development and application of a fluid–structure interaction (FSI) solver for compressible flows with large-scale flow-induced deformation of the structure. The FSI solver utilizes a partitioned approach to strongly couple a sharp interface immersed boundary method-based flow solver with an open-source finite-element structure dynamics solver. The flow solver is based on a higher-order finite-difference method using a Cartesian grid, where it employs the ghost-cell methodology to impose boundary conditions on the immersed boundary. Higher-order accuracy near the immersed boundary is achieved by combining the ghost-cell approach with a weighted least squares error method based on a higher-order approximate polynomial. We present validations for two-dimensional canonical acoustic wave scattering on a rigid cylinder at a low Mach number and for flow past a circular cylinder at a moderate Mach number. The second order spatial accuracy of the flow solver was established in a grid refinement study. The structural solver was validated according to a canonical elastostatics problem. The FSI solver was validated based on comparisons with published measurements and simulations of the large-scale deformation of a thin elastic steel panel subjected to blast loading in a shock tube. The solver correctly predicted the oscillating behavior of the tip of the panel with reasonable fidelity and the computed shock wave propagation was qualitatively consistent with the published results. In order to demonstrate the fidelity of the solver and to investigate the coupled physics of the shock–structure interaction for a thin elastic plate, we employed the solver to simulate a 6.4 kg TNT blast loading on the thin elastic plate. The initial conditions for the blast were taken from previously reported field tests. Using numerical schlieren, the shock front propagation, Mach reflection, and vortex shedding at the tip of the plate were visualized during the impact of the shock wave on the plate. We discuss the coupling between the nonlinear dynamics of the plate and blast loading. The plate oscillates under the influence of blast loading and the restoration of elastic forces. The time-varying displacement of the tip of the plate is the superimposition of two dominant frequencies, which correspond to the first and second modes of the natural frequency of a vibrating plate. The effects of the material properties and length of the plate on the flow-induced deformation are briefly discussed. The proposed FSI solver is a versatile computational tool for simulating the impact of a blast wave on thin elastic structures and the results presented in this study may facilitate the design of thin structures subjected to realistic blast loadings.     

带激波的一维非定常两相流动的数值模拟

本文将处理带激波的单相气体非定常流动问题的两种高分辨数值方法(随机取样法和二阶GRP有限差分法)推广应用于气固悬浮体(亦称含灰气体)两相情况,计算了含灰气体激波管中两相激波特性、波后流场结构及气固两相流动参数随时间的变化.数值结果表明:这两种方法均能给出带有尖锐间断阵面的两相激波松弛结构.二阶GRP方法在计算精度和机时耗用等方面优于随机取样法.     

Theory of the hypersonic viscous shock layer at high Reynolds numbers and intensive injection of foreign gases : PMM vol. 38, n 6, 1974, pp. 1015–1024

The hypersonic flow around smooth blunted bodies in the presence of intensive injection from the surface of these is considered. Using the method of external and internal expansions the asymptotics of the Navier-Stokes equations is constructed for high Reynolds numbers determined by parameters of the oncoming stream and of the injected gas. The flow in the shock layer falls into three characteristic regions. In regions adjacent to the body surface and the shock wave the effects associated with molecular transport are insignificant, while in the intermediate region they predominate. In the derivation of solution in the first two regions the surface of contact discontinuity is substituted for the region of molecular transport (external problem). An analytic solution of the external problem is obtained for small values of parameters ₁ = ρ_∞/ρs^* and δ = ρ_ω^*1/2ν_ω^*/ρ_∞1/2ν_∞, in the form of corresponding series expansions in these parameters. Asymptotic formulas are presented for velocity profiles, temperatures, and constituent concentration across the shock layer and, also, the shape of the contact discontinuity and of shock wave separation. The derived solution is compared with numerical solutions obtained by other authors. The flow in the region of molecular transport is defined by equations of the boundary layer with asymptotic conditions at plus and minus infinity, determined by the external solution (internal problem). A numerical solution of the internal problem is obtained taking into consideration multicomponent diffusion and heat exchange. The problem of multicomponent gas flow in the shock layer close to the stagnation line was previously considered in [1] with the use of simplified Navier-6tokes equations.The supersonic flow of a homogeneous inviscid and non-heat-conducting gas around blunted bodies in the presence of subsonic injection was considered in [2–7] using Euler's equations. An analytic solution, based on the classic solution obtained by Hill for a spherical vortex, was derived in [2] for a sphere on the assumption of constant but different densities in the layers between the shock wave and the contact discontinuity and between the latter and the body. Certain results of a numerical solution of the problem of intensive injection at the surface of axisymmetric bodies of various forms, obtained by Godunov's method [3], are presented. Telenin's method was used in [4] for numerical investigation of flow around a sphere; the problem was solved in two formulations: in the first, flow parameters were determined for the whole of the shock layer, while in the second this was done for the sutface of contact discontinuity, which was not known prior to the solution of the problem, with the pressure specified by Newton's formula and flow parameters determined only in the layer of injected gases. The flow with injection over blunted cones was numerically investigated in [5] by the approximate method proposed by Maslen. The flow in the shock layer in the neighborhood of the stagnation line was considered in [6, 8], and intensive injection was investigated by methods of the boundary layer theory in [8–12].     

Discrete Lagrangian algorithm for finding geodesics on triangular meshes

The present paper introduces an approximation method for finding open geodesics on triangular surfaces. The algorithm is specifically designed to be able to solve real world problems where geodesic paths are needed. We use the model of geodesic curvature flow for open curves in the Lagrangian formulation. The model is enriched with a tangential term in order to have a control over the quality of the discretization grid during the computation. The governing equation of the flow is solved by a numerical method based on a semi-implicit time discretization and a finite difference space discretization. The paper presents the numerical scheme and various implementation details as well as numerous experiments to demonstrate the performance of the method and to provide comparison with several other well known methods. We also present a Grasshopper component for Rhinoceros for finding optimal paths on surface meshes that we developed and that includes our algorithm.     

Fourth order accurate compact scheme with group velocity control (GVC )

For solving complex flow field with multi-scale structure higher order accurate schemes are preferred. Among high order schemes the compact schemes have higher resolving efficiency. When the compact and upwind compact schemes are used to solve aerodynamic problems there are numerical oscillations near the shocks. The reason of oscillation production is because of non-uniform group velocity of wave packets in numerical solutions. For improvement of resolution of the shock a parameter function is introduced in compact scheme to control the group velocity. The newly developed method is simple. It has higher accuracy and less stencil of grid points     

基于子网格边界近似Riemann解的人为粘性方法

在交错网格型Lagrange(拉格朗日)流体力学算法中,通常采用人为粘性捕捉激波,人为粘性的好坏对于计算结果至关重要.研究了一种基于子网格边界处近似Riemann解的新型人为粘性.新人为粘性能够满足动量守恒和熵不等式.利用子网格边界速度差中引入的限制器,新人为粘性能够区分激波和等熵压缩,并能满足球对称问题中的波面不变性.新人为粘性在典型模型数值模拟及惯性约束聚变黑腔整体数值模拟中取得了较好的结果.     

Continuum models of rarefied gas flows in problems of hypersonic aerothermodynamics

The limits of applicability of continuum flow models in the problem of the hypersonic rarefied gas flow over blunt bodies are determined by an asymptotic analysis of the Navier–Stokes equations, the numerical solution of the viscous shock layer equations and the numerical and asymptotic solution of the thin viscous shock layer equations for low Reynolds numbers. It is shown that the thin viscous shock layer model gives correct values of the skin friction coefficient and the heat transfer coefficient in the transitional to free-molecule flow regime. The asymptotic solutions, the numerical solutions obtained within the framework of different continuum models, and the results of a calculation by Direct Simulation Monte Carlo method are compared.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.     

Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface

In the present analysis, we study the steady mixed convection boundary layer flow of an incompressible Maxwell fluid near the two-dimensional stagnation-point flow over a vertical stretching surface. It is assumed that the stretching velocity and the surface temperature vary linearly with the distance from the stagnation-point. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. Analytical and numerical solutions of the derived system of equations are developed. The homotopy analysis method (HAM) and finite difference scheme are employed in constructing the analytical and numerical solutions, respectively. Comparison between the analytical and numerical solutions is given and found to be in excellent agreement. Both cases of assisting and opposing flows are considered. The influence of the various interesting parameters on the flow and heat transfer is analyzed and discussed through graphs in detail. The values of the local Nusselt number for different physical parameters are also tabulated. Comparison of the present results with known numerical results of viscous fluid is shown and a good agreement is observed.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.