论粘性激波层理论的理论基础及其进一步发展

本文将Davis的量级分析方法改用匹配渐近展开方法,作为一级近似推导出了高超音速化学反应粘性激波层方程。证明了粘性激波层方程是NS方程在匹配渐近意义上的一级近似方程。进一步讨论了这一方程的基本假设条件。本文首次推导出的二级近似方程,是对Davis粘性激波层方程的修正。这种修正可以提高数值解的精度,有助于对问题获得更全面的了解,对进一步发展与完善高超音速钝头体绕流问题的数值求解方法起一定的作用。     

An upwind compact mpact approach with group velocity control for compressible flow fields

In this paper we construct an upwind compact finite difference scheme with group velocity control for better simulation of compressible flow fields. Compared with traditional difference schemes, compact schemes have higher accuracy for the same stencil width. By means of the characteristic analysis of the operators, the group velocity of wave packets will be controlled to suppress the non‐physical oscillations in numerical solutions. In numerical simulation of the 3D compressible flow fields the third‐order accurate upwind compact operator is used to approximate the derivatives in the convection terms of the compressible N–S equations, the traditional finite difference scheme is used to approximate the viscous terms. Numerical solutions indicate that the method is satisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical simulation of boundary layer effects in a shock tube

A numerical scheme is used to investigate boundary layer effects in a shock tube. The method consists of a mixture of Roe's approximate Riemann solver and central differences for the convective fluxes and central differences for the viscous fluxes and is implicit in one space dimension. Comparisons are made with experimental data and with solutions obtained via boundary layer equations. Examination of the calculated flow field explains the observed behaviour and highlights the approximate nature of boundary layer solutions.     

Calculation of supersonic viscid flow near stagnation line of a blunt body

A numerical study is made of supersonic flow of a viscous gas in the vicinity of the stagnation line of plane and axisymmetric blunt bodies (cylinder, sphere). As in [1–5], which consider the compressed layer of a viscous gas in the vicinity of the stagnation point, use is made of the locally self-similar approximation, which is used to transform the Navier-Stokes equations into a system of ordinary differential equations. In the present paper the solution is sought with the simplifications of [5] and with more general conditions, which makes it possible to study a broad class of flows. The proposed numerical algorithm permits obtaining the structure of the compressed layer near the stagnation line, including the shock wave and the boundary layer. The calculations made on a computer for different flow conditions are illustrated by graphs.The author wishes to thank G. I. Petrov, G. F. Telenin, and L. A., Chudov for their interest in the study and for their helpful discussions. discussions.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Improvements to compressible Euler methods for low‐Mach number flows

In the present study improvements to numerical algorithms for the solution of the compressible Euler equations at low Mach numbers are investigated. To solve flow problems for a wide range of Mach numbers, from the incompressible limit to supersonic speeds, preconditioning techniques are frequently employed. On the other hand, one can achieve the same aim by using a suitably modified acoustic damping method. The solution algorithm presently under consideration is based on Roe's approximate Riemann solver [Roe PL. Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics 1981; 43 : 357–372] for non‐structured meshes. The numerical flux functions are modified by using Turkel's preconditioning technique proposed by Viozat [Implicit upwind schemes for low Mach number compressible flows. INRIA, Rapport de Recherche No. 3084, January 1997] for compressible Euler equations and by using a modified acoustic damping of the stabilization term proposed in the present study. These methods allow the compressible Euler equations at low‐Mach number flows to be solved, and they are consistent in time. The efficiency and accuracy of the proposed modifications have been assessed by comparison with experimental data and other numerical results in the literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Hypersonic viscous shock layer on sweptback wings of infinite span at different angles of attack

A study is made of the flow of a viscous compressible gas in a hypersonic shock layer on sweptback wings of infinite span with blunt leading edge at different angles of attack. The equations of the hypersonic viscous shock layer with modified Rankine-Hugoniot relations across the shock wave and boundary conditions on the surface of the body that take into account slip and discontinuity of the temperature are solved by a method of successive approximation which yields not only an analytic solution for the first approximations but also an exact numerical solution when the method is implemented on a computer. The analytic solution of the problem is found in the first approximation. Expressions are obtained for the coefficients of friction and heat transfer on the surface of the body, and also for the profiles of the velocities and the temperature across the shock layer. Comparison of the analytic solution with the numerical solution reveals a satisfactory accuracy of the analytic solution for not too large Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 91–102, March–April, 1979.We thank G. A. Tirskii for his interest in the work and valuable discussions.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Effect of a shock wave on heat propagation

The nature of the propagation of a thermal wave produced by a powerful explosion was described in a number of papers, for example, [1–6]. It was shown by a numerical method [4] that a shock wave is present together with the thermal wave. In this paper, the effect of a homothermal shock wave on heat propagation is evaluated by an approximate method.     

The propagation of shock waves in viscous media

The question of the thickness of shock waves in a viscous gas was treated in papers [1, 2]. The present paper derives general equations for solving problems concerning the flow of a medium inside a shock wave layer, and the change of this layer in viscous media. By way of an example we consider a problem of this type for a Kelvin medium.     

Stagnation region of a blunt body in two-phase hypersonic flow

The fluid-mechanics equations of a two-velocity, two-temperature medium are used to investigate flow near the stagnation point of a blunt body washed by a hypersonic stream of gas containing solid or liquid deformed particles. The effect of particles of the gasdynamic flow parameters is analyzed. A relaxation layer was found to occur near the body, with marked changes in the gas parameters. It is shown that the presence of particles in the flow reduces the shock stand-off distance. The results of computations on the dynamics and heating of particles in the shock layer are discussed. A solution in finite form is obtained in the limiting case of fine particles by the method of asymptotic expansions. The motion of solid or liquid particles in hypersonic shock layers has been the subject of several papers [1–6], in which particle dynamics was examined, assuming that the particles have a negligible influence on the gasdynamic flow parameters. The solutions obtained are therefore limited to the case of low mass particle concentration in the incident flow. A numerical solution not subject to this limitation was obtained in [7] for supersonic two-phase flow over a wedge.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Flow over blunt cones during entry into an atmosphere of carbon dioxide and nitrogen

An investigation has been made of hypersonic flow over spherically blunted cones in an atmosphere consisting of carbon dioxide and nitrogen. Local thermodynamic equilibrium is assumed in the shock layer. Account is taken of viscosity, diffusion, heat conduction, and radiative energy transport. The problem is solved using equations for dynamics of a viscous radiating gas without isolating inviscid flow and boundary-layer regions in the shock layer. The selectivity of the radiation is allowed for by using a two-stage approximation for the spectral dependence of the absorption coefficient, obtained by processing detailed data on absorption cross sections. The solution is found by a flow establishment method. Results are presented for flow over blunt cones with different semiangles.     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

Axisymmetric analogy for three-dimensional viscous flow problems

A method of solving three-dimensional flow problems with the aid of two-dimensional solutions, which can be used for any Reynolds numbers, is proposed. The method is based on the use of similarity relations obtained in the theoretical analysis of the approximate analytic solution of the equations of a three-dimensional viscous shock layer. These relations express the heat flux and the friction stress on the lateral surface of a three-dimensional body in terms of the values on the surface of an axisymmetric body. The accuracy is estimated by comparing the results with those of a numerical finite-difference calculation of the flow past bodies of various shapes. Similar similarity relations were previously obtained in [1] for the plane of symmetry of a blunt body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–118, November–December, 1991.The authors are grateful to G. A. Tirskii for his interest in their work.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.