Shock wave formation as a result of interaction with a weak discontinuity on the boundary of a local subsonic zone

A self-similar solution, which explains the formation of a strong-family shock wave (Mach number behind the wave less than unity) on the sonic line, is obtained for the Tricomi equation of plane potential flow in hodograph variables. A characteristic with a discontinuity of the derivatives of the gas dynamic parameters arrives at the formation (interaction) point, while the characteristic of the other family leaving this point does not contain a singularity. The intensity of the shock wave varies along its generator in accordance with a power law with an exponent close to unity. At the interaction point the discontinuity of the derivatives along the streamline is equal to infinity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 152–158, July–August, 1990.The results were presented at the G. G. Chernyi seminar. The author is grateful to the seminar director and the participants for useful discussions.     

Use of flows with straight sonic line in nozzles with corners

The optimal scheme of a Laval nozzle is discussed. In the case of a profiled nozzle with a corner it is possible to use in the region of mixed flow both flows of general form with curvilinear sonic line as well as the special case when the sonic line is straight. It is shown that the latter alternative is preferable: when the supersonic part of the profile is determined by the simple wave method, the velocity at the wall increases monotonically and the flow does not contain shock waves. In contrast, in nozzles with curvilinear sonic line, a section in which the velocity decreases is formed immediately behind the corner, which can lead to boundary layer separation. In addition, for values of the supersonic velocity at the nozzle exit near the velocity of sound it is proved that the characteristics of the simple wave intersect in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 168–170, January–February, 1981.     

On the starting point of curvature for reflected diffracted shock wave (subsonic and sonic cases)

Srivastava (1995) has considered the case of starting point of curvature for the reflected diffracted shock wave when the relative outflow behind the reflected shock wave before diffraction is supersonic. In the present note we have considered the case of the starting point of curvature for the reflected diffracted shock wave when the relative outflow behind the reflected shock wave before diffraction in subsonic and sonic. Skews (1967) has earlier determined the starting point of curvature for a normal shock wave diffracting around a corner.     

Theoretical and experimental investigations of the flow of a viscous fluid near the line of intersection of a cylindrical surface and a flat surface

Flow along a corner was investigated at large Reynolds numbers in, for example, [1–3]. The present author [4] considered flow in the neighborhood of a corner formed by the intersection of a plane and a concave cylindrical surface, the main attention being devoted to the formation of the three-dimensional boundary layer on the plane near the corner. It was shown that the curvature of one of the intersecting surfaces changes the flow pattern qualitatively. In the present paper, we report an investigation of the formation of the flow on a concave cylindrical surface near such a corner and consider how the flow is rearranged in the neighborhood of a corner in, for example, a channel of rectangular cross section that has an initial straight section and then a bend with a discontinuity of the curvature of the line of intersection of the concave and flat sides of the channel. The results are given of some experimental investigations of flow near the line of intersection of a flat wall and a curved (concave and convex) wall at a bend in a rectangular channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 64–68, January–February, 1983.I thank G. M. Bam-Zelikovich for constant interest in the work and A. I. Ruban for a number of extremely helpful comments.     

Surface cracks in a plate of finite width under extension or bending

In this paper the problem of a finite plate containing collinear surface cracks is considered. The problem is solved by using the line spring model with plane elasticity and Reissner's plate theory. The main purpose of the study is to investigate the effect of interaction between two cracks or between cracks and stress-free plate boundaries on the stress intensity factors and to provide extensive numerical results which may be useful in applications. First, some sample results are obtained and are compared with the existing finite element results. Then the problem is solved for a single (internal) crack, two collinear cracks and two corner cracks for wide range of relative dimensions. Particularly in corner cracks the agreement with the finite element solution is surprisingly very good. The results are obtained for semielliptic and rectangular crack profiles which may, in practice, correspond to two limiting cases of the actual profile of a subcritically growing surface crack.     

Vortex Systems in a Viscous Fluid in the Neighborhood of a Corner Point on the Boundary

The expansion of the solution of the Navier-Stokes equations for steady-state plane-parallel incompressible fluid flows in a series in powers of the Reynolds number and the subjection of this series to no-slip conditions on the rectilinear boundaries in the neighborhood of their intersection point leads to the asymptotic form of the solution in the neighborhood of that point. The use of the leading part of the asymptotic form obtained as the boundary condition at a certain distance from the corner point makes it possible to formulate boundary value problems for the Navier-Stokes equations in closed regions. Examples of the numerical solution of such problems illustrate the formation of infinite vortex systems in the neighborhood of the corner point on the boundary of the flow region.     

The laminar boundary layer near a body corner point

A method is developed for calculating the characteristics of a laminar boundary layer near a body contour corner point, in the vicinity of which the outer supersonic stream passes through a rarefaction flow. In the study we use the asymptotic solution of the Navier-Stokes equations in the region with large longitudinal gradients of the flow functions for large values of the Reynolds number, the general form of which was used in [1].The pressure, heat flux, and friction distributions along the body surface are obtained. For small pressure differentials near the corner the solution of the corresponding equations for small disturbances is obtained in analytic form.The conventional method for studying viscous gas flow near body surfaces for large values of the Reynolds number is the use of the Prandtl boundary layer theory. Far from the body the asymptotic solution of the Navier-Stokes equations in the first approximation reduces to the solution of the Euler equations, while near the body it reduces to the solution of the Prandtl boundary layer equations. The characteristic feature of the boundary layer region is the small variation of the flow functions in the longitudinal direction in comparison with their variation in the transverse direction. However, in many cases this condition is violated.The necessity arises for constructing additional asymptotic expansions for the region in which the longitudinal and transverse variations of the flow functions are quantities of the same order. The general method for constructing asymptotic solutions for such flows with the use of the known method of outer and inner expansions is presented in [1].In the following we consider the flow in a laminar boundary layer for the case of a viscous supersonic gas stream in the vicinity of a body corner point. Behind the corner the flow separates from the body surface and flows around a stagnant zone, in which the pressure differs by a specified amount from the pressure in the undisturbed flow ahead of the point of separation. A pressure (rarefaction) disturbance propagates in the subsonic portion of the boundary layer upstream for a distance which in order of magnitude is equal to several boundary layer thicknesses. In the disturbed region of the boundary layer the longitudinal and transverse pressure and velocity disturbances are quantities of the same order. In this study we construct additional asymptotic expansions in the first approximation and calculate the distributions of the pressure, friction stress, and thermal flux along the body surface.     

Influence of nozzle profile on the characteristics of a gas-dynamic laser

The results are given of numerical profiling and analysis of the influence of nozzle shape and the gas-dynamic parameters on the characteristics of gas-dynamic lasers. Investigation of the two-dimensional nonequilbrium flow in a family of similar nozzles and nozzles with different angles of inclination of the contracting part show that it is expedient to choose a shape of the subsonic part that ensures a straight sonic line. Relationships between the geometrical parameters of the subsonic and transonic part of the nozzle are recommended which ensure separationless flow and a shape of the sonic surface that is nearly flat. A parametric investigation was made of the supersonic section of two classes of planar gas-dynamic laser nozzles constructed on the basis of uniform and symmetric characteristics at the exit. The parametric investigations of the influence of the degree of expansion, the total pressure and the temperature, and also the gas composition show that the smallest losses of useful vibrational energy in the cavity are achieved for nozzles constructed on the basis of uniform characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 163–167, November–December, 1982.     

Calculation of viscous compressible gas flow past a corner

We consider the problem of calculating the parameters for supersonic viscous compressible gas flow past a corner (angle greater than ). The complete system of Navier-Stokes equations for the viscous compressible gas is solved in the small vicinity Q₁. (characteristic dimensionl~1/R) of the corner point. The conditions for smooth matching of the solution of the Navier-Stokes equations and the solution of the ideal gas or boundary layer equations are specified on the boundary of Q₁. All these solutions are a priori unknown, and the conditions for smooth matching reduce to certain differential equations on the boundary of Q₁. Here account is taken of the interaction of the flows near the wall surface and in the so-called outer region [1].We note that no a priori assumptions are made in Q₁ concerning the qualitative behavior of the solution, in contrast with other studies on viscous flow past a corner (for example, [2–4]).The Navier-Stokes system in Q₁ is solved numerically, using the difference scheme suggested in [5]. This scheme permits obtaining the steady-state solution by the asymptotic method for large Reynolds numbers R, and also has an approximation accuracy adequate to account for the effects of low viscosity and thermal conductivity.     

Boundary layer in the neighborhood of the intersection of a concave cylindrical surface and a plane

In studies devoted to the theoretical and experimental investigation of longitudinal flow of a viscous fluid past corner regions, a corner formed by the intersection of two planes is usually considered [1–3]. In contrast, the present paper is concerned with the flow in the neighborhood of the line of intersection of a plane and a concave cylindrical surface (see Fig. 1). The asymptotic behavior of the Navier-Stokes equations at large Re is investigated for such a flow. Estimates are obtained for the velocity and characteristic scales of the flow. It is shown that curvature of one of the surfaces qualitatively changes the pattern of the longitudinal flow of a viscous fluid past a corner. The development of a three-dimensional boundary layer on a plane in the domain of influence of a concave cylindrical surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 160–165, March–April, 1981.     

Calculation of mixed sub- and supersonic gas flow in a plane pressure nozzle

Plane vortex-free sub- and supersonic gas flow is considered without taking account of viscosity and heat conduction. For the system of equations of motion of the mixed elliptic-hyperbolic type in the potential plane, a particular solution is found which corresponds to gas motion in a nozzle with a curved sonic line. The system of equations used to construct the solution is neatly homogeneous, which made it possible to separate the principal part of the solution in the transonic region. The validity of the simplifications made is well confirmed by comparison with calculation, using the method of characteristics, and with experiment.The author wishes to thank S. V. Fal'kovich for valuable comment on this study.     

基于混合网格的低声爆反设计方法研究

低声爆设计方法已成为新一代军民用超声速飞机研制过程中必须解决的关键难题之一。针对传统SGD低声爆外形反设计方法无法对声爆近场非线性效应进行描述和分析的缺点,提出了利用CFD方法求解得到的声爆近场压力分布代替F函数进行低声爆反设计的方法。声爆近场预测采用点-点对接的结构/非结构混合网格,充分利用非结构网格对复杂外形适应性强和结构化网格计算效率高的优点。结果分析表明,基于改进后的低声爆反设计方法得到的方案在声爆超压以及感觉噪声级等方面都比基于原始SGD方法得到的方案有较大改善。     

BENDING SOLUTION OF A RECTANGULAR PLATE WITH ONE EDGE BUILT－IN AND ONE CORNER POINT SUPPORTED SUBJECTED TO UNIFORM LOAD

ＢＥＮＤＩＮＧ　ＳＯＬＵＴＩＯＮ　ＯＦ　Ａ　ＲＥＣＴＡＮＧＵＬＡＲ　ＰＬＡＴＥ　ＷＩＴＨ　ＯＮＥ　ＥＤＧＥ　ＢＵＩＬＴ－ＩＮ　ＡＮＤ　ＯＮＥ　ＣＯＲＮＥＲ　ＰＯＩＮＴ　ＳＵＰＰＯＲＴＥＤ　ＳＵＢＪＥＣＴＥＤ　ＴＯ　ＵＮＩＦＯＲＭ　ＬＯＡＤＸｕＱｉｌｏ...     

Transonic flow past a convex corner with free streamline

An asymptotic analysis in the limit of large Reynolds numbers Re → ∞ is made of the system of Navier—Stokes equations in the neighborhood of a corner in a profile past which there is a transonic gas flow. The flow with free streamline from the corner point at which the velocity of sound is reached is taken as a limiting case. In the first approximation, it is described by a self-similar solution to the Kármán—Fal'kovich equation with self-similarity exponent n = 6/5 [1]. In such a flow, the favorable pressure gradient becomes infinite as the corner is approached from the side of the oncoming flow.     

Counterexamples to the Sonic Criterion

We consider self-similar (pseudo-steady) shock reflection at an oblique wall. There are three parameters: wall corner angle, Mach number, angle of incident shock. Ever since Ernst Mach discovered the irregular reflection named after him, researchers have sought to predict precisely for which parameters the reflection is regular. Three conflicting proposals—the detachment, sonic and von Neumann criteria—have been studied extensively without a clear result. We demonstrate that the sonic criterion is not correct. We consider polytropic potential flow and prove that there is an open nonempty set of parameters that admit a global regular reflection with a reflected shock that is transonic. We also provide a clear physical reason: the flow type (sub- or supersonic) is not decisive; instead the reflected shock type (weak or strong) determines whether structural perturbations decay towards the reflection point.     

Contouring the nozzles producing a uniform supersonic flow or a thrust maximum in the presence of a curvilinear sonic line

Within the framework of the ideal, i.e., inviscid and non-heat conducting, gas model we consider the problem of designing the supersonic section of a two-dimensional or axisymmetric nozzle realizing a uniform supersonic flow limitingly similar with a sonic flow when the choked flow involves a curvilinear sonic line. Emphasis is placed on nozzles with abruptly or steeply converging subsonic sections and a strongly curved sonic line formed by the C ⁻-characteristics of the expansion fan with the focus at the lower bend point of the vertical section of the subsonic contour. In the two-dimensional case, the least possible greater-than-unity Mach number M _em at the nozzle exit corresponds to the flow in which the first intersection of the C ⁺-characteristics originated at the closing C ⁻-characteristic of the expansion fan falls on the unknown contour of its supersonic part. For a uniform flow with M _e < M _em the intersection of C ⁺-characteristics beneath the unknown contour make impossible its construction. A part of the contour realizing a uniform flow with M _em > 1 ensures a limitingly rapid flow acceleration and forms the initial region of the supersonic generator of a maximum-thrust nozzle. For this reason, in the case of a curvilinear sonic line the supersonic generators of these nozzles have two, rather than one, bends, which, however, is interesting only for the theory. At least, in the calculated examples the thrusts of the nozzles with one and two bends differ only by a hundredth or even thousandth fractions of per cent.     

A hodograph-based method for the design of shock-free cascades

A hodograph-based method, originally developed by the first author for the design of shock-free aerofoils, has been modified and extended to allow for the design of shock-free compressor blades. In the present procedure, the subsonic and supersonic regions of the flow are decoupled, allowing the solution of either an elliptic or a hyperbolic-type partial differential equation for the stream function. The coupling of both regions of the flow is carried out along the sonic line which adjoins both regions. For the subcritical portion of the flow considered here, the pressure distribution is prescribed in addition to the upstream and downstream flow conditions. For the supercritical portion of the flow, the stream function on the sonic line is given instead of the supercritical pressure distribution which is found as part of the solution. In the special hodograph variables used, the equation for the stream function is solved iteratively using a second-order accurate line relaxation procedure for the subsonic portion of the flow. For the supercritical portion of the flow, a characteristic marching procedure in the hodograph plane is used to solve for the supersonic flow. The results are then mapped back to the physical plane to determine the blade shape and the supercritical pressures. Examples of shock-free compressor blade designs are presented. They show good agreement with the direct computation of the flow past the designed blade.     

Interaction of a jet exhausting from a body with an opposing supersonic stream of rarefied gas

The experimental investigation of supersonic flow past a sphere with a jet exhausting from the front point of the sphere into the flow at large [1] and moderate [2] Reynolds numbers Re has revealed an effect of shielding from the oncoming stream, this leading to a decrease in the drag coefficient of the sphere and of the energy flux to it. A numerical simulation of the flow has been made in the case of supersonic flow past a sphere with a sonic jet from a nozzle situated on the symmetry axis in the continuum regime [3]. In the present paper, this problem is investigated for flow of a rarefied gas on the basis of numerical solution of a model kinetic equation for a monatomic gas.