Shock wave incidence on a V-shaped cavity

We study theoretically and experimentally the motion of metal arising from a plane shock wave striking a V-shaped cavity. Using the functionally invariant solutions of Sobolev, we write out the acoustic approximation for this problem and determine the region of its applicability. It is shown that in the region in which the acoustic approximation is not applicable, the flow in the principal term is described by the incompressible fluid equations for which the boundary conditions are defined by the acoustic region. The experimental technique is described and a comparison of the theoretical and experimental data is made.Translated from Zhurnal PrikladnoiMekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 57–61, November–December, 1969.The authors wish to thank A. A. Deribas for discussion on the problem formulation and experimental technique, and N. S. Kozin for carrying out the numerical calculations.     

Approximate method of solving the problem of characteristic oscillations of a liquid in cavities of revolution

The problem of the characteristic oscillations of a liquid in axisymmetric cavities of rotation has been fairly fully studied [1–5], its solution in the general case being found by the variational method. Analysis of numerical results using the variational method shows that to achieve acceptable accuracy it is necessary to retain an appreciable number of coordinate functions, which entails the solution of a matrix eigenvalue problem of high order, this applying especially to the case when it is necessary to determine several eigenfrequencies and the shapes of the oscillations. In the present paper, a method proposed earlier by Shmakov [6] is developed, the velocity potential being sought in the form of a sun of two potentials. The first (base) potential is a solution to the problem of the characteristic oscillations of a liquid in a cavity whose free surface coincides with the free surface of the original cavity, and the second (correcting) potential is chosen in the form of a system of harmonic functions, this system being complete and orthogonal on the wetted surface of the cavity. Cavities of revolution are analyzed as examples, and a detailed investigation of numerical results is made for a spherical cavity. The numerical analysis shows that a sufficiently accurate result in the determination of a frequency is obtained if one term of the base problem is retained and only the correcting potential is used to make this more accurate. As a result, it is only necessary to solve an algebraic equation of first degree in the square of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–8, September–October, 1983.     

Asymptotic theory of the flow around an obstacle by a sonic flow

The first investigation of the problem of the flow around an obstacle by a gas flow whose velocity is equal to the speed of sound at infinity was carried out in [1, 2], where it is shown in particular that the principal term of the appropriate asymptotic expansion is a self-similar solution of Tricomi's equation, to which the problem reduces in the first approximation upon a hodographic investigation. The requirement that the stream function be analytic as a function of the hodographic variables on the limiting characteristic was an important condition determining the selection of the self-similarity exponent n (xy^–n is an invariant of the self-similar solution). The analytic nature of the velocity field everywhere in the flow above the shock waves, which arise from necessity upon flow around an obstacle, follows from this condition. The latter was found in [3], where one of the branches of the solution obtained in [1] was used in the region behind the shock waves. The principal and subsequent terms of the asymptotic expansion describing a sonic flow far from an obstacle were discussed in [4], where the author restricted himself to Tricomi's equation. Each term of the series constructed in [4] contains an arbitrary coefficient (we will call it a shape parameter) which is not determined within the framework of a local investigation, and consideration of the problem of flow around a given obstacle as a whole is necessary in order to determine these shape parameters. It follows from the results of [4] that the problem of higher approximations to the solution of [1] coincides with the problem, of constructing a flow in the neighborhood of the center of a Laval nozzle with an analytic velocity distribution along the longitudinal axis (a Meyer-type flow). Along with the Meyer-type flow in the vicinity of the nozzle center, which corresponds to a self-similarity exponent n=2, two other types of flow are asymptotically possible with n=3 and 11, given in [5]. The appropriate solutions are written out in algebraic functions in [6]. The results of [5] show that the condition that the velocity vector be analytic on the limiting characteristic in the flow plane is broader than the condition that the stream function be analytic as a function of the hodographic variables, which is employed in [1, 2, 4]. Therefore, the necessity has arisen of reconsidering the problem of higher approximations for the obstacle solution of F. I. Frankl'. It has proved possible for the region in front of the shock waves to use a series which is more general than in [4], which implies the inclusion of an additional set of shape parameters. The solution is given in the hodograph plane in the form of the sum of two terms; the series discussed in [4] corresponds to the first one, and the series generated by the self-similar solution with n=3 or with n=11 corresponds to the second one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 99–107, May–June, 1979.The authors thank S. V. Fal'kovich for a useful discussion.     

Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]     

Pulsating regime corresponding to an obstacle in a stationary inhomogeneous flow

The pulsating regime produced by the presence of a cylindrical cavity in a stationary inhomogeneous supersonic flow is simulated mathematically. The system of equations for an inviscid thermally nonconducting gas is solved by a numerical method based on a two-step difference scheme of second order of approximation. This method makes it possible to calculate in each time step the complete flow field at once, which makes it possible to follow the development of the nonstationary flow, which in the present case is a pulsating flow. The flow pattern in the pulsating regime is studied in detail. The pressure pulsations in the cavity are due to the alternating passage through it of shock waves and rarefaction waves, and the pulsations are nonlinear. The influence of the basic parameters on the characteristics of the pulsating flow is studied and some estimates are made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 64–71, September–October, 1979.     

Calculation of the characteristics of a gas-dynamic laser

The dependence of the radiated power on the characteristics of optical cavities in the case of flow systems has been investigated in a number of papers [1–3], in which it is assumed that population inversion of the laser levels is obtained until entry into the cavity. The operation of a cavity is analyzed in [1] in the geometric-optical approximation with allowance for vibrational relaxation in the gas flow. A simplified system of relaxation equations is solved under steady-state lasing conditions and an expression derived for the laser output power on the assumption of constant temperature, density, and flow speed. The vibrational relaxation processes in the cavity itself are ignored in [2, 3]. It is shown in those studies that the solution has a singularity at the cavity input within the context of the model used. In the present article the performance characteristics of a CO₂-N₂-He gas-dynamic laser with a plane cavity are calculated. A set of equations describing the processes in the cavity is analyzed and solved numerically. Population inversion of the CO₂ laser levels is created by pre-expansion of the given mixture through a flat hyperbolic nozzle. The dependence of the output power on the reflectivities of the mirrors, the cavity length, the pressure, and the composition of the active gas medium is determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi FiziM, No. 5, pp. 33–40, September–October, 1972.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

The Stress State of an Elastic Orthotropic Medium with an Ellipsoidal Cavity

The stress-concentration problem for an elastic orthotropic medium containing an ellipsoidal cavity is solved. The stress state in the elastic space is represented as a superposition of the principal state and the perturbed state due to the cavity. The equivalent-inclusion method, the triple Fourier transform in spatial variables, and the Fourier-transformed Green function for an infinite medium are used. Double integrals over a finite domain are evaluated using the Gaussian quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of the geometry of the cavity and the elastic properties of the material on stress concentration is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 93–100, March 2005.     

Jet flow around a wedge-shaped stanchion intersecting a free space

A solution to the problem of a jet flow around a narrow wedge-shaped stanchion intersecting a free space is examined. The dimensions of the cavity formed behind the stanchion and the coefficient of resistance are determined. The results of the calculations are compared with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–143, July–August, 1976.     

Asymptotic representation of the solution in the hodograph plane in transonic flow problems around profiles

The flow around a slender profile by an ideal gas flow at a constant, almost sonic, velocity at infinity is considered. The behavior of the perturbed stream in the domain upstream of the compression shocks sufficiently remote from the streamlined body is studied. The question is investigated of what conditions the solution in the hodograph plane satisfies when it corresponds to a flow without singularities on the limit characteristic in the physical flow plane. It is known that cases are possible when a regular solution in the hodograph plane loses its regularity property upon being mapped into the physical plane [1]. A regular flow on the limit characteristic can be continued analytically downstream into the supersonic domain between the limit characteristic and the shock. The requirement of analyticity of the streamlined profile is essential for realizability of the flow under consideration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 84–88, January–February, 1976.In conclusion, the author is grateful to O. S. Ryzhov for discussing the research.     

Supersonic flow past a blunt wedge

A study is made of the asymptotic solution of the problem of flow past a blunt wedge by a uniform supersonic stream of perfect gas. By separation of variables it is shown that at large distances the disturbance of the flow is damped exponentially. In the case of subsonic flow behind the shock wave the exponent of the leading correction term in the expansion of the shock front is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1984.     

Slender axisymmetric cavities in a supersonic flow

Slender axisymmetric cavities in a subsonic flow of compressible fluid were investigated in [1–4]. In [5] a finite-difference method was used to calculate the drag coefficient of a circular cone, near which the shape of the cavity was determined for subsonic, transonic, and supersonic water flows; however, in the supersonic case the entire shape of the cavity was not determined. Here, on the basis of slender body theory an integrodifferential equation is obtained for the profile of the cavity in a supersonic flow. The dependence of the cavity elongation on the cavitation number and the Mach number is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1989.     

Isothermal viscous incompressible flow in a hydrodynamic model of an electrophoresis chamber

A numerical solution and approximate analysis of the system of Navier-Stokes equations averaged over the transverse coordinate has made it possible to obtain the dependence of the length of the hydrodynamic flow stabilization interval in a thin cell of rectangular cross section on the Reynolds number, the relative thickness of the cell, and the relative size of the inlet opening. The principal and secondary flow regimes are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–20, March–April, 1990.     

Simulation of the reduction of unsteadiness in a passively controlled transonic cavity flow

A 30 dB reduction of the peak pressure tone and a reduction by 6 dB of the background pressure found in an experiment of high-subsonic cavity flow controlled by a spanwise rod are retrieved numerically. The injection of deterministic upstream fluctuations in the large-eddy simulation (LES) domain is found to be of crucial importance, in contrast with the baseflow case. Reduction of the vortex impingement onto the aft edge of the cavity is confirmed, together with reduction of mass flow rate breathing through the grazing plane. Visual evidence of merging between the Kelvin–Helmholtz-type vortices shed downstream of the fore edge of the cavity and the von Kármán vortices shed behind the cylinder is provided. Shocklets downstream of the cylinder are also observed.     

Capillary cavity flow past a circular cylinder

Cavity flow past a circular cylinder is considered accounting for the surface tension on the cavity boundary. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. The solution is based on two derived governing expressions, which are the complex velocity and the derivative of the complex potential defined in an auxiliary parameter region. An integral equation in the velocity magnitude along the free surface is derived from the dynamic boundary condition. The Brillouin–Villat criterion is employed to determine the location of the point of flow separation. The cases of zero surface tension and zero cavitation number are obtained as limiting cases of the solution. Numerical results concerning the effects of surface tension and cavitation development on the cavity detachment, the drag force and the geometry of the free boundaries are presented over a wide range of the Weber and the cavitation numbers.     

Development of laminar flow of a liquid in the intertray space of a separator

The article discusses the problem of the development of the laminar flow of a liquid in a slit-type space, formed by the conical trays of a separator, rotating with a constant angular velocity, in the two principal cases of the feed of the liquid into the intertray space. For these cases, a common method of investigation is used, consisting in expansion of the solution in series in powers of a small parameter. Two terms of the expansion of the two principal components of the relative flow rate, taking account of the conditions at the inlet, are found. A formula for calculating the initial section is obtained.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 16–27, March–April, 1978.In conclusion, the author thanks N. A. Slezkin for his evaluation of the work and his critical observations.     

Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.     

Heat exchange of sphere with blowing for slow flows

An analytic solution is given for the problem of convective heat- and mass-exchange of a sphere with transverse flow of matter along the surface for values of Peclet numbers smaller than one and for blowing velocity smaller than that of the incoming gas flow. The solution for velocity field obtained by the authors in a previously published publication is employed on flow past a sphere with blowing; the method of asymptotic expansions of Acrivos and Taylor is also used. Expressions to the second approximation are determined for temperature field and for the values of local and averaged Nusselt numbers. It is shown that blowing reduces the temperature gradient or the concentrations at the surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 89–94, July–August, 1972.     

Solution of three-dimensional nonlinear problems of the theory of jets in ideal fluids

A method of successive approximations is proposed for solving three-dimensional nonlinear problems of the theory of jets in ideal fluids (see, for example, [1–3]). Each approximation includes the calculation of the flow over a known surface, i.e., the solution of the exterior Neumann problem for the Laplace equation in the velocity potential and the correction of part of that surface for the purpose of reducing the discrepancy in the constant-pressure condition at the surface of the jets. The correction takes the form of small deformations found from a system of integral equations; the shape of the cavity in plan is also refined. The results of calculating the flow past triaxial ellipsoids, obtained using the generalized Zhukovskii-Roshko method for closing the jets, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–179, March–April, 1989.The authors are grateful to V. P. Karlikov for useful comments.     

Cavitation flow over a plate with a source

Symmetric flow by an ideal incompressible imponderable fluid over a plate with a source is studied. The existence, within the framework of the adopted scheme, of a limiting value of the power of the source is established. The nature of the flow is investigated at a power of the source close to the limiting value. A numerical analysis of the solution was made on a computer, and the results are shown in the form of the graphical dependences of the cavity size on the power of the source for different cavitation numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 5, pp. 157–161, September–October, 1979.The author is grateful to V. P. Karlikov for constant interest in the work, to G. Yu. Stepanov for valuable suggestions relating to the qualitative analysis of the solution, and to Yu. L. Yakimov for discussing the results of the work.