Plane incompressible fluid flow past an array of arbitrary profiles vibrating with arbitrary phase shift

We consider the problem of the vibration of an array of arbitrary profiles with arbitrary phase shift. Account is taken of the influence of the vortex wakes. The vibration amplitude is assumed to be small. The problem reduces to a system of two integral Fredholm equations of the second kind, which are solved on a digital computer. An example calculation is made for an array of arbitrary form.A large number of studies have considered unsteady flow past an array of profiles. Most authors either solve the problem for thin and slightly curved profiles or they consider the flow past arrays of thin curvilinear profiles [1].In [2] a study is made of the flow past an array of profiles of arbitrary form oscillating with arbitrary phase shift in the quasi-stationary formulation. The results are reduced to numerical values. Other approaches to the solution of the problem of unsteady flow past an array of profiles of finite thickness are presented in [3–5] (the absence of numerical calculations in [3, 4] makes it impossible to evaluate the effectiveness of these methods, while in [5] the calculation is made for a symmetric profile in the quasi-stationary formulation).     

Analysis of ellipsoid compressed by two axial concentrated forces at two ends

Integral equation method and photoelastic experiment are used for the stress analysis of an axial compressive ellipsoid. Let the concentrated forces and the centers of compression, with symmetrical unknown intensive functions X₁(c)=X₁(–c) and X₂(c)=X₂(–c) respectively, be distributed symmetrically to =0 plane along the axis z(=–c) in [a,) and [–a,–) of the elastic space, in addition to a pair of equal and opposite axial forces acting on z=a and z=–a. We can reduce the problem of an axial compressive ellipsoid to two coupled Fredholm integral equations of the first kind. Furthermore, numerical calculation is then made. Two photo-elastic models of ellipsoid were analysed by Freezing and Cutting method, and the results, in which ₂ is quite nearly to those obtained by integral equation method, had been used in the analysis of the data of compressive rock specimens.     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Kinetic theory of recondensation in a binary gas mixture with arbitrary Knudsen numbers

A rigorous solution of the problem of recondensation between two surfaces with arbitrary Knudsen numbers is possible only on the basis of a consecutive kinetic consideration. For the single-component case, this problem was solved in [1] using the BGK model of the collision integral in the kinetic equation. In [2], for the same purpose, the method of moments for Maxwellian molecules was used. The case of a binary mixture, in which one of the components is a noncondensing gas was discussed in [3, 4]. Under these circumstances, in [3], a single-relaxation lumped model was used for each component; the model did not reflect many of the properties of the exact collision integral. A more rigorous model (the collision integral in the Hamel form) was applied in [4]. Here there was written a system of integral equations for the hydrodynamic quantities, and its numerical solution is examined in several specific partial cases. In the present article, the problem of recondensation in a binary mixture is treated by the method of moments for Maxwellian molecules. For the case of small relative difference in the temperature of the surfaces, analytical expressions are obtained for the rates of mass transfer and the heat fluxes, making it possible to shed light on the principal special characteristics of the process of recondensation in a binary mixture.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–155, July–August, 1975.The authors thank R. Ya. Kucherov for his useful observations.     

Calculation of boltzmann equation collision integral and accurate solution of relaxation problem

A method of solving the Boltzmann equation on the basis of expansion in Maxwell distributions (Maxwellians) is developed. The calculation of the collision integral consists of two stages: calculation of a two-Maxwellian collision integral and its subsequent expansion in Maxwellians. A special numerical procedure is proposed for approximating functions of one and two variables by a finite number of exponentials; the mean-square deviation of the accurate and approximate functions is minimized and certain moments of these functions are brought into agreement. For a solid-sphere model, the two-Maxwellian collision integral comprises a set of 24 functions over the whole range of the parameters investigated. The Euler method is used to examine the equalization of velocities and temperatures in the gas (quasishock) and resonance recharging of ions on atoms. The accuracy of the solutions obtained is estimated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 132–141, September–October, 1977.     

Evolution of cylindrical waves of finite amplitude diverging in heated gas

The behavior of weak cylindrical and spherical waves of finite amplitude in a dissipative gas close to the wave front is described by a generalized Burgers equation [1]. The construction of various types of solution of this equation for large Reynolds numbers is known [1–3]. For the evolution of diverging perturbations in heated gas, a study of this equation in the region Re < 1, where Re is the effective Reynolds number at the initial time, is of interest. The direct application of the method of successive approximations to this problem is restricted by the condition Re 1, and becomes more and more difficult as the Reynolds number grows and the form of the initial wave becomes more complex. This paper describes in explicit form the construction of an approximate solution of the Cauchy problem for the generalized Burgers equation in the case of cylindrical symmetry in the region Re < 1. The initial wave selected is the arbitrary perturbation represented by a function which is absolutely integrable on the real axis. An integral estimate of the error as a function of Re is given. The question of how the structure of the solution corresponds to the Cole-Hopf transformation is discussed. All the treatment can easily be extended to the spherically symmetric case.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–153, July–August, 1985.     

An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

Calculation of the flow of an ideal fluid past a wing of finite thickness

The problem of irrotational flow past a wing of finite thickness and finite span can be reduced by Green's formula to the solution of a system of Fredholm equations of the second kind on the surface of the wing [1]. The wake vortex sheet is represented by a free vortex surface. Besides panel methods (see, for example, [2]) there are also methods of approximate solution of this problem based on a preliminary discretization of the solution along the span of the wing in which the two-dimensional integral equations are reduced to a system of one-dimensional integral equations [1], for which numerical methods of solution have already been developed [3–6]. At the same time, a discretization is also realized for the wake vortex sheet along the span of the wing. In the present paper, this idea of numerical solution of the problem of irrotational flow past a wing of finite span is realized on the basis of an approximation of the unknown functions which is piecewise linear along the span. The wake vortex sheet is represented by vortex filaments [7] in the nonlinear problem. In the linear problem, the sheet is represented both by vortex filaments and by a vortex surface. Examples are given of an aerodynamic calculation for sweptback wings of finite thickness with a constriction, and the results of the calculation are also compared with experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 124–131, October–December, 1981.     

Methods of calculation and numerical analysis of conducting-gas flow in high-current electric arcs

The mechanism of conducting-gas acceleration in an electric arc by intrinsic magnetic field was first investigated in [1]. Further theoretical study of this question was associated with the numerical calculation of arcs [2–7]. A more general approach to the solution of the problem was realized in [4], where the finite-difference method was used. Integral calculational models were developed in [5–7]. The present work proposes a modified version of the difference method [4] and a series of integral methods for the calculation of the conducting-gas flow in a high-current electric arc. The development of integral methods is of interest in that they are usually associated with adequate accuracy in determining integral values and values averaged over the cross section by a relatively simple calculation, and also allow the solution of the problem to be obtained in a number of situations when the realization of a difference method is complicated. The results of different calculation methods are compared. The effect of conditions in the initial cross section of the calculation region of the arc on its characteristics is investigated and a numerical analysis of the heating and acceleration of conducting gas is carried out.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 103–110, September–October, 1978.     

A three-dimensional variational problem in the gasdynamic theory of a lubricant

In [1–3] optimal forms of the gap were found for one-dimensional aerodynamic sliding bearings. The coefficient of the bearing capacity is optimized under the condition that the one-dimensional Reynolds equation of a gas lubricant is used to determine the pressure in the bearing. In the present article the three-dimensional problem of finding the optimal profile of an aerodynamic sliding bearing in the case of small compressibility numbers is considered. The problem is solved by the methods of variational calculation. A qualitative investigation is made of the form of the optimal profile, the results of which are confirmed by a numerical solution of a system of Euler-Lagrange equations. The results of the calculations are given for different elongations of the bearing. On the basis of the profiles obtained, optimal profiles with a rectangular pocket, which are more practical to fabricate, are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 34–39, September–October, 1975.     

Acoustic resonance associated with vibrations of an array of plates in a subsonic flow

The acoustic resonance effect occurring in the vibrations of an array of profiles (cascade) in a gas flow has been studied by a number of authors [1–8]. Relying on assumptions of a heuristic nature [1, 2, 7, 8] and using rather crude models [4, 6], they have derived criteria governing the acoustic resonance regimes and given the effect a certain physical interpretation. However, many problems of a physical bearing with regard to the quantitative and qualitative principles of the effect have been left unresolved. For a more complete and rigorous solution of the problem the author has previously [9, 10] analyzed the natural modes of a gas flowing past an array of plates It was determined that in the array domain the vibrational modes of the gas are localized in the vicinity, of the array and the eigenvalues are determined by the characteristic dimensions of the interstitial channel (as an open resonator). Also, the eigenfrequences were determined for the gas in the flow plane with the array absent [9]. Under spatial periodic conditions, such that the flow in the plane can be considered as a certain model of flow in an annular duct, these eigenfrequencies concurred with those obtained earlier in [1, 2, 4–6]. The results of [9, 10] are used in the present study as a basis for investigating certain laws and relations governing the unsteady aerodynamic characteristics of arrayed profiles in or close to regimes such that the gas can execute natural vibrations in the array domain and in an annular duct in the absence of the array.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–144, September–October, 1972.     

Unsteady flows of an erosion plasma in irradiation of solid targets by annular beams

In recent times high-pressure physics has made ever wider application of constructions which use convergent shock waves [1–8]. The study of gas dynamic flows with convergent shock waves imposes the need for more careful calculation of the motions of a gas in regions whose dimensions are much less than the characteristic dimensions of the flow. In the present study the numerical method is used to study the gas dynamic phenomena accompanying the irradiation of solid obstacles by annular beams of monochromatic radiation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 179–182, November–December, 1988.In conclusion we note that at very short durations t t_k the solution to the problem is similar to the flow during separation of a gaseous toroid [19].     

Effect of nonisothermal conditions of skin friction in subsonic turbulent channel flows

A number of different approaches are available in current literature to the theoretical determination of skin friction in nonisothermal flows in channels and pipes. Some of these (e.g., [1–3]) predict an appreciably stronger dependence of skin friction on temperature ratio =T_w/T (T_w is the wall temperature; T is the core fluid temperature), than those indicated by experimental results [4–7]. Significantly better agreement with existing experimental results is achieved in [8, 9] based on comprehensive numerical analysis of a system of integrodifferential equations. However, the assumptions they make are not always sound nor physically clear. Besides, the use of numerical methods does not allow the authors to relate their analysis to known limiting laws and simultaneously develop reliable numerical expressions to generalize experimental data. Physically quite clear results have been obtained in [10–12] and, in particular, very simple limiting laws for skin friction have been established. At the same time, it appears that based on the same physically clear assumptions, it is possible to obtain even more general results which agree well with experimental data. Simultaneously, these results which coincide with the limiting values at infinite Reynolds number make it possible to indicate the limits of applicability of these laws and extend them to finite Reynolds number range.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 69–76, January–February, 1984.     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

Approximate integration op nonstationary equations of a diffusion or thermal boundary layer

An approximate method is proposed for integrating the nonstationary equations of a diffusion or thermal boundary layer using the known steady solution in the planar or axisymmetric case. It is shown that the proposed method is exact in problems involving mass or heat transfer of reacting drops and bubbles in a laminar flow of a viscous incompressible fluid and also particles moving in an ideal fluid. An integral equation is obtained for the local diffusion or heat flux in the case of abrupt activation of a reaction on the surface of a particle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 87–92, September–October, 1982.     

Advance of the interface of liquids with pressure filtration

The article is a continuation of the investigations of a number of authors [1–3] on the advance of the interface between different liquids: it gives a solution of the problem in the case of one-dimensional filtration of liquids with different viscosities in distorted layers of variable permeability; it indicates a method for determining the interface between multicolored liquids, and the method is generalized for the case of a series of two-dimensional flows, connected by a conformai transformation. The article discusses problems that reduce to formation of the functions determining the flow, and to calculation of the integrals.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 185–191, September–October, 1971.     

A New Model for the Dynamics of Dispersions in a Batch Reactor: Numerical Approach

In this paper, we develop some considerations concerning the structure of coalescence, breakage and volume scattering kernels appearing in the evolution equation related to a new model for the dynamics of liquid–liquid dispersions and show some numerical simulations. The mathematical model has been presented in [3, 4], where a proof of the existence and uniqueness for a classical solution to the integro–differential equation describing the physical phenomenon is provided as well as a complete analysis of the general characteristics of the integral kernels. Numerical simulations agree with experimental data and with the expected asymptotical behavior of the solution.     

Supersonic Flow Past a Bent Delta Wing and Its Aerodynamic Characteristics

The supersonic flow past a bent delta wing and its aerodynamic characteristics, both local and integral, are studied using approximate analytic estimates and numerical calculations. The flow regimes with the shock attached to the leading edges are considered. It is established that at M = 4–6 and angles of attack as high as 6° the lift-drag ratio of the wing can be increased by 10% by deflecting the wing nose downwards. This is confirmed by the results obtained earlier in the hypersonic thin-shock-layer approximation. The effect is also obtained in calculations made within the framework of the Navier-Stokes equations.     

Numerical investigation of the steady-state conditions of interaction of a supersonic underexpanded jet with a plane obstacle located perpendicular to its axis

The results are presented of the numerical investigation of the interaction of a supersonic axisymmetrical jet of a nonviscous and nonthermally conductive gas, flowing from a conical nozzle into a space with reduced pressure, with a plane obstacle. The presence of a triple point of intersection of the shock wave issuing from the obstacle with the trailing and reflected oblique compression shock is characteristic for the conditions considered in the paper. The solution of the problem is obtained by numerical integration of the gasdynamic equations by means of monotonic difference schemes of a straight-through calculation with first-order accuracy. The interaction of supersonic gas jets with surfaces is a vast problem and is one of the trends being developed intensively in the theory of jet streams. Of the whole multiplicity of problems of practical interest, the two-dimensional case of the normal collision between a supersonic axisymmetrical jet and a plane obstacle has been studied in most detail. As a result of the investigations carried out, many characteristic mechanisms of these flows have been revealed. Together with the numerous experimental papers, several reports have been published (for example, [1–4]) in which various numerical methods are employed to solve this problem. In addition to the method of integral relations used in [1], an implicit difference scheme [2] and explicit schemes of straight-through calculation [3, 4] have been used to calculate the subsonic zone of increased pressure in front of the obstacle. However, an extensive investigation of the special features of the action of a supersonic underexpanded jet on a plane obstacle, at a very small distance from the nozzle exit, still has not been carried out up to the present. In this paper, a solution of this problem is undertaken by the numerical method described in [4] using difference schemes [5, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 49–56, September–October, 1976.The authors express sincere thanks to A. N. Kraiko, é. A. Ashratov, and U. G. Pirumov for constant interest and support in carrying out this project.