Symmetric shapes of contact of a jet with the surface of a heavy liquid

The plane linearized problem of oblique impingement of a weightless jet of an ideal incompressible fluid on the surface of a heavy fluid is considered. Flows are sought with symmetric forms of the contact region. Mathematically we arrive at the problem of the eigenvalues and eigenfunctions of an integral equation; solving this problem we obtain various contact forms. The fundamental result for the infinitesimally thin jet of finite intensity is derived by passing to the limit, yielding a result analagous with the forms of free vibrations of a string. Some results are presented for the problem under consideration in the nonlinear formulation.The two-dimensional problem on (vertical) impingement of a jet on a liquid was solved by Olmstead and Raynor [1]. Some results for oblique impingement of a sufficiently thin, slightly curved jet are presented by Frolov [2], Information on other studies, primarily experimental, is presented in [3].This problem is related to the model of a jet curtain of an air-cushion vehicle; in this regard we note the study of Stepanov [4] in which, in particular, a result is obtained for an infinitesimally thin jet curtain.     

Flow in a viscous jet escaping through a supersonic nozzle into a semi-infinite ambient space

The problem of an axisymmetric gas flow in a supersonic nozzle and in the jet escaping from the nozzle to a quiescent gas is solved within the framework of Navier-Stokes equations. The calculated pressure distribution is compared with that measured in the jet by a Pitot tube. The influence of the jet pressure ratio, Reynolds number, and half-angle of the supersonic part of the nozzle on nozzle flow and jet flow parameters is studied. It is shown that the distributions of gas-dynamic parameters at the nozzle exit are nonuniform, which affects the jet flow. The flow pattern for an overexpanded jet shows that jet formation begins inside the nozzle because of boundary-layer displacement from the nozzle walls. This result cannot be obtained with the inviscid formulation of the problem.     

Solution of the problem of flow of a non-axisymmetric swirling submerged jets

This paper considers the problem of a non-axisymmetric swirling jet of an incompressible viscous fluid flowing in a space flooded with the same fluid. The far field of the jet is studied under the assumption that the angular momentum vector corresponding to the swirling of the jet is not collinear to the momentum vector of the jet. It is shown that the main terms of the asymptotic expansion of the full solution for the velocity field are determined by the exact integrals of conservation of momentum, mass, and angular momentum. An analytical solution of the problem describing the axisymmetric swirling jet is obtained.     

Flow of a conducting gas jet beyond the nozzle outlet of an electromagnetic accelerator

A solution is given for the problem of the motion of a conducting gas beyond the outlet of an accelerator. The form of the jet is found as well as the distribution of all jet parameters. The problem is solved assuming that the flow is plane, that there are no Hall currents, and that the velocity increase in the jet is small compared with the magnitude of the velocity at the exit of the accelerator channel.     

Stability of a viscous liquid microjet in DC and AC electric fields

The problem of the stability of a liquid electrolyte jet under the action of a tangential electric field is considered. The radii of these jets, usually observable in experiments, vary from nanoscales to microscales. In this study, we consider microjets with the characteristic thickness of the double ion layer near the interface much less than the jet radius. The stability problem is analytically solved with account for the presence of this small parameter. The assumption on the electric neutrality of the jet as a whole leads to an explicit expression for the surface electric charge induced by the external field. The solution of the hydrodynamic problem in the external domain closes the solution and gives the dependence of the disturbance growth rate on the wavenumber. The cases of DC and AC electric fields are qualitatively compared. The distinctive features of jet stabilization by an AC high-frequency electric field are discussed.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Solution to the problem of a swirling jet from an annular orifice

The problem of the propagation of a laminar immersed fan jet with swirling was considered in [1–3]. In [1], the jet source scheme was used to find a self-similar solution for a weakly swirling jet. An attempt to solve by an integral method the analogous problem for a jet emanating from a slit of finite size was made in [2]. In [3], the equations of motion for a jet with arbitrary swirling were reduced under a number of assumptions to the equations that describe the flow of a flat immersed jet. This paper gives the numerical solution to the problem of the propagation of a radial jet emanating with arbitrary swirling from a slit of finite size and an analytic solution for the main section of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 49–54, March–April, 1991.     

Instability of a liquid jet in a high-frequency alternating electric field

Stability of a liquid (electrolyte) jet in a tangential electric field harmonically oscillating with a high frequency is considered under an assumption of an ideal liquid. It is demonstrated that it is possible to solve the electrodynamic and hydrodynamic parts of the problem inside the jet separately if the Peclet number based on the Debye layer thickness is small. Linear stability of the trivial solution of the problem is studied. A dispersion relation is derived, which is used to study the effect of the amplitude and frequency of electric field oscillations on jet stability. An increase in the amplitude of oscillations is demonstrated to exert a stabilizing effect, whereas an increase in frequency leads to insignificant destabilization of the jet.     

Development of a two-dimensional turbulent jet in an infinite space

The second and third terms in the asymptotic expansion of the stream function in the nonsimilar problem of the development of a two-dimensional turbulent jet in an unbounded space are found in final form. Results of experimental investigations of free turbulent jets are cited, and the effect of the initial velocity profile on the aerodynamic characteristics of the jet is considered. The problem of the development of a two-dimensional turbulent jet in an unbounded space has been considered in [1–3]. The existing solution is similar, and is valid only at a sufficiently large distance from the slit. Allowance for the finite dimensions of the slit leads to a nonsimilar problem. The papers [4–6] are devoted to the experimental investigation of the free two-dimensional turbulent jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1971.     

Thermal multipoles for a jet in ambient spathermal multipoles for a jet in ambient space

A general solution of the thermal problem for a non-self-similar axisymmetric jet is determined on the basis of the multiple approach developed for problems of non-self-similar jets in ambient space [1, 2], and several problems of convective heat-transfer in simply- or doubly-connected domains are solved. The advantage of expanding the solution of the convective heat conduction equation in eigenfunctions of the problem is demonstrated. As an illustration, the solutions for a thermal dipole and quadrupole and for a jet flow in a heated tube are presented. The corresponding solutions are also obtained for a turbulent jet in ambient space. The most favorable heat-exchange regime for a jet in a heated tube is predicted on the basis of the particular behavior of the eigenfunctions of the thermal problem.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 40–46, January–February, 1996.     

金属射流发散性理论分析及数值模拟

分析经典射流理论和相关文献,给出了在汇聚点坐标系中金属Cu对碰形成射流的汇聚射流区、无射流激波封锁区、无射流强度封锁区和形成发散射流区域。对金属超声速射流形成的发散性问题进行了理论分析,诠释了Walker基于实验提出射流发散理论,证明了金属射流超声速部分可形成发散射流也可形成汇聚射流,且超过1.23倍体声速的金属射流必定是发散的。最后,应用自编的欧拉计算程序MEPH对金属Cu以不同速度、倾角对碰射流形成过程进行数值模拟,得到了分叉射流、空洞射流和密实稀疏射流等的典型射流发散模式图像,印证了理论分析的结果。     

Mechanism of self-oscillations in a supersonic jet impact onto an obstacle. 2. Obstacle with no spike

Results of the numerical solution of the problem of impingement of an overexpanded supersonic jet onto an obstacle are reported. The mass-flow-rate mechanism of self-oscillations is revealed. This mechanism consists of periodic changes in the regimes of gas inflow and outflow from the separation region to the jet around this region. It is shown that the shock-wave structure of the impinging supersonic jet exerts a significant effect on the amplitude of self-oscillations.     

Flow of a jet from a slit-type source into a longitudinal carrying flow

It is shown that the problem of the mixing of a jet flowing out of an infinitely long slit-type source, with a homogeneous stream flowing parallel to the source of a jet, is analogous to the problem of the propagation of heat in a plane jet.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 83–86, November–December, 1971.The author thanks Yu. Ya. Boldyrev for his aid in making the calculations in the present work.     

Flow and instability of capillary jets interacting with the surrounding medium

The flow of an axially symmetric capillary jet of a viscous incompressible liquid in the space occupied by another liquid is investigated. The problem of stationary flow in the jet and in the surrounding medium under the action of viscosity, capillary forces, and gravity was obtained numerically. The instability problem of this flow to small perturbations in the form of running waves is stated and solved numerically. The values of the dimensionless Reynolds, Weber, and Froude numbers are explained, as well as the effect of the initial velocity profile in the jet, its instability, and subsequent jet decay into drops.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–59, November–December, 1978.     

Stability of jet flows in a rotating shallow water layer

The problem of the stability of an isolated jet flow and two counter-streaming jet flows in a rotating shallow-water layer is considered. These flows are described by exact solutions of the Charny–Obukhov equation with one or two discontinuities of the potential vorticity, respectively. The isolated jet flow is shown to be stable. For the system consisting of two jet flows the dependence of the characteristics of the unstable wave modes on a geometric parameter, namely, the ratio of the spacing between the jet axes to the deformation radius, is determined. On the basis of the contour dynamics method a weakly-nonlinear model of the longwave instability is developed.     

Nonlinear problem of a thin jet impinging on the surface of a heavy fluid

In a previous note by the author [1] the problem of symmetric forms of contact with oblique incidence of a free jet on a liquid was posed as a problem of the eigenfunctions of a nonlinear integral equation.Here we consider a more general flow scheme-a model of the jet curtain of an air cushion vehicle above the water surface (Fig. 1); the jet of inviscid, incompressible, weightless fluid of density ₁ impinges from a nozzle on the surface of a stationary liquid of density ₂, where, generally speaking, the pressures p₀ and p₁ are different. The problem is two-dimensional. We derive nonlinear integral equations, one of which is analogous to the Nekrasov equation for exact wave theory [2], In the limiting case of a thin jet we obtain a simple differential equation and exact solutions of the problem are constructed.Some data from the numerical calculations for the nonlinear problem of a thin jet curtain are presented in [3]; the problem has been solved in linearized form in [4],The author wishes to thank M. I. Gurevich and G. Yu. Stepanov, to whom he is indebted for his interest in the problem on jet impingement on a liquid and whose advice has been of assistance in improving the present note.     

Some problems of turbulent electrohydrodynamic jets

The problem of an axisymmetric turbulent electrohydrodynamic jet exhausting from a nozzle into an interelectrode gap is formulated. A numerical method of integrating the system of equations describing this flow is developed. This method is used to investigate three-dimensional effects in the jet (expansion of the jet, reverse flows). The influence on the jet characteristics (currents of the charge carried out of the nozzle, jet diameter, etc.) of the geometrical and electrical parameters and also of purely hydrodynamic factors (level of turbulence, relative velocity of parallel flow, etc.) is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 144–149, September–October, 1980.     

Mathematical simulation of unsteady flow past a wing with jets

The various approximate approaches to the investigation of the unsteady aerodynamic characteristics of an airfoil with jet flap [1–3] are applicable only for an airfoil, low jet intensity, and low oscillation frequencies. In the present paper, the method of discrete vortices [4] is generalized to the case of unsteady flow past a wing with jets and arbitrary shape in plan. The problem is solved in the linear formulation; the conditions used are standard: no flow through the wing and jet, finite velocities at the trailing edges where there is no jet, and also a dynamical condition on the jet. The wing and jet are assumed to be thin and the medium inviscid and incompressible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 139–144, May–June, 1982.     

Calculations of axisymmetric jet leaving a nozzle at jet pressure lower than pressure in medium

The problem of discharge of a supersonic axisymmetric jet from a nozzle at a jet pressure less than the pressure in the surrounding medium is considered. A calculation method is presented with account for the subsonic flow downstream of the central shock which forms near the axis of symmetry. Comparison of the results of the computation on a digital computer with the experimental data show the effectiveness of this method for determining the structure of the initial section of such an axisymmetric jet.     

Viscous-fluid outflow from a vessel with account for jet formation

The results of a numerical simulation of slow free-boundary viscous-fluid outflow from a vessel are presented with account for jet formation. The problem is formulated in the creeping motion approximation. For solving the problem, a numerical algorithm for plane geometry, based on an indirect variant of the boundary-element method, is used. As a result of parametric studies, the evolution of the free surface inside the vessel and the jet shape is determined for different values of the governing parameters. Flow regimes with rapid funneling and film formation on the vessel walls are detected. The existence of an asymptotic flow regime is demonstrated using dimensional analysis and confirmed by calculation.