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.     

Radial jet in rotating outer flow

The rotating radial jet formed by fluid introduced radially outwards into a rotating environment is studied by the momentum integral method and by perturbation for weak swirl, assuming incompressible laminar flow. For non-swirling outer fluid, the results slightly extend previously published results. For swirling outer fluid, the jet is characterized by a finite radial penetration distance. Its development over that distance is divided into three regions: an initial adjustment, a central pseudo-similarity region, and a final decay.     

Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body

We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations.     

A self-similar solution for a weakly swirling jet

A study is made of the laminar flow of a viscous incompressible fluid in a swirling jet that is produced by the action of a point source which transmits to the medium surrounding it a finite momentum flux. The limit of large Reynolds numbers is investigated under the assumption that the circulation of the azimuthal component of the velocity is a constant quantity at large distances from the jet axis. The boundary layer equations are solved asymptotically for the case of small circulation. It is shown that in the case of weak swirling of the jet the interaction of the azimuthal and axial motions is basically nonlinear.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 45–50, July–August, 1984.     

Development of a swirling jet in infinite space

We examine the problem of swirling-jet development in an infinite space filled with the same fluid. The fourth term of the asymptotic expansion of the tangential-velocity component is obtained. The constant appearing in the solution is obtained semlempirically. Results are presented of calculations of the velocities and pressure in swirling jets and of experimental studies.Swirling jet flows play an important role in the process of combustion intensification and stabilization and are widely used in engineering.The formulation and first solution of the problem of swirling-jet development in an infinite space filled with the same fluid at rest were accomplished by Loitsyanskii [1], who found the first two terms of the asymptotic expansion of the solution of the boundary-layer equations. The third and fourth terms of the asymptotic expansion of the axial-velocity component were found in [2], which made it possible to study the effect of jet swirl on the axial-velocity-component profile.In the present study we obtain the fourth term of the asymptotic expansion of the tangential-velocity component and present results of experimental studies on swirling jets.The authors wish to thank L. G. Loitsyanskii for valuable comments.     

Construction of optimal laws of variation of the angular momentum vector of a rigid body

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

On a class of motions of a Zhukovskii gyrostat with variable gyrostatic moment

The paper deals with the problem of Zhukovskii gyrostat motions in the case of a constant angle between the gyrostat invariable axis and the gyrostat angular momentum vector. It is assumed that the gyrostatic moment depends on time. The obtained conditions on the gyrostat parameters and the expression for the gyrostatic moment characterize a new solution of the Zhukovskii equations with variable gyrostatic moment. We prove that it describes semiregular precession of the first type with respect to the angular momentum vector.     

Computer-aided calibration and measurements with a quadruple hotwire probe

A method for calibration and measurement with a four-wire probe is described. For each of the wires a three dimensional calibration field is determined, thus no assumption like King's law or the cosine law need to be made. The velocity vector can then be detected in a fairly large angular range (± 40°) with a numerical search algorithm. First measurements in a free jet and a confined, strongly swirling flow are presented.A version of this paper was presented at the 11th Symposium on Turbulence, University of Missouri-Rolla, Oct. 17–19, 1988     

Destabilisation of a compressed vortex by a round jet

This paper presents data and analysis related to the compression and the breakdown of a tumbling motion after radial disruption in a simple geometry of the compression chamber of a model engine with large optical access. The disruption is a round jet injection perpendicular to the vorticity tube. Two configurations of injection are selected. They correspond respectively to a straight jet that competes with the tumble and an inclined jet that adds angular momentum to the large-scale rotating motion. The ratio between the angular momentum brought by the spray and the initial angular momentum of the tumble is of the order of 30% and is representative of the direct-injection engine situation at moderate rotation rate. The injection is performed at bottom dead centre (BDC) in a well-defined and well-known tumbling motion. The data are obtained in the symmetry plane of a square chamber by using particle image velocimetry (PIV) and planar laser induced fluorescence (PLIF). A calibration is made in order to take account of acetone fluorescence yield during compression. The analysis of the injection phase at BDC shows that the mean topology of the flow after both injections differs significantly and that the vorticity tube is significantly distorted only in the vicinity of the injection plane. Strong transverse mean flows are detected by analysing the divergence of the mean velocity field. Although a mean rotation is still observed after injection during the compression phase, the authors show that no strong vortex core is evident. An important consequence of this finding, confirmed by the evolution of the global in-plane mean and fluctuating kinetic energy in the symmetry plane is that no vortex breakdown occurs during the compression after the injection event. Therefore, the global fluctuating kinetic energy at the end of the compression is much lower after an injection. During the first half of the compression, an inhomogeneous distribution of the jet fluid in the chamber is detected by the PLIF measurements. The transport of the jet fluid clearly results from both in-plane and out-of-plane motions triggered by the injection jet. This spatial repartition depends strongly on the injection strategy and can be very difficult to control accurately from cycle to cycle. The mixture is more homogeneous at top dead centre (TDC) with a low value of the spatial variance of the mean concentration field.     

On swirling jets

An exact solution to the Navier-Stokes equations is found for a jet emanating from the end of a vortex filament into a region filled with a fluid. Depending on the degree of swirling of the jet, a closed or open flow regime is realized. In the case of strong swirling, the solution is not unique. Approximate analytic solutions to problems as well as numerical solutions are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 26–35, January–February, 1979.I am grateful to M. Kh. Pravdin for the numerical calculations.     

Buoyancy effects on turbulent swirling flows

A three-parameter model of turbulence applicable to free boundary layers has been developed and applied for the prediction of axisymmetric turbulent swirling flows in uniform and stagnant surroundings under the action of buoyancy forces. The turbulent momentum and heat fluxes appearing in the time-averaged equations for the mean motion have been determined from algebraic expressions, derived by neglecting the convection and diffusion terms in the differential transport equations for these quantities, which relate the turbulent fluxes to the kinetic energy of turbulence, k, the dissipation length scale of turbulence, L, and the temperature covariance, T ′². Differential transport equations have been used to determine these latter quantities. The governing equations have been solved using fully implicit finite difference schemes. The turbulence model is capable of reproducing the gross features of pure jet flows, buoyant flows and swirling flows for weak and moderate swirl. The behaviour of a turbulent buoyant swirling jet has been found to depend solely on exit swirl and Froude numbers. The predicted results indicate that the incorporation of buoyancy can cause significant changes in the behaviour of a swirling jet, particularly when the buoyancy strength is high. The jet exhibits similarity behaviour in the initial region for weak swirl and weak buoyancy strengths only, and the asymptotic case of a swirling jet under the action of buoyancy forces is a pure plume in the far field. The predicted results have been found to be in satisfactory agreement with the available experimental data and in good qualitative agreement with other predicted results.     

Mass and momentum transport in the near field of swirling turbulent jets. Effect of swirl rate

Local transport of the flow momentum and scalar admixture in the near-field of turbulent swirling jets (Re = 5,000) has been investigated by using a combination of the particle image velocimetry and planar laser-induced fluorescence methods. Advection and turbulent and molecular diffusions are evaluated based on the measured distributions of the mean velocity and concentration and the Reynolds stresses and fluxes. As has been quantified from the data, the flow swirl intensifies the entrainment of the surrounding fluid and promotes mass and momentum exchange in the outer mixing layer. A superimposed swirl results in the appearance of a wake/recirculation region at the jet axis and, consequently, the formation of an inner shear layer. In contrast to the scalar admixture, the momentum exchange in the inner shear layer is found to be strongly intensified by the swirl. For the jet with the highest considered swirl rate, a substantial portion of the surrounding fluid is found to enter the unsteady central recirculation zone, where it mixes with the jet that is issued from the nozzle. The contribution of the coherent velocity fluctuations, which are induced by large-scale vortex structures, to the turbulent transport has been evaluated based on triple decomposition, which was based on proper orthogonal decomposition analysis of the velocity data sets. For the considered domain of the jet with the highest swirl rate and vortex breakdown, the contributions of detected helical vortex structures, inducing pressing vortex core, to the radial fluxes of the flow momentum and the scalar admixture are found to locally exceed 65% and 80%, respectively.     

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.     

An experimental study of the under-expanded swirling jet with secondary coaxial stream

The present study addresses experimental results for investigating the details of the near field flow characteristics produced in an under-expanded, dual, coaxial, swirling jet. The under-expanded swirling jet is discharged from a sonic inner nozzle. An outer annular nozzle produces co- and counter-swirling streams relative to the inner primary swirling jet. The interaction between both the outer annular swirling stream and inner under-expanded swirling jet is quantified by impact and static pressure measurements, and visualized by using the shadowgraph method. Experiments are performed for several different pressure ratios. The results show that the outer secondary co-swirling jet significantly changes the structure of the inner under-expanded swirling jet, such as the shock structures and the recirculation region generated at the jet axis. The effect of the outer secondary stream on the major structures of the inner primary swirling jet is strongly dependent on the pressure ratio of the inner swirling jet, regardless of the swirl direction of the outer stream.Received: 17 May 2004, Accepted: 27 September 2004, Published online: 26 November 2004[/PUBLISHED]H.D. Kim: Correspondence to     

Integrals of motion for the classical two-body problem with drag

Integrals of motion for the two-body problem with drag are obtained by operating on the second-order vector differential equation describing the motion. The force field consists of an inverse-square gravitational attraction and a drag force proportional to the velocity vector and inversely proportional to the square of the distance to the attracting center. The developed integrals are the analogs of the Keplerian scalar energy, the vector angular momentum, and the Laplace vector.     

Horizontal submerged jet in a stratified fluid

The problem of jet flow excited in a viscous density-stratified fluid by a point source of momentum acting horizontally is considered. Simplified asymptotic equations are obtained in the boundary layer approximation. It is shown that the vertical velocity component is small and the motion in the jet has a layered structure. The longitudinal velocity distributions in the jet are measured experimentally. It is shown that these distributions are affine and can be satisfactorily approximated by Schlichting's well-known boundary layer solution for a round submerged jet in a fluid uniform with respect to density.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 10–16, November–December, 1993.We are grateful to I. A. Filippov for assisting with the experiments.     

Existence of Weak Solutions for a Two-dimensional Fluid-rigid Body System

We consider the problem of a rigid body immersed in an inviscid incompressible fluid in two dimensional space. The motion of the fluid is described by the incompressible Euler equations and the motion of the rigid body is governed by the balance of linear and angular momentum. A global weak solution is obtained, without any assumption on the weighted norm of the initial vorticity.     

Distance effect on the behavior of an impinging swirling jet by PIV and flow visualizations

The present paper deals with the problem of an impinging swirling jet against a wall. The study concerned a detailed experimental investigation on the jet–wall interaction using PIV and flow visualizations over a range of operating conditions in which the distance of the ducted propeller from the wall was changed. The influence of the impingement distance and the swirl number (i.e., ratio between the axial fluxes of the swirl and the axial momentum) as well as the interaction between the jet deformation and the perturbation induced on the wall is discussed in this paper.     

Laminar swirling jet with allowance for buoyancy forces

There have been many investigations into swirling immersed viscous jets, and fairly detailed analyses of these studies have been given in monographs [1–3]. Nevertheless, swirling jets with free convection have not been studied. In the present paper, exact and approximate self-similar solutions are found for the problem of a swirling jet produced by a source of viscous fluid propagating vertically upward (or downward) under the influence of Archimedean forces and an initial thrust.     

Hypersonic viscous shock layer in a swirling gas flow on a permeable surface

A hypersonic swirling flow of viscous compressible gas past rotating axisymmetric blunt bodies is considered, its velocity vector being parallel to the axis of rotation of the body. The body surface is assumed permeable, while, in the general case, the gas is not injected (drawn off) along the normal to the body surface. An analytic solution of the problem, valid at small Reynolds numbers, is found in the first approximation of the integral method of successive approximations. On the basis of the results of the numerical solution, obtained in a wide range of variation of the determining parameters of the problem, we investigate the influence of the swirling of the free-stream flow, the angular velocity of rotation of the body, the Reynolds number and the injection (suction) parameter on the structure of the compressed layer, and the coefficients of friction and heat transfer on the body surface. The influence of the swirling of the flow on the nature of the asymptotic behavior of the viscous shock layer equations at large Reynolds numbers is studied. It is shown that the presence of a nonzero peripheral component for the velocity vector of the gas in the shock layer can lead to a qualitative change in the nature of the flow. Deceased Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–37, November–December, 1986. The authors thank G. G. Chernyi for his useful discussion of the results of the work.