M. Goldshtik

Collapse in conical viscous flows

A class of steady conically similar axisymmetrical flows of viscous incompressible fluid is studied. The motion is driven by a vortex half-line or conical vortex in the presence of a rigid conical wall or in free space. The dependence of the solutions on parameters (say, the vortex circulation) is analysed

Analysis of inviscid vortex breakdown in a semi-infinite pipe

Axisymmetric vortex breakdown in a steady, inviscid, incompressible flow in a semi-infinite circular pipe is considered analytically. We suggest a new perception of vortex breakdown and compare ours with other approaches. In our view, vortex breakdown occurs due to solution nonuniqueness in some range of inflow parameters when the entire steady flow experiences a jump to another metastable steady state with the same boundary conditions. These co-existing solutions are smooth along the pipe length; they have the same mechanical energy but, in general, different flow forces. Vortex breakdown necessarily occurs by a continuous change in flow parameters (usually the swirl number) when the solution fails to exist (locally) because of fold or similar catastrophe, but spontaneous jumps (in some range of parameters) between different metastable solutions (not on a fold) can also be caused by large flow perturbations. The folds can appear due to transcritical bifurcation, which is destroyed (in the case considered here) by the injection of azimuthal vorticity into the vortex core at the pipe entrance. A high level of the entrance swirl leads to separation zones (even for solid-body inflow!) where the steady flow is undetermined. We find that the nonuniqueness interval in parameter space is connected with the flow pattern inside the separation zone. We consider models for dealing with two such flow patterns: the traditional analytic continuation (leading to a recirculation zone) and a new stagnant separation zone model. We reveal serious defects of the analytic continuation approach. The stagnation zone model is superior in that solutions always exist and, for large enough inflow swirl, exhibit nonuniqueness and folds, thus explaining the experimentally observed hysteretic jump transitions in vortex breakdown. We also predict some new phenomena, which deserve experimental investigation.

Conical flows of fluid with variable viscosity

It is proved that a model of a turbulent swirling vortex near a plane, which was studied by Wu (Proc. R. Soc. Lond. A 403, 235–268 (1986)), is inconsistent. There is no regular solution for a swirling downward flow satisfying the adherence condition at the surface. If the flow is upward the solution existence is not excluded but it cannot appear owing to a bifurcation, because an initial solution for a non-swirling conically similar jet emerging from an origin on the plane does not satisfy the adherence condition for any angular distribution of viscosity that is physically meaningful. On the other hand, if a jet in an ambient medium is induced by a convergent motion of the plane matter then, firstly, the laminar solution ceases to exist when the Reynolds number exceeds a finite critical value, so the flow must become turbulent; and, secondly, for a jet flow with a turbulent core a supercritical bifurcation takes place if the rotation friction on the plane is zero. As a result, a self-swirling jet flow is developed together with a spiral motion of the plane matter. Such a scenario may serve as a simple hydrodynamical model for some astrophysical and geophysical phenomena.

The vortex liquid piston engine and some other vortex technologies

By exploiting three unique characteristics of confined swirling incompressible flows — centrifugal acceleration, internal separation or recirculation zones near the axis, andbistability (i.e. rarefied and condensed stable states) of multi-phase flows — we developed several innovativevortex machines which will revolutionize mechanical technologies in a variety of industries. The machines utilizing these features include:Vortex Engine, Vortex Thruster, Vortex Suction Device, Vortex Chemical Reactor, Bubbling Centrifuge andVortex Mill. As a specific example, we describe here in some detail the development of a liquid piston engine, including analysis of its hydrodynamic and thermodynamic features. We have designed a laboratory ‘cold’ model and performed detailed experimental, theoretical and numerical analyses to study the role of the controlling parameters and are now ready to test a ‘hot’ model. In addition, we mention a few other vortex technologies of interest to us.

Free convection near a thermal quadrupole

Abstract A conically self-similar solution of the Boussinesq equations is reported. Thermogravitational convection near a quadrupolar point singularity of a temperature field is studied. At a Prandtl number of zero the solution loses its existence when the Grashof number achieves some critical value. If the Prandtl number differs from zero, then the solution exists at any Grashof number but, when the Prandtl number tends to zero, the passage to the limit can become nontrivial. At subcritical Grashof numbers, a strong upward jet is developed. A number of problems are studied in which the flow region is bounded by a conical surface. These problems may serve as simple models of convection near a volcano, a glacier, and an iceberg.

Threshold Regimes in the Plane Channel Flow

Steady self-oscillations in the Poiseuille flow are calculated by using the approach concerning the resonance interaction between three waves: the plane wave with a certain fundamental frequency and three-dimensional half-frequency symmetrical couple. The problem is reduced to the eigenvalue problem for the system of nonlinear differential equations and is solved by the Newton method. The relation between the pulsation energy of the threshold regimes and the Reynolds number in the subcritical region has been obtained. The calculated results are in agreement with the experimental data for the transition in a plane channel.

Self-Similar Separation

Rather simple flows are studied for which a separation phenomenon may be investigated by analytic and semianalytic methods. Conically self-similar flows of viscous fluid are considered including effects of buoyancy and electric conductivity. Regions of near-wall separation or recirculation zones inside of a motion region appear which are bounded by conical surfaces. Particularly, problems on a vortex-source flow in free space and near a wall, multicell tornado models, swirling cosmic jets, suppression of the separation by magnetic field, and free convection are reported.

Supercritical Regimes in Axisymmetric Submerged Jets

The bifurcation of secondary regimes in jet flows of a viscous incompressible fluid is studied. Initial axisymmetric submerged jets having velocity distributions from Schlichting’s self-similar profile to the “top hat” form are considered as parallel. In the latter case the secondary regime was found to include stationary rotation around the jet axis. The rotation is generated in jets with high-gradient profiles and is absent in the case of the self-similar distribution.