Effect of volume energy supply on swirled flows in a subsonic cocurrent stream

The results of a numerical investigation of the effect of thermal energy supply on a swirling viscous heat-conducting gas flow in a subsonic cocurrent stream are presented. The initial stage of development of the swirling flow in the neighborhood of the vortex axis with constant circulation in the outer flow region is considered for two different distributions of the streamwise velocity vector component which simulate a swirling jet-type flow and a wake flow with a streamwise velocity deficit. The effect of local volume energy supply in the neighborhood of the vortex axis, the circulation of the azimuthal velocity component, and the longitudinal pressure gradient in the inviscid stream on the development of the swirling flow and the process of breakdown of cocurrent vortex flows is investigated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 47–53, November–December, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00586).     

Stability of an axisymmetric swirled flow in a supersonic cocurrent stream for volume energy supply in the viscous vortex core

The results of numerical calculations of the stability of axisymmetric swirled flows in a viscous vortex embedded in a supersonic cocurrent stream with a constant circulation of the azimuthal velocity component are presented. The stability characteristics of the swirled three-dimensional viscous flow in the streamwise vortex are determined on the basis of the linearized system of Navier-Stokes equations for a viscous heat-conducting gas under the assumption that the basic undisturbed flow is locally plane-parallel. The disturbed flow stability is studied in the temporal formulation with respect to both symmetric and asymmetric three-dimensional waves traveling along the vortex axis and corresponding to both positive and negative values of the azimuthal wavenumber. It is shown that at external inviscid flow Mach numbers M = 2 and 3 thermal energy supply in a small region near the vortex axis leads to considerable restructuring of the basic undisturbed flow in the vicinity of the vortex core, the growth of the adverse pressure gradient along the vortex axis, and a significant change in the small perturbation stability and behavior.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 71–80. Original Russian Text Copyright © 2005 by Kazakov.     

Effect of Volume Energy Supply on the Stability of a Subsonic Vortex Flow

Calculations of the stability of an axisymmetric vortex flow of viscous heat-conducting gas with volume energy supply are presented. The unperturbed axisymmetric vortex flow was found numerically using a quasi-cylindrical approximation of the Navier-Stokes equations under the assumption of constant peripheral-velocity circulation in the ambient co-current flow. The volume energy supply in the viscous vortex core was modeled by an additional source term in the energy equation. The stability characteristics of the viscous vortex flow in a longitudinal vortex with respect to both axisymmetric and non-axisymmetric three-dimensional waves traveling along the vortex axis and corresponding to both positive and negative values of the azimuthal wave number were found using the time-dependent formulation of the linear stability theory for compressible three-dimensional plane-parallel flows.     

Some features of the flow around rapidly rotating bodies made of cellular-porous materials

Two types of gas flows arising near a rapidly rotating cellular-porous disk are studied numerically and experimentally. Steady-state limits for the flow around a disk rotating in free space and the type and scenario of the loss of stability are determined. Transitional flows are characterized by formation of a vortex sheet at the boundary of the exhausting jet. Numerical simulations of the flow around a cellular-porous disk rotating near a flat screen show that it is possible to form a closed swirl flow responsible for redistribution of swirl in the gap between the disk and the flat screen. The computed results offer an explanation for the experimentally observed excess of tangential velocity of the flow in the gap over the velocity of disk rotation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 86–96, January–February, 2007.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

A cylindrical model for rotational MHD instabilities in aluminum reduction cells

Large-scale horizontal vortices associated with deformations of the aluminum-electrolyte interface have been observed in operating aluminum reduction cells as well as in physical and numerical models. To expose their importance, we analyze a particular class of magnetohydrodynamic (MHD) interfacial instabilities which are induced by rotation. As we focus on a single vortex, a cylindrical geometry is preferred. Two analytical models are proposed. In a first model based on the MHD shallow-water approximation, we consider a vortex that has a solid rotation profile to obtain a wave equation and a dispersion relation. A more realistic second model includes a viscous rotation profile and the treatment of the base-state interface deformation. Energetics of the flow gives further insight on how an initial perturbation evolves as an oscillatory or a non-oscillatory instability, depending on the direction of rotation. We find that the mechanism at the very origin of these instabilities is neither due to a shear between the two layers—and are therefore not Kelvin–Helmholtz instabilities—nor simply due to magnetic force alone, but rather to the indirect action of the centripetal pressure due to the rotation induced by magnetic force.      

Stability of a compressible swirling flow in a circular pipe

The stability of an axisymmetric flow of viscous gas in a circular pipe, which models the Burgers vortex in the pipe axis neighborhood, is studied within the linear theory framework. Neutral curves for the most unstable disturbances are calculated. The influence of the characteristic Mach number on the flow stability is investigated. It is shown that for a given model velocity distribution the Mach number affects only the temperature and pressure profiles of the main undisturbed flow. In this case, for the disturbance types considered, as the Mach number increases, the critical Reynolds number corresponding to loss of stability decreases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 35–41, January–February, 1999. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-00586).     

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r ^−m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–43, January–February, 1998. The work was financially supported by the Russian Foundation for Fundamental Research (project No. 95-01-00483).     

Numerical modeling of the rotation of a solid body with a liquid-filled cavity having radial ribs

The boundary-value problem of unsteady vortex flow of a viscous incompressible fluid in a cylindrical vessel with radial ribs rotating at a variable angular velocity is solved using a finite-difference method. The results of the solution are used to calculate the motion of a system of a solid body and a cavity filled with a liquid. The results are compared with available experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 135–139, March–April, 2007.     

On Self-Oscillations during the Outflow of a Swirled Jet

It is shown that the core of a swirled helical flow can be described using a novel exact nonstationary solution of the hydrodynamic equations for a viscous incompressible fluid, which generalizes the rigid-body asymptotics for the Burgers and Sullivan vortices in the form of rigid-body rotation with a finite helicity. An estimate of the pressure fluctuations corresponding to this nonstationary vortex regime, which is proportional to the frequency of the swirled-jet core rotation as a rigid body and also depends on the parameters of the initial velocity field structure, is obtained. It is noted that this frequency may correspond to the frequency observed in the pressure fluctuation spectrum, which is almost proportional to the swirled flow rate in vortex acoustic emitters.     

The shock–vortex interaction patterns affected by vortex flow regime and vortex models

We have used a third-order essentially non-oscillatory method to obtain numerical shadowgraphs for investigation of shock–vortex interaction patterns. To search different interaction patterns, we have tested two vortex models (the composite vortex model and the Taylor vortex model) and as many as 47 parametric data sets. By shock–vortex interaction, the impinging shock is deformed to a S-shape with leading and lagging parts of the shock. The vortex flow is locally accelerated by the leading shock and locally decelerated by the lagging shock, having a severely elongated vortex core with two vertices. When the leading shock escapes the vortex, implosion effect creates a high pressure in the vertex area where the flow had been most expanded. This compressed region spreads in time with two frontal waves, an induced expansion wave and an induced compression wave. They are subsonic waves when the shock–vortex interaction is weak but become supersonic waves for strong interactions. Under a intermediate interaction, however, an induced shock wave is first developed where flow speed is supersonic but is dissipated where the incoming flow is subsonic. We have identified three different interaction patterns that depend on the vortex flow regime characterized by the shock–vortex interaction.      

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

Numerical calculations of supersonic underexpanded jets in a cocurrent flow with the use of parabolized Navier-Stokes equations

Results of a numerical study of three-dimensional supersonic jets propagating in a cocurrent flow are described. Averaged parabolized Navier-Stokes equations are solved numerically on the basis of a developed scheme, which allows calculations in supersonic and subsonic flow regions to be performed in a single manner. A jet flow with a cocurrent flow Mach number 0.05 ⩽ M_∞ ⩽ 7.00 is studied, and its effect on the structure of the mixing layer is demonstrated. The calculated results are compared with available experimental and numerical data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 54–63, May–June, 2008.     

Stability and nonuniqueness of axisymmetric flows in rotating spherical layers (nonlinear theory)

A numerical analysis technique and its results for viscous incompressible flows in the annulus between two concentric coaxial spheres generated by the rotation of the boundary spheres in the same or opposite directions are presented. It is shown that in the course of its development, the main flow passes through three characteristic stages which differ significantly from each other (qualitatively as well as quantitatively) with respect to meridional circulation and azimuthal flow. Depending on the fluid layer thickness, different stages of main flow development may precede the loss of stability; this determines the differences in the mechanisms of loss of stability of the main flow and in the nature of the secondary flows. The calculated results are compared with the experimental data. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 75–86, January–February, 1998.     

Receptivity of the boundary layer on a flat plate with a blunted leading edge to steady nonuniformity of the free stream

The flow past a flat plate with a blunted leading edge by a flow of a viscous incompressible fluid with a small spanwise-periodic, steady nonuniformity of the velocity profile is considered. Such a flow simulates the interaction of one type of vortex disturbances of a turbulent external flow with the boundary layer. The solution obtained predicts generation of strong disturbances in the boundary layer, which are similar to the streaky structure observed in the case of high free-stream turbulence. It is shown that the boundary-layer flow on blunted bodies is more sensitive to vortex disturbances than on a plate with a sharp leading edge. Central Aerohydrodynamic Institute, Zhukovskii, 140160. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 93–100, July–August, 2000.     

A Numerical Study of the Flow Around a Model Winged Seed in Auto-Rotation

In this study the flow around a winged-seed in auto-rotation is characterized using direct numerical simulations (DNS) at Reynolds number in the range 80–240, based on the descent speed and a characteristic chord length. In this range, the flow is approximately steady when observed from a reference frame fixed to the seed. For all cases, the flow structure consists of a wing tip vortex which describes a helical path, a vortex shed behind the nut of the seed and a stable leading edge vortex above the wing surface which merges with the tip vortex. With increasing Reynolds number, the leading edge vortex becomes more intense and gets closer to the wing surface. The simulation results also show the formation of a spanwise flow on the upper surface of the wing, moving fluid towards the wing tip in a region downstream and beneath the leading edge vortex. This spanwise flow is rather weak inside the core of the leading edge vortex, and the analysis of the streamlines show a very weak transport of vorticity along the vortex for the cases under consideration. The analysis of the flow suggests that the stabilization of the leading edge vortex is mainly due to non-inertial accelerations, although viscous effects may contribute, specially at lower Re. Furthermore, the leading edge vortex has been characterized by analysing the flow variables averaged along cross-sections of the vortex. While some quantities, like the spanwise velocity or the pressure inside the vortex, are rather insensitive to the threshold used to define the leading edge vortex, the same is not true for the circulation of the vortex or its averaged spanwise vorticity, due to the viscous nature of the vortex. Finally, it is observed that the spanwise vorticity scales with the angular rotation of the seed for the different Re.     

Formation of a 2D vortex pair and its 3D breakup: an experimental study

 Vortex pairs are studied using a dye tracing visualisation technique and a particle tracking velocimetry system. The vortex pairs are produced by gravity induced inlets of water issued through a uniform gap. The inlet Reynolds number is Re=Ud/ν≈875 in all tests (d being the gap width and U the cross sectional mean velocity), i.e. the flow is in the laminar regime. Initially, the dipolar vortex structure is two-dimensional, but after travelling a distance of a few times its own width, the flow structure becomes unstable, breaks up and changes into a three-dimensional flow structure. The breakup appears to be caused by an axial flow in the core centres of each vortex of the dipolar structure. These axial flows are induced by boundary effects related to the von Karman viscous pump. After the breakup, it is believed that a vortex ring is formed through reconnection of rudiments from the dipolar structure mediated by the wall induced vorticity. Received: 20 November 1995/Accepted: 14 November 1996     

Phase-resolved flow characteristics between two shrouded co-rotating disks

Flow characteristics in the interdisk midplane between two shrouded co-rotating disks were experimentally studied. A laser-assisted particle-laden flow-visualization method was used to identify the qualitative flow behaviors. Particle image velocimetry was employed to measure the instantaneous flow velocities. The flow visualization revealed rotating polygonal flow structures (hexagon, pentagon, quadrangle, triangle, and oval) existing in the core region of the interdisk spacing. There existed a difference between the rotating frequencies of the polygon and the disks. The rotating frequency ratio between the polygonal flow structure and the disks depended on the mode shapes of the polygonal core flow structures—0.8 for pentagon, 0.75 for quadrangle, 0.69 for triangle, and 0.6 for oval. The phase-resolved flow velocities relative to the bulk rotation speed of the polygonal core flow structure were calculated, and the streamline patterns were delineated. It was found that outside the polygonal core flow structure, there existed a cluster of vortex rings—each side of the polygon was associated with a vortex ring. The radial distributions of the time-averaged and phase-resolved ensemble-averaged circumferential and radial velocities were presented. Five characteristic regions (solid-body rotation region, hub-influenced region, buffer region, vortex region, and shroud-influenced region) were identified according to the prominent physical features of the flow velocity distributions in the interdisk midplane. In the solid-body rotation region, the fluid rotated at the angular velocity of the disks and hub. In the hub-influenced region, the circumferential flow velocity departed slightly from the disks’ angular velocity. The circumferential velocities in the hub-influenced and vortex regions varied linearly with variation of radial coordinates. The phase-resolved ensemble-averaged relative radial velocity profiles in the interdisk midplane at various phase angles exhibited grouping behaviors in three ranges of polygon phase angles (θ = 0 and α/2, 0 < θ < α/2, and α/2 < θ < α) because three-dimensional flow induced similar flow patterns to appear in the same range of polygon phase angles.     

Impact of short-span strip on oscillating-wing tip vortex

The impact of 12 spoiler–tab configurations, of different heights and widths, on the tip vortex generated by an oscillating NACA 0015 wing was investigated experimentally. For an oscillating wing equipped with a spoiler, the peak tangential velocity and core and total circulation were greatly reduced compared to a tab, regardless of its width, while the core radius remained largely unaffected with its center displaced vertically above the baseline wing. The most noticeable impact of a spoiler with a reduced height was its potential in alleviating the blade–vortex interaction (BVI) strength. Meanwhile, the largest favorable impact on the critical vortex flow parameters was achieved via a 25%-span spoiler–tab combination with a height of 5 and 2.5% chord, respectively. A contrary effect on the BVI suppression, especially during pitch-up, was, however, observed. The impact on the BVI can be improved by reducing the height of the spoiler at the expense of unfavorable change in the vortex strength and displacement.     

Measurements of the aperiodic wake of a hovering rotor

 An experimental investigation was made to study the aperiodic flow characteristics of the tip vortices generated by one-bladed and two-bladed hovering rotors. Measurements of the tip vortex locations and accompanying aperiodicity statistics were established as a function of vortex age. Velocity field measurements were made using three-component laser Doppler velocimetry. The average amplitude of the aperiodicity was found to be a fraction of the measured viscous core radius, this being approximately 5% of blade chord or about 50% of the core radius for wake ages of less than two rotor revolutions. The aperiodicity appeared isotropic. A numerical analysis of the aperiodicity problem based on convolution with an assumed displacement probability function showed that for this experiment the measured tangential velocities in the tip vortices were underestimated by approximately 20% and the viscous core radii were overestimated by 20%. There was no evidence that the number of blades or the blade passage adversely influenced the aperiodicity of the rotor tip vortices. Received: 22 August 1997/Accepted: 4 February 1998