An experiment on a taylor vortex flow in a gap with a small aspect ratio bifurcation of flows in an asymmetric system

Summary Both Taylor vortex flows in a symmetric or asymmetric system exhibit various patterns (cell modes). They can be classified by the process of cell formation, the number of cells and the direction of flow for the cell, into primary modes or secondary modes, and normal modes or anomalous modes. Following the previous report in which flows in a symmetric end condition were classified, in the present work, for flows in an asymmetric end condition, the Reynolds number at which a secondary mode bifurcates into another mode is experimentally investigated, and the bifurcation of the Taylor vortex flows in an asymmetric system when the Reynolds number is gradually decreased is presented in a bifurcation diagram.List of symbols R1 Radius of inner cylinder (2R1=40.19±0.006 mm) - R2 Radius of outer cylinder (2R2=60.11±0.024 mm) - D Clearance between cylinders (R2-R1=9.96±0.025 mm) - L Height of working fluid - Aspect ratio=L/D - Rotational angular speed - Kinematic viscosity - Re Reynolds number=R1D/     

Hydrodynamics of a vortex ring

We considered the kinematics and dynamics of a vortex ring in an incompressible fluid in toroidal coordinates. We obtained the change in the pressure difference along the boundary between two flow regions in the case of a moving torus.Notation , , toroidal coordinates - (V ;V ;V ) velocity of a fluid particle and its projections in toroidal coordinates - g ,g ,g metric tensor components - the Jacobian of transition to curvilinear coordinates - V ₀ velocity at the center of a vortex ring on its symmetry axis - x, y, z Cartesian coordinates - z, y, cylindrical coordinates - a distance from the axis of a torus (V=0) to its axis of symmetry (Oz) - angle between the Oy axis and the line that connects a fluid particle on the streamline =const, which represents a circle [16], with the center of this circle - U _z,U _y velocities in the cylindrical system of coordinates - ₀ stream function of a stationary vortex ring - velocity circulation - U V ₁, velocity of a rectilinear flow at infinity - ₁ stream function of a rectilinear flow - = ₀ + ₁ superposition of two flows - n=k ⁴=V ₁/V ₀ velocity ratio coefficient - R radius of a vortical region - U velocity of fluid particles at the boundary in polar coordinates (r, ) with the center at the coordinate origin (point 0) - fluid density - p ₀,p pressure at infinity and at a certain point of flow - pressure difference Polotsk State University, Polotsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 531–536, July–August, 1995.     

Numerical study of disk driven rotating MHD flow of a liquid metal in a cylindrical enclosure

Summary A numerical study of the steady laminar MHD flow driven by a rotating disk at the top of a cylinder filled with a liquid metal is presented. The governing equations in cylindrical coordinates are solved by a finite volume method. The effect of an axial magnetic field on the flow is investigated for an aspect ratioH/R equal to 1. The magnetic Reynolds number is assumed to be small whereas the interaction parameter,N, is large compared to unity. This allows to derive asymptotic results for the flow solution which are found in good agreement with the numerical calculations. The effect of the top, botton and vertical walls conductivity on the flow is studied. Various combinations of these conductivities are considered. The results obtained showed that one can control the primary flow through a good choice of the electrical conductivity of both the disk and cylinder walls.Notation B Magnetic field - H Height of the cylinder - Ha Hartmann number - jz Axial electric current - N Interaction parameter - P Dimensionless pressure - R Radius of the cylinder - Re Reynolds number - R _m Magnetic Reynolds number - r Dimensionless radius - V _r Dimensionless radial velocity - V _z Dimensionless axial velocity - V Dimensionless azimuthal velocity - Z Dimensionless height Greek symbols Density of the fluid - v Kinematic viscosity - Dynamic viscosity - Electrical conductivity - Angular velocity - Dimensionless electric potential - Thickness of the Ekman layer - Laplacian operator - r Increment of the grid in the radial direction - Z Increment of the grid in the axial direction     

Order parameters and energies of analytic and singular vortex lines in rotating3He-A

We present the expressions of the generalized Ginzburg-Landau (GL) theory for the free energy and the supercurrent in terms of thed vector, the magnetic fieldH, and operators containing the spatial gradient and the rotation. These expressions are then specialized to the Anderson-Brinkman-Morel (ABM) state. We consider eight single-vortex lines of cylindrical symmetry and radiusR=[2m/]^–1/2: the Mermin-Ho vortex, a second analytic vortex, and six singular vortices, i.e., the orbital and radial disgyrations, the orbital and radial phase vortices, and two axial phase vortices. These eight vortex states are determined by solving the Euler-Lagrange equations whose solutions minimize the GL free energy functional. For increasing field, the core radius of the texture of the Mermin-Ho vortex tends to a limiting value, while the core radius of the texture goes to zero. The gap of the singular vortices behaves liker forr 0, where ranges between and . The energy of the radial disgyration becomes lower than that of the Mermin-Ho vortex for fieldsH6.5H*=6.5×25 G (atT=0.99T _c and forR=10L*=60 µm, or=2.9 rad/sec). ForR 2 _T ( _T is the GL coherence length) or _c2 (upper critical rotation speed), the energies of the singular vortices become lower than the energies of the analytic vortices. This is in agreement with the exact result of Schopohl for a vortex lattice at _c 2. Finally, we calculate the correction of order (1 -T/T _c ) to the GL gap for the axial phase vortex.     

The temperature of an adiabatic surface at a turbulent boundary layer

A calculation of the temperature decrease of an adiabatic surface at a supersonic turbulent boundary layer is conducted. It is shown that the temperature decrease is a consequence of the appearance of a vortex chain in the flow near the walls. Comparison of calculated data with experimental gives qualitative agreement.Notation V incident flow velocity - V_v vortex velocity - v local velocity - u induced velocity - T thermodynamic temperature - T_w, T recovery temperature and undisturbed flow temperature - L length of depression - h₀ depth of depression - h distance from vortex center to wall - b relative vortex velocity - l _v vortex spacing - r recovery coefficient - R₀ recovery coefficient on smooth surface - c_p gas heat capacity at constant pressure - n vortex passage frequency - Re Reynolds number - M Mach number - velocity potential - time - vortex intensity Translated from Ihzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 5, pp. 903–908, May, 1971.     

Electronic computer study of separation flow features around a vibrating cylinder

The nonstationary separation flow around a circular cylinder performing harmonic vibrations across-the stream by an incompressible viscous fluid is investigated in a numerical experiment.Notation d, y, yo diameter, transverse deflection, and amplitude of cylinder vibrations - l spacing between vortices - ₁,₂ angular location of the points of separation - U_o unperturbed stream velocity - V_y velocity of transverse cylinder motion - u velocity of vortex motion - f₁ cylinder vibrations frequency - f vortex shedding frequency - t time - kinematic velocity - Sh₁=f₁d/U_o dimensionless cylinder vibrations frequency, the kinematic Strouhal number - Sh=fd/U_o Strouhal number of vortex shedding - Re=dU_o/ Reynold number - =tU_o/d dimensionless time - ¯y=y/d, ¯y_o=yo/d, ¯l=l/d; /U_o, t₁ period of cylinder vibrations - T=t₁U_o/d dimensionless period of vibrations Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 41–47, July, 1984.     

Measurement of vortex diffusivity and lifetime below the Kosterlitz-Thouless transition in helium films

Using vortex pair motion, we have measured the vortex diffusivity below the Kosterlitz-Thouless transition temperature (T _KT) in atomically thin superfluid helium films on both argon and neon substrates. The diffusivities are on the order of/m nearT _KT and then fall rapidly to10 ^–3 /m near0.1 K for argon and to0.03 /m for neon. However, for the thinnest helium film on neon, the diffusivity is large and independent of temperature. At small flow velocities, dissipation due to unpaired vortices indicates a vortex lifetime on the order of seconds for a neon substrate near0.1 K.     

Hydrodynamic contact between a rotating roll and a semipermeable tray

A viscous fluid flow in the gap between a side surface of a rotating roll and a rectangular cavity with a semipermeable bottom surface is considered in the Reynolds approximation.Notation x, y, z Cartesian coordinates - U peripheral velocity of the roll - W translational velocity of the tray - h gap height - h ₀ minimum gap - side gap - v _x,v _y,v _z velocity components - K coefficient of channel wall permeability - P,504-1 dimensional and dimensionless pressures - R roll radius - S channel width - fluid viscosity - angle - x₀,₁, , dimensional and dimensionless coordinates of the flow zone boundaries - dimensionless permeability - friction - ,q geometrical simplexes - dimensionless variable - Q bulk flow rate of fluid - F buoyancy force - N power - I ₁,I ₂ integral parameters of flow - ₁, ₂ angular velocities of rotor and cylinder - _c fluid density - q gravity acceleration - Re Reynolds number - R ₁,R ₂ radii of cylinder and rotor in a granulator Volgograd Technical University. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 612–618, July–August, 1995.     

Longitudinal critical current in type II superconductors. I. Helical vortex instability in the bulk

The vortex lattice in type II superconductors is unstable against the growth of helical perturbations if the current along the vortices exceeds a critical value. The longitudinal critical current, the pitch, and the spatially varying amplitude of the elliptically polarized helices are calculated from the London theory at the onset of instability in planar current distributions far from the surface. For weak pinning (_L² c ₆₆) the wavelength and width of the mode extend over the entire specimen, and the critical current is 2H(c ₆₆/c ₁₁)^1/4. For moderate pinning (c _{66 L}² c ₁₁) the wavelength and width are close to Campbell's pinning length (c ₁₁/_L)^1/2, and the critical current times its mean density is 2H ²(_L/c ₁₁)^1/2. For strong pinning (_L² c ₁₁) helical instability occurs at pin-free vortex sections, the helix wavelength is 2.2d, and the critical current density is 0.47Hd/² (H, d, c ₁₁ and c ₆₆), and _L are the magnetic field, spacing, elastic moduli, and pinning parameter of the vortex lattice, and is the magnetic penetration depth).     

Transport results from untwinned YBCO: Washboard frequency of the vortex lattice,and the quasiparticle mean free path

The washboard frequency of the moving vortex lattice in untwinned YBa₂ Cu₃ O_6.93 may be observed through mode-locking to an externally applied ac current of frequency _ext. The interference between and _ext results in jumps in the dc current-voltage characteristics when and _ext are harmonically related¹. The interference effect disappears in the vortex liquid state. The Hall conductivity _xy below T_c in YBCO contains contributions² from a positive quasiparticle (qp) term (H) and a negative vortex term (1/H). The qp term is surprisingly large well below T_c and implies a large gap anisotropy and a long qp mean free path (mfp). The thermal Hall effect³ _xy is closely related to the qp _xy; _xy is produced by asymmetric scattering of qp by pinned vortices. The qp mfp at H = 0, extracted from _xy and extended to low T by _xy, increases remarkably from 90 Å at T_c to more than 0.5m at 22 K.     

Mutual friction and vortex motion near the superfluid transition

The mutual friction parametersB and B for a moving vortex are calculated near the superfluid transition. They are proportional to the kinetic coefficient associated with the order parameter and, asT , diverge as (T – T)^–1/3, in agreement with experiment. The nonlinear couplings between the order parameter and the entropym, both the reversible one and the one in the free energy, are found to be crucial in the mutual friction near the point. These couplings were neglected in a previous paper by the author. First, the reversible coupling in the dynamic equations makes the chemical potential deviation long-ranged and causes the dissipation to take place only near the vortex core. Second,B can diverge asT T only in the presence of the coupling of the formm||² in the free energy. Thus theE model of Halperin et al., where the latter coupling is absent, cannot explain the critical anomaly ofB. The helical mode of a single vertex line is also investigated and its dispersion relation is found to be quite different from that at low temperatures. This mode has the same time scale as that of the second-sound mode when the wave vectors are of the order of the inverse correlation length, thus obeying the usual dynamic scaling law. The time correlation functions of the displacement fluctuations of a vortex line are also obtained. The force acting on a moving vortex is calculated and is found to be equal to the classical Magnus force.     

Natural convection about two-dimensional bodies with uniform surface heat flux in a porous medium

Summary The natural convection boundary-layer flow on horizontal circular cylinders of arbitrary shape in a saturated porous medium is considered when a uniform heat flux boundary condition is applied on the cylinder. A series expansion is derived which is applicable to cylinders of general shape. The case of a horizontal circular cylinder is then considered in detail. A finite-difference solution of this problem is obtained and compared with a simple approximate solution given by the series expansion, and these are found to be in good agreement over the lower half of the cylinder.Nomenclature g acceleration due to gravity - k _m thermal conductivity of the porous medium - K permeability of the porous medium - L characteristic length - q _w heat transfer rate - R _a ^* modified Rayleigh number based on heat transfer - T ₀ ambient temperature - u, v velocity components - x distance along the body surface measured from the lower stagnation point - y distance normal to the body surface - equivalent thermal diffusivity - coefficient of thermal expansion - v viscosity - non-dimensional temperature - , transformed coordinates - stream function - non-dimensional temperature of the wall - _T boundary-layer thickness With 2 Figures     

Initiation of air core in a swirl nozzle using time-independent power-law fluids

Summary Theoretical and experimental analyses have been carried out for determining the injection condition below which the formation of air core does not take place in the course of flow of a time-independent power-law fluid through a swirl nozzle. Analytical solution lends one distinct value of generalized Reynolds number at the inlet to a nozzle below which the air core is not formed. Experiments reveal that there exist two limiting values of such generalized Reynolds number regarding the formation of air core in a nozzle. One value being the upper limit below which steady flow occurs without air core, the other one is the lower limit above which steady flow with fully developed air core persists. In between these two limiting values, there prevails a transition zone through which fully developed air core is set up within the nozzle. For all the nozzles, theoretical results are in fair agreement with the experimental values of upper limit of generalized Reynolds numbers with respect to steady flow without air core. Amongst all the pertinent independent geometrical parameters of a nozzle, the orifice-to-swirl chamber-diameter ratio has the remarkable influence on generalized Reynolds number describing the initiation of air core.Nomenclature D ₁ Swirl chamber diameter - D ₂ Orifice diameter - D _s Diameter of tangential entry ports - E A non-dimensional parameter defined by Eq. (9) - E _R A non-dimensional parameter defined by Eq. (25) - K Flow consistency index - L ₁ Length of the swirl chamber - n Flow behaviour index - P Static pressure inside the nozzle - P _b Back-pressure of the nozzle - Q Volume flow rate - R Radius vector or longitudinal coordinate with respect to spherical coordinate system (Fig. 3) - R ₁ Radius of the swirl chamber - R ₂ Radius of the orifice - Generalized Reynolds number at the inlet to the nozzle - Limiting value of generalized Reynolds number describing initiation of air core - R _z Radius at any section - r Radial distance from the nozzle axis - r _a Air core radius - u Longitudinal component of velocity with respect to spherical coordinate system (Fig. 3) - V _r Radial velocity component - V _z Axial velocity component - V Tangential velocity component - Tangential velocity at inlet to the nozzle - v Component of velocity in the axial plane perpendicular toR (Fig. 3) - w Component of velocity perpendicular to axial plane with respect to the spherical coordinate system (Fig. 3) - z Distance along the nozzle axis from its inlet plane - Half of the spin chamber angle - Boundary layer thickness measured perpendicularly from the nozzle wall - ₂ Boundary layer thickness at the orifice - Angle, which a radius vector makes with the nozzle axis, in spherical coordinate system (Fig. 3) - Density of the fluid - Running coordinate in the azimuthal direction with respect to the cylindrical polar coordinate system as shown in Fig. 3 - Circulation constant With 8 Figures     

Distribution of finely dispersed particles with respect to residence time in a vortex chamber

The motion of finely dispersed particles is described statistically with the use of the Fokker-Planck equation. An expression is obtained for the particle distribution function with respect to residence time. Results of the calculation illustrate the dependence of the average particle residence time in the apparatus on the process parameters.Notation A constant - C' parameter characterizing the intensity of random forces - d _p particle diameter, m - K drying rate coefficient - r radial coordinate of the particle, m - R ₀ radius of the outlet orifice, m - R radius of the chamber, m - u,u _in,u _eq instantaneous, initial, and equilibrium moisture contents of the particle, kg/kg - V _r radial gas velocity, m/sec - W tangential velocity of the particle, m/sec - x=r/R dimensionless variable - dynamic viscosity, Pa·sec - _p, _g density of the particles and gas, kg/m³ - time, sec - angular velocity of gas suspension, sec^–1 Academic Scientific Complex A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus, Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 552–558, July–August, 1995.     

Convective instability of a free-convection vortex

The stability of free-convection vortex formations obtained in the laboratory is compared with natural tropical cyclones.Nomenclature r radius - density of air - P pressure drop between the center of a vortex and its edge - latitude of the location of a natural tropical cyclone Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 58, No. 3, pp. 416–419, March, 1990.     

Vortex depinning in superconducting wire networks

The pinning of vortices at low inductions is considered within the Ginzburg-Landau theory for both dense and sparse superconducting wire networks with periods d (T)and d (T),respectively. For dense networks, the depinning current is calculated analytically and found to be exponentially small compared to the characteristic pair-breaking current. The depinning current for sparse networks is obtained numerically. The anomalous compression modulus for a vortex lattice in a thin superconducting film is calculated, and the effects of the vortex lattice rigidity on thermally activated vortex creep in a dense network are investigated.     

Effect of thermal radiation on natural convection over cylinders of elliptic cross section

Summary The effect of conduction-radiation on natural convection flow of an optically dense viscous incompressible fluid along an isothermal cylinder of elliptic cross section has been investigated. The boundary layer equations governing the flow are shown to be nonsimilar. Full numerical solutions of the governing equations are obtained using the implicit finite difference method. The solutions are expressed in terms of the Nusselt number Nu against the eccentric angle in the range [0, ]. The working fluid is taken to have unit value of the Prandtl number, Pr, and the effects of varying the Planck number,R _d, the surface temperature parameter, _w, and the parameterA _O representing the ratio of the major and minor axes of the cylinder are investigated. From the present analysis it is found that the rate of heat transfer from the slender body is higher than from the blunt body and that these higher values become even higher due to an increase in the effect of radiation in the flow field.Nomenclature a semi-major axis of the cylinder - a _r Rosseland mean absorption coefficient - b semi-minor axis of the cylinder - C _p specific heat at constant pressure - f dimensionless stream function - g acceleration due to gravity - Gr Grashof number - Nu Nusselt number - Q _w surface heat flux - Pr Prandtl number - R _d Planck number (radiation-conduction parameter) - T temperature of the fluid - T _w temperature of the heated surface - T temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x coordinate measuring distance round the cylinder - y coordinate measuring distance normal to the cylinder Greek symbols eccentric angle - coefficient of cubial expansion - coefficient of thermal diffusivity - v kinematic viscosity     

Structure of flows in vortices

The structure of a gradient vortical flow was studied experimentally.Notation v_x, v_y, v_z flow velocity components in a rectangular coordinate system - v, v_r, v_z flow velocity components in a cylindrical coordinate system - v₁ tangential velocity at the boundary of solid revolution at r = r₁ - l length of the vortex - kinematic viscosity - R radius of the forming cylinder - circulation in the region of potential flow - second air flow rate through the eddy of ascending flows - Re=v₁r₁/ tangential Reynolds number - N=Q/r_o radial Reynolds number - a=l/r₀ configuration ratio for the vortex model - s=r_o/2Q effective exchange coefficient - a ^*=l/r configuration ratio for the vortex generator - s^*=R/Q constructive exchange coefficient - p=p–p pressure drop in the vortex relative to atmospheric pressure p - r^*= r/r₁ dimensionless radius of the vortex - v^*=v/v₁ dimensionless tangential velocity - a ^*/a gradient ratio Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 611–618, October, 1980.     

Mixed convection boundary-layer on a vertical cylinder embedded in a saturated porous medium

Summary The mixed convection boundary layer on a vertical circular cylinder embedded in a saturated porous medium is considered. It is found that the flow depends on the parameter =R _a /P _e whereR _a andP _e are the Rayleigh number and Peclet number respectively. gives the ratio of the velocity scale for free convection to that for the forced convection. When is small the solution is, to a first approximation, obtained by a known heat conduction problem. The flow near the leading edge is considered and it is shown that a solution is possible only for ₀, ₀–1.354, and that a stable finite-difference solution away from the leading edge can be obtained only if –1; with <–1 there is a region of reversed flow near the cylinder. The finite-difference scheme is unable to give a satisfactory solution at very large distances from the leading edge, and to overcome this difficulty a simple approximate solution is developed. This solution shows that at large distances along the cyclinder, forced convection eventually becomes the dominant mechanism for heat transfer. This is also confirmed by an asymptotic solution of the full boundarylayer problem.Nomenclature a radius of cylinder - g acceleration of gravity - K permeability of the porous medium - N _u non-dimensional Nusselt number - r radial coordinate - non-dimensionalr=r/a - R _a Rayleigh number=(g T)Ka/ - P _e Peclet number=U ₀ a/ - T temperature - T _w temperature of the cylinder (constant) - T ₀ temperature of the ambient fluid (constant) - T temperature difference=T _w –T ₀ - u Darcy's law velocity in thex direction - U ₀ velocity of the outer flow - v Darcy's law velocity in ther-direction - x coordinate measuring distance along the cylinder - X non-dimensionalx,=x(aP _e )^–1 - equivalent thermal diffusivity - coefficient of thermal expansion - ratio of free to forced convection=R _a /P _e - viscosity of the convective fluid - density of the ambient fluid - non-dimensional temperature - stream function With 2 Figures     

Enhancement of heat transfer of mixed convection for heated blocks using vortex shedding generated by an oblique plate in a horizontal channel

Summary The problem of heat transfer enhancement of mixed convective flow past heated blocks in a horizontal channel is investigated. The heat transfer enhancement in this paper has been accomplished by the installation of an oblique plate to generate vortex shedding, which is used in flow modulation. Results for the details of the streamlines in the channel and the Nusselt number along the blocks with and without an oblique plate have been presented.Notation C _p pressure coefficient (2f Pds/f ds) - d length of an oblique plate - ds surface area increment along an oblique plate - f_s frequency of the vortex shedding - Gr Grashof number - H channel wall-to-wall spacing - h height of the block - k thermal conductivity - L channel length - Nu Nusselt number - time-mean Nusselt number (f Nudt/f dt) - average time-mean Nusselt number - n normal vector - P dimensionless pressure (p ^*/(u ² ) - p ^* pressure - Pr Prandtl number (/) - q heat flux at the block boundary - Re Reynolds number (u w/v) - St Strouhal number (df_ssin /u ) - T^* temperature - T uniform inlet temperature - t dimensionless time (t ^* / (w/u )) - t dimensionless time increment - t ^* time - u uniform inlet velocity - u, v dimensionless velocity components (u=u ^*/u ,v=v ^*/v ) - u ^*,v ^* velocity components - w width of the block - x,y dimensionlessx ^*,y ^* coordinates (x=x ^*/w,y=y ^*/w) - x ^*,y ^* physical coordinates - thermal diffusivity - angle of inclination for a plate - dimensionless temperature ((T^*–T ^* )/(qw/k)) - v kinematic viscosity of fluid