Frictional Moment of Flow Between Two Concentric Spheres, One of Which Rotates

1978 ◽  
Vol 100 (1) ◽  
pp. 97-106 ◽  
Author(s):  
K. Nakabayashi
Keyword(s):  
Reynolds Number ◽  
Empirical Formula ◽  
Outer Sphere ◽  
Basic Flow ◽  
Confined Space ◽  
Clearance Ratio ◽  
Spherical Annulus ◽  
Concentric Spheres ◽  
Annulus Flow ◽  
Rotating Bodies

Frictional moment is investigated experimentally for the flow between a rotating inner sphere and a stationary outer one, and for the flow between a rotating outer sphere and a stationary inner one, respectively. The existence of four basic flow regimes for both is verified, and their extent is mapped over a range of Reynolds number-clearance ratio combinations. An empirical formula for the coefficient of frictional moment is obtained for every flow regime. The coefficient of frictional moment obtained for spherical annulus flow is then correlated with that of the flow around a disk rotating in a confined space, and the effect of the surface curvature of rotating bodies is considered.

Torque Characteristics for Spherical Annulus Flow

1979 ◽  
Vol 101 (2) ◽  
pp. 284-286 ◽  
Author(s):  
A. M. Waked ◽  
B. R. Munson
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Reynolds Numbers ◽  
Flow Effect ◽  
Spherical Annulus ◽  
Concentric Spheres ◽  
Annulus Flow ◽  
Angular Velocities

The torque needed to rotate concentric spheres when the spherical annulus gap between them is filled with a viscous fluid depends on the Reynolds number and the ratio of the angular velocities of the two spheres. Experimental torque results for low to moderate Reynolds numbers are presented. The secondary flow effect is evident.

scholarly journals Numerical studies of viscous incompressible flow between two rotating concentric spheres

2004 ◽  
Vol 2004 (2) ◽  
pp. 91-106 ◽  
Author(s):  
E. O. Ifidon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Outer Sphere ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Stable Configuration ◽  
Axially Symmetric ◽  
Closed Curves ◽  
Concentric Spheres ◽  
Flow Variables

The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier-Stokes equations expressed in terms of a stream function-vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite-difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments

1975 ◽  
Vol 69 (4) ◽  
pp. 705-719 ◽  
Author(s):  
B. R. Munson ◽  
M. Menguturk
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Outer Sphere ◽  
Radius Ratio ◽  
Large Reynolds Number ◽  
Spherical Annulus ◽  
Flow Part ◽  
Linearized Theory ◽  
Flow Transitions ◽  
The Stability

The stability of flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.

Forced-Convection Heat Transfer in a Spherical Annulus Heat Exchanger

1982 ◽  
Vol 104 (4) ◽  
pp. 670-677 ◽  
Author(s):  
D. B. Tuft ◽  
H. Brandt
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Reynolds Number ◽  
Outer Sphere ◽  
Reynolds Numbers ◽  
Flux Rate ◽  
Convection Heat ◽  
Spherical Annulus ◽  
Inner Sphere ◽  
Sphere Radius

Results are presented of a combined numerical and experimental study of steady, forced-convection heat tranfer in a spherical annulus heat exchanger with 53°C water flowing in an annulus formed by an insulated outer sphere and a 0°C inner sphere. The inner sphere radius is 139.7 mm, the outer sphere radius is 168.3 mm. The transient laminar incompressible axisymmetric Navier-Stokes equations and energy equation in spherical coordinates are solved by an explicit finite-difference solution technique. Turbulence and buoyancy are neglected in the numerical analysis. Steady solutions are obtained by allowing the transient solution to achieve steady state. Numerically obtained temperature and heat-flux rate distributions are presented for gap Reynolds numbers from 41 to 465. Measurements of inner sphere heat-flux rate distribution, flow separation angle, annulus fluid temperatures, and total heat transfer are made for Reynolds numbers from 41 to 1086. The angle of separation along the inner sphere is found to vary as a function of Reynolds number. Measured total Nusselt numbers agree with results reported in the literature to within 2.0 percent at a Reynolds number of 974, and 26.0 percent at a Reynolds number of 66.

scholarly journals Transitional vortices in wide-gap spherical annulus flow

2005 ◽  
Vol 2005 (18) ◽  
pp. 2913-2932 ◽  
Author(s):  
E. O. Ifidon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Outer Sphere ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Asymmetric Vortex ◽  
Concentric Spheres ◽  
Wide Gap ◽  
High Reynolds

We develop a semianalytic formulation suitable for solving the Navier-Stokes equations governing the induced steady, axially symmetric motion of an incompressible viscous fluid confined in a wide gap between two differentially rotating concentric spheres. The method is valid for arbitrarily high Reynolds number and aids in the presentation of multiple steady-state flow patterns and their bifurcations. In the case of a rotating inner sphere and a stationary outer sphere, linear stability analysis is conducted to determine whether or not the computed solutions are stable. It is found that the solution transforms smoothly into an unstable solution beginning with asymmetric vortex pairs identified near the point of a symmetry-breaking bifurcation which occurs at Reynolds number 589. This solution transforms smoothly into an unstable asymmetric vortex solution as the Reynolds number increases. Flow modes whose branches have not been previously reported are found using this method. The origin of the flow modes obtained are discussed using bifurcation theory.

The almost-rigid rotation of viscous fluid between concentric spheres

1956 ◽  
Vol 1 (5) ◽  
pp. 505-516 ◽  
Author(s):  
Ian Proudman
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Distribution ◽  
Outer Sphere ◽  
Reynolds Numbers ◽  
Rotating Systems ◽  
Velocity Gradients ◽  
Concentric Spheres ◽  
Viscous Stresses

Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model.It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case.

Rotating elliptic cylinders in a viscous fluid at rest or in a parallel stream

1977 ◽  
Vol 79 (1) ◽  
pp. 127-156 ◽  
Author(s):  
Hans J. Lugt ◽  
Samuel Ohring
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Numerical Solutions ◽  
Elliptic Cylinder ◽  
The Body ◽  
Oscillatory Behaviour ◽  
Incompressible Fluid Flow ◽  
Transient Period ◽  
Parallel Stream ◽  
Rotating Bodies

Numerical solutions are presented for laminar incompressible fluid flow past a rotating thin elliptic cylinder either in a medium at rest at infinity or in a parallel stream. The transient period from the abrupt start of the body to some later time (at which the flow may be steady or periodic) is studied by means of streamlines and equi-vorticity lines and by means of drag, lift and moment coefficients. For purely rotating cylinders oscillatory behaviour from a certain Reynolds number on is observed and explained. Rotating bodies in a parallel stream are studied for two cases: (i) when the vortex developing at the retreating edge of the thin ellipse is in front of the edge and (ii) when it is behind the edge.

On the flow properties of a fluid between concentric spheres

1971 ◽  
Vol 47 (4) ◽  
pp. 799-809 ◽  
Author(s):  
S. G. H. Philander
Keyword(s):  
Outer Sphere ◽  
Azimuthal Velocity ◽  
Ekman Layer ◽  
Shear Layers ◽  
The Other ◽  
Axial Motion ◽  
Flow Properties ◽  
Axis Of Rotation ◽  
Concentric Spheres ◽  
Angular Velocities

Proudman (1956) and Stewartson (1966) analyzed the dynamical properties of a fluid occupying the space between two concentric rotating spheres when the angular velocities of the spheres are slightly different, in other words, when the motion relative to a reference frame rotating with one of the spheres is due to an imposed azimuthal velocity which is symmetric about the equator. The consequences of forcing motion across the equator are explored here. Whereas the flow inside the cylinder [Cscr ] circumscribing the inner sphere and having generators parallel to the axis of rotation is similar to that which results when the driving is symmetric, the flow outside [Cscr ] is quite different. The Ekman layer on the outer sphere persists outside [Cscr ] - its dynamics is modified in the vicinity of the equator - and is instrumental in transferring fluid from one hemisphere to the other. The divergence of this Ekman layer causes slow, axial motion in the inviscid region outside [Cscr ]. On [Cscr ], two shear layers of thicknessO(R−2/7) andO(R−1/3) (whereRis the Reynolds number, assumed large) remove discontinuities in the flow field and return fluid from one hemisphere to the other (rather than one Ekman layer to the other as is the case when the driving is azimuthal).

The development of Long's vortex

1995 ◽  
Vol 286 ◽  
pp. 359-377 ◽  
Author(s):  
P.G. Drazin ◽  
W.H.H. Banks ◽  
M.B. Zaturska
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Viscous Fluid ◽  
Numerical Integration ◽  
Basic Flow ◽  
Asymptotic Results ◽  
Swirling Jets ◽  
Spatial Stability ◽  
Rotationally Symmetric ◽  
Direct Numerical Integration

This paper describes the solution of Long's problem for steady rotationally symmetric swirling jets in a uniform viscous fluid. Long found these vortices in 1958 by assuming a similarity form of solution, and in 1961 solved the consequent problem in the boundary-layer limit, finding dual solutions. The overall pattern of the solutions to the problem for general values of the Reynolds number is described. The linear spatial stability of the flows to small steady disturbances is analysed and a few results presented. In particular, details of the solutions and their stability are given asymptotically for small and large values of the Reynolds number. The asymptotic results for the basic flow are linked by direct numerical integration of the flow at several finite positive values of the Reynolds number.

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