The nonlinear calculation of Taylor-vortex flow between eccentric rotating cylinders

1975 ◽  
Vol 67 (1) ◽  
pp. 85-111 ◽  
Author(s):  
R. C. Diprima ◽  
J. T. Stuart
Keyword(s):  
Nonlinear Stability ◽  
Angular Position ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Perturbation Scheme ◽  
Rotating Cylinders ◽  
Clearance Ratio ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Small Parameters

This paper is concerned with the nonlinear stability of the flow between two long eccentric rotating cylinders. The problem, which is of interest in lubrication technology, is an extension both of the authors’ earlier work on the linear eccentric case and of still earlier work by Davey and others on the nonlinear concentric analysis of Taylor-vortex development. There are four parameters which are assumed small in the analysis; they are the mean clearance ratio, the eccentricity, the amount by which the Taylor number exceeds its critical value; and the Taylor-vortex amplitude. Following the earlier work mentioned above, relation-ships are specified between these parameters in order to develop a satisfactory perturbation scheme. Thus a non-local solution is obtained to the nonlinear stability problem, in which the whole flow field is taken into account.Of some importance in the analysis is the fact that it is necessary to allow for the development of a pressure field substantially bigger than that associated with Taylor vortices in the concentric case, owing to the Reynolds lubrication effect in a viscous fluid moving through a converging passage. I n order to achieve this mathematically, it is necessary to solve the continuity equation to a higher order than is necessary for the momentum equations.It is found that the angular position for maximum vortex activity, which is 90° downstream of the maximum gap in the linear case, can taken on any value between 0 and 90°, depending on the value of the supercritical Taylor number. For a particular experiment of Vohr (1968) acceptable agreement is obtained for this angle (50°), though the ‘small’ parameters are somewhat outside the expected range of perturbation theory. Formulae are obtained for the torque and forces acting on the inner cylinder.

The effects of eccentricity on torque and load in Taylor-vortex flow

1978 ◽  
Vol 87 (2) ◽  
pp. 209-231 ◽  
Author(s):  
P.M. Eagles ◽  
J. T. Stuart ◽  
R. C. Diprima
Keyword(s):  
Vortex Flow ◽  
Nonlinear Effects ◽  
Critical Value ◽  
Taylor Number ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Small Parameters ◽  
Extended Analysis

This paper extends two earlier papers in which DiPrima & Stuart calculated first (1972b) the critical Taylor number to order ε2, where the eccentricity ε is proportional to the displacement of the axes of the circular cylinders, and second (1975) the torque and load to order ε associated with nonlinear effects of Taylor vortices. In the latter paper, it was shown that to order ε the torque arising from the Taylor vortices is identical with that for the concentric problem, which was first calculated, by a perturbation method, by Davey (1962). This deficiency is remedied in the present paper, where the calculation is taken to order ε2. It is found that, as ε rises, the torque associated with the Taylor vortices falls slightly when we keep constant the percentage elevation of the Taylor number above the ε-dependent critical value. This result is in accordance with experimental observations by Vohr (1967, 1968). In addition, results of calculations of the pressure field developed by the Taylor-vortex flow in association with the eccentric geometry are presented; this is larger than in the concentric case owing to a Reynolds lubrication effect. Also given are the associated components of the load on the inner cylinder, but only for Taylor numbers close to the critical value.One additional observation by Vohr, for cylinders with a mean ratio of the gap to the inner radius of 0·099, was that the maximum Taylor-vortex strength with ε = 0·475 occurred some 50° downstream of the maximum gap for a 20% elevation of the Taylor number above the critical value. Calculations in the two earlier papers (1972b, 1975) gave 90 and 76°, respectively, for that angle. Note that in the 1975 paper a geometrical correction of order ε was included. Here, with an additional modification of order ε due to the flow, this result is improved to 49° by the extended analysis presented, although the ‘small’ parameters are somewhat outside the range for which perturbation theory is expected to be valid.

scholarly journals On the finite Dean problem: linear theory

1988 ◽  
Vol 30 (2) ◽  
pp. 157-178
Author(s):  
B. J. Kachoyan ◽  
P. J. Blennerhassett
Keyword(s):  
Outer Cylinder ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Mean Flow ◽  
Dean Flow ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Concentric Cylinders ◽  
End Conditions ◽  
Pressure Driven Flow

AbstractThe Dean problem of pressure-driven flow between finite-length concentric cylinders is considered. The outer cylinder is at rest and the small-gap approximation is used. In a similar procedure to that of Blennerhassett and Hall [8] in the context of Taylor vortices, special end conditions are imposed in which the ends of the cylinder move with the mean flow, allowing the use of a perturbation analysis from a known basic flow. Difficulties specific to Dean flow (and more generally to non-Taylor-vortex flow) require the use of a parameter α which measures the relative strengths of the velocities due to rotation and the pressure gradient, to trace the solution from Taylor to Dean flow. Asymptotic expansions are derived for axial wavenumbers at a given Taylor number. The calculation of critical Taylor number for a given cylinder height is then carried out. Corresponding stream-function contours clearly show features not evident in infinite flow.

Marginal instability in Taylor–Couette flows at a very high Taylor number

1979 ◽  
Vol 94 (3) ◽  
pp. 453-463 ◽  
Author(s):  
A. Barcilon ◽  
J. Brindley ◽  
M. Lessen ◽  
F. R. Mobbs
Keyword(s):  
Vortex Flow ◽  
Outer Cylinder ◽  
Quantitative Agreement ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Centrifugal Instability ◽  
The One ◽  
Flow Experiments

We report on a set of turbulent flow experiments of the Taylor type in which the fluid is contained between a rotating inner circular cylinder and a fixed concentric outer cylinder, focusing our attention on very large Taylor number values, i.e. \[ 10^3 \leqslant T/T_c \leqslant 10^5, \] where Tc is the critical value of the Taylor number T for onset of Taylor vortices. At such large values of T, the turbulent vortex flow structure is similar to the one observed when T – Tc is small and this structure is apparently insensitive to further increases in T. These flows are characterized by two widely separated length scales: the scale of the gap width which characterizes the Taylor vortex flow and a much smaller scale which is made visible by streaks in the form of a ‘herring-bone’-like pattern visible at the walls. These are conjectured to be Görtler vortices which arise as a result of centrifugal instability in the wall boundary layers. Ideas of marginal instability by which we postulate that both the Taylor and Görtler vortex structures are marginally unstable on their own scale seem to provide good quantitative agreement between predicted and observed Görtler vortex spacings.

Wavy vortices in the flow between two long eccentric rotating cylinders II. Nonlinear theory

1977 ◽  
Vol 354 (1679) ◽  
pp. 459-489 ◽  
Keyword(s):  
Experimental Data ◽  
Nonlinear Stability ◽  
Nonlinear Theory ◽  
Stability Problem ◽  
Critical Value ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Rotating Cylinders ◽  
Parameter Expansion ◽  
Theoretical Results

.In this paper we are confronted with a nonlinear stability problem of the flow between two long eccentric rotating cylinders. We make an extension of part I, including an analysis of the influence exerted by a nonlinear Taylor vortex mode with amplitude BØ) on the result of part I. The problem, which is of interest in lubrication technology, can be seen as an extension of the paper of Davey, Di Prima & Stuart (1968), where a similar problem has been treated for the concentric case. We define a new parameter y = T — T c (0, e) ^ eT 1 which is the amount by which the Taylor number T exceeds its critical value, T c (O e ) (proportional to the square of the speed of the inner cylinder), with e the eccentricity. The problem is solved by using a double parameter expansion, first the expansion in y 1/2 from y° to the y order on the equations and afterwards the expansion in e from e° to e 2 in certain integrals. A formula for the critical Taylor number is obtained. Qualitative and consistent agreement is obtained with the experimental data of Vohr for the occurrence of wavy instability, though the theoretical results are a little lower, due perhaps to the small-gap assumption and the absence of the quintic terms in the nonlinear expansion.

scholarly journals Turbulent Taylor vortex flow

1979 ◽  
Vol 93 (3) ◽  
pp. 515-527 ◽  
Author(s):  
E. L. Koschmieder
Keyword(s):  
Experimental Error ◽  
Initial Conditions ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Narrow Gap ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Critical Wavelength ◽  
Wide Gap ◽  
Gap Width

The wavelength of turbulent Taylor vortices at very high Taylor numbers up to 40000Tc, has been measured in long fluid columns with radius ratios η = 0·896 and η = 0·727. Following slow acceleration procedures the wavelength (in units of the gap width) of turbulent axisymmetric vortices was found to be λ = 3·4 ± 0·1 with the small gap and about λ = 2·4 ± 0·1 with the larger gap, and thus in both cases substantially larger than the critical wavelength of laminar Taylor vortices. In the narrow and wide gap the wavelength was, within experimental error, independent of the Taylor number for T > 100Tc. In the experiments with the narrow gap a clear dependence of the value of the wavelength of the turbulent vortices on initial conditions was found. After sudden starts to Taylor numbers > 700Tc the wavelength of steady axisymmetric turbulent vortices was only 2·4 ± 0·05, being then the same as the wavelength of the vortices after sudden starts in the wide gap, and being, within the experimental error, independent of the Taylor number. In the narrow gap all values of the wavelength between λmax = 3·4 and λmin = 2·4 can be realized as steady states through different acceleration procedures. In the wide gap the dependence of the wavelength on initial conditions is just within the then larger experimental uncertainty of the measurements.

The Taylor Vortex Regime in the Flow Between Eccentric Rotating Cylinders

1974 ◽  
Vol 96 (1) ◽  
pp. 127-134 ◽  
Author(s):  
F. R. Mobbs ◽  
M. A. M. A. Younes
Keyword(s):  
Axial Flow ◽  
Journal Bearings ◽  
Transition To Turbulence ◽  
Taylor Vortex ◽  
Rotating Cylinders ◽  
Clearance Ratio ◽  
Taylor Vortices ◽  
Wide Range ◽  
Small Clearance

With the exception of very small clearance ratios, transition to turbulence in journal bearings is likely to be preceded by the appearance of Taylor vortices. The resultant regime may extend over a wide range of Taylor numbers and include transitions to several types of wavy vortex modes. The influence of eccentricity, clearance ratio, and axial flow on the critical Taylor numbers corresponding to the appearance of regular Taylor cells and their subsequent wavy mode transformations is reviewed.

Wavy vortices in the flow between two long eccentric rotating cylinders I. Linear theory

1977 ◽  
Vol 354 (1679) ◽  
pp. 441-457 ◽  
Keyword(s):  
Vortex Flow ◽  
Nonlinear Effects ◽  
Time Variable ◽  
Taylor Vortex ◽  
Wave Instability ◽  
Clearance Ratio ◽  
Taylor Vortices ◽  
Eccentric Cylinders ◽  
Taylor Vortex Flow ◽  
Mathematical Investigation

This analysis deals with the occurrence of Coles’ wavy vortices in the flow between two eccentric cylinders when no Taylor vortex flow is present, and where the eccentricity is small and the clearance ratio is very small. This mathematical investigation can be seen as an extension of the previous papers of Di Prima (1963), Roberts (1965), Davey, Di Prima & Stuart (1968) on the concentric problem and the paper of Di Prima & Stuart (19726) which includes eccentricity. An extension to the latter paper is made by the inclusion of the time variable r in the equation for the amplitude B((/),t) of the Taylor vortices, (j) being the azimuthal variable. The results are compared with the observed critical speeds of Vohr (1967) for development of wave instability on Taylor vortices, and are qualitatively similar; they differ in magnitude from experiment, however, probably due to the absence of nonlinear effects.

Development and Effects of Super-Critical Taylor-Vortex Flow in a Lightly Loaded Journal Bearing

1974 ◽  
Vol 96 (1) ◽  
pp. 28-35 ◽  
Author(s):  
R. C. DiPrima ◽  
J. T. Stuart
Keyword(s):  
Vortex Flow ◽  
Axial Direction ◽  
Basic Flow ◽  
Circular Cylinders ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Secondary Motion ◽  
The Stability ◽  
Unified Description

At sufficiently high operating speeds in lightly loaded journal bearings the basic laminar flow will be unstable. The instability leads to a new steady secondary motion of ring vortices around the cylinders with a regular periodicity in the axial direction and a strength that depends on the azimuthial position (Taylor vortices). Very recently published work on the basic flow and the stability of the basic flow between eccentric circular cylinders with the inner cylinder rotating is summarized so as to provide a unified description. A procedure for calculating the Taylor-vortex flow is developed, a comparison with observed properties of the flow field is made, and formulas for the load and torque are given.

Numerical investigation of supercritical Taylor-vortex flow for a wide gap

1984 ◽  
Vol 138 ◽  
pp. 21-52 ◽  
Author(s):  
H. Fasel ◽  
O. Booz
Keyword(s):  
Flow Field ◽  
Vortex Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Taylor Number ◽  
Taylor Vortex ◽  
Plane Boundary ◽  
Aspect Ratios ◽  
Taylor Vortex Flow ◽  
Wide Gap

For a wide gap (R1/R2= 0.5) and large aspect ratiosL/d, axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100Tac(or 10Rec). Prior to this study numerical solutions were reported up to only 16Tac. The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100Tac. Spectral analysis (with respect toz) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercriticalTa. For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

Visual Observations and Torque Measurements in the Taylor Vortex Regime Between Eccentric Rotating Cylinders

1971 ◽  
Vol 93 (1) ◽  
pp. 121-129 ◽  
Author(s):  
P. Castle ◽  
F. R. Mobbs ◽  
P. H. Markho
Keyword(s):  
Outer Cylinder ◽  
Taylor Vortex ◽  
Rotating Cylinders ◽  
Taylor Vortices ◽  
Torque Measurements ◽  
Good Agreement ◽  
Visual Observations

The instability of Taylor vortices in the flow between a stationary outer cylinder and an eccentric rotating inner cylinder has been investigated by visual observations and by torque measurements. It is shown that both a “weak” and “strong” wavy mode of instability can be detected by torque measurements, giving critical Taylor numbers in good agreement with visual observations.

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