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.

Turbulent flow between counter-rotating concentric cylinders: a direct numerical simulation study

2008 ◽  
Vol 615 ◽  
pp. 371-399 ◽  
Author(s):  
S. DONG
Keyword(s):  
Turbulent Flow ◽  
Large Scale ◽  
Outer Cylinder ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Taylor Vortex ◽  
Small Scale ◽  
Mean Flow ◽  
Concentric Cylinders ◽  
High Reynolds

We report three-dimensional direct numerical simulations of the turbulent flow between counter-rotating concentric cylinders with a radius ratio 0.5. The inner- and outer-cylinder Reynolds numbers have the same magnitude, which ranges from 500 to 4000 in the simulations. We show that with the increase of Reynolds number, the prevailing structures in the flow are azimuthal vortices with scales much smaller than the cylinder gap. At high Reynolds numbers, while the instantaneous small-scale vortices permeate the entire domain, the large-scale Taylor vortex motions manifested by the time-averaged field do not penetrate a layer of fluid near the outer cylinder. Comparisons between the standard Taylor–Couette system (rotating inner cylinder, fixed outer cylinder) and the counter-rotating system demonstrate the profound effects of the Coriolis force on the mean flow and other statistical quantities. The dynamical and statistical features of the flow have been investigated in detail.

Experimental Study on Oscillatory Couette-Taylor Flows Behaviour

2013 ◽  
Author(s):  
Emna Berrich ◽  
Fethi Aloui ◽  
Jack Legrand
Keyword(s):  
Mass Transfer ◽  
Angular Velocity ◽  
Time Evolution ◽  
Vortex Flow ◽  
Outer Cylinder ◽  
Taylor Vortex ◽  
Shear Rates ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Wall Shear

In the simplest and original case of study of the Taylor–Couette TC problems, the fluid is contained between a fixed outer cylinder and a concentric inner cylinder which rotates at constant angular velocity. Much of the works done has been concerned on steady rotating cylinder(s) i.e. rotating cylinders with constant velocity and the various transitions that take place as the cylinder(s) velocity (ies) is (are) steadily increased. On this work, we concentrated our attention in the case in which the inner cylinder velocity is not constant, but oscillates harmonically (in time) clockwise and counter-clockwise while the outer cylinder is maintained fixed. Our aim is to attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case. If the modulation amplitude is large enough to destabilise the circular Couette flow, two classes of axisymmetric Taylor vortex flow are possible: reversing Taylor Vortex Flow (RTVF) and Non-Reversing Taylor Vortex Flow (NRTVF) (Youd et al., 2003; Lopez and Marques, 2002). Our work presents an experimental investigation of the effect of oscillatory Couette-Taylor flow, i.e. both the oscillation frequency and amplitude on the apparition of RTVF and NRTVF by analysing the instantaneous and local mass transfer and wall shear rates evolutions, i.e. the impact of vortices at wall. The vortices may manifest themselves by the presence of time-oscillations of mass transfer and wall shear rates, this generally corresponds to an instability apparition even for steady rotating cylinder. On laminar CT flow, the time-evolution of wall shear rate is linear. It may be presented as a linear function of the angular velocity, i.e. the evolution is steady even if the angular velocity is not steady. At a “critical” frequency and amplitude, the laminar CT flow is disturbed and Taylor vortices appear. Comparing to a steady velocity case, oscillatory flow accelerate the instability apparition, i.e. the critical Taylor number corresponds to the transition is smaller than that of the steady case. For high oscillation amplitudes of the inner cylinder rotation, the mass transfer time-evolution has a sinusoidal evolution with non equal oscillation amplitudes. If the oscillation amplitude is large enough, it can destabilize the laminar Couette flow, Taylor vortices appears. The vortices direction can be deduced from the sign of the instantaneous wall shear rate time evolution.

Experimental Study on the Effects of Surface Roughness on the Behaviors of Hydrodynamic Instabilities of Couette-Taylor Flow

2017 ◽  
Author(s):  
Mostafa Monfared ◽  
Lamia Gaied ◽  
Emna Berrich ◽  
Ebrahim Shirani ◽  
Maxence Bigerelle ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Vortex Flow ◽  
Outer Cylinder ◽  
Wire Mesh ◽  
Taylor Vortex ◽  
Hydrodynamic Instabilities ◽  
Taylor Flow ◽  
Taylor Vortex Flow ◽  
Concentric Cylinders ◽  
Effect Of Surface

Couette-Taylor flow is a type of fluid flow that occurs in the annulus between differentially concentric cylinders, when the outer cylinder is fixed and the inner cylinder rotates and the rotation rate exceeds a critical value. In this research we have studied the effect of surface roughness on the hydrodynamic structures of Couette-Taylor Flow. The PIV technique has been applied for flow visualization. At first, for a smooth surface, the different flow patterns include Couette flow, Taylor vortex flow, wavy vortex flow, modulated wavy vortex flow and turbulent flow. They are investigated numerically and experimentally. The transition Taylor number for every flow regime is also taken into consideration. The results showed that the numerical results correspond quite well to the experimental data. Then, the different surface conditions for inner cylinder which are studied are: a smooth one, a sandpaper-type P180, a canvas plastic with different wire-mesh roughness. They are used to study the effects of surface roughness on the flow structures and critical Taylor numbers. The experimental results showed that the roughness causes a delay in the appearance of the first instabilities.

Centrifugal instabilities of circumferential flows in finite cylinders: nonlinear theory

1980 ◽  
Vol 372 (1750) ◽  
pp. 317-356 ◽  
Keyword(s):  
Vortex Flow ◽  
Outer Cylinder ◽  
Perturbation Expansion ◽  
Direct Consequence ◽  
Finite Amplitude ◽  
Taylor Vortex ◽  
Taylor Vortex Flow ◽  
Concentric Cylinders ◽  
Axisymmetric Disturbances ◽  
Finite Cylinders

In an earlier paper, Blennerhassett & Hall (1979) investigated the linear stability of the flow between concentric cylinders of finite length. The inner cylinder was taken to rotate, while the outer cylinder was fixed. Furthermore, the end walls rotated such that the flow was purely circumferential. In this paper the finite amplitude development of the unstable disturbances to the flow is considered. It is found that the usual perturbation expansion of nonlinear stability theory must be modified if the cylinders are not infinitely long. The bifurcation to a Taylor vortex flow in finite cylinders is shown to be two-sided. The latter effect is shown to be a direct consequence of the finiteness of the cylinders and by taking the cylinders to be very long, we recover the results obtained previously for the infinite problem. The interaction of the two most dangerous modes of linear theory is also investigated. For certain values of the length of the cylinders the initial finite amplitude Taylor vortex flow is shown to become unstable to another class of axisymmetric disturbances. The effect of perturbing the end conditions towards the no-slip conditions appropriate to most experimental configurations is also discussed. Some discussion of the instability problem in very long cylinders with fixed ends is given.

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.

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.

scholarly journals Hydrodynamic education with rheoscopic fluid

2019 ◽  
Vol 213 ◽  
pp. 02014
Author(s):  
Daniel Duda ◽  
Marek Klimko ◽  
Radek Škach ◽  
Jan Uher ◽  
Václav Uruba
Keyword(s):  
Vortex Flow ◽  
Maximum Stress ◽  
Fluid Motion ◽  
Shear Layers ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Taylor Vortex Flow ◽  
Concentric Cylinders ◽  
The Subject

We present a educational poster supporting the subject „Mechanics of fluids I“, which the students evaluate to be difficult mainly due to abstractness. Our goal is to show in vivo the behavior, especially the non-linearity, of various flows transiting into turbulence. The fluid motion is visualized by using the rheoscopic fluid, which consist of water and the dust of mica, whose particles are longitudinal and shiny resulting into easily observable reflections, when the particles coherently orient along the maximum stress. This happens mainly in shear layers, e.g. at the boundary between vortex core and envelope. An example of flow transiting into turbulence is the Taylor-Couette flow between two concentric cylinders, which with increasing Taylor number passes through various regimes from fully laminar bearing flow through the Taylor vortex flow (TVF) and later Wavy vortex flow (WVF) up to Turbulent Taylor vortices regime (TTV) and, finally, the regime of featureless turbulence.

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.

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.

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