Ekman vortices and the centrifugal instability in counter-rotating cylindrical Couette flow

The finite length of a Taylor–Couette cell introduces endwall effects that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman boundary layers and the vortical structures in a finite-length cavity with counter-rotating cylinders via direct numerical simulation using a three-dimensional spectral method. To analyze the nature of the interaction between the vortices and the endwall layers we consider four endwall boundary conditions: fixed endwalls, endwalls rotating with the outer cylinder, endwalls rotating with the inner cylinder, and stress-free endwalls. The vortical structure of the flow depends on the endwall conditions. The waviness of the vortices is suppressed only very near the endwall, primarily due to zero axial velocity at the endwall rather than viscous effects. In spite of their waviness and random behavior, the vortices generally stay inside of the v=0 isosurface by adjusting quickly to the radial transport of azimuthal momentum. The thickness and strength of the Ekman layer at the endwall match with that predicted from a simple theoretical approach.     

Influence of a homogeneous axial magnetic field on Taylor–Couette flow of ferrofluids with low particle–particle interaction

Experimental results concerning the stability of Couette flow of ferrofluids under magnetic field influence are presented. The fluid cell of the Taylor–Couette system is subject to a homogeneous axial magnetic field and the axial flow profiles are measured by ultrasound Doppler velocimetry. It has been found that an axial magnetic field stabilizes the Couette flow. This effect decreases with a rotating outer cylinder. Moreover, it could be observed that lower axial wave numbers are more stable at a higher axial magnetic field strength. Since the used ferrofluid shows a negligible particle–particle interaction, the observed effects are considered to be solely based on the hindrance of free particle rotation.     

Direct numerical simulation of turbulent Taylor–Couette flow

The direct numerical simulation (DNS) of the Taylor–Couette flow in the fully turbulent regime is described. The numerical method extends the work by Quadrio and Luchini [M. Quadrio, P. Luchini, Eur. J. Mech. B/Fluids 21 (2002) 413–427], and is based on a parallel computer code which uses mixed spatial discretization (spectral schemes in the homogeneous directions, and fourth-order, compact explicit finite-difference schemes in the radial direction). A DNS is carried out to simulate for the first time the turbulent Taylor–Couette flow in the turbulent regime. Statistical quantities are computed to complement the existing experimental information, with a view to compare it to planar, pressure-driven turbulent flow at the same value of the Reynolds number. The main source for differences in flow statistics between plane and curved-wall flows is attributed to the presence of large-scale rotating structures generated by curvature effects.     

Flow pattern and stability of pseudoplastic axial Taylor–Couette flow

The effect of an axial flow on the stability of the Taylor–Couette flow is explored for pseudoplastic fluids. The fluid is assumed to follow the Carreau–Bird model and mixed boundary conditions are imposed while the axial flow can be independent of rotational flow. The four-dimensional low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional non-linear terms in the velocity components originated from the shear-dependent viscosity. In absence of axial flow the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the pseudoplasticity effects increases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow, induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. Complete flow field together with viscosity maps are given for stability regions in the bifurcation diagram.     

Nonlinear Stability of Taylor Vortices in Infinite Cylinders

We consider the Taylor‐Couette problem in an infinitely extended cylindrical domain in the case when Couette flow is weakly unstable and a family of spatially periodic equilibria, called the Taylor vortices, has bifurcated from this trivial ground state. We show that those Taylor vortices which are not linearly unstable in the sense of Eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations. The main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the Taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis. Bloch‐wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime. (Accepted September 8, 1997)     

On the formation of Taylor–G?rtler vortices in RMF-driven spin-up flows

This paper is concerned with a liquid metal flow driven by a rotating magnetic field inside a stationary cylinder. We consider especially the secondary meridional flow during the time when the fluid spins up from rest. The developing flow is investigated experimentally and by direct numerical simulations. The vertical profiles of the axial velocity are measured by means of the ultrasound Doppler velocimetry. Evolving instabilities in the form of Taylor–G?rtler vortices have been observed just above the instability threshold (Ta ≥ 1.5· Ta ^cr). The rotational symmetry may survive over a distinct time even if a first Taylor–G?rtler vortex pair has been formed as closed rings along the cylinder perimeter. The transition to a three-dimensional flow in the side layers results from the advection or a precession and splitting of the Taylor–G?rtler vortex rings. The predictable behaviour of the Taylor–G?rtler vortices disappears with increasing magnetic field strength. The numerical simulations agree very well with the flow measurements.     

Numerical simulation of Taylor Couette flow of Bingham fluids

The Bingham fluid flow between two concentric cylinders is studied using numerical simulation. The cylinders are assumed to rotate independently, and with an imposed axial sliding. The flow field is decomposed with linearity arguments of the base circular Couette shear flow and corresponding deviation field. The numerical methods are based on the expression of the deviation field in terms of complete sets of orthogonal functions and Chebyshev series. The Galerkin projection method is used with the pressure term being eliminated. The Adams Bashforth scheme is adopted for time marching. The results show that the vortices are squeezed toward the inner cylinder due to the effect of yield stress. When the outer cylinder is held stationary, the yield stress plays a role in weakening the vortex flow. However, for the co-rotation situation, the vortex flow is initially strengthened with an increase of yield stress, and then weakened as the yield stress is raised large enough. The annular unyielded regions emerge and stick to the outer cylinder. In case of Taylor Couette flow with an imposed axial sliding, a spiral vortex flow is visible with spiral unyielded region being obtained.     

Experiments on the stability of an impulsively-initiated circular Couette flow

Experiments were performed to study the stability characteristics of an unsteady circular Couette flow generated by an impulsive stop of the outer cylinder; the initial condition was a state of rigid-body rotation. Instability of the unsteady basic state is manifested by Görtler vortices, which themselves become unstable to longer-wavelength disturbances, or Taylor vortices which persist indefinitely. The quantities of primary interest are the onset time of instability, the axial vortex wavelength at onset, and the time-evolution of this wavelength. A one-dimensional photodiode array is used to gather data from the flow, which is seeded with flow-visualization material. At sufficiently high values of the Reynolds number, the influence of the inner cylinder on the onset of instability is negligible, based on comparisons with previous experimental data.     

Spiral and Wavy Vortex Flows in Short Counter-Rotating Taylor–Couette Cells

Differentially rotating cylinders result in a rich variety of vortical flows for cylindrical Couette flow. In this study we investigate the case of a short, finite-length cavity with counter-rotating cylinders via direct numerical simulation using a three-dimensional spectral method. We consider aspect ratios ranging from 5 to 6. Two complex flow regimes, wavy vortices and interpenetrating spirals, occur with similar appearance to those found experimentally for much larger aspect ratios. For wavy vortices the wave speed is similar to that found for counter-rotating systems and systems in which the outer cylinder is stationary. For the interpenetrating spiral structure, the vortices are largely confined to the unstable region near the inner cylinder. The endwalls appear to damp and stabilize the flow as the aspect ratio is reduced to the point that in some cases the vortical flow is suppressed. At higher inner cylinder speeds, the interpenetrating spirals acquire a waviness and the vortices, while generally near the inner cylinder, can extend all of the way to the outer cylinder. Received 5 November 2001 and accepted 29 March 2002 Published online 2 October 2002 Communicated by H.J.S. Fernando     

Spiral vortices between concentric cylinders

Spiral vortices appearing in Couette-Taylor flows are studied by means of numerical simulation. Transition curves from Couette to spiral vortices for different radius ratios and wavenumbers have been calculated in order to test our technique. Critical Reynolds numbers, angular velocities and slopes of the spirals at the onset of the instability agree with previous results [1]. Non-linear solutions obtained by a pseudospectral collocation method are studied, and they show a weak net axial flow. In order to counteract this effect, which is absent in the usual experimental set-up, an axial pressure gradient has been included. This procedure has proved to be sufficient to make the axial flow negligible. The onset of a quasiperiodic flow for larger Reynolds numbers, corresponding to a secondary bifurcation is also presented.     

Nonnormal Effects in the Taylor–Couette Problem

This work is devoted to the study of transient growth of perturbations in the Taylor–Couette problem due to linear nonnormal mechanisms. The study is carried out for a particular small gap case and is mostly focused on the linearly stable regime of counter-rotation. The exploration covers a wide range of inner and outer angular speeds as well as axial and azimuthal modes. Significant transient growth is found in the regime of stable counter-rotation. The numerical results are in agreement with former analyses based on energy methods and other independent numerical studies. The optimal energy transient growth factor appears to be consistent with experimental observations. This study might shed some light on the subcritical transition to turbulence which is found experimentally in Taylor–Couette flow when the cylinders rotate in opposite directions. Received 13 February 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work was supported by the UK EPSRC under Grant GR/M30890. The author thanks Nick Trefethen for fruitful discussions. RID="*" ID="*" Present address: Departament de Fisica Aplicada, Univ. Politecnica de Catalunya, 08034 Barcelona, Spain (alvar@fa.upc.es) Communicated by H.J.S. Fernando     

Instability of a slip flow in a curved channel formed by two concentric cylindrical surfaces

Instability of a slip flow in a curved channel formed by two concentric cylindrical surfaces is investigated. Two cases are considered. In the first (Taylor–Couette flow) case the flow is driven by the rotation of the inner cylindrical surface; no azimuthal pressure gradient is applied. In the second case (Dean flow) both cylindrical surfaces are motionless, and the flow is driven by a constant azimuthal pressure gradient. The collocation method is used to find numerically the critical values of the Taylor and Dean numbers, which establish the instability criteria for these two cases. The dependencies of critical values of these numbers on the ratio between the radii of concave and convex walls and on the velocity slip coefficient are investigated.     

A lattice Boltzmann model for adsorption breakthrough

A lattice Boltzmann model is developed to simulate the one-dimensional (1D) unsteady state concentration profiles, including breakthrough curves, in a fixed tubular bed of non-porous adsorbent particles. The lattice model solves the 1D time dependent convection–diffusion–reaction equation for an ideal binary gaseous mixture, with solute concentrations at parts per million levels. The model developed in this study is also able to explain the experimental adsortption/desorption data of organic vapours (toluene) on silica gel under varying conditions of temperature, concentrations and flowrates. Additionally, the programming code written for simulating the adsorption breakthrough is modified with minimum changes to successfully simulate a few flow problems, such as Poiseuille flow, Couette flow, and axial dispersion in a tube. The present study provides an alternative numerical approach to solving such types of mass transfer related problems.     

A Numerical Study of the Recirculation Zones During Spin-Up and Spin-Down for Confined Rotating Flows<Superscript>*</Superscript>

This paper reports computational simulations of the Navier–Stokes equations for confined axisymmetric rotating flows induced by rotating the endwalls instantaneously at a different rate to the sidewall. The transient behavior of the recirculation zones in the meridional plane is investigated during the temporal evolution. The changes in the topological structure of the meridional-plane streamline pattern are significant and the temporal evolution from one pattern to another reveals similarities between spin-up and spin-down at the early stages but subsequently differs. As the onset bubble for the first recirculating period always sets out from a certain axial station, a recirculation factor, R_f, is suggested to predict the onset time and location for the first period of recirculation. Accordingly, a stagnation point is observed numerically from a central axial station for low Reynolds numbers around 70–80. The effect of changing the rotation of the sidewall is also discussed, but no substantial influence is observed on the characteristics of the recirculation zones if there is no appearance of the Taylor–Görtler vortices in the sidewall boundary layer.     

Oscillatory rotationally symmetric loss of stability of nonisothermal couette flow

A study is made of the stability of nonisothermal Couette flow — steady flow of a viscous heat conducting fluid between two rotating concentric cylinders heated to different temperatures. The methods of perturbation theory are used to establish conditions sufficient for bifurcation of a neutral curve of oscillatory instability from the neutral curve of monotonic instability. Computer calculations show that for certain values of the parameters of the problem these conditions are realized and there is an oscillatory loss of stability of the nonisothermal Couette flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–80, January–February, 1984.I thank V. I. Yudovich for constant interest in the work.     

From global to local bifurcations in a forced Taylor–Couette flow

The unfolding due to imperfections of a gluing bifurcation occurring in a periodically forced Taylor–Couette system is analyzed numerically. In the absence of imperfections, a temporal glide-reflection Z₂ symmetry exists, and two global bifurcations occur within a small region of parameter space: a heteroclinic bifurcation between two saddle two-tori and a gluing bifurcation of three-tori. As the imperfection parameter increase, these two global bifurcations collide, and all the global bifurcations become local (fold and Hopf bifurcations). This severely restricts the range of validity of the theoretical picture in the neighborhood of the gluing bifurcation considered, and has significant implications for the interpretation of experimental results. PACS 47.20.Ky, 47.20.Lz, 47.20.Ft     

Large Eddy Simulation of boundary layer transition over an isolated ramp-type micro roughness element

Boundary layer transition over an isolated surface roughness element is investigated by means of numerical simulation. Large Eddy Simulation (LES) flow-modeling approach is employed to study flow characteristics and transition phenomenon past a roughness element immersed within an incoming developing boundary layer, at a height-based Reynolds number of 1170. LES numerical results are compared to experimental data from literature showing the time-averaged velocity distribution, the velocity fluctuation statistics and the instantaneous flow topology.Despite slight difference in the intensity of streamwise velocity fluctuations, the present LES results and experimental data show very good agreement. The mean flow visualization shows streamwise counter-rotating vortices pairs formation downstream of the obstacle. The primary pair induces an upwash motion and a momentum deficit that creates a Kelvin-Helmholtz type flow instability. The instantaneous flow topology reveals the formation of coherent K-H vortices downstream that produce turbulent fluctuations in the wake of the roughness element. These vortices are streched and lifted up when moving downstream. The velocity fluctuations results show that the onset of the turbulence is dominated by the energy transfer of large-scale vortices.     

Reduction of Ekman circulation within Taylor-Couette flow

We demonstrate experimentally that independently rotating intermediary end-rings between the cylinders of a Taylor–Couette apparatus can be utilized to reduce friction-driven secondary flow, i.e. Ekman circulation. This allows for velocity profiles in a device of small aspect ratio to be less constrained by ‘end effects’, so that the resulting wide-gap flows can be made to have a radial distribution of circumferential velocity that resembles a narrow-gap Couette solution.     

The Dean instability in power-law and Bingham fluids in a curved rectangular duct

The laminar flow of power-law and yield-stress fluids in 180° curved channels of rectangular cross section was studied experimentally and numerically in order to understand the effect of rheological fluid behavior on the Dean instability that appears beyond a critical condition in the flow. This leads to the apparition of Dean vortices that differ from the two corner vortices created by the channel wall curvature.Flow visualizations showed that the Dean vortices develop first in the near-wall zone on the concave (outer) wall, where the shear rate is higher and the viscosity weaker; then they penetrate into the centre of the channel cross section where power-law fluids have high viscosity and Bingham fluids are unyielded in laminar flow. Based on the complete formation on the concave wall of the new pairs of counter-rotating vortices (Dean vortices), the critical value of the Dean number decreases as the power-law index increases for the power-law fluids, and the Bingham number decreases for the Bingham fluids. For power-law fluids, a diagram of critical Dean numbers, based on the number of Dean vortices formed, was established for different axial positions. For the same flow conditions, the critical Dean number obtained using the axial velocity gradient criterion was smaller then that obtained with the visualization technique.     

Investigation of the influence of the transverse sweepback of delta wings on their vortex structures and aerodynamic characteristics in the case of flow separation at low subsonic velocities

The results are given of theoretical and experimental investigations into the influence of transverse sweepback of delta wings of small aspect ratio on their lifting properties and the stability of the vortex structure in the case of flow separation. Theoretical investigations have been made using the method of discrete vortices. The physical experiment included weight tests and also investigations on the breakup of vortex filaments by means of aerometric and schlieren methods. The results of the investigations are given and the calculated and experimental data are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–175, September–October, 1982.