Coriolis effects on the stability of pulsed flows in a Taylor–Couette geometry

Results steming from the linear stability of time-periodic flows in a Taylor–Couette geometry with cylinders oscillating in phase or out-of-phase are presented. Our analysis takes into account the gap size effects and investigates the influence of a superimposed mean angular rotation of the whole system.In case of no mean rotation, the finite gap geometry is found to affect the shape of the stability diagrams (critical Taylor number versus the frequency parameter) which consist of two distinct branches as opposed to being continuous in the narrow gap approximation. In particular, in the out-of-phase configuration a new branch for low frequencies was found, thus enabling better agreement with available experimental results.When cylinders are co-rotating and subject to rotation effects, our calculations provide the evolution of the critical Taylor number versus the rotation number for two values of the frequency. The stability curves are found to be in qualitative agreement with available experimental data revealing a maximum of instability for a rotation number of about 0.3.In the high rotation regime, enhancement of the critical Taylor number is investigated through an asymptotic analysis and the value of the rotation number at which restabilization occurs is found to depend on the frequency parameter.A restabilization of the flow also occurs when the rotation number and the gap size are of the same order, a phenomenon already pointed out in the case of steady flows and attributed to the near cancellation of Coriolis and centrifugal effects. Our investigation proves that the same mechanism still holds for time-periodic flows.     

Identification of complex flows in Taylor–Couette counter-rotating cavities

The transition in confined rotating flows is a topical problem with many industrial and fundamental applications. The purpose of this study is to investigate the Taylor–Couette flow in a finite-length cavity with counter-rotating walls, for two aspect ratios L=5 or L=6. Two complex regimes of wavy vortex and spirals are emphasized for the first time via direct numerical simulation, by using a three-dimensional spectral method. The spatio-temporal behavior of the solutions is analyzed and compared to the few data actually available.     

Turbulent Couette–Poiseuille flow with zero wall shear

A particular pressure-driven flow in a plane channel is considered, in which one of the walls moves with a constant speed that makes the mean shear rate and the friction at the moving wall vanish. The Reynolds number considered based on the friction velocity at the stationary wall (u_τ,S) and half the channel height (h) is Re_τ,S = 180. The resulting mean velocity increases monotonically from the stationary to the moving wall and exhibits a substantial logarithmic region. Conventional near-wall streaks are observed only near the stationary wall, whereas the turbulence in the vicinity of the shear-free moving wall is qualitatively different from typical near-wall turbulence. Large-scale-structures (LSS) dominate in the center region and their spanwise spacing increases almost linearly from about 2.3 to 4.2 channel half-heights at this Re_τ,S. The presence of LSS adds to the transport of turbulent kinetic energy from the core region towards the moving wall where the energy production is negligible. Energy is supplied to this particular flow only by the driving pressure gradient and the wall motion enhances this energy input from the mean flow. About half of the supplied mechanical energy is directly lost by viscous dissipation whereas the other half is first converted from mean-flow energy to turbulent kinetic energy and thereafter dissipated.     

Taylor–Couette flow with independently rotating end plates

Results are presented from a combined numerical and experimental study of steady bifurcation phenomena in a modified Taylor–Couette geometry where the end plates of the flow domain are allowed to rotate independently of the inner cylinder. The ends rotate synchronously and the ratio between the rate of rotation of the ends _e and the inner cylinder _i defines a control parameter :=_e/_i. Stationary ends favour inward motion along the end walls whereas rotating walls promote outward flow. We study the exchange between such states and focus on two-cell flows, which are found in the parameter range between =0 and =1 for =2. Hence is used as an unfolding parameter. A cusp bifurcation is uncovered as the organizing centre for the stability exchange between the two states. Symmetry breaking bifurcations, which lead to flows that break the mid-plane symmetry are also revealed. Overall, excellent agreement is found between numerical and experimental results. PACS 47.20, 47.11, 47.54     

Numerical study of eccentric Couette–Taylor flows and effect of eccentricity on flow patterns

In this study, the differential quadrature (DQ) method was used to simulate the eccentric Couette–Taylor vortex flow in an annulus between two eccentric cylinders with rotating inner cylinder and stationary outer cylinder. An approach combining the SIMPLE (semi-implicit method for pressure-linked equations) and DQ discretization on a non-staggered mesh was proposed to solve the time-dependent, three-dimensional incompressible Navier–Stokes equations in the primitive variable form. The eccentric steady Couette–Taylor flow patterns were obtained from the solution of three-dimensional Navier–Stokes equations. The reported numerical results for steady Couette flow were compared with those from Chou [1], and San and Szeri [2]. Very good agreement was achieved. For steady eccentric Taylor vortex flow, detailed flow patterns were obtained and analyzed. The effect of eccentricity on the eccentric Taylor vortex flow pattern was also studied.     

Large-eddy simulation of counter-rotating Taylor–Couette flow: The effects of angular velocity and eccentricity

In the present work, turbulent flow in the annulus of a counter-rotating Taylor-Couette (CRTC) system is studied using large-eddy simulation. The numerical methodology employed is validated, for both the mean and second-order statistics, with the direct numerical simulation (DNS) data available in the literature, for a range of Reynolds numbers from 500 to 4000. Thereafter, turbulent flow occurring in this system at Reynolds numbers of 8000 and 16000 are studied, and the results obtained are analyzed using mean and second-order statistics, vortical structures, velocity vector plots and power energy spectra. Further, the spatio-temporal variation of azimuthal velocity, extracted near the inner cylinder, shows the existence of herringbone like patterns similar to that observed in the previous studies. The effect of eccentricity of the inner cylinder with respect to the outer cylinder is studied, on the turbulent flow in the CRTC system, for two different eccentricity ratios of 0.2 and 0.5 and for two different Reynolds numbers of 1500 and 4000. The results of the eccentric CRTC are analyzed using contours of pressure, mean and second-order statistics, velocity vectors, vortical structures, and turbulence anisotropy maps. It is observed from the eccentric CRTC simulations that the smaller-gap region seems to contain higher amplitude fluctuations and more vortical structures when compared with the larger-gap region. The mean turbulent kinetic energy contours do not change qualitatively with the Reynolds number, however, quantitatively a higher turbulent kinetic energy is observed in the higher Reynolds number case of 4000.     

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     

Stress gradient-induced migration effects in the Taylor–Couette flow of a dilute polymer solution

We present an investigation of the phenomenon of stress-induced polymer migration for dilute polymer solutions in the Taylor–Couette device, consisting of two infinitely long, concentric cylinders rotating at constant angular velocities. The underlying physical model is represented by the dilute limit of a two-fluid Hamiltonian system involving two components: one (the polymer) is viscoelastic and obeys the Oldroyd-B constitutive equation, and the other (the solvent) is viscous Newtonian. The two components are considered to be in thermal, but not mechanical equilibrium, interacting with each other through an isotropic drag coefficient tensor. This allows for stress-induced diffusion of polymer chains. The governing equations consist of the continuity and the momentum equations for the bulk velocity, the constitutive model for the polymer chain conformation tensor and the diffusion equation for the polymer concentration. The diffusion equation contains an extra source term, which is proportional to gradients in the polymer stress, so that polymer concentration gradients can develop even in the absence of externally imposed fluxes in the presence of stress inhomogeneities. The solution to the steady-state purely azimuthal flow is obtained first using a spectral collocation method and an adaptive mesh formulation to track the steep changes of the concentration in the flow domain. The calculations show the development of strong polymer migration towards the inner cylinder with increasing Deborah number (De) in agreement with experimental observations. The migration is enhanced for increasing values of the gap thickness resulting in concentration changes by several orders of magnitude in the area between the inner and outer cylinder walls. The extent of the migration also depends strongly on the ratio of the solvent to the polymer viscosity. In addition to a strongly inhomogeneous polymer concentration, significant deviations from the homogenous flow are also observed in the velocity profile. Next, results are reported from a linear stability analysis around the steady-state solution against axisymmetric disturbances corresponding to various wavenumbers in the axial direction. The calculations show that the steady-state solution remains stable up to moderate values of the Deborah number, explaining why some of the predicted stress-induced migration effects should be experimentally observable. The role of the Peclet number (Pe) on the stability of the system is elucidated.     

Bubble capture and migration in Couette–Taylor flow

 Bubble capture and migration under the effect of organized structures in weak turbulent Couette–Taylor flow between two concentric cylinders, the inner one rotating, has been investigated. Bubbles generated at the free surface for large enough angular velocities are sucked into the flow by the upper organized structures. Then they migrate progressively from top to bottom by jumping from cell to cell. With an upper solid stationary wall instead of the free surface, injected bubbles are trapped by the coherent vortices beyond a critical Taylor number. However, in this situation there is no migration mechanism carrying the bubbles from top to bottom. This particular migration and capture process, able to act against the forces of buoyancy, has been investigated by perturbing the flow by adding a vertical plate protruding from the inner surface of the solid stationary wall. The perturbation so introduced causes the deformation of the upper coherent structures and reinstalls the migration of the bubbles. Received: 27 October 1997/Accepted: 21 May 1998     

Direct numerical simulation of Taylor–Couette turbulent flow controlled by a traveling wave-like blowing and suction

In wall turbulence, a traveling wave-like control is known to decrease the skin-friction drag and induce the relaminarization phenomenon. Because it is noteworthy to investigate the control effect in other canonical flows, direct numerical simulations of fully developed turbulent Taylor–Couette flows are performed. The Reynolds number, based on the wall velocity of a rotating inner cylinder and the radius of a centerline between cylinders, is set to 84,000. The traveling wave-like blowing and suction is imposed on the inner or outer cylinder wall, and the control effect is parametrically investigated. In the inner cylinder control, the torque reduction is obtained when the wave travels in the co-rotating direction with the inner cylinder, and its wavespeed is faster than the rotation. In the outer cylinder control, in contrast, the torque reduction is obtained when the wave propagates in the opposite direction. While the control is imposed on one side wall (i.e., inner or outer cylinder), the control affects the entire flow region. The Taylor vortex remains, while the traveling wave affects its strength. The three-component decomposition analysis shows that the traveling wave creates the coherent contribution on the torque, while the random contribution on it is reduced. Accordingly, a major factor of the torque reduction in the Taylor–Couette flow is the reduction of the random contribution. In addition, for the faster wavespeed cases with the small wavenumber (i.e., the long wavelength), the drag reduction larger than 60% is obtained and the relaminarization occurs in these cases.     

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.     

Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille–Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170–178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.     

Energy redistribution dynamics in coupled Couette–Poiseuille flows using large-Eddy simulation

The problem of turbulent Couette flow driven by a statistically steady external wind is studied in the framework of spatially filtered Navier–Stokes equations. The phenomenon of wind-driven flow of water is represented by a layer of air modeled as Poiseuille flow (air sub-domain), coupled to a layer of water modeled as Couette flow (water sub-domain). We focus on changes in the statistics in either the air or the water sub-domain, due to the coupling with the other sub-domain. We also highlight dynamic flow structures forming near the air-water interface. Simulations based on different Reynolds numbers in the air and the water sub-domains are compared to computationally less demanding simulations with equal Reynolds numbers. Results of these simulations indicate strong similarities, i.e., the flow is well approximated by simulating air and water at the same Reynolds numbers. Further analysis shows that the flow in the water domain shares important features with classical Couette flows. The horizontal turbulent mixing renders a thinner boundary layer in the water sub-domain. Moreover, an increased intermittency in the flow velocities is observed, which may be linked to so-called splat events near the air-water interface. These splats characterize the interaction of coherent structures across the interface, being stronger in the water phase. An analysis of the pressure-strain correlation near the air-water interface on the water side shows that such splats are responsible for redistributing energy from the streamwise and spanwise directions, to the vertical direction. This behavior, although qualitatively similar to wall-bounded flows, differ mainly on the fact that most of the energy drained comes from the streamwise direction: in wall-bounded the main contributor is the spanwise direction. The boundary layers near the air-water interface show inclined vortical structures. Unlike in coupled Couette–Couette flow, the peak in the Reynolds stress is displaced from the channel’s center into the buffer region of the water sub-domain.     

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.     

Numerical study of Saffman–Taylor instability in immiscible nonlinear viscoelastic flows

In this paper, a numerical solution for Saffman–Taylor instability of immiscible nonlinear viscoelastic-Newtonian displacement in a Hele–Shaw cell is presented. Here, a nonlinear viscoelastic fluid pushes a Newtonian fluid and the volume of fluid method is applied to predict the formation of two phases. The Giesekus model is considered as the constitutive equation to describe the nonlinear viscoelastic behavior. The simulation is performed by a parallelized finite volume method (FVM) using second order in both the spatial and the temporal discretization. The effect of rheological properties and surface tension on the immiscible Saffman–Taylor instability are studied in detail. The destabilizing effect of shear-thinning behavior of nonlinear viscoelastic fluid on the instability is studied by changing the mobility factor of Giesekus model. Results indicate that the fluid elasticity and capillary number decrease the intensity of Saffman–Taylor instability.     

Channel, tube, and Taylor–Couette flow of complex viscoelastic fluid models

We show how to formulate two-point boundary value problems to compute laminar channel, tube, and Taylor–Couette flow profiles for some complex viscoelastic fluid models of differential type. The models examined herein are the Pom-Pom Model [McLeish and Larson 42:81–110, (1998)] the Pompon Model [Öttinger 40:317–321, (2001)] and the Two Coupled Maxwell Modes Model (Beris and Edwards 1994). For the two-mode Upper-Convected Maxwell Model, we calculate analytical solutions for the three flow geometries and use the solutions to validate the numerical methodology. We illustrate how to calculate the velocity, pressure, conformation tensor, backbone orientation tensor, backbone stretch, and extra stress profiles for various models. For the Pom-Pom Model, we find that the two-point boundary value problem is numerically unstable, which is due to the aphysical non-monotonic shear stress vs shear rate prediction of the model. For the other two models, we compute laminar flow profiles over a wide range of pressure drops and inner cylinder velocities. The volumetric flow rate and the nonlinear viscoelastic material properties on the boundaries of the flow geometries are determined as functions of the applied pressure drop, allowing easy analysis of experimentally measurable quantities.     

“Everything flows?”: elastic effects on startup flows of yield-stress fluids

It is now 30 years since Barnes and Walters published a provocative paper in which they asserted that the yield stress is an experimental artifact. We now know that the situation is far more complicated than understood at the time, and that the mechanics of the solid material prior to yielding must be considered carefully. In this paper, we examine the response of a well-studied “simple” yield-stress material, namely a Carbopol gel that exhibits no thixotropy, and demonstrate the significance of the pre-yielding behavior through a number of elementary measurements.