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The stability of pulsed bi-dimensional flow between two co-oscillating cylinders in a linear Maxwell fluid was studied by Riahi et al. [J. Soc. Rheol. 42, 321–327 (2014)]. In the present paper, we revisit this flow configuration with emphasis on the effect of the non-linear terms in the constitutive equation of the model, measured by the Weissenberg number, on the dynamics of the system. Under these assumptions and using the upper convected Maxwell derivative, we examine this model to large amplitude oscillatory shear giving rise to the appearance, in comparison to the linear Maxwell model, of the azimuthal normal stress in the basic state. Using the spectral method and the Floquet theory for the spatiotemporal resolution of the obtained eigenvalue problem, numerical results exhibit numerous classes of Taylor vortex flows depending on the order of magnitude of the fluid elasticity. The resulting stability diagram consists of several branches intersecting at specific frequencies where two different Taylor vortex flows simultaneously branch off from the basic state. This feature is accompanied by the occurrence of several co-dimension two bifurcation points besides jumps/drops in the corresponding critical wave number. In addition, it turns out that the elasticity produces strong destabilizing and stabilizing effects in the limit of high and low frequency regimes, respectively, attributed solely to the non-linearities considered by the rheological model.

Drag reduction in turbulent Taylor–Couette flows by axial oscillation of inner cylinder is investigated by direct numerical simulation. In the present study, the reference friction Reynolds number is R e τ = 210 based on the friction velocity at the inner cylinder in the no control cases and the half gap width. We have obtained the effects of the oscillation period and the radius ratio of the inner to outer cylinders on the drag reduction rate. Our analysis shows that as the radius ratio is getting larger, the maximum drag reduction rate is decreased and the optimal oscillating period is increased. Under the condition of the short oscillating period, a larger radius ratio leads to a lower drag reduction rate. However, when the oscillating period becomes long, the larger radius ratio triggers a higher drag reduction rate. With the help of Fukagata–Iwamoto–Kasagi identity, the wall shear stress has been linked to turbulent motions at different scales. It is found that the long-period oscillations primarily reduce the wall friction drag induced by the large-scale Taylor vortices while the short-period oscillations mainly decrease wall shear stress originating from the small-scale velocity streaks. Visualizations of Taylor vortices and velocity streaks, premultiplied spectra, and the weighted Reynolds shear stress indicate that such different effects are related to the Stokes layer. A thick Stokes layer under the condition of large-period oscillations penetrates to the core region of the flow and the Taylor vortices whose center is located near the middle plane between the cylinders is thus attenuated effectively. On the contrary, the influence range of a thin Stokes layer caused by the short-period oscillation concentrates on the near-wall region, hence, the small-scale velocity streaks there are weakened greatly.

This study investigates the Taylor–Couette flow (TCF) with a longitudinal corrugated surface on a stationary outer cylinder and a rotating smooth inner cylinder using large eddy simulation for three values of amplitude to wavelength ratios (A*) (0.1875, 0.2149, and 0.25) to explore the influence of the corrugated surface on the flow structures and the variation of torque for a wider range of Reynolds numbers (Re) (60–650). From the results, four flow regimes are observed. At Re = 60, initially, a pair of secondary vortices appears at the inner wall of the minimum gap region and it evolves to a pair of axisymmetric stationary wall induced vortices (ASSWIVs) in the maximum gap region. As Re increases to 80, 85, and 103 for the three values of A* (0.1875, 0.2149, and 0.25), respectively, another pair of axisymmetric stationary secondary vortices is seen at the minimum gap region of the inner wall. A further increase in Re (Re > 125, 130, and 138 for the three values of A*, respectively) results in the appearance of axisymmetric periodic secondary axial flow. Increasing Re further (Re > 225, 240, and 260 for A* = 0.25, 0.2149, and 0.1875, respectively) leads to the emergence of non-axisymmetric and non-periodic secondary axial flow (NANPSAF) with an azimuthal wave. Generally, the torque in TCF with the corrugated surface is found to be lower than TCF with a smooth surface except for the occurrence of the ASSWIV flow regime and weak axial secondary flow in the NANPSAF regime.

The flow between two concentric cylinders, one of which is rotating (Taylor-Couette flow), has been the focus of extensive research, due to the number of flow instabilities that may occur and its use in various industrial applications. We examine Taylor-Couette flow of Newtonian and shear-thinning fluids (solutions of xanthan gum in water/glycerol) using a combination of particle-image velocimetry and flow visualization for a wide range of Reynolds number, spanning the circular Couette flow, Taylor vortex flow, and wavy vortex flow regimes. Shear thinning is associated with an increase in the axial wavelength and has a nonmonotonic effect on the critical Reynolds number for transition to Taylor vortex flow and wavy vortex flow. The magnitude of vorticity and the strength of the radial jets transporting fluid away from the inner cylinder (“outward jets”) are both reduced in shear-thinning fluids relative to the Newtonian case; the vorticity in the shear-thinning fluids also tends to concentrate at the edges of vortices, rather than in the cores. In the wavy vortex flow regime for Newtonian fluids, the amplitudes of the waves at the “inward jets” (moving toward the inner cylinder) are low compared to those at the outward jets. However, for the shear-thinning fluids, the amplitudes of the waves at both the inward and outward jets tend to be significantly larger. Finally, shear thinning is associated with greater variations in time and space: we observe slow drifts in the axial positions of vortices and spatial variations in the amplitudes of the wavy instability, which are absent in Newtonian fluids.

This work presents an investigation of a new phenomenon of the Taylor-Couette flow: the onset of Taylor vortices in a cavitating fluid. This particular form of the Taylor-Couette flow develops if the shear flow between a rotating inner and a fixed outer cylinder approaches the critical Taylor number and the vapor pressure of the fluid simultaneously. This process is achieved by increasing the rotational speed of the inner cylinder, which causes an increase of the radial pressure gradient inside the laminar flow. The fully developed Taylor vortex flow is characterized by a pressure distribution in the azimuthal plane showing a local minimum adjacent to the wall of the inner cylinder between a pair of vortices that form a radial flow towards the outer cylinder. Thence, cavitation occurs simultaneously if the local pressure minimum drops below the vapor pressure of the fluid. This transition from a two-dimensional (Couette) into a three-dimensional (Taylor) flow triggered the idea to apply a newly developed unsteady 2-phase 3D-computational fluid dynamics code by computing the generation of vapor that is coinciding with the formation of Taylor vortices at the critical Taylor number. Whereas the results of a numerical simulation prove the existence of toroidal vapor caused by cavitation, the experimental validation demands additionally the development of a special fluid. Thus, the present work describes this specifically tailored fluid, which not only fulfills Taylor and pressure analogy but also features a favorable refractive index and a chemical suitability for the task.

In this paper, numerical simulations are performed on the interaction of vortices with a longitudinal corrugated wall in a Taylor-Couette (TC) setting with the inner smooth surface cylinder rotating and the outer corrugated surface cylinder stationary. The motivation of the study is to shed light on how such an interaction affects the drag/torque with respect to two geometric parameters of the corrugations, namely, the wavelength λc* and amplitude A*, where * indicates a normalization by the gap width d. Results show that in the circular Couette flow regime, the secondary vortices induced by the corrugations cause the torque to increase. When λc* 1, there is a steeper increase of torque due to the interaction of the growing secondary vortices and the opposite wall. In the Taylor-vortex flow regime, the interaction between the Taylor vortices and the corrugations produces three distinct behaviors characterized by λc*. As the wavelength increases, our results show that the stronger modulation effects can override the inherent TC flow dynamics, which in turn leads to a wide range of flow structures that can have a significant impact on the resulting drag/torque characteristics. Generally, a torque reduction is achieved when λc*≤1, while forcing the Taylor vortices to stay on the crests of the corrugations can lead to significant improvement in torque reduction. Finally, the geometrical shape of the corrugations mainly alters the wall shear stress distribution on the corrugated wall, with a negligible effect on the flow dynamics when compared to λc*.In this paper, numerical simulations are performed on the interaction of vortices with a longitudinal corrugated wall in a Taylor-Couette (TC) setting with the inner smooth surface cylinder rotating and the outer corrugated surface cylinder stationary. The motivation of the study is to shed light on how such an interaction affects the drag/torque with respect to two geometric parameters of the corrugations, namely, the wavelength λc* and amplitude A*, where * indicates a normalization by the gap width d. Results show that in the circular Couette flow regime, the secondary vortices induced by the corrugations cause the torque to increase. When λc* 1, there is a steeper increase of torque due to the interaction of the growing secondary vortices and the opposite wall. In the Taylor-vortex flow regime, the interaction between the Taylor vortices and the corrugations produces three distinct behaviors characterized by λc*. As the wavelength ...

A hybrid Reynolds Averaged Navier-Stokes (RANS)/Large Eddy Simulation (LES) model based on the k equation is proposed with the quadratic stresses to stimulate the quick growth of resolved turbulence around the RANS/LES interface and remove the artificial buffer layer. The proposed hybrid model is applied to simulate the Taylor-Couette (TC) flow with different RANS/LES interface locations and Taylor numbers. The model is verified by comparing with the direct numerical simulation results and validated by experimental torques with different Taylor numbers. The model is insensitive to the interface location owing to the effect of the quadratic stress in adjusting the energy backscatter. By means of the proposed model, the shift of the mean velocity profile around the RANS/LES interface is diminished. It is revealed that the contribution of the Reynolds shear stress to transporting angular velocity in the annular gap center region rises from 20% to 50% as the radius ratio reduces from 0.909 to 0.5 in the TC flow. The tilting angle of herringbone streaks in the TC flow decreases with increasing Ta number, while varying non-monotonically with respect to the radius ratio.

The effect of an axial homogeneous magnetic field on the turbulence in the Taylor-Couette flow confined between two infinitely long conducting cylinders is studied by the direct numerical simulation using a periodic boundary condition in the axial direction. The inner cylinder is rotating, and the outer one is fixed. We consider the case when the magnetic Reynolds number Rem ≪ 1, i.e., the influence of the induced magnetic field on the flow is negligible that is typical for industry and laboratory study of liquid metals. Relevance of the present study is based on the similarity of flow characteristics at moderate and high magnetic field for the cases with periodic and end-wall conditions at the large flow aspect ratio, as proven in the earlier studies. Two sets of Reynolds numbers 4000 and 8000 with several Hartmann numbers varying from 0 to 120 are employed. The results show that the mean radial induced electrical current, resulting from the interaction of axial magnetic field with the mean flow, leads t...

The shear-induced migration of concentrated non-Brownian monodisperse suspensions in combined plane Couette-Poiseuille (C-P) flows is studied using a lattice Boltzmann simulation. The simulations are mainly performed for a particle volume fraction of ϕbulk = 0.4 and H/a = 44.3, 23.3, where H and a denote the channel height and radius of suspended particles, respectively. The simulation method is validated in two simple flows, plane Poiseuille and plane Couette flows. In the Poiseuille flow, particles migrate to the mid-plane of the channel where the local concentration is close to the limit of random-close-packing, and a random structure is also observed at the plane. In the Couette flow, the particle distribution remains in the initial uniform distribution. In the combined C-P flows, the behaviors of migration are categorized into three groups, namely, Poiseuille-dominant, Couette-dominant, and intermediate regimes, based on the value of a characteristic force, G, where G denotes the relative magnitude o...

A two-fluid Taylor–Couette flow with a deformable interface separating two liquid layers is studied numerically by a combination of the finite volume and level set methods. Effect of the interfacial tension is accounted for. It is shown that if the layers are infinitely long, there exist stable steady states with Taylor vortices of finite strength and finite deformations of the interface. On the other hand, if the length of the layers is finite and no-slip conditions are imposed at the edges, the liquid–liquid interface becomes unstable near the edges. Data from the literature and experimental data acquired in the present work are used for comparison with the numerical predictions. A qualitative agreement between the experimental and numerical observations of this instability is obtained. The results are of potential importance for development of bioseparators employing Taylor vortices for enhancement of mass transfer of a passive scalar (say, a protein) through the interface.

The results of an experimental study on the bifurcation structure of oscillatory modes in counterrotating Taylor–Couette flow with stationary end plates are presented. It is shown that the cylinder length L acts as an important geometric control parameter of the system. As a result of a supercritical Hopf bifurcation it is found that for an aspect ratio Γ=L/d>16 (d gap width) only spiral vortices appear in basic laminar flow. For Γ

Motivated by recent theoretical and experimental work, we numerically investigate spherical Couette flow with a view to obtaining for the first time Taylor vortices at large aspect ratios σ such as 0.38, 0.42, and 0.48. It is found that Taylor vortices can exist, stable or time-dependent, in a range of Reynolds numbers [Re1, Re2] and their formation depends on the aspect ratio, on the imposition of various rotationary conditions on the boundaries, on the history of the flow and on the rate at which energy is transferred into the fluid to its final value. With increasing σ the range [Re1, Re2] manifests a clear tendency to shorten.

The wake of a cube placed in a uniform flow is numerically studied. The present study concentrates on the first two transitions, one spatial and the other a temporal one. Computations are carried out for a Reynolds number range of 20–300. Starting from a steady symmetric flow, the transition to asymmetric steady flow occurs at a Reynolds number between 216 and 218. The loss of symmetry increases with increasing Reynolds number. The asymmetric steady flow experiences a Hopf bifurcation at a Reynolds number between 265 and 270 and becomes unsteady but maintains the planar symmetry. The transitions of the present study are compared to the transitions of flow past a square cylinder and a sphere. The transition sequence and flow structures match closely to that of a sphere. The drag and side force coefficients are calculated and it is observed that they are in accordance with the spatial structures of the wake. The drag coefficient is found to decrease with Reynolds number in the steady flow regime. Similarly, one component of the side force coefficients increases while the other component remains zero with increasing Reynolds number for the asymmetric flow.

The time-independent plane Couette flow of a gas in the continuum limit is studied on the basis of kinetic theory as the limit of the cylindrical Couette flow of a rarefied gas between two rotating coaxial circular cylinders when the mean free path and the curvature (or the inverse of the radius) of the inner cylinder tend to zero simultaneously, keeping the difference of the radii of the two cylinders fixed. The fluid-dynamic-type equations and their boundary conditions governing the limiting state are derived for arbitrary circumferential speeds of rotation of the cylinders and for arbitrary temperature difference of the two cylinders. The resulting equations depend on the relative speed of decay of the two parameters and contain a term due to the infinitesimal curvature of the cylinder, as well as non-Navier–Stokes stress terms, when the curvature decays not faster than some function (generally, the square) of the mean free path. The bifurcation analysis of the plane Couette flow of a linear profile is carried out when the speeds of the walls and their temperature difference are small. The bifurcation point and the bifurcated flow fields, where the linear profile is considerably deformed by infinitesimal cross flows induced by the infinitesimal curvature, are obtained.

This paper reports velocity measurements of slow flow through a model brush. The flow field was created in the annulus between two concentric cylinders, which was filled with a viscous oil. The inner cylinder was stationary and the outer one was installed on a turntable and rotated at a constant speed. A brush was modeled by an array of uniformly spaced rods mounted horizontally onto the inner cylinder. With a generous gap between the rod ends and the outer cylinder, the flow outside the array was circular Couette flow. Three brushes were made and the external shear flow penetrated these because the solid volume fractions were small, namely, 0.025, 0.05, and 0.10. Particle image velocimetry was used to study the velocity field in the penetration region, and from these measurements the velocity at the interface, “the slip velocity,” was determined. The slip velocity was found to be close to the value predicted by Brinkman’s equation, and to be higher than velocities found previously for rod arrays in other...

Recent studies [Lathrop et al., Phys. Rev. A 46, 6390 (1992); Lewis et al., Phys. Rev. E 59, 5457 (1999)] on Taylor–Couette flow, where the inner cylinder is rotating and the outer one is at rest, show that, despite earlier predictions [Wendt, Ing. Arch. 4, 557 (1933); Tong et al., Phys. Rev. Lett. 65, 2780 (1990)], the non-dimensional torque (G=T/ρν2L) does not follow a fixed power-law scaling (i.e., G∼Reα, where α is a constant value) for 800

Recent theoretical [Al-Mubaiyedh et al., Phys. Fluids 11, 3217 (1999); J. Fluid Mech. 462, 111 (2002)] and experimental [White and Muller, Phys. Rev. Lett. 84, 5130 (2000); J. Fluid Mech. 462, 133 (2002)] studies have revealed that viscous heating causes significant destabilization of the Taylor–Couette flow of highly viscous and thermally sensitive fluids. In this work, the roles of thermal sensitivity of fluid properties and co-rotation on the thermo-mechanical stability of Taylor–Couette flow are investigated theoretically. In turn, our theoretical findings are compared with the recent experimental ones by White and Muller [Phys. Fluids 14, 3880 (2002)]. It is shown that a finite gap temperature is necessary to predict the time-dependent transitions observed in the experiments. A universal scaling between the critical Reynolds number and the Nahme number is obtained for intermediate values of Nahme number ranging from 0.01 to 1.0. Studies are also performed to determine the influence of co-rotation of the outer cylinder relative to the inner one on the thermo-mechanical stability. Overall, a very favorable comparison between theoretical and experimental results is obtained.

A linear stability analysis has been implemented for the Taylor–Dean flow between porous concentric rotating cylinders when radial flow is present. The stability equations with respect to both axisymmetric and nonaxisymmetric disturbances are derived and solved by a direct numerical procedure. Both types of radial flows, inward and outward flows, are considered. A parametric study covering wide ranges of α, the radial Reynolds number based on the radial velocity at the inner cylinder and inner radius, and β, a parameter characterizing the ratio of average pumping velocity due to azimuthal Poiseuille flow and rotation velocity due to inner cylinder rotation, is conducted for the situation of practical interest where the outer cylinder is stationary and the inner cylinder is rotating. The area where the onset mode is nonaxisymmetric is shown in the plane (β,α). A critical curve is discovered that the most stable state always occurs at the point on this curve for an assigned value of α. Moreover, a discontinuity of the critical axial wavenumber happens when the parameters α or β cross this curve. The critical mode transition of the onset of instability is demonstrated in detail and results for the variations of the critical Taylor number and axial wavenumber are presented. It is found that the superimposed radial flow may produce a stabilizing or destabilizing effect depending heavily on the value of β related to the basic state.

The stability of wavy supercritical cylindrical Couette flow has been studied extensively, but few measurements of the velocity field in flow have been made. Particle image velocimetry was used to measure the azimuthal and radial velocities in latitudinal planes perpendicular to the axis of rotation for wavy cylindrical Couette flow in the annulus between a rotating inner cylinder and a fixed outer cylinder. These measurements were matched to previous measurements of the axial and radial velocity measured in several meridional planes resulting in an experimentally measured, time-resolved, three-dimensional, three-component velocity field for wavy cylindrical Couette flow. Using this complete velocity field it is possible to evaluate details of the flow field. The vortical motion transports azimuthal momentum radially while the axial exchange of fluid between vortices in wavy flow transports azimuthal momentum axially. As the Reynolds number increases, these effects strengthen. Streams of net axial flow stretch axially along the length of the annulus and wind around the vortices from the inner cylinder to the outer cylinder and back while also winding azimuthally in the annulus. The azimuthal velocity measured at the center of a vortex is similar to the azimuthal wave speed. Measurements of the azimuthal velocity in cylindrical surfaces concentric with the axis of rotation suggest that the origin of the waviness is related to a jet-like azimuthal velocity profile rather than the radial outflow jet. Near both cylinder walls, the shear stress is quite large, decreasing to near zero at the middle of the annular gap.

We present the results of a combined experimental and numerical study of Taylor–Couette flow where both inner and outer cylinders rotate. Excellent quantitative agreement has been obtained between finite-element calculations and experimental measurements for a range of aspect ratios and rotation rates. Counter-rotation was found to enhance the breaking of the reflectional symmetry about the midplane of the apparatus, corotation to suppress symmetry breaking. For the region of parameter space explored, increasing the corotation of the outer cylinder had a qualitatively similar effect to increasing the aspect ratio, and vice versa.

The rotational states of three-dimensional nonspherical particles including cylindrical and disk-shaped, as well as prolate and oblate, ellipsoidal in a Couette flow are studied by using a lattice Boltzmann simulation. We discover that rotation of a nonspherical particle exhibits several different states, depending on the ranges of the particle Reynolds numbers and the geometric shape of the particle. As the Reynolds number increases, the rotation transits from one state to another state.

This work is devoted to the experimental study of the transition to turbulence of a flow confined in a narrow gap between a rotating and a stationary disk. When the fluid layer thickness is of the same order of magnitude as the boundary layer depths, the azimuthal velocity axial gradient is nearly constant and this rotating disk flow tends to be a torsional Couette flow. As in the plane Couette flow or the Taylor–Couette flow, transition to turbulence occurs via the appearance of turbulent domains inside a laminar background. In the rotating disk case, the nucleation of turbulent spirals, previously called “solitary waves” in the rotating disk flow literature, is connected to the birth of structural defects in a periodic underlying roll pattern. As the rotation rate is increased, the lifetime of these turbulent structures increases until a threshold is reached where they then form permanent turbulent spirals arranged nearly periodically all around a circumference. However, since the number of these turbul...

A number of questions regarding the effects of viscous dissipation on the stability of Newtonian Taylor–Couette flows are addressed through flow visualization experiments. Studies with three types of fluids of varying thermal sensitivity reveal that the instability is caused by a coupling of centrifugal destabilization and thermally induced gradients in viscosity. These tests also demonstrate that the Nahme number is the appropriate variable with which to describe dissipative effects on the hydrodynamic stability. Additional tests are performed to elucidate the effects of centrifugal destabilization by varying the amount of outer and inner cylinder co-rotation with the Reynolds number and Nahme number fixed. Lastly, a small hysteresis loop in the critical conditions is revealed by performing ramp tests in which the instability is approached from either above or below the critical condition; this indicates that the instability is weakly subcritical.

Transient growth due to non-normality is investigated for the Taylor-Couette problem with counter-rotating cylinders as a function of aspect ratio eta and Reynolds number Re. For all Re < 500, transient growth is enhanced by curvature, i.e. is greater for eta < 1 than for eta = 1, the plane Couette limit. For fixed Re < 130 it is found that the greatest transient growth is achieved for eta between the Taylor-Couette linear stability boundary, if it exists, and one, while for Re > 130 the greatest transient growth is achieved for eta on the linear stability boundary. Transient growth is shown to be approximately 20% higher near the linear stability boundary at Re = 310, eta = 0.986 than at Re = 310, eta = 1, near the threshold observed for transition in plane Couette flow. The energy in the optimal inputs is primarily meridional; that in the optimal outputs is primarily azimuthal. Pseudospectra are calculated for two contrasting cases. For large curvature, eta = 0.5, the pseudospectra adhere more closely to the spectrum than in a narrow gap case, eta = 0.99.

A spatial, viscous stability analysis of Poiseuille pipe flow with superimposed solid body rotation is considered. For each value of the swirl parameter (inverse Rossby number) L>0, there exists a critical Reynolds number Rec(L) above which the flow first becomes convectively unstable to nonaxisymmetric disturbances with azimuthal wave number n=−1. This neutral stability curve confirms previous temporal stability analyses. From this spatial stability analysis, we propose here a relatively simple procedure to look for the onset of absolute instability that satisfies the so-called Briggs–Bers criterion. We find that, for perturbations with n=−1, the flow first becomes absolutely unstable above another critical Reynolds number Ret(L)>Rec(L), provided that L>0.38, with Ret→Rec as L→∞. Other values of the azimuthal wave number n are also considered. For Re>Ret(L), the disturbances grow both upstream and downstream of the source, and the spatial stability analysis becomes inappropriate. However, for Re

In the laminar–turbulent transition of spherical Couette flow with only the inner sphere rotating in the case of gap ratio 0.14, we have studied experimentally the evolution of mean and fluctuating velocities with increasing Reynolds number at two meridian angles, θ=60° and 90° (the equator), respectively. For a reduced Reynolds number, R*=1.2, the phase of a fluctuating velocity component in the main-flow direction advances radially inward from the outer to the inner sphere at θ=90°, but delays radially inward at θ=60°. When R* is 4.2 and 6.0, the phase slightly advances radially inward at θ=90°, but does not change so much at θ=60°. The phase-averaged profile of fluctuating velocity in the presence of spiral Taylor–Gortler (TG) vortices differs from that in the presence of traveling azimuthal waves. Spiral TG vortices make the amplitude of fluctuating velocity large in the central part of the gap. But traveling azimuthal waves make it large somewhat near the inner sphere. For the flow with spiral TG vor...

We report the results of an experimental and numerical study of the bifurcation structure of steady cellular states in small aspect ratio Taylor–Couette flows. In particular we focus on Z2 symmetry-breaking bifurcations where asymmetric states which break the mid-plane symmetry arise. We find good quantitative agreement between numerical and experimental results for the bifurcation set. The surprising new observation we make is that the symmetry-breaking bifurcation sequences coalesce in a self-consistent way as the aspect ratio is reduced so that only symmetric steady flows exist for very short cylinders. The implications of this for the onset of disordered motion are discussed.

Linear stability of the Taylor–Couette (TC) flow of semidilute non-Brownian suspension is investigated by utilizing the fiber orientation model developed by Hinch and Leal [J. Fluid Mech. 76, 187 (1976)] in conjunction with a quadratic and hybrid closure proposed by Advani and Tucker [J. Rheol. 34, 367 (1990)]. It is found that irrespective of the closure approximation used the fiber additives suppress the centrifugal TC instability, i.e., the critical Reynolds number (Re) increases with the fiber volume fraction and aspect ratio as well as the interfiber interaction coefficient. This increase in the critical Re is significantly larger than that in the total viscosity, except for very small values of the volume fraction and the interaction coefficient. The enhanced stabilization can be attributed to the fact that the suspension develops negative first and second normal stresses in the TC flow when the inner cylinder rotates and the outer one is stationary, i.e., the fluid is in a state of compression. Mor...

The unfolding due to imperfections of a gluing bifurcation occurring in a periodically forced Taylor–Couette system is numerically analyzed. In the absence of imperfections, a temporal glide-reflection Z2 symmetry exists, and two global bifurcations occur within a small parameter region: a heteroclinic bifurcation between two saddle two-tori and a gluing bifurcation of three-tori. Due to the presence of imperfections, these two global bifurcations collide, strongly reducing the range of validity of the generic unfolding of the gluing bifurcation.

This work is devoted to the study of transient growth of perturbations in the Taylor–Couette problem due to linear 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. Similarities with transient growth mechanisms in plane Couette flow and in Hagen–Poiseuille flow are discussed. This is reflected in the modulation of the basic circular Couette flow by the presence of azimuthal streaks as a result of the nonmodal growth of initial axisymmetric perturbations. 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.

The transition from Taylor to wavy vortices is revisited for parameter values in the range of new laboratory experiments [Lin et al., Phys. Fluids 10, 3233 (1998)]. The dependence of the critical Reynolds number with the axial wavelength of the Taylor vortices is obtained for azimuthal wave numbers from 1 to 5, and for five different values of the radius ratio. We show how islands of stable Taylor vortices above the transition to wavy vortices form.

Recently, based on a linear stability analysis we demonstrated the existence of a new thermoelastic mode of instability in the viscoelastic Taylor–Couette flow [Al-Mubaiyedh et al., Phys. Fluids 11, 3217 (1999); J. Rheol. 44, 1121 (2000)]. In this work, we use direct time-dependent simulations to examine the nonlinear evolution of finite amplitude disturbances arising as a result of this new mode of instability in the postcritical regime of purely elastic (i.e., Re=0), nonisothermal Taylor–Couette flow. Based on these simulations, it is shown that over a wide range of parameter space that includes the experimental conditions of White and Muller [Phys. Rev. Lett. 84, 5130 (2000)], the primary bifurcation is supercritical and leads to a stationary and axisymmetric toroidal flow pattern. Moreover, the onset time associated with the evolution of finite amplitude disturbances to the final state is comparable to the thermal diffusion time. These simulations are consistent with the experimental findings.

A linear stability analysis was carried out for a dilute suspension of rigid spherical particles in cylindrical Couette flow. The perturbation equations for both the continuous fluid phase and the discontinuous particle phase were decomposed into normal modes resulting in an eigenvalue problem that was solved numerically. At a given radius ratio, the theoretical critical Taylor number at which Taylor vortices first appear decreases as the particle concentration increases. Increasing the ratio of particle density to fluid density above one decreases the stability. However, using an effective Taylor number based on the suspension density and viscosity largely accounts for this effect. The axial wave number is the same for a suspension as it is for a pure fluid. Experiments using neutrally buoyant particles in a Taylor–Couette apparatus show that the flow is more stable as the particle concentration increases. The reason that the theory does not fully capture the physics of the flow should be addressed in future research.

Turbulence statistics, including complete budget data for the six individual components of the Reynolds stress tensor, have been compiled from a direct numerical simulation of the three-dimensional flow in an open rotor–stator configuration. Comparisons between turbulence statistics near the two disks and in plane Couette flow showed that the stator-side data closely resemble those found in Couette flow. At the rotor side rotational production terms due to the Coriolis force made only modest contributions to the budget equations for the Reynolds stresses, but the subtle influence of system rotation was by no means negligible.

Linear stability of Couette flow over a wavy wall is considered. It is shown that centrifugal effects may create an instability that leads to the formation of streamwise vortices. The conditions leading to the onset of the instability depend on the amplitude and the wavelength of the waviness and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number and the associated wave number of wall waviness required to create the instability for the specified amplitude of the waviness are also given.

Electrorheological (ER) characteristics of Nematic Liquid Crystals (NLCs) have been a topic of immense interest in the field of soft matter physics owing to its rheological modulation capabilities. Here we explore the augmentation in rheological characteristics of the nematic fluid confined within the annular region of the concentric cylindrical space with an Electrical Double Layer (EDL) induced at the fluid-substrate interface due to certain physico-chemical interactions. Using a Taylor-Couette flow configuration associated with an EDL induced at the inner cylinder wall, we show that a spontaneous electrorheological effect is generated owing to the intrinsic director anisotropy and structural order of complex nematic fluids. We seek to find the enhancement in torque transfer capability due to the inherent electrorheological nature of the nematic medium, apart from exploiting the innate nature of such homogeneous media to remain free of coagulation, a fact which makes it an excellent candidate for the ap...

The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the sc...

Nonlinear evolution of pancake-like vortices in a uniformly rotating and stratified fluid is studied using a 3D Boussinesq numerical model at large Rossby numbers. After the initial stage of viscous decay, the simulations reveal exponential growth of toroidal circulation cells (aka Taylor vortices) at the peripheral annulus with a negative Rayleigh discriminant. At the nonlinear stage, these thin cells redistribute the angular momentum and density differently at the levels of radial outflow and inflow. Resulting layering, with a vertical stacking of sharp variations in velocity and density, enhances small-scale mixing and energy decay. Characteristic detectable stretching patterns are produced in the density field. The circulation patterns, induced by centrifugal instability, tend to homogenize the angular momentum in the vicinity of the unstable region. We demonstrate that the peak intensity of the cells and the vortex energy decay are dramatically reduced by the earth’s rotation due to conservation of t...

Accurate prediction of laminar to turbulent transition in compressible flows is a challenging task, as it can be affected by a combination of factors. Compressibility causes large variations in thermodynamic as well as transport properties of a gas, which in turn are known to affect flow stability. We study the stratification of individual transport properties and their combined behavior. We also examine the effect of a change in the magnitude of viscosity and conductivity on flow stability. The Couette flow of a perfect gas is our model problem and both modal and non-modal analyses are carried out. We notice a large destabilizing role of the increase in the conductivity value and a dramatic stabilizing effect of mean viscosity stratification, over a range of free-stream Mach number, Reynolds number, Prandtl number, and disturbance wavenumber. In the combined case, viscosity stratification plays a dominant role. We find this to be the case for finite-time transient growth in the parameter regime below linear instability as well as asymptotically at large time. A budget of the transient growth energy amplification is also shown to identify the effects of transport properties on the constituents of perturbation energy. The extensive results presented in this paper, we believe should motivate those studying more realistic flows to examine how these contrasting effects of stratification come together. Published by AIP Publishing.

The effect of steady base suction and blowing on the stability and dynamics of the cylinder wake is investigated at low Reynolds numbers, using numerical simulation and stability analysis. Simulation results show that, in the supercritical Reynolds number regime (Re>47), slight blowing or high enough suction stabilizes the wake; in the subcritical regime, suction destabilizes the wake and results in vortex shedding, whereas blowing has no detectable effect on the flow stability. At supercritical Reynolds numbers, the transition from unsteady to steady flow at a critical suction flow rate is accompanied by simultaneous symmetry breaking, resulting in strongly asymmetric steady flow. For finite flow domain, the flow undergoes another transition from steady asymmetric to steady symmetric flow at even higher suction flow rates. The dynamics of vortex shedding in the controlled flow can be strongly modified, in comparison to the uncontrolled flow. Global stability analysis confirms the results of numerical sim...

Results are presented from an experimental investigation of a novel shear flow. Two parallel sections of planar Couette flow are connected by two semicircular sections of circular Couette flow to give a flow domain with the shape of a running track. Driving the flow with a moving inner boundary leads to centrifugal instability in the curved regions as in conventional Taylor–Couette flow. This is in contrast to the planar regions, which are linearly stable and are characterized instead by finite-amplitude instability. In the steady regime, the entire flow field is dominated by structures akin to Taylor vortices. The mechanism of exchange between a four-cell and a six-cell flow over a range of aspect ratio is qualitatively the same as for the standard Taylor–Couette problem. In the unsteady regime, the flow is characterized by various spatiotemporal modes, the selection of which is dependent on the manner of flow evolution. Quasistatic increase of the Reynolds number from zero typically results in flow with a banded spatial structure and low-dimensional dynamics, both of which are associated with instability in the semicircular regions. However, an abrupt step-like increase of Reynolds number produces a persistent flow state with strong spatial disorder and a broadband dynamical spectrum. The results of this study have implications for the conventional distinctions between the properties of open and closed flows, and suggest the possibility of intermediate flows which are worthy of investigation in their own right.

A rigorous upper bound on the long-time-averaged vertical buoyancy flux is derived from the Navier–Stokes equations for a Boussinesq fluid confined between two parallel horizontal plates a distance d apart, maintained at a constant statically stabilizing temperature difference ΔT and driven at a constant relative velocity ΔU. The upper bound on the volume and long-time-averaged vertical buoyancy flux B≔limt→∞1/t∫0t 〈ρu3〉g/ρ0 dt is B⩽Bmax=(1−16√/Re)(ΔU)3/(64√d), where Re=ΔUd/ν and ρ0 is some reference density. Significantly, Bmax is independent of the bulk Richardson number of the flow and is achieved by an optimal solution with a mixing efficiency (or flux Richardson number) which approaches 0.5 as the Reynolds number becomes large. The time-averaged turbulent dissipation of kinetic energy and the time-averaged vertical buoyancy flux are then in equipartition for the optimizing flow.

We present the results of an experimental and numerical study of bifurcation phenomena in the flow between a rotating inner cylinder and a stationary outer one, the so-called Taylor–Couette problem. Novel asymmetric boundary conditions have been used where one end plate is rotated with the inner cylinder while the other is held fixed. The steady cellular flows consist of one or three vortices in the aspect ratio range considered. We have investigated the selection procedure for the preferred steady states experimentally and numerically and good quantitative agreement is found between the two sets of results. The sequence involves a fold in the solution surface which produces a cusp in the bifurcation set. Hopf bifurcations to three-dimensional oscillatory flow are also present and so the full bifurcation structure is revealed using the combined numerical and experimental approach in a complimentary manner.

The first portion of this two-part paper investigates the short time evolution of a low Reynolds number flow characterized by a spatially inhomogeneous viscosity within the annular domain between two widely separated concentric circular cylinders undergoing relative rotation. The viscosity is regarded as a material property and as such is convected with the fluid. Any possible “diffusion” of this viscosity is supposed negligible, at least in the short times of interest in our calculation. The initial viscosity field, assumed to be only slightly inhomogeneous, is expanded into a Fourier series with respect to the polar angle, and the contributions of the zeroth and first harmonics are subsequently addressed. Approximate short-time analytic solutions for the velocity and viscosity fields are obtained. In the second part of this paper the results of the preceding analysis are employed in an attempt to gain insight into the experimentally observed shear-induced migration of particles in suspensions being shea...

Results from the experimental study of fully developed turbulent plane Poiseuille–Couette flow (PCF) are presented. This study provides information that improves the physical descriptions of previous experimental works and sheds light on turbulence phenomena reported at low to moderate Reynolds numbers. Molecular tagging velocimetry (MTV) is employed for the first time on gas PCFs over the Poiseuille and Couette flow Reynolds number ranges of 3500

We computationally investigate the stability of a pair of radially stratified immiscible liquids undergoing countercurrent axial flow in the annular gap between rapidly corotating coaxial cylinders: two-fluid Taylor-Couette flow with counterflow. A simple analysis determines conditions under which a nearly cylindrical interface is maintained in the presence of counterflow (i.e., axial pressure gradients). Stability analysis reveals that for small axial Reynolds numbers, the flow is slightly stabilized against Taylor-Couette instability, consistent with results for a single phase. At axial Reynolds numbers greater than about ten, however, the flow is susceptible to a (generally nonaxisymmetric) Kelvin-Helmholtz instability, which precedes the Taylor-Couette mode. Furthermore, new results are presented for the case without axial flow. A bifurcation to vortices that corotate with their counterparts in the other phase is found. Finally, limitations of the generalized Rayleigh criterion developed in our earlie...

The flow in the gap between an inner rotating cylinder concentric with an outer stationary cylinder with an imposed pressure-driven axial flow was studied experimentally using particle image velocimetry (PIV) in a meridional plane of the annulus. The radius ratio was η=0.83 and the aspect ratio was Γ=47. Velocity vector fields for nonwavy toroidal and helical vortices show the axial flow winding around vortices. When the axially averaged axial velocity profile is removed from the velocity field in a meridional plane, the velocity field looks much like it would with no imposed axial flow except that the vortices translate axially and the distortion of the azimuthal velocity contours in meridional plane related to the vortices is shifted axially by the axial flow. The velocity vector fields for wavy vortices also show axial flow winding around the vortices. Again, removing the axial velocity profile results in a flow that appears similar to that with no axial flow. The path of the vortices is generally axial, but the vortices periodically move retrograde to the imposed axial flow due to the waviness of the vortices. The axial velocity of helical vortices, both nonwavy and wavy, is twice the rotational frequency of the inner cylinder indicating a coupling between the axial translation of the vortices and the cylinder rotation. Little fluid transport between vortices occurs for nonwavy vortices, but there is substantial transport between vortices for wavy vortex flow, much like supercritical cylindrical Couette flow with no axial flow.

Previously reported isothermal linear stability analyses of viscoelastic Taylor–Couette flow have predicted transitions to nonaxisymmetric and time-dependent secondary flows for elasticity numbers E≡De/Re>0.01. In contrast, recent experiments by Baumert and Muller using constant viscosity Boger fluids have shown that the primary flow transition leads to axisymmetric and stationary Taylor-type toroidal vortices. Moreover, experimentally observed onset Deborah number is an order of magnitude lower than that predicted by isothermal linear stability analyses. In this work, we explore the influence of energetics on the stability characteristics of the viscoelastic Taylor–Couette flow. Our analysis is based on a thermodynamically consistent reformulation of the Oldroyd-B constitutive model that takes into account the influence of thermal history on polymeric stress, and an energy equation that takes into account viscous dissipation effects. Our calculations reveal that for experimentally realizable values of Pe...

The dynamical behavior of wide-gap instabilities in spherical Couette flow is investigated experimentally with chaos analyzing techniques applied on time series from Laser-Doppler-Velocimetry (LDV) measurements. With an increasing Reynolds number of the rotating inner sphere, the flow undergoes two Hopf bifurcations and several mode changes. The transition scenarios for two different gap widths investigated can be described primarily as the Ruelle–Takens–Newhouse type and show a strong dependence on the meridional coordinate. The transition processes are accompanied by phenomena such as a locked torus in reconstructed phase space and periodic long wave modulations of chaotic flows. Although the investigated gap widths are all classified as wide, a comparison exhibits significant differences in the development of chaotic motion.