Unsteady circular Couette flow of a Bingham plastic with the Augmented Lagrangian Method

Numerical simulations are undertaken for unsteady flows of an ideal Bingham fluid in a circular Couette viscometer. The main difficulties in such simulations are caused by the non-differentiability of the constitutive equation and the need to determine the position and shape of the yield surface separating the yielded zones from the unyielded ones. In this work, these difficulties are overcome by using a numerical method based on variational inequalities, i.e. the augmented Lagrangian/Uzawa method. The start-up and cessation of circular Couette flows of a Bingham fluid are solved numerically assuming that only one of the cylinders is rotating. An improved theoretical upper bound for the stopping time in the case of cessation is derived. The numerical estimates for the stopping time compare well with the theoretical bounds. Moreover, with the adopted method the evolution of the velocity profiles and the locations of yielded/unyielded surfaces are accurately calculated. In flow cessation, we observe an interesting effect, namely the appearance of a small unyielded region adjoined to the outer cylinder shortly before cessation.     

Convective and absolute instabilities in counter-rotating spiral Poiseuille flow

We present results of an experimental study on the stability of Taylor–Couette flow in case of counter-rotating cylinders and an imposed axial through flow. We are able to confirm results form recent numerical investigations done by Pinter et al. [24] by measuring the absolute and convective stability boundaries of both propagating Taylor vortices (PTV) and spiral vortices (SPI). Thus our work shows that these theoretical concepts from hydrodynamic stability in open flows apply to experimental counter-rotating Taylor–Couette systems with an imposed axial through flow. PACS 47.20.-k, 05.45.-a, 47.15.fe     

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.     

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     

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.     

Radial variation of adiabatic and diabatic spiral vortex flow in a wide annular gap

Transitions occurring after the onset of spiral vortex flow in a wide concentric annular gap of radius ratio 0.8, formed by a stationary outer cylinder and a rotatable inner cylinder, have been studied experimentally. By isothermal heating of the annular surface, it was possible to consider diabatic as well as adiabatic conditions. At an axial Reynolds number of 500 and for a range of Taylor numbers up to 10⁷, power spectra and auto-correlograms were taken at two radial positions near to the inner and outer annular surfaces; these are compared with previous results taken at mid-gap under adiabatic conditions. Measurements of turbulence intensity across the gap were made also. Probability histograms and signal traces for diabatic flow near to the outer annular surface are presented. It has been shown that the vortex transitions affect the thermal boundary layer and, consequently, the heat transfer rates at the outer surface. A positive radial thermal gradient was seen to have little effect on the flow. The imposed axial flow was shown to be stabilising under adiabatic conditions but destabilising under diabatic conditions.     

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.     

Circular Couette flow with pressure-driven axial flow and a porous inner cylinder

In a rotating filter separator a suspension is introduced at one end of the annulus between a rotating porous inner cylinder and a fixed impermeable outer cylinder. The filtrate is removed through the inner cylinder and the concentrate is removed from the opposite end of the annulus from which the suspension entered. The flow in a rotating filter separator is circular Couette flow with a pressure-driven axial flow and a suction boundary condition at the inner cylinder. Flow visualization was used to determine the effect of the Taylor number, axial Reynolds number, and radial Reynolds number on the types of flows present in the annulus. A rich variety of secondary vortical flows appear, depending upon the flow parameters. The radial inflow at the inner cylinder delays the appearance of supercritical circular Couette flow and prevents the appearance of certain flow regimes that have a helical vortex structure. Nevertheless, the average azimuthal velocity measured using laser Doppler velocimetry indicates that the velocity profile is nearly the same for all supercritical flow regimes.This work was supported by a grant from The Whitaker Foundation     

Transition from vortex to wall driven turbulence production in the Taylor–Couette system with a rotating inner cylinder

Axisymmetrically stable turbulent Taylor vortices between two concentric cylinders are studied with respect to the transition from vortex to wall driven turbulent production. The outer cylinder is stationary and the inner cylinder rotates. A low Reynolds number turbulence model using the k‐ω formulation, facilitates an analysis of the velocity gradients in the Taylor–Couette flow. For a fixed inner radius, three radius ratios 0.734, 0.941 and 0.985 are employed to identify the Reynolds number range at which this transition occurs. At relatively low Reynolds numbers, turbulent production is shown to be dominated by the outflowing boundary of the Taylor vortex. As the Reynolds number increases, shear driven turbulence (due to the rotating cylinder) becomes the dominating factor. For relatively small gaps turbulent flow is shown to occur at Taylor numbers lower than previously reported. Copyright © 2002 John Wiley & Sons, Ltd.     

An asymmetrical periodic vortical structures and appearance of the self induced pressure gradient in the modified Taylor flow

An incompressible liquid flow in the gap between two coaxial cylinders, such that the inner rotating (wavy) cylinder has a periodically varying radius along the axial direction while the outer stationary cylinder has a constant radius, is studied experimentally and theoretically. Basic attention is focused on the symmetry-breaking phenomenon of the vortex flow arising from the rotation of the inner wavy cylinder. It is found that the symmetry-breaking phenomenon of the vortical flow structures in this geometry is accompanied by the occurrence of a self-induced axial pressure gradient. A theoretical formulation of the problem of periodic vortical flow prevailing in such a geometry having large axial length is presented. The comparison between the computed and the experimental results is presented and the underlying phenomena are discussed.     

A finite element formulation for global linear stability analysis of a nominally two‐dimensional base flow

A stabilized finite element method, to carry out the linear stability analysis of a two‐dimensional base flow to three‐dimensional perturbations that are periodic along span, is presented. The resulting equations for the time evolution of the disturbance requires a solution to the generalized eigenvalue problem. The analysis is global in nature and is also applicable to non‐parallel flows. Equal‐order‐interpolation functions for velocity and pressure are utilized. Stabilization terms are added to the Galerkin formulation to admit the use of equal‐order‐interpolation functions and to eliminate node‐to‐node oscillations that might arise in advection‐dominated flows. The proposed formulation is tested on two flow problems. First, the mode transitions in the circular Couette flow are investigated. Two scenarios are considered. In the first one, the outer cylinder is at rest, while the inner one spins. Two linearly unstable modes are identified. The primary mode is real and represents the axisymmetric Taylor vortices. The second mode is complex and consists of spiral vortices. For the counter‐rotating cylinders, the primary transition is via the appearance of spiral vortices. Excellent agreement with results from earlier studies is observed. The formulation is also utilized to investigate the parallel and oblique modes of vortex shedding past a cylinder for the Re = 100 flow. It is found that the flow is associated with a large number of unstable oblique shedding modes. The parallel mode of vortex shedding is a special case of this family of modes and is associated with the largest growth rate. Copyright © 2014 John Wiley & Sons, Ltd.     

An experimental study of diabatic spiral vortex flow

In spiral vortex flow, between concentric cylinders with the inner cylinder rotating and the outer stationary, the addition of a thermal gradient across the gap is a known complicating factor. The present diabatic study for narrow and wide gaps (radius ratios N=0.955 and N=0.8), with a heated outer and adiabatic inner cylinder, was undertaken to investigate this problem. The heat transfer characteristics and the modes of transition have been investigated together with the relationship between them. Using standard on-line digital computer techniques, the onset of vortex flow and its higher transitions have been shown to cause a sharp increase in Nusselt number. At higher Taylor numbers, of the order of 10⁶, a marked change in the Nusselt number occurs with the onset of the transition to periodic turbulent vortex flow. Outer wall heating is seen to affect the modes of transition. Diabatic critical Taylor numbers are much higher than those for adiabatic conditions and are found to depend on the close approach of the vortices to the outer wall     

MODELLING AND CALCULATION OF THE TURBULENT BOUNDARY LAYER ON A ROTATING CYLINDER SURFACE WITH STRONG SUCTION 1)

提出了湍流边界层的一种简单、快速计算方法, 用以求解强吸气作用下旋转圆筒表面边界层流动. 首先, 理论分析了同心圆筒间的旋转流体运动, 外筒静止、内筒旋转且为多孔吸气条件. 强吸气情况下旋转流动主要表现为内筒壁面附近的边界层流动, 基于这一事实得到了周向速度分布的解析表达式. 其次, 通过引入新参数扩展Cebeci-Smith代数湍流模型, 使其能考虑流线曲率、壁面吸气、低Reynolds数效应等因素. 针对这些因素的综合影响, 采用解析修正和经验参数对模型进行调整. 同时, 基于Reynolds应力湍流模型的仿真结果, 校准代数湍流模型中的经验参数. 最后, 给出基于广义Cebeci-Smith湍流模型的旋转壁面边界层流动的迭代算法, 该算法适用于需要特殊迭代过程的轴向及周向流动均匀情况. 计算了不同旋转速度和吸气强度组合工况下的边界层流动, 其周向速度和湍流强度分布与基于Reynolds应力湍流模型的计算结果非常接近. 并且表明, 当Reynolds应力湍流模型数值模拟预测内筒边界层为稳定层流时, 该方法也再现了相同初始条件下的层流边界层.     

强吸气旋转圆筒壁面湍流边界层建模及计算

提出了湍流边界层的一种简单、快速计算方法,用以求解强吸气作用下旋转圆筒表面边界层流动.首先,理论分析了同心圆筒间的旋转流体运动,外筒静止、内筒旋转且为多孔吸气条件.强吸气情况下旋转流动主要表现为内筒壁面附近的边界层流动,基于这一事实得到了周向速度分布的解析表达式.其次,通过引入新参数扩展Cebeci-Smith代数湍流模型,使其能考虑流线曲率、壁面吸气、低Reynolds数效应等因素.针对这些因素的综合影响,采用解析修正和经验参数对模型进行调整.同时,基于Reynolds应力湍流模型的仿真结果,校准代数湍流模型中的经验参数.最后,给出基于广义Cebeci-Smith湍流模型的旋转壁面边界层流动的迭代算法,该算法适用于需要特殊迭代过程的轴向及周向流动均匀情况.计算了不同旋转速度和吸气强度组合工况下的边界层流动,其周向速度和湍流强度分布与基于Reynolds应力湍流模型的计算结果非常接近.并且表明,当Reynolds应力湍流模型数值模拟预测内筒边界层为稳定层流时,该方法也再现了相同初始条件下的层流边界层.     

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.     

Orbital flow past a cylinder: A numerical approach

Orbital flow past a cylinder is relevant to offshore structures. The numerical scheme presented here is based on a finite-difference solution of the Navier–Stokes equations. Alternating-directional-implicit (ADI) and successive-over-relaxation (SOR) techniques are used to solve the vorticity-transport and stream-function equations. Theoretical simulations to low Reynolds number flows (up to 1000) are discussed for cases involving uniform flow past stationary and rotating cylinders and orbital flow past a cylinder. The separation points for cylinders that are rotating or immersed in an orbital flow are deduced from velocity profiles through the boundary layer using a hybrid mesh scheme. During the initial development of orbital flow surface vorticity on the impulsively started cylinder dominates the flow. A vortex then detaches from behind the cylinder and establishes the flow pattern of the orbit. After some time a collection of vortices circles the orbit and distorts its shape a great deal. These vortices gradually spiral outward as others detach from the cylinder and join the orbital path.     

Hydrodynamic forces on dual cylinders of different diameters in steady currents

Turbulent flow past two circular cylinders of different diameters is numerically investigated. The two-dimensional Reynolds-averaged Navier–Stokes equations are solved by using a finite element method with a k–ω turbulence closure. Following a relevant numerical model validation process, effects of cylinder gap-to-diameter ratio, the angular position of the smaller cylinder and the diameter ratio of cylinders on the vortex shedding and the forces on the cylinders are investigated using the numerical model. It is found that the relative position of the small cylinder has significant effects on the hydrodynamic force and vortex shedding characteristics of the cylinders.     

Computations with viscoplastic and viscoelastoplastic fluids

This numerical study focuses on regularised Bingham-type and viscoelastoplastic fluids, performing simulations for 4:1:4 contraction?Cexpansion flow with a hybrid finite element?Cfinite volume subcell scheme. The work explores the viscoplastic regime, via the Bingham?CPapanastasiou model, and extends this into the viscoelastoplastic regime through the Papanastasiou?COldroyd model. Our findings reveal the significant impact that elevation has in yield stress parameters, and in sharpening of the stress singularity from that of the Oldroyd/Newtonian models to the ideal Bingham form. Such aspects are covered in field response via vortex behaviour, pressure-drops, stress field structures and yielded?Cunyielded zones. With rising yield stress parameters, vortex trends reflect suppression in both upstream and downstream vortices. Viscoelastoplasticity, with its additional elasticity properties, tends to disturb upstream?Cdownstream vortex symmetry balance, with knock-on effects according to solvent-fraction and level of elasticity. Yield fronts are traced with increasing yield stress influences, revealing locations where relatively unyielded material aggregates. Analysis of pressure drop data reveals significant increases in the viscoplastic Bingham?CPapanastasiou case, O (12%) above the equivalent Newtonian fluid, that are reduced to 8% total contribution increase in the viscoelastoplastic Papanastasiou?COldroyd case. This may be argued to be a consequence of strengthening in first normal stress effects.     

A two-dimensional model of the flow of ordered suspensions of rods

Analysis of creeping flow over an array of freely-rotating cylinders sandwiched between two sliding parallel plates is studied using a finite-difference and a least-squares numerical technique. The flow pattern was found to be very much influenced by the cylinder-to-cylinder spacing and by the gap width of the parallel plates. The shear stress on the cylinder surface and on the parallel plates was found to be a strong function of position. The viscosity of a suspension composed of an array of freely-rotating cylinders was deduced from the applied shear rate and the evaluated shear stress on the parallel plates. Experimental results confirm the numerical findings.     

Identification of flow regimes around two staggered square cylinders by a numerical study

The flow over two square cylinders in staggered arrangement is simulated numerically at a fixed Reynolds number (\(Re =150\)) for different gap spacing between cylinders from 0.1 to 6 times a cylinder side to understand the flow structures. The non-inclined square cylinders are located on a line with a staggered angle of \(45^{\circ }\) to the oncoming velocity vector. All numerical simulations are carried out with a finite-volume code based on a collocated grid arrangement. The effects of vortex shedding on the various features of the flow field are numerically visualized using different flow contours such as \(\lambda _{2}\) criterion, vorticity, pressure and magnitudes of velocity to distinguish the distinctive flow patterns. By changing the gap spacing between cylinders, five different flow regimes are identified and classified as single body, periodic gap flow, aperiodic, modulated periodic and synchronized vortex shedding regimes. This study revealed that the observed multiple frequencies in global forces of the downstream cylinder in the modulated periodic regime are more properly associated with differences in vortex shedding frequencies of individual cylinders than individual shear layers reported in some previous works; particularly, both shear layers from the downstream cylinder often shed vortices at the same multiple frequencies. The maximum Strouhal number for the upstream cylinder is also identified at \({G}^{*}=1\) for aperiodic flow pattern. Furthermore, for most cases studied, the downstream cylinder experiences larger drag force than the upstream cylinder.