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.     

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.     

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.     

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.     

Influence of mass-flow ratio of inner to outer secondary air on gas–particle flow near a swirl burner

The influence of mass-flow-rate ratio of inner to outer secondary air on gas–particle flow characteristics was determined in the near-burner region of a centrally fuel-rich swirl coal combustion burner. Velocity and particle volume flux profiles and normalized particle number concentrations were obtained. Peaks in tangential mean velocity and three-dimensional root-mean-square fluctuation velocities were found to decrease as the mass-flow-rate ratio increased. Moreover, the peaks in the mean axial velocities and particle volume flux near the wall increased, whereas those near the chamber axis decreased. Simultaneously, both recirculation zone and swirl number decreased as the mass-flow-rate ratio increased.     

Effects of longitudinal grooves on the Couette–Poiseuille flow

The effects of longitudinal grooves on the flow resistance in a channel where the flow is driven by movement of one of the walls and modified by a streamwise pressure gradient have been studied. The reducedorder geometrymodel has been used to extract geometric features that are hydraulically relevant. Three distinct zones leading to the reduced resistance have been identified, depending on the flow pressure gradient and the groove wave number. Two of these zones correspond to grooves with long wavelengths and one to grooves with short wavelengths. Optimization has been used to determine shapes that provide the largest flow rate. In the case of the long-wavelength grooves, the optimal shapes depend on the constraints. These shapes are well approximated by a certain universal trapezoid for grooves that have the same height and depth. There exists an optimum depth which, combined with the corresponding shape, defines the optimal geometry in the case of the unequal-depth grooves; this shape is well approximated by a Gaussian function. No optimal shape exists for the short-wavelength grooves if the groove amplitude is sufficiently small; the shortest admissible wavelength dominates system performance under such conditions. The most effective groove wave number does exist for higher grooves, but the optimal shape cannot be determined due to numerical limitations.     

Influence of conicity on the stress–strain state of a flexible orthotropic conical shell in a nonstationary magnetic field

A problem of magnetoelasticity for a flexible conical shell in a nonstationary magnetic field is solved. The effect of conicity on the stress–strain state of the shell is analyzed     

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.     

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.     

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.     

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.     

Wavelet analysis on particle dynamics in a horizontal air–solid two-phase pipe flow at low air velocity

High-speed particle image velocimetry (PIV) is first used to measure two components of the fluctuating particle velocities for different particle sizes and solid mass flow rates at low air velocity in a horizontal pipe. Then, the continuous wavelet transform and orthogonal wavelet multi-resolution techniques are employed to analyze and decompose the fluctuating particle velocities to provide both quantitative and qualitative information on the particle fluctuation velocity of various frequencies. It is revealed that the fluctuating energy of axial particle velocity is mainly contributed from the wavelet components of low frequency, accounting for about 84%, near the bottom part of the pipe cross-section. However, the contribution to the fluctuating energy of vertical particle velocity accounts for about 82% from the wavelet components of high frequency. The auto-correlation analysis suggests a quasi-periodical large-scale axial particle fluctuating velocity. On the other hand, the spatial correlation analysis indicates that the low-frequency components of the axial particle velocity exhibit a large correlation near the bottom part of the pipe cross-section. From the probability density function (PDF) distribution, it is found that the low-frequency components of the axial particle velocity exhibit larger fluctuation, and this fluctuation reduces as the frequencies increase near the bottom part of the pipe cross-section. Near the top part of the pipe cross-section, however, a larger fluctuating axial particle velocity appears in the high-frequency range.     

Multi-scale particle dynamics of low air velocity in a horizontal self-excited gas–solid two-phase pipe flow

The particle fluctuation velocities of a horizontal self-excited gas–solid two-phase pipe flow with soft fins near MPD (minimum pressure drop) air velocity are first measured by high-speed PIV in the acceleration and fully-developed regimes. Then orthogonal wavelet multi-resolution analysis and power spectrum are used to reveal multi-scale characteristics of particle fluctuation velocity. It is observed that the pronounced peaks of the spectra of axial and vertical fluctuation velocities appear in the range of low frequency near the bottom of pipe. These peaks of spectra become larger and their frequencies decrease by using fins. In the range of low frequencies (3–25 Hz), the wavelet components of the fluctuating energy of axial particle velocity make the main contribution accounting for 87% and 93% respectively for non-fin and using fins near the bottom of pipe. In the range of relatively high frequency (50–400 Hz), however, the wavelet components of using fins, accounting for about 49%, become smaller than that of non-fin, accounting for about 72%, in the suspension flow regime near the top of pipe. The skewness factor of axial particle fluctuation velocity indicates that the wavelet components follow the Gaussian probability distribution as the central frequency decreases.     

Qualitative relation between reflected light intensity by Kalliroscope flakes and velocity field in the Couette–Taylor flow system

We have used particle image velocimetry adapted to the Couette–Taylor system in order to measure the flow characteristics (axial and radial velocity components, vorticity fields, kinetic energy, etc.) and their spatio-temporal dependence. By looking for similarity between spatio-temporal diagrams of reflected light intensity and those of velocity fields, we have established that the intensity of light reflected by Kalliroscope flakes is related to the radial velocity component when the outer cylinder is fixed.     

Measurement on particle rotation speed in gas–solid flow based on identification of particle rotation axis

When a two-dimensional (2D) imaging system is used to visualize particle motion in a 3D gas–solid flow, the particle rotation speed was found extremely difficult to be accurately measured due to the fact that the direction of rotation axis was usually random and hard to be distinguished. The paper presents a method to calculate the particle rotation speed from particle images based on the identification of its rotation axis using two or more characteristic points on its surface. The idea was analyzed and realized in a mathematical way and based on which a calculation program was given. The measurement method was verified with an experiment using a small sphere with known rotation axis and rotation speed. The effects of several factors, including the direction of the particle rotation axis, the particle image resolution, the types and positions of characteristic points, etc., on the measurement error are discussed. The error is found to be acceptable for most cases. The measurement method was finally applied to those small glass beads in a real 3D gas–solid flow inside a cold circulating fluidized bed (CFB) riser, which indicates that the problems of 2D imaging system applying to 3D particulate system could be solved by using this mathematical method.     

Particle size influence on effective permittivity of particle–gas mixture with particle clusters

The influence of particle size on the effective permittivity of a particle–gas mixture in the presence of particle clusters was studied using numerical analysis involving the three-dimensional finite element method. The effective permittivity of the mixture was obtained by calculating the electrostatic energy generated in the computation domain. Numerical results show that for fixed volume fraction of particles in the mixture, the effective permittivity of the mixture increases with decreasing particle size. Static experiments were carried out by using a differential capacitance sensor with parallel plates. The variation of the effective permittivity with particle size is shown by experimental data to agree with the numerical results. The methodology described and the results obtained in this paper may be used to help modify the measurement of particles volume fraction in the presence of particle clusters when a capacitance sensor is used.