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Transitions of flow between two corotating disks in an enclosure are investigated numerically. The outer cylindrical boundary of the flow field is fixed, whereas the inner cylinder rotates together with the two disks. The flow is not only symmetric with respect to the inter-disk midplane but also axisymmetric around the axis of rotation at small Reynolds numbers although it becomes unstable to disturbances at large Reynolds numbers. Two kinds of instability modes are known, one of which breaks the axisymmetry of the flow to yield a polygonal flow pattern in the radial–tangential plane. Such instability occurs for small length ratios, the ratio of the length of cylinders (gap between two disks) to the width of the annulus. The other instability is a symmetry-breaking instability with respect to the inter-disk midplane retaining the axisymmetry, which occurs for intermediate length ratios. We investigate the latter symmetry-breaking instability assuming the axisymmetry of the flow field, in which we obtain steady axisymmetric flows numerically and analyze their linear stability. It is found that oscillatory flow as well as steady asymmetric flow appears resulting from the first instability even in the intermediate length ratios where the flow remains axisymmetric. Numerical simulations are also performed to obtain time-periodic flows, and bifurcation diagrams of the flow are depicted for various values of length ratio. The critical Reynolds numbers for the Hopf and symmetry-breaking pitchfork instabilities are evaluated and a transition diagram is obtained.

The effect of transverse magnetic field on the stability of plane Poiseuille and Couette flow between two parallel flat plates is investigated for Oldroyd-B fluid. The classical Orr–Sommerfeld analysis is extended to electrically conducting fluid for Oldroyd-B model, assuming that the magnetic Prandtl number is sufficiently small. The Chebyshev collocation method with the QZ-algorithm is used for solving the stability equations to find the eigenvalues. The critical values of Reynolds number, wave number, and wave speed are computed. It is found that the magnetic field has a stabilizing effect on Poiseuille flow, while it has both destabilizing and stabilizing effects on the Couette flow.

We study the stability of Couette flow in the case in which the outer cylinder is held fixed and the speed of the inner cylinder oscillates harmonically with time around zero mean. We find that, if the amplitude of the oscillation is large enough, it is possible that the resulting time-dependent Taylor vortex flow rotates in a direction which does not depend on the direction of the azimuthal motion which drives it. The transition to this "non-reversing" Taylor vortex flow is examined as a function of frequency of modulation and radius ratio. We also find that "non-reversing" flow is present in the wavy modes regime.

A hierarchical structure (HS) analysis (β-test and γ-test) is applied to a fully developed turbulent pipe flow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10 < y+ < 3000. It is shown that at all locations the velocity fluctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, β and γ, are interpreted in terms of the variation of fluid structures. Intense anisotropic fluid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background fluctuations; this yields a more intermittent velocity signal with smaller γ and β. As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and β and γ are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which γ drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity effects of widely spread turbulent fluctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor flow at high Reynolds numbers.

Recent experiments of Weisberg et al. (1997. J. Fluid Mech. 348, 141{151) have demonstrated that the onset of the centrifugal instability leading to Taylor vortex ow can be delayed by harmonic oscillations of the inner cylinder in the axial direction. Marques and Lopez (1997. J. Fluid Mech. 348, 153{175) used Floquet analysis to describe the observed control of the instability over a wide range of frequencies and amplitudes of this oscillation; the dynamics remained axisymmetric and the response to the applied periodic control mechanism was synchronous over an extensive range of parameters. Here we implement an accurate and ecient spectral-projection scheme for solving the fully nonlinear axisymmetric Navier{Stokes equations to examine the eects of endwalls on the ow dynamics and the breaking of space-time symmetries. By varying a single parameter we have detected a route to chaos. As the parameter is increased the system undergoes a quasiperiodic Naimark{Sacker bifurcation, goes through a 1:9 resonance horn (Arnold’s tongue), and then chaos. c 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.

The stability of Couette flow between two cylinders with the outer one stationary and the inner one rotating and also moving with a constant axial velocity is investigated. Both axisymmetric and non-axisymmetric disturbances are considered. The perturbations equations governing the marginal stability state of the flow are derived and solved numerically for the situation when the gap spacing between the cylinders is small compared to the mean radius. For various values of the Reynolds number R, based on the axial velocity of the inner cylinder, the critical parameters at the onset of instability are determined. It is found that the constant axial movement of the inner cylinder stabilizes the flow. It is also predicted that the most critical mode of the disturbances changes from stationary axisymmetric when R = 0 to oscillatory axisymmetric when R≠0. However, for all calculated values of R, the axisymmetric modes prevail at the onset of instability of the flow and the corresponding non-axisymmetric modes were found to be more stable.

The stability against small disturbances of the plane laminar motion of an electrically conducting fluid between parallel plates in relative motion under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and non-ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number Rc, the critical wavenumber αc and the critical wave speed cc are obtained for wide ranges of the magnetic Prandtl number Pm and the Hartmann number M. It is found that there exists a stationary mode of instability in addition to a travelling-wave mode of instability, and that except for the case when Pm is sufficiently small, the fluid flow becomes more unstable to the stationary mode as Pm increases.

Second-moment closure predictions of fully developed turbulent Poiseuille and Couette flow subjected to spanwise rotation are verified against direct numerical and large eddy simulations in addition to recent experimental results. Near-wall effects are modelled by elliptic relaxation which is used in conjunction with a nonlinear, variable-coefficient pressure strain model consistent with the principle of material frame indifference (MFI) in the limit of two-dimensional turbulence. The dissipation rate model equation is modified in the near-wall region to become explicitly dependent on the imposed system rotation, however, without violating the MFI principle. The model predictions exhibit many of the features due to the Coriolis force on the turbulence and mean flow field, respectively. These include an almost irrotational region in the channel core, augmentation of the turbulence on the unstable side and a corresponding reduction on the stable side, relaminarisation caused not only by stabilising rotation but also by sufficiently high destabilising rotation in Couette flow and, finally, localised regions in the core of the Poiseuille flow where mean flow energy is extracted from the turbulence. The effects of rotational-induced secondary motions on the flow field are also addressed.

The structures and mechanisms for maintaining a high Reynolds shear stress throughout the core region of plane turbulent Couette flow were examined by means of databases originating from a direct numerical simulation. At the relatively low Reynolds number considered, the mean shear rate and some one-point statistics exhibited a non-negligible variation in the core region, which conflicted with the postulated homogeneity. The slope of the mean velocity profile was below a theoretically established lower bound for the limit of infinite Re. Analysis of the time-averaged and structural information obtained by conditional sampling showed that mean shear generation and velocity-pressure gradient correlations played a crucial role in the generation and annihilation of the Reynolds shear stress. It was moreover observed that strong, very localized velocity fluctuations in the wall-normal direction were essential in both processes. A simple conceptual model was proposed to explain the physical significance of the pressure field associated with quadrant 2 events.

Weakly nonlinear calculations are made to describe interactions of a pair of oblique waves inclined at equal but opposite angles with axisymmetric Taylor vortices in the flow between rotating cylinders with a narrow gap. The so-called wavy Taylor vortices observed in a simple case of the outer cylinder at rest and the inner cylinder rotating at a speed slightly above the critical value of the Taylor instability are shown to result from a cooperating interaction between the two oblique waves of the same amplitude and the axisymmetric vortices. In the case of counter-rotating cylinders, on the other hand, spiral vortices appear as the result of an exclusive interaction, in which one of the oblique waves wins their competition in growth and forces the other eventually to disappear, and are found to be stable even to axisymmetric vortices with positive growth rate of linear theory, because their interaction also belongs to the exclusive type. Sometimes both spiral and Taylor vortices form two independent equilibrium solutions of the exclusive interaction in a certain parameter range, and in that case, either of them can be realized in line with the initial amplitudes of the two vortices.

The effect of axial flow on bubbly drag reduction has been experimentally investigated in a vertical Couette–Taylor flow system. The water flow is combined from circumferential and axial flow. Flow condition is fully turbulence and Taylor vortices have appeared in the annulus gap. The shear stress modification in the simultaneous presence of air bubbles and axial flow in the system has been studied by measuring torque acting on the inner cylinder. The results show that axial flow improves the effect of bubbles on drag reduction by damping Taylor vortices and increasing upward velocity of bubbles. In this case, drag reduction of more than 25% has been achieved, which corresponds to lower tested and this amount is gradually decreased with increasing in each and . Increasing causes drag reduction enhancement which could be due to the effect of bubbles on flow density reduction, flow fluctuations and Taylor vortices. Moreover, it is observed that skin friction is affected by axial flow solely and by increasing its volume rates, drag reduction reaches 11%. It is concluded that when bubbles and axial flow are simultaneously applied into the Couette–Taylor flow, the amount of achieved drag reduction is more than when they are separately applied.

A numerical study of the flow between two coaxial cylinders, where one of the cylinders has a step change in radius, is carried out. The inner cylinder rotates and the outer cylinder is stationary. Computation is restricted to axisymmetric motion since instability in flow between coaxial cylinders is found to first occur in the form of axisymmetric Taylor vortices. In the presence of a step, Taylor vortices are found to appear first in the region where the gap between the cylinders is larger and approximately when the local Taylor number in this region reaches the critical Taylor number for onset of instability. Subsequently, Taylor vortices appear in the region where the gap is narrower, and when the local Taylor number in that region exceeds the critical Taylor number. The Taylor vortices have inward flow at a stationary end plate, and outward flow at an end plate which rotates with the same angular velocity as the inner cylinder. Similar results were obtained by Sprague et al (2008 Phys. Fluids 20 014102) for a step on inner cylinder configuration. The step functions as another end plate, if the step size is large. Whereas, it has no effect, if the step size is small. In most situations, these determine whether the number of Taylor vortices in the wide and narrow gap regions is even or odd. When the end plates rotate synchronously, but at a different speed from the inner cylinder, a change from even to odd or odd to even number of vortices in each region occurs at certain rotation rates of the end plates by sudden appearance or disappearance of a vortex at the end of the column. For a certain range of rotation rates of the end plates, the total number of vortices in the entire fluid column is odd, although the end conditions are symmetrical.

The flow around two circular cylinders in tandem arrangement with different diameters has been investigated numerically. The upstream to downstream cylinder diameter ratio was kept constant at a value of d/D=2/3 and the centre-to-centre distance was varied from 1.2D to 5D. The Reynolds number based on upstream cylinder diameter and free-stream velocity used in this study is equal to 200. Force coefficients and flow frequencies have been obtained from 2D fluid flow finite-element simulations. Single-wake shedding, reattachment and co-shedding regimes have been observed. The results show that, depending on initial perturbations, reattachment and co-shedding can be observed on a wide range of centre-to-centre distances (L/D[1.8, 3.8]), which defines the bi-stable flow range. The lower part of the bi-stable range is characterized by a lock-in phenomenon for which vortex-shedding from the upstream and downstream cylinders are synchronized when there are co-shedding flow conditions.

The flow between two concentric cylinders with the inner cylinder rotating with an imposed radial temperature gradient was studied using the digital particle image velocimetry method. Four models of a stationary outer cylinder without and with different numbers of slits (6, 9 and 18) were used. The Grashof number in this study was 1000. The results showed clearly that the buoyant force due to the temperature gradient caused the helical flow; this flow occurred when the Richardson number was larger than 0.0045. The constant temperature gradient considered in this research has little effect on the transition to a turbulent Taylor vortex flow in all models.

The effect of the presence of slits in the wall of the outer one of two concentric cylinders involved in creating a Taylor–Couette flow is experimentally investigated. Three models with different numbers of slits (6, 9 and 18) are used and the radius ratio and aspect ratio of the models are 0.825 and 48, respectively. The velocity fields are measured by using the digital particle image velocimetry method, and the refractive index matching technique is applied to remove image distortion. The flow regime in the Taylor–Couette flow is studied by increasing the Reynolds number. The results show that the presence of slits does not affect the laminar transition process and that the transition to turbulent Taylor vortex is accelerated as the number of slits increases.

Propagation of turbulence in the plane Couette flow is analysed based on the Nee-Kovasznay model. The flow state is assumed uniform in the main stream direction, and a turbulent region is assumed to propagate spanwise into the laminar region with a constant speed. The speed is obtained to be 0.08U0, where the velocity difference between two boundaries is 2U0. Possible applications of this result are discussed.

Results of experimental studies are presented on the evolution of the circular Couette flow with the outer cylinder at rest. In particular, several modes of modulated wavy vortical flows have been identified in a consistent way both from flow visualization and from signals of scattered laser beam. The latter technique has been extended to measure the spatial correlation between fluctuations as a function of the azimuthal separation of the points. It has been found that the two-point angular correlation reflects the pattern of modulation, and falls off as (Rd - R)σ when the `disorder Reynolds number' Rd is approached. The value of the exponent σ has been estimated to be about 1/3 in the experiment.