Search

A new type of travelling wave solution is obtained in plane Couette flow. The solution appears via a Hopf bifurcattion from the so-called stationary hairpin vortex state found recently by [1] and [2]. The most striking feature of the solution is an anti-symmetric mean flow profile. The fluctuating flow patterns propagate in the streamwise direction with the phase velocity being zero at its onset and increasing gradually as the Reynolds number is increased.

Direct numerical simulations (DNS) of stably-stratified, turbulent plane Couette flow are currently being performed on the XC-4000. The friction Reynolds number is kept approximately constant, Re τ ≃540 and the Richardson number Ri w varies between 0 and 0.1. The flow is divided into two regions: the region close to the wall and the core region. The region close to the wall presents strong velocity and density gradients and its structure is similar to unstratified wall-turbulence. In the core region, the gradients of mean velocity and density are approximately constant, and the structure of the flow is similar to homogeneous stratified turbulence with shear. With increasing stratification the formation of layers is clearly observed. The layers are inclined with respect to the wall plane and their thickness decreases with increasing stratification.

Turbulent Couette-Poiseuille and Couette flows inside a square duct at bulk Reynolds number 10,000 are investigated using the Large Eddy Simulations. The mean secondary flow is observed to be modified by the presence of moving wall where the symmetric vortex pattern vanishes. Secondary flow near the top corner shows a gradual change of vortex size and position as the moving wall velocity increased.

The mixing in Taylor-Couette flow was studied with 2D laser-optical methods (Particle Image Velocimetry PIV and Laser Induced Fluorescence LIF). Mixing processes are grouped into three stages (macro-, meso- and micromixing) according to the characteristic length scale of the inhomogeneities under consideration. Results of this work are correlations which describe the macro-, mesoand micromixing times as a function of the geometry and process parameters with the mixing iso- and non-isoviscous fluids. The calculation of the micromixing times is shown as a function of the angular velocity of inner cylinder, the fluid viscosity and the geometry. With the fluid viscosity of 1·10− 6 m2/s the mixing times are investigated as a function of the radial Reynolds number for three length scales. The macromixing times are calculated with help of two correlations for the axial dispersion coefficient in the turbulent and wave regime. The boundary between the regimes is applied to Reφ = 20 Reφ,crit at Reax = 1. The macromixing times are the longest from the mixing times. Mesomixing times are shorter and the micromixing runs fastest here. The Mesomixing limits the velocity therefore the micromixing and the macromixinge with which occurs the mesomixing.

We consider an incompressible viscous fluid with the kinematic viscosity v between two infinitely long concentric cylinders with radii a and b (b > a). The fluid experiences a shear motion produced by pulling the inner cylinder with the axial speed U while keeping the outer cylinder at rest (see Fig.1). The axial basis flow at the radius r, $$U_{B}(r) = R\frac{\ln(r(1 - \eta)/2)}{\ln(\eta)},$$ can be obtained as an exact solution of the Navier-Stokes equation, where \(R = U(b - a)/2v\) is the Reynolds number and \(\eta = a/b\) is the radius ratio.

The subject of our paper is the post-transition flow pattern formed in the plane channel flow on a rotating system when transition is induced by the Coriolis force acting when the rotation vector is aligned with the spanwise direction of the channel. It is known from previous work, see eg. [5] [6], that, when transition sets in in this flow, it does so at rather low Reynolds numbers of only around 50 and the flow in the post-transition stage exhibits stationary longitudinal vortices, an observation standing in sharp contrast to the case of no rotation. The object of our present work is to try to extract this salient flow feature in quantitative terms through studying the bifurcation characteristics of this flow on the verge of transition. The focus of our attention is on the spatial periodicity of the longitudinal vortices, for which the relevant properties of the bifurcating solution of the nonlinear equations of motion for disturbances to the basic flow are examined. The velocity profile of the basic flow here is simply the parabola itself. The parameters influencing transition in this flow are, in conventional notation, the Reynolds number \(Re = \frac{U_{ref} H}{v}\) and the rotation number \(Ro = \frac{\Omega H}{U_{ref}}\), where 2H is the channel-height.

Plane Couette flow has the peculiar property that transition to turbulence occurs despite the fact that its laminar profile is linearly stable. For high enough Reynolds numbers small amplitude perturbations are enough to initiate the growth of turbulent spots. Such spots have been subject to investigation for a few decades, see e.g. [1, 2]. In addition, it has been observed that around transitional Reynolds numbers laminar and turbulent regions can co-exist if this Reynolds number is approached from the turbulent side, e.g., [3, 4]. For both cases, distinct interfaces divide the regimes between turbulent and non turbulent flow. It is our main interest to study the evolution of such interfaces similar to the work of [5] where the shear free case with inhomogeneous forcing by an oscillating grid has been investigated in detail.

A numerical continuation method is carried out in a homotopy space connecting two different flows, the Plane Couette Flow (PCF) and the Laterally Heated Flow in a vertical slot (LHF). This numerical continuation method enables us to obtain an exact steady solution in PCF. The new solution has the shape of hairpin vortices (HVS: hairpin vortex solution), which is observed ubiquitously in turbulent shear flows.

The origin and the transition to turbulence in wall-bounded shear flows is one of the outstanding problems of classical physics. In order to shed light on this transition, recent investigations have analyzed the decay of localized turbulent structures. At low Reynolds numbers, an exponential distribution of survival times has been observed in pipe and plane Couette flows [1, 2, 3, 4]. In phase space this is related to the decay from a chaotic saddle. However, pipe flow is an open flow, posing many experimental and numerical challenges for the study of bifurcation events. In a closed system we have found a flow state that shows transient turbulent behavior also at low Reynolds numbers. It appears as turbulent bursting in the Taylor-Couette system without an external forcing for counter-rotating cylinders above the centrifugal instability. Therefore it competes with coherent states of the system. Here we present spatiotemporal properties and lifetime behavior of this flow.

Over a century, and thousands of articles, after Reynolds' description of the transition to turbulence in pipe flow, a predictive theory of transition is still unavailable. One of the most intriguing phenomena observed near transition ishe coexistence of well-defined and long-lived laminar and turbulent regions, first observed in counter-rotating Taylor-Couette flow in the 1960s [1]. In the 2000s, Prigent & Dauchot [2] showed that these coexisting regions were part of a regular pattern of stripes, whose wavelength and orientation are Reynolds-number-dependent and reproducible. Analogous phenomena have been observed experimentally [2] and numerically [3] in plane Couette flow, in stator-rotor experiments (the flow between a stationary and a rotating disk) [4], in plane Poiseuille simulations [5], and, most recently, in simulations of pipe flow [6].

Subcritical transition to turbulence can occur in a variety of wall-bounded shear flows when the laminar base state is not subject to linear instability. In this case, perturbations to the base state with a finite but very small amplitude can be amplified by non-normal effects, up to a level where nonlinear interactions come into play. Since transition is often undesired, leading to a dramatic increase in wall drag, it is important to know which kind of (weak) perturbations are susceptible to trigger it if we wish to delay it. A vast amount of litterature has described the linear mechanisms responsible for disturbance amplification. Optimisation methods have yielded linear optimal disturbances[1, 2], the disturbances which exhibit the largest linear growth. Yet the focus has been on linear amplification rather than on actual transition, which requires full nonlinearity to be taken into account. In this study, we go back to the original question of which perturbation is most likely to trigger transition to turbulence. We are hence investigating nonlinear optimal disturbances, those with smallest initial energy which effectively lead to a highly disordered flow. In the recent years, progress in the understanding of subcritical transition was made using the concept of ’edge states’, originating from dynamical systems theory. ’Edge state’ refers to the flow regime reached asymptotically by critical trajectories at the exact onset of transition. It is an unstable flow state and cannot be observed in experiments, but its stable manifold determines the basin of attraction of the base state. Direct numerical simulation in minimal domains of plane Couette flow has shown that the flow at the onset of transition asymptotically approaches an unstable finite-amplitude steady state solution, characterised by wavy streaks and streamwise rolls [3]. Using a shooting method, approach to such steady states is the way to characterise whether a transitional trajectory in phase space is actually an ’edge’ trajectory.

The onset of turbulence in shear flows cannot be explained by classical hydrodynamic stability theory. The linear instability threshold is usually far above the practically observed one. An example in liquid-metal MHD is plane channel flow with wall-normal magnetic field B0 (the Hartmann flow). Indeed, the critical Reynolds number for linear instability is about two orders of magnitude larger than the experimentally observed values [1]. Numerically, two problems were studied: the periodic Hartmann channel flow [2] and the complementary case of plane channel flow under spanwise magnetic field [3]. In the first case the basic Poiseuille flow is modified by the magnetic field (Hartmann profile) while it remains unaffected for the second one. Both studies were performed in the framework of a scenario based on transient growth and subsequent breakdown of optimal perturbations. In case of Hartmann flow, the optimal modes are purely streamwise rolls, while in the second case the optimal modes become rolls oriented at an oblique angle to the flow. In the present work we extend the previous studies to the case of a rectangular duct with insulating walls. The field B 0 is parallel to the sidewalls located at y = ±a, so that the Hartmann layers are present at z = ±d. On the side walls, the so-called Shercliff layers develop. In this geometry, transition mechanisms found in channel flow with either spanwise or wall-normal magnetic field can then be both relevant. The relative importance of these mechanisms is studied here by changing the aspect ratio r = a/d of the duct (see Fig. 1): when r « 1 the case of channel flow under spanwise magnetic field is approached, on the contrary, when r » 1 the geometry would get close to the Hartmann channel flow case.

We investigate the problem of tracer mixing in presence of boundaries by means of direct numerical simulation of a Couette flow at small Reynolds number and in kinematic simulations of a chaotic flow. Scalar profiles close to the boundary are computed. Their shape and time evolution are found to agree with published theoretical predictions.

Plane Couette flow is the flow between two counter-sliding plates of velocity U separated by a gap 2h. It has a simple laminar solution with constant shear which is stable for all values of the Reynolds number \(Re = \frac{Uh}{v}\) (v is the kinematic viscosity of the fluid). However, transition to turbulence is observed experimentally above a threshold value of Re between 300 and 400. Whereas featureless turbulence is seen for Re ≥ 400, lower-Re experiments have shown a turbulent regime displaying spatio-temporal intermittency effects, notably the appearance of turbulent stripes, inclined with respect to the direction of the base flow, interspersed with quiescent, nearly laminar regions [1], see Figure 1. It is the ‘fspiral turbulence’ regime, so-called because it has also been identified in Taylor-Couette experiments. Similar patterns have also been numerically reproduced in a minimal-like computational domain (of size 10h in length and 40h in width), oriented against the inclination of one stripe [2]. This reduced computational box allows to understand locally the appearance and teh stability of these localised structures.

The optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing is computed at Re = 750. The equations for the coherent perturbations are linearised near the turbulent mean flow and the effect of small scales is modelled with the standard eddy viscosity. The mean flow is linearly stable but it is shown to amplify coherent streamwise streaks from streamwise vortices. The largest amplifications are realized by streamwise uniform structures associated to spanwise wavelengths of 4.3h for largest amplifications of the energy of initial conditions, 5.1h for the maximum amplification of the variance of stochastic forcing. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realisations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.4h)..

Reverse transition of ‘turbulence → laminar’ in wall-bounded shear flows remains poorly understood and few studies have been reported. A plane Couette flow (CF) is conceptually one of the simplest non-trivial fluid dynamics systems, where the flow is solely driven by the shear. This flow is linearly stable for all Reynolds numbers, but experiences direct transition to turbulence through the development of localized perturbations (cf. [1] for a discussion). Above some threshold, turbulence is sustained with a complex behavior characterized by laminar-turbulent co-existence in the form of eturbulent stripe. The transition, though well-described experimentally [2], is far from being completely elucidated. On the other hand, a direct numerical simulation (DNS) of intermittently turbulent flows is challenging and requires a huge computational domain so that another strategy has been necessary. For instance, existing numerical studies are limited within frameworks of a semi-realistic model [3] or of a tilted geometry with a minimal domain [4]. However, owing to the recent development of computers, DNS of the subcritical CF is now possible to be performed.

Transition to turbulence can often be understood through analysis of instabilities and bifurcations. Plane Couette-flow and Hagen-Poiseuille flow, however, are well known to be linearly stable for all Reynolds numbers. Rather recently finite-amplitude states not bifurcating from the laminar base flow have been found [1, 2] and are now being discussed as possibly forming decisive structures in the high-dimensional phase space in which the transition to turbulence actually takes place [3, 4, 5]. It is noteworthy that most of the recently investigated finite-amplitude states in plane Couette and Hagen-Poiseuille flow are periodic and infinitely extended in the spatial directions in which the underlying system is homogeneous. One of the remaining questions seems to be, however, how the spatial localization of turbulence structures like puffs and patches can be explained in terms of a dynamical systems perspective. The existence of localized finite-amplitude states in the style of strongly nonlinear dissipative solitons is an attractive hypothesis in that context: The complex spatially localized turbulence could then be affected in one way or another by the phase-space properties and the phase space neighborhood of such localized solutions. The existence of these localized, solitary solutions in plane Couette flow is still being debated [6, 7, 8, 9]. In other pattern forming systems solitary solutions have, however, been found: in magnetohydrodynamics so-called ‘convectons’ have recently been detected [10, 11]; in Ekman type flows localized vortex solutions have been reported [12]; and also in granular materials spatially strongly localized patterns have been observed [13].

Subcritical instabilities in small gap Taylor-Couette (TCF) problem are studied numerically when both cylinders rotate in opposite directions. The computations are carried out for a radius ratio \(\eta = r_{{\rm i}}/r_{{\rm o}} = 0.883.\) A first exploration is focused on the study of spiral flows originated from subcritical Hopf bifurcations of the basic circular Couette solution. The second exploration addresses the transition from laminar flow to the usually termed as spiral turbulence regime characterized by alternating laminar and turbulent spiral bands which coexist even in regions of the parameter space where the circular Couette flow is linearly stable.

A special set of boundary conditions has been devised with the view to break the inflow-outflow coupling associated with periodic boundary conditions. This necessitated the replacement of the FFT routines with a multigrid solver. The inclination for roll cells to develop was reduced, but not eliminated, and firm conclusions concerning their questionable reality could not be drawn.

Solutions currently considered to provide the bench-mark for 2-D lid-driven cavity flow are shown here to be are not grid-independent. Using clustered grids and a time-accurate multi-grid code, grid-independent solutions are reported here at Reynolds number Re=100 and 400 (based on lid velocity and cavity size). Preliminary solutions are presented also at Re=3200 and 10000. It is found that fine resolution near the wall is necessary to obtain the solution accurately even in the core of the cavity. The solutions presented here call into question both the theories proposing an asymptotically constant core vorticity in the limit of infinite Reynolds numbers, and current interpretations of experimental data.

We have developed, tested, and implemented a numerical code for calculating the viscous, three dimensional flow between two differentially rotating cylinders (cylindrical Couette flow). In this paper we describe the methods that we have used to apply efficiently pseudo-spectral techniques to a complicated (i.e. non-Cartesian) geometry with real (i.e. viscous, no-slip) boundaries and the tests that we have used to determine the accuracy of our code.

The turbulent Couette flow is considered from the point of view of the asymptotic theory for turbulent shear flows at large Reynolds numbers. According to this theory the flow has a two-layer structure which can be treated by the method of matched asymptotic expansions. Without specifying a turbulence model the analysis leads to boundary conditions which have to be satisfied by the solution and hence by any turlence modelling. It is shown that some of the existing turbulence models do not satisfy these conditions. A new turbulence model is proposed which leads to analytical solutions.

The near flow fields of periodically perturbed circular air jets have been experimentally investigated for the Strouhal number range 0 ≤ StD ≤ 4.0 and the Reynolds number range 11,000–113,000. The largest total fluctuation intensity growth on the centerline occurs at StD ≌ 0.85 while the largest fundamental amplitude growth occurs at StD ≌ 0.30. At StD ≌ 0.85, most of the fluctuations is at the subharmonic frequency associated with vortex pairing in the ‘jet mode’, which is independent of the ‘shear layer mode’ of vortex pairing occurring at Ste ≌ 0.011. Smoke visualization at lower speeds provided further details of pairing in either mode. In the jet mode of vortex pairing, the Reynolds stress undergoes large excursions even though its time average values are not altered appreciably by the excitation.

We have conducted a numerical studv of spherical Couette Vlow -the flow between differentiallv rotatin~ concentric spheres. When the qa D ratio o{(R2-RI)/RI (where R 1 and R 2 are the radii of the inner and outer spheres) is small, the flow near the e~uator resembles that between cylinders, the classic Tavlor-Couette ~roblem. As in Tavlor-Couette flow, when the angular momentum gradient, measured by the Reynolds number Re ~RI2QI/~ (where Q1 is the anaular velocitv of the inner sphere and v the kinetic viscosit}7) exceeds a critical value, Taylor vortices form to redistribute anaular momentum between radial shells.

The instability and transitions in flow regimes of circular Couette flow with axial density stratification, characterized by the buoyancy frequency N, has been investigated experimentally for different values of the non-dimensional gap width e between the cylinders. It is shown that onset of instability and the structure of the resulting flow regimes, namely non-axisymmetric vortices, Taylor vortices and the related descrete layer formation strongly depend on N in addition to Reynolds number and e. The linear stability analysis of this problem has been performed for the asymptotic case of e → 0.

The importance of end effects in the cellular selection process in Taylor-Couette flow has been demonstrated. A universal law governing the primary flow exchange process has been demonstrated to hold for all of the flows studied. In addition, new types of flow viz. the anomalous modes, have been investigated and demonstrated to exhibit regular behaviour. Finally, end effects have beer shown to be of the utmost importance in the cell number selection process.

The non-newtonian properties of a polymer solution are a consequence of the deformability of the polymer molecules. The constitutive equation which models the viscoelastic behaviour of this non-newtonian liquid, was derived on thermodynamic grounds by LHUILLIER and OUIBRAHIM [1]. The resulting equations were recently solved by ROESNER [2] for the COUETTE-flow problem and the steady state velocity field was fully obtained analytically. Here it is used as a basic solution for a linear stability analysis. Preliminary results concerning the influence of polymer molecules on the onset of TAYLOR-GORTLER vortices are discussed.

We study liquid-liquid dispersions in a turbulent Taylor - Couette flow, produced between two counterrotating coaxial cylinders. In pure Water and in counterrotation, Reynolds numbers up to 1.4 10^5 are reached. We first characterize the single-phase flow, in terms of threshold for transition to turbulence, scaling of the torque and measurements of the mean flow and of the Reynolds stress by stereoscopic PIV. We then study the increase of the dissipation in the two-phase flows and find that the torque per unit mass can be twice the torque for a single-phase flow. Long-time behaviours are also reported.

We review the application of the two-fluid HVBK equations to helium II in Couette geometry, that is flow between concentric, rotating cylinders. This application is particularly interesting as a large number of experiments have been carried out in this geometry and also because Couette flow is an exact solution of the HVBK equations for both the normal fluid and superfluid.

In order to study the Reynolds number effects on the behaviors of asymmetric vortex flow over slender body the experimental investigations were carried out in D-4 Low Speed Wind Tunnel of BUAA with pointed tangent ogive/cylinder model with the angle of attack 50° at Reynolds numbers 0.1×l06~1.0×l06. A new transitional separation pattern in the inception regime was found at the Reynolds number 0.2×l06, which means the laminar boundary layer separation with the turbulent shear layer reattachment on one side and the laminar boundary layer separation without reattachment is still kept on the other side. And the pressure difference ΔCp = Cpsep - Cpmin can be selected as criterion to determine the lower-critical and upper-critical Reynolds numbers for inception regime quantitatively. In the post-critical regime it is found from the present study that the asymmetric vortices behaviors are no longer changed with Reynolds numbers. Moreover, the asymmetric vortices flow pattern at bistable state will switch over two cycle. And a close correlation between circumferential position of manual micro-perturbation block and the response of asymmetric vortices flow pattern is revealed over the post-critical Reynolds number region.