Turbulence in plane Couette flow

Experimental results for fully developed turbulent plane Couette flow are reported and compared to earlier experimental and numerical results. In addition some turbulent statistics not previously reported are shown.     

Modeling plane turbulent Couette flow

Among the salient features of shear-driven plane Couette flow is the constancy of the total shear stress (viscous and turbulent) across the flow. This constancy gives rise to a quasi-homogenous core region, which makes the bulk of the flow substantially different from pressure-driven Poiseuille flow. The present second-moment closure study addresses the conflicting hypotheses relating to turbulent Couette flow. The inclusion of a new wall-proximity function in the wall-reflection part of the pressure-strain model seems mandatory, and the greement with recent experimental and direct numerical simulation (DNS) results is encouraging. Analysis of model computations in the range 750 ≤ Re ≤ 35,000 and comparisons with low-Re DNS data suggest that plane Couette flow exhibits a local-equilibrium core region, in which anisotropic, homogeneous turbulence prevails. However, the associated variation of the mean velocity in the core, as obtained by the model, conflicts with the intuitively appealing assumption of homogeneous mean shear. The constancy of the velocity gradient exhibited by the DNS therefore signals a deficiency in the modeled transport equation for the energy dissipation rate.     

On the nonlinear stability behaviour of distorted plane Couette flow

This paper discusses the nonlinear stability behaviour of distorted plane Couette flowto 2-dimensional disturbances,and compares it with that of distorted plane Poiseuille flow.The results show that plane Couette flow is more unstable than plane Poiseuille flow tofinite-amplitude disturbances.     

3D-PTV measurements in a plane Couette flow

Genuine plane Couette flow is hard to realize experimentally, and no applications of modern spatially resolving measurement techniques have been reported for this flow so far. In order to resolve this shortcoming, we designed and built a new experimental facility and present our first results here. Our setup enables us to access the flow via 3D particle tracking velocimetry and therefore to obtain truly three-dimensional flow fields for the first time experimentally in plane Couette flow. Results are analyzed in terms of basic flow properties, and a clear distinction of flow regimes (laminar for Re < 320, transitional for 320 < Re < 400, and turbulent when Re > 400) could be made. Comparison with DNS data shows good agreement in the turbulent regime and builds trust in our data. Furthermore, vortical coherent structures are studied in detail with the additional help of kalliroscope imaging, and the typical vortex spacing is determined to be roughly one gap width. As a noteworthy result, we find that the onset of the turbulent regime coincides with the range of Reynolds numbers at which a distance of 100 wall units is comparable to the gap width.     

Newtonian plane Couette flow with dynamic wall slip

Meccanica - The occurrence of slip complicates the estimation of the viscosity in rheometric flows. Thus, special analyses and experimental protocols are needed in order to obtain reliable...     

Spanwise propagation of turbulence in the plane Couette flow

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.     

Instability of plane Couette flow of viscoelastic liquids

A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The first part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability.In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.     

Suppression or enhancement of interfacial instability in two-layer plane Couette flow of FENE-P fluids past a deformable solid layer

The linear stability of two-layer plane Couette flow of FENE-P fluids past a deformable solid layer is analyzed in order to examine the effect of solid deformability on the interfacial instability due to elasticity and viscosity stratification at the two-fluid interface. The solid layer is modeled using both linear viscoelastic and neo-Hookean constitutive equations. The limiting case of two-layer flow of upper-convected Maxwell (UCM) fluids is used as a starting point, and results for the FENE-P case are obtained by numerically continuing the UCM results for the interfacial mode to finite values of the chain extensibility parameter. For the case of two-layer plane Couette flow past a rigid solid surface, our results show that the finite extensibility of the polymer chain significantly alters the neutral stability boundaries of the interfacial instability. In particular, the two-layer Couette flow of FENE-P fluids is found to be unstable in a larger range of nondimensional parameters when compared to two-layer flow of UCM fluids. The presence of the deformable solid layer is shown to completely suppress the interfacial instability in most of the parameter regimes where the interfacial mode is unstable, while it could have a completely destabilizing effect in other parameter regimes even when the interfacial mode is stable in rigid channels. When compared with two-layer UCM flow, the two-layer FENE-P case is found in general to require solid layers with relatively lower shear modulii in order to suppress the interfacial instability. The results from the linear elastic solid model are compared with those obtained using the (more rigorous) neo-Hookean model for the solid, and good agreement is found between the two models for neutral stability curves pertaining to the two-fluid interfacial mode. The present study thus provides an important extension of the earlier analysis of two-layer UCM flow [V. Shankar, Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification, J. Non-Newtonian Fluid Mech. 125 (2005) 143–158] to more accurate constitutive models for the fluid and solid layers, and reaffirms the central conclusion of instability suppression in two-layer flows of viscoelastic fluids by soft elastomeric coatings in more realistic settings.     

Velocity measurements of streamwise roll cells in rotating plane Couette flow

For the first time, quantitative velocity measurements in rotating plane Couette flow are demonstrated. Particle image velocimetry is used at a low Reynolds number with anti-cyclonic, i.e. destabilising rotation, where the instability is expected to give rise to steady streamwise-oriented roll cells. The streamwise and spanwise velocities of the roll cells were measured on the centreplane of the flow and at two planes on either side. The streamwise velocity is spanwise periodic with an amplitude variation approximately ±42 % of half the velocity difference between the moving walls. The wall-normal velocity was estimated by assuming steady, laminar and streamwise-independent flow. Despite the large amplitude of the disturbance, both the spanwise wave length and amplitude ratio between the streamwise and wall-normal components were close to what is obtained from linear theory. A splitting event of a roll cell was also captured by the velocity measurements and its development followed in time.     

Ekman vortices and the centrifugal instability in counter-rotating cylindrical Couette flow

The finite length of a Taylor–Couette cell introduces endwall effects that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman boundary layers and the vortical structures in a finite-length cavity with counter-rotating cylinders via direct numerical simulation using a three-dimensional spectral method. To analyze the nature of the interaction between the vortices and the endwall layers we consider four endwall boundary conditions: fixed endwalls, endwalls rotating with the outer cylinder, endwalls rotating with the inner cylinder, and stress-free endwalls. The vortical structure of the flow depends on the endwall conditions. The waviness of the vortices is suppressed only very near the endwall, primarily due to zero axial velocity at the endwall rather than viscous effects. In spite of their waviness and random behavior, the vortices generally stay inside of the v=0 isosurface by adjusting quickly to the radial transport of azimuthal momentum. The thickness and strength of the Ekman layer at the endwall match with that predicted from a simple theoretical approach.     

Linear stability of plane creeping Couette flow for Burgers fluid

It is well known that plane creeping Couette flow of UCM and Oldroy-B fluids are linearly stable. However, for Burges fluid, which includes UCM and Oldroyd-B fluids as special cases, unstable modes are detected in the present work. The wave speed, critical parameters and perturbation mode are studied for neutral waves. Energy analysis shows that the sustaining of perturbation energy in Poiseuille flow and Couette flow is completely different. At low Reynolds number limit, analytical solutions are obtained for simplified perturbation equations. The essential difference between Burgers fluid and Oldroyd-B fluid is revealed to be the fact that neutral mode exists only in the former.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Spots and turbulent domains in a model of transitional plane Couette flow

A review of the globally subcritical transition to turbulence in shear flows is presented, with an emphasis on the cases of plane and circular Couette flows (pCf and cCf, respectively). A Swift–Hohenberg-like model is next proposed to interpret the behavior of plane Couette flow in the vicinity of its global stability threshold. We present results of numerical simulations supporting this proposal and helping us to raise good questions about the growth and decay of intermittent turbulent domains in this precise context, and more generally about the coexistence of laminar flow and turbulence in other spatio-temporally intermittent flows. PACS 47.27.-i, 47.54.-r, 05.45.-a     

Numerical study of turbulent plane Couette flow at low Reynolds number

A direct numerical simulation of fully developed turbulent plane Couette flow has been performed. Unsteady large-scale structures, which contributed to the instantaneous energy level, were observed. These evolving and drifting vortical structures vanished after time-averaging, and the resulting mean velocity and streamwise turbulence intensity compared favourably with recent laboratory data.     

Oscillatory instability of the Couette flow between two unidirectionally rotating cylinders

In the problem of stability of the Couette flow between unidirectionally rotating cylinders, bifurcation points of codimensions 1, 2, and 3 are calculated. These points are interesting because in their small neighborhood the sequences of fluid flow bifurcations, including the development of chaotic regimes, can be studied analytically by completely correct methods. It is found that at large rotation velocities of both cylinders the pattern of the neutral curves corresponding to bifurcation points of codimension 1 changes and various intersections of two and three neutral curves (bifurcation points of codimensions 2 and 3) appear.     

Threshold and decay properties of transient isolated turbulent band in plane Couette flow

By direct numerical simulations of the plane Couette flow(PCF) in a large computational domain, it is shown that an isolated turbulent band decays monotonically at low Reynolds numbers but experiences transient growth before the eventual relaminarization at moderate Reynolds numbers. The lower bound Reynolds number of the transient-growth regime is determined as 286. The width, length, and tilt angle of the isolated band structure are defined based on the disturbance kinetic energy in the mid-plane,and the geometric characteristics of the band can be described with a tilted rectangle. It is illustrated that before its eventual fragmentation, the isolated turbulent band decays in a style of longitudinal contraction, where the center, width, and tilt angle of the band keep almost constant but the band length contracts with a statistically constant velocity.     

Non-modal instabilities of two-dimensional disturbances in plane Couette flow of a power-law fluid

Instabilities of fluid flows have traditionally been investigated by normal mode analysis, i.e. by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In this paper we study the instabilities of two-dimensional Couette flow of a polymeric fluid in the framework of non-modal stability theory rather than normal mode analysis. A power-law model is used to describe the polymeric liquid. We focus on the response to external excitations and initial conditions by examining the pseudospectra structures and the transient energy growths. For both Newtonian and non-Newtonian flows, the results show that there can be a rather large transient growth even though the linear operator of Couette flow has no unstable eigenvalue. The effects of non-Newtonian viscosity on the transient behaviors are examined in this study. The results show that the “shear-thinning/shear-thickening” effect increases/decreases the amplitude of responses to external excitations and initial conditions.