Nonlinear vorticity-banding instability in granular plane Couette flow: higher-order Landau coefficients, bistability and the bifurcation scenario

2013 ◽  
Vol 718 ◽  
pp. 131-180 ◽  
Author(s):  
Priyanka Shukla ◽  
Meheboob Alam
Keyword(s):  
Couette Flow ◽  
Volume Fraction ◽  
Critical Density ◽  
Higher Order ◽  
Jump Discontinuity ◽  
Linear Growth Rate ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Dense Flows ◽  
The Mean

AbstractThe rapid granular plane Couette flow is known to be unstable to pure spanwise perturbations (i.e. perturbations having variations only along the mean vorticity direction) below some critical density (volume fraction of particles), resulting in the banding of particles along the mean vorticity direction: this is dubbed ‘vorticity banding’ instability. The nonlinear state of this instability is analysed using quintic-order Landau equation that has been derived from the pertinent hydrodynamic equations of rapid granular fluid. We have found analytical solutions for related modal/harmonic equations of finite-size perturbations up to quintic order in perturbation amplitude, leading to an exact calculation of both first and second Landau coefficients. This helped to identify the bistable nature of nonlinear vorticity-banding instability for a range of densities spanning from moderately dense to dense flows. For perturbations with small spanwise wavenumbers, the bifurcation scenario for vorticity banding unfolds, with increasing density from the dilute limit, as supercritical pitchfork $\rightarrow $ subcritical pitchfork $\rightarrow $ subcritical Hopf bifurcations. The transition from supercritical to subcritical pitchfork bifurcations is found to occur via the appearance of a degenerate/bicritical point (at which both the linear growth rate and the first Landau coefficient are simultaneously zero) that divides the critical line into two parts: one representing the first-order and the other the second-order phase transitions. Both subcritical oscillatory and stationary solutions have also been uncovered for dilute and dense flows, respectively, when the spanwise wavenumber is large. In all cases, the nonlinear solutions correspond to inhomogeneous states of shear stress and pressure along the vorticity direction, and hence are analogues of vorticity banding in other complex fluids. The quartic-order mean-flow resonance is evidenced in the parameter space for which the second Landau coefficient undergoes a jump discontinuity of infinite order. The importance of retaining higher-order terms to calculate the second Landau coefficient and their possible effects on the nature of bifurcations are elucidated.

Turbulent plane Couette flow subject to strong system rotation

1997 ◽  
Vol 347 ◽  
pp. 289-314 ◽  
Author(s):  
KNUT H. BECH ◽  
HELGE I. ANDERSSON
Keyword(s):  
Flow Field ◽  
Couette Flow ◽  
Turbulence Level ◽  
Streamwise Vortices ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Cell Pattern ◽  
System Rotation ◽  
The Mean ◽  
Turbulent Plane

System rotation is known to substantially affect the mean flow pattern as well as the turbulence structure in rotating channel flows. In a numerical study of plane Couette flow rotating slowly about an axis aligned with the mean vorticity, Bech & Andersson (1996a) found that the turbulence level was damped in the presence of anticyclonic system rotation, in spite of the occurrence of longitudinal counter-rotating roll cells. Moreover, the turbulence anisotropy was practically unaffected by the weak rotation, for which the rotation number Ro, defined as the ratio of twice the imposed angular vorticity Ω to the shear rate of the corresponding laminar flow, was ±0.01. The aim of the present paper is to explore the effects of stronger anticyclonic system rotation on directly simulated turbulent plane Couette flow. Turbulence statistics like energy, enstrophy and Taylor lengthscales, both componental and directional, were computed from the statistically steady flow fields and supplemented by structural information obtained by conditional sampling.The designation of the imposed system rotation as ‘high’ was associated with a reversal of the conventional Reynolds stress anisotropy so that the velocity fluctuations perpendicular to the wall exceeded those in the streamwise direction. It was observed that the anisotropy reversal was accompanied by an appreciable region of the mean velocity profile with slope ∼2Ω, i.e. the absolute mean vorticity tended to zero. It is particularly noteworthy that these characteristic features were shared by two fundamentally different flow regimes. First, the two-dimensional roll cell pattern already observed at Ro=0.01 became more regular and energetic at Ro=0.10 and 0.20, whereas the turbulence level was reduced by about 50%. Then, when Ro was further increased to 0.50, a disordering of the predominant roll cell pattern set in during a transient period until the flow field settled at a new statistically steady state substantially less affected by the roll cells. This was accompanied by a substantial amplification of the streamwise turbulent vorticity and an anomalous variation of the mean turbulent kinetic energy which peaked in the middle of the channel rather than near the walls. While the predominant flow structures of the non-rotating flow were longitudinal streaks, system rotation generated streamwise vortices, either ordered secondary flow or quasi-streamwise vortices. Eventually, at Ro=1.0, the turbulent fluctuations were completely suppressed and the flow field relaminarized.

scholarly journals Mean flow of turbulent–laminar patterns in plane Couette flow

2007 ◽  
Vol 576 ◽  
pp. 109-137 ◽  
Author(s):  
DWIGHT BARKLEY ◽  
LAURETTE S. TUCKERMAN
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Full Range ◽  
Force Balance ◽  
Computational Domain ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
The Mean ◽  
The Relationship ◽  
And Diffusion

A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared.

Optimal energy growth in stably stratified turbulent Couette flow

2021 ◽  
Author(s):  
Grigory Zasko ◽  
Andrey Glazunov ◽  
Evgeny Mortikov ◽  
Yuri Nechepurenko ◽  
Pavel Perezhogin
Keyword(s):  
Boundary Layer ◽  
Turbulent Flow ◽  
Couette Flow ◽  
Large Scale ◽  
Thin Layers ◽  
Plane Couette Flow ◽  
Turbulent Layer ◽  
Wide Range ◽  
The Mean ◽  
Optimal Disturbances

<p>In this report, we will try to explain the emergence of large-scale organized structures in stably stratified turbulent flows using optimal disturbances of the mean turbulent flow. These structures have been recently obtained in numerical simulations of turbulent stably stratified flows [1] (Ekman layer, LES) and [2] (plane Couette flow, DNS and LES) and indirectly confirmed by field measurements in the stable boundary layer of the atmosphere [1, 2]. In instantaneous temperature fields they manifest themselves as irregular inclined thin layers with large gradients (fronts), spaced from each other by distances comparable to the height of the entire turbulent layer, and separated by regions with weak stratification.</p><p>Optimal disturbances of a stably stratified turbulent plane Couette flow are investigated in a wide range of Reynolds and Richardson numbers. These disturbances were computed based on a simplified linearized system of equations in which turbulent Reynolds stresses and heat fluxes were approximated by isotropic viscosity and diffusion with coefficients obtained from DNS results. It was shown [3] that the spatial scales and configurations of the inclined structures extracted from DNS data coincide with the ones obtained from optimal disturbances of the mean turbulent flow.</p><p>Critical value of the stability parameter is found starting from which the optimal disturbances resemble inclined structures. The physical mechanisms that determine the evolution, energetics and spatial configuration of these optimal disturbances are discussed. The effects due to the presence of stable stratification are highlighted.</p><p>Numerical experiments with optimal disturbances were supported by the RSF (grant No. 17-71-20149). Direct numerical simulation of stratified turbulent Couette flow was supported by the RFBR (grant No. 20-05-00776).</p><p>References:</p><p>[1] P.P. Sullivan, J.C. Weil, E.G. Patton, H.J. Jonker, D.V. Mironov. Turbulent winds and temperature fronts in large-eddy simulations of the stable atmospheric boundary layer // J. Atmos. Sci., 2016, V. 73, P. 1815-1840.</p><p>[2] A.V. Glazunov, E.V. Mortikov, K.V. Barskov, E.V. Kadantsev, S.S. Zilitinkevich. Layered structure of stably stratified turbulent shear flows // Izv. Atmos. Ocean. Phys., 2019, V. 55, P. 312–323.</p><p>[3] G.V. Zasko, A.V. Glazunov, E.V. Mortikov, Yu.M. Nechepurenko. Large-scale structures in stratified turbulent Couette flow and optimal disturbances // Russ. J. Num. Anal. Math. Model., 2010, V. 35, P. 35–53.</p>

Effect of Number of Roughing Passes on the Dynamic Transformation of Austenite during Simulated Plate Rolling

2018 ◽  
Vol 941 ◽  
pp. 717-722
Author(s):  
Samuel F. Rodrigues ◽  
Fulvio Siciliano ◽  
Clodualdo Aranas Jr. ◽  
Gedeon S. Reis ◽  
Brian J. Allen ◽  
...  
Keyword(s):  
Grain Size ◽  
Volume Fraction ◽  
Austenite Phase ◽  
Mean Flow ◽  
Grain Sizes ◽  
Dynamic Transformation ◽  
Plate Rolling ◽  
Forward Transformation ◽  
The Mean ◽  
Rough Rolling

When austenite is deformed within the austenite phase field, it partially transforms dynamically into ferrite. Here, plate rolling simulations were carried out on an X70 steel using rough rolling passes of 0.4 strain each. The influence of the number of roughing passes on the grain size and volume fraction of induced ferrite was determined. Up to three roughing passes applied at 1100 °C followed by 5 finishing passes at 900 °C were employed. The sample microstructures were analysed by means of metallographic techniques. Both the critical strain to the onset of dynamic transformation as well as the grain size decreased with pass number during the roughing simulations. For the finishing passes, the mean flow stresses (MFS`s) applicable to each schedule decreased when a higher number of roughing passes was applied. The volume fraction of dynamically formed ferrite retained after simulated rolling increased with the roughing pass number. This is ascribed to the increased amount of ferrite retransformed into austenite and the finer grain sizes produced during roughing. The forward transformation is considered to occur displacively while the retransformation into austenite during holding takes place by a diffusional mechanism. This indicates that both dynamic transformation (DT) and dynamic recrystallization were taking place during straining.

scholarly journals Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number

2010 ◽  
Vol 643 ◽  
pp. 333-348 ◽  
Author(s):  
YONGYUN HWANG ◽  
CARLO COSSU
Keyword(s):  
Couette Flow ◽  
Large Scale ◽  
Eddy Viscosity ◽  
Initial Conditions ◽  
Viscous Sublayer ◽  
Mean Flow ◽  
Stochastic Forcing ◽  
Output Analysis ◽  
Optimal Response ◽  
The Mean

We compute the optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing at Re = 750. The equations for the coherent perturbations are linearized near the turbulent mean flow and include the associated eddy viscosity. The mean flow is found to be linearly stable but it has the potential to amplify steamwise streaks from streamwise vortices. The most amplified structures are streamwise uniform and the largest amplifications of the energy of initial conditions and of the variance of stochastic forcing are realized by large-scale streaks having spanwise wavelengths of 4.4h and 5.2h respectively. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realizations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing, related to the sensitivity to boundary conditions and artificial forcing, can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.7h). The optimal large-scale streaks are furthermore found proportional to the mean turbulent profile in the viscous sublayer and up to the buffer layer.

Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: analytical solution, comparison with numerics and bifurcation

2010 ◽  
Vol 666 ◽  
pp. 204-253 ◽  
Author(s):  
PRIYANKA SHUKLA ◽  
MEHEBOOB ALAM
Keyword(s):  
Couette Flow ◽  
Shear Banding ◽  
Base Flow ◽  
Plane Couette Flow ◽  
Weakly Nonlinear ◽  
Subcritical Bifurcation ◽  
Supercritical Bifurcation ◽  
Weakly Nonlinear Theory ◽  
Pitchfork Bifurcations ◽  
The Mean

A weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of the shear-banding instability in a rapid granular plane Couette flow using the amplitude expansion method. The nonlinear modes are found to follow certain symmetries of the base flow and the fundamental mode, which helped to identify analytical solutions for the base-flow distortion and the second harmonic, leading to an exact calculation of the first Landau coefficient. The present analytical solutions are used to validate a spectral-based numerical method for the nonlinear stability calculation. The regimes of supercritical and subcritical bifurcations for the shear-banding instability have been identified, leading to the prediction that the lower branch of the neutral stability contour in the (H, φ0)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and φ0 is the mean density or the volume fraction of particles, is subcritically unstable. The predicted finite-amplitude solutions represent shear localization and density segregation along the gradient direction. Our analysis suggests that there is a sequence of transitions among three types of pitchfork bifurcations with increasing mean density: from (i) the bifurcation from infinity in the Boltzmann limit to (ii) subcritical bifurcation at moderate densities to (iii) supercritical bifurcation at larger densities to (iv) subcritical bifurcation in the dense limit and finally again to (v) supercritical bifurcation near the close packing density. It has been shown that the appearance of subcritical bifurcation in the dense limit depends on the choice of the contact radial distribution function and the constitutive relations. The scalings of the first Landau coefficient, the equilibrium amplitude and the phase diagram, in terms of mode number and inelasticity, have been demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density (φ0) being the single control parameter that dictates the nature of the bifurcation. The predicted bifurcation scenario for the shear-band formation is in qualitative agreement with particle dynamics simulations and the experiment in the rapid shear regime of the granular plane Couette flow.

Analysis of Transitional and Fully Turbulent Plane Couette Flow

1983 ◽  
Vol 105 (3) ◽  
pp. 364-368 ◽  
Author(s):  
J. R. Missimer ◽  
L. C. Thomas
Keyword(s):  
Couette Flow ◽  
Flow Analysis ◽  
Wall Region ◽  
Turbulent Regime ◽  
Plane Couette Flow ◽  
Burst Frequency ◽  
Mean Velocity ◽  
The Mean ◽  
Turbulent Plane ◽  
Turbulent Burst

The two-dimensional, incompressible, fully-developed, turbulent plane Couette flow is a limiting case of circular Couette flow. As such, plane Couette flow analyses have been used in lubrication theory to analyze the lubrication flow in an unloaded journal bearings. A weakness of existing analyses, other than the turbulent burst analysis, is that they are not capable of characterizing the transitional turbulent regime. The objective of the proposed paper is to develop a model of the turbulent burst phenomenon for momentum in transitional turbulent and fully turbulent plane Couette flow. Model closure is obtained by specification of the mean turbulent burst frequency and, for moderate to high Reynolds numbers, by interfacing with classical eddy diffusivity models for the turbulent core. The analysis is shown to produce predictions for the mean velocity profile and friction factor that are in good agreement with published experimental data for transitional turbulent and fully turbulent flow. This approach to modeling the wall region involves a minimum level of empiricism and provides a fundamental basis for generalization. The use of the present analysis extends the applicability of plane Couette flow analysis in lubrication problems to the transitional turbulent regime.

An investigation of turbulent plane Couette flow at low Reynolds numbers

1995 ◽  
Vol 286 ◽  
pp. 291-325 ◽  
Author(s):  
Knut H. Bech ◽  
Nils Tillmark ◽  
P. Henrik Alfredsson ◽  
Helge I. Andersson
Keyword(s):  
Couette Flow ◽  
Reynolds Stress ◽  
Reynolds Numbers ◽  
Wall Turbulence ◽  
Turbulence Structure ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Low Reynolds Numbers ◽  
Low Reynolds ◽  
Near Wall

The turbulent structure in plane Couette flow at low Reynolds numbers is studied using data obtained both from numerical simulation and physical experiments. It is shown that the near-wall turbulence structure is quite similar to what has earlier been found in plane Poiseuille flow; however, there are also some large differences especially regarding Reynolds stress production. The commonly held view that the maximum in Reynolds stress close to the wall in Poiseuille and boundary layer flows is due to the turbulence-generating events must be modified as plane Couette flow does not exhibit such a maximum, although the near-wall coherent structures are quite similar. For two-dimensional mean flow, turbulence production occurs only for the streamwise fluctuations, and the present study shows the importance of the pressure—strain redistribution in connection with the near-wall coherent events.

Spanwise reflection symmetry breaking and turbulence control: plane Couette flow

2014 ◽  
Vol 745 ◽  
pp. 300-320
Author(s):  
G. Chagelishvili ◽  
G. Khujadze ◽  
H. Foysi ◽  
M. Oberlack
Keyword(s):  
Kinetic Energy ◽  
Shear Flow ◽  
Turbulent Kinetic Energy ◽  
Couette Flow ◽  
Promising Result ◽  
Spanwise Direction ◽  
Reflection Symmetry ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Turbulence Control

AbstractWe propose and analyse a new strategy of shear flow turbulence control that can be realized by the following steps: (i) imposing specially designed seed velocity perturbations, which are non-symmetric in the spanwise direction, at the walls of a flow; (ii) the configuration of the latter ensures a gain of shear flow energy and the breaking of turbulence spanwise reflection symmetry: this leads to the generation of spanwise mean flow; (iii) that changes the self-sustained dynamics of turbulence and results in a considerable reduction of the turbulence level and the production of turbulent kinetic energy. In fact, by this strategy the shear flow transient growth mechanism is activated and the formed spanwise mean flow is an intrinsic, nonlinear composition of the controlled turbulence and not directly introduced in the system. In the present paper, a weak near-wall volume forcing is designed to impose the velocity perturbations with required characteristics in the flow. The efficiency of the proposed scheme has been demonstrated by direct numerical simulation using plane Couette flow as a representative example. A promising result was obtained: after a careful parameter selection, the forcing reduces the turbulence kinetic energy and its production by up to one-third. The strategy can be naturally applied to other wall-bounded flows, e.g. channel and boundary-layer flows. Of course, the considered volume force is theoretical and hypothetical. Nevertheless, it helps to gain knowledge concerning the design of the seed velocity field that is necessary to be imposed in the flow to achieve a significant reduction of the turbulent kinetic energy. This is convincing with regard to a new control strategy, which could be based on specially constructed blowing/suction or riblets, by employing the insight gained by the comprehension of the results obtained using the investigated methodology in this paper.

Sign in / Sign up

Export Citation Format

Share Document