Nonlinear stability, bifurcation and vortical patterns in three-dimensional granular plane Couette flow

2013 ◽  
Vol 716 ◽  
pp. 349-413 ◽  
Author(s):  
Meheboob Alam ◽  
Priyanka Shukla
Keyword(s):  
Three Dimensional ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Mean Flow ◽  
Instability Modes ◽  
The Mean ◽  
Nonlinear Solutions

AbstractThe effects of three-dimensional (3D) perturbations, having wave-like modulations along both the streamwise and spanwise/vorticity directions, on the nonlinear states of five types of linear instability modes, the nature of their bifurcations and the resulting nonlinear patterns are analysed for granular plane Couette flow using an order-parameter theory which is an extension of our previous work on two-dimensional (2D) perturbations (Shukla & Alam, J. Fluid Mech., vol. 672, 2011b, pp. 147–195). The differential equations for modal amplitudes (the fundamental mode, the mean-flow distortion, the second harmonic and the distortion of the fundamental mode), up to cubic-order in perturbation amplitude, are solved using a spectral-based numerical technique, yielding an estimate of the first Landau coefficient that accounts for the leading-order nonlinear effect on finite-amplitude perturbations. In the near-critical regime of flows, we found evidence of mean-flow resonance, characterized by the divergence of the first Landau coefficient, that occurs due to the interaction/resonance between a linear instability mode and a mean-flow mode. The nonlinear solutions are found to appear via both pitchfork and Hopf bifurcations from the underlying linear instability modes, leading to supercritical nonlinear states of stationary and travelling wave solutions. The subcritical travelling wave solutions have also been uncovered in the linearly stable regimes of flow. It is shown that multiple nonlinear states of both stationary and travelling waves can coexist for a given parameter combination of mean density and Couette gap. The 3D nonlinear solutions persist for a range of spanwise wavenumbers up to ${k}_{z} = O(1)$ that originate from 2D instabilities which occur beyond a moderate value of the mean density. For purely 3D instabilities in dilute flows (having no analogue in 2D flows), the supercritical finite-amplitude solutions persist for a much larger range of spanwise wavenumber up to ${k}_{z} = O(10)$. For all instabilities, the vortical motion on the cross-stream plane has been characterized in terms of the fixed/critical points of the underlying flow field: saddles, nodes (sources and sinks) and vortices have been identified. While the cross-stream velocity field for supercritical solutions in dilute flows contains nodes and saddles, the subcritical solutions are dominated by large-scale vortices in the background of saddle-node-type motions. The latter type of flow pattern also persists at moderate densities in the form of supercritical nonlinear solutions that originate from the dominant 2D instability modes for which the vortex appears to be driven by two nearby saddles. The location of this vortex is found to be correlated with the local maxima of the streamwise vorticity.

scholarly journals Equilibrium and travelling-wave solutions of plane Couette flow

2009 ◽  
Vol 638 ◽  
pp. 243-266 ◽  
Author(s):  
J. F. GIBSON ◽  
J. HALCROW ◽  
P. CVITANOVIĆ
Keyword(s):  
Couette Flow ◽  
Three Dimensional ◽  
Physical Space ◽  
Space Velocity ◽  
Travelling Wave ◽  
Isotropy Group ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Equilibrium Solutions ◽  
Wave Solutions

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number Re and two new travelling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-Re turbulence. Projections of these solutions and their unstable manifolds from their ∞-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

Tertiary and quaternary solutions for plane Couette flow

1997 ◽  
Vol 344 ◽  
pp. 137-153 ◽  
Author(s):  
R. M. CLEVER ◽  
F. H. BUSSE
Keyword(s):  
Periodic Solutions ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Plane Couette Flow ◽  
Longitudinal Vortices ◽  
Wave Solutions ◽  
Time Periodic Solutions ◽  
Steady Solutions ◽  
Time Periodic

The plane Couette system does not exhibit any secondary solutions bifurcating from the primary solution of constant shear. Since the work of Nagata (1990) it has been well known that three-dimensional steady solutions exist. Here the manifold of those steady solutions is explored in the parameter space of the problem and their instabilities are investigated. These instabilities usually lead to time-periodic solutions whose properties do not differ much from those of the steady solutions except that the amplitude varies in time. In some cases travelling wave solutions which are asymmetric with respect to the midplane of the layer are found as quaternary states of flow. Similarities with longitudinal vortices recently observed in experiments are discussed.

scholarly journals Large-scale mean patterns in turbulent convection

2015 ◽  
Vol 776 ◽  
pp. 96-108 ◽  
Author(s):  
Mohammad S. Emran ◽  
Jörg Schumacher
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Turbulent Fluctuations ◽  
The Mean ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

Large-scale patterns, which are well-known from the spiral defect chaos (SDC) regime of thermal convection at Rayleigh numbers $\mathit{Ra}<10^{4}$, continue to exist in three-dimensional numerical simulations of turbulent Rayleigh–Bénard convection in extended cylindrical cells with an aspect ratio ${\it\Gamma}=50$ and $\mathit{Ra}>10^{5}$. They are revealed when the turbulent fields are averaged in time and turbulent fluctuations are thus removed. We apply the Boussinesq closure to estimate turbulent viscosities and diffusivities, respectively. The resulting turbulent Rayleigh number $\mathit{Ra}_{\ast }$, that describes the convection of the mean patterns, is indeed in the SDC range. The turbulent Prandtl numbers are smaller than one, with $0.2\leqslant \mathit{Pr}_{\ast }\leqslant 0.4$ for Prandtl numbers $0.7\leqslant \mathit{Pr}\leqslant 10$. Finally, we demonstrate that these mean flow patterns are robust to an additional finite-amplitude sidewall forcing when the level of turbulent fluctuations in the flow is sufficiently high.

scholarly journals Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190018
Author(s):  
E. Yim ◽  
P. Meliga ◽  
F. Gallaire
Keyword(s):  
Turbulent Flow ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Backward Facing Step ◽  
Eddy Viscosity Model ◽  
Coherent Interaction ◽  
Self Consistent ◽  
The Mean

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent–incoherent interaction is lumped into a Reynolds averaged Navier–Stokes (RANS) eddy viscosity model, and the mean–coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793 , 777–797. ( doi:10.1017/jfm.2016.109 )). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

scholarly journals Divergent versus Nondivergent Instabilities of Piecewise Uniform Shear Flows on the f Plane

2009 ◽  
Vol 39 (7) ◽  
pp. 1685-1699
Author(s):  
Nathan Paldor ◽  
Yona Dvorkin ◽  
Eyal Heifetz
Keyword(s):  
Numerical Solutions ◽  
Delta Function ◽  
Relative Vorticity ◽  
Linear Instability ◽  
Large Radius ◽  
Mean Flow ◽  
Uniform Shear ◽  
Linear Shear ◽  
The Mean ◽  
Inner Region

Abstract The linear instability of a piecewise uniform shear flow is classically formulated for nondivergent perturbations on a 2D barotropic mean flow with linear shear, bounded on both sides by semi-infinite half-planes where the mean flows are uniform. The problem remains unchanged on the f plane because for nondivergent perturbations the instability is driven by vorticity gradient at the edges of the inner, linear shear region, whereas the vorticity itself does not affect it. The instability of the unbounded case is recovered when the outer regions of uniform velocity are bounded, provided that these regions are at least twice as wide as the inner region of nonzero shear. The numerical calculations demonstrate that this simple scenario is greatly modified when the perturbations’ divergence and the variation of the mean height (which geostrophically balances the mean flow) are retained in the governing equations. Although a finite deformation radius exists on the shallow water f plane, the mean vorticity gradient that governs the instability in the nondivergent case remains unchanged, so it is not obvious how the instability is modified by the inclusion of divergence in the numerical solutions of the equations. The results here show that the longwave instability of nondivergent flows is recovered by the numerical solution for divergent flows only when the radius of deformation is at least one order of magnitude larger than the width of the inner uniform shear region. Nevertheless, even at this large radius of deformation both the amplitude of the velocity eigenfunction and the distribution of vorticity and divergence differ significantly from those of nondivergent perturbations and vary strongly in the cross-stream direction. Whereas for nondivergent flows the vorticity and divergence both have a delta-function structure located at the boundaries of the inner region, in divergent flows they are spread out and attain their maximum away from the boundaries (either in the inner region or in the outer regions) in some range of the mean shear. In contrast to nondivergent flows for which the mean shear is merely a multiplicative factor of the growth rates, in divergent flows new unstable modes exist for sufficiently large mean shear with no shortwave cutoff. This unstable mode is strongly affected by the sign of the mean shear (i.e., the sign of the mean relative vorticity).

Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity

1990 ◽  
Vol 217 ◽  
pp. 519-527 ◽  
Author(s):  
M. Nagata
Keyword(s):  
Couette Flow ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Narrow Gap ◽  
Plane Couette Flow ◽  
Bifurcation Problem ◽  
Bifurcation From Infinity ◽  
Flow Bifurcation ◽  
Average Rotation

Finite-amplitude solutions of plane Couette flow are discovered. They take a steady three-dimensional form. The solutions are obtained numerically by extending the bifurcation problem of a circular Couette system between co-rotating cylinders with a narrow gap to the case with zero average rotation rate.

Experimental Study of Flow Inside a Centrifugal Fan Using Magnetic Resonance Velocimetry

2019 ◽  
Author(s):  
Davis W. Hoffman ◽  
Laura Villafañe ◽  
Christopher J. Elkins ◽  
John K. Eaton
Keyword(s):  
Three Dimensional ◽  
Leading Edge ◽  
Suction Side ◽  
Reynolds Numbers ◽  
Centrifugal Fan ◽  
Mean Flow ◽  
Magnetic Resonance Velocimetry ◽  
Flow Features ◽  
Outlet Flow ◽  
The Mean

Abstract Three-dimensional, three-component time-averaged velocity fields have been measured within a low-speed centrifugal fan with forward curved blades. The model investigated is representative of fans commonly used in automotive HVAC applications. The flow was analyzed at two Reynolds numbers for the same ratio of blade rotational speed to outlet flow velocity. The flow patterns inside the volute were found to have weak sensitivity to Reynolds number. A pair of counter-rotating vortices evolve circumferentially within the volute with positive and negative helicity in the upper and lower regions, respectively. Measurements have been further extended to capture phase-resolved flow features by synchronizing the data acquisition with the blade passing frequency. The mean flow field through each blade passage is presented including the jet-wake structure extending from the blade and the separation zone on the suction side of the blade leading edge.

scholarly journals Three-dimensional spatial structure of the macro-pores and flow simulation in anthracite coal based on X-ray μ-CT scanning data

2020 ◽  
Vol 17 (5) ◽  
pp. 1221-1236
Author(s):  
Hui-Huang Fang ◽  
Shu-Xun Sang ◽  
Shi-Qi Liu
Keyword(s):  
Spatial Structure ◽  
Flow Velocity ◽  
Single Channel ◽  
Three Dimensional ◽  
Bedding Plane ◽  
Mean Flow ◽  
Equivalent Radius ◽  
Flow Pressure ◽  
The Mean ◽  
Mean Flow Velocity

Abstract The three-dimensional (3D) structures of pores directly affect the CH4 flow. Therefore, it is very important to analyze the 3D spatial structure of pores and to simulate the CH4 flow with the connected pores as the carrier. The result shows that the equivalent radius of pores and throats are 1–16 μm and 1.03–8.9 μm, respectively, and the throat length is 3.28–231.25 μm. The coordination number of pores concentrates around three, and the intersection point between the connectivity function and the X-axis is 3–4 μm, which indicate the macro-pores have good connectivity. During the single-channel flow, the pressure decreases along the direction of CH4 flow, and the flow velocity of CH4 decreases from the pore center to the wall. Under the dual-channel and the multi-channel flows, the pressure also decreases along the CH4 flow direction, while the velocity increases. The mean flow pressure gradually decreases with the increase of the distance from the inlet slice. The change of mean flow pressure is relatively stable in the direction horizontal to the bedding plane, while it is relatively large in the direction perpendicular to the bedding plane. The mean flow velocity in the direction horizontal to the bedding plane (Y-axis) is the largest, followed by that in the direction horizontal to the bedding plane (X-axis), and the mean flow velocity in the direction perpendicular to the bedding plane is the smallest.

Coherent structures and dominant frequencies in a turbulent three-dimensional diffuser

2012 ◽  
Vol 699 ◽  
pp. 320-351 ◽  
Author(s):  
Johan Malm ◽  
Philipp Schlatter ◽  
Dan S. Henningson
Keyword(s):  
Coherent Structures ◽  
Large Scale ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Low Frequency ◽  
Bulk Velocity ◽  
Time Signal ◽  
Mean Flow ◽  
The Mean ◽  
Dominant Frequencies

AbstractDominant frequencies and coherent structures are investigated in a turbulent, three-dimensional and separated diffuser flow at $\mathit{Re}= 10\hspace{0.167em} 000$ (based on bulk velocity and inflow-duct height), where mean flow characteristics were first studied experimentally by Cherry, Elkins and Eaton (Intl J. Heat Fluid Flow, vol. 29, 2008, pp. 803–811) and later numerically by Ohlsson et al. (J. Fluid Mech., vol. 650, 2010, pp. 307–318). Coherent structures are educed by proper orthogonal decomposition (POD) of the flow, which together with time probes located in the flow domain are used to extract frequency information. The present study shows that the flow contains multiple phenomena, well separated in frequency space. Dominant large-scale frequencies in a narrow band $\mathit{St}\equiv fh/ {u}_{b} \in [0. 0092, 0. 014] $ (where $h$ is the inflow-duct height and ${u}_{b} $ is the bulk velocity), yielding time periods ${T}^{\ensuremath{\ast} } = T{u}_{b} / h\in [70, 110] $, are deduced from the time signal probes in the upper separated part of the diffuser. The associated structures identified by the POD are large streaks arising from a sinusoidal oscillating motion in the diffuser. Their individual contributions to the total kinetic energy, dominated by the mean flow, are, however, small. The reason for the oscillating movement in this low-frequency range is concluded to be the confinement of the flow in this particular geometric set-up in combination with the high Reynolds number and the large separated zone on the top diffuser wall. Based on this analysis, it is shown that the bulk of the streamwise root mean square (r.m.s.) value arises due to large-scale motion, which in turn can explain the appearance of two or more peaks in the streamwise r.m.s. value. The weak secondary flow present in the inflow duct is shown to survive into the diffuser, where it experiences an imbalance with respect to the upper expanding corners, thereby giving rise to the asymmetry of the mean separated region in the diffuser.

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