scholarly journals Non-modal stability in sliding Couette flow

2012 ◽  
Vol 710 ◽  
pp. 505-544 ◽  
Author(s):  
R. Liu ◽  
Q. S. Liu
Keyword(s):  
Reynolds Number ◽  
Modal Analysis ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Initial Conditions ◽  
Streamwise Vortex ◽  
Optimal Time ◽  
Energy Growth ◽  
Wide Range ◽  
Axisymmetric Disturbances

AbstractThe problem of an incompressible flow between two coaxial cylinders with radii$a$and$b$subjected to a sliding motion of the inner cylinder in the axial direction is considered. The energy stability and the non-modal stability have been investigated for both axisymmetric and non-axisymmetric disturbances. For the non-modal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the$\epsilon $-pseudospectrum, and the latter by examining the energy growth function$G(t)$. Unlike the results of the modal analysis in which the stability of sliding Couette flow is determined by axisymmetric disturbances, the energy analysis shows that a non-axisymmetric disturbance has a critical energy Reynolds number for all radius ratios$\eta = a/ b$. The results for non-modal stability show that rather large transient growth occurs over a wide range of azimuthal wavenumber$n$and streamwise wavenumber$\ensuremath{\alpha} $, even though the Reynolds number is far below its critical value. For the problem of response to external excitations, the response is sensitive to low-frequency external excitations. For all values of the radius ratio, the maximum response is achieved by non-axisymmetric and streamwise-independent disturbances when the frequency of external forcing$\omega = 0$. For the problem of response to initial conditions, the optimal disturbance is in the form of helical streaks at low Reynolds numbers. With the increase of$\mathit{Re}$, the optimal disturbance becomes very close to straight streaks. For each$\eta $, the maximum energy growth of streamwise-independent disturbances is of the order of${\mathit{Re}}^{2} $, and the optimal time is of the order of$\mathit{Re}$. This relation is qualitatively similar to that for plane Couette flow, plane Poiseuille flow and pipe Poiseuille flow. Direct numerical simulations are applied to investigate the transition of the streamwise vortex (SV) scenario at$\mathit{Re}= 1000$and 1500 for various$\eta $. The initial disturbances are the optimal streamwise vortices predicted by the non-modal analysis. We studied the streak breakdown phase of the SV scenarios by examining the instability of streaks. Moreover, we have investigated the sustainment of the energy of disturbances in the SV scenario.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

scholarly journals Dynamics of spatially localized states in transitional plane Couette flow

2019 ◽  
Vol 867 ◽  
pp. 414-437 ◽  
Author(s):  
Anton Pershin ◽  
Cédric Beaume ◽  
Steven M. Tobias
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Initial Conditions ◽  
Reynolds Numbers ◽  
Localized States ◽  
Transition To Turbulence ◽  
Dynamical Structure ◽  
Plane Couette Flow ◽  
Homoclinic Snaking ◽  
Chaotic Transients

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider et al., Phys. Rev. Lett., vol. 104, 2010, 104501). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re<175$) to transitional ones ($Re\approx 325$), these spatially localized states traverse various regimes where their relaminarization time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterized in this paper for a $4\unicode[STIX]{x03C0}$-periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarizing. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarizing. Very long chaotic transients ($t>10^{4}$) can be observed without difficulty for relatively low values of the Reynolds number ($Re\approx 250$).

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

2000 ◽  
Vol 407 ◽  
pp. 291-314 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Phase Velocity ◽  
Poiseuille Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Flexible Tube ◽  
Moderate Reynolds Number ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

Shear stress-driven flow: the state space of near-wall turbulence as

2019 ◽  
Vol 874 ◽  
pp. 606-638 ◽  
Author(s):  
Patrick Doohan ◽  
Ashley P. Willis ◽  
Yongyun Hwang
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Wall Turbulence ◽  
Phase Portraits ◽  
Wall Region ◽  
Velocity Statistics ◽  
Near Wall Turbulence ◽  
Near Wall

An inner-scaled, shear stress-driven flow is considered as a model of independent near-wall turbulence as the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}\rightarrow \infty$. In this limit, the model is applicable to the near-wall region and the lower part of the logarithmic layer of various parallel shear flows, including turbulent Couette flow, Poiseuille flow and Hagen–Poiseuille flow. The model is validated against damped Couette flow and there is excellent agreement between the velocity statistics and spectra for the wall-normal height $y^{+}<40$. A near-wall flow domain of similar size to the minimal unit is analysed from a dynamical systems perspective. The edge and fifteen invariant solutions are computed, the first discovered for this flow configuration. Through continuation in the spanwise width $L_{z}^{+}$, the bifurcation behaviour of the solutions over the domain size is investigated. The physical properties of the solutions are explored through phase portraits, including the energy input and dissipation plane, and streak, roll and wave energy space. Finally, a Reynolds number is defined in outer units and the high-$Re$ asymptotic behaviour of the equilibria is studied. Three lower branch solutions are found to scale consistently with vortex–wave interaction (VWI) theory, with wave forcing localising around the critical layer.

On the stability of pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances

1985 ◽  
Vol 158 ◽  
pp. 289-316 ◽  
Author(s):  
P. K. Sen ◽  
D. Venkateswarlu ◽  
S. Maji
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Finite Amplitude ◽  
Higher Order ◽  
Equilibrium States ◽  
Minimum Threshold ◽  
Threshold Amplitude ◽  
The Stability ◽  
The One ◽  
Axisymmetric Disturbances

The stability of fully developed pipe-Poiseuille flow to finite-amplitude axisymmetric and non-axisymmetric disturbances has been studied using the equilibrium-amplitude method of Reynolds & Potter (1967). In both the cases the least-stable centre-modes were investigated. Also, for the non-axisymmetric case the mode investigated was the one with azimuthal wavenumber equal to one. Many higher-order Landau coefficients were calculated, and the Stuart-Landau series was analysed by the Shanks (1955) method and by using Padé approximants to look for the existence of possible equilibrium states. The results show in both cases that, for each value of the Reynolds number R, there is a preferred band of spatial wavenumbers α in which equilibrium states are likely to exist. Moreover, in both cases it was found that the magnitude of the minimum threshold amplitude for a given R decreases with increasing R. The scales of the various quantities obtained agree very well with those deduced by Davey & Nguyen (1971).

scholarly journals Study of the instability of the Poiseuille flow using a thermodynamic formalism

2015 ◽  
Vol 112 (31) ◽  
pp. 9518-9523 ◽  
Author(s):  
Jianchun Wang ◽  
Qianxiao Li ◽  
Weinan E
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Thermodynamic Formalism ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Action Function ◽  
Continuous Transition ◽  
Wide Range ◽  
The Stability

The stability of the plane Poiseuille flow is analyzed using a thermodynamic formalism by considering the deterministic Navier–Stokes equation with Gaussian random initial data. A unique critical Reynolds number, Rec≈2,332, at which the probability of observing puffs in the solution changes from 0 to 1, is numerically demonstrated to exist in the thermodynamic limit and is found to be independent of the noise amplitude. Using the puff density as the macrostate variable, the free energy of such a system is computed and analyzed. The puff density approaches zero as the critical Reynolds number is approached from above, signaling a continuous transition despite the fact that the bifurcation is subcritical for a finite-sized system. An action function is found for the probability of observing puffs in a small subregion of the flow, and this action function depends only on the Reynolds number. The strategy used here should be applicable to a wide range of other problems exhibiting subcritical instabilities.

scholarly journals Disruption of states by a stable stratification

2015 ◽  
Vol 784 ◽  
pp. 548-564 ◽  
Author(s):  
T. S. Eaves ◽  
C. P. Caulfield
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Coherent Structures ◽  
Initial Conditions ◽  
Edge State ◽  
Stable Stratification ◽  
Plane Couette Flow ◽  
Moderate Reynolds Number ◽  
The Waves ◽  
Laminar State

We identify ‘minimal seeds’ for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density $E_{c}$ that trigger turbulence from the laminar state, in stratified plane Couette flow, the flow between two horizontal plates of separation $2H$, moving with relative velocity $2{\rm\Delta}U$, across which a constant density difference $2{\rm\Delta}{\it\rho}$ from a reference density ${\it\rho}_{r}$ is maintained. To find minimal seeds, we use the ‘direct-adjoint-looping’ (DAL) method for finding nonlinear optimal perturbations that optimise the time-averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system’s ‘edge state’. Therefore, the trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (Phys. Fluids, vol. 9, 1997, pp. 883–900), or equivalently finite Reynolds number solutions of the vortex–wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666). The stratified coherent states we identify at moderate Reynolds number display an altered form from their unstratified counterparts for bulk Richardson numbers $\mathit{Ri}_{B}=g{\rm\Delta}{\it\rho}H/({\it\rho}_{r}{\rm\Delta}U^{2})=O(\mathit{Re}^{-1})$, and exhibit chaotic motion for larger $\mathit{Ri}_{B}$. We demonstrate that at hith Reynolds number the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when $\mathit{Ri}_{B}=O(\mathit{Re}^{-2})$.

Instability due to viscosity stratification

1967 ◽  
Vol 27 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Hydrodynamic Stability ◽  
Poiseuille Flow ◽  
Plane Couette Flow ◽  
Plane Poiseuille Flow ◽  
Neutral Mode ◽  
Usual Theory

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow

1960 ◽  
Vol 9 (3) ◽  
pp. 353-370 ◽  
Author(s):  
J. T. Stuart
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Unstable Equilibrium ◽  
Wave Number ◽  
Neutral Stability ◽  
Plane Poiseuille Flow ◽  
Wide Range ◽  
Non Linear ◽  
Infinitesimal Disturbance

This paper considers the nature of a non-linear, two-dimensional solution of the Navier-Stokes equations when the rate of amplification of the disturbance, at a given wave-number and Reynolds number, is sufficiently small. Two types of problem arise: (i) to follow the growth of an unstable, infinitesimal disturbance (supercritical problem), possibly to a state of stable equilibrium; (ii) for values of the wave-number and Reynolds number for which no unstable infinitesimal disturbance exists, to follow the decay of a finite disturbance from a possible state of unstable equilibrium down to zero amplitude (subcritical problem). In case (ii) the existence of a state of unstable equilibrium implies the existence of unstable disturbances. Numerical calculations, which are not yet completed, are required to determine which of the two possible behaviours arises in plane Poiseuille flow, in a given range of wave-number and Reynolds number.It is suggested that the method of this paper (and of the generalization described by Part 2 by J. Watson) is valid for a wide range of Reynolds numbers and wave-numbers inside and outside the curve of neutral stability.

Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations

2000 ◽  
Vol 404 ◽  
pp. 289-309 ◽  
Author(s):  
PAOLO LUCHINI
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Flat Surface ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Adjoint Equations ◽  
Mode Analysis ◽  
Energy Growth ◽  
Wavenumber Range ◽  
Blasius Boundary Layer

The dependence on initial conditions of the three-dimensional algebraic spatial instability of the Blasius boundary layer is examined by a recently developed method of receptivity analysis based on the upstream integration of adjoint equations. This method allows us to determine optimal perturbations, i.e. initial perturbations that maximize the energy growth, even in the wavenumber range where the problem is not amenable to a mode analysis, and thus to complement a previous paper in which the small-wavenumber regime was described.

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