scholarly journals Transient growth in linearly stable Taylor–Couette flows

2014 ◽  
Vol 742 ◽  
pp. 254-290 ◽  
Author(s):  
Simon Maretzke ◽  
Björn Hof ◽  
Marc Avila
Keyword(s):  
Linear Stability ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Energy Growth ◽  
Transient Energy ◽  
Taylor Couette ◽  
Semi Empirical ◽  
Essential Prerequisite ◽  
Cylinder Rotation

AbstractNon-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor–Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers$\mathit{Re}$, the optimal transient energy growth always follows a$\mathit{Re}^{2/3}$scaling, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows, the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations, we show that linear stability and transient growth are independent of the cylinder rotation ratio and we derive a universal$\mathit{Re}^{2/3}$scaling of optimal energy growth using Wentzel–Kramers–Brillouin theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.

scholarly journals Modal and non-modal stability analysis of electrohydrodynamic flow with and without cross-flow

2015 ◽  
Vol 770 ◽  
pp. 319-349 ◽  
Author(s):  
Mengqi Zhang ◽  
Fulvio Martinelli ◽  
Jian Wu ◽  
Peter J. Schmid ◽  
Maurizio Quadrio
Keyword(s):  
Stability Analysis ◽  
Electric Field ◽  
Linear Stability ◽  
Poiseuille Flow ◽  
Cross Flow ◽  
Transient Growth ◽  
Unstable Flow ◽  
Electrohydrodynamic Flow ◽  
Stability Threshold ◽  
Energy Growth

We report the results of a complete modal and non-modal linear stability analysis of the electrohydrodynamic flow for the problem of electroconvection in the strong-injection region. Convective cells are formed by the Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in electrohydrodynamics, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable flow. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$\mathit{Re}$Poiseuille flow yields a more unstable flow in both modal and non-modal stability analyses. Even though the energy analysis and the input–output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centreline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centres of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.

scholarly journals Turbulent Taylor–Couette flow with stationary inner cylinder

2016 ◽  
Vol 799 ◽  
Author(s):  
R. Ostilla-Mónico ◽  
R. Verzicco ◽  
D. Lohse
Keyword(s):  
Angular Momentum ◽  
Large Scale ◽  
Outer Cylinder ◽  
Reynolds Numbers ◽  
Law Of The Wall ◽  
Large Scale Structures ◽  
Large Gradient ◽  
The Mean ◽  
Taylor Couette ◽  
Cylinder Rotation

A series of direct numerical simulations were performed of Taylor–Couette (TC) flow, the flow between two coaxial cylinders, with the outer cylinder rotating and the inner one fixed. Three cases were considered, where the Reynolds number of the outer cylinder was $Re_{o}=5.5\times 10^{4}$, $Re_{o}=1.1\times 10^{5}$ and $Re_{o}=2.2\times 10^{5}$. The ratio of radii ${\it\eta}=r_{i}/r_{o}$ was fixed to ${\it\eta}=0.909$ to mitigate the effects of curvature. Axially periodic boundary conditions were used, with the aspect ratio of vertical periodicity ${\it\Gamma}$ fixed to ${\it\Gamma}=2.09$. Being linearly stable, TC flow with outer cylinder rotation is known to have very different behaviour than TC flow with pure inner cylinder rotation. Here, we find that the flow nonetheless becomes turbulent, but the torque required to drive the cylinders and level of velocity fluctuations was found to be smaller than those for pure inner cylinder rotation at comparable Reynolds numbers. The mean angular momentum profiles showed a large gradient in the bulk, instead of the constant angular momentum profiles of pure inner cylinder rotation. The near-wall mean and fluctuation velocity profiles were found to coincide only very close to the wall, showing large deviations from both pure inner cylinder rotation profiles and the classic von Karman law of the wall elsewhere. Finally, transport of angular velocity was found to occur mainly through intermittent bursts, and not through wall-attached large-scale structures as is the case for pure inner cylinder rotation.

scholarly journals Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

2018 ◽  
Vol 840 ◽  
pp. 5-24 ◽  
Author(s):  
Junho Park ◽  
Paul Billant ◽  
Jong-Jin Baik ◽  
Jaemyeong Mango Seo
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Couette Flow ◽  
Linear Stability Analysis ◽  
Reynolds Numbers ◽  
Axisymmetric Mode ◽  
Stability Threshold ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

scholarly journals Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos

2012 ◽  
Vol 702 ◽  
pp. 415-443 ◽  
Author(s):  
Chris C. T. Pringle ◽  
Ashley P. Willis ◽  
Rich R. Kerswell
Keyword(s):  
Variational Problem ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Initial Perturbation ◽  
Transition To Turbulence ◽  
Critical Threshold ◽  
General Strategy ◽  
Transient Growth ◽  
Initial Amplitude ◽  
Energy Growth

AbstractWe propose a general strategy for determining the minimal finite amplitude disturbance that triggers transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude which respect the boundary conditions, incompressibility and the Navier–Stokes equations, to maximize a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approach using the ratio of the final to initial perturbation kinetic energies (energy growth) as the functional and the energy norm to measure amplitudes in the context of pipe flow. Our results indicate that the variational problem yields a smooth converged solution provided that the initial amplitude is below the threshold for transition. This optimal is the nonlinear analogue of the well-studied (linear) transient growth optimal. At the critical threshold, the optimization seeks out a disturbance that is on the ‘edge’ of turbulence during the period. Above this threshold, when disturbances trigger turbulence by the end of the period, convergence is then practically impossible. The first disturbance found to trigger turbulence as the amplitude is increased identifies the ‘minimal seed’ for the given geometry and forcing (Reynolds number). We conjecture that it may be possible to select a functional such that the converged optimal below threshold smoothly converges to the minimal seed at threshold. Our choice of the energy growth functional is shown to come close to this for the pipe flow geometry investigated here.

Instabilities and transient growth of the stratified Taylor–Couette flow in a Rayleigh-unstable regime

2017 ◽  
Vol 822 ◽  
pp. 80-108 ◽  
Author(s):  
Junho Park ◽  
Paul Billant ◽  
Jong-Jin Baik
Keyword(s):  
Couette Flow ◽  
Density Stratification ◽  
Rotational Instability ◽  
Transient Growth ◽  
Unstable Regime ◽  
Stable Regime ◽  
Energy Growth ◽  
Centrifugal Instability ◽  
Taylor Couette ◽  
Taylor Couette Flow

The stability of the Taylor–Couette flow is analysed when there is a stable density stratification along the axial direction and when the flow is centrifugally unstable, i.e. in the Rayleigh-unstable regime. It is shown that not only the centrifugal instability but also the strato-rotational instability can occur. These two instabilities can be explained and well described by means of a Wentzel–Kramers–Brillouin–Jeffreys asymptotic analysis for large axial wavenumbers in inviscid and non-diffusive limits. In the presence of viscosity and diffusion, numerical results reveal that the strato-rotational instability becomes dominant over the centrifugal instability at the onset of instability when the axial density stratification is sufficiently strong. Linear transient energy growth is next investigated for counter-rotating cylinders in the stable regime of the Froude number–Reynolds number parameter space. We show that there exist two types of transient growth mechanism analogous to the lift up and the Orr mechanisms in homogeneous fluids but with the additional effect of density perturbations. The dominant mechanism depends on the stratification: when the stratification is strong, non-axisymmetric three-dimensional perturbations achieve the optimal energy growth through the Orr mechanism while for moderate stratification, axisymmetric perturbations lead to the optimal transient growth by a lift-up mechanism involving internal waves.

scholarly journals Taylor vortices versus Taylor columns

2014 ◽  
Vol 750 ◽  
pp. 1-4 ◽  
Author(s):  
Laurette S. Tuckerman
Keyword(s):  
Couette Flow ◽  
Parameter Space ◽  
Maximum Energy ◽  
Linear Instability ◽  
Transition To Turbulence ◽  
Transient Growth ◽  
Taylor Vortices ◽  
Toroidal Vortices ◽  
Taylor Couette ◽  
Taylor Couette Flow

AbstractTaylor–Couette flow is inevitably associated with the visually appealing toroidal vortices, waves, and spirals that are instigated by linear instability. The linearly stable regimes, however, pose a new challenge: do they undergo transition to turbulence and if so, what is its mechanism? Maretzke et al. (J. Fluid Mech., vol. 742, 2014, pp. 254–290) begin to address this question by determining the transient growth over the entire parameter space. They find that in the quasi-Keplerian regime, the optimal perturbations take the form of Taylor columns and that the maximum energy achieved depends only on the shear.

Energy growth in viscous channel flows

1993 ◽  
Vol 252 ◽  
pp. 209-238 ◽  
Author(s):  
Satish C. Reddy ◽  
Dan S. Henningson
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Operator ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Energy Methods ◽  
Transient Growth ◽  
Energy Growth

In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille–Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

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