scholarly journals A doubly localized equilibrium solution of plane Couette flow

2014 ◽  
Vol 750 ◽  
Author(s):  
E. Brand ◽  
J. F. Gibson
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Equilibrium Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Edge States ◽  
Plane Couette Flow ◽  
Turbulent Spot ◽  
Navier Stokes Equations ◽  
Exponential Decay Rate

AbstractWe present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier–Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

scholarly journals Doubly localized states in plane Couette flow

2014 ◽  
Vol 758 ◽  
pp. 1-4 ◽  
Author(s):  
Bruno Eckhardt
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Localized States ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

AbstractMuch of our understanding of the transition to turbulence in flows without a linear instability came with the discovery and characterization of fully three-dimensional solutions to the Navier–Stokes equation. The first examples in plane Couette flow were periodic in both spanwise and streamwise directions, and could explain the transitions in small domains only. The presence of localized turbulent spots in larger domains, the spatiotemporal decoherence on larger scales and the ability to trigger turbulence with pointwise perturbations require solutions that are localized in both directions, like the one presented by Brand & Gibson (J. Fluid Mech., vol. 750, 2014, R3). They describe a steady solution of the Navier–Stokes equations and characterize in unprecedented detail, including an analytic computation of its localization properties. The study opens up new ways to describe localized turbulent patches.

On the instability of vortex–wave interaction states

2016 ◽  
Vol 802 ◽  
pp. 634-666 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Reynolds Number ◽  
Coherent Structures ◽  
Time Scale ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Navier Stokes ◽  
Edge States ◽  
Asymptotic Approach ◽  
Navier Stokes Equations ◽  
High Reynolds

In recent years it has been established that vortex–wave interaction theory forms an asymptotic framework to describe high Reynolds number coherent structures in shear flows. Comparisons between the asymptotic approach and finite Reynolds number computations of equilibrium states from the full Navier–Stokes equations have suggested that the asymptotic approach is extremely accurate even at quite low Reynolds numbers. However, unlike the situation with an approach based on solving the full Navier–Stokes equations numerically, the vortex–wave interaction approach has not yet been developed to study the instability of the structures it describes. In this work, a comprehensive study of the different instabilities of vortex–wave interaction states is given and it is shown that there are three different time scales on which instabilities can develop. The most dangerous type is a rapidly growing Rayleigh instability of the streak part of the flow. The least dangerous type is a slow mode operating on the diffusion time scale of the roll–streak part of the flow. The third mode of instability, which we will refer to as the edge mode of instability, occurs on a time scale midway between those of the other two modes. The existence of the latter mode explains why some exact coherent structures can act as edge states between the laminar and turbulent attractors. These stability results are compared to results from Navier–Stokes calculations.

scholarly journals A streamwise constant model of turbulence in plane Couette flow

2010 ◽  
Vol 665 ◽  
pp. 99-119 ◽  
Author(s):  
D. F. GAYME ◽  
B. J. McKEON ◽  
A. PAPACHRISTODOULOU ◽  
B. BAMIEH ◽  
J. C. DOYLE
Keyword(s):  
Couette Flow ◽  
Small Amplitude ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Theoretic Approach ◽  
Plane Couette Flow ◽  
Navier Stokes Equations ◽  
Turbulent Plane ◽  
3C Model

Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behaviour of fully turbulent plane Couette flow using a streamwise constant projection of the Navier–Stokes equations. This results in a two-dimensional three-velocity-component (2D/3C) model. We first use a steady-state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the 2D/3C model under small-amplitude Gaussian forcing of the cross-stream components are compared to direct numerical simulation (DNS) data. The results indicate that a streamwise constant projection of the Navier–Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A systems-theoretic approach is used to demonstrate the presence of large input–output amplification through the forced 2D/3C model. It is this amplification coupled with the appropriate nonlinearity that enables the 2D/3C model to generate turbulent behaviour under the small-amplitude forcing employed in this study.

Bounds for threshold amplitudes in subcritical shear flows

1994 ◽  
Vol 270 ◽  
pp. 175-198 ◽  
Author(s):  
Gunilla Kreiss ◽  
Anders Lundbladh ◽  
Dan S. Henningson
Keyword(s):  
Couette Flow ◽  
Half Plane ◽  
Stokes Equations ◽  
Model Problem ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Threshold Curve ◽  
Navier Stokes Equations ◽  
Threshold Amplitude ◽  
Norm Of The Resolvent

A general theory which can be used to derive bounds on solutions to the Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R−21/4. Examples, obained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R−1. The discrepancy is discussed in the light of a model problem.

On the behaviour of small disturbances in plane Couette flow

1962 ◽  
Vol 13 (1) ◽  
pp. 91-100 ◽  
Author(s):  
A. P. Gallagher ◽  
A. McD. Mercer
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Wave Number ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Approximate Methods ◽  
Navier Stokes Equations ◽  
Sommerfeld Equation ◽  
Small Disturbances

The problem considered here is concerned with small disturbances of plane Couette flow. As is usual in such problems it is assumed that the disturbance velocities are sufficiently small to allow the Navier-Stokes equations to be linearized. There results a special case of the well-known Orr-Sommerfeld equation and this is solved by an exact method using a digital computer. The problem has previously been considered by several authors, mostly using approximate methods and their results have been compared where possible with those obtained here. It was possible to proceed to values of αR not in excess of 1000 (α being the wave-number of the disturbance and R the Reynolds number of the basic flow), and the results tend to confirm the belief that Couette flow is stable at all Reynolds numbers.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

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