Solutions Of Plane Couette Flow Research Articles

Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Modified snaking in plane Couette flow with wall-normal suction

Abstract

Origin of localized snakes-and-ladders solutions of plane Couette flow.

Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.

Localized travelling waves in the asymptotic suction boundary layer

We present two spanwise-localized travelling-wave solutions in the asymptotic suction boundary layer, obtained by continuation of solutions of plane Couette flow. One of the solutions has the vortical structures located close to the wall, similar to spanwise-localized edge states previously found for this system. The vortical structures of the second solution are located in the free stream far above the laminar boundary layer and are supported by a secondary shear gradient that is created by a large-scale low-speed streak. The dynamically relevant eigenmodes of this solution are concentrated in the free stream, and the departure into turbulence from this solution evolves in the free stream towards the walls. For invariant solutions in free-stream turbulence, this solution thus shows that the source of energy of the vortical structures can be a dynamical structure of the solution itself, instead of the laminar boundary layer.

Homoclinic snaking in plane Couette flow: bending, skewing and finite-size effects

Invariant solutions of shear flows have recently been extended from spatially periodic solutions in minimal flow units to spatially localized solutions on extended domains. One set of spanwise-localized solutions of plane Couette flow exhibits homoclinic snaking, a process by which steady-state solutions grow additional structure smoothly at their fronts when continued parametrically. Homoclinic snaking is well understood mathematically in the context of the one-dimensional Swift–Hohenberg equation. Consequently, the snaking solutions of plane Couette flow form a promising connection between the largely phenomenological study of laminar–turbulent patterns in viscous shear flows and the mathematically well-developed field of pattern-formation theory. In this paper we present a numerical study of the snaking solutions of plane Couette flow, generalizing beyond the fixed streamwise wavelength of previous studies. We find a number of new solution features, including bending, skewing and finite-size effects. We establish the parameter regions over which snaking occurs and show that the finite-size effects of the travelling wave solution are due to a coupling between its fronts and interior that results from its shift-reflect symmetry. A new winding solution of plane Couette flow is derived from a strongly skewed localized equilibrium.

Mirror-symmetric exact coherent states in plane Poiseuille flow

AbstractTwo new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top–bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517–1534).

Edge states for the turbulence transition in the asymptotic suction boundary layer

AbstractWe demonstrate the existence of an exact invariant solution to the Navier–Stokes equations for the asymptotic suction boundary layer. The identified periodic orbit with a very long period of several thousand advective time units is found as a local dynamical attractor embedded in the stability boundary between laminar and turbulent dynamics. Its dynamics captures both the interplay of downstream-oriented vortex pairs and streaks observed in numerous shear flows as well as the energetic bursting that is characteristic for boundary layers. By embedding the flow into a family of flows that interpolates between plane Couette flow and the boundary layer, we demonstrate that the periodic orbit emerges in a saddle–node infinite-period (SNIPER) bifurcation of two symmetry-related travelling-wave solutions of plane Couette flow. Physically, the long period is due to a slow streak instability, which leads to a violent breakup of a streak associated with the bursting and the reformation of the streak at a different spanwise location. We show that the orbit is structurally stable when varying both the Reynolds number and the domain size.

Snakes and Ladders: Localized Solutions of Plane Couette Flow

We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in one spatial dimension. Solutions of different size are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations. These new solutions are a step towards extending the dynamical systems view of transitional turbulence to spatially extended flows.

Micro-macro numerical simulation of rod-like polymeric solutions

利用微宏观耦合方法模拟了棒状分子聚合物溶液在平板Couette流动中的复杂流变行为，其中微宏观模型通过非均匀Doi理论来描述.数值模拟中，应用有限体积方法耦合求解了介观尺度上的Smoluchowski方程和宏观尺度上的流场守恒方程.数值结果不仅得到了若干种典型的流动类型，而且还预测了另外两种新的复合瑕疵结构.数值试验表明：棒状分子聚合物的流变结构主要依赖于De数、分子相对尺度以及溶液浓度常数的取值；并且De数对分子指向矢的翻滚周期、随流取向角等微观特性也均有明显影响.

Equilibrium and travelling-wave solutions of plane Couette flow

We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds numberReand two new travelling-wave solutions. The solutions are continued under changes ofReand 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-Returbulence. 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.

Recurrent motions within plane Couette turbulence

The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low-speed fluid away from the wall, has been studied experimentally, theoretically and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. Five new three-dimensional solutions of turbulent plane Couette flow are produced, one of which is periodic while the other four are relative periodic. Each of these five solutions demonstrates the breakup and re-formation of near-wall coherent structures. Four of our solutions are periodic, but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic, but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. It is argued that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the breakup and re-formation of coherent structures. A new periodic solution of plane Couette flow is also computed that could be related to transition to turbulence.The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. This computationally demanding requirement is addressed with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas – namely the Newton–Krylov iteration and the locally constrained optimal hook step. Each of the six solutions is accompanied by an error estimate.Dynamical principles are discussed that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier–Stokes equation.

Characterization of near-wall turbulence in terms of equilibrium and “bursting” solutions

Near-wall turbulence in the buffer region of Couette and Poiseuille flows is characterized in terms of recently-found nonlinear three-dimensional solutions to the incompressible Navier–Stokes equations for wall-bounded shear flows. The data suggest that those solutions can be classified into two families, of which one is dominated by streamwise vortices, and the other one by streaks. They can be associated with the upper and lower branches of the equilibrium solutions for Couette flow found by Nagata [“Three-dimensional finite-amplitude solutions in plane Couette flow: Bifurcation from infinity,” J. Fluid Mech. 217, 519 (1990)]. The quiescent structures of near-wall turbulence are shown to correspond to the vortex-dominated family, but evidence is presented that they burst intermittently both in minimal and in fully turbulent flows. The intensity and period of the bursts are Reynolds-number dependent, but they saturate at high enough Reynolds numbers. The time-periodic exact solution found for Couette flow by Kawahara and Kida [“Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst,” J. Fluid Mech. 449, 291 (2001)] can be used as a simplified model for the bursting process.

Three-dimensional traveling-wave solutions in plane Couette flow

Nonlinear three-dimensional time-dependent solution branches are obtained in plane Couette flow modified by plane Poiseuille flow component. It is found that as the Poiseuille component is added a branch of time-dependent solutions is produced from the time-independent solution branch in plane Couette flow, and that there exists a second branch of time-dependent solutions in the form of a closed loop inside the primary time-dependent solution branch. The second branch intersects the line of zero plane Poiseuille flow component at two points with nonvanishing phase velocity for higher Reynolds numbers, creating shape preserving nonlinear traveling-wave solutions in plane Couette flow.

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

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.

On wavy instabilities of the Taylor-vortex flow between corotating cylinders

At least four wavy instabilities are found numerically by analysing the linear stability of Taylor-vortex flow (TVF) in the limit of a small gap between two concentric cylinders which rotate differentially in the same direction. Two of the wavy instabilities, including the one leading to conventional wavy vortex (WVF), have the same axial wavelength as TVF at the onset of instability, while the other two are characterized by subharmonic modes with axial wavelengths twice as long as those of TVF. The two subharmonic instabilities appear to correspond to the wavy-inflow-boundary flow (WIB) and the wavy-outflow-boundary flow (WOB) observed in the experiment of Andereck, Liu & Swinney (1986). The phase velocities, measured in the rotating frame of reference, of all the wavy instabilities are non-zero at the onset except that the phase velocity of WVF vanishes in the region where the average rotation rate Ω of the cylinders is small. By using this simple bifurcation property of WVF for small Ω, time-independent finite-amplitude non-axisymmetric solution branches bifurcating from TVF are followed numerically. The most interesting findings are that some of the solution branches cross the line Ω = 0, producing three-dimensional nonlinear solutions in plane Couette flow.