Genta Kawahara

Periodic motion embedded in plane Couette turbulence : regeneration cycle and burst

Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with strong variation in spatial and temporal structure exhibits a full regeneration cycle, which consists of the formation and breakdown of streamwise vortices and low-velocity streaks; the other one, of gentle variation, represents a spanwise standing-wave motion of low-velocity streaks. These two solutions are unstable and the corresponding periodic orbits in the phase space are connected with each other. A turbulent state wanders around the strong one for most of the time except for occasional escapes from it. As a result, the mean velocity profile and the root-mean-squares of velocity fluctuations of the plane Couette turbulence agree very well with the temporal averages of those of this periodic motion. After an occasional escape from the strong solution, the turbulent state reaches the gentle periodic solution and returns. On the way back, it experiences an overshoot accompanied by strong turbulence activity like an intermittent bursting phenomenon.

The Significance of Simple Invariant Solutions in Turbulent Flows

Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. A significant advance in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier-Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e., the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.

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 flo...

Turbulent shear flow over active and passive porous surfaces

The behaviour of turbulent shear flow over a mass-neutral permeable wall is studied numerically. The transpiration is assumed to be proportional to the local pressure fluctuations. It is first shown that the friction coefficient increases by up to 40% over passively porous walls, even for relatively small porosities. This is associated with the presence of large spanwise rollers, originating from a linear instability which is related both to the Kelvin–Helmholtz instability of shear layers, and to the neutral inviscid shear waves of the mean turbulent profile. It is shown that the rollers can be forced by patterned active transpiration through the wall, also leading to a large increase in friction when the phase velocity of the forcing resonates with the linear eigenfunctions mentioned above. Phase-lock averaging of the forced solutions is used to further clarify the flow mechanism. This study is motivated by the control of separation in boundary layers.

Marginally turbulent flow in a square duct.

A direct numerical simulation of turbulent flow in a straight square duct was performed in order to determine the minimal requirements for self-sustaining turbulence. It was found that turbulence can be maintained for values of the bulk Reynolds number above approximately 1100, corresponding to a friction-velocity-based Reynolds number of 80. The minimum value for the streamwise period of the computational domain is around 190 wall units, roughly independently of the Reynolds number. We present a characterization of the flow state at marginal Reynolds numbers which substantially differs from the fully turbulent one: the marginal state exhibits a four-vortex secondary flow structure alternating in time whereas the fully turbulent one presents the usual eight-vortex pattern. It is shown that in the regime of marginal Reynolds numbers buffer-layer coherent structures play a crucial role in the appearance of secondary flow of Prandtls second kind.

Periodic motion representing isotropic turbulence

Temporally periodic solutions are extracted numerically from forced box turbulence with high symmetry. Since they are unstable to small perturbations, they are not found by forward integration but can be captured by Newton–Raphson iterations. Several periodic flows of various periods are identified for the micro-scale Reynolds number Rλ between 50 and 67. The statistical properties of these periodic flows are compared with those of turbulent flow. It is found that the one with the longest period, which is two to three times the large-eddy-turnover time of turbulence, exhibits the same behaviour quantitatively as turbulent flow. In particular, we compare the energy spectrum, the Reynolds number dependence of the energy-dissipation rate, the pattern of the energy-cascade process, and the magnitude of the largest Lyapunov exponent. This periodic motion consists of high- and low-activity periods, which turbulence approaches, more often around its low-activity part, at the rate of once over a few eddy-turnover times. With reference to this periodic motion the Kaplan–York dimension and the Kolmogorov–Sinai entropy of the turbulence with high symmetry are estimated at Rλ = 67 to be 19.7 and 0.992, respectively. The significance of such periodic solutions, embedded in turbulence, for turbulence analysis is discussed.

Laminarization of minimal plane Couette flow: Going beyond the basin of attraction of turbulence

Laminarization of minimal plane Couette turbulence is achieved numerically through short-time imposition of weak spanwise system rotation. A laminarization strategy presented in this Letter is inspired by investigation of the phase-space structure in the vicinity of a recently found unstable periodic orbit [G. Kawahara and S. Kida, “Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst,” J. Fluid Mech. 449, 291 (2001)]. The periodic orbit, which a turbulent state occasionally approaches, and its local stable manifold are found to form the separatrix between the basin of attraction of turbulent and laminar flows. The introduction of the slight rotation during its approach to the periodic orbit enables the state to go beyond the basin of attraction of the turbulence toward the laminar flow. The global stabilization of the unstable periodic orbit by the method of controlling chaos is also performed to accomplish the laminarization without waiting until the natural approach.

Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow

The mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation Γ starting with a vortex filament in a simple shear flow ( U = SX 2 Xˆ 1 , S being a shear rate) is investigated analytically. An asymptotic expression for the vorticity field is obtained at a large Reynolds number Γ/ ν [Gt ]1, ν being the kinematic viscosity of fluid, and during the initial time St [Lt ]1 of evolution as well as St [Lt ](Γ/ ν ) 1/2 . The vortex tube, which is inclined from the streamwise ( X 1 ) direction both in the vertical ( X 2 ) and spanwise ( X 3 ) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. The simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. The magnitude of the azimuthal vorticity increases up to O ((Γ/ ν ) 1/3 S ) at distance r = O ((Γ/ ν ) 1/3 ( νt ) 1/2 ) from the vortex tube. The spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. As a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. The axial vorticity increases in time as S 2 t , the direction of which is opposite to that of the vortex tube at r = O ((Γ/ ν ) 1/2 ( νt ) 1/2 ) where the vorticity magnitude is strongest. In the near region r [Lt ](Γ/ ν ) 1/3 ( νt ) 1/2 , on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. Only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. As a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. The growth rate of vorticity due to this effect attains a maximum (or minimum) value of ± S 2 /3 3/2 when the vortex tube is oriented in the direction of Xˆ 1 + Xˆ 2 ∓ Xˆ 3 . The present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.

Linear instability of a corrugated vortex sheet - a model for streak instability

The linear inviscid instability of an infinitely thin vortex sheet, periodically corrugated with finite amplitude along the spanwise direction, is investigated analytically. Two types of corrugations are studied, one of which includes the presence of an impermeable wall. Exact eigensolutions are found in the limits of very long and of very short wavelengths. The intermediate-wavenumber range is explored by means of a second-order asymptotic series and by limited numerical integration. The sheets are unstable to both sinuous and varicose disturbances. The former are generally found to be more unstable, although the difference only appears for finite wavelengths. The effect of the corrugation is shown to be stabilizing, although in the wall-bounded sheet the effect is partly compensated by the increase in the distance from the wall. The controlling parameter in that case appears to be the minimum separation from the sheet valley to the wall. The instability is traced to a pair of oblique Kelvin–Helmholtz waves in the flat-sheet limit, but the eigenfunctions change character both as the corrugation is made sharper and as the wall is approached, becoming localized near the crests and valleys of the corrugation. The study is motivated by the desire to understand the behaviour of lifted low-speed streaks in wall-bounded flows, and it is shown that the spatial structure of the fundamental sinuous eigenmode is remarkably similar to previously known three-dimensional nonlinear equilibrium solutions in both plane Couette and Poiseuille flows.

Homoclinic tangle on the edge of shear turbulence.

Experiments and simulations lend mounting evidence for the edge state hypothesis on subcritical transition to turbulence, which asserts that simple states of fluid motion mediate between laminar and turbulent shear flow as their stable manifolds separate the two in state space. In this Letter we describe flows homoclinic to a time-periodic edge state that display the essential properties of turbulent bursting. During a burst, vortical structures and the associated energy dissipation are highly localized near the wall, in contrast with the familiar regeneration cycle.