Sadayoshi Toh

Pulse interactions in an unstable dissipative‐dispersive nonlinear system

An attempt is made to understand several features of the wave evolutions in an unstable dissipative‐dispersive nonlinear system in terms of the interactions of localized solitonlike pulses. It is found that the wave evolutions can be qualitatively well described by weak interactions of pulses, each of which is the steady solution to the original evolution equation. The oscillatory structure of a tail of the pulse for weakly dispersive cases is responsible for the existence of bound states of pulses, which explains the numerical result that the interpulse distances in the initial value problem take certain fixed values or values in the definite regions. In cases of monotone tails for strongly dispersive cases, the effects of pulse interactions become repulsive, which explains the result that the pulses asymptotically tend to be arranged periodically, adjusting to the periodic boundary conditions in the numerical simulation.

Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?

We have examined bursting processes observed in turbulent channel flow by direct numerical simulation of the incompressible Navier-Stokes equations in a minimal flow unit. A traveling wave solution (TWS) is obtained by a shooting method. The TWS corresponds to a saddle point in a two-dimensional phase space. A low-dimensional dynamics confined to the near-wall region is proposed in terms of the TWS and its manifolds. The characteristics of the coherent structures constituting the TWS are investigated dynamically and statistically in detail. The dynamics well describes an elementary process of intermittent turbulent regeneration in wall turbulence.The instability of a streak and its nonlinear evolution are investigated by direct numerical simulation (DNS) for plane Poiseuille flow at Re=3000. It is suggested that there exists a traveling-wave solution (TWS). The TWS is localized around one of the two walls and notably resemble to the coherent structures observed in experiments and DNS so far. The phase space structure around this TWS is similar to a saddle point. Since the stable manifold of this TWS is extended close to the quasi two dimensional (Q2D) energy axis, the approaching process toward the TWS along the stable manifold is approximately described as the instability of the streak (Q2D flow) and the succeeding nonlinear evolution. Bursting corresponds to the escape from the TWS along the unstable manifold. These manifolds constitute part of basin boundary of the turbulent state.

A periodic-like solution in channel flow

We search channel flow for unsteady solutions for different Reynolds numbers and configurations by extending a shooting method which was previously used to obtain a travelling-wave solution. A general initial condition is considered. A periodic-like solution to the incompressible Navier–Stokes equations in a minimal flow unit is found. One cycle of the solution consists of two typical intervals: a single-streak period and a double-streak period. The solution seems to be periodic; however, it cannot be distinguished from a heteroclinic cycle which consists of two heteroclinic orbits connecting two single-streak solutions, because the solution is tracked only for one and half periods.

Interaction between a large-scale structure and near-wall structures in channel flow

Direct numerical simulation of a turbulent channel flow in a periodic domain of relatively wide spanwise extent, but minimal streamwise length, is carried out at Reynolds numbers

Cylindrical quasi-solitons of the Zakharov-Kuznetsov equation

\Rey_\tau\,{=}\,137

Statistical Model with Localized Structures Describing the Spatio-Temporal Chaos of Kuramoto-Sivashinsky Equation

and 349. The large-scale structures previously observed in studies of turbulent channel flow using huge computational domains are also shown to exist even in the streamwise-minimal channels of the present study. Moreover, the limitation of the streamwise length of the domain enforces the interaction between large-scale structures and near-wall structures, which consequently makes it tractable to extract a simple cycle of processes sustaining the structures in the present channel flow. It is shown that the large-scale structures are generated by the collective behaviour of near-wall structures and that the generation of the latter is in turn enhanced by the large-scale structures. Hence, near-wall and large-scale structures interact in a co-supporting cycle.

On the Stability of Soliton-Like Pulses in a Nonlinear Dispersive System with Instability and Dissipation

Abstract Evolutions and interactions of two-dimensional solitary waves of the Zakharov-Kuznetsov equation are investigated numerically. Formations of cylindrical bell-shaped pulses are observed in the initial value problems. A single bell-shaped pulse propagates stably without any deformation like a soliton. Two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision. After a collision of two pulses, the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples, so that the collision process is slightly inelastic. Generated ripples are very small for center-to-center collision of similar pulses. The cylindrically symmetric pulses of the Zakharov-Kuznetov equation are thus found to behave approximately soliton-like. Some properties of collision process are interpreted in terms of the conservation laws.

Interactions of two-dimensionally localized pulses of the regularized-long-wave equation

Statistical properties of the chaos of the Kuramoto-Sivashinsky equation are investigated numerically and theoretically. It is found that the chaos consists of spatially localized structures (pulses) and the distances between adjacent pulses have the distribution which is localized around a single peak through fate mechanism of creation and annihilation of pulses. The energy spectrum is calculated by a statistical model in which the pulses with a fixed shape are lined up in the way that each distance is independent of others. This model reproduces a peak near the wavenumber \(k=1/\sqrt{2}\) as well as the flat part near k =0 in the energy spectrum. The linear dependence of the amount of chaos on the system parameter is discussed with this model.

Enstrophy and momentum fluxes in two-dimensional shear flow turbulence

The stability of various equilibrium solutions of a strongly dispersive nonlinear system with instability and dissipation is investigated both numerically and analytically. Periodic trains of soliton-like pulses are found to be stable when the distance between adjacent pulses becomes smaller than a critical value. This critical value is determined by linear stability analysis. A modulational type instability is also observed for a very long string of soliton-like pulses even when the fundamental distance is within the stable regime.

An approximate equation for nonlinear cross-field instability

Abstract Interactions between positive and/or negative cylindrical bell-shaped pulses are investigated numerically for a two-dimensional regularized-long-wave equation. Overtaking collisions of two same-signed pulses are found to be fairly elastic. Head-on collisions between positive and negative pulses are inelastic and their overall features are analogous to those in the one-dimentional case. Some properties of collision processes are discussed in terms of the conversed quantities.