Mitsuaki Funakoshi

Reflection of Obliquely Incident Solitary Waves

Initial-value problem of two-dimensional reflection of solitary waves in shallow water is studied with a finite-difference method. If \(\alpha{>}\sqrt{3\varepsilon} (\alpha{=}\)the angle of incidence, e = a / h , a =wave amplitude, h =undisturbed depth), the solution represents a regular reflection pattern which is composed of incident and reflected waves. On the other hand, if \(\alpha{<}\sqrt{3\varepsilon}\), a Mach reflection pattern which consists of three obliquely oriented solitary waves is obtained as an asymptotic solution. This solution agrees very well with the resonantly interacting solitary wave solution predicted by Miles. The run-up at the wall is considerably larger than that in linearized theory when α is close to \(\sqrt{3\varepsilon}\), and this result is consistent with that predicted by Miles. It requires the time of order \(100h/\sqrt{ag}(g{=}\)gravitational acceleration) to reach stationary Mach reflection patterns when e =0.05.

SURFACE WAVES DUE TO RESONANT HORIZONTAL OSCILLATION

Experiments on surface waves were made using a cylindrical container oscillated horizontally with the period T close to that associated with the two known degenerate modes. Outside a certain region in the ( T, x 0 )-plane, where x 0 is the amplitude of the forcing displacement, surface waves exhibit either of the two kinds of regular motions whose amplitudes are constant. Within this region, however, the wave amplitude slowly changes, expressing the irregular or periodic motion of surface waves. In order to analyse these motions in detail, the slow evolution of four variables associated with the amplitudes and phases of the two modes is computed from the free-surface displacement at two measuring points. It is shown that the most common attractor corresponding to the irregular wave motion is the strange attractor with a positive maximum Liapunov exponent and a correlation dimension of 2.1–2.4. Furthermore, another kind of chaotic attractor and a few periodic orbits are found in a small parametric region. The route to chaos associated with period-doubling bifurcation is also observed. The above experimental results are compared with the solutions to weakly nonlinear evolution equations derived by Miles. We find that the equations can explain well many of the experimental results on regular and irregular wave motions. In particular, the most common chaotic attractors both in the experiments and in the theory have similar shapes in a phase space, and also yield similar values for maximum Liapunov exponents and correlation dimensions.

Long Internal Waves of Large Amplitude in a Two-Layer Fluid

Long internal waves of large amplitude are studied for a two-layer fluid with a rigid upper boundary. Steady periodic waves are calculated numerically assuming that mean velocities for each layer are equal. In the limit of long wavelength, the waves asymptote to either of a non-dissipative internal bore or a solitary wave of elevation or depression. The conditions for the appearance of each solution are obtained numerically, and one of the conditions is complemented by an analysis based on the conservations of mass, momentum and wave energy. Then it is shown that the interface for the solitary wave of large amplitude is not allowed to cross a critical level similarly to the case of small amplitude. Here the critical level is defined by the distance \(\sqrt{\Delta}h/(1+\sqrt{\Delta})\) from the bottom, and h is the interval between two boundaries, Δ=ρ 2 /ρ 1 and ρ 1 and ρ 2 are densities of upper and lower layers. This result gives a criterion for the largest amplitude of a solitary wave.

Two-Dimensional Resonant Interaction between Long and Short Waves

Two-dimensional resonant interaction between an internal gravity wave and a surface gravity wave packet in a two-layer fluid is investigated. The equations describing this interaction are derived. Modulational instability of a plane wave solution of the quations is discussed. Some interesting solutions which follow from a two-soliton solution to these equations are given.

Chaotic mixing due to a spatially periodic three-dimensional flow

The chaotic mixing of a fluid due to a slow flow in a spatially periodic system called the partitioned-pipe mixer is studied. This system, originally composed of alternate horizontal and vertical plates of the same length in a duct, is generalized so that both the ratio a of the lengths of these plates and the angle between neighboring plates can be changed. Using the Poincare plots of the locations of fluid particles after every period, we find that the mixing performance in many periods can be improved to a considerable extent by choosing appropriate values of a and . Furthermore, it is shown that the mixing performance in a few periods can be estimated from the distribution of the lines of separation, defined as the set of cross-sectional initial locations of fluid particles which move to one of the leading edges of the plates within a specified period. Using this distribution, we find that this mixing performance also can be improved by the above generalization.

Evolution of a vortex street in the far wake of a cylinder

The evolution of a primary vortex street shed from a circular cylinder in the far wake is experimentally examined for 70 < R < 154 (R is the Reynolds number). According to the vorticity fields obtained using digital image processing for visualized flow fields, the primary vortex street breaks down into a nearly parallel shear flow of Gaussian profile at a certain downstream distance, before a secondary vortex street of larger scale appears further downstream. The process leading to the nearly parallel flow can be explained as the evolution of the vortex regions of an inviscid fluid if we invoke the observation that the distance between the two rows in the primary vortex street increases with the downstream distance, although the viscous effect probably contributes to this increase. Numerical computations with the discrete vortex method also support this explanation. The wavelengths and speeds of the primary and secondary vortex street are also measured.

Stability of Flow between Eccentric Rotating Cylinders

The linear stability of two-dimensional steady flows between two long, eccentric, rotating circular cylinders is studied numerically under the condition that the inner cylinder rotates uniformly while the outer one is at rest. By using the pseudospectral method it is found that the critical Reynolds number increases with the eccentricity e . The critical axial wave number is found to remain nearly constant for small e and to increase with larger e . The eigenfunctions are distributed in the region from the position of the maximum gap to 180° downstream of that position. The Taylor-vortexlike three-dimensional steady flows are computed for several supercritical Reynolds numbers. The torques acting on the cylinders and the position of maximum vortex activity are calculated.

Long Internal Waves in a Two-Layer Fluid

Steady long periodic internal waves of small amplitude in a two-layer fluid with a rigid upper boundary are examined under the assumption of no difference between mean horizontal velocities in each layer. When the depth ratio in quiescent state is close to a certain value σ c determined by density ratio, the Korteweg-de Vries equation does not describe well such waves. For this case an equation including both quadratic and cubic nonlinear terms as well as dispersive term is derived. By taking the long-wave limit of periodic solutions to this equation, a shock like solution as well as a solitary wave solution of elevation or depression is obtained. One of the depth ratios at two flat regions for the shock like solution is larger than σ c , and the other is smaller than σ c . The velocity of this solution is independent of its amplitude. The amplitude of the solitary wave solution has a certain upper limit.

Chaotic motion of fluid particles around a rotating elliptic vortex in a linear shear flow

Abstract The motion of fluid particles around a rotating elliptic vortex in an external irrotational linear shear flow is examined both numerically and analytically. When the strain rate of the external flow, s, is small, fluid particles move chaotically only within two narrow regions. These regions are near the heteroclinic orbits of the Poincare map of particle locations after every vortex rotation period in two special flows obtained in the limit s → 0 . The existence of these chaotic regions is confirmed by computing the Melnikov function for perturbed systems with small s. The widths of these chaotic regions are also estimated using this function. As s increases, these chaotic regions become larger and finally merge at s = s c . For s>sc, fluid particles near the vortex can move toward infinity. This critical value sc is larger for a vortex closer to a circular shape. By examining the residence time of fluid particles near the vortex for s>sc, we find its sensitive and complicated dependence on the particle initial positions.

Orbital instability and chaos in the Stokes flow between two eccentric cylinders

Abstract The motion of fluid particles due to the slow flow between two eccentric cylinders rotating alternately is examined experimentally and numerically. In ‘return experiments’ composed of alternate rotations of the cylinders by N periods and their time reversal, the dye starting from one region almost returns to its initial position even for large N, whereas the deviation of the dye starting from the other region from its initial position is large and rapidly increases with N. These two regions correspond to the regular and chaotic regions in the numerically computed Poincare plot for the alternate rotations of the cylinders. These results suggest the significance of orbital instability in the chaotic region in the experiments with unavoidable inperfections. A part of the experimental results can be explained qualitatively using a loccl Lyapunov exponent (L.L.E.) for finite evolution time. The importance of the stagnation point of the flow due to the rotation of a cylinder in the orbital instability is also suggested using the L.L.E.