Tsutomu Kambe

Nonlinear Dynamics and Chaos in Parametrically Excited Surface Waves

Surface waves in a closed container subject to vertical oscillation are studied. Nonlinear dynamical equations of two nearly degenerate subharmonic modes responding to the external forcing are derived, using the averaged Lagrangian method for slowly varying amplitudes. Stability and bifurcation diagrams are shown for the system with linear damping. Period-doubling bifurcation and chaotic solutions with one positive Lyapunov characteristic exponent are obtained numerically. It is shown that some of the period-doubling bifurcations are related to the symmetry of the dynamical system.

Interaction of Two Vortex Rings Moving along a Common Axis of Symmetry

Interaction of two vortex rings is examined by experiments and numerical analysis. Experimental observations are carried out in a water tank using the electrolysis method and it is found that the interaction patterns are classified into three types. Numerical simulations are performed by integrating the Navier-Stokes equation and the results of the experiments and calculations are found in good agreement. The game of the passing through each other does not happen in this region of low Reynolds number.

Generation and Decay of Viscous Vortex Rings

Decay of a vortex ring in a viscous fluid is discussed by using a solution in the form of an asymptotic expansion for large time t . It is found that the velocity of the vortex ring varies as t -1.5 in the final state of low Reynolds number. The asymptotic expansion is not uniformly valid, and an improvement is made by using the method of matched asymptotic expansions. Generation and development of vortex rings are simulated by numerical integration of the Navier-Stokes equation as an initial and boundary value problem. Time variations of physical quantities such as total energy, impulse and velocity of the vortex rings, etc. are obtained, and have been shown to approach asymptotically to those obtained from the asymptotic expansion. Comparison with experimentally produced vortex rings is also given briefly.

Axisymmetric Vortex Solution of Navier-Stokes Equation

An exact solution of a viscous incompressible flow is presented for a general initial condition. This flow represents an axisymmetric shear layer superimposed on an irrotational straining flow. The solution incorporates the three representative features of vortex motion: stretching, convection and viscous diffusion. In a particular case of constant straining, the flow approaches to a steady state in which the above three effects are in equilibrium. However if the initial state is composed of the same amount of opposite vorticities, the shear layer disappears exponentially in time. Spectral analysis of the solution shows the cascade of vorticity fluctuations to smaller scales. The general solution includes the vortex solutions given by Oseen (1911) and by Burgers (1948), and partly overlaps the similarity solution found by Bellamy-Knights (1970).

A new formulation of equations of compressible fluids by analogy with Maxwell's equations

A compressible ideal fluid is governed by Eulers equation of motion and equations of continuity, entropy and vorticity. This system can be reformulated in a form analogous to that of electromagnetism governed by Maxwells equations with source terms. The vorticity plays the role of magnetic field, while the velocity field plays the part of a vector potential and the enthalpy (of isentropic flows) plays the part of a scalar potential in electromagnetism. The evolution of source terms of fluid Maxwell equations is determined by solving the equations of motion and continuity. The equation of sound waves can be derived from this formulation, where time evolution of the sound source is determined by the equation of motion. The theory of vortex sound of aeroacoustics is included in this formulation. It is remarkable that the forces acting on a point mass moving in a velocity field of an inviscid fluid are analogous in their form to the electric force and Lorentz force in electromagnetism. The significance of the reformulation is interpreted by examples taken from fluid mechanics. This formulation can be extended to viscous fluids without difficulty. The Maxwell-type equations are unchanged by the viscosity effect, although the source terms have additional terms due to viscosities.

Self-intensification in shock wave and vortex interaction

Computer simulation has been performed for the interaction between a shock wave and a vortex ring moving toward the wave. The computed density contours are compared with the pattern of shadowgraphs. A remarkable property found in the simulation is that, during the passage of the shock wave over the vortex ring, the part of the wave propagating through the inside of the ring-vortex is intensified spontaneously at a localized region. Maximum pressure occurs inside the vortex and attains a high value, about several times that of the impinging shock for incident Mach numbers of around 1.2 with the vortex translation Mach number 0.60. This is due to a double-step mechanism of intensification within the flow field by the shock-vortex interaction.

Scattering of Sound by a Vortex Ring

Following the theory of aerodynamic sound scattering, a general formula of the scattered sound pressure is presented for a case of a plane harmonic wave incident upon a flow field localized in space. The expression is valid even when the flow is time-dependent and applied to the scattering by a vortex ring in translational motion. The scattering amplitude and cross-section are determined numerically under the condition of slow motion. They depend not only on the incident wavelength, but also on the direction of the vortex motion with respect to the incident wave. The scattering power of the vortex ring is compared with that of a solid sphere of the same size. In the present analysis, no restriction is imposed about the magnitude of the incident wavelength so far as the scattered wave is a small perturbation to the incident one.

Observation and analysis of scattering interaction between a shock wave and a vortex ring

Interaction of a shock wave with a vortex ring is investigated experimentally and computationally. The experimental observation is made by the shadowgraph method, using a spark light of very short duration of about 20 ns. The shadowgraphs are transformed into digital images by an image processor, and the intensity distributions are processed digitally. Compressive (longitudinal) waves generated (scattered) by the shock-vortex interaction are observed experimentally and compared with a computer simulation. The speeds of the shock wave, vortex and scattered wave obtained from the digitized images are compared with the simulation, and agreement is obtained between them. It is found that the scattered wave is regarded as an acoustic wave whose source is identified at the position and instant of the crossing of the shock wave over the core of the vortex ring.

Acoustic emission from interaction of a vortex ring with a sphere

Acoustic waves emitted by a vortex ring interacting with a fixed solid sphere are studied experimentally and theoretically. The experiments are carried out for two kindsof vortex-sphere arrangement: (A) a vortex ring passes over the sphere, and (B) a vortex ring passes by the sphere. The vortex motion is examined optically by means of a photosensor system, and the pressure signals of the emitted wave are detected by ½-inch microphones in the far field. In case A, the measured diameter of the vortex ring after passing the sphere increases from its initial diameter. The observed acoustic wave is dominated mainly by a dipole emission, and some contribution from a quadrupole radiation is present. In case B, the emitted wave is characterized by a rotating dipole emission in which the dipole axis rotates as the vortex position changes relative to the sphere.

Geodesics and Curvature of a Group of Diffeomorphisms and Motion of an Ideal Fluid

Motion of an ideal fluid is represented as geodesics on the group of all volume-preserving diffeomorphisms. An explicit form of the geodesic equation is presented for the fluid flow on a three-torus Riemannian connection, commutator and curvature tensor are given explicitly and applied to a couple of simple flows with the Beltrami property. It is found that the curvature is non-positive for the section of two ABC flows with different values of the constants (A, B and C). The study is an extension of the Arnolds results (1989) in the two-dimensional case to three-dimensional fluid motions.