A review is given on the similarity laws of wind waves and related problems, with special reference to our recent studies. Growing wind waves have statistical similarity laws. One of these is expressed as the 3/2-power law. Experimental evidence indicates that the boundary layers above and below wind waves are both turbulent boundary layers, which are similar to those on rough solid walls. Combination of the characteristics of turbulent boundary layers of air and water with the 3/2-power law requires that several characteristic velocities, which are concerned with the air-water boundary processes including wind waves, are all proportional to one another. Experimental observations are consistent with the above reasoning, together with the empirical value of the coefficient of the 3/2-power law. An interpretation of the physical situation is given as a concept of the breaking-adjustment of wind waves. The relation between the power law and the form of energy spectra of waves, and practical applications of the similarity laws are also given.

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There are two methods available to predict gas-particle two-phase flows : Eulerian and Lagrangian approaches. In the Eulerian method, various turbulence closure models are proposed in the same manner as in the single phase flow. In the Lagrangian method, particle-to-wall collision is important in the simulation, and irregular bouncing should be taken into consideration in horizontal wall-bounded flows. Interaction between particles and fluid is described by the PSI-Cell model.

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In the previous paper the authors have observed the formation of spoon-shaped vortex chain in a wake behind a circular cylinder as a coherent structure in turbulence. In this report numerical simulation is carried out based on the assumption that the structure is formed by deformation of the Kármán vortices. Basic equation is the localized induction equation for a single vortex filament with an influence of the background mean flow. The vortex filament is given an initial deformation within a plane at an angleθ to x-z plane (x is the mean flow direction and z the spanwise direction) with the widthZw, and the further deformation process of the filament is numerically traced. The first calculation is made with fixed Zw, and various values of θ. The result shows that the vortex filament finally reaches a structure lying on a plane with a constant angle of 30°-45° to x-z plane irrespective of the initial values of θ. The second calculation is made with fixed θ and various values of Zw. In this case the final spanwise scale of the deformed region of the filament has an almost constant values of about 4d-6d (d is diameter of the cylinder). These results indicate that the final structure of the vortex filament is stable and definite irrespective of the initial disturbances.

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In the previous paper the authors have reported that the two-dimensional Kármán vortices behind a circular cylinder are deformed, until they form chains of spoon-shaped vortex couples whose spanwise scale is about 8d, which is a new type of coherent structure. In this report experimental evidence of this structure is presented. Formation process of the structure and the turbulence in it were investigated for the wake behind a circular cylinder with Re=2100 and 4200 by means of the flow visualization technique, simultaneous hot wire measurements, spanwise correlation measurements, construction of instantaneous velocity field by the conditional sampling method, etc.

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The methods of obtaining exact solutions for nonlinear diffusion equations are surveyed. Included are linearization technique, bilinear transformation, Painlevé analysis and method of similarity solution. The equations considered are Burgers' type equations, Fisher's type equations, diffusion equations with integral terms and equations with density dependent diffusion. Explicit solutions are also shown for some of the equations.

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Magma is generated by partial melting of rocks and ascends due to its buoyancy in interstitial conduits through the country rock. When the country rock creeps, the magma conduits are deformable. A vertical cylindrical conduit that is deformable with time and space is considered for the present analysis, assuming that both country rock and magma are viscous fluids. Here the fluid representing the country rock has a significantly higher viscosity than magma. This analysis can be generalized to permeable flow with a network of deformable magma paths. The theory gives such a solution that a bulge of magma conduit propagates upward at a constant speed without change of wave form. Such a stationary wave resumes its form after collision with another wave so that it may be called magma soliton. The propagation velocity of a magma soliton increases with increasing amplitude. If magma flux is changed to a higher value at a certain depth, a new state of greater flux is established upward, creating new magma solitons at the moving tip. This process of soliton creation might be applicable for explaining the episodicity of volcanic eruptions.

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