This paper brings together recent information on flow development in curved pipes. In the case of curved pipes of circular cross section, the important flow properties are a crossover between shear maxima on the inside and outside of the pipe, a boundary-layer collision, subsequent formation of a separation bubble on the inside wall of the pipe, etc. In the case of curved pipes of square cross section, an additional pair of counter-rotating vortices is found to develop in the inlet region within a definite range of the Dean number.
Attention may specially be called to the occurrence of dual solutions of the Navier-Stokes equations for flow in curved pipes of circular and semi-circular cross sections, where a two-vortex solution bifurcate into a two-and four-vortex solution above some definite Dean number.

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Motions of 1000 point vortices in an infinite fluid region without a solid boundary are computed numerically on the basis of the Biot-Savart law. Three cases for their circulations and initial distributions are treated, one with the same circulations and within a circular domain, one with the same circulations and within a square domain, and one with 500 vortices of positive and 500 vortices of negative circulations within a square domain. The initial distributions are determined by the use of the uniform random numbers. The Runge-Kutta-Gill method is used, because the small scale behavior is the main interest in this work and an accuracy is needed. At each time step of computation the conserved quantities (such as the Hamiltonian), the configuration temperature introduced by Nobikov, the fractal dimension of the point distribution and the enstrophy defined properly are calculated. Computation is made up to 20 time steps. The quantities, which should be conserved, proved to be conserved actually during the computation. The fractal dimension showed a tendency to decrease from 2 to a value near 1. 7. In the third case the configuration temperature continued to increase. The present work shows that the fractal dimension and the configuration temperature, which are geometrical parameters, are affected by the dynamics of the vortex motion.

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The β-model is known to give a clear-cut description of the intermittent structure of fully developed turbulence. By considering the dynamics of the eddy breaking in the β-model theory, Kida [Prog. Theor. Phys. 67 (1982) 1630] showed that in the cascade process of disintegration of eddies the total volume of eddies (active regions) after a disintegration is greater than that before, nevertheless since smaller eddies break down faster, the mean volume occupied by eddies decreases with their size, which is just the characteristics of the intermittent structure of turbulence. Later, Mori and Fujisaka [Prog. Theor. Phys. 68 (1982) 2180] proposed another model, which is similar to the, β-model, to criticise Kida's theory. In their model, however, there are at least two fundamental mistakes found, namely, misunderstanding of the notion of intermittency of turbulence and overlooking of the lifetime of eddies in calculation of the energy spectrum. Here these fatal mistakes are explained by comparing the two theories in the hope that they would reconsider their model.

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A phenomenological model is proposed which explain that in a three dimensional invisid flow field point singularities appear within a finite time where vorticity diverges. The contribution of a point singularity to kinetic energy of flow vanishes and that to enstropy diverges. The model is based on vortex filament approximation of flow. To support the basic idear of the model, deformation and stretching of a straight vortex filament due to mutual interaction with another straight vortex filament is simulated numerically.

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The nonlinear stability of the natural convection in a vertical fluid layer between two flat plates with different temperatures is investigated by a direct method to find the equilibrium states of the secondary convection. We confine ourselves to two-dimensional flows and assume that the aspect ratio of the fluid layer is very large. Since the limit of small Prandtl number is also assumed, buoyancy effect caused by temperature disturbances is negligible. As a result we obtained a neutral surface of the energy of the fundamental mode of the secondary convection. It is concluded that there is no finite amplitude instability below the critical Grashof number derived from linear stability theory, and that both the unstable equilibrium solution (threshold amplitude solution) and the stable equilibrium solution (finite amplitude solution) are found outside the neutral curve of the linear stability. Our results are almost consistent with those of Nagata and Busse (1983), but are more accurate and more thorough.

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Studies of internal gravity waves and their interaction with the general circulation form the recent frontiers of the middle-atmosphere dynamics, a new field opened between meteorology and aeronomy. Theories of weak winds generated and maintained near the mesopause and in the middle stratosphere have triggered international observations, which provide quantitative information on the spectrum of internal gravity waves. However, we have not been connect the observational evidences with the theories, partly because any sounding techniques have each dimensional restrictions. In this article those theoretical and observational studies in Sections 1 and 2, and we notice that some observations suggest a monochromatic feature like simple theoretical models rather than a continuous spectrum. It is hypothesized in Section 3 that a locally-monochromatic wave in slowly-varying media could take a continuous spectrum in an appropriate time-space series. As discussed in Section 4, studies in the future must direct to organizations of many observational techniques and also to establishment of some fluid-dynamical theories in order to clarify any criteria of such a quasi-monochromatic feature of the middle-atmospheric internal gravity waves.

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