A variety of random and intricate patterns seen in nature and many fields in science and technology are now recognized to be fractals. In order to characterize them more unambiguously, the concept of fractals has to be extended from self-similar to self, affine fractals and even to multifractals. Taking the simplest examples possible, their definitions and fundamental properties are discussed, and the importance of clear distinction among them is stressed.

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Oscillation of a nearly circular hydraulic jump in the radial liquid film flow was investigated both experimentally and theoretically. The form of hydraulic jump deformed from the circular shape was obtained by the asymmetric back pressure distribution produced by downstream obstacles. The static and oscillating modes of the hydraulic jump were investigated by means of visualization technique and digital image processing. The oscillation consists of mode 2, 0and the same mode as the original static mode. Frequency of each static mode depends on the deformation degrees from the real circle.

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Compressibility is important in a low-Mach-number flow with a large temperature variation. However, it is well known that time-dependent compressible flow schemes become ineffective at low Mach numbers. This ineffectiveness occurs because the wide disparity exists between the time scales associated with convection and the propagation of acoustic waves. For this reason, scale analysis modifies the compressible equations in order to remove acoustic waves from them. Pressure is divided into the thermodynamic and dynamic parts. The density variation caused by the variation in the dynamic part is neglected. The scale analysis shows the restrictions on the modified equations. Examination of small-amplitude waves shows that the modified equations contain internal gravity waves (buoyant effects) while exclude acoustic waves. A finite difference scheme solves the modified equations. The scheme is essentially the MAC method. First, thermal convection of air in a square cavity is simulated in order to validate the simulation method. The result is in good agreement with a bench mark solution. Secondly, thermal convection is simulated in a vertical pipe furnace used for heat treatment of semiconductor wafers. An axisymmetric steady flow is obtained.

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