In this paper, we apply a scaling analysis of the maximum of the probability density function（pdf） of velocity increments, i.e., max（） = max（）up p u, for a velocity field of turbulent Rayleigh-Bénard convection obtained at the Taylor-microscale Reynolds number Re60. The scaling exponent is comparable with that of the first-order velocity structure function, （1）, in which the large-scale effect might be constrained, showing the background fluctuations of the velocity field. It is found that the integral time T（x/ D） scales as T（x/ D）（x/ D）, with a scaling exponent =0.25 0.01, suggesting the large-scale inhomogeneity of the flow. Moreover, the pdf scaling exponent （x, z） is strongly inhomogeneous in the x（horizontal） direction. The vertical-direction-averaged pdf scaling exponent （x） obeys a logarithm law with respect to x, the distance from the cell sidewall, with a scaling exponent 0.22 within the velocity boundary layer and 0.28 near the cell sidewall. In the cell＇s central region, （x, z） fluctuates around 0.37, which agrees well with （1） obtained in high-Reynolds-number turbulent flows, implying the same intermittent correction. Moreover, the length of the inertial range represented in decade（）IT x is found to be linearly increasing with the wall distance x with an exponent 0.65 0.05.
High-precision measurements of the Nusselt number Nu for Rayleigh-B6nard （RB） convection have been made in rectangular cells of water （Prandtl number Pr ≈ 5 and 7） with aspect ratios （F~, Fy） varying between （1, 0.3） and （20.8, 6.3）. For each cell the data cover a range of a little over a decade of Rayleigh number Ra and for all cells they jointly span the range 6x105 〈 Ra 〈1011. The two implicit equations of the Grossmann-Lohse （GL） model together with the empirical finite conductivity cor- rection factorf（X） were fitted to obtain estimates of Nu∞ in the presence of perfectly conducting plates, and the obtained Nu∞ is independent of the cells＇ aspect ratios. A combination of two-power-law, Nu∞= O.025Ra0.357＋O.525Ra0.168, can be used to de- scribe Nu∞（Ra）. The fitted exponents 0.357 and 0.168 are respectively close to the predictions 1/3 and 1/5 of the 11μ. and 1Vμ re- gimes of the GL model. Furthermore, a clear transition from the II. regime to the IVμ regime with increasing Ra is revealed.
Self-similar behavior for the multicomponent coagulation system is investigated analytically in this paper. Asymptotic self-similar solutions for the constant kernel, sum kernel, and product kernel are achieved by introduction of different generating functions. In these solutions, two size-scale variables are introduced to characterize the asymptotic distribution of total mass and individual masses. The result of product kernel （gelling kernel） is consistent with the Vigli-Ziff conjecture to some extent. Furthermore, the steady-state solution with injection for the constant kernel is obtained, which is again the product of a normal distribution and the scaling solution for the single variable coagulation.
