This paper considers the breakup of liquid drops at the large Weber number within the framework of Kolmogorov’s scenario of breakup. The population balances equation for droplet radius distribution is written to be an invariant under the group of scaling transformations. It is shown that due to this symmetry, the long-time limit solution of this equation is a power function. When the standard deviation of droplet radius strongly increases and, consequently, the characteristic length scale disappears, the power asymptotic solution can be viewed as a further evolution of Kolmogorov’s log-normal distribution. This new universality appears to be consistent with the experimental observation of fractal properties of droplets produced by air-blast breakup. The scaling properties of Kolmogorov’s model at later times are also demonstrated in the case where the breakup frequency is a power function of instantaneous radius. The model completes the Liouville equation for distribution function of liquid particles in the phase space of droplet position, velocity, and radius. The numerical scheme is proposed for stochastic modeling of droplets production. Lagrangian simulation of the spray under air-blast atomization is performed using KIVA II code, which is a frequently used code for computation of turbulent flows with sprays. The qualitative agreement of simulation with measurements is demonstrated.

This paper analyses statistical universalities that arise over time during constant frequency fragmentation under scaling symmetry. The explicit expression of particle-size distribution obtained from the evolution kinetic equation shows that, with increasing time, the initial distribution tends to the ultimate steady-state delta function through at least two intermediate universal asymptotics. The earlier asymptotic is the well-known log-normal distribution of Kolmogorov (1941 Dokl. Akad. Nauk. SSSR 31 99–101). This distribution is the first universality and has two parameters: the first and the second logarithmic moments of the fragmentation intensity spectrum. The later asymptotic is a power function (stronger universality) with a single parameter that is given by the ratio of the first two logarithmic moments. At large times, the first universality implies that the evolution equation can be reduced exactly to the Fokker–Planck equation instead of making the widely used but inconsistent assumption about the smallness of higher than second order moments. At even larger times, the second universality shows evolution towards a fractal state with dimension identified as a measure of the fracture resistance of the medium.

On the basis of a recently discovered collision group [V. L. Saveliev, in Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. Gallis, AIP Conf. Proc. No. 585 (AIP, Melville, NY, 2001), p. 101], the Boltzmann collision integral is exactly rewritten in two parts. The first part describes the scattering of particles with small angles. In this part the infinity due to the infinite cross sections is extracted from the Boltzmann collision integral. Moreover, the Boltzmann collision integral is represented as a divergence of the flow in velocity space. Owing to this, the role of collisions in the kinetic equation can be interpreted in terms of the nonlocal friction force that depends on the distribution function.

On the basis of the Euler equation and its symmetry properties, this paper proposes a model of stationary homogeneous developed turbulence. A regularized averaging formula for the product of two fields is obtained. An equation for the averaged turbulent velocity field is derived from the Navier-Stokes equation by renormalization-group transformation.