# Anomalous scalings for fluctuations of inertial particles concentration and large-scale dynamics Academic Article

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### abstract

• Small-scale fluctuations and mean-field dynamics of the number density of inertial particles in turbulent fluid flow are studied. Anomalous scaling for the second-order correlation function of the number density of inertial particles is found. The mechanism for the anomalous scaling is associated with the inertia of particles that results in a divergent velocity field of particles. The anomalous scaling appears already in the second moment when the degree of compressibility $\ensuremath{\sigma}>1/27$ (where $\ensuremath{\sigma}$ is the ratio of the energies in the compressible and the incompressible components of the particles velocity). The $\ensuremath{\delta}$-correlated in time random process is used to describe a turbulent velocity field. However, the results remain valid also for the velocity field with a finite correlation time, if all moments of the number density of the particles vary slowly in comparison with the correlation time of the turbulent velocity field. The mechanism of formation of large-scale inhomogeneous structures in spatial distribution of inertial particles advected by a low-Mach-number compressible turbulent fluid flow with a nonzero mean temperature gradient is discussed as well. The effect of inertia causes an additional nondiffusive turbulent flux of particles that is proportional to the mean temperature gradient. Inertial particles are concentrated in the vicinity of the minimum (or maximum) of the mean temperature of the surrounding fluid depending on the ratio of the material particle density to that of the surrounding fluid. The equation for the turbulent flux of particles advected by a low-Mach-number compressible turbulent fluid flow is derived. The large-scale dynamics of inertial particles is studied by considering the stability of the equilibrium solution of the derived equation for the mean number density of the particles. A modified Rayleigh-Ritz variational method is used for the analysis of the large-scale instability.

### publication date

• September 1, 1998