Cascade model for particle concentration and enstrophy in fully developed turbulence with mass-loading feedback

Abstract

A cascade model is described based on multiplier distributions determined from three-dimensional (3D) direct numerical simulations (DNS) of turbulent particle laden flows, which include two-way coupling between the phases at global mass loadings equal to unity. The governing Eulerian equations are solved using psuedospectral methods on up to 512(3) computional grid points. DNS results for particle concentration and enstrophy at Taylor microscale Reynolds numbers in the range 34-170 were used to directly determine multiplier distributions on spatial scales three times the Kolmogorov length scale. The multiplier probability distribution functions (PDFs) are well characterized by the beta distribution function. The width of the PDFs, which is a measure of intermittency, decreases with increasing mass loading within the local region where the multipliers are measured. The functional form of this dependence is not sensitive to Reynolds numbers in the range considered. A partition correlation probability is included in the cascade model to account for the observed spatial anticorrelation between particle concentration and enstrophy. Joint probability distribution functions of concentration and enstrophy generated using the cascade model are shown to be in excellent agreement with those derived directly from our 3D simulations. Probabilities predicted by the cascade model are presented at Reynolds numbers well beyond what is achievable by direct simulation. These results clearly indicate that particle mass loading significantly reduces the probabilities of high particle concentration and enstrophy relative to those resulting from unloaded runs. Particle mass density appears to reach a limit at around 100 times the gas density. This approach has promise for significant computational savings in certain applications.

Turbulent Clustering of Protoplanetary Dust and Planetesimal Formation

“…The back-reaction cannot be neglected in regions with C 10 3 , because the local mass loading factor Φ m is much larger than one (assuming an average dust-to-gas ratio of 0.01). Therefore, the back-reaction may significantly affect the high tails of P (C) (Hogan & Cuzzi 2007). This will be discussed in more detail in Section 4.5.…”

“…In the model, a flow parcel breaks up into two equal-sized subdivisions in each cascade step, and the partitioning of the particle concentration and the flow enstrophy in the two subdivisions is controlled by a probability distribution, called the multiplier PDF. Hogan & Cuzzi (2007) computed the multiplier PDF for the step from 3η to 1.5η in their simulations, and found that the multiplier PDF becomes narrower as the mass loading factor, Φ m , exceeds ∼10. When Φ m becomes larger than ∼100, the multiplier PDF is essentially a delta function, meaning that the bifurcation of the particle density stops in these highly loaded regions.…”

“…Hogan & Cuzzi (2007) studied the back-reaction effect on the concentration PDF for particles with St = 1. They built up a model assuming that the development of the fluctuations in the particle concentration and the flow enstrophy (defined as vorticity squared, ω 2 ) can be described as a joint cascade process.…”

“…Therefore the prediction of the probability requires understanding of turbulent clustering at inertial-range scales, which, however, has not been well explored. In their quantitative predictions for that probability, Chambers (2010) and Cuzzi et al (2010) made use of the cascade mode for the joint PDF of the particle concentration and the flow enstrophy developed by Hogan & Cuzzi (2007).…”

“…The prediction of the cascade model depends on the multiplier PDF that controls each cascade step (Section 4.5). The multiplier PDF used in C08 and the follow-up studies was measured from a cascade step in the dissipation range, from 3η to 1.5η (see Hogan & Cuzzi 2007). This multiplier PDF was assumed to apply to all cascade steps including those in the inertial range.…”