2 results
The pollution of pristine material in compressible turbulence
- Liubin Pan, Evan Scannapieco, John Scalo
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- Journal:
- Journal of Fluid Mechanics / Volume 700 / 10 June 2012
- Published online by Cambridge University Press:
- 01 May 2012, pp. 459-489
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The first generation of stars had very different properties than later stellar generations, as they formed from a ‘pristine’ gas that was completely free of heavy elements. Normal star formation took place only after the first stars had polluted the surrounding turbulent interstellar gas, increasing its local heavy-element mass concentration, , beyond a ‘critical’ threshold value, (). Motivated by this astrophysical problem, we investigate the fundamental physics of the pollution of pristine fluid elements in statistically homogeneous and isotropic compressible turbulence. Turbulence stretches the pollutants, produces concentration structures at small scales, and brings the pollutants and the unpolluted flow in closer contact. The pristine material is polluted when exposed to the pollutant sources or the fluid elements polluted by previous mixing events. Our theoretical approach employs the probability distribution function (p.d.f.) method for turbulent mixing, as the fraction of pristine mass corresponds to the low tail of the density-weighted concentration p.d.f. We adopt a number of p.d.f. closure models and derive evolution equations for the pristine fraction from the models. To test and constrain the prediction of theoretical models, we conduct numerical simulations for decaying passive scalars in isothermal turbulent flows with Mach numbers of 0.9 and 6.2, and compute the mass fraction, , of the flow with . In the Mach 0.9 flow, the evolution of is well-described by a continuous convolution model and goes as , if the mass fraction of the polluted flow is larger than If the initial pollutant fraction is smaller than an early phase exists during which the pristine fraction follows an equation derived from a nonlinear integral model: . The time scales and are measured from our simulations. When normalized to the flow dynamical time, the decay of in the Mach 6.2 flow is slower than at Mach 0.9 because the time scale for scalar variance decay is slightly larger and the low tail of the concentration p.d.f. broadens with increasing Mach number. We show that in the Mach 6.2 flow can be well fitted using a formula from a generalized version of the self-convolution model.
Relative velocity of inertial particles in turbulent flows
- LIUBIN PAN, PAOLO PADOAN
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- Journal:
- Journal of Fluid Mechanics / Volume 661 / 25 October 2010
- Published online by Cambridge University Press:
- 27 July 2010, pp. 73-107
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We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.