%0 Generic %1 Pan2011 %A Pan, Liubin %A Scannapieco, Evan %A Scalo, John %D 2011 %K metal mixing turbulence %T The Pollution of Pristine Material in Compressible Turbulence %U http://arxiv.org/abs/1110.0571 %X Motivated by the astrophysical process of primordial star formation, we study the general problem of the pollution of pristine fluid elements in compressible turbulence. The faction of unpolluted or slightly-polluted fluid mass corresponds to the low tail of the concentration probability distribution function (PDF) with density weighting, and we derive evolution equations for this faction from a number of PDF closure models for turbulent mixing. To test and constrain these predictions, 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, $P(Z_c, t)$, of fluid elements with pollutant concentration below a small threshold, $Z_c$. In the Mach 0.9 flow, the evolution of $P(Z_c, t)$ goes as $P(Z_c, t)= P(Z_c, t) łnP(Z_c, t)/\tau_con$ if the mass fraction of the polluted flow is larger than $0.1.$ If the initial pollutant fraction is smaller than $0.1,$ an early phase exists during which $P(Z_c, t) = P(Z_c, t) P(Z_c, t)-1/\tau_int.$ These equations are obtained from the adopted closure models, and the timescales $\tau_con$ and $\tau_int$, for a continuous convolution model and a nonlinear integral model, respectively, are measured from our simulation data. When normalized to the flow dynamical time, the decay of $P(Z_c, t)$ in the Mach 6.2 flow is slower than at Mach 0.9 because the timescale for scalar variance decay is slightly larger and the low tail of the concentration PDF broadens with increasing Mach number. A modified fitting formula is proposed for $P(Z_c, t)$ in highly supersonic turbulence, which agrees well with the simulation results from the Mach 6.2 flow.

@misc{Pan2011, abstract = { Motivated by the astrophysical process of primordial star formation, we study the general problem of the pollution of pristine fluid elements in compressible turbulence. The faction of unpolluted or slightly-polluted fluid mass corresponds to the low tail of the concentration probability distribution function (PDF) with density weighting, and we derive evolution equations for this faction from a number of PDF closure models for turbulent mixing. To test and constrain these predictions, 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, $P(Z_{\rm c}, t)$, of fluid elements with pollutant concentration below a small threshold, $Z_{\rm c}$. In the Mach 0.9 flow, the evolution of $P(Z_{\rm c}, t)$ goes as $\dot P(Z_{\rm c}, t)= P(Z_{\rm c}, t) \ln[P(Z_{\rm c}, t)]/\tau_{\rm con}$ if the mass fraction of the polluted flow is larger than $\approx 0.1.$ If the initial pollutant fraction is smaller than $\approx 0.1,$ an early phase exists during which $\dot P(Z_{\rm c}, t) = P(Z_{\rm c}, t) [P(Z_{\rm c}, t)-1]/\tau_{\rm int}.$ These equations are obtained from the adopted closure models, and the timescales $\tau_{\rm con}$ and $\tau_{\rm int}$, for a continuous convolution model and a nonlinear integral model, respectively, are measured from our simulation data. When normalized to the flow dynamical time, the decay of $P(Z_{\rm c}, t)$ in the Mach 6.2 flow is slower than at Mach 0.9 because the timescale for scalar variance decay is slightly larger and the low tail of the concentration PDF broadens with increasing Mach number. A modified fitting formula is proposed for $P(Z_{\rm c}, t)$ in highly supersonic turbulence, which agrees well with the simulation results from the Mach 6.2 flow. }, added-at = {2011-10-06T01:17:14.000+0200}, author = {Pan, Liubin and Scannapieco, Evan and Scalo, John}, biburl = {https://www.bibsonomy.org/bibtex/24a91bb7b1f1ca13912cb5f3d21c1e340/miki}, description = {[1110.0571] The Pollution of Pristine Material in Compressible Turbulence}, interhash = {ee3d9399e0c2764d060685f22054c888}, intrahash = {4a91bb7b1f1ca13912cb5f3d21c1e340}, keywords = {metal mixing turbulence}, note = {cite arxiv:1110.0571Comment: 30 pages, 8 figure. Submitted to Journal of Fluid Mechanics}, timestamp = {2011-10-06T01:17:14.000+0200}, title = {The Pollution of Pristine Material in Compressible Turbulence}, url = {http://arxiv.org/abs/1110.0571}, year = 2011 }