Passive scalar conditional statistics in a model of random advection

Abstract

We study numerically a model of random advection of a passive scalar by an incompressible velocity field of different prescribed statistics. Our focus is on the conditional statistics of the passive scalar and specifically on two conditional averages: the averages of the time derivative squared and the second time derivative of the scalar when its fluctuation is at a given value. We find that these two conditional averages can be quite well approximated by polynomials whose coefficients can be expressed in terms of scalar moments and correlations of the scalar with its time derivatives. With the fitted polynomials for the conditional averages, analytical forms for the probability density function ͑pdf͒ of the scalar are obtained. The variation of the coefficients with the parameters of the model result in a change in the pdf. Three different kinds of velocity statistics, ͑i͒ Gaussian, ͑ii͒ exponential, and ͑iii͒ triangular, are studied, and the same qualitative results are found demonstrating that the one-point statistics of the velocity field do not affect the statistical properties of the passive scalar.

Mixing and scalar dissipation rate statistics in a starting gas jet

“…The above observations are corroborated by additional measurements in boundary layers, 15 jets, 15,22,23 opposed flows, 24 and in calculations. 25 The same picture emerges from measurements in non-premixed reacting flows. In these cases, the shape of the pdf of the mixture fraction determines the dependence of the dissipation on the scalar.…”

Intermittency of a passive scalar advected by a quasifrozen velocity field

“…It was found 15 that the passive scalar fluctuation becomes non-Gaussian for a certain range of parameters of the model. Moreover, such a change was shown 16 to be independent of the statistics prescribed for the velocity field. More recently, with the stream function suitably modified, the effects of a a͒ Author to whom correspondence should be addressed; electronic mail: ching@phy.cuhk.edu.hk b͒ Present address: Department of Physics, University of Maryland, College Park, Maryland 20742.…”

“…In this paper, we report our numerical study of the intermittency problem of a passive scalar when the advecting velocity field has a finite correlation time. We use a twodimensional lattice model [15][16][17] in which the stream function generating the incompressible velocity field is modeled by a random Gaussian noise that is identically independently distributed at each lattice point and is updated every certain finite time interval. The velocity field remains the same within the time intervals between the updates and thus has a finite correlation time equals to the time between updates.…”

Conditional statistics for a passive scalar with a mean gradient and intermittency

“…One proposed strategy to obtain such a formula is to fit the conditional dissipation by a polynomial of appropriate degree and substitute that polynomial in the integral above. 5,7,19,20 Mostly quadratic fits were used for the conditional dissipation in those experiments. As noted before, this is indeed the most natural choice for the inner core of the conditional dissipation in our model, in particular at low or moderate turbulence level.…”

“…Understanding the fundamental mechanisms that can potentially lead to such behavior is a question of great theoretical interest and has motivated a number of studies with a passive scalar advected by flows of various complexity. [5][6][7][8][9][10][11][12] That such complex intermittent behavior can be studied unambiguously via judiciously chosen idealized models has been demonstrated in Refs. 13-17 for the case of a decaying scalar at long times, and in Ref.…”