Analytical Solution of the Equation for the Probability Density Function of a Scalar in Decaying Grid-Generated Turbulence with a Uniform Mean Scalar Gradient

Abstract

An asymptotic self-similar, closed-form solution of the equation for the one-point probability density function (pdf) of a passive scalar in decaying grid-generated turbulence with a uniform mean cross-stream scalar gradient is obtained. The present solution generalizes the well-known asymptotic (time t → ∞) closed-form solution of Y.Sinai and V.Yakhot (Phys. Rev. Lett. 63, 1962 (1989)) for the scalar pdf equation for homogeneous turbulence with no mean gradient. It is shown that when the conditional expectation of transverse velocity is a linear function of a scalar (experimental data of K.S.Venkataramany and R.Chevray (J.Fluid Mech. 86, 513(1978)) indicate that this is a reasonable assumption) the present solution is solely a function of the conditional expectation of scalar dissipation and looks exactly the same as the asymptotic solution for the case with no mean scalar gradient. This result explains why the pdf’s measured by Jayesh and Z.Warhaft (Phys. Fluids A 4, 2292 (1992)) in a decaying grid-generated turbulence with a uniform scalar mean gradient are accurately represented by the asymptotic solution of Y.Sinai and V.Yakhot if the measured conditional expectation of scalar dissipation is substituted in the latter solution. An exact expression that relates the conditional expectation of molecular diffusion to the conditional expectation of transverse velocity is derived. This expression generalizes the result of L.Valiño et al. (Phys. Rev. Lett. 72, 3518 (1994)) for the conditional molecular diffusion in the absence of a mean scalar gradient. The case of the temperature fluctuations in grid-generated turbulence under the action of a stable (negatively buyoant), linear, mean temperature profile is briefly studied. Here, the results are less definite because the strict self-similar solution of the equation does not exist. The analysis show, however, that approximate (quasi) self-similar solution can be obtained for the above case.