Conditional Scalar Dissipation Rate Research Articles

Quasistationary probability density functions in the turbulent mixing of a scalar field.

The assumption of the existence of an asymptotic shape when t\ensuremath{\rightarrow}\ensuremath{\infty} for the probability density function (PDF) of an inert scalar field undergoing turbulent convection and molecular transport [Y. G. Sinai and V. Yakhot, Phys. Rev. Lett. 63, 1962 (1989)] is generalized to include the time evolution. Statistically quasistationary PDFs and conditional scalar dissipation rates compare surprisingly well with direct numerical simulation data at different times. Closed form PDFs are parametrized in terms of the flatness factor solely.

Conditional scalar dissipation rates in turbulent wakes, jets, and boundary layers

The expected value Eχ≡E(χ,θ=θ0) of the dissipation rate χ of a passive scalar θ conditioned on the scalar value θ=θ0 has been measured in three varieties of turbulent shear flows: heated wakes, dyed liquid jets, and the atmospheric surface layer. The quantity Eχ depends fairly strongly on θ0 and on the flow. For the wake, Eχ exhibits two peaks—one on the low-temperature end and the other on the high-temperature end—and the peaks are separated by an approximately flat region. The relative strength of the two peaks varies with the spatial position. Measured in the turbulent part alone, Eχ tends to have only one peak on the hot side, but is still nonuniform. The related quantity, Eθ″≡(∇2θ,θ=θ0), which is the expected value of the Laplacian of the scalar conditioned on the scalar concentration, has also been measured on the wake centerline and shows a simpler dependence on θ0 than Eχ. For jets, Eχ has a single peak on the high-concentration side. This feature appears to be essentially independent of the use of Taylor’s hypothesis and on whether or not the dissipation rate χ is approximated by only one of its components. It is, however, sensitive to the resolution of measurement. For the temperature fluctuation in the atmospheric boundary layer, the peak in Eχ on the cold side is far weaker than that on the hot side. From this combination of experiments, it is argued that the different shapes of Eχ in different flows are related to differences in the nature of the scalar pdf itself and, for the high-Schmidt-number dyes in water flows, on whether or not the finest scales of the scalar are resolved.

Probability distribution, conditional dissipation, and transport of passive temperature fluctuations in grid-generated turbulence

The evolution of the scalar probability density function (pdf), the conditional scalar dissipation rate, and other statistics including transport properties are studied for passive temperature fluctuations in decaying grid-generated turbulence. The effect of filtering and differentiating the time series is also investigated. For a nonzero mean temperature gradient it is shown that the pdf of the temperature fluctuations has pronounced exponential tails for turbulence Reynolds number (Rel) greater than 70 but below this value the pdf is close to Gaussian. The scalar dissipation rate, conditioned on the fluctuations, shows that there is a high expectation of dissipation in the presence of the large, rare fluctuations that produce the exponential tails. Significant positive correlation between the mean square scalar fluctuations and the instantaneous scalar dissipation rate is found when exponential tails occur. The case of temperature fluctuations in the absence of a mean gradient is also studied. Here, the results are less definite because the generation of the fluctuations (by means of fine heated wires) causes an asymmetry in the pdf. The results show, however, that the pdf is close to Gaussian and that the correlation between the mean square temperature fluctuations and the instantaneous scalar dissipation rate is very weak. For the linear profile case, measurements over the range 60≤Rel≤1100 show that the dimensionless heat flux Nu is proportional to Rel0.88 and that the transition from a Gaussian pdf to one with exponential tails occurs at Nu∼31, a value close to transitions observed in other recent mixing experiments conducted in entirely different turbulent flows.

A Generalized Conditional Scalar Dissipation-Mixture Fraction Joint PDF for Flamelet Modeling of Non-Premixed Flames

Abstract One problem in flamelet modeling of turbulent non-premixed flames is that of formulating a scalar dissipation rate-mixture fraction joint probability density function (pdf). While the individual flamelet calculation provides a conditional scalar dissipation rate, the turbulent flame calculation provides a marginal scalar dissipation rate. Thus, linking the laminar and turbulent flame calculations through the joint pdf is not straightforward, and, for simplicity, it is often assumed that the marginal and conditional probability densities are equal. The present paper assesses the validity of this assumption and examines its consequences in a numerical example. A conditional, scalar dissipation rate-mixture fraction joint pdf is developed which is based upon numerical solutions of the laminar counterflow diffusion flame and functional relationships among the scalar dissipation rate, the conditional scalar dissipation rate, and the mixture fraction. Interestingly, the result, when applied to a turbulent non-premixed flame, results in predictions of NO concentration which are not significantly different. The analysis also suggests experiments which may provide insight into flamelet structure, thereby assessing the validity of the idealized, underlying laminar counterflow configuration used in the flamelet modeling of non-premixed flames.

Direct numerical simulations of the turbulent mixing of a passive scalar

The evolution of scalar fields, of different initial integral length scales, in statistically stationary, homogeneous, isotropic turbulence is studied. The initial scalar fields conform, approximately, to ‘‘double-delta function’’ probability density functions (pdf ’s). The initial scalar-to-velocity integral length-scale ratio is found to influence the rate of the subsequent evolution of the scalar fields, in accord with experimental observations of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. On the other hand, the pdf of the scalar is found to evolve in a similar fashion for all the scalar fields studied; and, as expected, it tends to a Gaussian. The pdf of the logarithm of the scalar-dissipation rate reaches an approximately Gaussian self-similar state. The scalar-dissipation spectrum function also becomes self-similar. The evolution of the conditional scalar-dissipation rate is also studied. The consequences of these results for closure models for the scalar pdf equation are discussed.