On passive scalar derivative statistics in grid turbulence

Abstract

The probability density function, and related statistics, of scalar (temperature) derivative fluctuations in decaying grid turbulence with an imposed cross-stream, passive linear temperature profile, is studied for a turbulence Reynolds number range, Rel, varying from 50 to 1200, (corresponding to a Taylor Reynolds number range 30<Rλ<130). It is shown that the temperature derivative skewness in the direction of the mean gradient, Sθy has a value of 1.8±0.2 (twice the value observed in shear flows), and has no significant variation with Reynolds number. The ratio of the temperature derivative standard deviation along the gradient to that normal to it is approximately 1.2±0.1 also, with no variation with Re. The kurtosis of the derivatives increases approximately as Re0.2l. The results show that the rare, intense temperature deviations that produce the skewed scalar derivative, increase in frequency, but their area fraction (of the total field) becomes smaller as the Reynolds number increases. Thus, since Sθy remains constant, they become sharper and more intense, occurring deeper in the tails of the probability density function. Measurements in a thermal mixing layer, which has a nonlinear mean temperature profile, are also presented, and these show a similar value of Sθy to the linear profile case. The experiments broadly confirm the two-dimensional numerical simulations of Holzer and Siggia [Phys. Fluids (in press)], as well as other recent simulations, although there are some differences.

Reynolds-number effects and anisotropy in transverse-jet mixing

“…Such scalar plateaux and cliffs in physical space (or, equivalently, ramp-cliffs for timeseries measurements) have previously been observed in experiments on mixing in turbulent shear flows such as wakes (Gibson, Friehe & McConnell 1977), boundary layers (Mestayer et al 1976;Gibson et al 1977), and jets (Uberoi & Singh 1975;Sreenivasan, Antonia & Britz 1979;Antonia et al 1986;Dahm & Dimotakis 1990;Yoda et al 1992). Such ramp-cliff structures have also been found in grid turbulence with a superimposed mean-temperature profile (Budweig, Tavoularis & Corrsin 1985;Thoroddsen & Van Atta 1992;Tong & Warhaft 1994), despite the absence of mean shear, entrainment and large-scale anisotropy of the velocity field. Numerical simulations in two- (Holzer & Siggia 1994) and three- (Pumir 1994) dimensions show the same basic structure of the scalar field in physical space: relatively wellmixed regions separated by cliffs where the scalar gradient is very large.…”

“…There has long been evidence of a lack of scalar isotropy in turbulent mixing flows at both inertial and dissipation scales. A persistent (even at high Reynolds numbers) non-zero skewness of the scalar derivative has been reported in shear flows (Mestayer et al 1976;Gibson et al 1977;Sreenivasan et al 1979) as well as isotropic grid turbulence with a linear temperature profile (Tavoularis & Corrsin 1981;Budwig et al 1985;Tong & Warhaft 1994). The skewness of the scalar derivative is a consequence of the ramp-cliff structures in the scalar field, which can be preferentially aligned in a turbulent flow.…”

“…Experiments (Antonia et al 1986) and simulations (Pumir 1994) indicate that cliffs tend to develop where the flow is hyperbolic (at a diverging separatrix), while elliptic regions are better mixed. Based on experiments in heated grid turbulence, Tong & Warhaft (1994) concluded that the cliffs persist and become even sharper and more intense with increasing Reynolds number up to at least Re λ = 130. In doing so, the cliff-plateau structures contribute deeper and deeper into the tails of the scalar-derivative PDF with increasing Reynolds number.…”

On conditional scalar increment and joint velocity–scalar increment statistics

“…Such a conditional increment PDF shape has been observed previously for dissipationscale separations [11], but not for an inertial-range separation. It may be closely related to the presence of ramp-cliff (diffusion-layer) structure in the scalar field previously observed (see e.g., [16,31]). In such a structure two portions of well-mixed scalar are separated by a sharp interface (cliff); therefore the scalar increment has a fixed magnitude while its sign depends on the orientation of the interface.…”

“…These studies also show that the bimodal FDF for large local variance is due to the ramp-cliff (or diffusion layer) structure in the local SGS scalar. Such a structure has also been found to be the cause of the small-scale anisotropy observed in scalar fields [16,17].…”