Mixed velocity–passive scalar statistics in high-Reynolds-number turbulence

Abstract

Statistics of the mixed velocity–passive scalar field and its Reynolds number dependence are studied in quasi-isotropic decaying grid turbulence with an imposed mean temperature gradient. The turbulent Reynolds number (using the Taylor microscale as the length scale), Rλ, is varied over the range 85 [les ] Rλ [les ] 582. The passive scalar under consideration is temperature in air. The turbulence is generated by means of an active grid and the temperature fluctuations result from the action of the turbulence on the mean temperature gradient. The latter is created by differentially heating elements at the entrance to the wind tunnel plenum chamber. The mixed velocity–passive scalar field evolves slowly with Reynolds number. Inertial-range scaling exponents of the co-spectra of transverse velocity and temperature, Evθ(k1), and its real-space analogue, the ‘heat flux structure function,’ 〈Δv(r)Δθ(r)〉, show a slow evolution towards their theoretical predictions of −7/3 and 4/3, respectively. The sixth-order longitudinal mixed structure functions, 〈(Δu(r))2(Δθ(r))4〉, exhibit inertial-range structure function exponents of 1.36–1.52. However, discrepancies still exist with respect to the various methods used to estimate the scaling exponents, the value of the scalar intermittency exponent, μθ, and the effects of large-scale phenomena (namely shear, decay and turbulent production of 〈θ2〉) on 〈(Δu(r))2(Δθ(r))4〉. All the measured fine-scale statistics required to be zero in a locally isotropic flow are, or tend towards, zero in the limit of large Reynolds numbers. The probability density functions (PDFs) of Δv(r)Δθ(r) exhibit roughly exponential tails for large separations and super-exponential tails for small separations, thus displaying the effects of internal intermittency. As the Reynolds number increases, the PDFs become symmetric at the smallest scales – in accordance with local isotropy. The expectation of the transverse velocity fluctuation conditioned on the scalar fluctuation is linear for all Reynolds numbers, with slope equal to the correlation coefficient between v and θ. The expectation of (a surrogate of) the Laplacian of the scalar reveals a Reynolds number dependence when conditioned on the transverse velocity fluctuation (but displays no such dependence when conditioned on the scalar fluctuation). This former Reynolds number dependence is consistent with Taylor’s diffusivity independence hypothesis. Lastly, for the statistics measured, no violations of local isotropy were observed.

Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence

“…Hence, the k −2 scaling reported in previous works seems to be a low Reynolds number effect, that occurs around Re λ = 100, typical for DNS. The k −7/3 scaling has also been obtained experimentally (Mydlarski 2003) An interesting point to mention, that has not been explicitly reported so far, is about the infrared range of the cospectrum. Indeed, as F = 0 in the initial isotropic flow, one can wonder how it evolves at very large scales.…”

On the turbulent Prandtl number in homogeneous stably stratified turbulence

“…Venayagamoorthy & Stretch (2006) and authors' unpublished observations showed that T L /T ρ is relatively insensitive to Ri with γ 0.7, using DNS results and experiment data. Their dataset included DNS results of homogeneous sheared stably stratified turbulence of Shih et al (2000), unstratified homogeneous sheared DNS data of Rogers, Mansour & Reynolds (1989) and experimental data on gridgenerated turbulence (Srivat & Warhaft 1983;Itsweire, Helland & Atta 1986;Yoon & Warhaft 1990;Mydlarski 2003). The insensitivity of the mechanical-to-scalar time scale ratio to stratification has important simplifying implications for modelling the turbulent Prandtl number.…”

The Fine Structure of a Slender Scalar Plume in Sheared Turbulence

“…The conditional expectations of the velocity components conditioned upon the scalar value in the present plume were distinctly nonlinear (figure 4), unlike those in flows with homogeneous scalar fields in which the joint velocity-scalar pdfs were nearly Gaussian [2,3,4]. The variation of u 2 /u 2 |c/c illustrates that negative scalar fluctuations (i.e., mostly undyed fluid) are associated with motions that originated outside the plume, whereas positive scalar fluctuations are associated with motions that originated largely in the core of the plume.…”

“…Although these equations have certain advantages over conventional Reynolds-averaged balance equations of velocity and scalar moments, they are complicated by the appearance of conditional expectations of the velocity and the scalar dissipation values, conditioned upon the scalar fluctuations. Most previous studies have focused on flows in which the velocity and scalar are nearly homogeneous and jointly-Gaussian, in which case these conditional expectations would be linear functions of the scalar value [2,3,4]; however, this would not necessarily be the case for inhomogeneous scalar fields as they would appear in indus- …”

“…A popular approach for analyzing reactive flows is to consider the balance equation of the scalar probability density function (pdf) [1,2,3]. Although these equations have certain advantages over conventional Reynolds-averaged balance equations of velocity and scalar moments, they are complicated by the appearance of conditional expectations of the velocity and the scalar dissipation values, conditioned upon the scalar fluctuations.…”