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.
Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles