Abstract
Passive scalars advected by turbulence with a uniform mean scalar gradient have been investigated with great interest in the atmospheric and oceanic contexts. One of the key quantities is the scalar flux spectrum $E_{u\theta}(k)$ which is a measure of how the scalar flux is distributed over the scales. Lumley (1964,1967) predicted the scaling behaviour of scalar flux spectrum in the inertial convective range (ICR) on dimensional grounds as $E_{u\theta}(k) = C_{u\theta} G \bar{\epsilon}^{1/3} k^{-7/3}$, where $G$ represents mean scalar gradient, and $C_{u\theta}$ is a non-dimensional constant expected to be of order unity. Although many studies by experiments, computations and theories have been made for the scalar flux spectrum (or structure functions) with a mean scalar gradient for various Reynolds numbers, there is no conclusive agreement on the existence of Lumley's scaling due to the insufficient width of ICR.
In order to examine the scaling behaviour of scalar flux spectrum, we have performed very high resolution direct numerical simulations (DNSs) of the passive scalar turbulence with uniform mean scalar gradient up to $2048^3$ grid points and $R_\lambda\approx 600$, and analysed the various statistical functions. It is found that the scalar flux spectrum tends to obey the power law $k^{-7/3}$, as predicted by Lumley, with a nondimensional constant of $C_{u\theta} = 1.50 \pm 0.08$ at $R_\lambda \simeq 600$. The $R_\lambda$-dependence of $C_{u\theta}$ is also compared with the results of previous studies, and its asymptotic state at an infinite Reynolds number is discussed. In addition, we discuss the scaling of scalar flux in terms of the mixed velocity-scalar structure function, and the scaling behaviour associated with intermittency in the ICR are examined.
Moreover to get more insight into the behaviour of scalar flux spectrum, we studied the spectral equation for the scalar flux. We examined using DNS data how the terms appeared in the spectral equation contribute in the wavenumber space. It is found that the production term has a large positive contribution in the ICR scale, where the scalar-pressure gradient covariance term shows the negative contribution with the comparable intensity to the production one. The possibility of crossover of the scalar flux spectrum from $k^{-7/3}$ to $k^{-2}$ range is also discussed.