Scalar dissipation anomaly and scalar-gradient scaling in turbulence: A joint velocity-scalar multifractal view

Abstract

We revisit the problem of scalar dissipation anomaly and scaling of scalar gradients in passive scalar turbulence using theory and data from well-resolved direct numerical simulations (DNS) on grid sizes of up to $8192^3$, spanning Taylor-scale Reynolds numbers $Re_λ=140-1000$ and Schmidt numbers $Sc = 1-512$. The theory is based on a joint multifractal description of longitudinal velocity increments and scalar increments, constrained by Yaglom's law and extended to gradients via a fluctuating Batchelor cutoff scale. The DNS data show that the normalized mean scalar dissipation approaches a single asymptotic value as both $Re_λ$ and $Sc$ increase, although larger $Sc$ requires larger $Re_λ$ to reach this state. In the multifractal framework, this corresponds to an effective scalar Hölder exponent tending to zero, associated with sharp cliff-like scalar fronts, and saturation of inertial-range scaling scalar structure-function exponents. The joint velocity-scalar fractal dimension of the dissipative structures is inferred to approach $7/3$, indicating a non-space-filling support. The framework further predicts that for fixed $Re_λ$, higher-order central moments of scalar gradients are independent of $Sc$. This prediction is confirmed by DNS data and by the collapse of standardized probability distributions of scalar-gradient across Schmidt numbers. These results suggest that the $Sc$-scaling of scalar gradients is dictated solely by scalar dissipation anomaly. In contrast, their $Re_λ$-dependence reflects strong intermittency, which can be directly related to mixed velocity-scalar structure function exponents.