Passive scalar cascade in the intermediate layer of turbulent channel flow for $Pr\leq 1$

TL;DR

The paper analyzes how a passive scalar cascades across scales in the intermediate layer of turbulent channel flow for $Pr ≤ 1$ by marrying matched asymptotics with DNS data. It extends the KHMH/Yaglom framework to the scalar, derives outer/inner scale budgets, and shows that a Kolmogorov-like equilibrium is reached only near the scale $r_min$, which scales as $r_min ∼ \lambda_T$ with $\lambda_T = \lambda Pr^{-1/3}$. The inner scalar behavior aligns with Batchelor-scale physics, while the outer/inner matching predicts specific $S_{T2}$ and $S_{T12}$ scalings that are borne out by DNS across several $Pr$. Although the scalar and velocity cascades share qualitative features, their aligned/anti-aligned contributions differ in magnitude and timing, reflecting non-homogeneous effects and flow organization in the intermediate layer.

Abstract

Similarities and differences between Kolmogorov scale-by-scale equilibria/non-equilibria for velocity and scalar fields are investigated in the intermediate layer of a fully developed turbulent channel flow with a passive scalar/temperature field driven by a uniform heat source. The analysis is based on intermediate asymptotics and direct numerical simulations at different Prandtl numbers lower than unity. Similarly to what happens to the velocity fluctuations, for the fluctuating scalar field Kolmogorov scale-by-scale equilibrium is achieved asymptotically around a length scale $r_{min}$, which is located below the inertial range. The lengthscale $r_{min}$ and the ratio between the inter-scale transfer and dissipation rates at $r_{min}$ vary following power laws of the Prandtl number, with exponents determined by matched asymptotics based on the hypothesis of homogeneous two-point physics in non-homogeneous turbulence. The interscale transfer rates of turbulent kinetic energy and passive scalar variance are globally similar but show evident differences when their aligned/anti-aligned contributions are considered.

Figures (3)

• Figure 1: Top panels: $S_{T2} u_\tau^2 /(\theta_\tau^2 u_\eta^2)$ and $S_{T12} u_\tau /(\theta_\tau u_\eta^2)$ as functions of $r/\eta_B$ and $r/\eta$ respectively (left); $\varPi^v_{T}/\varepsilon^v_T$ (green lines), $\mathcal{P}^v_T/\varepsilon^v_T$ (red lines), and transport terms $\varXi_T^v/\varepsilon^v_T=(\varPi_{Tm}^v+\mathcal{T}_{T}^v)/\varepsilon^v_T$ (blue lines) as a function of $r/\lambda_T$ for $Pr=1$. Wall-normal distance is increased from light to dark colours (right).
• Figure 2: Top: Scale $r_{min}$ of $\varPi_T^v/\varepsilon_T^v$ minima as a function of wall distance $y^+$ (left) and values of $1 + \varPi_T^v/\varepsilon_T^v$ as a function of $Re_\lambda$ (right). Top panels: quantities not scaled with $Pr$; Bottom panels: quantities scaled with $Pr$. The dashed line in the right panels is proportional to $Re_\lambda^{-2/3}$.
• Figure 3: Decomposition of the inter-scale transfer rate into anti-aligned (blue lines) and aligned (red lines) contributions for $Pr=1$. Left panels: Velocity; Right panels: Temperature.