Hessian-based Lagrangian closure theory for passive scalar turbulence

Self-consistent closure theory for passive scalar turbulence has been developed on the basis of the Hessian of the scalar field. As a primitive indicator of spatial structure of the scalar, we employ the Hessian into the core of the theory to properly characterize the time scale intrinsic to the scalar field itself. The resultant closure model is now endowed with several realistic features, i.e., the scale locality of the interscale interaction, the detailed conservation, and the memory-fading effect. Applying the current theory to the inertial-convective range eventually leads to self-consistent derivation of the Obukhov-Corrsin spectrum with its universal constant consistent with numerical and experimental data.

Figure 1

Configuration of the dimensionless function $g\left(\tau \right)$ obtained from LRA equations of Ref. [17]. $g\left(\tau \right)$ is monotonically decaying as the dimensionless time $\tau$ passes.

Figure 2

Configuration of the universal function $h\left(\tau \right)$ which rapidly decays as $\left|\tau \right|$ increases. Unlike the velocity autocorrelation, $h\left(\tau \right)$ is asymmetric in time reversing.

Figure 3

Configuration of the universal function $g_H\left(\tau \right)$ which rapidly decays within finite time scale. Unlike the scalar field preserving long-time memory, the Hessian field loses its memory in the inertial-range time scale.

Figure 4

Integration domain of Eq. (3.20) in the $p\text{−}q$ space.

Figure 5

$W_{\theta }\left(\alpha \right)$ peaks at $\alpha \approx 2$ while $F_{\theta }\left(\alpha \right)$ monotonically increases, tending to unity as $\alpha \to \infty$. Their asymptotic behaviors $W_{\theta }/\langle \chi \rangle \sim 1-F_{\theta }/\langle \chi \rangle \sim {\alpha }^{-4/3}\ln \alpha$ (HBLRA) and $W_{\theta }/\langle \chi \rangle \sim 1-F_{\theta }/\langle \chi \rangle \sim {\alpha }^{-2/3}\ln \alpha$ (scalar-based LRA) are obtained.

Figure 6

Distributions of integrand in Eq. (3.21) for (a) HBLRA and (b) scalar-based LRA. More divergent distribution is observed around $(\widehat{q},\widehat{p})=(1,0)$ in (b), where $Q(p;t,s)$ in Eq. (3.16) diverges as ${\widehat{p}}^{-11/3}$. Red dotted lines show $\alpha =max(1,\widehat{p},\widehat{q})/min(1,\widehat{p},\widehat{q})=2$.

Figure 7

For symmetry of $S_{\theta }(k;p,q)=S_{\theta }(k;q,p)$ in the scalar-transport function $T_{\theta }(k,t)$, the above gray area may be sufficient in calculating the high-wave-number contribution of Eq. (3.15).