Extraction of invariant manifolds and application to turbulence with a passive scalar

The reduction of dimensionality of physical systems, especially in fluid dynamics, leads in many situations to nonlinear ordinary differential equations which have global invariant manifolds with algebraic expressions containing relevant physical information on the original system. We present a method to identify such manifolds, and we apply it to a reduced model for the Lagrangian evolution of field gradients in homogeneous and isotropic turbulence with a passive scalar.

Figure 1

(a) Logarithm of the joint PDF of $Q$ and $R$ for fluid elements in the DNS (in colors). The Vieillefosse tail $Q=-{(27/4R^2)}^{1/3}$ is shown by the solid line. (b) Same as (a) for the joint PDF of $T$ and $S(U+SQ)$. The solid line indicates the manifold in Eq. (13), and the dashed lines indicate the projection of the fixed points in Eq. (12) on this plane. All quantities are dimensionless. In both panels the arrows indicate the projection of the direction and speed of evolution of fluid elements in the DNS on these planes in phase space.

Figure 2

Logarithm of the joint PDF (in colors) of $9RT$ and $2Q(3U+2QS)$ for (a) all fluid elements in the DNS and (b) fluid elements in locations where $\left|Q\right|>0.5$, $\left|R\right|>0.5$, $\left|S\right|>0.5$, $\left|T\right|>0.5$, and $\left|U\right|>0.5$. The manifold in Eq. (14) is shown by the solid line, and the projections of the fixed points in Eq. (12) on this plane are indicated by the dashed lines. References for the arrows are the same as in Fig. 1.

Figure 3

Logarithm of the joint PDF of $-QS$ and $U$ for (a) only fluid elements in the DNS with $S\approx S_0=1$ and (b) all fluid elements. The local invariant manifold $U+Q{S}_0=0$ is shown by the solid line as a reference. In (b) we also show $QS=0$ with dashed lines. Arrows have the same representation as in Fig. 1.