Three dimensional branching pipe flows for optimal scalar transport between walls

TL;DR

The paper addresses the maximal rate of heat transport between hot and cold walls under an enstrophy constraint, framing the problem via a variational principle that splits transport, nonlocal, and dissipation effects. It shows that in 3D, self-similar three-dimensional branching pipe flows can saturate the $Q_{ ext{max}} \\sim \\mathscr{P}^{1/3}$ scaling without the log correction that appears in 2D, thereby establishing the exact asymptotic behavior under enstrophy constraints. The construction relies on a careful assembly of pipe-like flow segments in a hierarchical, self-similar tree, with scalar fields designed to be constant along streamlines away from a thin boundary layer. The results have implications for anomalous dissipation of passive scalars and for understanding heat transfer limits in Rayleigh–Bénard-type settings when momentum is constrained by enstrophy rather than the full Navier–Stokes dynamics, suggesting the scaling is robust in three dimensions and potentially improvable under geometric roughness or altered boundary conditions.

Abstract

We consider the problem of "wall-to-wall optimal transport" in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation subject to an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an a priori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Tobasco & Doering (Phys. Rev. Lett. vol.118, 2017, p.264502}) and Doering & Tobasco (Comm. Pure Appl. Math. vol.72, 2019, p.2385--2448}) constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction. This logarithmic correction appears to arise due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional "branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara & Shimizu (J. Fluid Mech. vol.851, 2018, p.R4}). We also discuss the implications of our result to the heat transfer problem in Rayleigh--Bénard convection and the problem of anomalous dissipation in a passive scalar.

Figures (4)

• Figure 1: Panel (a) illustrates good and bad strategies to maximize term $I$ defined in (\ref{['terms var prin']}). In the good scenario $\xi$ is positive (indicated by red color) where the flow is moving upward (positive $z$-direction) and is negative (blue color) where the flow is moving downward. Therefore, $u_z$ and $\xi$ are positively correlated. This is not the case in the bad scenario. Panel (b) illustrates good and bad strategies to minimize term $II$. In the good scenario $\nabla \xi$ is perpendicular to $\boldsymbol{u}$, hence $\boldsymbol{u} \cdot \nabla \xi \equiv 0$, i.e., $\xi$ is constant along the streamlines and therefore the term $II$ zero. In the bad scenario $\nabla \xi$ is parallel to $\boldsymbol{u}$, so the term $II$ is nonzero.
• Figure 2: Panel (a) shows the streamlines of a set of typical convective rolls. Panel (b) shows the streamlines of a steady two dimensional branching flow. In both figures, the streamlines have been overlayed with a $\xi$ field according to the good scenario described in figure \ref{['strategies I II a']} (i.e. $\xi$ is positive whenever $u_z$ is positive, and $\xi$ is negative whenever $u_z$ is negative). The red color indicates a positive value of $\xi$, whereas the blue color indicates a negative value. The dashed circles in both the figures show a region where $\boldsymbol{u} \cdot \nabla \xi$ is nonzero.
• Figure 3: Illustration of the branching pipe flow. Panel (a): the parent construct $\overline{\boldsymbol{u}}$. It consists of red and blue pipes which are the part of pipelines $\boldsymbol{P}_{up}$ and $\boldsymbol{P}_{down}$, respectively. In panel (a), arrows are used in some pipes to show the direction of the flow. The reducer region of a pipe is also shown using a dashed circle. Panel (b): the branching skeleton. To build the main copy $\overline{\boldsymbol{u}}_N$ away from the boundary layer, we place the appropriately dilated version of the parent construct $\overline{\boldsymbol{u}}$ along the skeleton up to $N$ levels. Panel (c): the parent construct $\overline{\boldsymbol{u}}_b$, which we use in the boundary layer. In the construct $\overline{\boldsymbol{u}}_b$, the flow from red pipes turn back to blue pipes (shown in the pink color). A 2D cartoon of the resultant pipe flow is shown in figure \ref{['2D cartoon']}.
• Figure 4: shows a 2D cartoon of the main copy $\overline{\boldsymbol{u}}_N$. The pipeline $\boldsymbol{P}_{up}$ is shown in red color and the pipeline $\boldsymbol{P}_{down}$ is shown in blue color. In the blow-up figure of a section of the pipeline, the graph of $\overline{\xi}_N$ is also shown. Notice is that $\overline{\xi}_N$ is constant in the support of $\overline{\boldsymbol{u}}_N$.

Theorems & Definitions (51)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1: Steady three-dimensional case
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Proposition 1.2: doering2019optimal
• Conjecture 1.3: Weak
• Conjecture 1.4: Strong