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Optimal Stirring Strategies for Passive Scalars in a Domain with a General Shape and No-Flux Boundary Condition

Sirui Zhu, Zhi Lin, Liang Li, Lingyun Ding

TL;DR

The paper develops a general framework to quantify and maximize mixing of a passive scalar in bounded domains with no-flux boundaries by introducing a mix norm $\| heta\|_m$ that is equivalent to the negative Sobolev norm $H^{-1}$. It derives explicit local-in-time optimal stirring flows under fixed energy and fixed enstrophy constraints, showing exponential or faster decay of the mix norm and linking the decay rate to the smallest nonzero Laplacian eigenvalue $\lambda_1$. Numerical simulations across rectangles, circles, and general domains reveal geometry-dependent decay rates, with periodic boundaries generally enhancing decay under energy constraints and no-flux boundaries enhancing decay under enstrophy constraints; even initial data yield identical optimals across boundary types. A quasi-optimal flow formula is provided for general domains under enstrophy constraint, with exact optimality established for rectangles via even extension. The work highlights the role of domain geometry (via $\lambda_1$) and boundary conditions in mixing efficiency, and sets the stage for global-in-time optimization and extensions to other advection-diffusion phenomena.

Abstract

Multiscale metrics such as negative Sobolev norms are effective for quantifying the degree of mixedness of a passive scalar field advected by an incompressible flow in the absence of diffusion. In this paper we introduce a mix norm that is motivated by Sobolev norm $H^{-1}$ for a general domain with a no-flux boundary. We then derive an explicit expression for the optimal flow that maximizes the instantaneous decay rate of the mix norm under fixed energy and enstrophy constraints. Numerical simulations indicate that the mix norm decays exponentially or faster for various initial conditions and geometries and the rate is closely related to the smallest non-zero eigenvalue of the Laplace operator. These results generalize previous findings restricted for a periodic domain for its analytical and numerical simplicity. Additionally, we observe that periodic boundaries tend to induce a faster decay in mix norm compared to no-flux conditions under the fixed energy constraint, while the comparison is reversed for the fixed enstrophy constraint. In the special case of even initial distributions, two types of boundary conditions yield the same optimal flow and mix norm decay.

Optimal Stirring Strategies for Passive Scalars in a Domain with a General Shape and No-Flux Boundary Condition

TL;DR

The paper develops a general framework to quantify and maximize mixing of a passive scalar in bounded domains with no-flux boundaries by introducing a mix norm that is equivalent to the negative Sobolev norm . It derives explicit local-in-time optimal stirring flows under fixed energy and fixed enstrophy constraints, showing exponential or faster decay of the mix norm and linking the decay rate to the smallest nonzero Laplacian eigenvalue . Numerical simulations across rectangles, circles, and general domains reveal geometry-dependent decay rates, with periodic boundaries generally enhancing decay under energy constraints and no-flux boundaries enhancing decay under enstrophy constraints; even initial data yield identical optimals across boundary types. A quasi-optimal flow formula is provided for general domains under enstrophy constraint, with exact optimality established for rectangles via even extension. The work highlights the role of domain geometry (via ) and boundary conditions in mixing efficiency, and sets the stage for global-in-time optimization and extensions to other advection-diffusion phenomena.

Abstract

Multiscale metrics such as negative Sobolev norms are effective for quantifying the degree of mixedness of a passive scalar field advected by an incompressible flow in the absence of diffusion. In this paper we introduce a mix norm that is motivated by Sobolev norm for a general domain with a no-flux boundary. We then derive an explicit expression for the optimal flow that maximizes the instantaneous decay rate of the mix norm under fixed energy and enstrophy constraints. Numerical simulations indicate that the mix norm decays exponentially or faster for various initial conditions and geometries and the rate is closely related to the smallest non-zero eigenvalue of the Laplace operator. These results generalize previous findings restricted for a periodic domain for its analytical and numerical simplicity. Additionally, we observe that periodic boundaries tend to induce a faster decay in mix norm compared to no-flux conditions under the fixed energy constraint, while the comparison is reversed for the fixed enstrophy constraint. In the special case of even initial distributions, two types of boundary conditions yield the same optimal flow and mix norm decay.
Paper Structure (29 sections, 6 theorems, 61 equations, 10 figures, 2 tables)

This paper contains 29 sections, 6 theorems, 61 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Governing equation for a stirred passive scalar
  4. Mix norm as an equivalent negative Sobolev norm
  5. Flow constraints and cost function
  6. Leray-Helmholtz projection operator
  7. Optimal stirring strategy
  8. Lower bounds of the mix norm
  9. Optimal stirring strategy with fixed energy constraint
  10. Optimal stirring strategy with fixed enstrophy constraint
  11. Numerical simulation
  12. Energy constrained simulations with no-flux boundary conditions
  13. Optimal stirring in a square under no-flux boundary conditions
  14. Optimal stirring in a circle under no-flux boudary conditions
  15. Enstrophy constrained simulations with no-flux boundary conditions
  16. ...and 14 more sections

Key Result

Theorem 2.2

Suppose the spatially mean-zero function $\theta$ is bounded uniformly in $L^{2} (\Omega)$, then $\lVert \theta \rVert_{H^{-q}}=0$ ($q>0$) if and only if $\dfrac{1}{|\Omega|}\int\limits_{\Omega}^{} f (\mathbf{x}) \theta (\mathbf{x}) \mathrm{d} \mathbf{x}= 0$ for all $f \in L^{2} (\Omega)$.

Figures (10)

  • Figure 1: Evolution of the scalar field in $[-1, 1]^2$ with fixed energy constraint \ref{['eq:fixed energy']} and $U=1$ with initial condition \ref{['eq:energy_theta']} that has higher scalar concentration on the right half of the square.
  • Figure 2: Semi-logarithmic plot of mix norms under fixed energy constraint for (a). the square $[-1,1]^2$ and for (b). the circular region (\ref{['eq:circular domain']}) with the same initial condition \ref{['eq:energy_theta']}. The red, solid curves represent the renormalized mix norm. The blue, dashed lines represent the fitted exponential function of the mix norm. The black, dot-dashed curves represent the theoretical lower bound (\ref{['eq:lower bound energy']}).
  • Figure 3: Evolution of the scalar field with fixed energy constraint \ref{['eq:fixed energy']} and $U=1$ in a circle of radius $2/\sqrt{\pi}$. The initial condition is provided in equation \ref{['eq:energy_theta']}.
  • Figure 4: Evolution of the scalar field in $[-1, 1]^2$ with fixed enstrophy constraint \ref{['eq:fixed enstrophy']} and $\tau^{-1}=15$, the initial condition is \ref{['eq:energy_theta']}.
  • Figure 5: Semi-logarithmic plot of mix norm in the square with the initial scalar \ref{['eq:energy_theta']} under fixed enstrophy constraint. The red solid line represents the mix norm. The dashed blue line represents the fitted exponential function of the mix norm.
  • ...and 5 more figures

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: Mix norm
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 7.1
  • proof
  • Proposition 7.2
  • ...and 3 more