Quantifying mixing in arbitrary fluid domains: a Padé approximation approach

Abstract

We consider the model problem of mixing of passive tracers by an incompressible viscous fluid. Addressing questions of optimal control in realistic geometric settings or alternatively the design of fluid-confining geometries that successfully effect mixing requires a meaningful norm in which to quantify mixing that is also suitable for easy and efficient computation (as is needed, e.g., for use in gradient-based optimization methods). We use the physically inspired reasonable surrogate of a negative index Sobolev norm over the complex fluid mixing domain $\Omega$, a task which could be seen as computationally expensive since it requires the computation of an eigenbasis for $L^{2}(\Omega )$ by definition. Instead, we compute a representant of the scalar concentration field in an appropriate Sobolev space in order to obtain an equivalent definition of the Sobolev surrogate norm. The representant, in turn, can be computed to high-order accuracy by a Padé approximation to certain fractional pseudo-differential operators, which naturally leads to a sequence of elliptic problems with an inhomogeneity related to snapshots of the time-varying concentration field. Fast and accurate potential theoretic methods are used to efficiently solve these problems, with rapid per-snapshot mix-norm computation made possible by recent advances in numerical methods for volume potentials. We couple the methodology to existing solvers for Stokes and advection equations to obtain a unified framework for simulating and quantifying mixing in arbitrary fluid domains. We provide numerical results demonstrating the convergence of the new approach as the approximation order is increased.

Appendices

Appendix 1. Eigenfunction expansion in circular or annular domains

Circular domain

Let $\Omega=B(0,a)=\{x(\rho,\theta),\;0\leq\rho<a,\,0\leq\theta<2\pi\}$ be the disk of radius a. The Dirichlet Laplace eigenfunctions for $\Omega$ are

$$\begin{aligned} \varphi _{0m} = \gamma _{0m}J_{0}(z_{0m} \rho /a), \quad \varphi ^{(1)}_{nm} = \gamma _{nm}J_{n}(z_{nm}\rho /a)\cos n\theta , \quad \varphi ^{(2)}_{mn} = \gamma _{nm}J_{n}(z_{nm}\rho /a)\sin n\theta \end{aligned}$$

(28)

where $J_{n}$ is the Bessel function of first kind and integer order n and $z_{nm}$ ($m= 1,2,\ldots$) are the (real, positive) zeros of $J_{n}$ (the excluded zero $z=0$ of $J_{n}$ ($n\ge 1$) not producing nonzero eigenfunctions). Setting the normalization constants to $\gamma _{0m} = \sqrt{\pi }\big [ aJ_{1}(z_{0m}) \big ]^{-1}$ and $\gamma _{nm} = \sqrt{\pi /2}\big [ aJ_{n+1}(z_{nm}) \big ]^{-1}$ $(n\ge 1)$, the eigenfunctions (28) are $L^{2}(\Omega )$-orthonormal and satisfy

$$\begin{aligned} -\Delta \varphi _{0m} = \lambda _{0m}\varphi _{0m}\quad \text {and}\;-\Delta \varphi ^{(1,2)}_{nm} = \lambda _{nm}\varphi ^{(1,2)}_{nm}, \qquad \text {with }\; \lambda _{nm} = z^{2}_{nm}/a^{2} \quad (n,m\ge 0). \end{aligned}$$

(29)

As (Dirichlet) Laplace eigenfunctions, the functions (28) define a Hilbert basis of $L^{2}(\Omega )$, so that any $c\in L^{2}(\Omega )$ admits the expansion

$$\begin{aligned} c(\rho ,\theta ) = \sum _{m\ge 0} \Big \{ c_{0m}\varphi _{0m}(\rho ,\theta ) + \sum _{n\ge 1} \big ( c^{(1)}_{nm}\varphi ^{(1)}_{nm}(\rho ,\theta ) + c^{(2)}_{nm}\varphi ^{(2)}_{nm}(\rho ,\theta ) \big ) \Big \} \end{aligned}$$

(30)

with $c_{0m} = \big ( \varphi _{0m},c \big )_{L^{2}(\Omega )}$ and $c^{(1,2)}_{nm} = \big ( \varphi ^{(1,2)}_{nm}, c \big )_{L^{2}(\Omega )}$. The $H^{-r}(\Omega )$ norm of c is therefore given by

$$\begin{aligned} \Vert c\Vert ^{2}_{-r} = \sum _{m\ge 0} \Big \{ (1+\lambda _{0m})^{-r}[c_{0m}]^{2} + \sum _{n\ge 1} (1+\lambda _{nm})^{-r}\,\big ( [c^{(1)}_{nm}]^{2} + [c^{(2)}_{nm}]^{2} \big )\Big \}. \end{aligned}$$

(31)

Annular domain

Let now $\Omega = B(0, \rho _{1})\setminus \overline{B(0, \rho _{2})}$ be the annulus of internal radius $\rho _{2}$ and external radius $\rho _{1}$. The (unnormalized) radial Dirichlet eigenfunctions for $\Omega$ are given by the expression

$$\begin{aligned} f_{nm}(\rho ) = \frac{-1}{J_{n}(\zeta _{nm} \rho _{1})} J_{n}(\zeta _{nm} \rho ) + \frac{1}{Y_{n}(\zeta _{nm} \rho _{1})} Y_{n}(\zeta _{nm} \rho ). \end{aligned}$$

(32)

so that eigenfunctions are given (analogously to the disc case, and before normalization) via $\varphi _{nm}^{(1,2)}(\rho , \theta ) = f_{nm}(\rho ) \left\{ \begin{array}{c} \cos (n\theta )\\ \sin (n\theta )\end{array}\right\}$. While the functional form (32) of the eigenfunctions is clearly known, the corresponding annular eigenvalues $\lambda _{nm} = \zeta _{nm}^{2}$ are required for the basis to be fully determined. We solve for the eigenvalues using a Newton iteration on the eigenvalue equation, with the method bootstrapped using an approximate eigenvalue obtained using the chebfun system [23] (whose values, at least for larger eigenvalues fail to provide adequate accuracy but are still highly useful to start a Newton iteration).

Appendix 2. Derivation of Padé approximations

Let f have the ($2n-1$)-th order Taylor expansion

$$\begin{aligned} f(X) = a_{0} + a_{1}X \ldots + a_{2n-1}X^{2n-1} + o(X^{2n-1}) \end{aligned}$$

(33)

about $X=0$. In particular, we have $a_{0}=1$ and $a_{k+1}=a_{k}(2k+ r)/(2k+2)$ for $f(X)=(1- X)^{-r/2}$. The coefficients of the polynomials

$$\begin{aligned} P_{n-1}(X) = p_{0} + p_{1}X \ldots + p_{n-1}X^{n-1}, \qquad Q_{n}(X) = 1 + q_{1}X \ldots + q_{n} X^{n}, \end{aligned}$$

(34)

such that $\Pi _{n}[f]=P_{n-1}/Q_{n}$ (with the adopted normalization $q_{0}=1$ ensuring uniqueness of $P_{n-1},\,Q_{n}$) are found from the linear relations

$$\begin{aligned}& \text {(a)}\; a_{n+k} + a_{n+k-1}q_{1} + \ldots + a_{k} q_{n} = 0 \quad&(0\le k\le n-1), \nonumber \\ &\text {(b)}\; p_{k} = a_{0} q_{k} + a_{1} q_{k-1} \ldots + a_{k} q_{0} \quad&(0\le k\le n-1) \end{aligned}$$

(35)

where $q_{1},\ldots q_{n}$ solve the n linear equations (a) and $p_{0},\ldots ,p_{n-1}$ are then given explicitly by relations (b). Numerical experiments for $f(X)=(1- X)^{-1/2}$ indicate however that the linear system (35a) becomes ill-conditioned for n larger than about 10. We therefore solved (35) using symbolic computation.