Péclet-number dependence of optimal mixing strategies identified using multiscale norms

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Highlights
P´eclet-number dependence of optimal mixing strategies identified
using multiscale norms
Conor Heffernan, Colm-cille P. Caulfield
•Direct-adjoint-looping method used to compute fluid flows to optimize
mixing.
•Questions of interest relate to the structure of the underlying passive
scalar.
•Qualitative differences between a rectilinear and disc geometry are
found.
•The disc geometry problem is scaled up and we find the base structure
is preserved.
arXiv:2210.00555v1 [physics.flu-dyn] 2 Oct 2022
P´eclet-number dependence of optimal mixing strategies
identified using multiscale norms
Conor Heffernana, Colm-cille P. Caulfielda,b
aDepartment of Applied Mathematics and Theoretical Physics, Wilberforce Road,
Cambridge, CB3 0WA, Cambridgeshire, United Kingdom
bBP Institute, Bullard Laboratories, Madingley Road, Cambridge, CB3 0EZ,
Cambridgeshire, United Kingdom
Abstract
The optimization of the mixing of a passive scalar at finite P´eclet number
P e =Uh/κ (where U, h are characteristic velocity and length scales and κ
is the scalar diffusivity) is relevant to many significant flow challenges across
science and engineering. While much work has focused on identifying flow
structures conducive to mixing for flows with various values of P e, there has
been relatively little attention paid to how the underlying structure of initial
scalar distribution affects the mixing achieved. In this study we focus on two
problems of interest investigating this issue. Our methods employ a nonlinear
direct-adjoint looping (DAL) method to compute fluid velocity fields which
optimize a multiscale norm (representing the ‘mixedness’ of our scalar) at a
finite target time. First, we investigate how the structure of optimal initial
velocity perturbations and the subsequent mixing changes between initially
rectilinear ‘stripes’ of scalar and disc-like ‘drops’. We find that the ensuing
stirring of the initial velocity perturbations varies considerably depending on
the geometry of the initial scalar distribution. Secondly, we examine the case
of lattices of multiple initial ‘drops’ of scalar and investigate how the struc-
ture of optimal perturbations varies with appropriately scaled P´eclet number
defined in terms of the drop scale rather than the domain scale. We find that
the characteristic structure of the optimal initial velocity perturbation we
observe for a single drop is upheld as the number of drops and P e increase.
However, the characteristic vortex structure and associated mixing exhibits
some nonlocal variability, suggesting that rescaling to a local P e will not
capture all the significant flow dynamics.
Keywords: keyword one, keyword two
Preprint submitted to Nonlinear Physics D October 4, 2022
PACS: 0000, 1111
2000 MSC: 0000, 1111
1. Introduction
A robust understanding of fluid mixing is of huge importance in the study
of various physical phenomena. Irreversible scalar mixing is characterized
by a relatively large scale ‘stirring’ combined with smaller scale diffusion
to homogenize an initially inhomogeneous distribution of a scalar field [1].
One of the many difficulties of studying irreversible mixing lies in the fact
that there is no unified theory that rigorously defines it. For example, an
interesting question from a mathematical viewpoint is how exactly does one
define the ‘mixedness’ of a substance? Intuitively, the variance of a mean
zero passive scalar θon a domain Ω, defined as
||θ(·, t)||2
L2=1
µ(Ω) ZΩ|θ(x, t)|2dx, (1)
where µdenotes Lebesgue measure (area or volume) is a natural measure of
the mixedness of a scalar field. In the particularly simple situation when the
domain is a 2D torus Ω = T2, it can be shown that Equation (1) reduces to
||θ(·, t)||2
L2=X
k|ˆ
θk|2,(2)
where ˆ
θkare the Fourier coefficients of θand the sum is taken over all
wavenumbers k.
While this measure is intuitive, there are non-trivial issues which need
to be addressed if the scalar field satisfies an advection-diffusion equation
with appropriate boundary conditions and a divergence-free velocity field u
defined with periodic boundary conditions with spatial period Ldefined at
points (x, y)∈Ω
∂θ
∂t +u· ∇θ=1
P e∇2θ, (3)
∇ · u= 0,(4)
θ(x, y, t) = θ(x+L, y, t),(5)
θ(x, y, t) = θ(x, y +L, t).(6)
2
First in the limit when the diffusivity goes to zero, or equivalently when
P e → ∞, the variance is actually a conserved quantity. Even when P e is
finite yet large, the decay of the variance is generically expected to be rela-
tively slow, which introduces computational challenges to flow optimization
calculations [2]. A multiscale measure has been developed to overcome this
problem, based around the use of Sobolev norms of negative index −s[3]:
||θ(·, t)||2
H−s=X
k6=0|k|−2s|ˆ
θk|2.(7)
This measure is similar to Equation (2) but weights wavenumbers differently
(depending on s), with smaller wavenumbers (corresponding to larger scale
structures) leading to larger values of this measure. This measure is com-
monly referred to as a ‘mix-norm’ [4], as if a scalar field is ‘mixed’ (in the
intuitive sense of having no large scale coherent structure) then the mix-norm
is expected to be small. A mix-norm is also advantageous as it is consistent
with the mathematically rigorous ergodic mixing and any choice of s > 0 is
consistent with the theory [5].
Mixing quantification is a necessary component for mixing optimization
subject to a finite initial energy constraint. One successful method in par-
ticular has been to use a cost functional-based approach and maximize the
energy growth at a target time T[6, 7]. To solve this type of optimiza-
tion problem a ‘direct-adjoint-looping’ (DAL) method has been developed
[8, 9, 10]. The algorithm computes initial flows uwhich optimize a given
cost functional at a prescribed target time T. It is advantageous in that it
enforces the fully nonlinear Navier-Stokes equations as a constraint of the
problem. A benchmark of its success has been established by producing op-
timal perturbations which effectively homogenize a scalar field undergoing
mixing. Interestingly, optimal perturbations which minimize the mix-norm
(with given index) at relatively short target times can be excellent prox-
ies for perturbations which minimize variance at longer target times (and
hence ensure thorough mixing). Furthermore, for a given initial energy, such
mix-norm-minimizing perturbations are much more efficient at mixing than
perturbations which are chosen to maximize (perturbation) energy growth
[2, 11, 12]. More recently, the DAL method has also been used to investigate
how variations in the mix-norm index saffects the ensuing mixing dynamics
in a toroidal geometry of the associated ‘optimal’ perturbations [13, 14].
The connections (and contrasts) between variance and mix-norm based
strategies has been investigated for flows with passive and dynamic (i.e. the
3
flow is density-stratified in a gravitational field) scalars [2, 11] as well as
at high and low values of finite P´eclet number in three-dimensional flows
[12]. All these studies have demonstrated that the mix-norm is indeed a use-
ful proxy for computing optimal mixing. A common theme of the previously
considered mixing optimization problems has been to consider relatively sim-
ple rectilinear structures for the initial passive (or dynamic) scalar, typically
with one or two interfaces between areas of high and low concentration, with
θ≈1 and θ≈ −1 respectively such that the spatial mean is zero. Such con-
siderations were natural first steps in demonstrating that the DAL method
can identify optimal perturbations, and also that the use of the mix-norm is
appropriate and computationally efficient.
Now that we have confidence in the DAL method, in this paper we build
on such previous studies [2, 11, 12, 13, 14] to investigate mixing optimization
for initial scalar distributions with more detailed structure. We wish to gain
insight into the particular fluid-dynamical processes which lead to mixing
for different structures in the initial scalar distribution. In particular, we
attempt to answer two questions. First, we ask the question:
Q1: what differences may be found in mixing results for rectilinear ‘stripes’
of scalar as compared to disc-like ‘drops’?.
To answer this question, we perform a comparative study between an ini-
tial scalar distribution of a rectilinear stripe used previously [13, 14] and an
initial scalar distribution of one disc-like drop. We find that there is a qual-
itative difference between flows associated with these initial structures, with
the stripe-like structure undergoing more mixing compared with the drop-like
structure for the same initial perturbation energy. For both structures, the
optimal initial perturbation takes the form of multiple vortices that localize
along the interfaces (i.e. the locations where θvary rapidly) and hence en-
hance mixing. These vortices have a characteristic scale significantly smaller
than both the stripe and the (single) drop.
Motivated by the generic observation of vortices localized at interfaces,
the second question we ask is:
Q2: at what scale, if any, do these vortices cease to align along drop interfaces
for smaller and smaller drops and if they do cease to appear what structures
replace them?
In other words, if the initial structure is a lattice of sufficiently many
and hence sufficiently small drops, does the initial optimizing velocity per-
turbation (in the form of interfacially-localized vortices) change from what
has been found in [2, 11, 12, 13, 14] and if so, what sort of mixing dynamics
4
摘要：

HighlightsPeclet-numberdependenceofoptimalmixingstrategiesidenti edusingmultiscalenormsConorHe ernan,Colm-cilleP.Caul eldˆDirect-adjoint-loopingmethodusedtocomputeuidowstooptimizemixing.ˆQuestionsofinterestrelatetothestructureoftheunderlyingpassivescalar.ˆQualitativedi erencesbetweenarectilinearand...

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