Characterizing scale dependence of effective diffusion driven by fluid flows
CCharacterizing scale dependence of effective diffusion driven by fluid flows
Yohei Kono, Yoshihiko Susuki, and Takashi Hikihara Department of Electrical Engineering, Kyoto University, Kyoto, Japan Department of Electrical and Information Systems, Osaka Prefecture University, Sakai, Japan
We study the scale dependence of effective diffusion of fluid tracers, specifically, its dependenceon the P´eclet number, a dimensionless parameter of the ratio between advection and moleculardiffusion. Here, we address the case that length and time scales on which the effective diffusion canbe described are not separated from those of advection and molecular diffusion. For this, we proposea new method for characterizing the effective diffusivity without relying on the scale separation. Fora given spatial domain inside which the effective diffusion can emerge, a time constant related tothe diffusion is identified by considering the spatio-temporal evolution of a test advection-diffusionequation, where its initial condition is set at a pulse function. Then, the value of effective diffusivityis identified by minimizing the L ∞ distance between solutions of the above test equation and thediffusion one with mean drift. With this method, for time-independent gyre and time-periodic shearflows, we numerically show that the scale dependence of the effective diffusivity changes beyond theconventional theoretical regime. Their kinematic origins are revealed as the development of themolecular diffusion across flow cells of the gyre and as the suppression of the drift motion due to atemporal oscillation in the shear. I. INTRODUCTION
Effective diffusion is a phenomenological concept fordescribing mixing and dispersion of fluid tracers (e.g.temperature and chemicals) driven by fluid flows [1–4].In this paper, we study the scale dependence of the ef-fective diffusion with a new formulation and numericalsimulations of rudimentary flow models.Here is briefly introduced the concept of effective dif-fusion as follows. Let X ⊆ R n ( n = 2 ,
3) be configurationspace, x ∈ X be location, and t ≥ θ ( x , t ) of fluid tracers at x and t is governed by the following advection-diffusion equation: ∂ t θ ( x , t ) + u ( x , t ) · ∇ θ ( x , t ) = D ∆ θ ( x , t ) , (1)where u ( x , t ) represents a pre-defined velocity field on R n and satisfies the incompressibility condition. The con-stant D represents the molecular diffusivity of a medium, ∂ t the differential operator in time, ∇ and ∆ the vec-tor differential and Laplace operators on R n . Follow-ing Refs. [5, 6], if the mean-squared displacement offluid parcels asymptotically increases with t , then theirmacroscopic dispersion can be described by the simplediffusion equation ∂ t ¯ θ ( x , t ) = D eff ∆¯ θ ( x , t ) , (2)where we call the new concentration profile (function)¯ θ ( x , t ) the up-scaled field, and D eff is known as the effec-tive diffusivity . The meaning of up-scaling in this paperis to determine a pair of finite-volume, connected domainΩ ⊂ X and time-interval I := [0 , τ ] , τ > P e := U L/D :the ratio between advection and molecular diffusion,where U and L are the characteristic velocity and lengthof the field u ( x , t ). The dependence of effective diffusiv-ity on P e , known as a scaling law , has been studied byseveral groups of researchers, as reviewed in the followingthree paragraphs.For the steady, two-dimensional periodic gyre flow, theeffective diffusivity D eff exhibits different types of the P e dependence. The
P e dependence is theoretically estab-lished in cases of sufficiently small and large
P e [13–18].This is exactly expressed as D eff ∝ DP e for a suffi-ciently small P e [13–15]. The dependence also behaveslike D eff ∝ D √ P e for a sufficiently large
P e [16–18],where the √ P e dependence is explained by the slow dif-fusive motion of fluid parcels from one periodic flow cellto another. Moreover, the dependence in the large
P e can be enhanced from O ( √ P e ) onto O ( P e ) by adding amean drift or a time-periodic perturbation to the steadygyre flow [17–20], which is called the maximal diffusivity .In addition to the above limiting cases, the P e depen-dence is studied numerically for a finite magnitude of a r X i v : . [ phy s i c s . f l u - dyn ] S e p P e . The authors of Ref. [13] numerically computed theeffective diffusivity D eff without mean drift and showedthat its P e dependence is expressed as an arc-like curve,well fit with both the O ( P e ) and O ( √ P e ) scaling lawsfor sufficiently small and large
P e , respectively. The au-thors of Refs. [19, 21] numerically showed that by addinga mean drift to the gyre, the
P e dependence can exhibitan exponent lower than O ( √ P e ) at a finite
P e , referredto as the crossover effect . Although these numerical stud-ies contain significant progress, their kinematic originsare not necessarily clarified.For time-invariant and time-periodic shear flows, theeffective diffusivity D eff has been calculated as closed-form functions of P e without limitation of the magnitudeof
P e [16, 22, 23]. Its kinematic origin is stated in [21]as the combination of the molecular diffusion betweenstreamlines and of the drift motion along the shear. The D eff can take O ( P e ) only for the time-invariant shear[20].Most of the above theoretical and numerical studiesare conducted with the so-called homogenization [5, 6],which is the framework of asymptotic analysis and buildson the assumption that the original length-scale (or time-scale) of mixing and dispersion characterized by U, L , and D is clearly separated from the length L Ω of the domainΩ (or the time constant τ of the interval I ). The as-sumed scale separation is crucial to deriving the aboveexplicit formulae of the P e dependence of effective diffu-sion. The author of Ref. [24] shows finite time-scales onwhich a limit theorem of the homogenization holds forsufficiently large
P e . Also, recent studies [25–29] haveelaborated on the homogenization framework for comput-ing the effective diffusivity where no scale separation isassumed. However, these studies on the finite time-scalesand the elaboration are conducted without reference tothe kinematic origin of the
P e dependence.To the best of our survey, the kinematic study on the
P e dependence of effective diffusion is limited where thescale separation does not necessarily hold, that is, howthe effective diffusion is affected by wide ranges of themagnitudes of L Ω and τ . As mentioned above, the au-thors of Ref. [21] state that the effective diffusion for theshear flows can be explained by assuming that τ is solarge that a fluid parcel can transit from one streamlineof the shear to another by molecular diffusion. Thus, theshear-induced transport for a finite τ has not been con-sidered in the context of effective diffusion. It should beemphasized that the need of investigating the effectivediffusion for such a finite τ is pointed out in simulationstudies on oceanography and atmospheric science [30–32]and motivated by the engineering of thermal dynamics inoffice buildings as mentioned above.The purpose of this paper is to characterize the scaledependence of effective diffusivity D eff for wider rangesof parameters than the theoretical regime. More specifi-cally, we numerically investigate how the P e dependenceof D eff is affected by the parameters L Ω and τ with finitemagnitudes. To this end, as the first part, we determine the values of L Ω and τ for which the macroscopic disper-sion of Eq. (1) is dominant in some sense. In this paper,we will show that the macroscopic dispersion is domi-nant for a given L Ω if τ takes the same order as the timewhen it takes for a fluid parcel to travel over Ω. Then,as the second part, we focus on the two rudimentarymodels—time-independent gyre and time-periodic shearflows—and show computational results on their effectivediffusivity D eff so that its P e dependence is numericallydescribed. The computation scheme is formulated andconducted in an engineering framework.The contributions of this paper are twofold. First, wedevelop a new method for identifying the effective dif-fusion beyond the theoretical regime building upon thescale separation. For given Ω with L Ω , using techniquesfrom control and optimization, we propose to identifythe time-scale τ and then the effective diffusivity D eff .Here, τ is identified by considering the spatio-temporalevolution of a test advection-diffusion equation, where itsinitial field is set at a pulse function, whose definition ispresented in Sec. II. This is analogue to the identifica-tion of dynamic responses of linear time-invariant sys-tems using an impulsive input [33]. The identification of τ is novel especially for a finite magnitude of P e , whereadvection and molecular diffusion are compatible so thatthe identification based on vanishing molecular diffusion[18, 24] is not available. Using the identified τ , D eff isidentified by minimizing the L ∞ distance between solu-tions of the above test equation and the diffusion one (2)with mean drift. To the best of our survey, the identi-fication method for D eff is novel. Second, for the tworudimentary models, we reveal the kinematic origins ofthe P e dependence of D eff , where no scale separation isassumed. For the gyre flow, we numerically show thatthe P e dependence can change from O ( √ P e ) at a finite
P e . Based on a finite magnitude of τ , we newly estimatethe length of finite-time dispersion of fluid parcels dueto the effective diffusion, by which the change of the P e dependence is explained as the development of moleculardiffusion between flow cells of the gyre. For the shearflow, we numerically show that a finite magnitude of τ causes the deviation of D eff from the closed-form func-tion of P e . The deviation is explained as the degree ofinsufficiency of molecular diffusion between streamlines.The rest of this paper is organized as follows. Sec. II isdevoted to developing the method for characterizing theeffective diffusion where no scale separation is assumed.Sec. III shows computational results of the effective dif-fusion for the gyre flow. By modulating the moleculardiffusivity D , we will validate our method with the √ P e scaling and show its breakdown when no scale separa-tion holds. In Sec. IV, we study the
P e dependence ofthe effective diffusion for the shear flow, where the time-constant of molecular diffusion between streamlines is onthe same order as drift motion by the shear. Sec. V is theconclusion with brief summary and discussion on gener-ality of the proposed method and the physical findings.
II. PULSE-BASED METHOD FORCHARACTERIZING EFFECTIVE DIFFUSION
This section is devoted to developing a method forcharacterizing the effective diffusion without assumptionof scale separation. Here, for a given Ω ∈ X , the time-scale τ , effective diffusivity D eff , and associated errorterm as the result of the approximation of effective dif-fusion are determined via the spatio-temporal evolutionof a pulse function, which is defined in the next para-graph. Below, we assume that u ( x , t ) is periodic in bothspace and time. Precisely, it is assumed that the signal u ( x , t ) for any fixed x ∈ Ω has a finite number of peaksin Fourier spectrum, for which we use τ to represent thefundamental period. It is also assumed that τ and thefundamental period L of u ( x , t ) in space are known a pri-ori, and that L is smaller than L Ω . These assumptionshold for our rudimentary models in this paper.Let us introduce a test equation used for this method.Recalling that effective diffusivity is not sensitive to aninitial field θ (see, e.g., Ref. [30]) and, in certain cases,can be a functional of u ( x , t ) [5], we evaluate it via thefollowing test partial differential equation (PDE): ∂ t ρ ( x , t ) + u ( x , t ) · ∇ ρ ( x , t ) = D ∆ ρ ( x , t ) . (3)The initial field is fixed at a certain class of functionsthat we call the “pulse” function, ρ ( x ,
0) = ρ ( x ). Forour method, ρ ( x ) should be taken as a function suchthat it is supported in the interior of Ω and localized(in x ) in terms of L Ω so that the diffusion phenomenonclearly develops in space. Regarding this, there are mul-tiple choices of the pulse function; e.g., the Dirac’s deltafunction can be used in context of mathematical analysis.This choice is used in terms of the Lagrangian approach[20, 21, 34], where effective diffusion is described via thelong-term evolution of the density of fluid parcels thatstart from the support of the delta function. In this pa-per, in order to gain better regularity of the problem fornumerics, we use the Gauss function as ρ ( x ) and controlits length scale by the variance parameter σ : ρ ( x ) = exp (cid:18) (cid:107) x − c Ω (cid:107) σ (cid:19) , (4)where c Ω stands for a geometric center (centroid) of Ω,and (cid:107) · (cid:107) for the vector norm. To clearly investigate theeffect of advection with its spatial period L , we fix σ suchthat its order of magnitude is equal to and smaller thanthat of L .As the first step of the method, for given Ω, we deter-mine the time-scale τ relevant to the dispersion of fluidparcels in Ω. The τ is identified via the parameter τ Ω ,α that is a function of Ω and a small parameter α for judg-ing if a fluid parcel reaches a given position or not. Let ∂ Ω be the boundary curve or surface of Ω. For given x ∈ Ω, if a fluid parcel reaches x from an initial position closeto c Ω , then there exists an onset t , denoted by ˜ τ Ω ,α ( x ),such that ρ ( x , t ) = α (cid:82) Ω ρ ( y ) µ (d y ) / | Ω | holds, where µ ( (cid:116) ) is a standard measure on Ω, and | Ω | = (cid:82) Ω µ (d y )coincides with the volume or area of Ω. Here, by sup-posing that a fluid parcel can reach the boundary ∂ Ω bythe advection and diffusion, it is possible to estimate thetime τ Ω ,α given by τ Ω ,α = inf x ∈ ∂ Ω ˜ τ Ω ,α ( x ) . (5)This is an approximation of the first time when the pulseof Eq. (4) hits ∂ Ω.Here, we comment on how to determine the parameter α . As α becomes large, it is possible to clearly detectthe hitting of the pulse; however, it requires long timefor the computation of Eq. (5). Also, for avoiding thetrivial case τ Ω ,α = 0, the initial ρ ( x ) should be smallerthan the threshold α (cid:82) Ω ρ ( y ) µ (d y ) / | Ω | at every x ∈ ∂ Ω.Thus, α needs to satisfy the following inequality: α > | Ω | sup x ∈ ∂ Ω ρ ( x ) (cid:82) Ω ρ ( y ) µ (d y ) =: Λ Ω ,σ . Below, we will fix α at a small value satisfying the aboveinequality, and hence τ Ω ,α can be computed in practicaltime. Regarding this, we will also show the L Ω depen-dence of Λ Ω ,σ (see Fig. 3).Next, we identify the drift-oriented transport of ρ dur-ing the interval [0 , τ ] in order to eliminate it for estimat-ing the effective diffusivity. Inspired by the averagingmethod [5], we quantify the so-called bulk movement offluid parcels by c ( t ) := 1 C (cid:90) Ω x ρ ( x , t ) µ (d x ) , t ∈ [0 , τ ] , (6)where C := (cid:82) Ω ρ ( y ) µ (d y ). The meaning of c ( t ) is de-scribed below in terms of the averaging method. As inRef. [5] we suppose that Ω is point-symmetric with re-spect to its center c Ω . Then, by combining this with theperiodicity of u , the equality (cid:82) ∂ Ω x { ( ρ u ) · n } µ (d x ) = holds, where n ( x ) is the normal vector at point x on ∂ Ω.Also, when the pulse of Eq. (4) does not hit ∂ Ω at t < τ ,the gradient ∇ ρ ( x , t ) is negligible at any x ∈ ∂ Ω. Then,the time derivative of c ( t ) is derived asd c d t = 1 C (cid:90) Ω x {− ∇ · ( ρ u ) + D ∆ ρ } µ (d x ) , = − C (cid:90) ∂ Ω x { ( ρ u ) · n } µ (d x )+ 1 C (cid:90) Ω ( ρ u ) µ (d x ) + DC (cid:90) ∂ Ω ( ∇ ρ ) · n µ (d x ) , ∼ C (cid:90) Ω u ( x , t ) ρ ( x , t ) µ (d x ) , (7)where we use the integration by parts for each elementto move from the second line to the third. Eq. (7) corre-sponds to the classical effective velocity [5] if ρ ( x , t ) /C is regarded as a probability density function on Ω. Thisclearly shows that c ( t ) in Eq. (6) represents the averaged(in space) movement of fluid parcels in Ω. Since the effec-tive velocity in [5] represents the spatio-temporal meanof u ( x , t ), we define¯ U Ω ,α := c ( τ Ω ,α ) − c Ω τ Ω ,α , (8)as the effective velocity to describe the drift transportover [0 , τ ].Finally, we develop the concrete step of identifying theeffective diffusivity. To do this, the following two testPDEs are introduced: (cid:0) ∂ t + ¯ U Ω ,α · ∇ (cid:1) ˆ ρ ¯ D, Ω ,α ( x , t ) = ¯ D ∆ˆ ρ ¯ D, Ω ,α ( x , t ) , (9) ∂ t ¯ ρ ¯ D ( x , t ) = ¯ D ∆¯ ρ ¯ D ( x , t ) , (10)where ¯ D is a candidate of the effective diffusivity. Eq. (9)is used to identify the effective diffusivity under the meanflow quantified by ¯ U Ω . Eq. (10) is used to generate thepure diffusive solution that approximates ρ ( x , t ), param-eterized by ¯ D . Using Eq. (9) to consider the presence ofmean flow, through the L ∞ distance between solutionsof advection-diffusion and effective diffusion equations inRef. [5], we introduce the L ∞ distance to identify the ef-fective diffusivity as follows:ˆ d ¯ D, Ω ,α ( t ) := sup x ∈ Ω | ρ ( x , t ) − ˆ ρ ¯ D, Ω ,α ( x , t ) | . (11)This ˆ d ¯ D, Ω ,α can change in not only ¯ D, Ω but also t . Inparticular, it can have a wide range of time-frequencyspectrum beyond that of ˆ ρ ¯ D, Ω ,α ( x , t ) since ρ ( x , t ) is af-fected by the time-dependent u ( x , t ) but ˆ ρ ¯ D, Ω ,α ( x , t ) bythe constant ¯ U Ω ,α . However, by the original notion of ef-fective diffusion, such high-frequency components shouldbe filtered out for the modeling of macroscopic mixingand dispersion. Thus, for estimating better ¯ D in the L ∞ sense, it is necessary to filter out high-frequency com-ponents whose time-scale is smaller than a pre-definedconstant denoted as τ . There exist many methods forthis low-pass filtering in signal-processing textbooks: seee.g., Ref. [35]. In this paper, for simplicity of implemen-tation, we use the first-order filter to derive a smoothed error d ¯ D, Ω ,α ( t ) from ˆ d ¯ D, Ω ,α ( t ) as follows: (cid:18) τ dd t + 1 (cid:19) d ¯ D, Ω ,α ( t ) = ˆ d ¯ D, Ω ,α ( t ) , (12)where the smoothed error d ¯ D, Ω ,α ( t ) is initialized as d ¯ D, Ω ,α (0) = ˆ d ¯ D, Ω ,α (0). With this, we search a value of¯ D that minimizes the cost function defined bysup t ∈I d ¯ D, Ω ,α ( t ) , (13)where its minimizer is referred to as ¯ D Ω ,α , correspondingto an estimated value of the effective diffusivity D eff forgiven Ω and α .Moreover, using Eq. (10), we introduce a metric forthe effective diffusion to investigate its performance and application limit, which we will compare with an errormetric given by the homogenization in order to validatethe proposed method. For this, we consider the differencebetween the two initial fields: θ ( x ) assumed by the clas-sical homogenization and ρ ( x ) by our method. In [5], θ ( x ) is assumed to be a periodic function with its periodsufficiently smaller than L Ω , implying that fluid parcelsare homogeneously located on Ω. However, in this pa-per, the initial field is set at the pulse function ρ ( x ) sothat fluid parcels stay near its center c Ω for small t . Thelocal dispersion does not appear in the homogenizationapproach and therefore should be excluded for investigat-ing the performance of the effective diffusion. By takingthis into account, it is desirable to introduce a metricbased on the concentration profile after fluid parcels aresufficiently dispersed over Ω. In this paper, we use thefollowing metric E Ω ,α at the onset t = τ Ω ,α : E Ω ,α := sup x ∈ Ω (cid:12)(cid:12)(cid:12) ρ ( x , τ Ω ,α ) − ¯ ρ ¯ D Ω ,α ( x , τ Ω ,α ) (cid:12)(cid:12)(cid:12) , (14)where ¯ ρ ¯ D Ω ,α ( x , τ Ω ,α ) is derived from Eq. (10) by setting¯ D at ¯ D Ω ,α . The method developed above is summarizedas a schematic diagram in Fig. 1.Before the concrete applications, it is valuable to men-tion the computational aspect of the proposed method.The optimization poses, unfortunately, a non-convexproblem, and hence its local minimizer does not implythe global one. In the applications below, the explicitformulae of rudimentary flow models are available so thatthe order of “true” effective diffusivity D eff can be esti-mated a priori. Restricted to the true order, the costfunction (13) likely exhibits convex property. Then, wewill conduct the grid search algorithm [36, 37] in orderto search the minimizer of Eq. (13). However, when theproposed method is used in the combination with thecomputational fluid dynamics (CFD) (see, e.g., Ref. [2]),the true order of D eff cannot be estimated because of thecomplex nature of u ( x , t ). In this case, the optimizationrequires global techniques such as meta-heuristics [38] forlocating a global minimum of Eq. (13) while avoiding alocal one. III. SCALE DEPENDENCE FOR GYRE FLOW
In this section, we apply the proposed method to asimple gyre flow and characterize its effective diffusionas a function of the molecular diffusivity D . The pur-poses of this application are two-fold. The first one isto validate the method by comparison with the tradi-tional theory of D -dependence of the effective diffusivity D eff (see Eq. (16) below), which is based on the scaleseparation. The second one is to show a breakdown ofthe theory beyond the regime where the scale separationholds, which we will refer to as the transition of effectivediffusion.The model flow is time-invariant and represented bythe following two-dimensional vector field: for x = Real PDE systemsTest PDE systems
Eq.(1) Eq.(2)Eq.(3) Eq.(9) Eq.(10)Identify Consider the lack ofzero-mean assumption Investigatemodeling performancevia the error metric (14)Set initial fieldas a pulse Up-scaled under scale-separationand zero-mean assumptionCaluculate andSet Identification (13) Computingerror series (11)Removingfluctuation (12) Error estimation (14)Eq.(3)Eq.(9) Eq.(10)
FIG. 1. Schematic diagram of the proposed characterization of effective diffusion. [ x y ] (cid:62) [39], u ( x ) = (cid:20) − U sin(2 πx/L ) cos(2 πy/L ) U cos(2 πx/L ) sin(2 πy/L ) (cid:21) , (15)where X = [ − . , . × [ − . , . U and L are thecharacteristic velocity and the length of the flow field.Figure 2(a) illustrates streamlines of the vector field(15) with L = 0 . − . , . × [ − . , . − L Ω / , L Ω / × [ − L Ω / , L Ω /
2] with a control-lable length L Ω .Under the scale separation (i.e. L Ω (cid:29) L ), the scalinglaw and error convergence of the effective diffusivity forEq. (15) are well-known. For P e := U L/D , the √ D scal-ing law of D eff is given in Refs. [16, 18]: For sufficientlylarge P e , D eff ∝ D √ P e = √ DU L. (16)Also, for the scaling ratio
L/L Ω , the upper bound of thefollowing error ˜ E Ω ,α is given in Ref. [5] by virtue of thehomogenization: Under the assumption that the initialfield θ ( x ) of Eq. (1) is periodic, we have˜ E Ω ,α := sup x ∈ Ω | θ ( x , τ Ω ,α ) − ¯ θ ( x , τ Ω ,α ) |≤ sup x ∈ Ω ,t ∈ [0 ,τ Ω ,α ] | θ ( x , t ) − ¯ θ ( x , t ) | < L/L Ω . (17)Below, we will validate the proposed method by not onlyreproducing these results but also exploring their irrele-vance. In this paper, it is referred to that the scaling law (16)becomes irrelevant for explaining the underlying trans-port phenomenon, as the transition of effective diffusion.The occurrence of the transition is numerically shown inliterature, e.g., Refs. [19, 21]. Here, we point out that itresults from the enhancement of the molecular diffusivity D . The scaling law is originated from the formation of adiffusive boundary layer with width W b , implying a smallneighborhood of separatrix of the flow [16, 18]; see verti-cal and horizontal lines in Fig. 2(a). Because advectionis dominant in a flow cell, the two time constants—thetime τ a := L/U to go around a cell by advection and thetime τ d := W /D to traverse diffusively the boundarylayer—can be balanced, i.e., τ a (cid:39) τ d . Thus, the width W b is determined as W b = (cid:112) DL/U . (18)Then, D eff is estimated by multiplying an usual random-walk expression L / ( L/U ) by the ratio of particles W b /L lying in the boundary layer, leading to the scaling (16)in Ref. [16]. Here, it naturally follows for large D that τ a and τ d are not compatible. Then, the balance betweenadvection and molecular diffusion can break down andcause the transition of the scaling law. This will be nu-merically clarified below.Let us summarize the current setting of numerical sim-ulations. We used the parameters σ = 0 . U = 1, and L = 0 .
1. Also, as shown in Tab. I, we varied L Ω and D in order to confirm whether or not the scaling law(16) and the error convergence (17) are reproduced by -2 -1 -2 -1 -6 -5 -4 -3 (c-1)(c-2)
105 104 103 Pe (a) (b) FIG. 2. Simulation results of effective diffusion for gyre flow (15): (a) Illustration of streamlines of the target vector field; (b) L Ω -dependencies of E Ω ,α (red) and τ Ω ,α (blue); (c-1) D -dependencies of ¯ D Ω ,α (red) and τ Ω ,α (blue); and (c-2) D -dependencyof σ Ω ,α .TABLE I. Setting of of numerical simulations for gyre flowSetting D L Ω − { . , . , . . . , . } { − , − . , . . . , − } the proposed method. Moreover, to show that the ef-fective diffusion is not sensitive to the selection of initialfields, we computed the error metric ˜ E Ω ,α in Eq. (17)with a certain initial field θ . Since the effective diffusiv-ity ¯ D Ω ,α and its time constant τ Ω ,α are computed by thedispersion from the single center c Ω , any peak of θ ( x )should not be located at the center c Ω for distinguish-ing E Ω ,α from ˜ E Ω ,α . Thus, we set θ ( x ) at the followingmixed-Gaussian distribution: θ ( x ) = exp (cid:18) (cid:107) x − c (cid:107) σ (cid:19) − exp (cid:18) (cid:107) x − c (cid:107) σ (cid:19) , (19)where the positions of peaks are denoted by c =[0 . . (cid:62) and c = [0 . . (cid:62) . With this, we computed -50 -40 -30 -20 -10 FIG. 3. Numerically computed Λ Ω ,σ for each L Ω . Here thevariance parameter σ is fixed at 0.04. The blue cross marksdenote the calculated values, and the broken line denotesΛ Ω ,σ = 0 . ˜ E Ω ,α by setting τ = τ Ω ,α and D eff = ¯ D Ω ,α in Eqs. (1) and(2). Here, by varying L Ω , we plot the lower bound of α ,namely Λ Ω ,σ in Fig. 3, which takes its maximum 0.0154at L Ω = 0 .
2. Thus we fixed α at 0.05, which was on thesame order as (and larger than) the above maximum,and by which we could compute τ Ω ,α in practical time.All numerical simulations were conducted by the forward-time centered-space scheme [40], where the discretizationsteps were set at 0.005 in space and 0.001 in time. Theminimizer of Eq. (13) was searched by the grid searchwith the candidates ¯ D ∈ { − , − . , . . . , − . } .Figure 2 summarizes simulation results of the effectivediffusion for the gyre flow. In the panel (b) of Fig. 2,the time constant τ Ω ,α and error E Ω ,α are shown as thesetting E Ω ,α , denoted by theblue circles, becomes small as L Ω increases, so that thedominant phenomenon is transited from mixing and dis-persion to diffusion. The rate of change (decay) of E Ω is approximately L − . , which is faster than L − inEq. (17) derived under the scale separation. The error˜ E Ω ,α computed by setting the initial field at θ ( x ) inEq. (19), denoted by blue x-marks in the panel (b), is ina good agreement with E Ω ,α , showing that both ρ ( x , t )and θ ( x , t ) become diffusive regardless of their differentinitial fields. Thus, no consideration of the dependenceon initial fields is needed to capture well the effectivediffusion for the gyre flow. Also in the panel (b), thetime constant τ increases with ( L Ω /L ) . , which is rel-evant in comparison with the well-known diffusive scaling τ ∝ ( L Ω /L ) in Ref. [5].The panel (c-1) of Fig. 2 shows the results of ¯ D Ω ,α and τ as the setting D Ω ,α increases differs for the two regimes D ≤ − . and D > − . . For D ≤ − . , a linear approxima-tion gives us an estimated rate D . , which implies thescaling law (16). It is here noted that the estimation er-ror from D . depends on the choice of samples used forthe linear approximation. Indeed, when four samples at D ∈ { − , − . , − . , − . } are chosen and a linearapproximation is utilized to them, the estimated rate be-comes D . . On the other hand, the rate for D > − . is smaller than for D ≤ − . , showing the transition ofthe √ D scaling law between the two regimes. As statedabove, it suggests that the large value of the moleculardiffusivity D causes a breakdown of the balance betweenmolecular diffusion and advection, in other words, the as-sociated boundary layer does not work in the transportphenomenon dominantly, i.e. W b (cid:28) (cid:112) DL/U .This mechanism is justified by the following observa-tion. For quantifying how long is the distance of move-ment of a fluid parcel across the periodic flow cells, wedenote by σ Ω ,α the dispersion length in the x - (or y -)direction governed by the effective diffusion as σ Ω ,α := (cid:113) ¯ D Ω ,α τ Ω ,α /n, (20)where n is the dimension of Ω and introduced in thedenominator for representing the x - (or y -) directionalmovement of fluid parcels. The panel (c-2) of Fig. 2 shows σ Ω ,α for each D , where the horizontal broken line corre-sponds to the path length per one rotation of a circula-tion, σ Ω ,α = πL/
2. Clearly, σ Ω ,α is larger than πL/ D > − . , implying that a fluid parcel visits multiplecells before circulating a single cell. The discontinuous change at D = 10 − . suggests that the fluid parcels arenot trapped in the boundary layer, and that the effectivediffusion is governed by their molecular diffusion that de-velops over multiple cells. IV. SCALE DEPENDENCE FOR SHEAR FLOW
In this section, we address a simple model of time-periodic shear flow and investigate the effective diffusionarising there, especially its scale dependence caused bya temporal oscillation in the shear. The model flow isgiven in [16, 22] as follows: u ( y, t ) = (cid:34) U cos(2 πy/L ) cos(2 πt/τ )0 (cid:35) , (21)where x = [ x y ] (cid:62) ∈ X = [ − . , . × [ − . , . x -direction can be produced bythe interaction of shear and molecular diffusion. Theassociated effective diffusivity is analytically determinedvia spatio-temporal Fourier analysis in Ref. [21]. Let Ωbe the rectangle [ − . , . × [ − . , .
5] with L Ω = 0 . L (cid:28) L Ω and τ (cid:28) τ in Ref. [5], the effective diffusivity inthe x -direction, denoted by ¯ D xx , is described with thefollowing analytic formula [16, 21, 22]:¯ D xx = D + D U ( L/τ ) + (2 πD/L ) . (22)This leads to the associated PDE for the effective diffu-sion as ∂ t ¯ θ ( x , t ) = ( ¯ D xx ∂ x + D∂ y )¯ θ ( x , t ) , (23)where ∂ x and ∂ y stand for the differential operators in x and y .To clarify the scale dependence to be studied here, weexplain the kinematic origin of the analytic formula (22)based on Ref. [21]. Without the presence of moleculardiffusion, a fluid parcel would move along the stream-lines of the shear, which are straight lines parallel tothe x -axis, at a ballistic rate (implying that the distancefor the movement grows linearly in time). The presenceof molecular diffusion enables the parcel to move fromits initial streamline onto others with velocities in theopposite direction of the initial one, thereby suppress-ing the ballistic motion and making a diffusive transportdominant instead. In contrast, the temporal oscillationsin the shear induce bounded oscillations of fluid parcelsrather than the unidirectional ballistic motion, thereforedisturbing the transport of parcels. Thus, the effectivediffusivity ¯ D xx depends on D and decays as the period τ decreases; see Eq. (22) and the black line in Fig. 4(c)for details. The dependence of ¯ D xx on not only D butalso τ is crucial to our current study.The scale dependence which we will study is relatedto how deviated is the evaluation of effective diffusivityby the method in Sec. II from the analytic formula (22).Regarding this, it should be emphasized that the mech-anism explained above implicitly relies on the assump-tion of scale separation; namely, τ should be so largethat a fluid parcel can diffusively move between differentstreamlines in the shear. In this, it is not clear to inves-tigate the diffusive behavior of fluid parcels for a smalltimescale τ . By taking a large value of the period τ ofthe shear, τ can be small such that fluid parcels can reachthe boundary ∂ Ω by the ballistic motion before movingbetween the streamlines diffusively. For such a small τ ,we will estimate the effective diffusivity and investigatehow the estimated value is affected by the dominance ofthe ballistic motion. Below, in order to consider not only D but also multiple values of τ , we regard as the length-scale not L (the interval between the streamlines) but U τ (the length of the ballistic movement) and rewritethe P´eclet number as the leading parameter like P e := U τ D . (24)Let us summarize the setting of numerical simulations.We used σ = 0 . U = 1, L = 0 .
02, and D = 10 − , andvaried τ through 0 . , . , . . . , . α was set at 0.05 as in Sec. III.To explicitly search ¯ D xx , the diffusion operator ¯ D ∆ hasbeen modified into ¯ D∂ x + D∂ y in Eqs. (2), (3), and (10).The simulation and optimization schemes were the sameas in Sec. III, and the discretization steps were set at0.002 in space and at 0.001 in time.Figure 4 summarizes simulation results of the effec-tive diffusion for the shear flow. The P´eclet number P e is denoted for each τ at the bottom of Fig. 4. In thepanel (a), the time-scale τ Ω ,α is shown with the blue lineand takes small values as τ increases. This implies that afast oscillatory component in the shear makes the lengthof the ballistic motion U τ small so that few fluid parcelscan reach the boundaries x = ± .
4. Also, τ Ω ,α has avalue with the same order as L /D = 4, which representsthe diffusive time-constant in the y -direction. Thus, τ Ω ,α becomes so small that the ballistic motion is dominantby comparison with the diffusive movement between thestreamlines.The panel (b) of Fig. 4 shows the error term E Ω ,α with the blue line. For comparison, the L ∞ normsup x ∈ Ω ρ ( x , τ Ω ,α ) is also shown with the broken line.The order of E Ω ,α is smaller than that of the L ∞ norm.Thus, the solution ¯ ρ ¯ D Ω ,α ( x , τ Ω ,α ) of Eq. (10) is in a goodagreement with ρ ( x , τ Ω ,α ) of Eq. (3), showing the valid-ity of the approximation as the effective diffusion. Also,this implies that in the proposed method we regard thedispersion mainly caused by the time-periodic ballisticmotion as the effective diffusion, which is general andnever clarified with the homogenization. Here, while E Ω ,α decreases with almost all τ , it takes larger value at τ = 0 .
03 than τ = 0 .
04. This is related to that the driftlength
U τ = 0 .
03 is compatible with (or smaller than) -4 -3 Identified valueAnalytic formula= 0.2= 0.4= 0.55= 0.7= 0.8= 0.9 -3 -2 -1 (a)(b)(c) FIG. 4. Simulation results of effective diffusion for theshear flow (21): (a) τ Ω ,α (solid) (b) E Ω ,α (solid) andsup x ∈ Ω ρ ( x , τ Ω ,α ) (broken); (c) ¯ D Ω ,α (blue, solid), ¯ D xx (black, solid) and ¯ D β (broken) for each time-period τ . the variance σ = 0 .
04 and hence that not only the bulktransport of the initial pulse ρ but also its deformation(i.e. the geometric change from the Gaussian pulse) areemergent and affect the validity of the approximation asthe effective diffusion. Indeed, for different values of σ ,we confirmed that E Ω ,α took such a large value at a cer-tain τ with a smaller magnitude than σ/U , showing theeffect of the above deformation.The panel (c) shows the effective diffusivity for each τ and P e . The blue circles represent the values of ¯ D Ω ,α identified with our method, and the black solid line does¯ D xx computed with the analytical formula (22). Asstated above, the analytic ¯ D xx decays as τ decreases;however, the determined ¯ D Ω ,α becomes larger than ¯ D xx for the small τ . This indicates that the ballistic motionis not well suppressed by the diffusive transport. To ver-ify it, let us modify the analytical formula (22) of effectivediffusivity into˜ D β = D + D U ( βL/τ ) + (2 πD/βL ) , (25)where the degree of insufficiency of the diffusive transportbetween streamlines is represented by shortening the dif-fusive length L via a controllable coefficient β ∈ (0 , D β for β ∈ { . , . , . , . , . , . } with the broken lines. By increasing β with τ (or P e ),it is possible to fit ˜ D β into the identified ¯ D Ω ,α for each τ . This implies that a fast oscillation in the shear sup-presses the unidirectional ballistic motion instead of themolecular diffusion, and that the effective diffusion for afinite magnitude of τ is mainly governed by the shear. V. CONCLUDING REMARKS
This paper is devoted to the scale dependence of the ef-fective diffusivity D eff for the two rudimentary flow mod-els. Technically, we investigated how the P e dependenceof D eff was affected by the parameters L Ω and τ with fi-nite magnitudes. To do this, for given L Ω , we developeda pulse-based method for identifying τ and D eff basedon finite-time evolution of the Gaussian pulse function inSec. II. For the time-invariant gyre flow in Sec. III, theproposed method successfully reproduced the well-knownerror convergence and scaling law, showing its validity.Also, by enhancing the molecular diffusivity D for thegyre flow, we show that the scaling exponent of D eff canchange according to the breakdown of a balance betweenadvection and diffusion in a single flow cell. For the time-periodic shear flow in Sec. IV, we show that the effectivediffusivity D eff can deviate from the Fourier-based ana-lytic formula (22), which has not been clearly reportedin literature to the best of our survey. We point out thatthe diffusive transport between streamlines can be insuf-ficient in a finite time-scale of the effective diffusion forthe shear, and hence the deviation from Eq. (22) origi-nates from the suppression of a ballistic motion due tothe temporal oscillation in the shear.Here, we revisit the kinematic origins of the scale de-pendence of effective diffusivity delineated in the two models. The origins are commonly related to how fluidparcels are trapped in the flow structures, i.e., flow cells ofthe gyre in Sec. III and streamlines of the shear in Sec. IV.It is universal beyond the two models and central to theresearch on mixing and dispersion by fluid flows [41]. Toshow this concretely, we refer to Ref. [42] that investigatesthe effective diffusion for a time-dependent shear modeldifferent from Eq. (21). In this, as a metric of the effectivediffusion, the authors have computed the variance of fluidparcels starting from a certain initial position. Then, byvarying the initial position, they have illustrated the spa-tial distribution of the variances, which shows the spatialpattern associated with the finite-time Lyapunov expo-nents (FTLEs) (see, e.g., Ref. [43]) of the vector field.Clearly, the formation of such a spatial pattern dependson how fluid parcels are trapped in the flow structuresdescribed by FTLE, showing the similarity with what weshowed in this paper. Also, the authors of Ref. [42] havereported that the FTLE-induced pattern disappears at alarge value of the molecular diffusivity. The delineatedmechanism in Sec. III that a fluid parcel can move acrossmultiple cells diffusively helps to explain the disappear-ing process of the pattern. We contend in this paperthat the finite-scale effective diffusion is governed by theinterplay between the fluid parcels (passive tracers) andthe flow structures.Finally, we discuss the generality of the method pro-posed in Sec. II. Except for the parameters L and τ , theproposed method does not require any information onthe velocity field u ( x , t ). This implies that the proposedmethod can be applied to general (non-periodic in spaceand time) advection-diffusion systems if the fundamentalspace and time scales of a non-periodic fluid motion canbe determined (that is, if L and τ are available). Also,by simulating the spatio-temporal evolution of fluid flowswith CFD techniques and by using it as the pre-definedfield u ( x , t ), the effective diffusion can be characterizedeven when the concentration profile θ ( x , t ) affects thevelocity field u ( x , t ) (i.e., the fluid tracers are not neces-sarily passive). In addition to the computational applica-tion, if the initial pulse ρ ( x ) is realized experimentallyand its evolution ρ ( x , t ) measured (sampled), the pro-posed method makes it possible to characterize the ef-fective diffusion of passive tracers from the measurementdata on ρ ( x , t ), while avoiding the computational burdenin terms of CFD. [1] G. T. Csanady, Turbulent Diffusion in the Environment ,volume 3 (Springer, Netherlands, 2012).[2] S. B. Pope,
Turbulent Flows (Cambridge UniversityPress, Cambridge, 2000).[3] U. Hornung, editor,
Homogenization and Porous Media (Springer-Verlag, New York, 1997).[4] J. A. Knauss and N. Garfield,
Introduction to PhysicalOceanography (Waveland Press, Illinois, 2016). [5] G. A. Pavliotis and A. M. Stuart,
Multiscale Methods:Averaging and Homogenization (Springer-Verlag, NewYork, 2008).[6] A. Bensoussan, J.-L. Lions, and G. Papanicolaou,
Asymptotic Analysis for Periodic Structures , volume 374(American Mathematical Society, Rhode Island, 2011).[7] A. M. Soward, Fast dynamo action in a steady flow, Jour-nal of Fluid Mechanics , 267 (1987). [8] T. Gerz, T. D¨urbeck, and P. Konopka, Transport andeffective diffusion of aircraft emissions, Journal of Geo-physical Research: Atmospheres , 25905 (1998).[9] D. Mu, Z.-S. Liu, C. Huang, and N. Djilali, Determina-tion of the effective diffusion coefficient in porous mediaincluding Knudsen effects, Microfluidics and Nanofluidics , 257 (2008).[10] Y. Kono, Y. Susuki, M. Hayashida, I. Mezi´c, and T. Hik-ihara, Multiscale modeling of in-room temperature dis-tribution with human occupancy data: A practical casestudy, Journal of Building Performance Simulation ,145 (2018).[11] Y. Kono, Y. Susuki, M. Hayashida, and T. Hikihara, Ap-plications of Koopman mode decomposition to modelingof heat transfer dynamics in building atrium—I: Effectiveheat diffusion by small-scale air movement, Transactionsof the Society of Instrument and Control Engineers ,123 (2017), (in Japanese).[12] Y. Kono, Y. Susuki, and T. Hikihara, Modeling of advec-tive heat transfer in a practical building atrium via koop-man mode decomposition, In The Koopman Operator inSystems and Control: Concepts, Methodologies, and Ap-plications , edited by A. Mauroy, I. Mezi´c, and Y. Susuki(Springer Nature, Switzerland, 2020), pp. 481–506.[13] P. McCarty and W. Horsthemke, Effective diffusion coef-ficient for steady two-dimentional convective flow, Phys-ical Review A , 2112 (1988).[14] F. Sagu´es and W. Horsthemke, Diffusive transport in spa-tially periodic hydrodynamic flows, Physical Review A , 4136 (1986).[15] H. K. Moffatt, Transport effects associated with turbu-lence with particular attention to the influence of helicity,Reports on Progress in Physics , 621 (1983).[16] M. B. Isichenko, Percolation, statistical topography, andtransport in random media, Reviews of Modern Physics , 961 (1992).[17] M. Avellaneda and A. J. Majda, An integral represen-tation and bounds on the effective diffusivity in passiveadvection by laminar and turbulent flows, Communica-tions in Mathematical Physics , 339 (1991).[18] A. Fannjiang and G. Papanicolaou, Convection enhanceddiffusion for periodic flows, SIAM Journal on AppliedMathematics , 333 (1994).[19] A. J. Majda and R. M. McLaughlin, The effect of meanflows on enhanced diffusivity in transport by incompress-ible periodic velocity fields, Studies in Applied Mathe-matics , 245 (1993).[20] I. Mezi´c, J. F. Brady, and S. Wiggins, Maximal effectivediffusivity for time-periodic incompressible fluid flows,SIAM Journal on Applied Mathematics , 40 (1996).[21] A. J. Majda and P. R. Kramer, Simplified models forturbulent diffusion: Theory, numerical modelling, andphysical phenomena, Physics Reports , 237 (1999).[22] Y. B. Zel’dovich, Exact solution of the problem of diffu-sion in a periodic velocity field, and turbulent diffusion,Doklady Akademii Nauk , 821 (1982), (in Russian).[23] I. Frankel and H. Brenner, On the foundations of general-ized Taylor dispersion theory, Journal of Fluid Mechanics , 97 (1989).[24] A. Fannjiang, Time scales in homogenization of periodic flows with vanishing molecular diffusion, Journal of Dif-ferential Equations , 433 (2002).[25] H. Owhadi and L. Zhang, Metric-based upscaling, Com-munications on Pure and Applied Mathematics: A Jour-nal Issued by the Courant Institute of Mathematical Sci-ences , 675 (2007).[26] H. Owhadi and L. Zhang, Homogenization of parabolicequations with a continuum of space and time scales,SIAM Journal on Numerical Analysis , 1 (2008).[27] L. Berlyand and H. Owhadi, Flux norm approach to finitedimensional homogenization approximations with non-separated scales and high contrast, Archive for RationalMechanics and Analysis , 677 (2010).[28] I. Babuska and R. Lipton, Optimal local approximationspaces for generalized finite element methods with ap-plication to multiscale problems, Multiscale Modeling &Simulation , 373 (2011).[29] P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoˇs, Dif-fusion and mixing in fluid flow, Annals of Mathematics , 643 (2008).[30] E. Shuckburgh, H. Jones, J. Marshall, and C. Hill, Ro-bustness of an effective diffusivity diagnostic in oceanicflows, Journal of Physical Oceanography , 1993 (2009).[31] E. Shuckburgh, H. Jones, J. Marshall, and C. Hill, Under-standing the regional variability of eddy diffusivity in thePacific sector of the Southern Ocean, Journal of PhysicalOceanography , 2011 (2009).[32] F. ´dOvidio, E. Shuckburgh, and B. Legras, Local mixingevents in the upper troposphere and lower stratosphere.Part I: Detection with the Lyapunov diffusivity, Journalof the Atmospheric Sciences , 3678 (2009).[33] L. Ljung, System Identification: Theory for the User (PTR Prentice Hall, Upper Saddle River, N.J., 1999).[34] I. Mezi´c and S. Wiggins, On the dynamical origin ofasymptotic t dispersion of a nondiffusive tracer in in-compressible laminar flows, Physics of Fluids , 2227(1994).[35] S. W. Smith, The Scientist and Engineer’s Guide to Dig-ital Signal Processing (California Technical Pub., SanDiego, 1997).[36] C.-W. Hsu, C.-C. Chang, and C.-J. Lin, A practical guideto support vector classification. Technical Report, De-partment of Computer Science, National Taiwan Univer-sity (2003).[37] D. Chicco, Ten quick tips for machine learning in com-putational biology, BioData mining , 35 (2017).[38] E.-G. Talbi, Metaheuristics: From Design to Implemen-tation , volume 74 (John Wiley & Sons, Danvers, MA,2009).[39] The symbol (cid:62) stands for transpose operation of vectors.[40] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P.Flannery,
Numerical Recipes in C (Cambridge UniversityPress, Cambridge, 1988).[41] J. M. Ottino,
The Kinematics of Mixing: Stretching,Chaos, and Transport (Cambridge University Press,Cambridge, 1989).[42] W. Tang and P. Walker, Finite-time statistics of scalardiffusion in Lagrangian coherent structures, Physical Re-view E , 045201 (2012).[43] G. Haller, Lagrangian coherent structures, Annual Re-view of Fluid Mechanics47