Effective transport in random shear flows

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Effective transport in random shear flows

Marco Dentz, Tanguy Le Borgne, Jesus Carrera

To cite this version:

Marco Dentz, Tanguy Le Borgne, Jesus Carrera. Effective transport in random shear flows. Physical

Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2008, 77,

pp.020101. �10.1103/PhysRevE.77.020101�. �insu-00372061�

Effective transport in random shear flows

Marco Dentz

*

Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia (UPC), Barcelona, Spain Tanguy Le Borgne

Géosciences Rennes, UMR 6118, CNRS, Université de Rennes 1, Rennes, France Jesus Carrera

Institute of Earth Sciences Jaume Almera, CSIC, Barcelona, Spain

共Received 6 July 2007; revised manuscript received 11 October 2007; published 6 February 2008兲

We obtain an effective transport description for the superdiffusive motion of random walkers in stratified flow by projection of the process on the direction of stratification. The effective dimensionally reduced motion is shown to describe a correlated random walk characterized by the Lagrangian velocity correlation. We analyze the projected motion through exact analytical solutions for the distribution density for an arbitrary correlated Gaussian noise and derive an evolution equation for the one-point and conditional two-point dis-placement densities. The latter gives an explicit effective equation for superdiffusive transport in stratified random flow and demonstrates that the displacement density has a Gaussian scaling form for all times. DOI:10.1103/PhysRevE.77.020101 PACS number共s兲: 05.40.Fb, 05.10.Gg, 05.60.⫺k, 92.40.Kf

Non-Markovian transport and non-Fickian and anomalous diffusion are frequently observed in disordered dynamic sys-tems共e.g., 关1–3兴兲. Many studies have focused on the

devel-opment of effective non-Markovian transport frameworks based on the assumption of heavy-tailed uncorrelated disor-der distributions共e.g., 关4,3兴兲. Here we study the

complemen-tary scenario of non-Markovian processes originating from 共strong兲 correlations by focusing on effective transport in stratified random flow共e.g., 关5–7兴兲. Matheron and de Marsily

关5兴 studied the latter as a model for solute transport in

strati-fied porous media and found superdiffusive growth of the distribution density in the direction of stratification. Such anomalous and in general non-Fickian transport features dominate solute transport in geological formations.

Many applications, ranging from performance assessment of nuclear waste repositories in geological media to ground-water remediation, require identifying and quantifying the effective large scale transport dynamics caused by the inter-action of medium heterogeneities and microscale transport dynamics关e.g., 关8,9兴兴. The latter implies spatial or stochastic

averaging of the microscale transport equations, i.e., dimen-sional reduction or in general projection of the process共e.g., 关10,11兴兲. Such a contraction of information in general leads

to a non-Markovian property of the projected process关11兴

that can be cast into formal effective equations by using the projector formalism proposed by Zwanzig and Mori关12,13兴,

see, e.g.,关14兴 for an application to transport in random

me-dia. Here, we derive an explicit effective transport equation and explicit solutions for the distribution densities of the projected process by exploiting the fact that the projected motion of random walkers in stratified flow describes a cor-related random walk characterized by its Lagrangian velocity correlation. It has been shown that the latter plays an impor-tant role for characterizing non-Fickian transport in

hetero-geneous media in general共e.g., 关15兴兲. In the following, we

pose the projection problem for transport in stratified random flow, which is solved by共i兲 demonstrating that the projected motion of a random walker describes a correlated random walk;共ii兲 derivation of a 共non-Markovian兲 transport equation for correlated random walks. These steps render an explicit effective transport framework for stratified random media.

Transport of a passive scalar c共x,t兲 in a stratified flow field u共y兲 is described by the Fokker–Planck equation 共e.g., 关5兴兲, ⳵c共x,t兲 ⳵t = − u共y兲 ⳵c共x,t兲 ⳵x + D ⳵2_c共x,t兲 ⳵y2 , 共1兲

where the coordinate vector is given by x =共x,y兲T_{; D is the}

diffusion coefficient. For simplicity, we restrict this study to an infinite d = 2 dimensional domain, no diffusion in flow direction. The spatial density c共x,t兲 and its gradient normal to the boundaries at infinity are zero. The normalized initial distribution c共x,t=t0兲=␳共y兲␦共x兲, where␳共y兲 is normalized to

one, its variance is denoted by a2_{. For a
1,␳共y兲 is assumed}

to have the scaling form,

␳共y兲 =1

af␳

冉

y

a

冊

, 共2兲

where f_␳共0兲 is constant. The stratified flow field u共y兲 is a realization of the stationary spatial Gaussian random process 兵u共y兲其 characterized by the constant ensemble mean velocity

uE= u共y兲 and the correlation of the velocity fluctuations

u

⬘

共y兲=u共y兲−uE, which is given by CE共y−y

⬘

兲=u

⬘

共y兲u

⬘

共y

⬘

兲.

The overbar denotes the ensemble average over all realiza-tions of兵u共y兲其.

In order to obtain a simplified effective transport descrip-tion, the spatial density is projected on the x direction by, *marco.dentz@upc.edu

c

¯共x,t兲 ⬅

冕

−⬁ ⬁

dyc共x,t兲. 共3兲

We determine a non-Markovian evolution equation and ex-plicit analytical expressions for the average density c¯共x,t兲 by

formulating the projected transport problem as a correlated random walk.

The starting point is the Langevin equation equivalent to the Fokker-Planck equation共1兲, which is given by 共e.g., 关6兴兲,

dX共t兩b兲

dt = uE+ u

⬘

关Y共t兩b兲兴,

dY共t兩b兲

dt =␰共t兲. 共4兲 We used here the decomposition u共y兲=uE+ u

⬘

共y兲; ␰共t兲 is a

Gaussian white noise with zero mean and correlation 具␰共t兲␰共t

⬘

兲典=2D␦共t−t

⬘

兲, where the angular brackets denote the white noise average. The initial particle position at time

t = t0 is given by 关X共t0兩b兲,Y共t0兩b兲兴=共0,b兲. Particles are

ini-tially distributed along the y axis according to ␳共b兲. The distribution density c共x,t兲 in this picture is given by

c共x,t兲 =

冕

−⬁ ⬁

db␳共b兲具␦关x − X共t兩b兲兴␦关y − Y共t兩b兲兴典. 共5兲 Application of共3兲 to 共5兲 gives for the average density,

c

¯共x,t兲 =

冕

−⬁ ⬁

db␳共b兲具␦关x − X共t兩b兲兴典 ⬅ 具␦关x − X共t兲兴典_␨. 共6兲 The trajectory X共t兲 describes the d=1 dimensional random walk,

dX共t兲

dt = uE+␨共t兲, 共7兲

with the initial position X共t=t0兲=0 and noise ␨共t兲

= u

⬘

关y共t,b兲兴. The latter is Gaussian distributed because u

⬘

共y兲 is a Gaussian random field. The noise average, denoted by 具·典␨, is defined in共6兲 and consists of taking the average over

the white noise␰共t兲 and integration over the source distribu-tion␳共b兲. Thus, noise mean and correlation are given by

具␨共t兲典␨=

冕

−⬁ ⬁ dy

冋

冕

−⬁ ⬁

db␳共b兲u

⬘

共y + b兲

册

g共y,t兩0兲, 共8兲

具␨共t兲␨共t

⬘

兲典␨=

冕

−⬁ ⬁ dy

冕

−⬁ ⬁ dy

⬘

冋

冕

−⬁ ⬁

db␳共b兲u

⬘

共y + b兲u

⬘

共y

⬘

+ b兲

册

⫻g共y,t − t

⬘

兩y

⬘

兲g共y,t

⬘

兩0兲, 共9兲 where g共y,t−t

⬘

兩y

⬘

兲=具␦关y−y共t,b兲兴典Y共t⬘兲=y⬘ is the conditional

density of the transverse diffusion process in 共4兲. For the

infinitely extended medium under consideration here, it is given by

g共y,t兩y

⬘

兲 =

_冑

1

4␲Dtexp

冋

−

共y − y

⬘

兲2

4Dt

册

. 共10兲

In the limiting case of a
1 and using the scaling form 共2兲, the expressions in the square brackets in 共8兲 and 共9兲 can

be substituted by the respective ensemble averages,

lim a→⬁ 1 a

冕

_−⬁ ⬁ dbf

冉

b a

冊

u

⬘

共y + b兲 ⬅ u

⬘

共y兲 = 0, 共11兲 lim a→⬁ 1 a

冕

−⬁ ⬁ dbf

冉

b

a

冊

u

⬘

共y + b兲u

⬘

共y

⬘

+ b兲

= u

⬘

共y兲u

⬘

共y

⬘

兲

= CE共y − y

⬘

兲, 共12兲

where we used共2兲 and the ergodicity of 兵u共y兲其. Thus, in this

limit the Lagrangian mean具␨共t兲典_␨⬅0. Using 共12兲 in 共9兲, we

obtain for the Lagrangian velocity correlation, 具␨共t兲␨共t

⬘

兲典␨⬅ CL共t − t

⬘

兲 =

冕

−⬁ ⬁

dyCE共y兲g共y,t − t

⬘

兩0兲.

共13兲 Thus, for sufficiently large source distributions, the noise␨共t兲 can be approximated by a stationary correlated Gaussian noise. The projected motion of a random walker in stratified random flow is a correlated random walk characterized by the correlated Gaussian noise ␨共t兲, with zero mean and cor-relation共13兲. As such, we can obtain an effective transport

framework and a complete effective description for transport in stratified random flow by deriving solutions for the n-point densities of correlated random walks, which are non-Markovian by nature. In the following, we derive explicit solutions for correlated random walks.

Discretizing共7兲 in time, we obtain

X共tN+␶兲 = X共tN兲 + uE␶+␨共tN兲␶, 共14兲

with the constant time increment␶and discrete time tN= N␶.

The random walk 共14兲 has the sharp initial position

X共t₀兲=x₀at time t₀= N₀␶. The random force␨共tN兲 kicks in at

time tN₀. In order to make the process well defined, for

tN⬍tN₀ we consider the process deterministic or driven by

white noise. Thus, for tN⬎tN₀, the evolution of X共tN兲 does

not depend on the system states previous to tN₀. The series of

random velocities␨共tn兲 共n=N0, . . . ,⬁兲 describes the

stochas-tic process兵␨共tn兲其. The latter can be described by its

charac-teristic function, which is defined by the Fourier transform of the multivariate distribution density共e.g., 关10兴兲,

G共兵␬共tn兲其兲 =

冓

exp

冋

i

兺

n

␬共tn兲␨共tn兲

册

冔

␨

. 共15兲

The兵␬共tn兲其 denote the Fourier variables conjugate to 兵␨共tn兲其.

For the multi-Gaussian correlated processes␨共t兲, it is given by

G共兵␬共tn兲其兲 = exp

冋

−

1

2

兺

_nn⬘␬共tn兲CL共tn− tn⬘兲␬共tn⬘兲

册

. 共16兲 Note that 兵␨共tn兲其 is Gaussian white noise for CL共tn− tn⬘兲

⬀␦nn⬘. Systems driven by strongly correlated noise have

been studied in various contexts 共e.g., 关16兴兲 ranging from

Brownian motion in fractals关e.g. 关17,18兴兴 to the description

of animal movements共e.g., 关19兴兲.

DENTZ, LE BORGNE, AND CARRERA PHYSICAL REVIEW E 77, 020101共R兲 共2008兲

The correlated random walk共14兲 is non-Markovian. Note

that a non-Markovian process is not sufficiently character-ized by its one-point density and transition probability. Here, for simplicity we focus only on the latter two distributions.

In order to derive an explicit expression for c¯共x,t兲, we

perform a Fourier transform of共6兲, which gives for c¯˜共k,t兲 in

discrete time,

c ¯

˜共k,tN兲 = 具exp关ikX共tN兲兴典␨, 共17兲

where k is frequency and the tilde denotes Fourier-transformed quantities. From共14兲 we obtain for X共tN兲,

X共tN兲 = x0+

兺

l=N0 N−1 uE␶+

兺

l=N0 N−1 ␨共tl兲␶. 共18兲

Inserting the latter into 共17兲, and subsequently performing

the noise average, gives

c ¯ ˜共k,tN兲 = exp

冋

ik

冉

x0−

兺

i=N0 N−1 uE␶

冊

册

⫻G共兵␬共ti兲 = k其i=NN−1₀;兵␬共ti兲 = 0其iⱖN兲, 共19兲

which can be verified by inserting共15兲 into 共19兲.

For process times tN that are large compared to the

con-stant time increment, tN
␶, we take the limit to continuous

time, set tN= t, and identify

兺

i=N0 N−1 ␸共ti兲␶⬅

冕

t₀ t dt

⬘

␸共t兲 共20兲

for an arbitrary function␸共t兲. Thus, we obtain from 共19兲 by

using共16兲 and taking the limit of continuous time,

c ¯ ˜共k,t兲 = exp

冋

− k2

冕

0 t−t0 dt

⬘

D共t

⬘

兲 + ikuE共t − t0兲

册

, 共21兲

where we defined the time-dependent diffusion coefficient,

D共t兲 =

冕

0

t

dt

⬘

CL共t

⬘

兲. 共22兲

A similar method was used in Ref.关18兴 for the derivation of

the evolution equation for the one-point density for a gener-alized Langevin equation with power-law correlated noise. Note that with the same method as outlined above all n-point densities of the correlated random walk共7兲 can be calculated

关23兴.

Derivation of共21兲 with respect to time and transforming

the resulting equation back to real space, we obtain the fol-lowing evolution equation for the average concentration,

⳵ ⳵t¯c共x,t兲 + uE ⳵ ⳵x¯c共x,t兲 − D共t − t0兲 ⳵2 ⳵x2¯c共x,t兲 = 0 共23兲

with the initial condition c¯共x,t=t₀兲=␦共x−x₀兲. Note that the

average concentration is identical to the two-point condi-tional density,

P共x,t兩x

⬘

,t

⬘

兲 = 具␦关x − X共t兲兴典_␨

兩

X共t⬘兲=x⬘ 共24兲

for t

⬘

= t0. As outlined above, for consistency reasons, we

assume that the correlated noise is switched on at t0while for

t⬍t₀ the noise is uncorrelated or zero. Thus,共23兲 expresses

the fact that the system remembers the point in time when the correlated random force is switched on, i.e., the non-Markovianity of the projected process. Seen as the evolution equation for P共x,t兩x

⬘

, t

⬘

兲, 共23兲 expresses the system’s

memory of a measurement at time t

⬘

.

The solution of共23兲 is a Gaussian pulse, i.e.,

c

¯共x,t兲 = 关2␲␴2_共t兲兴−1/2_{exp关− 共x − u}

Et兲2/␴2共t兲兴, 共25兲

where we set t0= 0 and defined the variance␴2共t兲 by

␴2_{共t兲 =}

冕

0

t

dt

⬘

D共t

⬘

兲, 共26兲

where D共t兲 is given by the general expression 共22兲. In terms

of the Eulerian velocity correlation function, D共t兲 is given by 共e.g., 关5,20兴兲, D共t兲 =

冕

0 t dt

⬘

冕

−⬁ ⬁ dyCE共y兲g共y,t

⬘

兩0兲, 共27兲

which can be obtained by inserting共13兲 into 共22兲. Note that

the average density共25兲 is always Gaussian, irrespective of

the noise correlation CL共t兲 and thus of the Eulerian velocity

correlation CE共y兲. Figure1shows the average concentration

c

¯共x,t兲 obtained by random walk simulations compared to the

analytical solution 共25兲. Analytical and numerical solutions

are in perfect agreement.

For times t
␶D= l2/D and a CE共y兲 that decays sharply on

the correlation scale l, it is easy to show that D共t兲⬀

冑

t; thus

we find for the variance共26兲␴2_共t兲⬀t3/2_{. For delta-correlated}

velocity, i.e., CE共y兲⬀␦共y兲, this is exactly so for all times 关6兴.

0 0.005 0.01 0.015 0.02 0.025 400 450 500 550 600 650 700 750 800 eff ect ive d ens ity x

FIG. 1. Effective distributions obtained from共triangles兲 projec-tion of the random walk共4兲 to the direction of stratification using

numerical random walk simulations and共solid兲 expression 共25兲, at

two different times. The velocity mean and autocorrelation are u_E = 10 and CE共y兲⬀␦共y兲.

Thus, for zero bias, i.e., uE= 0, and delta-correlated velocity,

the projected density c¯共x,t兲, 共25兲, has the exact scaling form,

c

¯共x,t兲 = t−3/4f共x/t3/4兲, 共28兲

where f⬀exp共−u2兲. The scaling form 共28兲 has been

conjec-tured by 关6,21兴. However, unlike argued there, the scaling

function f共u兲 has Gaussian shape for all times as the solution

c

¯共x,t兲 of 共23兲 for uE= 0 is a Gaussian pulse with variance

proportional to t3/2 according to 共25兲 and 共26兲. This is

con-firmed by comparison with explicit random walk simulation as illustrated in Fig.1.

The analysis of effective transport in stratified random flow in terms of correlated random walks is conditioned by

共i兲 the velocity distribution and 共ii兲 the initial distribution of random walkers, which determine the statistical characteris-tics of the effective correlated noise. For source distributions with an extension of the order of the correlation scale, pre-asymptotic transport is dominated by the details of the local velocities, which can induce nonlocal terms in the effective transport equation 共e.g., 关22兴兲. This memory is expected to

persist until the distribution is spread out vertically by diffu-sion over a distance that is much larger than the correlation scale and has sampled a representative part of the flow vari-ability. A line source with a transverse extension that is large compared to the correlation scale wipes this memory out from the very beginning, thus giving the Gaussian scaling form共25兲 at all times.

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DENTZ, LE BORGNE, AND CARRERA PHYSICAL REVIEW E 77, 020101共R兲 共2008兲