J. Fluid Mech. (2012), vol. 699, pp. 94–114.
c
Cambridge University Press 2012 94
doi:10.1017/jfm.2012.85
Self-similar ity in coupled
Br inkman/Navier–Stokes flows
Ilenia Battiato†
Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, G
¨
ottingen, Germany
(Received 2 April 2011; revised 21 January 2012; accepted 10 February 2012;
first published online 24 April 2012)
In this paper we derive self-similar solutions of flows through both a porous medium
and a pure fluid. Self-similar filtration velocity and hydrodynamic shear profiles are
obtained by means of asymptotic analysis in the limit of infinitely small permeability,
and for both laminar and turbulent regimes over the porous medium. We show that a
spatial length scale, related to the porous layer thickness, naturally emerges from the
limiting process and suggests a more formal definition of thick and thin porous media.
We finally specialize the analysis to porous media constituted of patterned cylindrical
obstacles, which can freely deflect under the aerodynamic shear exerted by the fluid
flowing through and over the forest. A self-similar solution for the bending profile of
the elastic cylindrical obstacles is obtained as intermediate asymptotics, and applied
to carbon nanotube (CNT) forests’ response to aerodynamic stresses. This self-similar
solution is successfully used to estimate flexural rigidity of CNTs by linear fit of
appropriately rescaled maximum deflection and average velocity measurements.
Key words:
flow–structure interactions, porous media
1. Introduction
Coupled flows through and over porous layers occur in a variety of natural
phenomena, biological systems and industrial processes. Some examples include
flow over sediment beds (Goharzadeh, Khalili & Jorgensen 2005), coral reefs and
submerged vegetation canopies (Ghisalberti & Nepf 2009), crop canopies and forests
(Kruijt et al. 2000), endothelial glycocalyx of blood vessels (Weinbaum et al. 2003),
and polymer brushes (Tachie, James & Currie 2004). Coupled flows also occur in
packed-bed heat exchangers, thermal insulation, geothermal engineering, and nuclear
waste repositories (Tien & Vafai 1990, and references therein), just to mention a few.
The main focus of the majority of works on the subject has been on the formulation
of appropriate conditions at the interface separating the pure fluid from the porous
medium flow. This has been an area of active research since the seminal works
of Beavers & Joseph (1967) and Taylor (1971). Two main approaches have been
used to couple flow over and through permeable layers: single- and multiple-domain
methods. While the former treats the system as a single domain with spatially variable
permeability, the latter employs two different mathematical models for the porous
medium and the free-fluid region, and enforces boundary conditions for the tangential
† Present address: Department of Mechanical Engineering, Clemson University, Clemson,
SC 29634, USA. Email address for correspondence: [email protected]
Self-similarity in coupled Brinkman/Navier–Stokes flows 95
Channel flow
PM flow
0
–
H
2L
FIGURE 1. Schematic of the problem. Fluid flows in a channel. The bottom of the channel
is occupied by a porous medium of permeability K. The computational domain is divided
into two regions: the channel flow region for ˆy 2 (0, 2L) and the porous medium region for
ˆy 2(H, 0).
velocity and shear at the liquid–porous matrix interface. A classical (multiple-) domain
approach consists of solving (Navier–) Stokes equations in the free-fluid region and
Darcy’s law in the porous medium. Many types of boundary conditions have been
postulated, e.g. discontinuity of the interfacial velocity (Beavers & Joseph 1967;J
¨
ager
& Mikeli
´
c 2000), continuity of tangential velocity and discontinuity of the shear stress
(Ochoa-Tapia & Whitaker 1995), discontinuity of both tangential velocity and shear
stress (Cieszko & Kubik 1999), and continuity of the velocity vector at the viscous
transition zone interface as defined in Bars & Worster (2006). Multiple-domain models
also include more complex formulations for the flow in the porous matrix, such
as the Brinkman equation (Brinkman 1947), where higher-order (viscous) correction
terms are retained in the upscaling process (Auriault 2009). A rigorous analysis of
differences and commonalities between the various formulations is given by Bars &
Worster (2006).
While the question of proper boundary conditions still remains open, in the present
paper we focus on a rather different aspect of the problem. Specifically, our main
objective is to establish self-similarity of coupled flows inside and over a permeable
layer. To this end, we employ a multiple-domain approach to describe the system
under consideration (see figure 1). Such a model allows us to naturally couple the
porous medium flow to both laminar and turbulent flow in the pure-fluid domain with
the significant advantages of (i) increased generality of our results, and (ii) reducing
the problem to an analytically tractable case, for which closed-form solutions are
available (Battiato, Bandaru & Tartakovsky 2010). Self-similar behaviour is obtained
as an intermediate asymptotic of the exact solution in the limit of infinitely small
permeability. A classification between thin and thick porous media, based on a well-
defined spatial length scale ⇤, naturally arises from the asymptotic analysis. The
dynamical response of flows through thin (⇤ 1) and thick (⇤ ⌧ 1) porous media
exhibits different asymptotic behaviours. This renders the suggested classification
quantitative and reproducible. We then extend the analysis to a forest of deformable
cylindrical obstacles. Self-similar deflection profiles are derived for both thin and
thick forests. While our main motivation stems from predicting elastic deformations of
forests of aerodynamically-sheared carbon nanotubes from our theoretical (self-similar)
solutions, the proposed framework can be straightforwardly applied to completely
different systems including flow inside and above canopies (e.g. Ghisalberti & Nepf
2009) and the endothelium glycocalyx in blood vessels (Weinbaum et al. 2003), just to
mention a few.
96 I. Battiato
The manuscript is organized as follows. After a brief description of the model (§ 2),
we discuss some limitations of dimensional analysis as a tool to identify a self-similar
solution for the problem under consideration (§ 3). In § 4, we look for self-similarity
of the velocity and shear profiles by asymptotic analysis. We then apply such results
to flow through a forest of vertically aligned deformable cylindrical obstacles (§ 5). We
show that the elastic response of the forest to the aerodynamic shear exerted by the
flowing fluid admits a self-similar solution. Finally in § 6, our theoretical predictions
are employed to estimate the flexural rigidity of carbon nanotubes (CNTs) by a linear
fit of (appropriately rescaled) data collected by Deck et al. (2009). The main results
and conclusions are summarized in § 7.
2. F low over a porous layer: model formulation and analytical solutions
We consider a two-dimensional fully developed incompressible fluid flow in a
channel formed by two impermeable walls separated by the distance of H + 2L
(figure 1). The bottom part of the flow domain, H < ˆy < 0, is occupied by a porous
medium of permeability K. The flow is driven by an externally imposed (mean)
constant pressure gradient dˆp/dˆx < 0. We use a two-domain approach since it allows
us to couple the Darcy–Brinkman equation in the porous layer with both laminar
and turbulent flow in the rest of the channel in a quite straightforward manner. We
here employ classical continuity conditions of both tangential velocity and shear stress
(Vafai & Kim 1990; Kim & Choi 1996) as they have been shown to be sufficiently
accurate for our modelling purposes (Battiato et al. 2010).
2.1. Two-domain approach
The flow in the porous medium region, ˆy 2 (H, 0), can be described by the Brinkman
equation for the horizontal component of the intrinsic average velocity ˆu (Auriault
2009),
µ
e
d
2
ˆu
dˆy
2
µ
K
ˆu
dˆp
dˆx
= 0, ˆy 2 (H, 0), (2.1)
where µ is the fluid’s dynamic viscosity, and µ
e
is its ‘effective’ viscosity that
accounts for the slip at the solid–liquid boundary between the porous matrix and
the fluid if µ
e
6= µ. In the rest of the flow domain, ˆy = (0, 2L), we use steady-state
Reynolds or Stokes equations to describe fully developed flow in turbulent ( = 1)
(Pope 2000, (7.8)), or laminar regimes ( = 0)
µ
d
2
ˆu
dˆy
2
⇢
dhˆu
0
ˆv
0
i
dˆy
dˆp
dˆx
= 0, ˆy 2 (0, 2L), (2.2)
where ⇢ is the fluid density and ˆu(ˆy) is the horizontal component of flow velocity
ˆ
u(ˆu, ˆv). In laminar flow, ˆu(ˆy) is the actual velocity and ˆv( ˆy) = 0. In a turbulent
regime,
ˆ
u denotes the mean velocity, ˆu
0
and ˆv
0
are the velocity fluctuations about their
respective means, and hˆu
0
ˆv
0
i is the Reynolds stress. Fully-developed turbulent channel
flow has velocity statistics that depend on ˆy only. The no-slip condition is imposed at
ˆy = H and ˆy = 2L, and the continuity of velocity and shear stress is prescribed at the
interface, ˆy = 0, between the free and filtration flows (Vafai & Kim 1990):
ˆu(H) = 0, ˆu(2L) = 0, ˆu(0
) =ˆu(0
+
) =
ˆ
U,µ
e
✓
dˆu
dˆy
◆
ˆy=0
= µ
✓
dˆu
dˆy
◆
ˆy=0
+
,
(2.3)
Self-similarity in coupled Brinkman/Navier–Stokes flows 97
where
ˆ
U is an unknown matching velocity at the interface between the channel flow
and porous medium.
2.2. Analytical solutions
Choosing the height of the porous layer, H, the velocity q = (H
2
/µ) dˆp/dˆx, and
the fluid viscosity, µ, as the repeating variables, the problem can be formulated in
dimensionless form. Inside the porous medium, the solution for the dimensionless
velocity distribution u =ˆu/q is given by (Battiato et al. 2010)
u(y) =
1
M
2
+ C
1
e
y
+ C
2
e
y
, y 2 [1, 0], (2.4a)
where
C
1,2
=±
1
M
2
(M
2
U 1)e
±
+ 1
e
e
, U =
1 sech
M
2
+
M
tanh , (2.4b)
y =
ˆy
H
, M =
µ
e
µ
,
2
=
H
2
MK
, =
L
H
, (2.4c)
with U =
ˆ
U/q the dimensionless interfacial velocity, and = 1 or = 1 +
(tanh )/(2M) for turbulent or laminar regime in the channel, respectively. The
different values of for laminar and turbulent regimes reflect the coupling of
flow fields in and above the porous medium at the interface y = 0
+
. In the
laminar regime, the flow velocity above the porous layer (0 6 y 6 2) is given by
u(y) = y
2
/2 + ( U/2)y + U. In the turbulent regime, assuming the surface of
the porous medium is hydrodynamically smooth, the dimensionless mean velocity u
in the viscous sublayer of (dimensionless) width
⌫
obeys the law of the wall (Pope
2000, pp. 270–271). This yields (Battiato et al. 2010) u(y) = y + U for 0 6 y 6
⌫
.
The parameter
2
is inversely proportional to the Darcy number, usually defined as
a dimensionless permeability, i.e. K/H
2
. Velocity profiles in the porous layer for
different values of are shown in figure 2 (inset a) and in the inset of figure 4. The
drag force per horizontal unit area exerted by the fluid on any cross-section y 2 [1, 0)
in the porous medium is given by the xy-component of the dimensionless stress tensor
=ˆ H/qµ,
(y) = M
du
dy
= M|C
1
e
y
C
2
e
y
|, y 2 [1, 0]. (2.5)
Drag force profiles are plotted in inset (b) of figures 3 and 5 for different values of
. In the following, we intend to identify self-similarity behaviour of the velocity (2.4)
and drag force (2.5).
3. Buckingham’s ⇧ theorem and its limits
A standard approach to find self-similar solutions to boundary value problems is
to use dimensional analysis. Assuming the solution to (2.1)–(2.3) was unknown and
choosing (q,µ,H) as a set of variables with independent units, dimensional analysis
and Buckingham’s ⇧ theorem would lead to
ˆu = q
F
1
(y, , M, Re
p
, ) and ˆ = µqH
1
F
2
(y, , M, Re
p
, ), (3.1)
where Re
p
= ⇢qH/µ. However, in (3.1) the number of dimensionless groups is too big
to suggest any self-similarity by simple inspection. While for a laminar regime ( = 0)
98 I. Battiato
and M = 1(3.1) simplifies to
F
i
(
y, , 1, 0,
)
=
G
i
(
y, ,
)
, i ={1, 2}, (3.2)
nonetheless the identification of a self-similar behaviour, or even its existence, still
remains a challenging task. Further progress could be made by looking at !1, as
we are mostly interested in the system response in the low permeability limit. In this
case the asymptotic solution requires that ! +1 so that
lim
!+1
G
i
(y, , ) ⌘ g
i
(y, ) (3.3)
exists, is finite and does not vanish. However, the assumption that lim
!+1
G
i
6= 0
is violated, and, if a self-similar solution of the second type exists, it has the more
general form (Barenblatt & Zel’dovich 1972)
G
i
(y, , ) =
↵
˜g
i
(
y, ) as ! +1, (3.4)
where ↵ and are anomalous exponents and ˜g
i
, i 2 {1, 2}, is finite. The turbulent case
still remains a more challenging task due to the higher number of parameters.
Since exact expressions for the velocity and shear are available for both laminar
and turbulent regimes, we take advantage of the connection between self-similarity and
intermediate asymptotics to identify a self-similar behaviour. Such connection can be
summarized in the following statement: quoting from Barenblatt & Zel’dovich (1972),
the self-similar solutions correspond to degenerate problems, which are obtained when
some of the parameters tend to zero or infinity, and are simultaneously the exact special
solutions of these degenerate problems with a fewer number of governing parameters and
the asymptotic representations of non-self-similar solutions.
In § 4 we seek self-similar behaviour by studying the asymptotics of the solution in
the limit ! +1. This approach has the advantage of allowing a unifying treatment
of both the laminar and turbulent regimes. Also, we show that a spatial length scale,
⇤, related to the porous layer thickness, naturally emerges from the limiting process
and suggests a more formal definition of thick and thin porous media.
4. Self-similarity as intermediate asymptotics
We aim to determine the behaviour of the system in the limit ! +1 if such
a limit exists that is finite and different from zero, or the asymptotic behaviour,
otherwise. We say that f (x) is asymptotic to g(x) in some neighbourhood of x
0
,
I
x
0
, and we write f (x) ⇠ g(x) for x ! x
0
if lim
x!x
0
f (x)/g(x) = 1 and g(x) does not
vanish in I
x
0
\{x
0
}. For convenience, we introduce a rescaled variable y
?
= y, with
y
?
2 [, 0]. Furthermore, the notation ! +1 will be always implied whenever an
asymptotic behaviour is calculated. We first determine lim
!+1
u(y
?
;) which requires
a preliminary calculation of lim
!+1
U() and lim
!+1
C
i
(), i = 1, 2. By simple
inspection,
lim
!+1
U() = 0. (4.1)
The limits for C
1
and C
2
can be calculated as follows:
lim
!+1
C
1
() = lim
!+1
U 1/M
2
+ e
/M
2
1 e
2
= 0, (4.2a)
lim
!+1
C
2
() = lim
!+1
(U 1/M
2
)e
2
+ e
/M
2
1 e
2
= 0. (4.2b)
Self-similarity in coupled Brinkman/Navier–Stokes flows 99
Substitution of (4.1) and (4.2) in (2.4a) leads to the indeterminate form [0 · 1].
Therefore, a calculation of the asymptotic behaviour is required to make progress in
the calculation. Since sech ⇠ 2/e
and tanh ⇠ 1, and keeping the leading-order
terms, then
U ⇠
1
M
2
(⇤ + 1), (4.3)
where ⇤ =  and = 1 or (⇤) ⇠ 1 + 1/2M⇤ for a turbulent or laminar regime,
respectively. Since 1 e
2
⇠ 1, at the leading order we obtain
C
1
⇠
1
M
2
(⇤ + 1 ), (4.4a)
C
2
⇠
1
M
2
e
. (4.4b)
Equations (4.3) and (4.4) exhibit two different limits depending on whether ⇤ ⌧ 1
or ⇤ 1. The parameter ⇤ naturally introduces a vertical length scale associated
with the porous medium thickness relative to the channel height. We therefore refer to
systems with ⇤ 1 and ⇤ ⌧ 1 as thin and thick porous media, respectively. In §§ 4.1
and 4.2 we investigate these two limits separately.
4.1. Thin porous media
4.1.1. Flow velocity
Keeping the leading-order term in (4.3)–(4.4) gives
U ⇠
M
and C
1
⇠
M
, (4.5)
with = 1 for both laminar and turbulent regimes if ⇤ 1. Inserting (4.5) in (2.4a),
leads to the following asymptotic:
u
?
(y
?
;) := M
u(y
?
/;)
1
M
2
⇠ e
y
?
✓
1
1
⇤
e
2y
?
◆
, (4.6)
where u
?
is a rescaled velocity. We stress that u
?
exhibits two different limits, one of
which being self-similar, depending on the sign of the exponent. Specifically
u
?
(y
?
) ⇠ e
y
?
, y
?
2 (/2, 0], (4.7a)
u
?
(y
?
;) ⇠ e
y
?
✓
1
1
⇤
e
2y
?
◆
, y
?
2 (, /2]. (4.7b)
In figure 2 the rescaled velocity u
?
, defined by (4.7a) (dashed line (green online))
and (4.7b) (crosses (magenta online)), is plotted for seven different values of
. The black points represent the full velocity profile given in (2.4) (plotted
in inset a of figure 2). Figure 2 shows that u
?
indeed exhibits a self-similar
behaviour for 1 and away from the bottom boundary of the channel, i.e.
y
?
2 (/2, 0]. The internal boundary separating porous medium and free flow
does not destroy the self-similarity of the solution. We also stress that the
self-similar (dimensionless) solution (4.7) for flow in a thin porous layer is
independent of the type of flow regime (laminar or turbulent) over the porous
medium.
100 I. Battiato
0
20
40
60
80
100
120
–10
3
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
4
–10
–3
(a)
y
–0.8
–0.6
–0.4
–0.2
–1.0
0
u(y)
10
–5
10
0
0
–0.05
0.05
(b)
–50 –5
FIGURE 2. (Colour online available at journals.cambridge.org/flm) Dimensionless velocity
profile u
?
(y
?
) inside a thin (⇤ 1) porous medium for M = 1 and = 100. For this set of
parameters ⇠ 1, and the velocity profile in the porous medium is independent of whether
the flow in the rest of the channel is laminar or turbulent. The black points represent the full
solution of the velocity profile in the porous layer, for seven different values of , as given
by (2.4) and plotted in inset (a). The dashed line (green online) represents the self-similar
solution of the rescaled velocity u
?
given in (4.7a). Inset (a): dimensionless velocity profile
u(y) for several values of . Inset (b): a zoom of the non-self-similar tails (black points) of
the solution. Such tails appear for y
?
< /2 (small dotted (red online) vertical lines) and are
perfectly captured by the function defined by (4.7b) (crosses (magenta online)).
4.1.2. Shear stress
The calculation of the limit of the dimensionless drag force leads to the
indeterminate form [0 · 1] and, therefore, the study of its asymptotic behaviour is
needed. Combining (4.4b) and (4.5) with (2.5) leads to
(y
?
) = e
y
?
, y
?
2 (/2, 0], (4.8a)
(y
?
;) = e
y
?
✓
1 +
1
⇤
e
2y
?
◆
, y
?
2 (, /2], (4.8b)
i.e. exhibits a self-similar behaviour in the proximity of the interface separating
the channel flow from the Brinkman flow. Figure 3 shows the asymptotic behaviour
described by (4.8a) (dashed line (green online)) and (4.8b) (crosses (magenta online))
of the drag force for different values of . Inset (a) in figure 3 is a zoom of the
main plot, showing that the asymptotic behaviour requires > 1. Inset (b) in figure 3
represents the drag force as a function of the physical y-coordinate for seven different
values of . These lines collapse onto the predicted line (4.8a) (dashed line (green
online)) for y
?
2 (/2, 0], while they start diverging and following (4.8b) (crosses
(magenta online)) for y
?
2 (, /2].
Self-similarity in coupled Brinkman/Navier–Stokes flows 101
10
–80
10
–60
10
–40
10
–20
10
0
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
–3
–10
3
–10
–4
10
2
10
0
(a)
(b)
y
–10
1
–10
0
0
40
80
–10
0
–10
–1
–10
–2
–10
–3
FIGURE 3. (Colour online) Dimensionless drag force in a thin porous medium (⇤ 1) for
M = 1 and = 100, and several values of . The black points represent the full solution of the
drag force as given by (2.5) and plotted in the inset (b) at the bottom right of the figure. The
dashed line (green online) represents the universal behaviour of the drag force given in (4.8a)
for y
?
> /2 (dotted (red online) vertical lines). For y
?
2 (, /2] the non-self-similar
tails of the solution are captured by the function given in (4.8b) (crosses (magenta online)).
Inset (a): a zoom, showing that self-similarity is attained only for 1.
4.2. Thick porous media
For thick porous media (⇤ =  ⌧ 1), we only investigate the case of a laminar
regime over the porous layer: if the porous layer occupies a significant portion of the
channel, then viscous effects will dominate everywhere in the computational domain.
In more quantitative terms, if ˆu
b
is the average velocity across the channel, then we
can define Re
pm
=ˆu
b
H⌫
1
and Re =ˆu
b
L⌫
1
to be the Reynolds numbers associated
with the porous medium and the channel flow, respectively. Therefore, Re
pm
< Re
c
,
with Re
c
the critical Reynolds number, implies Re < Re
c
, where ⌧
1
⌧ 1.
4.2.1. Flow velocity
For ⇤ ⌧ 1 and laminar flow over the porous layer, (4.3) and (4.4) give
U ⇠
1
M
2
and C
1
⇠
1
M
2
, (4.9)
with ⇠ 1/(2M⇤). Combining (4.9) with (2.4a) leads to the following asymptotic
behaviour:
u
?
1
(y
?
;) := M
2
u(y
?
/;)
1
M
2
⇠e
y
?
(1 + e
2y
?
), (4.10)
102 I. Battiato
–0.8
–0.6
–0.4
–0.2
–1.0
0
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
–3
–10
3
–10
–4
u(y)
–0.8
–0.6
–0.4
–0.2
y
–1.0
0
10
–6
10
–4
10
–2
FIGURE 4. (Colour online) Dimensionless velocity profile u
?
1
(y
?
) inside a thick porous
medium (⇤ ⌧ 1) for M = 1, = 10
5
and laminar flow over the porous layer. The black
points represent the full solution of the velocity profile in the porous layer, for six different
values of , as given by (2.4) and plotted in the inset. The dashed line (green online)
represents the self-similar solution of the rescaled velocity u
?
1
given in (4.11a). Non-self-
similar tails appear for y
?
< /2 (dotted (red online) vertical lines) and are perfectly
captured by the function defined by (4.11b) (crosses (magenta online)). Inset: dimensionless
velocity profile u(y) for several values of .
where u
?
1
is a rescaled velocity. Similarly to what obtained before, u
?
1
exhibits two
different limits, one of which being self-similar. Non-self-similar tails emerge close to
the bottom boundary of the channel. Specifically
u
?
1
(y
?
) ⇠e
y
?
, y
?
2 (/2, 0], (4.11a)
u
?
1
(y
?
;) ⇠e
y
?
(1 + e
2y
?
), y
?
2 (, /2]. (4.11b)
In figure 4 we plot u
?
1
for = 10
5
and six different values of such that ⇤ is in
the interval [10
5
, 10
5/2
]. The dashed line (green online) and crosses (magenta online)
in figure 4 represent the asymptotics described by (4.11a) and (4.11b), respectively.
Equations (4.11) capture well the full solution (2.4) for sufficiently large values of
(i.e. >
p
10). The non-rescaled velocity u(y) is plotted in the inset.
4.2.2. Shear stress
Combining (4.9) and (4.4b) with (2.5), we obtain
?
(y
?
;) :=  ⇠ e
y
?
1 + e
2y
?
, (4.12)
Self-similarity in coupled Brinkman/Navier–Stokes flows 103
(a)
(b)
y
–10
1
–10
0
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
3
–10
–3
10
–60
10
–40
10
–20
10
–80
10
0
10
–10
10
–5
10
0
0.1
0.2
0.3
0
0.4
–1.0 –0.6 –0.2
FIGURE 5. (Colour online) Dimensionless shear profile
?
(y
?
) inside a thick porous medium
(⇤ ⌧ 1) for M = 1, = 10
5
and laminar flow over the porous layer. The black points
represent the full solution of the shear profile in the porous layer, for six different values of
, as given by (2.5) and plotted in the inset (b). The dashed line (green online) represents
the self-similar solution of the rescaled shear
?
given in (4.13). Non-self-similar tails appear
for y
?
< /2 (dotted (red online) vertical lines) and are perfectly captured by the function
defined by (4.13b) (crosses (magenta online)). Inset (a): zoomed view of the main plot
showing that the asymptotic analytical solution does not describe the data well if is not
sufficiently bigger than one. Inset (b): dimensionless shear profile (y) for several values of .
with
?
a rescaled shear. As previously observed, self-similarity is attained sufficiently
close to the interface separating the free and porous medium flow, that is
?
(y
?
) ⇠ e
y
?
, y
?
2 (/2, 0], (4.13a)
?
(y
?
;) ⇠ e
y
?
1 + e
2y
?
, y
?
2 (, /2]. (4.13b)
The full and asymptotic solutions given by (2.5) and (4.13), respectively, are plotted in
figure 5.
5. Flow over deformable cylindrical obstacles
We now apply a similar analysis to flow through porous media constituted of
vertically aligned cylindrical obstacles, free to deflect under the aerodynamic stress
exerted by the flowing fluid.
Flows over and through layers of vertically aligned (deformable) obstacles have
received much attention from the scientific community due to their ubiquity to
many environmental, biological and technological systems, e.g. the endothelial
glycocalyx (Weinbaum et al. 2003) of blood vessels, polymer brushes (Tachie et al.
2004), CNT forests in shear sensors (Deck et al. 2009) and mechanical actuators (Kim
104 I. Battiato
R
0
R
1
(a)(b)
FIGURE 6. Axonometric (a) and top (b) view of the square-patterned forest of cylindrical
obstacles. R
0
and R
1
represent the post radius and the midway distance between aligned
cylinders, respectively.
& Lieber 1999), submerged vegetation canopies (Ghisalberti & Nepf 2009), and crop
canopies and forests (Kruijt et al. 2000), just to mention a few.
In the following we derive self-similar solutions for the elastic bending of
cylindrical obstacles in a forest by means of intermediate asymptotic analysis.
5.1. Model formulation and analytical solution
We assume that the bottom part of the channel, H < ˆy < 0, is occupied by (hexagon-
or square-) patterned arrays of elastic cylindrical obstacles of height H, such that
the flow is orthogonal to their axes. The cylinders are free to deflect due to the
aerodynamic shear exerted by the fluid flowing through and above the forest. Such
a forest can be treated as a porous medium with permeability K and porosity
= 1 (R
0
/R
1
)
2
, where R
0
and R
1
are the radius of and the half-distance between
aligned cylinders, respectively (see figure 6). Assuming that the maximum horizontal
deflection of the pillars’ tips is small compared to their length H, we can treat K as
a constant and decouple an analysis of the flow from that of the mechanics of the
bending. The permeability of arrays of infinite cylinders (Happel 1959, (19)) in terms
of porosity is
K = R
2
1
f (), (5.1a)
where
f () =
1
8
ln
(
1
)
(1 )
2
1
(1 )
2
+1
. (5.1b)
The parameter can be calculated as
1
= ✏
p
Mf (), where ✏ = R
1
/H is the relative
spacing of the cylindrical obstacles.
The horizontal deflection
ˆ
l(ˆy) of an individual cylinder inside the forest is caused
by the force (drag)
ˆ
D(ˆy) exerted by the fluid on the obstacle at the elevation ˆy,
ˆy 2 [H, 0]. The deflection
ˆ
l(ˆy) can be found as a solution of
d
2
dˆy
2
ˆ
E
ˆ
I
d
2
ˆ
l
dˆy
2
!
=
ˆ
D(ˆy), (5.2)
Self-similarity in coupled Brinkman/Navier–Stokes flows 105
–10
2
–10
1
–10
0
–10
3
–10
–1
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
–3
–10
–4
–10
3
–10
–5
l
10
–5
10
–4
10
–3
10
–2
–10
–1
–10
–2
–10
–3
y
–10
0
–10
–4
FIGURE 7. (Colour online) Rescaled bending profile l
?
of cylindrical obstacles in a thin
forest (⇤ 1) for ✏ = 0.001, = 100, and several values of . The black points represent
the full solution of the bending profile as given by (5.3) and plotted in the inset. The dashed
line (green online) represents the self-similar solution of the rescaled deflection given in (5.9)
for y
?
! 0. Boundary effects become important in the limit y
?
! and tailing emerges
(crosses (magenta online)) as described by (5.11).
where
ˆ
D =ˆ /
ˆ
N ,
ˆ
N = 1/R
1
is the number of cylinders per unit length,
ˆ
E the Young
modulus,
ˆ
I the moment of inertia of the cylinder’s cross-section and the product
ˆ
E
ˆ
I the
flexural rigidity. Treating each cylinder as a cantilever beam enforces zero deflection
and slope at the cylinder’s fixed end ˆy = H (i.e.
ˆ
l = 0, d
ˆ
l/dˆy = 0), and zero bending
moment and shear at the free end ˆy = 0 (d
2
ˆ
l/dˆy
2
= 0 and d
3
ˆ
l/dˆy
3
= 0, respectively).
The dimensionless bending profile of each individual cylinder, l =
ˆ
l/H, is (Battiato
et al. 2010)
l(y) =
1
2EI
2I
4
(y)
I
1
(0)
3
y
3
I
2
(0)y
2
+ Ay +
B
3
, (5.3)
where EI =
ˆ
E
ˆ
I/(H
3
µq), A = I
1
(0) 2I
2
(0) 2I
3
(1), B = 2I
1
(0) 3I
2
(0) 6I
3
(1)
6I
4
(1) and
I
n
(y) = ✏M
1n
⇥
C
1
e
y
+ (1)
n+1
C
2
e
y
⇤
, n = 1,...,4, (5.4)
with C
1
and C
2
defined by (2.4b). The corresponding dimensionless bending profiles
are shown in the inset of figure 7 for several values of . In the following section, we
proceed with an asymptotic study of the bending profile (5.4).
106 I. Battiato
5.2. Self-similar solution: bending profile and maximum deflection
We rewrite (5.3) in the following form:
l(y
?
;) =
1
2EI
2I
4
(
y
?
)
I
1
(0)
3
3
y
?3
I
2
(0)
2
y
?2
+
A
?
y
?
+
B
?
3
, (5.5)
where y
?
= y, A
?
= I
1
(0) 2I
2
(0) 2I
3
(), B
?
= 2I
1
(0) 3I
2
(0) 6I
3
()
6I
4
(), and I
n
(y
?
;) = ✏M
1n
⇥
C
1
e
y
?
+ (1)
n+1
C
2
e
y
?
⇤
exhibits different asymptotic
behaviours for thin or thick porous media. They are derived in §§ 5.2.1 and 5.2.2,
respectively.
5.2.1. Thin porous media
We combine (4.4b) and (4.5) with (5.4) and obtain
I
n
(y
?
;) ⇠
✏e
y
?
n
1
(1)
⇤
n+1
e
2y
?
, n = 1,...,4. (5.6)
Inserting (5.6) in (5.5) and dropping the exponentially decaying terms of the form e
leads to
l
?
(y
?
) ⇠
0
(y
?
1) +
1
✓
2
⇤
2y
?
◆
+
2
2e
y
?
✓
1 +
1
⇤
e
2y
?
◆
1
3
y
?3
y
?2
+
2
⇤
y
?
, (5.7)
where
l
?
(y
?
) := 2
EI
✏
l(y)
1
3
(5.8)
is a rescaled deflection. If y
?
! 0, then a self-similar solution is obtained in the form
l
?
(y
?
) ⇠ y
?
1. (5.9)
When y
?
!, terms in (5.7) can be grouped as follows
l
?
(y
?
) ⇠ y
?
✓
1
y
?2
3
2
◆
✓
1 +
y
?
◆
2
+
2
⇤
✓
1 +
y
?
◆
+ O(
2
), (5.10)
where the first, second and third terms on the right-hand side are of order ,
0
and
1
, respectively. Keeping the terms of order and
0
gives
l
?
(y
?
) ⇠ y
?
✓
1
y
?2
3
2
◆
✓
1 +
y
?
◆
2
, (5.11)
which describes the non-self-similar tails of (5.3), shown in figure 7 (crosses (magenta
online)).
5.2.2. Thick porous media
For thick porous media, I
n
(y
?
;) scales as follows:
I
n
(y
?
;) ⇠✏
1n
e
y
?
[1 + (1)
n+1
e
2y
?
], n = 1,...,4. (5.12)
Combining (5.12) with (5.5), dropping the exponentially decaying terms of the form
e
and defining
l
?
1
(y
?
) := 2
EI
✏
2
l(y) +
1
3
, (5.13)
Self-similarity in coupled Brinkman/Navier–Stokes flows 107
10
0
10
1
10
2
10
3
–10
3
–10
2
–10
1
–10
0
–10
–1
–10
–2
–10
–3
l
–10
–5
–10
–10
–10
–2
y
–10
0
–10
–4
FIGURE 8. (Colour online) Rescaled bending profile l
?
1
, defined in (5.13), for a thick forest
(⇤ ⌧ 1) of cylindrical obstacles and ✏ = 0.001, = 10
5
, and several values of . The ⇤
values range from 10
5
to 0.1 for = 1 and = 10
4
, respectively. The black points represent
the full solution of the velocity profile as given by (5.3) and plotted in the inset at the top
right of the figure. The dashed line (green online) represents the self-similar solution of the
rescaled deflection given in (5.15) for y
?
! 0. Boundary effects become important in the limit
y
?
! and tailing effects, described by (5.16), emerge (crosses (magenta online)).
we obtain
l
?
1
(y
?
) ⇠
0
(1 y
?
) +
1
(
2y
?
+ 2
)
+
2
[2 + 2y
?
+ y
?2
+
1
3
y
?3
2e
y
?
(1 e
2y
?
)], (5.14)
which provides the self-similar solution
l
?
1
(y
?
) ⇠ 1 y
?
, (5.15)
when y
?
! 0. For y
?
!,(5.14) can be rearranged as follows:
l
?
1
(y
?
) ⇠ y
?
✓
y
?2
3
2
1
◆
+
✓
1 +
y
?
◆
2
+ O(
1
). (5.16)
Equation (5.16) describes the tails (crosses (magenta online) in figure 8) of the
solution close to the lower boundary of the channel. While (5.15) captures well
the solution behaviour in the interior of the computational domain, tailing effects are
observed close to the interface separating free and porous media flow, contrary to the
thin-forest case. This could be attributed to the fact that the solution at the internal
boundary (y
?
= 0) still feels the effects of the upper boundary of the channel, since the
distance between the former and the latter is now very small. For thick porous media,
108 I. Battiato
R
1
= 0.08 (µm) ⇢
air
= 1.2 (kg m
3
) = 0.9735
R
0
= 0.02 (µm) ⌫
air
= 1.5 ⇥ 10
5
(m
2
s
1
) =[1.1–1.6] ⇥ 10
3
H = 40–60 (µm)µ
air
= 1.8 ⇥ 10
5
(kg m
1
s
1
) = 24.83–37.75
L = 1.5 (mm) ⇢
Ar
= 1.784 (kg m
3
) M = 1
K = 1.4 ⇥ 10
3
(µm
2
) ⌫
Ar
= 1.18 ⇥ 10
5
(m
2
s
1
) ✏ =[1.3–2] ⇥ 10
3
ˆu
b
= 5–55 (ms
1
)µ
Ar
= 2.1 ⇥ 10
5
(kg m
1
s
1
) ⇤ =[2.7–6]⇥ 10
4
Re = 640–7000
TABLE 1. Parameter values used in the experiment of Deck et al. (2009) and
corresponding dimensionless quantities.
self-similarity is only achieved in the bulk region of the porous layer, i.e. far away
from both the bottom and internal boundary.
6. Application to carbon nanotube forests
We apply the asymptotic solutions derived in § 5.2 to the experimental data,
collected by Deck et al. (2009), on flow past forests of carbon nanotubes (CNTs). We
show that our self-similar solution can be conveniently used to determine mechanical
properties of the CNTs by linear fit of properly rescaled data.
A significant number of works have focused on the study of CNTs since
they possess a remarkable combination of mechanical characteristics, including
exceptionally high elastic moduli (Treacy, Ebbesen & Gibson 1996), reversible
bending and buckling characteristics (Falvo et al. 1997), and superplasticity (Huang
et al. 2006). Such properties mean that complex interactions between fluid flow and
patterned nanostructures composed of CNTs play an important role in a variety
of applications, including mechanical actuators (Kim & Lieber 1999), chemical
filters (Srivastava et al. 2004), and flow sensors (Ghosh, Sood & Kumar 2003). Most
experiments dealing with these phenomena assemble CNTs into macroscopic sheets or
forests (Zhang et al. 2005; Deck et al. 2009). When placed on a body’s exterior, CNT
‘forests’ can act as superhydrophobic surfaces that significantly reduce drag (Wilson
2009; Joseph et al. 2006).
The experiments performed by Deck et al. (2009) consist of CNTs, with typical
heights H 2 [40–60] µm and diameters 2R
0
2 [30–50] nm, grown in square arrays of
sizes 5–10 µm on quartz substrates. The CNT samples were placed inside a quartz
tube with inner diameter of 6.2 mm (Deck et al. 2009, figure 1). The samples were
then exposed to fluid (air or argon) at various pressures. A linearly polarized He–Ne
laser was used to illuminate the CNT forests and the transmitted light intensity was
monitored as a function of fluid flow. The horizontal deflections of the CNT ensembles
(initially oriented parallel to the polarised laser beam) were translated into a change of
the light intensity and sampled by a photodetector or a charged coupled device camera.
The data consist of measurements of maximum (dimensional) horizontal deflection of
the CNT tips,
ˆ
X =
ˆ
l(0), and bulk velocity across the wind tunnel, ˆu
b
. The experiments
were used to estimate the flexural rigidity
ˆ
E
ˆ
I of four CNT samples. The parameter
values of the experiment are summarized in table 1. For a full description of the
experiment, including a detailed description of the synthesis and patterning of the
arrays of vertically aligned, multi-walled CNTs, we refer the interested reader to Deck
et al. (2009).
Self-similarity in coupled Brinkman/Navier–Stokes flows 109
0
2
4
6
8
Air, sample 1
Air, sample 2
Air, sample 3
Ar, sample 3
Air, sample 4
0
2
4
6
8
Air, sample 1
Air, sample 2
Air, sample 3
Ar, sample 3
Air, sample 4
20 30 40 60
2345 7
(× 10
–6
)
(× 10
–6
)
(× 10
–21
)
(a)
(b)
10 50
16
FIGURE 9. (a) Measurements of CNT tip deflection,
ˆ
X, and bulk velocity across the wind
tunnel, ˆu
b
, from Deck et al. (2009). (b) Plotting data in (
ˆ
X, G )-space allows determination of
ˆ
E
ˆ
I by linear fit.
Data sets collected by Deck et al. (2009) are reported in figure 9(a). Since
ˆ
X is
10 % of the CNT height, H, we can treat the forest’s permeability, given by (5.1),
as constant. The typical values of (⇡10
3
) and ⇤ (⇡10
4
) suggest that the forest
(i) should behave according to the self-similar solution derived for ! +1, and
(ii) can be treated as a thin porous medium (⇤ 1). The asymptotic solutions for
thin porous layers have the additional advantage of being valid for both laminar and
turbulent regimes over the forest, which makes their applicability quite robust for
systems spanning both flow regimes. The maximum dimensionless deflection X
?
can
110 I. Battiato
Sample 1 2 3 (Air) 3 (Ar) 4
H (m) 40 ⇥ 10
6
40 ⇥ 10
6
50 ⇥ 10
6
50 ⇥ 10
6
60 ⇥ 10
6
↵ (-) 6.2811 ⇥10
7
6.2811 ⇥10
7
4.021 ⇥10
7
4.021 ⇥10
7
2.793 ⇥10
7
ˆ
E
ˆ
I (Nm
2
)
(linear fit)
2.29 ⇥ 10
22
2.53 ⇥ 10
22
2.80 ⇥ 10
22
2.83 ⇥ 10
22
2.74 ⇥ 10
22
R
2
(-)
(linear fit)
0.957 0.985 0.959 0.988 0.986
TABLE 2. Values of CNT height, H, parameter ↵,
ˆ
E
ˆ
I linear fit and R
2
for each sample.
Data for sample 3 correspond to experiments performed with two different fluids, i.e. air
and Ar. The predicted values of
ˆ
E
ˆ
I (boldface values) for sample 3, obtained from linear fit
of the two sets of data, are in very good agreement.
be easily calculated from (5.9) as X
?
= l
?
(0) ⇠1. In dimensional form, the maximum
deflection,
ˆ
X = HX
?
, can be written as follows:
ˆ
X ⇠ ↵
⇣
ˆ
E
ˆ
I
⌘
1
G , (6.1)
where ↵ = (2 3)✏
2
/6 is a purely geometric parameter, and G = µH
4
q. We use
the measurements of average bulk velocity ˆu
b
to determine the scaling factor q =ˆu
b
,
where is defined by the implicit equation (Battiato et al. 2010, (10))
p
Re
✓
1 +
1
2
H
◆
1/2
1
2
ln =
1
ln(
p
Re) + 5.9
1
, (6.2)
where
H = (2M
2
)
1
[1 + (coth csch )( tanh
1
1
sech )],
Re =
Lˆu
b
⌫
(6.3)
is the Reynolds number, and = 0.41 the von K
´
arm
´
an constant. Equation (6.1)
provides a closed-form expression to estimate the flexural rigidity of carbon nanotubes
from their elastic response to hydrodynamic loading by linear fit of data on a (
G ,
ˆ
X)
plot. Alternatively, for a known value of
ˆ
E
ˆ
I,(6.1) can be used to predict the CNT tip
deflections due to aerodynamic loading.
Contrary to figure 9(a) where experimental points are scattered, data aligns if
properly rescaled (see figure 9b). Linear interpolation gives values of R
2
greater
than 0.96 (see table 2). Also, the data collected from sample 3 in two different
experiments performed with air (filled circles) and argon (filled triangles) collapse onto
each other, as expected. The second moment of inertia,
ˆ
I, for CNTs with diameter
2R
0
= 40 nm and a typical wall thickness of 0.34 nm (Lu 1997; Meyyappan 2005, p.
33) is approximately 8.3 ⇥ 10
33
m
4
. Hence our estimate of the flexural rigidity
ˆ
E
ˆ
I
predicts the Young modulus of individual CNTs to be
ˆ
E ⇡ 0.034 TPa, which is in
agreement with data reported in Poncharal et al. (1999), Fig. 3A, for single CNTs of
comparable diameters.
7. Summary and concluding remarks
Coupled flows through and over porous layers are ubiquitous in a number of natural
phenomena and industrial processes. The focus of many studies on the topic has
Self-similarity in coupled Brinkman/Navier–Stokes flows 111
been the identification of the proper conditions to apply at the interface separating
the porous medium and the pure fluid flow. Generally, the two main approaches for
coupling are single- and multiple-domain methods. While the former treats the system
as a single domain with spatially variable permeability, the latter employs two different
mathematical models for the porous medium and the free fluid – e.g. Brinkman and
Stokes equations, respectively – and enforces boundary conditions for the tangential
velocity and shear at the liquid–porous matrix interface.
In this paper we derive self-similar solutions of flows through both a porous medium
and a pure fluid. The analytical solutions, obtained from a multiple-domain approach
(Battiato et al. 2010), are here employed to identify self-similar behaviour of filtration
velocity and shear stress in the porous matrix, while the pure-fluid flow is allowed
to span both laminar and turbulent regimes. Self-similarity is obtained by means
of asymptotic analysis in the infinitely small permeability limit (i.e. !1). We
show that a spatial length scale, ⇤ (=), related to the porous layer dimensionless
thickness, , and permeability, naturally emerges from the limiting process and
suggests a more formal definition of thick (⇤ ⌧ 1) and thin (⇤ 1) porous media.
Depending on the magnitude of ⇤, two different self-similar behaviours emerge, which
render a classification between thin and thick porous media fully quantitative. We
finally specialize the analysis to porous media constituted of patterned cylindrical
obstacles, which can freely deflect under the aerodynamic shear exerted by the
fluid flowing through and over the forest. A self-similar solution for the bending
profile of the elastic pillars is obtained as intermediate asymptotics for both thin
and thick forests. This self-similar solution is finally applied to CNT forests, and
successfully used to estimate their flexural rigidity by a linear fit of appropriately
rescaled maximum deflection and average velocity measurements.
Our analysis leads to the following main conclusions.
(a) Self-similar solutions for coupled flow over and through a porous layer are
obtained from asymptotic analysis in the low permeability limit, and derived
for both laminar and turbulent regimes over the porous medium.
(b) The asymptotic analysis allows us to formally classify thin and thick porous media
based on the magnitude of a length scale parameter, ⇤. Different asymptotic
solutions for the velocity and shear profiles arise if ⇤ 1 (thin porous medium)
or ⇤ ⌧ 1 (thick porous medium).
(c) Self-similarity of appropriately rescaled quantities is achieved in the bulk of the
porous medium and close to the interface separating the porous layer from the
pure fluid. Boundary effects lead to non-self-similar tailing in the proximity of the
bottom wall of the channel.
(d) Thin porous media exhibit the same (dimensionless) asymptotic solutions for both
laminar and turbulent regime flows above the porous layer.
(e) Self-similar asymptotic solutions are obtained for the bending profile of forests
of deformable cylindrical obstacles. Such formulae are successfully applied to
model deformations of aerodynamically sheared carbon nanotubes’ forests. CNTs’
flexural rigidity is obtained by linear fit of appropriately rescaled quantities,
derived from asymptotic analysis.
While the identification of a self-similar solution is generally valuable since it
enables one to reduce the number of dimensionless parameters (Breugem, Boersma
& Uittenbogaard 2006) and to identify dynamical similarities between systems at
different scales, some recent works (Ghisalberti 2009; Manes, Poggi & Ridolfi 2011)
112 I. Battiato
have explicitly focused on the existence of (self-similar) scaling laws of laminar and
turbulent flows over permeable layers. Specifically, Ghisalberti (2009) shows that a
wide range of environmental flows (from laminar to turbulent) above canopies, packed
beds, coral reefs, etc. are inherently dynamically similar. The connection between the
data reported in Ghisalberti (2009), and references therein, and our scaling laws
is the object of current investigations. The results of the present study are also
directly comparable with the data pertaining to flow over low-permeability layers,
including porous mats (e.g. Manes et al. 2011), aquatic and atmospheric canopies (e.g.
Ghisalberti & Nepf 2002; Poggi et al. 2004; Finnigan, Shaw & Patton 2009), etc.
Our results could be beneficial in addressing a number of other open questions
in the study of flows over permeable layers, an area of active research due to
the implications for near-wall turbulence control and skin friction reduction. Such
studies are generally hampered by the difficulty of differentiating between the effects
of permeability and roughness (Manes et al. 2009, 2011). Our theoretical results
isolate the effect of permeability from that of roughness since they are based on the
assumption of a hydrodynamically smooth interface between the porous medium and
channel flow. Additionally, Breugem et al. (2006) and Manes et al. (2011) observed
that, with increasing permeability, the near-wall structure evolves towards a more
organized state until it reaches a perturbed mixing layer where the scale of the eddies
is dominated by the shear instability of the inflectional mean velocity profile. We plan
to use the proposed framework and scaling laws to quantify such a transition, and
establish if and how it correlates with the existence of a self-similar solution in the
inflectional mean velocity profile.
As a final remark, large-scale results might be influenced by the postulated
conditions at the pure fluid–porous layer interface as discussed in (Bars & Worster
2006). Therefore, in follow-up studies we will also investigate the effect of different
boundary conditions on the asymptotic behaviour of velocity, shear stress and
deflection profiles.
Acknowledgements
The author would like to thank the three anonymous referees for their invaluable
insight and feedback. Funding from BP International within the ExploRe program is
gratefully acknowledged.
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