J. Fluid Mech. (2006), vol. 562, pp. 35–72.
c
2006 Cambridge University Press
doi:10.1017/S0022112006000887 Printed in the United Kingdom
35
The influence of wall permeability on turbulent
channel flow
By W. P. BREUGEM
1,3
,B.J.BOERSMA
1
AND R. E. UITTENBOGAARD
2,1
1
J. M. Burgers Center for Fluid Dynamics, Delft University of Technology,
Leeghwaterstraat 21, 2628 CA Delft, The Netherlands
2
WL | Delft Hydraulics, PO Box 177, 2600 MH Delft, The Netherlands
3
Royal Netherlands Meteorological Institute (KNMI), PO Box 201,
3730 AE De Bilt, The Netherlands
(Received 25 November 2004 and in revised form 25 January 2006)
Direct numerical simulations (DNS) have been performed of turbulent flow in a plane
channel with a solid top wall and a permeable bottom wall. The permeable wall is a
packed bed, which is characterized by the mean particle diameter and the porosity.
The main objective is to study the influence of wall permeability on the structure and
dynamics of turbulence. The flow inside the permeable wall is described by means of
volume-averaged Navier–Stokes equations. Results from four simulations are shown,
for which only the wall porosity (
c
) is changed. The Reynolds number based on
the thickness of the boundary layer over the permeable wall and the friction velocity
varies from Re
p
τ
= 176 for
c
=0 to Re
p
τ
= 498 for
c
=0.95. The influence of wall
permeability can be characterized by the permeability Reynolds number, Re
K
,which
represents the ratio of the effective pore diameter to the typical thickness of the
viscous sublayers over the individual wall elements. For small Re
K
, the wall behaves
like a solid wall. For large Re
K
, the wall is classified as a highly permeable wall near
which viscous effects are of minor importance. It is observed that streaks and the
associated quasi-streamwise vortices are absent near a highly permeable wall. This is
attributed to turbulent transport across the wall interface and the reduction in mean
shear due to a weakening of, respectively, the wall-blocking and the wall-induced
viscous effect. The absence of streaks is consistent with a decrease in the peak value
of the streamwise root mean square (r.m.s.) velocity normalized by the friction velocity
at the permeable wall. Despite the increase in the peak values of the spanwise and
wall-normal r.m.s. velocities, the peak value of the turbulent kinetic energy is therefore
smaller. Turbulence near a highly permeable wall is dominated by relatively large
vortical structures, which originate from a Kelvin–Helmholtz type of instability. These
structures are responsible for an exchange of momentum between the channel and
the permeable wall. This process contributes strongly to the Reynolds-shear stress
and thus to a large increase in the skin friction.
1. Introduction
Turbulent flows over permeable walls, i.e. rigid porous walls with interconnected
pores through which fluid may flow, are encountered in a wide range of problems.
Examples are: flows in oil wells, catalytic reactors, heat exchangers of open-cell metal
foam (Lu, Stone & Ashby 1998), and porous river beds (Vollmer et al. 2002). To
some extent, densely built-up urban areas and plant canopies can be considered as
36 W. P. Breugem, B. J. Boersma and R. E. Uittenbogaard
permeable wall layers as well. Related research topics are dispersion of pollutants in
metropoles, the exchange of energy and oxide and carbon dioxide between forests and
the atmosphere (Finnigan 2000), and the propagation of forest fires (S
´
ero-Guillaume &
Margerit 2002).
Despite its relevance to the aforementioned applications, only a few experimental
studies report on the effect of wall permeability on turbulence. Zagni & Smith (1976)
conducted experiments on open-channel flow over permeable beds composed of
spheres. It was found that the friction factor was higher than for flows over imper-
meable walls with the same surface roughness. Furthermore, after having reached
a constant value, at typically Re = O(10
5
) the friction factor showed a tendency to
increase again with Reynolds number. Also Kong & Schetz (1982) reached the con-
clusion that in their experiments wall permeability alone could be responsible for an
increase in skin friction by as much as 30–40 % relative to an impermeable wall with
similar surface roughness. A rise in friction factor at high Reynolds numbers was
also observed by Zippe & Graf (1983) in wind-tunnel experiments on boundary-layer
flow over a permeable bed composed of grains. Zagni & Smith (1976) attributed
the increase in friction factor to additional energy dissipation caused by exchange of
momentum across the bed interface. Evidence of this exchange has been provided by
experiments of Ruff & Gelhar (1972) on turbulent flow in a pipe lined with highly
porous foam.
The experiments mentioned above clearly indicate that the effect of wall permeabil-
ity on turbulence is different from wall roughness. This implies that wall permeability
alters the structure and dynamics of turbulence. In this paper we aim to improve
our understanding of this alteration by means of direct numerical simulation (DNS)
of turbulent flow over a permeable wall. Unlike experiments, in DNS it is relatively
easy to isolate the effect of wall permeability from wall roughness. Furthermore, these
simulations provide very detailed information on the flow field both above and inside
a permeable wall, which is difficult to obtain from measurements. In literature two
different methods can be found for simulating flow over and through a permeable wall.
The first and computationally most simple method is the specification of boundary
conditions that incorporate the effect of wall permeability. This approach was followed
by Hahn, Je & Choi (2002) in DNS of turbulent flow in a plane channel with
permeable walls. The boundary conditions used were similar to those proposed by
Beavers & Joseph (1967) for laminar flow over a permeable wall, and allow for a slip-
velocity at the wall. The wall-normal velocity was put to zero, which is, however, not
realistic for high wall permeabilities at which exchange of momentum across the wall
interface takes place (Ruff & Gelhar 1972). As pointed out by Hahn, Je & Choi (2002),
a wall can be classified as highly permeable when the permeability Reynolds number
Re
K
≡
√
Ku
τ
/ν 1, where K is the wall permeability of the order of the square of
the characteristic pore diameter, ν is the kinematic viscosity and u
τ
is the wall friction
velocity. Therefore, for an accurate description of flow over a permeable wall at high
Re
K
, the flow inside the wall itself also has to be described.
The second method is the continuum approach where the flow inside the permeable
wall is modelled as a continuum, which is coupled to the flow over the wall. The
theoretical basis for this approach is provided by the volume-averaging method
(Whitaker 1999). In this method the flow is averaged over a small spatial volume
with dimensions sufficiently large to smooth inhomogeneities at pore scales, but on
the other hand sufficiently small to retain the flow dynamics of interest. The volume-
averaged flow field is governed by the volume-averaged Navier–Stokes (VANS) equa-
tions (Whitaker 1996). In order to solve the VANS equations, closures are required for
Influence of wall permeability on turbulent channel flow 37
the subfilter-scale stress and the drag force. The continuum approach has been used
in a number of recent large-eddy simulation (LES) studies of flows over forests (e.g.
Shaw & Schumann 1992; Dwyer, Patton & Shaw 1997; Watanabe 2004). Other studies,
in which a Reynolds-averaged form of the VANS equations was used, considered not
only flow over vegetation (Wilson 1988; Uittenbogaard 2003), but also flow over a
permeable wall layer with a porosity significantly lower than that of vegetation (De
Lemos & Pedras 2000; Silva & De Lemos 2003).
In a recent publication (Breugem & Boersma 2005), we verified that the VANS
equations can be used for an accurate simulation of turbulent flow over and through
a permeable wall. The VANS equations were solved in a DNS of turbulent channel
flow with a lower permeable wall consisting of a three-dimensional Cartesian grid
of cubes. The turbulence statistics agreed very well with the results of a different
DNS in which, by means of the standard Navier–Stokes equations, the flow field in
between the cubes was fully resolved. This gives us confidence for using the VANS
equations in the present study where we consider again flow in a plane channel with
a lower permeable wall. Different from the grid of cubes in our previous study, the
permeable wall in the present study is a packed bed, which is encountered in many
applications. In the DNS, the packed bed is characterized by the porosity and the
mean particle diameter. In order to isolate the effect of wall permeability from wall
roughness, we consider packed beds with relatively high wall porosities and small
mean particle diameters. Results from four simulations will be shown, each with a
different porosity.
This paper is organized as follows. Section 2 discusses the continuum approach for
flows through porous media. The next section deals with the coupling between the
flow in the channel and the flow inside the permeable wall. In §4, a discussion is given
of the implications of wall permeability for the scaling of turbulence. Section 5 deals
with the numerical method. The DNS results are presented in §6. In §7, the results
are summarized and discussed.
2. Continuum approach for flows in porous media
In this section, we briefly discuss the continuum approach for flows in porous
media. See Breugem (2004) for a detailed discussion.
The first step in the derivation of the governing equations for the volume-averaged
flow is the definition of the superficial volume average, denoted by ···
s
:
u
s
x
≡
V
γ (r)m( y)u(r)dV, (2.1)
where the subscript x means that the volume average is evaluated at the centroid x
of the averaging volume V , y = r −x is the relative position vector, γ is the phase-
indicator function that equals unity when r points in the fluid phase and zero when
r points in the solid phase, and m is a weighting function. The volume-averaging
technique is illustrated in figure 1. Notice that the volume-averaging operator acts as
a filter, which passes information only on the large-scale structure of the flow field.
Furthermore, we note that the volume-averaged flow field is continuous in the sense
that it is defined both in the fluid and the solid phase, provided of course that the
averaging volume is sufficiently large. This is the basis of the continuum approach
for flows in porous media.
In principle, the weighting function can be chosen freely, but it is desirable that the
volume-averaged flow field contains negligible variations on scales smaller than the
38 W. P. Breugem, B. J. Boersma and R. E. Uittenbogaard
d
p
Averaging volume V
d
f
r
x
Surface A of solid phase
inside volume V
n
Fluid phase
Solid phase
r
0
y
Figure 1. Illustration of the volume-averaging technique for a disordered porous medium.
dimensions of the averaging volume. A customary averaging volume for a disordered
porous medium as sketched in figure 1, is a sphere with radius r
0
and a top-hat
distribution for the weighting function (Quintard & Whitaker 1994):
m( y)=
3
4
πr
3
0
, |y| 6 r
0
,
0, |y| >r
0
.
(2.2)
The velocity at a certain point in the porous medium can be decomposed into a
contribution from the volume-averaged velocity at this point and a subfilter-scale
velocity
˜
u according to (Gray 1975):
u = u +
˜
u, (2.3)
where u≡u
s
/ is the intrinsic volume average and is the porosity. The latter is
defined according to:
(x) ≡
V
γ (r)m( y)dV. (2.4)
Later on in this paper we will also make use of a temporal decomposition according
to (Tennekes & Lumley 1999):
u =
u + u
, (2.5)
where the overbar denotes the Reynolds- or ensemble-averaged value and the prime
denotes the deviation from the Reynolds-averaged value. It is easy to show that
the ensemble- and spatial-averaging operators commute (Pedras & De Lemos 2001):
u= u, u
= u
,
˜
u =
˜
u and
˜
u
=
u
.
The spatial averaging theorem (Whitaker 1969) relates the volume average of a
spatial derivative to the spatial derivative of the volume average:
∇p
s
x
= ∇p
s
x
+
A
m( y)np(r)dA, (2.6)
where A is the contact area between the fluid and the solid phase inside the averaging
volume V ,andn is the unit normal at A that points from the fluid into the solid
phase, see figure 1.
Influence of wall permeability on turbulent channel flow 39
Application of the spatial filter (2.1) and the spatial averaging theorem (2.6) to the
Navier–Stokes equations yields the volume-averaged Navier–Stokes (VANS) equa-
tions (Whitaker 1996):
∂u
∂t
+
1
∇·[uu]+
1
∇·[τ ]=−
1
ρ
∇[p]+
ν
∇
2
[u]+ f , (2.7a)
∇·[u]=0, (2.7b)
where τ is the subfilter-scale stress, which in the LES literature is known as the
subgrid-scale stress, and f is the drag force per unit mass that the solid phase exerts
on the fluid phase. The expressions for τ and f are given by:
τ ≡uu−uu≈
˜
u
˜
u, (2.7c)
f ≡
1
A
mn
−
p
ρ
I + ν∇u
dA
≈
1
A
mn
−
˜
p
ρ
I + ν∇
˜
u
dA +
p
ρ
∇ −
ν
∇ ·∇u, (2.7d)
where the approximations in the last two equations are valid when u≈u.
In order to solve the VANS equations, closures are required for the subfilter-scale
stress and the drag force in terms of the volume-averaged flow quantities. In
Appendix A, we argue that, in porous media, subfilter-scale dispersion is normally
negligible with respect to the drag force and/or the Reynolds-shear stress of the
volume-averaged flow field.
Whitaker (1996) gives theoretical support to the following customary parameteriza-
tion of the drag force:
1
A
mn
−
˜
p
ρ
I + ν∇
˜
u
dA = −νK
−1
u−νK
−1
Fu, (2.8)
where K and F are, respectively, the permeability and the Forchheimer tensor. The
first term on the right-hand side of (2.8) represents the drag force in the limit of Stokes
flow in the pores, whereas the second term is a correction for inertial effects at higher
Reynolds numbers. In general, the permeability tensor depends only on the geometry
of the porous medium. The Forchheimer tensor depends on the Reynolds number
|u|d
f
/ν, with d
f
the typical pore diameter, on the geometrical parameters of the
porous medium and on the orientation of the solid obstacles relative to the direction
of the volume-averaged flow. Generally valid expressions for the permeability and
the Forchheimer tensor do not exist, as they are strongly related to the geometry
of the porous medium and the Reynolds number. They must be determined from
experiments or numerical calculations of flow through a representative region of
the porous medium. Numerical calculations of the permeability and the Forchheimer
tensor for several geometries are presented by Zick & Homsy (1982), Larson & Higdon
(1986, 1987), Sahraoui & Kaviany (1992), Ma & Ruth (1993), Lee & Yang (1997)
and Breugem, Boersma & Uittenbogaard (2004). A few references to experiments are
MacDonald et al. (1979), Fand et al. (1987), Kececioglu & Jiang (1994) and Lage,
Antohe & Nield (1997). For flows through packed beds, which are considered in the
present study, a widely used relation for the drag force is the modified Ergun equation
(Bird, Stewart & Lightfoot 2002). This equation can be written in the form of (2.8)
![Table 3. Fit parameters. λ is equivalent to the von Kármán constant for a solid wall. d and z0 are the length scales in the log law (4.5), normalized by, respectively, √ Kc , dp , δw and ν/u p τ . The value of dp+ in the penultimate column is obtained from Jackson’s model (JM), equation (6.1). α is obtained from a least-squares fit of (4.5) to the mean velocity profile in the region z/H ∈ [−0.85 : −0.02].](/figures/table-3-fit-parameters-l-is-equivalent-to-the-von-karman-1uaiavj5.webp)
