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1
TURBULENT FLOW OVER AND WITHIN A POROUS BED
by Panayotis Prinos1, Dimitrios Sofialidis2 and Evangelos Keramaris 3
ABSTRACT
The characteristics of turbulent flow in open channels with porous bed are studied
numerically and experimentally. The "microscopic" approach is followed, by which the
Reynolds–Averaged Navier–Stokes equations are solved numerically in conjunction with a
low–Reynolds k–0� WXUEXOHQFH� PRGHO� DERYH� DQG� ZLWKLQ� WKH� SRURXV� EHG�� 7KH� ODWWHU� LV�
represented by a bundle of cylindrical rods of certain diameter and spacing, resulting in
permeability, K, ranging from 5.5490×10−7 to 4.1070×10−4 m2 and porosity, φ , from 0.4404
to 0.8286. Mean velocities and turbulent stresses are measured for φ =0.8286 using hot–film
anemometry. Emphasis is given to the effect of Darcy number, Da, on the flow properties
over and within the porous region. Computed and experimental velocities in the free flow are
shown to decrease with increasing Da due to the strong momentum exchange near the porous
medium/free flow interface and the corresponding penetration of turbulence into the porous
layer for highly permeable beds. Computed discharge indicates the significant reduction of the
channel capacity, compared to the situation with smooth impermeable bed. On the contrary,
laminar flow computations, along with analytical solutions and measurements, indicate
opposite effects of the porous medium on the free flow.
1Professor, Hydraulics Lab., Dept. of Civil Eng., Aristotle University of Thessaloniki, 54124
Thessaloniki, Greece.
2CFD Engineer, SimTec Ltd., 54622 Thessaloniki, Greece.
3Research Associate, Hydraulics Lab., Dept. of Civil Eng., Aristotle University of
Thessaloniki, 54006 Thessaloniki, Greece.
2
KEYWORDS
Turbulent flow, porous medium, Darcy number, rods bundle, low−Re turbulence model.
INTRODUCTION
Flow phenomena and the associated momentum transfer near a porous medium/free flow
interface (termed "porous/fluid interface" or "interface" hereafter) are encountered in various
fields (environmental hydraulics, geophysical fluid dynamics and mechanical engineering
among others). In all problems the knowledge of the flow interaction above and inside the
porous medium and the momentum transfer across the interface is very important and is
required for water quality aspects. For example, the hydrodynamics of the Ekmann layer in
the benthic boundary layer affect the oxygen flux at the water/sediment interface (Svensson
and Rahm, 1991). Also, sediment oxygen demand increases linearly with water velocity
above the sediments when the velocities are low (Mackentum and Stefan, 1998).
The effect of a porous medium on the flow above it and the flow characteristics near its
interface with the free flow have been studied by several investigators (Beavers and Joseph,
1967, Poulikakos and Kazmierczak, 1987, Rudraiah, 1985, Vafai and Thiyagaraja, 1987,
Sahraoui and Kaviany, 1992, Ochoa–Tapia and Whitaker, 1995a and b, Choi and Waller,
1997, Gupte and Advani, 1997 , James and Davis, 2001) for relatively low Reynolds numbers
(laminar flow). Most of the above studies deal with: (a) the determination of proper porous
fluid interfacial conditions at a ″macroscopic″ level, (b) the computation of momentum
transport phenomena at the interface using ″macroscopic″ equations for the porous region
with appropriate interfacial conditions (Brinkman–Forchheimer–extended Darcy model; Vafai
and Tien, 1981) and, (c) the computation of flow characteristics within the fluid and porous
regions using ″microscopic″ equations. Using the latter the porous region is simulated by an
3
array of circular cylinder and the Navier–Stokes equations are solved in the whole
computational domain occupied by the fluid.
In their experimental study, Beavers and Joseph (1967) used a porous block of high
permeability in a closed channel and found an empirical relationship for the interfacial slip
velocity, which takes into account the Darcy velocity inside the porous layer, the permeability
of the porous medium and a slip parameter assumed to be independent of velocity. The
empirical relationship used is dUf/dy=(.�.1/2)(Uint−UD), where K=porous medium
SHUPHDELOLW\�� . VOLS� SDUDPHWHU�� 8f=flow velocity over the porous surface, Uint=interfacial
velocity, UD=Darcy velocity within the porous bed (UD �.����−G3�G[���� G\QDPLF�YLVFRVLW\��
−dP/dx=streamwise pressure gradient). They used the momentum equation for fully
developed laminar flow and the above–mentioned relationship as an interfacial condition for
calculating: (a) the velocity distribution over the porous media, (b) the interfacial velocity Uint
and, (c) the increase in mass flow over the permeable bed with regard to that with
impermeable bed.
Poulikakos and Kazmierczak (1987), Rudraiah (1985), and Vafai and Thiyagaraja (1987)
have used the Navier−Stokes equations for the flow above the porous medium together with
the Darcy−Brinkman equations for the flow inside the porous medium (″macroscopic
approach″). They used "continuity" conditions at the interface for both velocity and shear
stresses and calculated analytically the interfacial velocity and the velocity distribution above
and inside the porous region.
Also, Sahraoui and Kaviany (1992) have shown that a variable "effective" viscosity has to be
used in the Darcy−Brinkman equations for the accurate computation of the velocity near the
interface and inside the porous layer. Ochoa−Tapia and Whitaker (1995a and b) have
developed a stress jump condition based on the non−local form of the volume averaged
Stokes equations and explored the use of a variable porosity model as a substitute for the
4
jump condition. This approach did not lead to a successful representation of all experimental
data, but provided some insight into the complexities of the interfacial region between a
porous medium and a homogeneous fluid.
Choi and Waller (1997) have investigated numerically the momentum transport phenomena
through a porous/fluid interface using a "macroscopic" single fluid domain approach with
matching boundary conditions. The flow was laminar and the results showed the importance
of viscous shear in the channel flow. They concluded that the Darcy law is inappropriate to
describe the flow in the interfacial region.
James and Davis (2001) presented "microscopic" computations for Stokes flow in a channel
partially filled with an array of circular cylinders. The porosity was 0.9 or greater, simulating
fibrous porous media. They found that the external flow penetrates the porous layer very little
even for sparse arrays and that the apparent slip velocity at the interface is about one quarter
of the velocity predicted by the Brinkman model.
Studies of turbulent flow near a porous/fluid interface are rather limited since there are
additional difficulties due to turbulence effects. Most of the studies have presented
experimental data and findings about the effect of a porous bed on turbulent flow above it
(Munoz−Goma and Gelahr, 1968, Ruff and Gelhar, 1970 and Chu and Gelhar, 1972, Nezu,
1977, Zippe and Graf, 1983). Very few studies (Mendoza and Zhou, 1992, Zhou and
Mendoza, 1993, Li and Garga, 1998) have presented analytical results, however with various
shortcomings. Computational studies at "macroscopic" level are very limited for cases
without turbulence penetration into the porous layer (Hahn et al., 2002) and, to the authors’
knowledge, no studies have been performed yet for turbulence penetration. In the latter case a
"macroscopic" turbulence model for porous media flow is needed for the solution of the
momentum equations inside the porous media, which is at present under development (Lage,
1998, Nield, 2001). Computational studies at "microscopic level" (similar to that of James and
5
Davis, 2001 for Stokes flow) for turbulent flow conditions have not presented yet and to the
author’s knowledge this is the first study of this kind.
Studies in MIT (Munoz−Goma and Gelahr, 1968, Ruff and Gelhar, 1970 and Chu and Gelhar,
1972) have shown that the logarithmic law of the wall is valid for turbulent flow over a
porous bed but the von–.DUPDQ�FRQVWDQW�����KDV�WR�EH�UHGXFHG�IURP�������IORZ�RYHU�D�VPRRWK�
impermeable bed) to 0.26. Nezu (1977) conducteG� VRPH� H[SHULPHQWV� DQG� IRXQG� WKDW� ��
decreases with increasing permeability. Zippe and Graf (1983), based on experimental
findings, concluded that the boundary resistance of the tested permeable surface is higher than
that of the non−permeable boundary with identical roughness.
Mendoza and Zhou (1992) and Zhou and Mendoza (1993) presented analytical results for the
turbulent flow characteristics and for the velocity distribution, over and within a porous bed,
respectively. Mendoza and Zhou (1992) found a logarithmic law of the wall for the turbulent
flow over a porous bed, using a general expression for the turbulent shear stress. However, the
constant of the law of the wall was a function of the interfacial velocity, which is not known a
priori. Hence, application of such a law of the wall is rather difficult.
Li and Garga (1998) presented analytical results for the turbulent seepage flow occurring at
the transition zone (between the fluid zone and the pressure seepage zone) of gravel river
reaches or non−conventional rockfill spillways. However, the whole analysis was based on
known velocity and shear stress at the interface (top of the transition seepage zone) from the
main channel flow characteristics (Li, 1990).
Hahn et al. (2002) applied DNS to the fluid region only for turbulent flow conditions, using
an extended version of the interfacial condition suggested by Beavers and Joseph (1967) for
laminar flow, which is also appropriate for turbulent flow. They found significant skin–
friction reductions at the permeable wall, decrease of the viscous sub–layer thickness and
weakening of the near–wall vortical structures.
6
The hydrodynamics effects of such an interface on water quality and mass transfer have been
studied by Svensson and Rahm (1991), Nakamura and Stefan (1994), Nakamura et al. (1996)
and Mackentum and Stefan (1998) among others.
Svenson and Rahm (1991) presented a mathematical model of the benthic boundary layer and
the porous bottom and examined the vertical exchange of oxygen in the benthic boundary
layer for different porosities and consumption rates. They found that such an exchange may
be considerably enhanced in a thin layer near the sediment/water interface due to a dispersion
mechanism.
Nakamura and Stefan (1994) presented a model of sediment oxygen demand (SOD) that
relates SOD to flow velocity over the sediments. The effect of the diffusive boundary layer in
the water above the sediment on the SOD was shown quantitatively. At very low velocities
the SOD can be simply expressed in terms of velocity. In a companion paper Mackentum and
Stefan (1998) verified experimentally that SOD increases linearly with the velocity of the
water above the sediments for low velocities. Both the rate of increase with velocity as well as
the upper bound of SOD were found to depend strongly on the sediment material, the benthic
biology and the temperature.
Nakamura et al. (1996) proposed a general model for predicting oxygen flux at the
sediment−water interface. The oxygen flux is described as a function of DO (?) concentration
in the bulk water, the shear velocity, Schmidt number, equivalent sand roughness, volumetric
consumption rate of oxygen in the sediment and apparent diffusion coefficient.
In the present study the characteristics of turbulent flow in a two–dimensional (2D) open
channel with a porous bed are studied numerically using the "microscopic" approach. The
porous layer is simulated as a bundle of cylindrical rods (their axes being normal to the flow
direction) of certain diameter and spacing resulting in porosity, which ranges from 0.4404 to
0.8286. The Reynolds Averaged Navier−Stokes (RANS) equations are solved in conjunction
7
with a low−Reynolds number k−0� WXUEXOHQFH�PRGHO��/DXQGHU�DQG�6KDUPD��������DERYH�DQG�
within the porous region, as the flow may exhibit laminar regions inside the porous medium.
Two arrangements for the bundle of rods, staggered and non–staggered, are used for
investigating the effects of the porous layer configuration on the channel flow characteristics.
In addition experimental results are presented mainly for comparison purposes, from an
experimental study with hot film anemometry measurements, conducted in the hydraulics
laboratory of Aristotle University of Thessaloniki (Keramaris, 2001).
Emphasis is given on the effects of relative porous depth (hf/H, hf=free flow depth,
H=hf+hp=total depth and hp=porous depth) and the permeability K which can be expressed in
terms of a dimensionless parameter called Darcy number, Da(=K/H2) on the flow
characteristics over the porous bed. The equations were solved with a finite−volume method
using the commercial CFD code FLUENT5 (1998). Computed and measured mean velocities
and turbulence characteristics indicate the significant influence of the above–mentioned
factors on the flow characteristics. Also, the discharge over the porous region is compared
with the corresponding discharge of channels with impermeable bed. Computed flow
characteristics for laminar flow are compared with analytical solutions of Poulikakos and
Kazmierczak (1987) extracted by the "macroscopic" approach and experimental findings of
Beavers and Joseph (1967).
To the author’s knowledge for the first time in the present study:
(a) The "microscopic" approach is used for investigating turbulent flow in a channel with a
porous bed.
(b) Detailed flow characteristics above and within the porous region are presented for cases
in which penetration of turbulence occurs into the porous layer.
8
(c) The discharge capacity of channels with porous bed is estimated for turbulent flow
conditions and is compared with the respective one of channels with smooth and rough
impermeable beds.
(d) The role of turbulence is identified by comparing flow characteristics for respective
laminar and turbulent cases.
THEORETICAL CONSIDERATIONS – GOVERNING EQUATIONS
Either the "macroscopic" or "microscopic" approach can be used for the problem under
consideration (Fig. 1). The two approaches differ in the way they treat the porous layer. The
former uses macroscopic characteristics for describing the variation of flow characteristics
within the porous bed, while in the latter the flow is resolved at a local level. The governing
equations for each approach are presented in the following paragraphs together with the
advantages and drawbacks of each methodology.
Macroscopic Approach
Using this approach the turbulent flow over the porous bed is described by the RANS
equations, while the flow within the porous region is described by the extended Darcy
equation including the Forchheimer (microscopic form drag) term and the Brinkman (viscous
diffusion) term (Vafai and Tien, 1981, Vafai and Kim, 1995). The latter equation has been
extended by Antohe and Lage (1997), Nakayama and Kuwahara (1999) and Getachew et al.
(2000) for turbulent, incompressible flow in porous media. These equations, namely
continuity (eqs. (1) and (3)) and momentum (eqs. (2) and (4)), for both free flow and porous
regions (Fig. 1) are:
Free Flow Region:
0x
U
i
i =∂∂
(1)
9
−
∂∂
+∂∂
∂∂+
∂∂−=
∂∂
uux
U
x
U�
xx
P
!
1
x
UU ji
i
j
j
i
jij
ij (2)
Porous Region:
0x
U
i
i =∂∂
(3)
+φ−νφ−
−
∂∂
+∂∂
∂∂+
∂∂−=
∂∂
jijj
jijj
F2i
jii
j
j
i
jij
ij
uuUU
UUUU
K
cU
K
uux
U
x
UJ�xx
P
!
1
x
UU
(4)
where Ui=time−averaged fluid velocity in the xi direction, P=effective pressure (the difference
between the static and the hydrostatic pressure), jiuu− =Reynolds stresses, ! IOXLG�GHQVLW\��
� ���!� IOXLG� NLQHPDWLF� YLVFRVLW\�� φ =porosity, K=permeability, cF=Forchheimer (inertia)
FRHIILFLHQW�� - YLVFRVLW\� UDWLR� � �eff�����eff=effective viscosity). Ratio J can be assumed to be
equal to unity, although it was indicated that its value deviates from unity for high porosity
media (Givler and Altobelli, 1994). Eq. (4) consists of the convective inertia term in the left–
hand side and the following terms in the right–hand side: (a) pressure gradient term, (b)
Brinkman (viscous diffusion) and turbulent diffusion terms, (c) Darcy (microscopic viscous
drag) term and (d) Forchheimer (microscopic form drag) first and second order terms,
respectively. The Einstein convention is adopted for repeated indices in all equations. It
should be noted that for 2D uniform, open channel flow, the pressure gradient (–dP/dx) is
HTXDO� WR� !J6o (g=acceleration due to gravity, So=channel slope). Finally (Fig. 1),
x=streamwise and y=normal coordinates, while U and V are the corresponding velocity
components.
In order for Eqs. (1) to (4) to "close", the Reynolds stresses that appear in Eqs. (2) and (4)
have to be estimated with the aid of a turbulence model. While turbulence models for the free
10
flow region are well−established and applied in numerous cases (Rodi, 1980), macroscopic
turbulence models for flow in porous media are rather scarce. Recently, theoretical
developments of such models have been attempted by several investigators (Masuoka and
Takatsu, 1996, Antohe and Lage, 1997, Nakayama and Kawahara, 1999 and Getachew et al.,
2000) but their application is very limited.
In cases where momentum transfer from the free flow to the porous region are weak and the
penetration of turbulence into the porous region is restricted by the structure of the solid
matrix (low porosity), the flow in the porous medium is laminar and a macroscopic turbulence
model for the porous region is not required. In this case the turbulent diffusion and second
order Forchheimer terms in equation (4) are omitted and hence equation (4) can be solved in
conjunction with equations (1) (2) and (3) and appropriate matching conditions at the
porous/fluid interface (Svensson and Rahm, 1991). Also, in the case of laminar flow in both
regions the above equations are simplified into the respective Navier−Stokes equations for the
fluid region and can be solved either numerically or analytically.
Analytical solutions to the problem of laminar flow over and within a porous bed have been
derived by several investigators. Poulikakos and Kazmierczak (1987) used Eqs. (1) to (4)
(omitting the turbulent and Forchheimer terms) together with continuity interfacial conditions
for velocity and shear stress. They found analytically the velocity distribution in both fluid
and porous regions for 2D, fully developed channel flow. Such a distribution is given by the
following relationships (Fig. 1):
Fluid Region (0 ≤ y ≤ hf):
−
−
+
−−
−
−
=−
− 1
DaHh1cosh
1DaDa
Hh1tanh
HhDa
Hh
H
yh2
1
A
U
2/1f
2/1ff2/122
ff (5)
11
Porous Region (−hp ≤ y ≤ 0):
Da
DaHh1cosh
DaH
yh1sinhHhDaDa
H
ycoshDa
A
U
2/1f
2/1ff2/12/1
−
−
−−−
=−
−−
(6)
where A=(H2����G3�G[�@� DQG�'D .�+
2. In the present study, the velocity distribution in the
free flow region and the discharge capacity of the channel for laminar flow, derived by Eq.
(5), are compared with computed values calculated by the "microscopic" approach.
Microscopic Approach
Using the "microscopic" approach the RANS eqs. (1) and (2) are solved in the whole flow
region, above the porous medium, as well as inside it. The latter, of given permeability K and
porosity φ , is simulated as a bundle of cylindrical rods (Fig. 2).
For modeling the Reynolds stresses appearing in Eq. (2), the eddy−viscosity concept is used:
k3
2
x
U
x
Uuu ij
i
j
j
itji δ−
∂∂
+∂∂ν=− (7)
ZKHUH��t HGG\�YLVFRVLW\��/ij=Kronecker delta and k=(1/2)u2i =turbulent kinetic energy.
The Launder and Sharma (1974) low Reynolds k−0�PRGHO�LV�XVHG�IRU�FDOFXODWLQJ��t. The use
of such a turbulence model is necessary in regions where turbulence is damped (near–wall
regions or low permeability porous media). The wall regions are resolved down to the solid
boundaries without using any boundary conditions for the first grid point near the wall (wall
functions). The first grid point lies well inside the viscous sublayer and hence flow
characteristics are calculated in both the viscous sub−layer and the fully turbulent flow region.
The eddy viscosity �t� LV� UHODWHG� WR� N� DQG� LWV� UDWH� RI� GLVVLSDWLRQ�� 0�� WKURXJK� WKH�
Kolmogorov−Prandtl relationship.
( )ε=ν µµ /kfc 2t (8)
12
where c�=0.09 and f�=exp[−3.4/(Rt/50)2]=damping function accounting for low−Re and
wall−proximity effectV��7KH�IROORZLQJ�WUDQVSRUW�HTXDWLRQV�IRU�N�DQG�0�DUH�VROYHG�
ε−
∂∂
σν+ν
∂∂+=
∂∂
x
k
xP
x
kU
jk
t
jk
jj (9)
kfc
xxP
kfc
xU
2
22j
t
jk11
jj
ε−
∂
ε∂
σν+ν
∂∂+ε=
∂ε∂
εεε
εε (10)
where Pk= uu ji− (∂Ui/∂xj)=production rate of k due to shear, f0�=1, f0�=1−0.3[exp(−Rt2)];
Rt=k2���0� WXUEXOHQW�5H\QROGV�QXPEHU��F0�=1.44, c0� ������1k �����DQG�10=1.3.
The interaction of the free turbulent flow with that inside the solid matrix may result in the
penetration of turbulence into the upper part of the porous medium and to increased
turbulence levels (shear stresses and intensities). Increased shear is expected to bring a
reduction of the mean velocity in the flow above the porous region and a respective increase
in the upper porous area. In other words, turbulence promotes the momentum exchange across
the interface, compared with the situation where an impermeable wall is located at the
interface. The reduction of the velocity above the porous layer implies a corresponding
reduction of the channel discharge capacity.
Opposite trends have been observed in the case of laminar flow over and within a porous bed.
Beavers and Joseph (1967) used the momentum equation for fully developed laminar flow
and an empirical porous/fluid interfacial condition and estimated an increase in mass flow
over the porous medium with regard to that with impermeable bed.
NUMERICAL PROCEDURE − BOUNDARY CONDITIONS
Eqs. (1), (2), (7), (8), (9) and (10) are solved with the CFD code FLUENT5 (1998). The grid
was dense enough near the walls, as indicated by the low s+ (=sU*/�, s=normal distance from
solid boundaries, U*=friction velocity, �=kinematic viscosity) values at the mesh nodes
13
adjacent to the walls. Provision was made in the mesh spacing in order to have at least 2÷5
nodes located inside the viscous sub–layer (s+<2).
The computational domain, as well as the notation, co–ordinate system, boundary conditions
and fluid/solid zones are shown in Fig. 3. The geometric characteristics are the same with the
ones used in the experiments. The geometry is assumed to repeat infinitely in the streamwise
direction, x. The used computational meshes consisted of quadrilateral (adjacent to all walls)
and triangular (in the core flow regions) elements (a sample is shown in Fig. 4). The mesh
size for all cases is given in Table 1.
Periodicity conditions apply in the direction of the mean flow, extracted for a given channel
slope (given streamwise pressure gradient). At the free surface symmetry conditions are
applied, as the Froude number is relatively small and there is no surface distortion in the flow
direction. At the solid boundaries (bottom wall and rod surfaces) no−slip boundary conditions
are employed.
The main characteristics of the numerical procedure used in FLUENT5, which is an elliptic
iterative solver, are: (a) Discretization Schemes. Combination of bounded QUICK (at
quadrilateral mesh elements; Leonard and Mokhtari, 1990) and Second Order Upwind (at
WULDQJXODU�PHVK�HOHPHQWV��%DUWK�DQG�-HVSHUVHQ��������IRU�8��9��N�DQG�0��6HFRQG�2UGHU�IRU�3�
(analogous to Second Order Upwind for convection terms above). PISO for velocity–pressure
coupling. (b) Number of Iterations Required. Approximately 50,000 for complete
convergence. (c) Convergence. The periodic condition poses a difficulty in convergence,
hence the large number of iterations. The problem may be considered quasi−elliptic (the
boundary condition at the periodic faces is not fixed, rather it changes from iteration to
iteration and progresses to its converged state very slowly). The double precision solver was
used and all normalized residuals were reduced to less than 10–8 before convergence was
accepted. Also, convergence was monitored at selected locations (e.g. rod surface and
14
periodic planes), in terms of achieving invariable (with iteration) average values of P, U, k
and drag coefficient.
EXPERIMENTAL PROCEDURE AND MEASUREMENTS
The experiments were conducted in an open channel of the Hydraulics Lab. of the Civil Eng.
Dept., in Aristotle University of Thessaloniki. The channel was 12 m long, 25 cm wide and 50
cm high. For simulating the porous bed a bundle of rods placed perpendicular to the flow with
a non-staggered arrangement was constructed. Geometrical characteristics and other details
are those of Fig. 3. The bundle had a length equal to half of the channel length and with such
an arrangement the porosity φ found equal to 0.8286. For this type of porous bed the relative
depth, hf/H, was varied from 0.35 to 0.48 approximately. A part of the model porous medium
used in the experiments is shown in Fig. 5. The slope, So, of the channel was kept constant
and equal to 2×10−3 in all the experiments. The velocity distribution in the fluid region (over
the porous region) was measured initially with a Pilot tube (4 mm internal diameter)
connected to an inclined manometer and subsequently with a hot−film anemometry (TSI cross
hot film connected with IFA 100). Measurements were made at a distance 8 m from the
channel inlet, where the flow was fully developed and uniform (constant flow depth in the
flow direction). The uniformity and the development of the flow were checked using classical
laboratories procedures. The measurements were conducted at the vertical central line of the
cross section where the flow was two−dimensional since in all experiments the flow aspect
ratio (width/depth) was always higher than 5. Local velocities were measured within the flow
depth and especially near the porous/fluid interface (the closest point was 2 mm from the
interface).
The total discharge was measured at the channel outlet through a triangular weir. From the
velocity distribution over the porous bed the corresponding discharge could be evaluated and
15
hence the velocity and the discharge within the porous region could be estimated by
subtracting it from the total measured discharge.
The Reynolds number, Ref, (based on the mean free flow velocity, Uf, and on hf) varied from
7,000 to 20,000 (fully turbulent flow). The main characteristics of the flow conditions are
shown in Table 1.
ANALYSIS OF RESULTS
Three permeable beds with porosity φ (volume of fluid over total porous medium volume)
equal to 0.4404 (cases 135−30, −50 and −70), 0.7144 (150−30, −50 and −70) and 0.8286
(250−30, −50 and −70) were studied for both laminar and turbulent flow conditions. In the
150 and 250 cases three rods were employed (having diameter, D=10 mm), while for cases
135 four rods (D=11.5 mm). The height of the porous region, hp, was kept constant and equal
to 55 mm. The permeability K was estimated by a method described in Bird et al. (1960). The
geometric and hydrodynamic characteristics are shown in Fig. 3 and Table 1, respectively.
Table 1 also displays the friction velocity, U*(= Shg of ), corresponding to channels with
smooth impermeable bed at the location of the interface, Darcy number, Da=K/H2 and
Reynolds number, Ref �!KfUf�����ZKHUH�! ������NJP−3�DQG�� �����×10−3 kgm−1s−1 (water).
Table 2 presents the results for the interfacial velocities; computed Uint,po and Uint,m, being the
point velocity at the intersection of the interface and the periodic plane and the
interface−averaged value (Fig. 3), respectively and analytical, Uint,ana, calculated by Eq. (5) or
(6) for laminar flow only. Also, the depth−averaged velocities for the free flow region are
presented; computed, Uf,com, analytical for laminar flow only, Uf,ana, integrating Eq. (5) and
the ones for impermeable wall, Uf,imp (calculated by the Poisseuille parabolic solution for
laminar flow and the logarithmic�ODZ�GLVWULEXWLRQ�ZLWK�� �����DQG�& �����IRU�WXUEXOHQW�IORZ���
Finally, the depth−averaged velocities for the porous region are given; computed, Up,com,
16
analytical for laminar flow only, Up,ana, integrating Eq. (6) and the Darcy one for laminar flow
only, UD �.����−dP/dx).
The effect of cylinders arrangement (structure of the model porous medium) on the flow
characteristics was studied initially by using a staggered (case 135–30s)and non–staggered
(case 135–30) arrangement (Figure 6a) with different rods diameter but the same porosity
(φ =0.4404). Figure 6b shows the velocity distribution above the porous region in wall
coordinates for both arrangements. In both cases velocities are much lower than the respective
ones for flow over a smooth impermeable bed with flow depth hf. There is a small difference
in velocities between the two arrangements which is attributed to the slightly different Darcy
numbers (different permeabilities) due to the different rods diameter. Similar conclusions can
be derived from the comparison of the computed turbulence kinetic energy and shear stress
for both arrangements (Figures 6c and 6d).
In the following figures computed results using the non-staggered arrangement are presented.
Figs. 7a and 7b show the contour plots for U/Uint,po for turbulent and laminar flow conditions,
respectively, for three Da numbers ranging from 5.0331×10−5 to 3.7252×10−2. The velocities
have been made dimensionless with the interfacial velocity at the periodic plane for indicating
the interaction between the flow over and within the rods. Contours of turbulent U/Uint,po
indicate that mixing (interaction) increases with increasing Da number, resulting in the
decrease of the maximum U/Uint,po from 4.2 to 2.8. Also, velocities below the interface
increase with increasing Da number. (Note that the solution domain was graphically repeated
in the streamwise direction for clarity reasons and that white areas in the region of the rods'
wake indicate negative streamwise velocities, i.e. recirculation).
Similarly, contours of laminar U/Uint,po indicate increasing mixing with increase of Da
number, however, it is much lower than that of turbulent flow. The maximum value of the
ratio for the lowest Da number is approximately 26, an order of magnitude higher than for the
17
corresponding turbulent case. With increasing Da number this value drops to approximately
13, being again much higher than the respective value for turbulent flow.
In Fig. 8 the contour plots for the normalized turbulent kinetic energy, k/U2* , are plotted for
hf/H=0.4762. Turbulent kinetic energy has been made dimensionless with the shear velocity at
the interface for comparing its values with the respective ones for flow in a channel with an
impermeable bed at the position of the interface. Turbulent kinetic energy is shown to be of
significant magnitude in the region above and in the upstream part of the rods and then
decreases continuously as the free surface is approached. The highest value of k/U2*
approaches 3.84, much lower than 4.78 which is the asymptotic for flow over an impermeable
bed with flow depth hf (Nezu and Nakagawa, 1993). The size of the region in which k is
significant increases with Da. Penetration of k into the rods region is also considerable and
increases with Da (increasing gap between the rods). In areas close to the impermeable bed,
levels of k are very low (laminar flow), while for the smaller Da, k is almost zero in the whole
porous region.
Fig. 9 shows the contour plots for the normalized turbulent shear stress, uv− / U2*
( uv− �t(∂U/∂y+∂V/∂x)) again for hf/H=0.4762. The shear stress has been normalized with
the shear velocity at the interface for the same reasons as for k.As in the case of k, shear stress
increases significantly above and in the upstream part of the rods and its maximum value is
much higher than unity (the asymptotic value for flow over an impermeable bed). The size of
the areas with high values of uv− , as well as the penetration of uv− within the rods,
increases with Da. (Note that white areas indicate small negative values of uv− ).
Fig. 10 presents the turbulent U profile in wall co–ordinates y+(=yU*���� DQG�8+(=U/U*) for
the three hf/H studied (Re varying from 4918 to 26523) at the periodic vertical line (Fig. 3). In
the same plot the laws for the viscous sub−layer, U+=y+, and the fully–turbulent (logarithmic)
18
layer, U+ �����OQ\
+�&� �� 0.41, C=5.29), for 2D open channel flow over a smooth
impermeable bed are also plotted.
In addition, the logarithmic law for flow over an impermeable bed with two-dimensional
roughness (semi–cylindrical roughness elements of 0.5 cm height) is included,
U+=�����OQ\++C−(ûU/U*) where (ûU/U*) is the roughness function (Raupach et al., 1991).
The roughness function is related to the roughness Reynolds number e+ (=eU*/�, e=height of
roughness element) through the ratio � (=e/lw, lw=streamwise ″wavelength″). Hence, for the
arrangements considered � takes values from 0.20 (for porosity φ =0.8286) to 0.33 (for
φ =0.4404) and the respective constants of the law of the wall have values of C=–7.57 and
C=–2.61 (Raupach et al., 1991).
It is shown that the velocities above the interface are much lower than the respective ones for
flow over a smooth impermeable bed. This is due to the action of turbulent shear stress and
the penetration of turbulence inside the porous region, which reduces the mean velocities
above the porous region. Near the interface the velocities deviate from the logarithmic law,
indicating that the flow near the rods is rather of a mixing layer type than of boundary layer
type. Similar trends are observed in the experimental velocity distribution, which indicates
higher velocities than the computed ones but still much lower than those over a smooth
impermeable bed. This may be due to the limited length of the porous region in the laboratory
channel. It is shown that with a reduction of Darcy the velocity above the porous region
increases and approaches the one over an impermeable bed. In addition, a decrease of Darcy
(i.e. of permeability) the penetration of turbulence into the porous matrix also decreases,
hence the effect of turbulent shear stresses on the mean flow field is weaker.
Velocities above the porous region are also lower than the respective ones for flow over a
rough impermeable bed having �=0.33 (small spacing between the roughness elements) while
they are similar with the ones for flow over a rough impermeable bed having �=0.20 (large
19
spacing between the roughness elements). For low porosity of the porous layer (small spacing
between the roughness elements) turbulence is capable of penetrating the interface and hence
mean velocities in the fluid region are reduced. For high porosity of the porous layer (large
spacing between the roughness elements) the turbulence structure behind the rods is similar
with that behind the roughness elements and this results in similar mean velocity distribution
above the porous region with that over the rough impermeable bed.
Computed velocities over the porous region (φ =0.8286, Da=3.7252×10–2) are also compared
with computed velocities, for the case that an impermeable bed exists at a distance (hf+D/2)
below the free surface with roughness elements of 5 mm height and �=0.2 (Fig. 11). In the
same figure the law of the wall for �=0.2 is shown as in the previous figure. The two velocity
distributions are comparable and so is the discharge capacity.
Opposite trends are observed when the flow is laminar in both the free fluid and the porous
regions (Ref ranging from 9.4 to 119.8). Fig. 12 shows the velocity distribution above and
below the porous/fluid interface for the three values of hf/H examined. The velocities are
presented in normalized form, Un=U/[(H2����−dP/dx)], in order to be directly compared with
analytical solutions derived by Poulikakos and Kazmierczak (1987). In the same figure half
the velocity distribution for laminar flow between two impermeable plates is plotted (from
one plate up to the symmetry plane), hence the effect of porous layer on velocity distribution
in the free flow region is shown. Computed velocities above the interface are higher than
those over an impermeable bed and in close agreement with those predicted analytically by
the macroscopic approach of Poulikakos and Kazmierczak (1987) but only for the smaller Da
number (=3.5514×10−5, 5.0331×10−5 and 7.6803×10−5 for hf/H=0.5600, 0.4762 and 0.3529,
respectively). For the other Da numbers the analytical solution, Eqs. (5) and (6), indicates
much higher velocities in both the fluid and the porous regions, as form drag effects
(Forchheimer term) are not taken into account. These cases correspond to the cylinder
20
configuration with large gaps (cases 150 and 250). For laminar flow conditions the interaction
between porous flow and free flow results in an interfacial slip velocity, which in turn affects
the velocity distribution above the interface. These findings are in agreement with
experiments performed by Beavers and Joseph (1967) for laminar flow conditions. They
measured increased volume flow rates for the region above the porous matrix. The latter
findings are also clear from the investigation of the discharge capacity of such channels and in
comparison with the respective one for impermeable channels (shown in Fig. 15 below).
Fig. 13 displays the variation of the computed interfacial velocity Uint,m with the Da number
for both laminar and turbulent flow conditions. The interfacial velocity has been normalized
with Uf for indicating its strength with regard to the mean flow velocity. Interfacial velocities
for turbulent flow conditions are quite considerable (24−34 % of Uf) for all values of Da
number, while for laminar flow they decrease significantly (3−6 % of Uf ) for low values Da
number (low permeabilities). Hence, the assumption of no−slip velocity at the interface is
only valid for laminar flow and low permeability.
Fig. 14 shows the effect of Da number on channel discharge Qf (in the free flow region),
normalized with the corresponding value for channel with smooth impermeable bed, Qf,imp,
for both laminar and turbulent flow conditions. In addition, experimental results of Beavers
and Joseph (1967) for laminar flow conditions are included together with those predicted
analytically by Poulikakos and Kazmierczak (1987) only for laminar flow. Also, the present
measurements for turbulent flow are included in the plot. In turbulent regime, the discharge
above the porous region is highly reduced (being 36−55% of Qf,imp) especially for high
permeabilities. On the contrary, computed, experimental and analytical discharge for laminar
flow for conditions studied are shown to be higher than that over an impermeable bed,
ranging from 102−114 % of Qf,imp for computed and 103−420 % for analytical results. The
21
higher rates predicted analytically by Poulikakos and Kazmierczak (1987) for some cases are
due to the assumptions involved in their analytical derivation (no form drag effects included).
Fig. 15 presents the vertical variation at the periodic plane (Fig. 3) of k+(=k/ U2* ) above and
within the porous region for the turbulent cases. It is shown that the k distribution in the free
flow region is more uniform than that observed over impermeable beds (empirical
relationship developed by Nezu and Nakagawa, 1993; k+=4.78exp(−2y/hf)). Computed and
experimental k are in satisfactory agreement, except near the free surface where experimental
measurements indicate higher levels. It should be noted that experimental k has been
estimated from independent measurements of the three normal stresses, k=0.5(wvu 222 ++ ).
Significant levels of k are computed at the interface and within the porous layer. For the
highest Da number (highest permeability) the penetration of turbulence is shown to extend up
to y/hf=−0.1. Again the distribution of k over the porous region indicates mixing layer flow
character in the interfacial region, rather than those over an impermeable bed (boundary layer
characteristics).
Figure 16 shows the distribution of the measured turbulence intensities (uï/U*, vï/U*, wï/U*,
where 2u'u = , 2v'v = , 2w'w = ) above the porous region for the experimental
conditions. In addition the semi-empirical distributions of Nezu and Nakagawa (1993)
(uï/U*=2.30exp(−y/hf), vï/U*=1.27exp(−y/hf), wï/U*=1.63exp(−y/hf)) for the flow over an
impermeable bed are shown for comparison purposes. The experimental distribution of
intensities is more uniform than that over an impermeable bed indicating the significant
″mixing″ occurring near the fluid/porous interface which results in reduced levels of
intensities above the interface, similar to those of turbulence kinetic energy k.
Fig. 17 presents the profile of the normalized turbulent shear stress uv− +(= uv− / U2* ) above
and within the porous region at the periodic plane (Fig. 3). It is shown that turbulent shear
22
stresses are increased above the porous region. In particular, uv− is higher than U2* ,
indicating again that the flow is of the mixing layer rather than of the boundary layer type.
Experimental measurements also indicate similar trends. Significant shear stresses are
computed up to y/hf equal to –0.1 indicating penetration of turbulence in the rods region.
CONCLUSIONS
Turbulent 2D flow in an open channel with a permeable bed of porosity varying from 0.4404
to 0.8286 has been studied numerically and experimentally for Da numbers ranging from
3.5514×10−5 to 5.6844×10−2. The characteristics of the flow over and within the porous region
were computed using a "microscopic" approach, while mean and turbulence flow parameters
over the permeable region were measured with hot–film anemometry.
The following conclusions can be derived:
(a) The porous layer structure (staggered and non-staggered arrangement) has a very small
effect on the flow characteristics near the fluid/porous interface.
(b) For turbulent flow over the permeable bed, mean velocities inside the free flow region
decrease significantly, with regard to those of flow over a smooth impermeable bed due to the
penetration of turbulence and the associated momentum transfer in the upper part of the
porous matrix. Such a decrease is augmented with the increase of Da (increase of
permeability). Velocities over the porous bed are also lower that the respective ones over a
rough impermeable bed. The level of reduction is related with the porosity of the permeable
layer and the respective characteristics of the roughness elements (parameter �). On the other
hand, for laminar flow over the permeable bed computed results and analytical solutions of
other investigators indicate opposite trends (increase of mean velocities over the porous
region).
23
(b) Discharge capacity of channels with a permeable region is reduced for turbulent flow and
high permeabilities when compared with the respective one for channels with smooth and
rough impermeable bed. Conversely, for laminar flow over the porous bed, discharge capacity
increases when compared with the respective one over an impermeable bed.
(c) Mean turbulent velocities over the permeable bed are shown to deviate from the
logarithmic law of the wall, not only in terms of the constant of integration, C (being lower
WKDQ������IRU�VROLG�ZDOO���EXW�DOVR�LQ�WHUPV�RI�WKH�YRQ�.DUPDQ�FRHIILFLHQW�����7KLV�LPSOLHV�WKDW�
the flow is rather of a mixing layer type than of a boundary layer type.
(d) Penetration of turbulence to the upper part of the porous region results in significant levels
of turbulence kinetic energy up to y/hf=−0.1. The distribution of k further indicates the mixing
layer character of the flow at the interfacial region.
(e) Levels of the turbulent shear stress uv− are increased within the interfacial region and its
value at the porous/fluid interface is higher than the asymptotic maximum value of U2* for
impermeable wall (derived from momentum balance considerations).
24
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28
APPENDIX II: NOTATION
The following symbols are used in this paper:
C = constant of the logarithmic law;
Da = Darcy number = �/þ2;
cF = Forchheimer (inertia) coefficient;
c�, f� = coefficients for the definition of �t;
c01, c02, f01, f02 = turbulent model coefficients;
D = rod's diameter;
e = roughness height;
g = gravitational acceleration;
h = height;
H = total channel depth;
J = viscosity ratio =�eff/�;
K = permeability;
k = turbulent kinetic energy;
L = vertical distance between rods;
l = vertical distance of the last rod from the solid bed;
lw = streamwise wavelength;
P = effective pressure;
Pk = production rate of k;
Q = volumetric discharge;
Re = Reynolds number;
Rt = turbulent Reynolds number;
So = channel slope;
U, V = streamwise and vertical velocity components;
29
UD = Darcy velocity;
Un = normalized streamwise velocity = U/[(H2/�)(dP/dx)] = U/A;
U* = friction velocity = (ghfSo)0.5;
uï, vï, wï = turbulence intensity components;
jiuu− = Reynolds stress tensor;
W = periodic width;
x, y = streamwise and vertical directions;
. = slip parameter;
/ij = Kronecker delta tensor;
0 = dissipation rate of k;
� = von Karman constant;
� = e/lw;
� = fluid's dynamic viscosity;
� = fluid's kinematic viscosity;
�t = turbulent (eddy) kinematic viscosity;
! = fluid's density;
1k, 10 = turbulent Prandtl numbers; and
φ = porosity.
Subscripts
ana = analytical value;
com = computed value;
f = mean free (unblocked) flow value;
i, j = index for the Cartesian coordinates;
imp = value for the impermeable bed;
30
int = interfacial value;
m = (horizontal) mean value between periodic planes;
p = mean flow value inside the porous medium; and
po = point value at the periodic plane.
Superscript
+ = wall coordinates (logarithmic law).
31
Table 1. Geometric and Hydrodynamic Characteristics.
Case W [m]
H [m]
hf [m]
hp [m]
hf/H D [m]
No of Rods
φ K [m2]
Da −dP/dx [Nm−3]
So U*
[ms−1] Ref
No of Cells
LAMINAR FLOW −COMPUTATIONS 135-30 0.0850 0.03 0.3529 7.6803×10−5 1.7336×10−4 9.4436×10+0 9225 135-50 0.0135 0.1050 0.05 0.4762 0.0115 4 0.4404 5.5490×10−7 5.0331×10−5 2.2381×10−4 4.2758×10+1 9282 135-70 0.1250 0.07 0.5600 3.5514×10−5 2.6481×10−4 1.1619×10+2 9576 150-30 0.0850 0.03 0.3529 4.9412×10−3 1.7336×10−4 9.5558×10+0 10779 150-50 0.0150 0.1050 0.05 0.055 0.4762 0.01 3 0.7144 3.5700×10−5 3.2381×10−3 0.001 1.0212×10−7 2.2381×10−4 4.3055×10+1 11073 150-70 0.1250 0.07 0.5600 2.2848×10−3 2.6481×10−4 1.1611×10+2 10154 250-30 0.0850 0.03 0.3529 5.6844×10−2 1.7336×10−4 1.0165×10+1 12025 250-50 0.0250 0.1050 0.05 0.4762 0.01 3 0.8286 4.1070×10−4 3.7252×10−2 2.2381×10−4 4.4689×10+1 12853 250-70 0.1250 0.07 0.5600 2.6285×10−2 2.6481×10−4 1.1980×10+2 13815
TURBULENT FLOW −COMPUTATIONS 135-30s 0.006934 11 2.0174×10−7 2.7922×10−5 5.6047×10+3 9306 135-30 0.0850 0.03 0.3529 7.6803×10−5 2.4283×10−2 6.1000×10+3 9225 135-50 0.0135 0.1050 0.05 0.4762 0.0115 4 0.4404 5.5490×10−7 5.0331×10−5 3.1349×10−2 1.4792×10+4 9282 135-70 0.1250 0.07 0.5600 3.5514×10−5 3.7093×10−2 2.6523×10+4 9576 150-30 0.0850 0.03 0.3529 4.9412×10−3 2.4283×10−2 5.7366×10+3 10779 150-50 0.0150 0.1050 0.05 0.055 0.4762 0.01 3 0.7144 3.5700×10−5 3.2381×10−3 19.62 2.0036×10−3 3.1349×10−2 1.3762×10+4 11073 150-70 0.1250 0.07 0.5600 2.2848×10−3 3.7093×10−2 2.3887×10+4 10154 250-30 0.0850 0.03 0.3529 5.6844×10−2 2.4283×10−2 4.9182×10+3 12025 250-50 0.0250 0.1050 0.05 0.4762 0.01 3 0.8286 4.1070×10−4 3.7252×10−2 3.1349×10−2 1.2105×10+4 12853 250-70 0.1250 0.07 0.5600 2.6285×10−2 3.7093×10−2 2.2008×10+4 13815
TURBULENT FLOW −EXPERIMENTS 250-30 0.0850 0.03 0.3529 5.6844×10−2 2.4283×10−2 7.5805×10+3 250-40 0.0250 0.0950 0.04 0.055 0.4211 0.01 3 0.8286 4.1070×10−4 4.5507×10−2 19.62 2.0036×10−3 2.8040×10−2 1.0784×10+4 250-50 0.1050 0.05 0.4762 3.7252×10−2 3.1349×10−2 1.4366×10+4
32
Table 2. Results for Mean Streamwise Velocities (in ms−1).
Case Uint,po Uint,m Uint,ana Uf,imp Uf,com Uf,ana UD Up,com Up,ana LAMINAR FLOW
135-30 2.9957×10−5 1.8330×10−5 2.1731×10−5 2.9910×10−4 3.1630×10−4 3.2171×10−4 7.1818×10−7 9.6992×10−7 135-50 4.9359×10−5 3.0193×10−5 3.6585×10−5 8.3084×10−4 8.5928×10−4 8.6791×10−4 5.5324×10−7 1.0291×10−6 1.6154×10−6 135-70 6.8467×10−5 4.1906×10−5 5.1438×10−5 1.6284×10−3 1.6679×10−3 1.6798×10−3 1.3545×10−6 2.8048×10−6 150-30 2.9281×10−5 2.2607×10−5 1.4518×10−4 2.9910×10−4 3.2006×10−4 5.1318×10−4 5.7764×10−6 5.1260×10−5 150-50 6.3408×10−5 3.6962×10−5 2.6404×10−4 8.3084×10−4 8.6525×10−4 1.1637×10−3 3.5593×10−5 6.1973×10−6 6.4644×10−5 150-70 8.7343×10−5 5.0719×10−5 3.8295×10−4 1.6284×10−3 1.6667×10−3 2.0798×10−3 6.6318×10−6 7.8573×10−5 250-30 7.2267×10−5 4.4850×10−5 2.7162×10−4 2.9910×10−4 3.4045×10−4 1.2552×10−3 9.9545×10−6 4.5391×10−4 250-50 1.1891×10−4 7.1981×10−5 6.6125×10−4 8.3084×10−4 8.9808×10−4 2.1871×10−3 4.0947×10−4 1.1021×10−5 5.8360×10−4 250-70 1.6236×10−4 9.7908×10−5 1.0530×10−3 1.6284×10−3 1.7196×10−3 3.3846×10−3 1.1918×10−5 7.1384×10−4
TURBULENT FLOW 135-30s 5.7498×10−2 5.0415×10−2 1.8772×10−1 7.6500×10−2 135-30 6.6747×10−2 5.7059×10−2 4.5839×10−1 2.0431×10−1 4.3436×10−3 135-50 8.5912×10−2 7.5654×10−2 6.5120×10−1 2.9727×10−1 4.7018×10−3 135-70 1.0247×10−1 9.1709×10−2 8.1652×10−1 3.8072×10−1 5.0155×10−3 150-30 5.5346×10−2 5.5003×10−2 4.5839×10−1 1.9214×10−1 2.8926×10−2 150-50 8.5414×10−2 7.0670×10−2 6.5120×10−1 2.7656×10−1 2.9085×10−2 150-70 9.6829×10−2 8.1210×10−2 8.1652×10−1 3.4289×10−1 2.8913×10−2 250-30 6.8583×10−2 5.5223×10−2 4.5839×10−1 1.6473×10−1 2.1502×10−2 250-50 8.7406×10−2 7.1331×10−2 6.5120×10−1 2.4326×10−1 2.1844×10−2 250-70 1.0436×10−1 8.5473×10−2 8.1652×10−1 3.1591×10−1 3.2024×10−2
33
List of Figures
Figure 1. Definition sketch for 2D open channel flow over a porous bed.
Figure 2. Arrangement of the rods bundle and geometrical parameters for all cases.
Figure 3. Computational domain with 3 rods, notation and boundary conditions.
Figure 4. Representative mesh used in the computations.
Figure 5. Part of the model porous medium used in the experiments (φ =0.8286).
Figure 6. Comparison of the non-staggered and staggered rod arrangement for hf/H=0.3529.
(a) Geometry setup and dimensions; (b) Streamwise velocities; (c) k/U2* profiles; (d)
U/uv 2*− profiles.
Figure 7a. Contours of turbulent U/Uint,po for hf/H=0.4762.
Figure 7b. Contours of laminar U/Uint,po for hf/H=0.4762.
Figure 8. Contours of U/k 2* for hf/H=0.4762.
Figure 9. Contours of U/uv 2*− for hf/H=0.4762.
34
Figure 10. Turbulent velocity distribution over the porous region in wall coordinates. Lines;
computations, symbols; measurements.
Figure 11. Comparison of turbulent velocity distribution for porous and rough bed (φ =0.8286
and Da=3.7252×10−2). Lines; computations, symbols; measurements.
Figure 12. Laminar velocity distribution over the whole depth. Lines; computations, symbols;
analytical solution (Poulikakos and Kazmierczak, 1987).
Figure 13. Effect of bed permeability on the interfacial velocity.
Figure 14. Effect of Darcy number on channel discharge.
Figure 15.Variation of U/k 2* above and within the porous bed.
Figure 16. Variation of turbulent intensities above the porous bed. Lines; empirical relations
of Nezu and Nakagawa (1993) for impermeable bed, symbols present measurements.
Figure 17.Variation of U/uv 2*− above and within the porous bed.
35
)UHH�)ORZ�
3RURXV�5HJLRQ�
,PSHUPHDEOH�:DOO�
KI�
KS�
+�[��8��
\��9��
LQWHUIDFH�
Figure 1. Definition sketch for 2D open channel flow over a porous bed.
36
&DVHV ���
0 ������
: ������ >P@
' ������ >P@
/ ������ >P@
O ������ >P@
KS ����� >P@
&DVHV ���
0 ������
: ������ >P@
' ������ >P@
/ ������ >P@
O ������ >P@
KS ����� >P@
&DVHV ���
0 ������
: ������ >P@
' ������ >P@
/ ������ >P@
O ������ >P@
KS ����� >P@
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
KI � �� � � > P @
Figure 2. Arrangement of the rods bundle and geometrical parameters for all cases.
37
wall
H
hp
hfD
L
L
W
xy
free surface (symmetry)
pe
rio
dic
pe
rio
dic
wall
wall
wall
interface
lwall
H
hp
hfD
L
L
W
xy
free surface (symmetry)
pe
rio
dic
pe
rio
dic
wall
wall
wall
interface
l
Figure 3. Computational domain with 3 rods, notation and boundary conditions.
41
13.5 13.5
85.0
30.0
11.5
1.5
2.5
1.06.934
6.934
2.0
7.86
3
3.283
3.39
62.
0
non-staggered staggered
13.5 13.5
85.0
30.0
11.5
1.5
2.5
1.06.934
6.934
2.0
7.86
3
3.283
3.39
62.
0
non-staggered staggered (a)
1 1 0 1 00 1 00 0y
0
5
10
15
U+
+
����
� & ��
��
N on -S tag gered
S tagg ered
(b)
0 1 2 3 4N
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y/hf
+
N o n-S taggered
S tag gered
(c)
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2�XY
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y/hf
+
N on-S taggered
Staggered
(d)
Figure 6. Comparison of the non-staggered and staggered rod arrangement for hf/H=0.3529: (a) Geometry setup and dimensions; (b) Streamwise
velocities; (c) k/U2* profiles; (d) U/uv 2
*− profiles.
42
Da=5.0331×10−5 Da=3.2381×10−3 Da=3.7252×10−2
Figure 7a. Contours of turbulent U/Uint,po for hf/H=0.4762.
Da=5.0331×10−5 Da=3.2381×10−3 Da=3.7252×10−2
Figure 7b. Contours of laminar U/Uint,po for hf/H=0.4762.
45
1 1 0 1 00 1 00 0y
0
5
1 0
1 5
U+
+
h /H = 0 .3 529
'D ������ ��
'D ������ ��
'D ������ ��
f0
5
1 0
1 5
U
h /H = 0 .4 762
'D ������ ��
'D ������ ��
'D ������ ��
+ f
0
5
1 0
1 5
2 0
U
h /H = 0 .5 600
'D ������ ��
'D ������ ��
'D ������ ��
+ f
����
� & ����
����
� & ����
����
� & ����
-5
-3
-2
-5
-3
-2
-5
-3
-2
����
� & ��
����
�����
����
� & ��
����
�����
����
� & ��
����
�����
����
� & ��
����
�����
����
� & ��
����
�����
����
� & ��
����
�����
Figure 10. Turbulent velocity distribution over the porous region in wall coordinates. Lines;
computations, symbols; measurements.
46
+1 1 0 1 00 1 00 0y
0
5
1 0
1 5
2 0
U+
����
� & ����
po rou s b ed
ro ugh b ed
����
� & ��
����
�����
Figure 11. Comparison of turbulent velocity distribution for porous and rough bed (φ =0.8286
and Da=3.7252×10−2). Lines; computations, symbols; measurements.
47
0 .0 0 0 .0 4 0 .0 8 0 .1 2 0 .1 6 0 .2 0
8
-2 .0
-1 .5
-1 .0
-0 .5
0 .0
0 .5
1 .0
y/h f
-1 .0
-0 .5
0 .0
0 .5
1 .0
y/h f
-0 .5
0 .0
0 .5
1 .0
y/h f
h /H = 0 .560 0
Im perm eab le W all
D a= 3.5 5 1 4 10
D a= 2.2 8 4 8 10
D a= 2.6 2 8 5 10
f
-5
-3
-2
h /H = 0 .476 2
Im perm eab le W all
D a= 5.0 3 3 1 10
D a= 3.2 3 8 1 10
D a= 3.7 2 5 2 10
f
-5
-3
-2
h /H = 0 .352 9
Im perm eab le W all
D a= 7.6 8 0 3 10
D a= 4.9 4 1 2 10
D a= 5.6 8 4 4 10
-5
-3
-2
f
n
Figure 12. Laminar velocity distribution over the whole depth. Lines; computations, symbols;
analytical solution (Poulikakos and Kazmierczak, 1987).
48
1E -5 1E -4 1E -3 1E -2 1E -1
D a
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
U
/
Uin
t,m
f
L am in ar; C ompu ted
L am in ar; A n a ly tica l
T u rb u le n t; C ompu ted
Figure 13. Effect of Darcy number on the interfacial velocity.
49
1E -6 1 E -5 1 E -4 1 E -3 1 E -2 1E -1
D a
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
Q /
Qf
f,im
p L am in ar; C ompu ted
L am in ar; A na ly tic a l
L am in ar; B eav ers & Jo seph (196 7 )
T u rb u len t; C o mpu ted
T u rb u len t; P resen t E xpe rim en t
Figure 14. Effect of Darcy number on channel discharge.
50
0 1 2 3 4 5N
-2 .0-1 .8-1 .6-1 .4-1 .2-1 .0-0 .8-0 .6-0 .4-0 .20 .00 .20 .40 .60 .8
y/h f
+
-1 .2
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
y/h f
-1 .2
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y/h f
h /H = 0 .4 76 2
E m pir ica l
D a= 5 .0 33 1 10
D a= 3 .2 38 1 10
D a= 3 .7 25 2 10
f
-5
-3
-2
h /H = 0 .3 52 9
E m pir ica l
D a= 7 .6 80 3 10
D a= 4 .9 41 2 10
D a= 5 .6 84 4 10
-5
-3
-2
f
h /H = 0 .5 60 0
E m pir ica l
D a= 3 .5 51 4 10
D a= 2 .2 84 8 10
D a= 2 .6 28 5 10
f
-5
-3
-2
Figure 15.Variation of U/k 2* above and within the porous bed.
51
0 .0 0 .5 1 .0 1 .5 2 .0 2 .5Xï�8��Yï�8��Zï�8
0 .0
0 .2
0 .4
0 .6
0 .8
y/hf
*
0 .0
0 .2
0 .4
0 .6
0 .8
y/hf
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y/hf
h /H = 0 .4 211D a= 4 .55 07 10
u ´
v ´
w ´
f -2
h /H = 0 .3 529D a= 5 .68 44 10
u ´
v ´
w ´
f
h /H = 0 .4 762D a= 3 .72 52 10
u ´
v ´
w ´
f -2
��
Figure 16. Variation of turbulent intensities above the porous bed. Lines; empirical relations
of Nezu and Nakagawa (1993) for impermeable bed, symbols present measurements.
52
-0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0�XY
-2 .0-1 .8-1 .6-1 .4-1 .2-1 .0-0 .8-0 .6-0 .4-0 .20 .00 .20 .40 .60 .8
y/h f
+
-1 .2
-1 .0
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
y/h f
-0 .8
-0 .6
-0 .4
-0 .2
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y/h f
h /H = 0 .4 76 2
E m pir ica l
D a= 5 .0 33 1 10
D a= 3 .2 38 1 10
D a= 3 .7 25 2 10
f
-5
-3
-2
h /H = 0 .3 52 9
E m pir ica l
D a= 7 .6 80 3 10
D a= 4 .9 41 2 10
D a= 5 .6 84 4 10
-5
-3
-2
f
h /H = 0 .5 60 0
E m pir ica l
D a= 3 .5 51 4 10
D a= 2 .2 84 8 10
D a= 2 .6 28 5 10
f
-5
-3
-2
Figure 17.Variation of U/uv 2*− above and within the porous bed.