Journal of Membrane Science 231 (2004) 81–90
Numerical simulation of the flow in a plane-channel containing
a periodic array of cylindrical turbulence promoters
C.P. Koutsou, S.G. Yiantsios, A.J. Karabelas∗
Department of Chemical Engineering, Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki,
P.O. Box 361, GR 570 01, Thermi, Thessaloniki, Greece
Received 26 August 2003; accepted 14 November 2003
Abstract
This work is aimed at obtaining a better understanding of transport phenomena in membrane elements, where feed-flow spacers (employed
to separate membrane sheets and create flow channels) tend to enhance mass transport characteristics, possibly mitigating fouling and
concentration polarization phenomena, while augmenting pressure drop. A model flow geometry is considered consisting of a plane-channel,
where a regular array of cylinders is inserted, acting as turbulence promoters. Direct numerical simulations using the Navier–Stokes equations
are performed over a range of Reynolds numbers typical of such membrane modules. The results show that the flow becomes unstable at a
critical Reynolds number of 60, and progressively tends to a chaotic state. Qualitative flow features, such as the development and separation of
boundary layers, vortex formation, the presence of high-shear regions and recirculation zones, and the underlying mechanisms are examined.
In addition, quantitative statistical characteristics such as time-averaged velocities, Reynolds stresses, wall-shear rates and pressure drop are
obtained, which are directly related to mass transport enhancement.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Membrane spacers; CFD; Direct numerical simulation
1. Introduction
Membrane processes are among the most advanced methods in water treatment and desalination. Spirally wound
membrane modules are predominantly employed in reverse
osmosis and nanofiltration, and they also find use in ultrafiltration and microfiltration. A characteristic of this type of
modules is the presence of spacers in the feed-flow and permeate channels. The feed-flow spacers, in particular, which
have the form of non-woven crossed cylinders, serving to
separate adjacent membrane leaves and create flow passages,
tend to promote turbulence and enhance mass transport.
The term “turbulence promoters” has been accepted in the
literature, although in reality the unstable flows may not necessarily manifest the characteristics of fully developed turbulence. In this way, the undesirable fouling and concentration
polarization phenomena are mitigated. In parallel, however,
the presence of spacers leads to increased pressure drop, as
∗ Corresponding author. Tel.: +30-2310-996201;
fax: +30-2310-996209.
E-mail address: karabaj@cperi.certh.gr (A.J. Karabelas).
0376-7388/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.memsci.2003.11.005
well to the formation of localized stagnation or dead zones,
where the above phenomena may be intensified. Thus, in recent years the important role of membrane spacers has been
recognized, and several experimental and theoretical studies have appeared aiming at understanding the underlying
phenomena and optimizing spacer configurations.
Shock and Miquel [1] studied experimentally various
commercial spacers, both for the feed and permeate channels, and obtained correlations for the friction and the mass
transfer coefficients. On the basis of the results obtained,
they analyzed the effect of permeate flow channel length
and concluded that there is an optimum number of membrane leaves to minimize pressure losses in the permeate
channel. Zimmerer and Kottke [2] studied experimentally
the effect of geometrical spacer characteristics on flow patterns, fluid residence times, pressure drop and mass transfer
coefficients and proposed optimal configurations that would
improve mass transfer at acceptable energy costs due to
increased pressure drop.
Today, with the development of powerful computational
tools and high accuracy numerical methods, direct numerical simulation of turbulent flows in complex geometries
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C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
is feasible for Reynolds numbers on the order of a few
thousands. Thus, CFD is becoming a useful tool to assist in
such studies. Karniadakis et al. [3] in a theoretical study of
turbulence promoters for heat transfer enhancement studied the flow in a plane-channel, where a periodic array of
small-diameter cylinders is placed. They report that the
presence of the small cylinders leads to the destabilization
of the flow by essentially the same mechanisms as in a
plane-channel, namely formation of Tollmien–Schlichting
waves, but at greatly reduced Reynolds numbers. These
are close to 150, as compared to 5772 for a plane-channel
according to linear hydrodynamic stability theory [4]. Relevant to turbulence promotion are also the studies of Chen
et al. [5] and Zovatto and Pedrizzetti [6], who treat various
geometric configurations of a single cylinder located in a
plane-channel and analyze flow features and stability.
Several theoretical studies have focused closer to
membrane spacer configurations. Karode and Kumar [7]
performed three-dimensional simulations. However, they
employed a k–ε turbulence model, which is of questionable
usefulness in such a complex geometry. Furthermore, spatial resolution appeared not to be adequate. Cao et al. [8]
studied two-dimensional flow in a short channel containing
two cylinders at various arrangements. Turbulence modeling
was employed in that study as well. Schwinge et al. [9,10]
performed direct numerical simulations of flow and mass
transfer in channels containing five cylinders. The effect of
various configurations on flow characteristics such as pressure drop, eddy formation, wall-shear stresses, and mass
transfer was examined. Li et al. [11] presented a study of
flow and mass transfer by performing a three-dimensional
direct numerical simulation in a geometry closely representing membrane spacers. Periodic boundary conditions were
employed, thus enabling them to simulate just one cell of
the pattern formed by the spacer. The effect of spacer geometrical characteristics was studied and optimal parameters
were proposed. Although no information about the spatial
and temporal resolution of these results was provided, limited experimental data on average mass transfer coefficients
appeared to support their simulations.
The aim of the present study is to obtain a better understanding of the flow behavior, features and structures, as
well as statistical characteristics in a model two-dimensional
geometry which resembles membrane spacer configurations. This understanding is expected to facilitate spacer
configuration optimization studies, where three-dimensional
simulations and more realistic geometries need to be con-
sidered. In the next section the model problem and the
numerical techniques as well as computational resources
are described. Subsequently, the results obtained as regards
the dominant flow features and the underlying mechanisms
are discussed. In addition, statistical flow characteristics
relevant to mass transfer optimization are described. Furthermore, the temporal and spatial resolution of the numerical procedures employed is discussed. Finally, a short
discussion is provided of the issue of periodic boundary
conditions employed in the flow simulations.
2. Problem formulation and numerical method
The flow geometry consists of a plane-channel of thickness H, in which an array of equally spaced cylinders is inserted, as shown in Fig. 1. The cylinders, of diameter D, are
separated by a distance L, and are located symmetrically in
the channel. The channel is infinite in extend in the streamwise and spanwise directions and the flow is assumed to be
fully developed and two-dimensional. The ratios L/D and
H/D are nominal parameters of the problem under consideration. However in this study, taking into account typical
characteristics of membrane spacers, these ratios are fixed
to the values of 6 and 2, respectively.
The fluid is assumed to be Newtonian and incompressible and the flow is governed by the Navier–Stokes and the
continuity equations:
∂u
1 2
+ u · ∇u = −∇P +
∇ u
∂t
Re
(1a)
∇ ·u=0
(1b)
The cylinder diameter, D, which is half of the channel thickness, is chosen as a length scale, and the average velocity
U0 as the velocity scale, defined by
1 H/2
u(x, y, t)dy
(2)
U0 =
H −H/2
Then, the time scale t0 is D/U0 , and the Reynolds number is
Re = U0 D/ν, where ν is the kinematic viscosity of the fluid.
It may be noted that various definitions of Reynolds number
have been used in the literature, e.g. taking into account the
porosity of the channel due to the presence of the inserts [1].
The boundary conditions are the no-slip and no-penetration
conditions on the cylinder and channel surfaces. In addition, the flow is assumed to be periodic in the streamwise
Fig. 1. Schematic of the flow geometry of a plane-channel and a periodic array of turbulence-promoting cylinders.
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
direction with periodicity L. Thus, for the velocity
u(x + L, y, t) = u(x, y, t)
(3)
Similarly, the pressure is composed of a constant mean pressure gradient and a periodic part:
dP
P(x, y, t) =
x + P̂(x, y, t)
(4a)
dx
P̂(x + L, y, t) = P̂(x, y, t)
(4b)
The assumption of periodicity, which is a typical practice in
direct numerical simulations of turbulent flow (i.e. [12]), as
well as in other applications of computational methods (i.e.
molecular dynamics), allows the computational domain to
be restricted to just one cell containing one cylinder. This
issue is further discussed in the next section.
83
It may be emphasized that the results in this study were
obtained through a direct numerical simulation (DNS) of the
Navier–Stokes equations, without the introduction of any
approximation by turbulence modeling. The numerical simulations covered the range of Reynolds numbers up to 200,
which is representative of the operation of spirally wound
membrane modules. In typical desalination membrane applications the feed-flow channel Reynolds number does not
exceed 1000. However, this is defined on the basis of the
channel hydraulic diameter, which is four times the cylinder
diameter used in the present definition of Reynolds number.
A CFD code (FLUENT, v. 6.0.12), which employs the
finite-volume method, was used. During each simulation the
governing equations were integrated in time by imposing a
constant mean pressure gradient until the flow reached a statistically steady state, that is when all mean variables such as
velocities and Reynolds stresses reach steady values. Time
was advanced by a second-order Adams–Moulton scheme
and the convective terms were discretized by a second-order
upwind scheme. In all the simulations the dimensionless
time step was less than 0.0015. The computational grid was
chosen to be finer near the cylinder and the solid surfaces,
in order to closely capture boundary layers and high-shear
zones. By numerical experimentation it was found that a
grid containing 5400 nodes was adequate for the range of
Reynolds numbers covered in this study. This issue is discussed in more detail in the next section.
3. Results and discussion
3.1. Flow features and structures
For Reynolds numbers up to 60 the flow is steady and
symmetric with respect to the channel symmetry plane. As
may be seen from Fig. 2, the boundary layers which develop
Fig. 2. Steady flow-streamfunction (a and b) and vorticity (c and d)
contours at Reynolds numbers of 55 (a and c) and 60 (b and d).
Fig. 3. Unsteady flow instantaneous streamfunction (a) and vorticity (b)
contours at Re = 70.
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C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
2
Re=70
Re=90
Re=110
Re=170
v
1
0
-1
-2
Fig. 4. Instantaneous streamfunction contours of the fluctuating part of
the flow field, at two different instances in time (a and b), at Re = 70.
Red and purple points correspond to the minimum and maximum values,
respectively. (Please refer to colour version on Science Direct).
0
2
4
6
8
10
12
14
t
Fig. 5. Fluctuations of the normal velocity component as a function of
time for various Reynolds numbers, at a position midway between two
consecutive cylinders on the channel symmetry plane.
Fig. 6. Instantaneous streamfunction contours at various time intervals during an eddy turnover period T, for Re = 170.
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
0.5
wall shear rate
0.4
0.3
0.2
0.1
0
-0.1
0
1
2
3
4
5
6
4
5
6
x
(a)
0.5
0.4
wall shear rate
around the cylinder are separated, thus resulting in the formation of two symmetric standing eddies behind the cylinder. As the Reynolds number increases, the standing eddies
increase in length and the boundary layer separation points
move towards the cylinder front stagnation point. Looked
from the point of view of vorticity transport, as presented in
Fig. 2(c) and (d), the wake of the cylinder consists of two
layers of vorticity, shed in a similar fashion as in the case
of a cylinder in unbounded flow. An additional important
feature here is the presence of two additional vorticity layers bound on the channel walls. Another characteristic of
the flow is the two high-velocity jets formed at the two flow
constriction sections above and below the cylinder. These
jets merge after the recirculation zone, leading to a velocity profile with an inflection point at the channel symmetry
plane.
As the Reynolds number is increased beyond the critical
value of 60, the flow is destabilized, apparently through a
Hopf bifurcation, and becomes periodic. As may be seen
in Fig. 3, the recirculation eddies cease to be symmetric,
they oscillate and are periodically detached. In parallel, there
is an oscillation in the strength of the two high-velocity
jets, as well as in the free vorticity layers shed from the
cylinder.
The above features suggest that the mechanism of the first
instability of this flow is the same as that for a cylinder in
unbounded flow. Indeed, if the mean flow is subtracted, the
streamfunction of the oscillating part, as shown in Fig. 4, is
very similar to the center- or cylinder-modes observed by
Karniadakis et al. [3] and Chen et al. [5] (see their Figs. 9(c)
and 8, respectively). In addition, the critical Reynolds number found by Chen et al. [5] for the case of a single cylinder in a channel is 83, which is not very different from 60
obtained in this study.
Progressively, as the Reynolds number is increased, the
flow instability is intensified. The waveforms of the fluctuating velocities, shown in Fig. 5, are enriched in harmonics,
become more complex and tend to a chaotic state. Another
characteristic observed is that the basic frequency of the oscillations changes rapidly over a narrow range of Reynolds
numbers from 70 to 90.
In Fig. 6 instantaneous streamfunctions are shown during
an oscillation cycle at Re = 170. As may be observed, a
characteristic of the flow is that the two high-velocity jets
periodically increase in strength, collide and break through
each other. Another distinct characteristic of the flow at
higher Reynolds numbers is the periodic appearance of recirculation eddies at the channel walls. These eddies form
near the point where the recirculation zone ends behind the
cylinder, move along the channel walls, and disappear as
they approach the constrictions close to the next cylinder.
Similar wall eddies were observed in the study of Karniadakis et al. [3] when the small-diameter cylinders were
placed close to one of the channel walls.
The critical Reynolds number above which wall eddies
first appear, was found to be close to 78. This was determined
85
0.3
0.2
0.1
0
-0.1
0
(b)
1
2
3
x
Fig. 7. Dimensionless instantaneous shear rate as a function of position
along the lower wall for various time intervals during an eddy turnover
period, at Reynolds numbers 77 (a) and 78 (b). At Re = 78 negative
shear rates appear.
by calculating instantaneous shear rate profiles along one of
the walls. As may be seen from Fig. 7, at Re = 77 shear
rates are positive throughout an oscillation cycle, whereas
at Re = 78 instantaneous negative shear rates appear.
It is instructive to observe flow vorticity contours, as
shown in Fig. 8. As the Reynolds number is increased, the
free vorticity layers oscillate and gain strength locally to
form separate vortices. These vortices interact with the layers of wall-bound vorticity forcing it to move away from the
walls. As a result of this interaction the free vortices move
away from the walls, as well. This behavior bears resemblance with the one presented in the numerical study of the
interaction of a free vortex with a flat plate boundary layer
by Luton et al. [13]. Thus, a distinct characteristic of the
flow around cylinders in a channel is that, in addition to the
vortices shed from the cylinders, conjugate vortices appear,
originating from the wall-bound vorticity. Additionally, as
86
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
Fig. 8. Instantaneous vorticity contours at Reynolds numbers of 70 (a), 90 (b), 110 (c), and 170 (d). At the higher Reynolds numbers vortices originating
from the walls appear.
pointed out by Zovatto and Pedrizzetti [6], in contrast to the
case of a cylinder in unbounded flow, counterclockwise vortices shed from the lower side of the cylinder move upward
and vice versa.
Finally, another characteristic feature of the flow is the
frequency of oscillations. By taking velocity traces at various
locations in the channel and applying FFT transforms the
frequency spectrum of the oscillations can be calculated. As
shown in Fig. 9, over a narrow range of Reynolds numbers
from 70 to 90 the dominant dimensionless frequency, which
is equal to the Strouhal number (St = ΩD/U0 ), shifts from
0.295 to 0.495. This is the same Reynolds number range
where wall eddies appear. Further increase of the Reynolds
number results in enrichment of the frequency spectrum with
harmonics and subharmonics, but without significant change
in the dominant frequency. It may be mentioned here that
a Strouhal number of 0.369 at the onset of instability was
obtained by Chen et al. [5] for an analogous case of a single
cylinder in a channel, whereas for a cylinder in unbounded
flow the corresponding number is 0.138.
3.2. Statistical flow characteristics
Fig. 9. Dimensionless frequency spectra of the fluctuations of the normal
velocity for various Reynolds numbers, at a position midway between
two consecutive cylinders on the symmetry plane.
In Fig. 10 time-averaged flow velocity and Reynolds
stress contours are shown. The velocity contours, apart from
their obvious symmetry characteristics (i.e. u is symmetric
and v is anti-symmetric with respect to the channel centerline), reveal that at relatively low Reynolds numbers the
two high-velocity jets retain their identity, whereas as the
Reynolds number increases the intense oscillations result
in a smooth mean profile approaching plug flow. From the
Reynolds stress contours, it may be observed that intense
fluctuations move closer to the cylinder as the Reynolds
number is increased and progressively occupy a larger
extent of the flow domain on both sides of the cylinder.
Apart from flow mixing, a characteristic of spacers
and turbulence promoters in channels is the increase of
wall-shear rates, which is directly related to the increase of
mass transfer coefficients in high Schmidt number transport
processes, such as salt transport in membrane desalination. As mentioned already, optimization of mass transfer
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
87
Fig. 10. Contours of time-averaged velocity components and Reynolds stresses for various Reynolds numbers. Red and purple points correspond to the
minimum and maximum values, respectively. (Please refer to colour version on Science Direct).
mean wall-shear rate
40
Re=70
Re=90
Re=110
Re=145
Re=170
30
20
10
0
0
1
2
(a)
3
4
5
6
x
characteristics is beyond the scope of this paper, since this
would require a more realistic representation of the membrane spacer geometry and a three-dimensional simulation.
However, it is interesting to observe some relevant characteristics of the flow studied. In Fig. 11 time-averaged wall-shear
rates and their fluctuations (r.m.s. values) are shown. Given
D and U0 , as length and velocity scales, the shear rates are
normalized by U0 /D. As expected, for all Reynolds numbers
the maximum mean shear rate occurs near the constrictions
above and below the cylinder. This maximum is significantly
larger than that in a plane-channel, where the corresponding
dimensionless shear rate for laminar flow is uniform and
equal to 3. Fluctuations in shear rate appear to be significant
as well. In all cases a local maximum occurs right before
the cylinder where velocity fluctuations are more intense.
Above a Reynolds number of 90, secondary local maxima
appear downstream of the cylinder, which are related to wall
boundary layer separation and the appearance of wall eddies.
Finally, in Fig. 12 the dimensionless pressure drop,
which is proportional to a friction coefficient, is shown
1
Re=70
Re=90
Re=110
Re=145
Re=170
6
Steady flow
Unsteady flow
Plane-channel
-dP/dx
wall shear rate fluctuations
8
4
0.1
2
0
0
(b)
1
2
3
4
5
6
x
0.01
50
60
70
80
90 100
200
Re
Fig. 11. Dimensionless time-averaged shear rate (a) and r.m.s. values
of the fluctuations (b) for various Reynolds numbers, as a function of
position along the lower wall.
Fig. 12. Dimensionless mean pressure gradient as a function of Reynolds
number. Plane-channel laminar flow correlation: −dP/dx = 3Re−1 .
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
as a function of Reynolds number. This is also an important parameter in spacer optimization, since it is related
to the additional energy costs incurred to achieve improved mass transfer benefits. The correlation found for the
present geometry in the unstable flow regime is −dP/dx =
3.1Re−0.47 . As expected, the presence of the cylinders
leads to a significant increase in pressure drop compared
with that in a plane-channel, where for laminar flow
−dP/dx = 3Re−1 .
0.25
N=5400
N=10000
N=21600
0.2
Amplitude
88
0.15
0.1
3.3. Validation of the results
0.05
To test the validity of the simulation results, tests with
grids containing approximately twice and four times the
original number of nodes were performed. In Fig. 13 timeaveraged wall-shear rates and r.m.s. values of their fluctuations are shown. In Fig. 14 the frequency spectra of the
N=5400
N=10000
N=21600
mean shear rate
1
1.5
2
Frequency
velocity fluctuations at a certain position in the channel are
shown. Inspection of Figs. 13 and 14 suggests that not only
statistical quantities but the transient flow behavior is also
adequately resolved, for the range of Reynolds numbers covered in this study.
30
20
3.4. Effect of periodic conditions
10
0
0
1
2
(a)
3
4
5
6
x
8
wall shear rate fluctuations
0.5
Fig. 14. Dimensionless frequency spectra of the fluctuations of the normal
velocity for three different grids comprised of 5400, 10 000 and 21 600
nodes, at Re = 170, at a position midway between two consecutive
cylinders on the channel symmetry plane.
40
N=5400
N=10000
N=21600
6
4
2
0
0
(b)
0
1
2
3
4
5
6
x
Fig. 13. Dimensionless time-averaged shear rate (a) and r.m.s. values of
the fluctuations (b), as a function of position along the lower wall, for three
different grids comprised of 5400, 10 000 and 21 600 nodes, at Re = 170.
As mentioned in the previous section, periodic boundary
conditions are usually employed in turbulent flow numerical
simulations. The presence of the periodic array of cylinders
imposes a strong constraint on the flow to acquire spatially
periodic characteristics. However, the periodicity of geometry does not necessarily coincide with that of the flow, especially in the unstable flow regime. Ideally, a large number
of cells, each containing one cylinder, should be considered.
It is assumed in this study that considering only a single
periodic cell adequately captures the dominant flow features
and the main statistical quantities. To test this assumption,
the flow in a periodic cell of twice the original length,
containing two cylinders was simulated. The same spatial
resolution was kept resulting in a grid containing twice the
number of nodes. As may be seen from Fig. 15 a qualitative
difference exists in that the oscillations in two neighboring
cells are now exactly out of phase, whereas the single-cell
simulations constrain neighboring cells to oscillate in phase.
It may thus be hypothesized that in a simulation containing
more cells, phase differences would exist between neighboring cells. However, as may be observed, the dominant flow
features remain the same as in the single-cell simulations.
In Fig. 16 time-averaged wall-shear rates and r.m.s. values
are shown. It is evident that there is qualitative similarity
of the flow features, and that the statistical quantities retain
the basic periodicity of the geometry. Moreover, there is
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
89
satisfactory quantitative agreement between the single and
the double-cell results. On the basis of these test results,
it is suggested that simulations considering a flow periodicity coinciding with that of the geometry are capable
of adequately capturing the dominant flow features and
may also be used to obtain statistical quantities of interest,
such as pressure drop, wall-shear rates and mass transfer
coefficients.
4. Concluding remarks
Fig. 15. Streamfunction (a) and vorticity contours (b) at Re = 170,
obtained from a simulation where the flow periodicity interval is taken
to be twice that of the geometry.
mean wall shear rate
40
30
20
10
0
0
2
4
(a)
6
8
10
12
8
10
12
x
10
mean wall shear rate
8
6
4
2
0
(b)
The flow in a model two-dimensional geometry, consisting of a plane-channel in which an array of cylinders
is located, has been studied in order to obtain a better
understanding of the behavior, the dominant features and
structures, as well as the statistical characteristics. Above
a critical Reynolds number of 60 the flow becomes unstable due to mechanisms similar to those operating in the
case of a cylinder in unbounded flow. Above a Reynolds
number of 78 wall eddies appear, conjugate to those shed
by the cylinder. This distinct characteristic of the flow in
the channel wall is due to the interaction of the vorticity
shed by the cylinders with the vorticity layers created on
the channel walls. Three-dimensional effects (i.e. along the
cylinder axis) have not been considered in this study. It is
known, for example, that above a Reynolds number close
to 180 three-dimensional instabilities appear in the case of
a cylinder in unbounded flow. However, the dominant flow
features remain unaltered [5].
By employing periodic boundary conditions in the
streamwise direction, significant savings in computational
costs may be made since the computational domain can
be restricted to just one cell containing one cylinder. The
approach adopted, which is usual practice in turbulent flow
numerical simulations, may be also applied to tackle the
mass transfer problem in order to obtain local mass transfer
coefficients. The interested reader may consult Karniadakis
et al. [3] for a similar study of heat transfer. Although
the flow periodicity does not coincide with that of the geometry, its statistical characteristics do so. Thus reliable
information on features important for membrane spacer
optimization studies can be obtained in this way. The understanding obtained in this study is expected to facilitate
optimization studies of membrane spacer configuration,
where three-dimensional simulations and more realistic
geometries need to be considered.
0
2
4
6
x
Fig. 16. Time-averaged shear rate (a) and r.m.s. values of the fluctuations
(b), as a function of position along the lower wall, for Re = 170. The
continuous and dotted lines correspond to simulations where the flow
periodicity interval is the same and twice that of the geometry, respectively.
Nomenclature
dP/dx
D
H
L
pressure gradient (Pa/m)
diameter of cylinders (m)
channel height (m)
cylinder spacing (m)
90
N
P
P̂
Re
St
t
T
u
u
u′ u′
U0
v
v
v′ v′
x
y
C.P. Koutsou et al. / Journal of Membrane Science 231 (2004) 81–90
number of nodes
pressure (Pa)
periodic part of pressure (Pa)
Reynolds number
Strouhal number
time
turnover period (s)
u-velocity
mean u-velocity
Reynolds stress (or r.m.s. u-velocity)
average velocity (m/s)
v-velocity
mean v-velocity
Reynolds stress (or r.m.s. v-velocity)
x-coordinate
y-coornidate
Greek letters
v
kinematic viscosity (m2 /s)
Ω
shedding frequency (Hz)
References
[1] G. Shock, A. Miquel, Mass transfer and pressure loss in spiral wound
modules, Desalination 64 (1987) 339–352.
[2] C.C. Zimmerer, V. Kottke, Effects of spacer geometry on pressure
drop, mass transfer, mixing behavior, and residence time, distribution,
Desalination 104 (1996) 129–134.
[3] G.E. Karniadakis, B.B. Mikic, A.T. Patera, Minimum-dissipation
transport enhancement by flow destabilization: Reynolds’ analogy
revisited, J. Fluid Mech. 192 (1988) 365–391.
[4] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1982.
[5] J.-H. Chen, W.G. Pritchard, S.J. Tavener, Bifurcation for flow past
a cylinder between parallel planes, J. Fluid Mech. 284 (1995) 23–
41.
[6] L. Zovatto, G. Pedrizzetti, Flow about a circular cylinder between
parallel walls, J. Fluid Mech. 440 (2001) 1–25.
[7] S.K. Karode, A. Kumar, Flow visualization through spacer filled
channels by computational fluid mechanics. I. Pressure drop and
shear rate calculations for flat sheet geometry, J. Membr. Sci. 193
(2001) 69–84.
[8] Z. Cao, D.E. Wiley, A.G. Fane, CFD simulations of net-type turbulence promoters in a narrow channel, J. Membr. Sci. 185 (2001)
157–176.
[9] J. Schwinge, D.E. Wiley, D.F. Fletcher, Simulation of the flow around
spacer filaments between narrow channel walls. 1. Hydrodynamics,
Ind. Eng. Chem. Res. 41 (2002) 2977–2987.
[10] J. Schwinge, D.E. Wiley, D.F. Fletcher, Simulation of the flow around
spacer filaments between narrow channel walls. 2. Mass transfer
enhancement, Ind. Eng. Chem. Res. 41 (2002) 4879–4888.
[11] F. Li, W. Meindersma, A.B. de Haan, T. Reith, Optimization of commercial net spacers in spiral wound membrane modules, J. Membr.
Sci. 208 (2002) 289–302.
[12] J. Jimenez, P. Moin, The minimal flow unit in near-wall turbulence,
J. Fluid Mech. 225 (1991) 213–240.
[13] A. Luton, S. Ragab, D. Telionis, Interaction of spanwise vortices
with a boundary layer, Phys. Fluids 7 (1995) 2757–2765.