21ème Congrès Français de Mécanique
Bordeaux, 26 au 30 août 2013
Modeling a weak turbulent flow in a narrow and wavy channel:
case of micro irrigation
Jafar AL-MUHAMMADa , Séverine TOMASa and Fabien ANSELMETb
a. IRSTEA, UMR G-EAU, 361 Rue Jean-François Breton BP 5095, 34196 Montpellier Cedex 05, France
b. IRPHE, 49 Rue Frédéric Joliot-Curie, BP 146, 13384 Marseille Cedex 13, France
Abstract
This study aims to propose a flow modeling in a baffle-fitted one millimeter labyrinth-channel. It appears that the classical
turbulent models can not reproduce the characteristics of such a low Reynolds number flow. One of the main applications
of this work is for studying the hydrodynamics of micro-irrigation emitters. Indeed, one of the major drawbacks of this
technique is the clogging of the emitters; and this clogging is strongly related with the flow conditions. In this paper,
the results of three low Reynolds number models are compared with those of the standard k − , and the RNG k − .
Remarkable differences are observed between the different low Reynolds number models. This work raises questions
about the dissipation process and its role in such a flow.
Résumé
Le but de ce travail est de proposer un modèle pour l’écoulement au sein d’un labyrinthe millimétrique. Il semble que
les modèles de turbulence classiques ne peuvent pas reproduire les caractéristiques d’un tel écoulement à bas nombre
de Reynolds. L’une des principales applications de ce travail est d’étudier l’hydrodynamique des goutteurs de microirrigation. En effet, l’un des inconvénients majeurs de cette technique est le colmatage de gouteurs. Ce colmatage est
fortement lié aux conditions d’écoulement. Dans cet article, les résultats de trois modèles à bas nombre de Reynolds
sont comparés à ceux des modèles k − standard et RNG. Des différences notables sont observées entre les différents
modèles à bas nombre de Reynolds. Ce travail soulève des questions sur le processus de dissipation et son rôle dans un
tel écoulement.
Keywords : Emitters; Low Reynolds number; Narrow channel.
1
Introduction
The micro irrigation is a technique characterized by low water flows. The water drops near the plants through emitters.
This type of irrigation improves efficiency by reducing energy consumption, evaporation, drift, runoff and deep percolation losses when compared with the other techniques such as sprinkler irrigation. In this technique, the emitters are the
most important and critical component. They work with weak flow rate between 0.5 and 8 l/h for pressure between 0.5
and 4 bars. The emitter flow rate increases with static pressure in a lateral pipe in keeping with an exponential relation
(Karmeli.,1977) [2]: q = KP x , where q is the flow rate of emitter, K is the constant of proportionality that characterizes
each emitter, P is the pressure head and x is the emitter discharge exponent. The value of this exponent depends on the
conception of the emitter. Indeed, manufacturers try to design emitters which flow rate is not directly dependent on the
pressure head (x < 0.5). To reach with this goal, they introduce some elements in the emitter such as a labyrinth-channel
which generates local pressure head losses. This problem stems from the fact that this labyrinth-channel is very sensitive
to the clogging phenomena, which is governed partially by hydrodynamics. Therefore, it is necessary to analyze the
flow. This analysis in the labyrinth-channel is focused on the swirl zones at the downstream side of the baffles where the
velocities are very low, thus favoring the deposition mechanism. Therefore, when designing emitters, clogging can be
prevented or at least significantly reduce by decreasing, as much as possible the size of the swirl zones. The shape of the
swirling region can be predicted by computational fluid dynamic (CFD) modeling. However, the CFD modeling differs
from one study to another. For example Palau Salvador et al.(2004)[6] employ the laminar model, Wei et al.(2006) [8]
choose the standard k- model while Wei et al.(2012) [9] and Dazhuang et al.( 2007) [1] employ the RNG k- model and
the realizable k- model respectively. The Reynolds number, based on the mean velocity and on the hydraulic diameter,
is low (around 500), so the flow should be laminar. But as the channel is narrow and wavy, some authors postulate a laminar to turbulent transition at lower Reynolds number : around 350 (Nishimura et al.,1984 [5]) or between 100 and 700
1
(Pfahler et al.,1990 [7]). Thus, it appears that a question is still open on the characteristic of such a flow and consequently
on the choice of the model,i.e., turbulent or not. Considering that the flow rate is weak and then that the Reynolds number is low, this study introduces the low Reynolds k- models in order to model the flow in labyrinth-channel. Several
turbulence models are examined in this paper: the RNG k − , standard k − and Low-Reynolds k − turbulence models.
The numerical results are then discussed and compared with the experimental data before concluding. It clearly appears
that dissipation is correlated with pressure losses.
2
2.1
Materials and methods
Turbulence models
In this study, five models are selected for flow modeling : the standard k-, RNG k- and three low Re k- models. The
standard k − model was initially proposed by Launder and Spalding (1972) [4]. This model was derived by assuming
that the flow is fully turbulent, so the effects of molecular viscosity are negligible except near walls [4]. The RNG
k- model uses a technique, namely renormalization group theory, described by Yakhot and Orszag (1986) [10]. The
effect of swirl is accounted for the RNG model thereby enhancing the accuracy of swirling flows. This model cessemes
appropriate treatment of the near wall region. The RNG model uses an analytically derived differential formula for the
effective turbulent viscosity which can be used for low Reynolds number flows. The standard and RNG models require
additional semi-empirical parameterization to take into account the effects of near-wall molecular viscosity . For this
study, as y + = ρuτ y/µ =p2 which was calculated only for the first point of meshing near the wall, where uτ is the
friction velocity defined as τw /ρ, the enhanced wall treatment is used.
In the low Reynolds number k- models, the standard model k- is modified to account for the low-Reynolds-number
effects thanks to the damping function and extra source terms in the turbulent kinetic energy and the dissipation rate
equations. There are about a hundred low-Reynolds-number models. The following low-Reynolds-number models are
used in this paper :[Abid] Abid (1991), [LS] Launder and Sharma (1974)and [CHC] Chang, Hsieh and Chen (1995). The
modelled equations of the turbulence kinetic energy k and the dissipation rate are :
∂k
∂
∂k
ρuj
=
(Dk )
+ Gk − ρ −
D
;
(1)
|{z}
|{z} |{z}
∂xj
∂xj
∂xj
| {z } |
{z
} prodk destk source termk
advk
dif fk
∂
2
∂
∂
=
+ C1 f1 Gk − C2 f2 ρ +
E
(D )
ρuj
|{z}
∂xj
∂xj
∂xj
k
k
{z
} | {z } | {z } source term
| {z } |
adv
dif f
prod
.
(2)
dest
Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients. The turbulent viscosity can
be written as a general term multiplied by a damping function, fµ :
k2
.
(3)
In the standard k- and low Reynolds models, Dk = µ + σµkt , D = µ + σµt , C1 = 1.44, C2 = 1.92, σk = 1,σ = 1.3
and Cµ = 0.09. In the RNG k- model, Dk = αk µef f , D = α µef f , C1 = 1.42, C2 = 1.68 and Cµ = 0.0845. The
damping functions fµ ,f1 , and f2 equal 1 for standard and RNG k- models. For the standard k- models, there are no
source terms, (D = E = 0). In the RNG k- model, D=0, but E 6= 0 [10]. The damping functions and source terms for
each low Reynolds number model are detailed in the paper of Karvinen et al. (2005) [3].
µt = ρfµ Cµ
2.2
Geometry of the study and numerical assumption
The emitter selected, called GR emitters, is integrated and is characterized by labyrinth-channel features. In addition,
it is characterized by an emitter discharge exponent of 0.6. For this numerical study, only the repeating pattern of the
labyrinth-channel is simulated (Fig. 1). This study is performed using commercial computational fluid dynamic software
(CFD) where the models studied are implemented: ANSYS/Fluent V14. Firstly, the geometry is designed by workbench
design modeler. Secondly, the mesh is performed by meshing. Mesh quality is examined to ensure that mesh is good.
For all models, we opted for a quadratic-dominant mesh type. several meshs are perform for each model. The mesh,
then, is verified to ensure that the results are not dependent of meshing. Finaly, the flow , in Fluent, is assumed to be
two-dimensional, steady and incompressible. Buoyancy and gravity are not taken into account. The boundary conditions
are the flow rates and to make sure that the conditions at the inlet and outlet do not affect the modeling results, they are
offset from the studied region. In addition, a low turbulent intensity (5%) and hydraulic diameter (2 mm) are chosen
for the specification method. The velocity at the channel walls is set to zero thereby satisfying the no-slip condition.
2
The simulation is performed with an initial inlet flow rate of 1.4 l/h, then it is regularity increased until 2.9 l/h; which
correspond to a Reynolds number ranging from 400 to 800. The results obtained are plotted and compared following two
transversal lines which cross through a swirling region at its center.
outlet
line 4
line 3
line 2
line 1
inlet
F IGURE 1− The emitter and the geometry studied. The green lines and the red rectangle define the zones
where the mean velocity and turbulent quantities are deeper analyzed in section 3.
3
3.1
Results and discussion
The pressure and discharge
The discharge-pressure curves are plotted for each turbulence model and compared with the experimental data (Fig. 2).
In the paper of Karmeli.(1977) [2], it can be noted that for long-path emitters, which are used for our study, the emitter
discharge exponent is between 0.5 and 1, respectively for a fully turbulent and laminar flow. The exponent of standard
k − , RNG k − and [LS] models is that of fully turbulent regime and it is close to the exponent of the experiment (0.57).
[Abid] and [CHC] seem to be closer to experimental curves, but the exponents of these models are close to 1, which is the
exponent of a laminar flow. In this study, the numerical results of three models are more deeply analyzed: the standard
k − model as a high Reynolds number model, the [LS] model as a low Reynolds model which behaves behavior of a
high Reynolds model and the [CHC] model as a low Reynolds number model which tends to a laminar behavior.
3
Abid
2.8
LS
2.6
CHC
Standard
2.4
RNG
q [l/h]
2.2
experimental data
2
q=2.245∆P1.02
1.8
q=2.771∆P0.56
q=2.295∆P0.89
1.6
q=2.747∆P0.49
1.4
q=2.718∆P0.48
1.2
1
0.2
q=2.114∆P0.56
0.4
0.6
0.8
1
∆P [bar]
1.2
1.4
1.6
F IGURE 2− Discharge-pressure curves
3.2
The mean velocity fields
The velocity fields obtained from the CFD simulations for the minimum and the maximum flow rates 1.4 l/h and 2.9 l/h
respectively are shown on Fig. 3, for the three turbulence models chosen. The velocity fields show that there are two
regions: one is the main flow and the other is the swirl region characterized by a low value and negative velocity vectors.
It appears that the swirl zone is bigger for [CHC] for a flow rate of 1.4l/h (Fig.3(c)). Therefore, the transition from the
mean flow to the swirl zone occurred over short distance, which explains why the velocity is significantly changed. It
can also be observed that the maximum velocity is at the corner of the labyrinth-channel, where the water hits the wall
in the different baffles. Fig. 4(a) and Fig. 4(b) show that the velocity profiles, on the lines 2 and 3, are identical for high
Reynolds models, likewise for [LS] model. This means that a stationary state seems to be reached just after the first
baffle. This is not the case for the [CHC] model, where the velocity profile evolves between lines 2 and 3 (Fig.4). All the
turbulence models studied give the same velocity profiles on the line 3, and for a flow rate of 2.9 l/h (Fig. 4(b)).
3
q=1.4 l/h
(a) Standard.
(b) LS.
(c) CHC.
(d) Standard.
(e) LS.
(f) CHC.
q=2.9 l/h
F IGURE 3− The velocity fields [m.s−1 ]
1.6
1.6
LS−line2
LS−line3
CHC−line2
CHC−line3
Standard−line2
Standard−ine3
1.4
1.2
1.2
1
U [m/s]
U [m/s]
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
LS−line2
LS−line3
CHC−line2
CHC−line3
Standard−line2
Standard−line3
1.4
0.2
0
0.5
1
1.5
y [mm]
2
0
2.67
(a) q=1.4 l/h
0
0.5
1
1.5
y [mm]
2
2.67
(b) q=2.9 l/h
F IGURE 4− The mean velocity fields
3.3
Variation in turbulence kinetic energy and the dissipation rate
The fields of the turbulence kinetic energy (k) and the dissipation rate () are shown on Fig. 5 and Fig. 6 for flow rate
of 1.4 l/h and 2.9 l/h. It can be observed that k and for standard and [LS] models no longer evolve just after the first
baffle, which has already been observed for the mean velocity fields (Fig. 4). For [CHC] model, k and are not impacted
by the first baffle. The increase of k and can be observed after the third and the second baffle for 1.4 l/h and 2.9 l/h
respectively (Fig. 5(c) and Fig. 5(f)). This could be linked to the value of mean velocity in the principal flow which is
high as the turbulence is not well developed (Fig. 4). The other models dissipate flow energy from the first baffle whereas
the [CHC] model dissipates a huge amount of energy after the second baffle.
There are two ways to dissipate flow energy, the first is by turbulent dissipation or the internal dissipation and the second
is by friction at the wall. In general, this can happens in two ways. The pressure drop is proportional to the length of the
tube. So, it can be written:
Z
dP
ρ
τw
=
× dA +
(4)
dx
Qv
Rh
The term, on the left-hand side, is the absolute value of the pressure drop per unit length. On the right-hand side, the first
term is the integral of the dissipation rate on the cross section, where Qv = Um A is the volume flow rate [m3 /s] and the
second term is the energy dissipated at the wall, by friction, divided by the hydraulic radius Rh . The friction at the wall
is given by formula : τw = µS, where S is the strain rate. By multiplying equation 4 by the total area A, we obtain:
∆P × Dinlet = ρA
+ τw × Lwall
U
(5)
The dissipation rate and the friction at the wall for the standard, [LS] and [CHC] models are calculated and presented in
table [ 1]. It can be observed that the dissipation rate for the [CHC] model is higher than for the other models (Fig. 6(c)
and Fig. 6(f)). Therfore, dissipation is the main phenomenon which explains the pressure drop, unlike what is observed
for standard channel flows. This dissipation is due to the large swirling regions where wall friction is minimal.
The evolutions of each term of k equations are then plotted on the line 3 (Fig 7(d)). These terms are normalized by
ρU 3
d , where U is the mean velocity and d is the labyrinth-channel diameter. In the middle of the flow, the diffusion term
normalized for [CHC] and standard models give the same profile for the turbulence kinetic energy. Far from the wall,
4
q=1.4 l/h
(a) standard.
(b) LS.
(c) CHC.
q=2.9 l/h
(d) standard.
(e) LS.
(f) CHC.
F IGURE 5− The turbulence kinetic energy [m2 .s−2 ]
q=1.4 l/h
(a) Standard.
(b) LS.
(c) CHC.
(d) Standard.
(e) LS.
(f) CHC.
q=2.9 l/h
F IGURE 6− The dissipation rate [m2 .s−3 ]
model
flow rate [l/h]
∆P × Dinlet [Pa.m]
× ρA
U [Pa.m]
τw × Lwall [Pa.m]
standard
2.9
1.4
6.18 1.58
5.26 1.2
0.87 0.31
LS
2.9
6.2
5.45
1.12
1.4
1.84
1.41
0.38
CHC
2.9
1.4
7.45 3.23
6.27 1.76
1
0.48
TABLE 1− Comparison of pressure drop ∆P with and τw , where A is the total area
the damping functions are equal to 1. Therefore, these factors do not affect the budget, hence they are the same for all
models. The production term for k equation is identical for the different models. It has two peaks (Fig. 7): one is in the
middle of the main flow and the other is at the contact between the main flow and the swirl region due to the shear rate.
The advection term is also the same for all models: it is positive between the main flow and the center of the swirl region.
That is due to the negative values of the velocity components and the gradients of k. Otherwise, the energy of turbulence
increases from the wall and the main flow to the center of the swirling region (ys ). The main difference is the treatment
√ 2
at the wall: in [CHC] model, the boundary conditions imposed at the wall are k = 0, = 2ν ∂∂yk , therefore the
dissipation at the wall has a high value. It can be linked with the high value of the destruction term for k equation which
is related to . Consequently, the diffusion term of the turbulence kinetic equation for [LS] and [CHC] models gives an
significant value at the wall (Fig. 7(c) and Fig. 7(e)), as this term is in equilibrium with the destruction term.
4
Conclusion
CFD simulations of the flow in a narrow labyrinth-channel are carried out. The results of three low Reynolds number
models are compared with those of the standard and RNG k- models. The standard and RNG k − models give the same
results. It appears that there are two trends for low Reynolds number models: [LS] model follows high Reynolds models
(Standard and RNG k − ), whereas [CHC] and [Abid] follow the laminar model. The predicted curve exponents for [LS]
5
terms of k equation / (ρ U3/d)
1
0
−1
−2
2
1
0
−1
−2
−0.2
−0.1
0
0.1
y−ys/ys
0.2
0.3
0.4
1
0
−1
−2
−0.1
0
0.1
y−ys/ys
0.2
0.3
−0.1
0
0.1
y−ys/ys
0.2
0.3
0.4
(d) Standard with q=2.9 l/h.
−0.2
2
−0.1
0
0.1
y−ys/ys
0.2
0.3
0.4
(c) CHC with q=1.4 l/h.
advection term
diffusion term
production term
destruction term
3
1
0
−1
−2
−0.2
0
−1
0.4
advection term
diffusion term
production term
destruction term
source term
3
terms of k equation / (ρ U3/d)
terms of k equation / (ρ U3/d)
2
1
(b) LS with q=1.4 l/h
advection term
diffusion term
production term
destruction term
3
2
−2
−0.2
(a) Standard with q=1.4 l/h.
advection term
diffusion term
production term
destruction term
3
terms of k equation / (ρ U3/d)
terms of k equation / (ρ U3/d)
2
advection term
diffusion term
production term
destruction term
source term
3
terms of k equation / (ρ U3/d)
advection term
diffusion term
production term
destruction term
3
2
1
0
−1
−2
−0.2
−0.1
0
0.1
y−ys/ys
0.2
0.3
(e) LS with q=2.9 l/h
0.4
−0.2
−0.1
0
0.1
y−ys/ys
0.2
0.3
0.4
(f) CHC with q=2.9 l/h.
F IGURE 7− Terms of turbulence kinetic energy equation normalized, where ys is the radical position of the center of the swirl.
and high Reynolds models are close to the experimental data. Finally, the pressure drop is due to turbulent dissipation
which is due to the large swirling regions. This dissipation rate is high for [CHC] model, and it dissipates most of the
energy at the wall for a low flow rate compared to the standard and [LS] models. Unlike flows in straight channel, it
appears that dissipative processes in these labyrinths are responsible for the pressure drop. More developments, such as
three dimensional simulations, are needed in order to determine which model is the best for describing such a flow. PIV
experiments are under development to visualize the velocity field. The aim of such experiments is to characterize the
swirl region and to determine the mean velocity and turbulent quantities.
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6