21eme Congres Francais de Mecanique Bordeaux, 26 au 30 aout 2013

Modeling a weak turbulent flow in a narrow and wavy channel:case of micro irrigation

Jafar AL-MUHAMMADa, Severine TOMASa and Fabien ANSELMETb

a. IRSTEA, UMR G-EAU, 361 Rue Jean-Francois Breton BP 5095, 34196 Montpellier Cedex 05, France

b. IRPHE, 49 Rue Frederic Joliot-Curie, BP 146, 13384 Marseille Cedex 13, France

AbstractThis study aims to propose a flow modeling in a baffle-fitted one millimeter labyrinth-channel. It appears that the classicalturbulent models can not reproduce the characteristics of such a low Reynolds number flow. One of the main applicationsof this work is for studying the hydrodynamics of micro-irrigation emitters. Indeed, one of the major drawbacks of thistechnique 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 − ε, and the RNG k − ε.Remarkable differences are observed between the different low Reynolds number models. This work raises questionsabout the dissipation process and its role in such a flow.

ResumeLe but de ce travail est de proposer un modele pour l’ecoulement au sein d’un labyrinthe millimetrique. Il semble queles modeles de turbulence classiques ne peuvent pas reproduire les caracteristiques d’un tel ecoulement a bas nombrede Reynolds. L’une des principales applications de ce travail est d’etudier l’hydrodynamique des goutteurs de micro-irrigation. En effet, l’un des inconvenients majeurs de cette technique est le colmatage de gouteurs. Ce colmatage estfortement lie aux conditions d’ecoulement. Dans cet article, les resultats de trois modeles a bas nombre de Reynoldssont compares a ceux des modeles k − ε standard et RNG. Des differences notables sont observees entre les differentsmodeles a bas nombre de Reynolds. Ce travail souleve des questions sur le processus de dissipation et son role dans untel ecoulement.Keywords : Emitters; Low Reynolds number; Narrow channel.

1 IntroductionThe 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 percola-tion losses when compared with the other techniques such as sprinkler irrigation. In this technique, the emitters are themost important and critical component. They work with weak flow rate between 0.5 and 8 l/h for pressure between 0.5and 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 characterizeseach emitter, P is the pressure head and x is the emitter discharge exponent. The value of this exponent depends on theconception of the emitter. Indeed, manufacturers try to design emitters which flow rate is not directly dependent on thepressure head (x < 0.5). To reach with this goal, they introduce some elements in the emitter such as a labyrinth-channelwhich generates local pressure head losses. This problem stems from the fact that this labyrinth-channel is very sensitiveto the clogging phenomena, which is governed partially by hydrodynamics. Therefore, it is necessary to analyze theflow. This analysis in the labyrinth-channel is focused on the swirl zones at the downstream side of the baffles where thevelocities are very low, thus favoring the deposition mechanism. Therefore, when designing emitters, clogging can beprevented or at least significantly reduce by decreasing, as much as possible the size of the swirl zones. The shape of theswirling region can be predicted by computational fluid dynamic (CFD) modeling. However, the CFD modeling differsfrom 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-ε model while Wei et al.(2012) [9] and Dazhuang et al.( 2007) [1] employ the RNG k-ε model andthe realizable k-ε 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 lam-inar to turbulent transition at lower Reynolds number : around 350 (Nishimura et al.,1984 [5]) or between 100 and 700

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(Pfahler et al.,1990 [7]). Thus, it appears that a question is still open on the characteristic of such a flow and consequentlyon the choice of the model,i.e., turbulent or not. Considering that the flow rate is weak and then that the Reynolds num-ber is low, this study introduces the low Reynolds k-ε models in order to model the flow in labyrinth-channel. Severalturbulence models are examined in this paper: the RNG k−ε, standard k−ε and Low-Reynolds k−ε turbulence models.The numerical results are then discussed and compared with the experimental data before concluding. It clearly appearsthat dissipation is correlated with pressure losses.

2 Materials and methods

2.1 Turbulence modelsIn this study, five models are selected for flow modeling : the standard k-ε, RNG k-ε and three low Re k-ε models. Thestandard k − ε model was initially proposed by Launder and Spalding (1972) [4]. This model was derived by assumingthat the flow is fully turbulent, so the effects of molecular viscosity are negligible except near walls [4]. The RNGk-ε model uses a technique, namely renormalization group theory, described by Yakhot and Orszag (1986) [10]. Theeffect of swirl is accounted for the RNG model thereby enhancing the accuracy of swirling flows. This model cessemesappropriate treatment of the near wall region. The RNG model uses an analytically derived differential formula for theeffective turbulent viscosity which can be used for low Reynolds number flows. The standard and RNG models requireadditional semi-empirical parameterization to take into account the effects of near-wall molecular viscosity . For thisstudy, as y+ = ρuτy/µ = 2 which was calculated only for the first point of meshing near the wall, where uτ is thefriction velocity defined as

√τw/ρ, the enhanced wall treatment is used.

In the low Reynolds number k-ε models, the standard model k-ε is modified to account for the low-Reynolds-numbereffects thanks to the damping function and extra source terms in the turbulent kinetic energy and the dissipation rateequations. There are about a hundred low-Reynolds-number models. The following low-Reynolds-number models areused in this paper :[Abid] Abid (1991), [LS] Launder and Sharma (1974)and [CHC] Chang, Hsieh and Chen (1995). Themodelled equations of the turbulence kinetic energy k and the dissipation rate ε are :

ρuj∂k

∂xj︸ ︷︷ ︸advk

=∂

∂xj

[(Dk)

∂k

∂xj

]︸ ︷︷ ︸

diffk

+ Gk︸︷︷︸prodk

− ρε︸︷︷︸destk

− D︸︷︷︸source termk

; (1)

ρuj∂ε

∂xj︸ ︷︷ ︸advε

=∂

∂xj

[(Dε)

∂ε

∂xj

]︸ ︷︷ ︸

diffε

+C1εf1ε

kGk︸ ︷︷ ︸

prodε

−C2εf2ρε2

k︸ ︷︷ ︸destε

+ E︸︷︷︸source termε

. (2)

Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients. The turbulent viscosity canbe written as a general term multiplied by a damping function, fµ :

µt = ρfµCµk2

ε. (3)

In the standard k-ε and low Reynolds models, Dk = µ + µtσk

, Dε = µ + µtσε

, C1ε = 1.44, C2ε = 1.92, σk = 1,σε = 1.3and Cµ = 0.09. In the RNG k-ε model, Dk = αkµeff , Dε = αεµeff , C1ε = 1.42, C2ε = 1.68 and Cµ = 0.0845. Thedamping functions fµ,f1, and f2 equal 1 for standard and RNG k-ε models. For the standard k-ε models, there are nosource terms, (D = E = 0). In the RNG k-ε model, D=0, but E 6= 0 [10]. The damping functions and source terms foreach low Reynolds number model are detailed in the paper of Karvinen et al. (2005) [3].

2.2 Geometry of the study and numerical assumptionThe 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 thelabyrinth-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 workbenchdesign 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 betwo-dimensional, steady and incompressible. Buoyancy and gravity are not taken into account. The boundary conditionsare the flow rates and to make sure that the conditions at the inlet and outlet do not affect the modeling results, they areoffset from the studied region. In addition, a low turbulent intensity (5%) and hydraulic diameter (2 mm) are chosenfor the specification method. The velocity at the channel walls is set to zero thereby satisfying the no-slip condition.

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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; whichcorrespond to a Reynolds number ranging from 400 to 800. The results obtained are plotted and compared following twotransversal lines which cross through a swirling region at its center.

line 3 line 2

line 4

line 1

inlet

outlet

FIGURE 1− The emitter and the geometry studied. The green lines and the red rectangle define the zoneswhere the mean velocity and turbulent quantities are deeper analyzed in section 3.

3 Results and discussion

3.1 The pressure and dischargeThe 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 emitterdischarge exponent is between 0.5 and 1, respectively for a fully turbulent and laminar flow. The exponent of standardk−ε, RNG k−ε 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 theexponent of a laminar flow. In this study, the numerical results of three models are more deeply analyzed: the standardk − ε model as a high Reynolds number model, the [LS] model as a low Reynolds model which behaves behavior of ahigh Reynolds model and the [CHC] model as a low Reynolds number model which tends to a laminar behavior.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.61

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

∆P [bar]

q [l/

h]

Abid

LS

CHC

Standard

RNG

experimental data

q=2.245∆P1.02

q=2.771∆P0.56

q=2.295∆P0.89

q=2.747∆P0.49

q=2.718∆P0.48

q=2.114∆P0.56

FIGURE 2− Discharge-pressure curves

3.2 The mean velocity fieldsThe velocity fields obtained from the CFD simulations for the minimum and the maximum flow rates 1.4 l/h and 2.9 l/hrespectively are shown on Fig. 3, for the three turbulence models chosen. The velocity fields show that there are tworegions: 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 themean flow to the swirl zone occurred over short distance, which explains why the velocity is significantly changed. Itcan also be observed that the maximum velocity is at the corner of the labyrinth-channel, where the water hits the wallin the different baffles. Fig. 4(a) and Fig. 4(b) show that the velocity profiles, on the lines 2 and 3, are identical for highReynolds models, likewise for [LS] model. This means that a stationary state seems to be reached just after the firstbaffle. This is not the case for the [CHC] model, where the velocity profile evolves between lines 2 and 3 (Fig.4). All theturbulence 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.

q=2.9 l/h

(d) Standard. (e) LS. (f) CHC.

FIGURE 3− The velocity fields [m.s−1]

0 0.5 1 1.5 2 2.670

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y [mm]

U [m

/s]

LS−line2LS−line3CHC−line2CHC−line3Standard−line2Standard−ine3

(a) q=1.4 l/h

0 0.5 1 1.5 2 2.670

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y [mm]

U [m

/s]

LS−line2LS−line3CHC−line2CHC−line3Standard−line2Standard−line3

(b) q=2.9 l/h

FIGURE 4− The mean velocity fields

3.3 Variation in turbulence kinetic energy and the dissipation rateThe fields of the turbulence kinetic energy (k) and the dissipation rate (ε) are shown on Fig. 5 and Fig. 6 for flow rateof 1.4 l/h and 2.9 l/h. It can be observed that k and ε for standard and [LS] models no longer evolve just after the firstbaffle, which has already been observed for the mean velocity fields (Fig. 4). For [CHC] model, k and ε are not impactedby the first baffle. The increase of k and ε can be observed after the third and the second baffle for 1.4 l/h and 2.9 l/hrespectively (Fig. 5(c) and Fig. 5(f)). This could be linked to the value of mean velocity in the principal flow which ishigh as the turbulence is not well developed (Fig. 4). The other models dissipate flow energy from the first baffle whereasthe [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 secondis by friction at the wall. In general, this can happens in two ways. The pressure drop is proportional to the length of thetube. So, it can be written:

dP

dx=

ρ

Qv

∫ε× dA+

τwRh

(4)

The term, on the left-hand side, is the absolute value of the pressure drop per unit length. On the right-hand side, the firstterm is the integral of the dissipation rate on the cross section, where Qv = UmA is the volume flow rate [m3/s] and thesecond term is the energy dissipated at the wall, by friction, divided by the hydraulic radius Rh. The friction at the wallis given by formula : τw = µS, where S is the strain rate. By multiplying equation 4 by the total area A, we obtain:

∆P ×Dinlet = ερA

U+ τw × Lwall (5)

The dissipation rate and the friction at the wall for the standard, [LS] and [CHC] models are calculated and presented intable [ 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 observedfor 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 byρU3

d , where U is the mean velocity and d is the labyrinth-channel diameter. In the middle of the flow, the diffusion termnormalized 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.

FIGURE 5− The turbulence kinetic energy [m2.s−2]

q=1.4 l/h

(a) Standard. (b) LS. (c) CHC.

q=2.9 l/h

(d) Standard. (e) LS. (f) CHC.

FIGURE 6− The dissipation rate [m2.s−3]

model standard LS CHCflow rate [l/h] 2.9 1.4 2.9 1.4 2.9 1.4∆P ×Dinlet [Pa.m] 6.18 1.58 6.2 1.84 7.45 3.23ε× ρA

U [Pa.m] 5.26 1.2 5.45 1.41 6.27 1.76τw × Lwall [Pa.m] 0.87 0.31 1.12 0.38 1 0.48

TABLE 1− Comparison of pressure drop ∆P with ε 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 allmodels. The production term for k equation is identical for the different models. It has two peaks (Fig. 7): one is in themiddle 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 turbulenceincreases from the wall and the main flow to the center of the swirling region (ys). The main difference is the treatment

at the wall: in [CHC] model, the boundary conditions imposed at the wall are k = 0, ε = 2ν(∂√k

∂y

)2

, therefore thedissipation at the wall has a high value. It can be linked with the high value of the destruction term for k equation whichis related to ε. Consequently, the diffusion term of the turbulence kinetic equation for [LS] and [CHC] models gives ansignificant value at the wall (Fig. 7(c) and Fig. 7(e)), as this term is in equilibrium with the destruction term.

4 ConclusionCFD simulations of the flow in a narrow labyrinth-channel are carried out. The results of three low Reynolds numbermodels are compared with those of the standard and RNG k-εmodels. The standard and RNG k−εmodels give the sameresults. It appears that there are two trends for low Reynolds number models: [LS] model follows high Reynolds models(Standard and RNG k−ε ), whereas [CHC] and [Abid] follow the laminar model. The predicted curve exponents for [LS]

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−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection termdiffusion term production term destruction term

(a) Standard with q=1.4 l/h.

−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection termdiffusion termproduction termdestruction termsource term

(b) LS with q=1.4 l/h

−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection termdiffusion term production termdestruction term

(c) CHC with q=1.4 l/h.

−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection termdiffusion termproduction termdestruction term

(d) Standard with q=2.9 l/h.

−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection termdiffusion termproduction termdestruction termsource term

(e) LS with q=2.9 l/h

−0.2 −0.1 0 0.1 0.2 0.3 0.4

−2

−1

0

1

2

3

y−ys/y

s

term

s of

k e

quat

ion

/ (ρ

U3 /d

)

advection term diffusion term production term destruction term

(f) CHC with q=2.9 l/h.

FIGURE 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 dissipationwhich is due to the large swirling regions. This dissipation rate is high for [CHC] model, and it dissipates most of theenergy at the wall for a low flow rate compared to the standard and [LS] models. Unlike flows in straight channel, itappears that dissipative processes in these labyrinths are responsible for the pressure drop. More developments, such asthree dimensional simulations, are needed in order to determine which model is the best for describing such a flow. PIVexperiments are under development to visualize the velocity field. The aim of such experiments is to characterize theswirl region and to determine the mean velocity and turbulent quantities.

References[1] Dazhuang, Yan., Peiling, Yang., Shumei, Ren., Yunkai, Li., Tingwu, Xu. 2007 Numerical study on flow property in

dentate path of drip emitters. New Zealand Journal of Agricultural Research . 50 705-712.

[2] Karmeli, D. 1977 Classification and flow regime analysis of drippers. Journal of Agricultural Engineering Research,22 165-173.

[3] Karvinen, A., Ahlstedt, H. 2005 Comparison of turbulence models in case of jet in crossflow using commercialCFD code. Engineering Turbulence Modeling and Experiments. 6 399-408.

[4] Launder, B. E., and Spalding, D. B. 1972 Lectures in mathematical models of turbulence. Academic Press. London,New York.

[5] Nishimura, T., Ohori, Y., Kawamura Y. 1984 Flow characteristics in a channel with symmetric wavy wall for steadyflow. Journal of Chemical Engineering of Japan. 17 466-471.

[6] Palau Salvador, G., Arviza Valverde, J., Bralts, V.F. 2004 Hydraulic flow behavior through an in-line emitterlabyrinth using CFD techniques. ASAE/CSAE Annual International Meeting. Paper Number: 042252 Ottawa,Ontario, Canada.

[7] Pfahler, J., Harley, J., Bau, H.H., Zemel, J. 1990 Liquid and gas transport in small channels. AMSE DSC. 19 149-157.

[8] Qingsong Wei , Yusheng Shi , Wenchu Dong , Gang Lu, Shuhuai Huang. 2006 Study on hydraulic performance ofdrip emitters by computational fluid dynamics. Agricultural Water Management. 84 30-136.

[9] Wei Zhengying, CAO Meng, LIU Xia, TANG Yiping, and LU Bingheng. 2012 Flow Behaviour Analysis andExperimental Investigation for Emitter Micro-channels. Chinese journal of mechanical engineering. 25 729-737.

[10] Yakhot , V., Orszag, S.A. 1986 Renormalization group analysis of turbulence. I. Basic theory. Journal of ScientificComputing. 1 3-51.

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