numerical study of turbulent forced convection nanofluid ... · numbers for steady asymmetric jet...

Int. J. Nano Dimens., 7(1):57-70, Winter 2016

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ABSTRACT: Research in convective heat transfer using suspensions of nanometer-sized solid particles in baseliquids started only over the past decade. Recent investigations on nanofluid, suspensions are often called, show thatthe suspended nanoparticles remarkably change the transport properties and heat transfer characteristics of thesuspension. Bending walls can also improve heat transfer by raising the total heat transfer from a surface and changingthe behavior of the flow. In this paper two-dimensional incompressible nanofluid flow in a confined sinusoidal convergingjet in turbulent flow regime is numerically investigated. Results showed for the flow structure at different Reynoldsnumbers for steady asymmetric jet development at various values of the duct-to-jet width ratio (aspect ratio), differentamplitudes of surface undulation and different volume fractions of nanoparticles. For considering unsteady treatment ofthe flow, the streamlines and temperature contours result for the unsteady problem is presented and compared with thesteady results. The present computations are in a good agreement with experimental results in open literature. Theresults show that by increasing the Reynolds number, aspect ratio, amplitude and volume fraction the average theNusselt number will increase.

Keywords: Aspect ratio; Nanoparticles; Nusselt number; Recirculation region; Wavy wall.


DOI: 10.7508/ijnd.2016.01.007

*Corresponding Author: Mazaher Rahimi-Esbo

Email: [email protected]

Tel.: (+98) 98911 6277073

Fax: (+98) 98911 6277073

Received 10 March 2015; revised 26 July 2015; accepted 10 October 2015; available online 04 December 2015

INTRODUCTIONConvective heat transfer can be enhanced passively

by changing the flow geometry, the boundaryconditions, or by enhancing the thermal conductivityof the fluid. Various techniques have been proposedto enhance the heat transfer properties of fluids.Changing geometry, by raising heat transfer surfacethe behavior of the flow will increase the total heattransfer. Using porous media, micro scale channels,increasing the surface and placing a shackle in theway of the fluid are common ways of increasing heattransfer by changing the geometry. Researchers havealso tried to raise the thermal conductivity of base fluidsby suspending micro- or larger-sized solid particles influids, since the thermal conductivity of a solid istypically higher than that of liquids. However, due tothe large size and high density of the particles, there is

Numerical study of turbulent forced convection nanofluid jet flow in aconverging sinusoidal channe

A. Taheri 1; E. Aboukazempour 2; A. Ramiar 3; A. A. Ranjbar 3; M. Rahimi-Esbo *, 3

1School of Mechanical Engineering, Mazandaran University of Science and Technology, Babol, Iran2Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

3School of Mechanical Engineering, Babol University of Technology, P. O. Box 484, Babol, Iran

not adequate way to prevent the solid particles fromsettling out of suspension. The lack of stability of suchsuspensions induces added flow resistance andpossible erosion. Therfore, fluids with dispersedcoarse-grained particles have not yet beencommercialized. Modern nanotechnology provides newopportunities to process and produce materials withaverage crystallite sizes below 50 nm. Fluids withsuspended nanoparticles are called nanofluid, a termproposed by Choi [1] of the Argonne NationalLaboratory, USA nanofluids can be considered to bethe next generation heat transfer fluids because theyoffer exciting new possibilities to enhance the heattransfer performance compared to pure liquids. Theyare expected to have superior properties compared toconventional heat transfer fluids, as well as fluidscontaining micro-sized metallic particles. The muchlarger relative surface area of nanoparticles, comparedto those of conventional particles, should not onlysignificantly improve heat transfer capabilities, but also

Research Paper


58

A. Taheri et al.

should raise the stability of the suspensions. Also,nanofluid can improve abrasion-related properties ascompared to the conventional solid/fluid mixtures. Toexplain the reasons for the anomalous increase of thethermal conductivity in nanofluid, Keblinski et al. [2]and Eastman et al. [3] proposed four possiblemechanisms, e.g., Brownian motion of thenanoparticles, molecular-level layering of the liquid atthe liquid/particle interface, the nature of heat transportin the nanoparticles, and the effects of nanoparticlesclustering.

Many researchers applied nanofluid in differentapplications. Zhang et al [4] numerically investigatedheat transfer enhancement of Al

2O

3-water nanofluids

in natural convection in heated wavy cavities. Theystudied the effects of Rayleigh number, amplitude ofwavy wall and volume fraction of nanofluids on theflow field and temperature distribution. They reportedthat the effects of this parameter are more effective onthe flow field than on temperature distribution. A setof experimental data and some correlation for nanofluidproperties are available in the paper of Ravikanth et al.[5] for a fully developed turbulent regime. Primary fluidwas a mixture of 60% ethylene glycol and 40% waterby mass. Aluminum oxide, copper oxide and silicondioxide were used as nanoparticles. An 80%enhancement has been seen with a 10% volumetricfraction aluminum oxide. However a fourfold increscentof pressure drop happens for this nanofluid. Sundarand Sharma [6] reported a 30% enhancement in heattransfer cause by adding nanoparticles. They usedaluminum oxide as nanoparticles and performed theirexperiments for Reynolds number range of 10000-22000.They also defined equations for Nusselt number andfriction coefficient for nanofluid in turbulent flowregime. Another set of experimental data fornanoparticles is reported by Duangthongsuk andWongwises [7] for a water based fluid containing 0.2% volume fraction of TiO

2. With 21nm diameter

nanoparticles were used in their research. Resultsshowed that convective heat transfer coefficient ofnanofluid is about 6-11 % greater than the base fluid.Mean while adding nanoparticles has a little penaltyon the pressure drop. However Fotukian and Esfahany[8] did not see a remarkable enhancement in heattransfer due to adding nanoparticles. Their experimentswere much like to the Duangthongsuk and Wongwises[7] work except with the nanoparticles which was Al

2O

3.

Their measurement showed that pressure drop for dilutenanofluid than that of base flow. Ghaffari et al. [9]

numerically investigated turbulent mixed convectionheat transfer of a nanofluid consisting of water andAl

2O

3 through a horizontal curved tube. They carried

out two-phase mixture model to study such a flow field.They reported that the nanoparticle volume fractiondoes not have a direct effect on the secondary flowand the skin friction coefficient; however, its effectson the thermal parameters and flow turbulence intensityare significant.

Also many works are recorded at the field of wavychannel and converging channel. Nishimura andMatsune [10] numerically investigated pulsated flowfor low Reynolds number in a wavy wall channel. Theycompared their numerically generated results withvisualized flow behavior that they presented in theirwork. They have reported that time-average vortexstrength and wall shear stresses first increase, asfrequency of the oscillation raises under a fixed massflow rate and then, for above a certain value offrequency decreases because of viscous effects.Tolentino et al. [11] presented experimental flowvisualization for a converging channel with wavy walls.They studied the flow behavior for different Reynoldsnumbers and converging/diverging angles. They foundthat a diverging channel has a better effect in a chaoticmixture as flow is more stable for a converging channeleven with a higher Reynolds number. Anothersimulation of turbulent flow in wavy channel wasperformed by Habib et al. [12] for a wide range ofReynolds numbers. Their research shows goodagreement with the experimental data. They found thatthe local Nusselt number takes its greatest value in thepeak of waves where the velocity of the fluid ismaximum. However friction coefficient also shows thesame behavior. They reported enhancement of heattransfer due to bending walls, especially for greateramplitude to wavelength ratios. Also Pham et al. [13]employed large-eddy simulation to address a majorpoint in heat transfer by studying fluid flow behaviorin sinusoidal wavy channels for a wide range ofReynolds number levels. They investigated the effectof spacing ratio, waviness parameter, Reynolds andPrandtl number on flow structure and Nusselt number.Zhang et al. [14] compared their numerical predictionsof the f (Fanning friction factor) and j (Colburn factor)with experimental investigations in the range of 10Re 1500 and according to the authors, the numericaland experimental data are in excellent agreement. Intheir study, numerical simulations were carried out for


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two sinusoidal geometries in the given Reynoldsnumber range while experimental tests were performedfor similar geometries in a wider Reynolds range (Re7000).

Furthermore, there are papers in the literature in theturbulence modeling in backward facing step andconverging channel. For examples Yang et al. [15]presented a numerical study of homogeneous turbulentshear flow and turbulent flow past a backward-facingstep using different linear and non-linear turbulencemodels. Driver et al [16] experimentally investigatedturbulent flow past a backward-facing step. In theirexperiment the ratio of step height to the tunnel exitwas 9. Shariati et al. [17] investigated the flow throughthe converging zone of a headbox, numerically andexperimentally. The experimental headbox that was usedis a laboratory scale model of a typical headbox withthe size reduced by a factor of 5. The main focus oftheir study was on the turbulence quantities andsecondary flows in the outlet jet. They reported theturbulence kinetic energy computed numerically ismuch higher than that measured experimentally. Zhang[18] measured the fiber orientations along the centerlineof an asymmetric headbox. He also predicted the fiberorientation in both symmetric and asymmetricheadboxes using a fiber model, but only using the meanflow field got from the k “ε model. His comparisonbetween the measured and predicted results showedthat the simulated orientations tend to align more withthe flow than the experimentally observed resultsbecause the turbulence was not taken intoconsideration for his calculations. Feng et al. [19]carried out a large eddy simulation (LES) to model theflow field in a converging section. They used aLagrangian tracking scheme to simulate the motion offlexible or rigid individual fibers.

There are also many works in the field of jet flow.Sarma et al. [20] investigated the 2D incompressible jetdevelopment inside a duct in the laminar flow regimefor cases with and without entrainment of ambientfluid. However, they didn’t solve the energy equation.The variation of critical Reynolds number forbifurcation with aspect ratio has been investigated indetails by Battaglia et al. [21]. Al-Aswadi et al. [22]investigated laminar forced convection flow of differentnanofluids over a 2D horizontal backward facing stepplaced in a duct. They reported that the reattachmentpoint moves downstream far from the step as Reynoldsnumber increases, and nanofluid containing SiO

2

nanoparticles has the highest velocity among the othernanofluid types, while nanofluid of Au nanoparticleshas the lowest velocity.

Although some researchers have investigatednanofluids in turbulent flow, none of them investigatedthe turbulent nanofluid jet flow in a converging wavychannel. In this paper the contemporaneous effect ofadding Al

2O

3 nanoparticles and jet flow on the

improvement of heat transfer properties in a convergingwavy channel is investigated. Also the effect of aspectratios (duct width / jet width) and amplitude on flowfeatures and heat transfer properties is investigated.Jet flows are encountered in a wide variety ofapplications, such as gas turbine combustors,industrial burners, ejector systems, rocket nozzles andthe like. The aim of this study is to get an understandingof the velocity and temperature distribution, Nusseltnumber along top and bottom walls downstream fromthe jet and velocity profile at different sections.Moreover, the performance of the jet flow at variousReynolds numbers is investigated. Also the effect ofnanoparticles on thermal and flow characteristics andthe effect of aspect ratios on flow features and heattransfer properties are investigated. In this researchAL

2O

3-H

2O is used. The thermal conductivity of Al

2O

3

which is used in this research is roughly one tenth ofCu; however, the unique property of Al

2O

3 is its low

thermal diffusivity. The reduced value of thermaldiffusivity leads to higher temperature gradients andtherefore, more heat transfer enhancement

EXPERIMENTALComputational Methods

2D incompressible turbulent jet flow in a wavyconverging channel is considered. For simplicity, thejet is taken as isothermal. The velocity profile at the jetinlet is taken as uniform. The geometry considered andthe flow configurations used in this work are shown inFig. 1. The nanoparticles and the base fluid (i.e. water)are assumed to be in thermal equilibrium and no slipcondition occurs between them. The other boundaryconditions are mentioned in Fig. 1.

Aspect ratio is taken as:

AR= h/d (1)

The continuity, momentum and energy equationsfor the 2D flow problem are given by:


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1. Continuity:

(2)

2. Momentum:

(3)

3. Energy:

(4)

where Γ and Γt are the molecular thermal diffusivity

and turbulent thermal diffusivity, respectively, and aregiven by:

(5)

The Reynolds-averaged approach to turbulencemodeling requires that the Reynolds stresses, i.e.

i jnf u u needs to be modeled. For this purpose the

k model is chosen. A common method employsthe Boussinesq hypothesis to relate the Reynoldsstresses to the mean velocity gradients:

(6)

The turbulent viscosity term is to be computed. It isdefined by the following equation:

(7)

4. Transport equation for k and

(8)

and

(9)

where Gk is the rate of generation of the TKE and ρεis its destruction rate, and is written as:

(10)

Sk , S are source terms for k and in this research are

taken as zero. The boundary values for the turbulentquantities near the wall are specified with the standardwall-treatment method. The values of Cμ= 0.009, C1ε=1.44, C2ε=1.92, σ

k=1, σε =1.3 and Pr

t =0.9 are chosen to

be the empirical constants in the turbulent transportequations. It is showed by Launder and Spalding [23].According to Rostamani et al. [24] suggestion thefollowing equations are used to calculate the turbulentintensity (I), turbulent kinetic energy (k) and turbulentdissipation rate (ε) at the inlet section of the headbox.

(11)

Thermophysical properties of nanofluid1. Thermal conductivity:

The thermal conductivity of the nanofluid iscalculated from Chon et al. [25], which is expressed inthe following form:

(12)(13)(14)(15)

A. Taheri et al.

Fig. 1: Geometry and boundary condition of the problem


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kb is the Boltzmann constant, 1.3807×10-23 and l

f is

the free average distance of water molecules thataccording to suggestion of Chon et al. is taken as 17nm. Minista et al. [26] approved the accuracy of thismodel.

2. Viscosity:The viscosity of the nanofluid is approximated as

viscosity of the base fluid f containing dilute

suspension of fine spherical particles as given byMasoumi et al. [27].

(11)

N is a parameter for adapting the results withexperimental data where c

1=-1.133×10-6, c

2=-2.771×10-6,

c3=-9.0×10-8 and c

1=-3.93×10-7.

3. Density and specific heat:The density and specific heat of the nanofluid are

calculated by using the Pak and Cho [28] correlations,which are defined as:

(17)

(18)

(19)

The nondimentional flow and heat transferparameters are defined by:

(20)

(21)

Numerical procedure and code validationA finite volume technique on a collocated grid is

implemented for discretizing the governing equationsinside the computational domain. The SIMPLEalgorithm is used to link the pressure and velocityfields. For the stability of the solution, the diffusionterm in the momentum equations is approximated bythe second order central difference scheme. Moreover,a deferred correction scheme is adopted for the

convective terms. The second order upwind schemeis used for discretizing the convection term in turbulentequations. For the unsteady term implicit Euler methodis applied. No quadrilateral elements and non-uniformgrid systems are employed in the simulations. Asdepicted in Fig. 2 the grid is highly concentrated closeto the jet to ensure the accuracy of the numericalsimulations and for saving both the grid size and thecomputational time. At the end of any iteration, theresidual sum for each of the conserved variables iscomputed and stored, thus recording the convergencehistory. The convergence criterion required that themaximum sum of the error for each of the conservedvariables be smaller than 1×10-5. Grid densities of300×60, 450×80, 550×100 and 650×120 were selected toperform a grid independence test and they show lessthan a 1% difference in velocity compared to the chosengrid (550×100).This is shown in Fig. 3.

Fig. 2: Employed mesh geometry in the modeling

Fig. 3: velocity profile at x=0.5 for mesh indepencyinvestigation at Re=10000, AR=5 and = 0%


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furthermore, the simulations are carried out for differentaspect ratios of 1, 2, 5, 10 and 20 and differentamplitudes of 0.02,0.03 and 0.04. The flow expands fromuniform profile and separates to form a primaryrecirculation region at the upper and lower plate. Afterthat, the velocity profile reattaches and redevelopsapproaching a fully developed flow as fluid flowstoward the channel exit.

For the case of AR=20, Re=15000 unsteady solutionis applied and the streamline and temperature contoursimultaneous is depicted in Fig. 5. It is observed thatafter t=400 seconds the streamlines remain fixed. Afterthis time, both steady and unsteady solutions havethe same result and asymmetric flow occurs for bothcases. It should be noted that the asymmetry in theflow pattern corresponds to a bimodal steadyconfiguration with the jet turning upwards ordownwards in a random way, during any computationalrun. Sarma et al. [20] presented this behavior in theirpapers. According to the above proof, the steadysolution can be generalized to the unsteady problem.From now on results are reported for the steady statecondition.

In Fig. 6 the steady state streamline patterns atdifferent aspect ratios are depicted. In this figure theinfluence of aspect ratio on flow structure at Re=15000and = 0% is shown. It is observed that for a lowaspect ratio such as AR=2, the jet develops almostsymmetrically and counter-rotating velocities are seenimmediately after the sudden expansion. The jet decayis rapid and transition from jet-to-duct flow occurs in ashort distance. At a higher aspect ratio of AR=20, it isobserved that the steady state jet flow development isasymmetric. the transition from jet-to-duct flow occursover a longer distance as well.

In Fig. 7, the influence of amplitude on flowstructure at Re=15000 and = 0% is shown. It isobserved that for low amplitude there isn’t anyrecirculation zone in the dip of the wall. At higheramplitude of a = 0.04 , it is observed that a recirculationzone forms in any dip of the sinusoidal wall. Thesezones have a great effect on increasing the local Nusseltnumber. Fig. 8 shows the distribution of the Nusseltnumber at the lower wall for Re = 10000 using differentnanoparticles volume fractions. The figure illustratesan enhancement in Nusselt number by increasing thevolume fraction of nanoparticles. This behavior can beinferred from Eq. (21). The effect of nanoparticles onthe temperature difference term is negligible.

As no work on nanofluid in a converging jet isrecorded in literature for validating of the code twodifferent cases are used. The present results arecompared with the simulation of Zhang Xiaosi [18] andexperimental data of Shariati et al. [17] in convergingjet. As it can be seen from Fig. 4 (a) there is a veryexcellent agreement between results. Also for the caseof sinusoidal channel the results are compared withexperimental data of Zhang et al. [14] and numericalresults of Pham et al. [13]. As it can be seen from Fig. 4(b) there is an acceptable accuracy.

Fig. 4: (a) Velocity distributions at midline for a convergingchannel at u

in=1.22 (m/s); (b) Global variations of j heat

transfer coefficient versus Reynolds numbers for a wavychannel configuration

RESULTS AND DISCUSSIONSimulations are performed for different Reynolds

numbers of 4000, 10000, 20000, 30000 and 40000, andvarious volume fraction of nanofluid 1%, 2%, 3%, 4%and 5% are compared to their pure base fluid (water)


63

a) t = 10 s

b) t = 50 s

c) t = 100 s

d) t = 400 s

e) Temperature legend T(K)

Fig. 5: Streamline contours for unsteady solution at AR=20, Re=15000 and =0.02


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Fig. 6: Streamline contours for different aspect ratio at Re=15000, = 0%


65

Fig. 7: Streamline contours for different amplitude at Re=10000, = 0%

Fig. 8: Effect of volume fraction of nanoparticles () on Nusselt number at Re=10000 and AR=5


66

Effects of volume fraction of nanoparticles on thetemperature gradient term and on the thermalconductivity ratio term are more pronounced. Beforethe point of reattachment as the volume fractionpercentage of nanoparticles intensifies the temperaturegradient at the top wall raises. This is related to theaddition inertia forces as depicted by Eq. (3). Theequations show that any increase in volume fractionraises inertia forces because

nf will be increased and

accordingly increases the temperature gradient.Besides, the nanoparticles increase the thermalconductivity ratio term as it can be seen from Eq. (12).So the temperature gradient term and thermalconductivity ratio term rise by increasing the volumefraction of nanoparticles. Therefore, it can be seen thatby increasing volume fraction the Nusselt number willbe increased, because the heat transfer properties isimproved. Two sudden growths is sawn arrangementat the position of vortex zone.

The present computations show that when thevolume fraction increases gradually from 0.0 to 0.03and 0.05 the averaged Nusselt numbers will be raisefrom 369.96 to 407.183 and 422.223, respectively. Asmentioned above, nanoparticles improvethrermophisycal properties of the fluid. They directlyincrease the Nusselt number.

The effect of aspect ratio on Nusselt number isdepicted in Fig. 9. As aspect ratio increases theinfluences of the jet and recirculation zones areincreased. Also chaotic behavior of the fluid flow willbe increased and will have more of an impact betweenmolecules of the fluid and the walls. These impactslead to more energy absorption by the fluid .This leadsto an average Nusselt number increase on both topand bottom walls.

The effect of amplitude of wall undulation on Nusseltnumber is depicted in Fig. 10. By increasing amplituderecirculation zones are formed in the dips of the wall.Chaotic behavior of the fluid flow will be raised andwill have more impact between molecules of the fluidand the walls. These impacts cause more energyabsorption by the fluid .These cause an averageNusselt number increase on top and bottom walls.

When, amplitude of the wavy wall is 0.0, 0.01, 0.02,0.03, and 0.04, the averaged Nusselt number on bottomwall will gain the value of 321.711, 329.82, 361.439 and392.727, respectively. These result are got at Re=15000and = 0% . High amplitude enhance heat transfersurface and changing the behavior of the flow regimeenhance the total heat transfer. In any dip of the wall acirculation zone is formed that has a great impact onincreasing the local Nusselt number.

Fig. 9: Effect of aspect ratio on Nusselt number at Re=15000 and = 0%

A. Taheri et al.


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Fig. 10: Effect of amplitude on Nusselt number at Re=10000 and = 0%

Fig. 11 shows the particle volume concentrationand flow Reynolds number effects on the averageNusselt number. It can be seen that the average Nusseltnumber increases with a rise in the nanoparticle volumeconcentration as well as an increase in the flowReynolds number. This behavior is expected since thehigher nanoparticle concentration increases thethermal conductivity of the nanofluid and higherReynolds numbers raise the length of the recirculationzones thus causing an increase in the heat transferrate.

Also, the figure shows that the average Nusseltnumber is more sensitive to volume fraction than toReynolds number. For example at Re=40000 increasingvolume fraction of 0 % to 5 % is caused 13.63 %

enhancement on average Nusselt number and at volumefraction of 5 % increasing Reynolds number 4000 to20000 is caused more than 80 % enhancement.

The pressure drops at different amplitudes of wallundulations and volume fractions are depicted in Fig.12. An increase in pressure drop is detected by raisingthe amplitude and volume fraction. In the Fig. 12adifferent amplitudes have been considered and the ratioof inlet width to the outlet width was 2.5. On the otherhand, in Fig. 12b different volume fractions have beenconsidered and the ratio of inlet width to the outletwidth was 5. It can be seen that these parameters havenegligible effect on the pressure drop. So, increasingin the pressure drop is negligible compared to heattransfer enhancement.


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Fig. 11: The effect of nanoparticles volume fraction and Reynolds number on averaged Nusselt number at AR=10.

Fig. 12: (a) Pressure drop for different amplitude at Re=10000, = 0%; (b) Pressure drop for differentamplitude at Re=10000, AR=5

A. Taheri et al.


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CONCLUSIONForced convection of a turbulent nanofluid flow in

a converging sinusoidal channel numerically has beenstudied. The results showed that by raising theReynolds number, aspect ratio, volume fraction andamplitude the average Nusselt number will increase.Enhancement aspect ratio from 1 to 20, as a result ofthe simultaneous increase in the developing length,penetration of the jet flow in to the duct, and length ofthe recirculation zones, more than 12 percentenhancement in averaged Nusselt number on the lowerwall is detected. Furthermore, it was showed in theinvestigations that increasing the Reynolds number,aspect ratio and amplitude the length of the recirculationzones rise.

This has a direct impact on the local Nusselt numberwhich consequently increases the average Nusseltnumber. The present computations reveal that whenthe volume fraction increases gradually from 0.0 to 0.05the averaged Nusselt numbers will be increase 14percent. Also 24 percent enhancement in averagedNusselt number is seen by increasing amplitude from0.0 to 0.04. It should be noted the intensification ofturbulence eddy transport, thinning of the boundarylayer, dispersion of the suspended particles, andaugmentation of thermal conductivity and heatcapacity of the fluid were suggested to be the possiblereasons for heat transfer enhancement.

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How to cite this article: (Vancouver style)Taheri A., Aboukazempour E., Ramiar A., Ranjbar A. A., Rahimi-Esbo M., (2016), Numerical study of turbulent forcedconvection nanofluid jet flow in a converging sinusoidal channe. Int. J. Nano Dimens. 7(1): 57-70.

A. Taheri et al.

DOI: 10.7508/ijnd.2016.01.007 URL: http://ijnd.ir/article_15585_2444.html

numerical study of turbulent forced convection nanofluid ... · numbers for steady asymmetric jet...

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