advances in mechanical engineering 2013 zheng 2013_795937(1)

Hindawi Publishing CorporationAdvances in Mechanical EngineeringVolume 2013, Article ID 795937, 13 pageshttp://dx.doi.org/10.1155/2013/795937

Research ArticleModeling Asymmetric Flow of Viscoelastic Fluid inSymmetric Planar Sudden Expansion Geometry Based onUser-Defined Function in FLUENT CFD Package

Zhi-Ying Zheng, Feng-Chen Li, and Juan-Cheng Yang

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Feng-Chen Li; [email protected]

Received 3 May 2013; Revised 24 June 2013; Accepted 24 June 2013

Academic Editor: Waqar Khan

Copyright © 2013 Zhi-Ying Zheng et al.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Through embedding an in-house subroutine into FLUENTcode by utilizing the functionalization of user-defined function providedby the software, a new numerical simulation methodology on viscoelastic fluid flows has been established. In order to benchmarkthis methodology, numerical simulations under different viscoelastic fluid solution concentrations (with solvent viscosity ratiovaried from 0.2 to 0.9), extensibility parameters (100 ≤ 𝐿2 ≤ 500), Reynolds numbers (0.1 ≤ Re ≤ 100), andWeissenberg numbers(0 ≤ Wi ≤ 20) are conducted on unsteady laminar flows through a symmetric planar sudden expansion with expansion ratio of1 : 3 for viscoelastic fluid flows. The constitutive model used to describe the viscoelastic effect of viscoelastic fluid flow is FENE-P(finitely extensive nonlinear elastic-Peterlin) model. The numerical simulation results show that the influences of elasticity, inertia,and concentration on the flow bifurcation characteristics are more significant than those of extensibility. The present simulationresults including the critical Reynolds number for which the flow becomes asymmetric, vortex size, bifurcation diagram, velocitydistribution, streamline, and pressure loss show good agreements with some published results. That means the newly establishedmethod based on FLUENT software platform for simulating peculiar flow behaviors of viscoelastic fluid is credible and suitable forthe study of viscoelastic fluid flows.

1. Introduction

Adding polymer or some certain surfactant into water orother Newtonian fluid, the solution has the behaviors of bothviscous Newtonian fluid and elastic solid simultaneouslyand is called viscoelastic fluid. The presence of elasticitybrings viscoelastic fluid special rheological characteristicswhich are different from those of Newtonian fluid, such asshear thinning in viscosity, nonzero normal stress difference,Weissenberg effect (rod-climbing), extrusion swell, tubelesssiphon, elastic recoil [1], and turbulent drag reductioneffect [2–7]. Among these characteristics, turbulent dragreduction is an exciting phenomenon and has greatpotential in energy saving for industrial application systems.Therefore, since Toms [8] reported the drag-reducingeffect in viscoelastic fluid turbulent flow for the first time,this has motivated researchers to conduct studies on it bytheoretical, experimental, and numerical methods. However,

the up-to-date experimental techniques cannot obtain theinformation of the molecular or micelle conformation tensorand elastic stress field in viscoelastic fluid flow, which isa key parameter for studying the mechanism of turbulentdrag reduction. Besides, the development of computertechnology prompts researchers to pay more attention tonumerical simulation studies on viscoelastic fluid turbulentdrag-reducing flows. Direct numerical simulation (DNS), asone of numerical simulation methods solving the governingequations directly, can obtain the most detailed informationof the turbulent flow field and microstructure conformationfield, and thus it is widely adopted to study the turbulent dragreduction mechanisms of viscoelastic fluid flow. However,the extremely large temporal and spatial resolutions inducedby the multiscale characteristics of turbulence bring aboutextremely high demands for computer’s memory andtime consumption. According to the exponential relation-ship between the number of grid and Reynolds number,

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2 Advances in Mechanical Engineering

the computable Reynolds number for DNS is confined tobe small or moderate, which cannot satisfy the need ofengineering practice. Compared with DNS, the currentcommercial computational fluid dynamics (CFD) softwareusingRANS (Reynolds averagedNavier-Stokes)method suchas FLUENT (used in the present study as the major tool torealize our method for numerical simulation on viscoelasticfluid flow) can providemature, reliable, and robust numericalsimulation algorithms for turbulent flows, which is suitablefor engineering applications. However, there is no constitu-tive model for viscoelastic fluid in those commercial CFDcodes for turbulent flow simulation. The functionalizationof user-defined function (UDF) provided by the FLUENTpackage makes it achievable to embed user-defined scalarequations and the source terms of governing equationsinto the calculations. With considerations of the above-mentioned issues, we have been then motivated to developa new numerical simulation methodology for viscoelasticfluid flow based on FLUENT CFD package. Our final goalis, mainly for the engineering purpose, to realize numericalsimulation on turbulent drag-reducing flows of viscoelasticfluid at Reynolds number as high as in a real applicationsystem.We have developed an in-house subroutinemodelingviscoelastic fluid flow and embedded this subroutine intothe FLUENT code through utilizing the functionalizationof UDF in the software package. Before the development ofsubroutine modeling turbulent flow for the next step, theestablished viscoelastic fluid subroutine for laminar flowwithout consideration of turbulence modeling needs to bebenchmarked, which is the purpose of the present paper.Flow through abrupt expansion geometry is a classical

problem and involves fundamental characteristics includingseparation, recirculation, and reattachment. It attracts manynumerical and experimental studies because of its practicalityin many areas such as the design of water channels, ducts,and the air flow around the buildings. For viscoelasticfluid, sudden expansion flow is a common phenomenon inchemical and food industries, such as extrusion process,moldfilling, and food processing. Exploring and understandingthe different flow behaviors of viscoelastic fluid from thatof Newtonian fluid in the expansion geometry is of greatimportance in practical application. Therefore, the flowsof Newtonian and non-Newtonian fluids through suddenexpansion geometry have attracted considerable attentions,especially for laminar flow through the expansion of planarpattern, where the asymmetric flows are known to occurabove a critical Reynolds number even though the expansionis symmetric. However, the published numerical researchesfor sudden expansion flow of viscoelastic fluid are scarce.One of the earlier numerical researches on laminar flow ofviscoelastic fluid through symmetric planar sudden expan-sion geometry was performed by Townsend and Walters[11] with adopting the linearised form of Phan-Thien andTanner (PTT) viscoelastic constitutivemodel and using finiteelement method. Flows through two-dimensional expansionwith expansion ratio of 3 : 40 at low Reynolds number weresimulated for comparison with some cases in their exper-iments, and satisfactory agreement was obtained. Balochet al. [12] used the same method and viscoelastic constitutive

model as used in Townsend and Walters [11] to simulate thetwo-dimensional expansion flows with expansion ratios of3 : 40 and 1 : 80 at low Reynolds numbers (Re ≤ 4). Theresults for the flows with expansion ratio of 3 : 40 showedgood qualitative agreement with some experimental casesconducted by Townsend and Walters [11]. Darwish et al. [13]and Missirlis et al. [14] applied upper-convected Maxwell(UCM) model and developed finite volume method tosimulate the creeping flows of viscoelastic fluid in a 1 : 4sudden expansion at Re = 0.1. Poole et al. [15, 16] numericallyinvestigated the creeping flow of viscoelastic fluid with threedifferent models (UCM, Oldroyd-B, and the linear formof PTT model) through a 1 : 3 planar sudden expansionand studied the effect of expansion ratio on the creepingflow of viscoelastic fluid obeying UCM model. The resultsof numerical simulations with different constitutive modelsshowed that the reduction of both the length and intensityof the vortex, which was caused by the effect of elasticity,is much lower than that reported in previous studies, anda significant region of recirculation still exists for all themodels at high Deborah number [15].The study of expansionratio effect on the flow characteristics showed that both thelength and intensity of the recirculation zone are increasedwith increasing expansion ratio at the sameDeborah number,and for large expansion ratios the recirculation size andstrength are decreased with increase of Deborah number,while for small expansion ratios the recirculation length isinitially decreased at low Deborah numbers, followed byan increase as Deborah number increases [16]. Since theReynolds number in above numerical researches is lower thana certain critical Reynolds number, the flow in sudden expan-sion geometry is symmetry. Oliveira [10] employed modifiedfinitely extensive nonlinear elastic-Chilcott-Rallison (FENE-CR)model to simulate the behaviors of viscoelastic fluid flowsin a symmetric planar expansion geometry with expansionratio of 1 : 3 under laminar state and is likely to be the firstone who reported the asymmetric flows at high Reynoldsnumbers. Somedetailed studieswere conducted on the effectsof Weissenberg numbers, Reynolds numbers, concentrationparameters, and extensibility parameters on the flow char-acteristics. The results showed that extensibility had a rathersmall effect upon the flow pattern, while concentration hada very strong effect that the asymmetry was completelyremoved with the increase of solvent viscosity ratio. Theeffect of elasticity mainly occurred for small Weissenbergnumbers with tendency to reduce the vortex asymmetry.Theincrease of inertia prompted the formation of asymmetrywith larger onset Reynolds number than that of Newtonianfluid. Afterwards, Rocha et al. [17] employed the modifiedFENE-CR model to study the bifurcation phenomena inviscoelastic fluid flow through a symmetric planar suddenexpansionwith expansion ratio of 1 : 4 by numericalmethods.Similar results to those in Oliveira’s research were obtainedfor the effects of Weissenberg number, Reynolds number,concentration parameter, and extensibility parameter on theflow characteristics. Besides, the smaller the onset Re was,larger and more intense the vortices acquired at largerexpansion ratio by comparison with Oliveira’s result were,and a new recirculation regionwas formed at the downstream



Advances in Mechanical Engineering 3

U y

xO

d

Xr1

Xr2

L1 L2

D

Figure 1: The schematic diagram of asymmetric flow in symmetricsudden expansion geometry.

of small vortex with Re higher than 64 and 73.5 for theNewtonian and viscoelastic fluids, respectively.In this paper, considering the comprehensive discussions

and analyses on the flow behaviors in the numerical studyperformed by Oliveira [10], the unsteady viscoelastic fluidand Newtonian fluid laminar flows through a symmetricplanar sudden expansion with expansion ratio of 1 : 3 withmain focus on the flow behavior of asymmetric flow areselected as the test case to benchmark the subroutine con-stituting viscoelastic fluid embedded into FLUENT software.The constitutive model used to simulate viscoelastic fluidflow is finitely extensive nonlinear elastic-Peterlin (FENE-P) model. By comparing our simulation results with thatby Oliveira [10], the established new simulation method forviscoelastic fluid flows using FLUENT software is verified.Besides, some special phenomena induced by elasticity inviscoelastic fluid are also studied. In Section 2, the descriptionof the flow problem and numerical method, including thealgorithm and the way to embed the constitutive modelof viscoelastic fluid into FLUENT software, is presented.The results, for Newtonian fluid flow at different Reynoldsnumber and for viscoelastic fluid flow with one changingparameter and three fixed parameters among the Reynoldsnumber, Weissenberg number, concentration parameter, andextensibility parameter, are given in Section 3. The analysesof flow characteristics and the effect of those four parameterson the flow behavior, comparison between the results ofNewtonian fluid case, and that of viscoelastic fluid case andcomparison with Oliveira’s results are also elucidated. Finally,the main conclusions are given in Section 4.

2. Numerical Simulation Procedures

2.1. Problem Description. The asymmetric flow phenomenonin symmetric planar sudden expansion geometry is schemat-ically shown in Figure 1. For both Newtonian fluid andviscoelastic fluid, there exists a critical Reynolds number atwhich the symmetric flow in the symmetric expansion geom-etry becomes asymmetric with a larger and a smaller recir-culation zone behind the step change. This bifurcation effectis possibly caused by Coanda effect as explained by Oliveirain which fluid tends to flow along the protruding surface,and hence any perturbation will push the main flow to oneside where larger velocity and lower pressure are brought

Figure 2: Partial view of mesh (−2𝑑 ≤ 𝑥 < 4𝑑, −1.5𝑑 ≤ 𝑦 ≤ 1.5𝑑).

about, and ultimately the asymmetry is naturally formed[10]. In order to compare with Oliveira’s earlier work, thesymmetric abrupt expansion channel with expansion ratio𝐸 = 𝐷/𝑑 = 3 is selected as the computational model. Herein,𝑑 and 𝐷 are the height of the smaller channel and the largerchannel, respectively, as shown in Figure 1. According to therecommendation from the study of Hawa and Rusak [18] andOliveira [10], the lengths of the smaller channel and the largerchannel are set to be 𝐿1 = 2𝑑 and 𝐿2 = 50𝑑, respectively.With 𝐿2 = 50𝑑, the flow at the outlet will not affect the flowin the expansion geometry, and the accuracy and facticity offlow are guaranteed. Considering that too small velocity at theinlet is not conducive to the calculation for the case at smallReynolds number, small value of the height of the smallerchannel 𝑑 is adopted (𝑑 = 1mm is used herein). Besides, theset of coordinate system is also shown in Figure 1. Numericalsimulations are conducted on two-dimensional isothermalflows of incompressible fluid for Newtonian and viscoelasticfluid cases, respectively.The flow characteristics under differ-ent Reynolds numbers, Weissenberg numbers, concentrationparameters, and extensibility parameters are obtained andanalyzed, including the recirculation zone, bifurcation effect,streamline, velocity distribution, and flow resistance. Thedifferences between Newtonian fluid and viscoelastic fluidand those between the present and previously reported resultsare compared.

2.2. NumericalMethod. Nonuniform collocatedmesh, whichis the same as used by Oliveira [10], is adopted in the com-putational domain, as shown in Figure 2. The meshes behindthe step change are arranged to be denser.The grid size alongthe 𝑥-direction is gradually increased with a constant factorfor 𝑥 ≥ 0, and the mesh along the 𝑦-direction is uniformand of the same size with the minimum cell size along the𝑥-direction.For a two-dimensional unsteady isothermal flow of in-

compressible fluid, the continuity and momentum equationsare as follows:

𝜕𝑢𝑖𝜕𝑥𝑖 = 0, (1)

𝜌𝜕𝑢𝑖𝜕𝑡 + 𝜌𝑢𝑗𝜕𝑢𝑖𝜕𝑥𝑗 = −

𝜕𝑝𝜕𝑥𝑖 +

𝜕𝜏𝑠𝑖𝑗𝜕𝑥𝑗 +

𝜕𝜏𝑝𝑖𝑗𝜕𝑥𝑗 , (2)

where 𝑢𝑖 is the velocity, 𝜌 is the fluid density and constantlyequals 1000 kg/m3, 𝑝 is the pressure, 𝜏𝑠𝑖𝑗 (𝑖 = 1, 2; 𝑗 = 1, 2) is




the stress caused by the solvent, and 𝜏𝑝𝑖𝑗 is the stress induced bythe elasticity. For Newtonian fluid; the third term on the rightside of (2) disappears. Consider 𝜏𝑠𝑖𝑗 = 𝜇𝑠(𝜕𝑢𝑖/𝜕𝑥𝑗 + 𝜕𝑢𝑗/𝜕𝑥𝑖),where 𝜇𝑠 is the solvent viscosity and constantly equals0.001 Pa ⋅ s. For viscoelastic fluid, FENE-P model is adoptedas the constitutive equation for elastic stress:

𝜏𝑝𝑖𝑗 = 𝑛 (𝑥, 𝑡) 𝜇𝑝𝜆 [𝑓 (𝑟) 𝐶𝑖𝑗 − 𝛿𝑖𝑗] , (3)

where 𝑛(𝑥, 𝑡) is the unit concentration and equals 1 whenthe concentration of polymer or surfactant is homogeneous(assumed in this paper), 𝜇𝑝 is the viscosity of viscoelasticpolymer or surfactant solution, 𝜆 is the relaxation time ofviscoelastic fluid, 𝛿𝑖𝑗 is the Kroneker symbol, 𝑓(𝑟) is thePeterlin function and for FENE-P model it is described as

𝑓 (𝑟) = 𝐿2 − 3𝐿2 − 𝑟2 , (4)

where 𝑟 is themolecular ormicellar length under equilibriumstate and equals the square root of the trace of conformationtensor and 𝐿 is the molecular or micellar extensibility inrelation to its equilibrium size. In (3) 𝐶𝑖𝑗 is the conformationtensor of polymer molecule or surfactant micelle; for 𝑖 ̸= 𝑗,there exists 𝐶𝑖𝑗 = 𝐶𝑗𝑖, and its transport equation is as follows:𝜕𝐶𝑖𝑗𝜕𝑡 + 𝑢𝑘

𝜕𝐶𝑖𝑗𝜕𝑥𝑘 = 𝐶𝑖𝑘

𝜕𝑢𝑗𝜕𝑥𝑘 + 𝐶𝑘𝑗

𝜕𝑢𝑖𝜕𝑥𝑘− 1𝜆 [𝑓 (𝑟) 𝛿𝑖𝑘𝐶𝑘𝑗 − 𝛿𝑖𝑗] + 𝜅

𝜕𝜕𝑥𝑘 (

𝜕𝐶𝑖𝑗𝜕𝑥𝑘 ) .

(5)

The last term in (5) is the artificial viscosity term, which isadded to enhance the stability and convergence of calculation.𝜅 is a nonzero value in the calculation, while, in the presentstudy, converged solution can be obtained with 𝜅 = 0 byadjusting the underrelaxation factors, and so the artificialviscosity term is actually not used.The Reynolds number (Re) is defined as follows:

Re = 𝜌𝑈𝑑𝜇𝑠 + 𝜇𝑝 , (6)

where 𝑈 is the average velocity at the inlet. The decisivesimilarity criterion for viscoelastic flow is the Weissenbergnumber (Wi) defined as follows:

Wi = 𝜆𝑈𝑑 . (7)

As mentioned above, the viscoelastic constitutive equa-tion is embedded into the FLUENT code through UDF func-tionality. In UDF, the general user-defined scalar transportequation is shown as follows, in which there are four termsto be customized: transient term, convection term, diffusionterm, and source term, respectively:

𝜕𝜌𝜙𝜕𝑡 +

𝜕𝜕𝑥𝑖 (𝐹𝑖𝜙 − Γ

𝜕𝜙𝜕𝑥𝑖) = 𝑆𝜙, (8)

where 𝜙 is the scalar, corresponding to components of theconformation tensor𝐶𝑖𝑗 in (5) herein, and Γ is diffusion coef-ficient, corresponding to the coefficient of artificial viscosityterm 𝜅, and equals 0 in the present calculation as mentionedabove. In (5), the transient term corresponds to the first termon the left side; the convection term is the second term on theleft side; the remaining terms in (5) belong to the source term𝑆𝜙. The customizations of these terms are realized by writingcorresponding functions with 𝐶 language, connecting thefunctions with interfaces through the corresponding settingsin the code. By customizing these terms, the viscoelasticconstitutive model is then embedded into the FLUENT code,and the viscoelastic fluid flow problem can be solved.The boundary condition of entrance is set to be velocity

inlet with a fully developed velocity profile (as shown inFigure 1) which is defined by the following expression:

𝑢 = 1.5𝑈[1 − ( 𝑦𝑑/2)2] . (9)

The exit is set as outflow condition, and all the walls haveno slip boundary conditions. The algorithm provided byFLUENT software is based on finite volume method. Doubleprecision is employed for our calculations. Pressure implicitwith splitting of operators (PISO) scheme, which is moresuitable for the calculation of transient flow, is adopted tosolve the coupling of velocity and pressure; standard schemeis utilized to discretize the pressure equation; QUICK schemeis used to discretize the momentum equation and user-defined scalar equations. The convergence criteria (residualsbelow which the calculation is considered to be converged)and underrelaxation factors are shown in Tables 1 and 10,respectively. Note that different underrelaxation factors areadopted for different conditions. The time step is set to be0.001 s for transient calculation. The velocity of whole flowfield is initialized by the average velocity at the inlet 𝑈, andthe conformation tensor is initialized to be 𝐶𝑥𝑥 = 1, 𝐶𝑦𝑦 = 1,and 𝐶𝑥𝑦 = 0.3. Results and Discussions

Numerical simulations are performed on the flow throughsudden expansion geometry for Newtonian fluid at differentReynolds numbers, while, for viscoelastic fluid, three moreparameters, Weissenberg number, concentration (measuredby the solvent viscosity ratio defined as 𝛽 = 𝜇𝑠/(𝜇𝑠+𝜇𝑝)), andextensibility parameters (𝐿2), are taken into consideration.The numerical simulations are conducted at one changingparameter and the other three unchanged parameters tostudy the effects induced by these four parameters individua-lly. Note that the existence of the dynamic viscosity of vis-coelastic polymer or surfactant solution 𝜇𝑝 in the definitionof Reynolds number makes the average velocity at the inlet𝑈 different for Newtonian fluid and viscoelastic fluid cases atthe same Reynolds number.

3.1. Mesh Independence Verification. In order to guaranteethe numerical accuracy, the mesh independence is studied




Table 1: Convergence criteria.

Variable Continuity 𝑢 V 𝐶𝑥𝑥 𝐶𝑦𝑦 𝐶𝑥𝑦Residual 10−8 10−6 10−6 10−6 10−6 10−6

Table 2: Verification of mesh independence.

Newtonian case Viscoelastic case𝑋𝑟1/𝑑 𝑋𝑟2/𝑑 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑

Coarse mesh 3.9314 7.5749 4.1452 7.4470Medium mesh 3.9498 7.5820 4.0891 7.4800Fine mesh 3.9515 7.6044 4.0588 7.5136Richardson’s extrapolation 3.9521 7.6119 4.0487 7.5248Discretization error (%) 0.06 0.39 1.00 0.60

Table 3: Recirculation size for Newtonian fluid cases.

Re 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑0.1 0.502 0.5021 0.548 0.54810 1.195 1.19520 2.096 2.09630 3.065 3.06540 4.057 4.05750 5.062 5.06252 5.265 5.26554 5.467 5.46755 4.841 6.19756 4.480 6.62858 4.132 7.17560 3.950 7.58264 3.770 8.23070 3.670 9.01175 3.650 9.57280 3.656 10.07485 3.677 10.53190 3.706 10.95095 3.741 11.334100 3.779 11.686

at first. Three meshes are selected, including the coarsemesh with 1550 grids and minimum cell size of 0.1𝑑, themedium mesh with 6200 grids and minimum cell size of0.05𝑑, and the fine mesh with 24800 grids and minimumcell size of 0.025𝑑, which are the same as those in Oliveira’swork. Numerical simulations are conducted at Re = 60 forNewtonian fluid flow and at Re = 60, Wi = 2, 𝛽 = 0.9,and 𝐿2 = 100 for viscoelastic fluid flow with three differentmeshes. To demonstrate themesh independence of solutions,Richardson’s extrapolation technique [10] is applied to ana-lyze the recirculation size (𝑋𝑟 shown in Figure 1), as listed inTable 2.The basic principle of Richardson’s extrapolation is that

any result which is directly evaluated from the numericalsolution or any integrated result, 𝜑, is assumed to convergewith mesh refinement as 𝜑 = 𝜑ℎ + 𝐶ℎ𝑝, where ℎ is the

Table 4: Critical Reynolds numbers obtained by previous resear-ches.

Sources Methods Re𝑐Oliveira [10] Finite volume method 54

Hawa and Rusak [18] Vorticity/stream functionfinite difference 53.8

Fearn et al. [9] Finite element method 53.9

Drikakis [19] Fourth-order finitedifference method 53.3

Mishra and Jayaraman [20] Finite element method 54

Table 5: Comparisons of the sizes of recirculation zones at Re = 187.

Sources 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑 𝑋𝑟3/𝑑 𝑋𝑟4/𝑑Present results 4.48 15.24 12.50 25.46De Zilwa et al. [21] 5.0 15.2 12.6 26Oliveira [10] 4.5 15.1 12.4 25.3

Table 6: Sizes and asymmetry of recirculation zone for differentWeissenberg numbers.

Wi 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑 𝐷𝑋/𝑑0 3.950 7.582 3.6321 4.094 7.397 3.3032 4.089 7.480 3.3913 4.083 7.590 3.5074 4.087 7.679 3.5925 4.099 7.747 3.64810 4.149 7.951 3.80215 4.108 8.105 3.99720 4.041 8.182 4.141

Table 7: Sizes and asymmetry of recirculation zone for differentextensibility parameters.

𝐿2 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑 𝐷𝑋/𝑑100 4.094 7.397 3.303200 4.153 7.351 3.198300 4.189 7.329 3.140400 4.215 7.315 3.100500 4.236 7.303 3.067

mesh size, 𝐶 is a constant, 𝑝 is the order of the scheme,and 𝜑ℎ is the value of 𝜑 obtained based on the mesh withsize of ℎ. If the cell size of fine mesh is denoted as ℎ,then the cell size of medium mesh would be 2ℎ, and thereexists 𝜑 = 𝜑2ℎ + 𝐶(2ℎ)𝑝. Therefore, the constant 𝐶 equalsΔ𝜑/[ℎ𝑝(2𝑝 − 1)] (Δ𝜑 = 𝜑ℎ − 𝜑2ℎ), and the correspondingdiscretization errors will be 𝜀ℎ = 𝜑 − 𝜑ℎ = Δ𝜑/(2𝑝 − 1) onfine mesh and 𝜀2ℎ = 𝜑 − 𝜑2ℎ = 2𝑝Δ𝜑/(2𝑝 − 1) on mediummesh. By assumption of second-order accuracy (𝑝 = 2), theextrapolated values are calculated from 𝜑 = (4𝜑ℎ − 𝜑2ℎ)/3.Note that the discretization error in Table 2 is based on

the medium mesh. It can be obtained that the discretization




Table 8: Sizes and asymmetry of recirculation zone for differentconcentrations.

𝛽 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑 𝐷𝑋/𝑑0.9 4.089 7.480 3.3910.8 4.221 7.358 3.1370.7 4.363 7.218 2.8550.6 4.511 7.067 2.5560.5 4.669 6.907 2.2380.4 4.834 6.744 1.9100.3 4.976 6.601 1.6250.2 5.146 6.409 1.263

Table 9: Recirculation sizes for different Reynolds numbers.

Re 𝑋𝑟1/𝑑 𝑋𝑟2/𝑑1 0.544 0.54410 0.850 0.85120 1.902 1.90230 2.931 2.93140 3.936 3.93650 4.933 4.93355 5.434 5.43456 5.535 5.53557 5.635 5.63558 5.322 6.11059 4.910 6.58960 4.669 6.90765 4.192 7.91270 4.035 8.63875 3.974 9.26180 3.956 9.81485 3.961 10.32790 3.979 10.79995 4.006 11.238100 4.039 11.643

errors are all within 1%. Considering the economy of the cal-culation and convenience of comparison with Oliveira’sresults at the same time, the medium mesh with 6200 gridsis used in the present numerical simulations.

3.2. Results for Newtonian Fluid Cases. The bifurcation effect(recirculation size asymmetry 𝐷𝑋 = (𝑋𝑟2 − 𝑋𝑟1) versus Re)obtained in this paper, together with that of Fearn et al. [9]and Oliveira [10], is shown in Figure 3. Recirculation sizesfor different Reynolds numbers are given in Table 3, and thecomparison with Oliveira’s results is shown in Figure 4. FromTable 3, we can obtain that the critical Reynolds numberat which the flow becomes asymmetric is 55. It is in goodagreement with those of previous numerical researches withdifferent numericalmethods, which are shown inTable 4.Thebifurcation effect (Figure 3) and recirculation size (Figure 4)also show excellent agreement with previous researches.Besides, the velocity distribution in the flow field is

compared with previous work at Re = 80 in Figure 5.

0 20 40 60 80 100

−8

−6

−4

−2

0

2

4

6

8

Fearn et al.OliveiraPresent result

DX

/d

Re

Figure 3: Comparisons of the bifurcation effect for Newtonianfluid case with the experimental data from Fearn et al. [9] and thenumerical results from Oliveira [10].

0 20 40 60 80 1000123456789

101112

Xr/d

Xr1/d (Oliveira)Xr2/d (Oliveira)

Xr1/d (present results)Xr2/d (present results)

Re

Figure 4: Comparison of the recirculation size for Newtonian fluidcase with results in Oliveira [10].

The profiles of dimensionless velocity in the 𝑥-direction atfour different positions along the streamwise direction areplotted. It can be obtained that the results in this paper areclosely consistent with experimental data of Fearn et al. [9]and numerical results of Oliveira [10].For further verification, we consider a case with Re = 187

at which a new recirculation zone (measured by 𝑋𝑟3 atdetachment point and 𝑋𝑟4 at reattachment point) appearsadjacent to the smaller recirculation zone behind the stepchange, and the streamlines obtained in this paper areplotted in Figure 6. The result, sizes of three recirculationzones, is compared with previous results and shows excellentagreement with them, as shown in Table 5.




Table 10: Underrelaxation factors.

Variable Pressure Density Body forces Momentum 𝐶𝑥𝑥 𝐶𝑦𝑦 𝐶𝑥𝑦Value 0.005∼0.3 0.02∼1 0.02∼1 0.01∼0.7 0.02∼0.9 0.02∼0.9 0.02∼0.9

−0.5

0.0

0.5

1.0

1.5

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

y/d

u/U

x/d = 1.25

(a)

−0.5

0.0

0.5

1.0

1.5

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5y

/du/U

x/d = 5

(b)

−0.5 0.0 0.5 1.0 1.5

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Present resultOliveiraFearn et al.

y/d

u/U

x/d = 10

(c)

−0.5

0.0

0.5

1.0

1.5

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Present resultOliveiraFearn et al.

y/d

u/U

x/d = 20

(d)

Figure 5: Comparison of velocity profiles with experimental data of Fearn et al. [9] and numerical results of Oliveira [10].

Figure 6: Streamline for Newtonian fluid case at Re = 187.

3.3. Results for Viscoelastic Fluid Cases. In this section,in order to study the effect induced by elasticity, extensi-bility, concentration, and inertia on the sudden expansionflow, numerical simulations are systematically conducted on

viscoelastic fluid flows through sudden expansion geometryat different Weissenberg numbers, extensibility parameters,solvent viscosity ratios, and Reynolds numbers, respectively.

3.3.1. Effect of Elasticity:Weissenberg Number Dependence. Inorder to study the effect of elasticity on the flow, numericalsimulations are conducted at changing Weissenberg numbersand constant Reynolds number, concentration, and extensi-bility. In order for comparison, the base values of the threeconstant parameters are selected to be the same with those in




0 5 10 15 203.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

Xr/d



Wi

Figure 7:Comparison of recirculation sizes at differentWeissenbergnumbers with results of Oliveira [10].

Oliveira [10]; that is, Re = 60, 𝛽 = 0.9, and 𝐿2 = 100. Asanalyzed by Oliveira, the extensibility parameter of 𝐿2 = 100is typical for FENE models, and solvent viscosity ratio of𝛽 = 0.9 corresponds to the concentration parameter of𝑐 ≈ 0.1 (concentration parameter 𝑐 is defined as 𝑐 = 𝜇𝑝/𝜇𝑠,and hence 𝛽 = 1/(1+ 𝑐)), which is typical for dilute solutions.Reynolds number is set to be 60, which is not far away fromthe critical Reynolds number 55 for Newtonian fluid flow,and is chosen for the convenience of studying the effect onthe flow characteristics, especially on the flow asymmetry.However, because the convergence of calculation cannot beguaranteed by only adjusting the underrelaxation factors forlarger Weissenberg numbers (Wi > 20), the Weissenbergnumbers in the present simulations only vary from 0 to20, while in Oliveira’s work Weissenberg number is up to100. Study on the viscoelastic fluid flow through the suddenexpansion geometry for larger Weissenberg numbers will beperformed and reported in the next step of our continuousstudies.

The effects of elasticity on the recirculation size and flowasymmetry are given in Table 6. And the comparison withOliveira’s results is shown in Figure 7. One can obtain that,for viscoelastic fluid, the size of the larger vortex slightlyincreases with the increase of Wi, while the size of thesmaller vortex decreases. The increase of the elasticity doesnot remove the asymmetry of the flow, which is consistentwith Oliveira’s result qualitatively. But the variation trend ofthe sizes of two recirculation zones is different from that ofOliveira’s result, especially for the size of the larger vortex,as shown in Figure 7. For this Wi-dependent phenomenonappeared in the asymmetry of planar sudden expansion flow,elaborated experiments are called for in order to quantify thesizes of two corner vortices as a function of viscoelasticity.

3.3.2. Effect of Extensibility: 𝐿2 Dependence. Numerical simu-lations on the effect of extensibility are conducted at Re = 60,

100 200 300 400 5004.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

L2

Xr/d



Figure 8: Comparison of recirculation sizes at different extensibilityparameters with results of Oliveira [10].

Wi = 1, and 𝛽 = 0.9. The extensibility parameter 𝐿2 variesfrom 100 to 500 with increment step of 100. The sizes andasymmetry of vortices are given in Table 7 and are comparedwith those of Oliveira’s results in Figure 8. It shows that theextensibility has little influence on the variation of vortexlength. With the increase of extensibility, the flow asymmetrydecreases, but, in Oliveira’s research, the asymmetry size(𝐷𝑋/𝑑) is almost the same with just a slight decrease exceptfor the point with 𝐿2 = 500. And the variation trendof the size of larger recirculation zone is opposite to thatin Oliveira’s work though there is only small quantitativedifference between them. The sizes of smaller recirculationzone are in both qualitative and quantitative agreement withOliveira’s results, as shown in Figure 8.

3.3.3. Effect of Concentration: 𝛽Dependence. Numerical sim-ulations are conducted at fixed Re = 60, Wi = 2, and𝐿2 = 100 and changing solvent viscosity ratios from 0.9 to 0.2(the corresponding concentration of polymer or surfactantsolution increases) to study the effect of concentration. Notethat we do not reduce the time step with the decrease of 𝛽as Oliveira did. For each concentration, the time steps in thenumerical simulations are all set to be 0.001 s.Theunderrelax-ation factors are adjusted to be smaller for smaller 𝛽 (higherconcentration parameter 𝑐) since the coupling between themomentum and constitutive equations is enhanced with theincrease of 𝑐. The sizes and asymmetry of recirculation zoneand comparisons with Oliveira’s resulet are shown in Table 8and Figure 9, respectively. We can obtain that concentrationexhibitsmuch stronger effect on the flowby comparing it withthat of extensibility. The elongation of the smaller vortex andthe reduction of the larger vortex make the flow asymmetryshrink gradually with the reduction of 𝛽 from Newtoniancase (𝛽 = 1), which is induced by the enhancement ofelasticity and in accordance with Oliveira’s result. But theasymmetry is not removed and we do not capture the change




0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.94.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

𝛽

Xr/d



Figure 9: Comparison of recirculation sizes at different concentra-tions with results of Oliveira [10].

from asymmetry to symmetry as Oliveira did in the presentlyinvestigated range of 𝛽. However, the variation trend of theasymmetry of recirculation zone can still illustrate that theincrease of concentration facilitates the evolvement fromasymmetric state to symmetric state.

3.3.4. Effect of Inertia: Reynolds Number Dependence. Inthis section, we perform the numerical simulations at fixedWi = 2, 𝐿2 = 100, and 𝛽 = 0.5 and changing Reynoldsnumber from 1 to 100 to study the effect of inertia on theflow characteristics of viscoelastic fluid.The results, includingthe recirculation sizes at different Re and the bifurcationbehaviors, are shown in Table 9 and Figures 10 and 11,respectively, together with Oliveira’s numerical results forcomparison. Note that all of the flows simulated in this paperare finally developed into steady state, except for the case withRe = 1, Wi = 2, 𝐿2 = 100, and 𝛽 = 0.5 in this section. In thiscase, the flow will change from steady state to unsteady stateagain and ultimately keep the periodical variation, which ispossibly caused by the elastic instability and will be studied inour further research. The detailed results given in this paperare only those for steady states. One can obtain that the onsetReynolds number at which the flow becomes asymmetric isdelayed to Re𝑐 = 58 compared with that of Newtonian fluid(Re𝑐 = 55), which is also obtained in Oliveira’s research.But the critical Reynolds number for viscoelastic fluid inthis paper is smaller than that in Oliveira’s work (Re𝑐 =64). The delay of onset Re for viscoelastic fluid is induced byelasticity which acts as a stabilizing factor for the occurrenceof bifurcation. The recirculation sizes and bifurcation effectare in qualitative agreement with Oliveira’s results: the sizesof two vortices and asymmetry extent for viscoelastic fluidare smaller than those for Newtonian fluid at the same Re forthe whole range of Re (except for the size of smaller vortexat higher Reynolds numbers above its critical onset value),as shown in Figures 10 and 11. The difference between the

0 20 40 60 80 1000

2

4

6

8

10

12

Re

Xr/d

Viscoelastic Xr1/d(present results)

Viscoelastic Xr2/d(present results)

Viscoelastic Xr1/d(Oliveira)

Viscoelastic Xr2/d(Oliveira)

Newtonian Xr1/d(present results)

Newtonian Xr2/d(present results)

Figure 10: Comparisons of recirculation sizes at different Reynoldsnumbers for viscoelastic fluid with those of Newtonian fluid in thispaper and results of Oliveira [10] for viscoelastic fluid.

0 20 40 60 80 100

0

2

4

6

8

Viscoelastic (present results)Viscoelastic (Oliveira)Newtonian (present results)

−2

−4

−6

−8

DX

/d

Re

Figure 11: Comparisons of bifurcation effect for viscoelastic fluidwith that of Newtonian fluid in this paper and results of Oliveira[10] for viscoelastic fluid.

vortex sizes of viscoelastic fluid and those of Newtonian fluidcan be considered as a swelling-like phenomenon similar toextrusion swell as explained by Oliveira [10].

Figure 12 shows the streamlines at different Re for vis-coelastic fluid and clearly depicts the changes of flow state,recirculation sizes, and bifurcation with the increase of Re.For small Reynolds numbers of Re = 1 and Re = 10, we




Re = 1

Re = 10

Re = 20

Re = 40

Re = 55

Re = 58

Re = 60

Re = 70

Re = 80

Re = 90

Re = 100

Figure 12: Streamlines at different Reynolds numbers for viscoelas-tic fluid.

capture the streamline’s convergence towards the centerlinebefore the expansion, followed by a divergence towards thewalls of larger channel after the expansion, which was alsocaptured by Oliveira with viscoelastic case at Re = 0.1 [10],although the phenomenon is not so obvious in the figure.Furthermore, the profiles of velocity in the 𝑥-directionalong the centerline at those two Reynolds numbers areplotted in Figure 13. And an overshoot of the profile with anundershoot at the downstream is observed at the expansion

for viscoelastic fluid flow, while there is no such kind ofphenomena for Newtonian fluid flow. The cause of thisdifference is that inertia is not necessary for diverging flowbehavior to occur, which is proposed and explained by Alvesand Poole [22] for the contraction flow, and the suddenexpansion flow of viscoelastic fluid is somewhat similar to theextrusion swell, as mentioned above, with enhancing swelleffect as Re decreases. Besides, the overshoot reduces with theincrease of Re, as shown in Figure 13, and finally disappears.

We consider a localized loss coefficient 𝐶𝐼 which is inde-pendent of the channel length to measure the energy losses,and its definition is described in detail by Oliveira [10]. Notethat for the part of fully developed pressure drop coefficient(𝐶𝐹) in 𝐶𝐼, the fully developed regions Δ𝑋 (Δ𝑋 = Δ𝑥/𝑑)in the definition of Fanning friction factor 𝑓 (𝑓 = Δ𝑃/Δ𝑋and Δ𝑃 = Δ𝑝/0.5𝜌𝑈2), whose choice is important for theaccurate calculation of 𝐶𝐼, are selected to be −2 < 𝑥/𝑑 < −1for smaller upstream channel and 30 < 𝑥/𝑑 < 50 for largerdownstream channel, respectively. The length of −2 < 𝑥/𝑑 <−1 in the upstream channel is not enough for fully developedflow; thus𝐶𝐼 is an estimation value. The relationship between𝐶𝐼 and Re is plotted in Figure 14 for viscoelastic fluid andNewtonian fluid flows and compared with the numerical dataof Oliveira [10]. It is seen that 𝐶𝐼 decreases with the increaseof Re for viscoelastic fluid flow, while, for Newtonian fluidcase, 𝐶𝐼 decreases at first and then increases slightly andattains the minimum at the critical Reynolds number. 𝐶𝐼of viscoelastic fluid flow is larger than that of Newtonianfluid flow in the whole range of Re. These results are all inqualitative agreement with those obtained by Oliveira. ForNewtonian fluid, the results are also quantitatively consistentwith Oliveira’s. That means the choice of fully developedregion within the upstream channel in this paper does notaffect the accuracy of 𝐶𝐼 so much, for the length of thesmaller upstream channel in Oliveira’s numerical simulationson Newtonian cases is 𝐿1 = 20𝑑, which can guarantee theaccurate evaluation of 𝐶𝐼.

Apart from localized loss coefficient, the flow resistancecoefficient 𝑓Re, another measure of pressure loss, is alsostudied in this paper. The definition of Fanning friction factoris given by Oliveira [10]. Figure 15 depicts the changes of𝑓Rewith the increase of Re for Newtonian fluid and viscoelasticfluid flows and results of Oliveira. From this figure, we cansee that our results show qualitative agreement with thoseof Oliveira. The flow resistance coefficient decreases withincreasing Re. 𝑓Re for viscoelastic fluid flow is greater thanthat for Newtonian fluid flow in the whole range of Re. Thechange of slope is also captured at critical Reynolds numberfor both Newtonian fluid and viscoelastic fluid flows. Thequantitative difference from Oliveira’s results for Newtoniancases is possibly induced by the difference of the length of theupstream channel.

3.3.5. Summary of the Effects of Four Parameters and Dis-cussions. According to the results of numerical simulationson viscoelastic fluid flows, the concentration and inertiaeffects on the flow characteristics are greatly dependent onthe results of recirculation sizes and bifurcation effect. While




−2 0 2 4 6 80.0

0.5

1.0

1.5

ViscoelasticNewtonian

u/U

x/d

Re = 1

(a)

−2 0 2 4 6 80.0

0.5

1.0

1.5

ViscoelasticNewtonian

u/U

x/d

Re = 10

(b)

Figure 13: Profile of velocity in the 𝑥-direction along the central plane (𝑦 = 0) for Newtonian fluid and viscoelastic fluid at small Reynoldsnumbers.

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

CI

Newtonian (Oliveira)Viscoelastic (Oliveira)

Newtonian (present results)Viscoelastic (present results)

Re

Figure 14: Change of localized loss coefficient with Reynoldsnumber for Newtonian fluid and viscoelastic fluid and comparisonwith results of Oliveira [10].

the effect of extensibility is small in the range of extensibilityparameter discussed in this paper.

Through comparisons, the numerical simulation resultsfor Newtonian fluid are in both qualitative and quantitativeagreements with previous experimental data and numericalresults. While for viscoelastic fluid, most of the results arequalitatively consistent with Oliveira’s numerical results. Thecause of quantitative discrepancy and other qualitative differ-ence is possibly due to the difference of viscoelastic fluid con-stitutive model except for the reason mentioned above. Theconstitutive model of FENE-CR adopted by Oliveira in the

0 20 40 60 80 100

−5

0

5

10

15

20

Newtonian (Oliveira)Viscoelastic (Oliveira)

Newtonian (present results)Viscoelastic (present results)

Re

fRe

Figure 15: Variation of flow resistance coefficient with Reynoldsnumber for Newtonian fluid and viscoelastic fluid and comparisonwith results of Oliveira [10].

numerical simulations on viscoelastic fluid flows is an empir-ical extension of FENE-P model (adopted in this paper),which has a constant shear viscosity for a simple shear flow.

4. Conclusion

Numerical simulations are conducted on the flows througha symmetric planar sudden expansion geometry with anexpansion ratio of 1 : 3 at different Reynolds numbers forNewtonian fluid and at different Weissenberg numbers,extensibility parameters, concentrations, and Reynolds num-bers for viscoelastic fluid modeled by FENE-P model.




The results are compared with previous published ones,especially with numerical data of Oliveira [10]. The mainconclusions drawn from this study are given below.

(1) The critical Reynolds number at which the flowbecomes asymmetric forNewtonian fluid is Re𝑐 = 55,in agreement with the previous experimental andnumerical results. All of the results for Newtonianfluid are in excellent agreement with previous pub-lished results except for slight quantitative differenceof flow resistance coefficient 𝑓Re from Oliveira’snumerical data, which is possibly caused by thedifference of the upstream channel length.

(2) The critical Reynolds number at which bifurcationoccurs for viscoelastic fluid with fixed Wi = 2,𝐿2 = 100, and 𝛽 = 0.5 is delayed to Re𝑐 = 58,which is smaller than that in Oliveira’s research. Someof the results for viscoelastic fluid in this paper are inqualitative agreement with numerical data of Oliveirawith some quantitative discrepancy. The differencesare possibly induced by the different constitutivemodels for viscoelastic fluid.

(3) The comparisons with the previous experimentaland numerical results with present numerical resultsprove the correctness and feasibility of the establishednumerical simulation methodology for viscoelasticfluid flow by embedding user-defined function ofviscoelastic fluid constitutive model into FLUENTsoftware in the present paper.

Nomenclature

𝑐: Concentration parameter𝐶𝑖𝑗: Components of conformation tensor𝐶𝐼: Localized loss coefficient𝐶𝐹: Fully developed pressure drop coefficient𝑑: Height of upstream channel𝐷: Height of downstream channel𝐷𝑋: Asymmetry of vortex size𝐸: Expansion ratio (= 𝐷/𝑑)𝑓: Fanning friction factor𝑓(𝑟): Peterlin function𝐿: Molecular or micellar extensibility inrelation to its equilibrium size𝐿1, 𝐿2: Lengths of upstream and downstreamchannels𝑛(𝑥, 𝑡): Unit concentration𝑝, 𝑃: Pressure, dimensionless pressure(= 𝑝/0.5𝜌𝑈2)𝑟: Molecular or micellar length underequilibrium state

Re: Reynolds number (= 𝜌𝑈𝑑/(𝜇𝑠 + 𝜇𝑝))𝑡: Time𝑢𝑖(𝑢, V): Velocity components𝑈: Average velocity at the inletWi: Weissenberg number (= 𝜆𝑈/𝑑)𝑥𝑖(𝑥, 𝑦): Cartesian coordinates𝑋𝑟: Size of vortex𝛿𝑖𝑗: Kronecker symbol

𝜏𝑝𝑖𝑗 , 𝜏𝑠𝑖𝑗: Components of stress tensors caused by thesolvent and elasticity𝛽: Solvent viscosity ratio (= 𝜇𝑠/(𝜇𝑠 + 𝜇𝑝))𝜆: Relaxation time𝜅: Coefficient of artificial viscosity term𝜙: User-defined scalarΓ : Coefficient of diffusion term𝑆𝜙: Source term𝜇𝑝: Dynamic viscosity of viscoelastic polymeror surfactant solution𝜇𝑠: Dynamic viscosity of solvent.

Subscripts and Superscripts

𝑐: Critical value𝑖, 𝑗, 𝑘: Cartesian components (from 1 to 2)𝑝: Polymer or surfactant𝑟: Recirculation𝑠: Solvent.

Acknowledgments

This work is supported by Foundation for InnovativeResearch Groups of the National Natural Science Foundationof China (51121004), National Natural Science Foundation ofChina (51076036, 51276046), and Specialized Research Fundfor the Doctoral Program of Higher Education of China(20112302110020). The authors would like to appreciate thevaluable discussions with the members of Complex Flow andHeat Transfer Lab and thankDr. J. L. Yin of Shanghai JiaotongUniversity for his help and guidance on writing user-definedfunction.

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