Four-Field Galerkin/Least-Squares

Formulation for Viscoelastic Fluids

Oscar M. Coronado a, Dhruv Arora a, Marek Behr b,a,∗∗and Matteo Pasquali a,∗

aDepartment of Chemical and Biomolecular Engineering andComputer and Information Technology Institute

Rice University, MS 362, 6100 Main St, Houston, TX 77005, USAbChair for Computational Analysis of Technical Systems

CCES, RWTH Aachen University, 52056 Aachen, Germany

Submitted to the Special Issue of the Journal of non-Newtonian FluidMechanics on 14th International Workshop on Numerical Methods for

Non-Newtonian Flows.

Abstract

A new Galerkin/Least-Squares (GLS) stabilized finite element method is presentedfor computing viscoelastic flows of complex fluids described by the conformationtensor; it extends the well-established GLS method for computing flows of incom-pressible Newtonian fluids. GLS methods are attractive for large-scale computationsbecause they yield linear systems that can be solved easily with iterative solvers(e.g., the Generalized Minimum Residual method) and because they allow simplecombinations of interpolation functions that can be conveniently and efficiently im-plemented on modern distributed-memory cache-based clusters.

Like other state-of-the-art methods for computing viscoelastic flows (e.g., DEVSS-TG/SUPG), the new GLS method introduces a separate variable to represent thevelocity gradient; with the aid of this variable, the conservation equations of mass,momentum, conformation, and the definition of velocity gradient are converted intoa set of first-order partial differential equations in four unknown fields—pressure,velocity, conformation, and velocity gradient. The unknown fields are representedby low-order (continuous piecewise linear or bilinear) finite element basis functions.

The method is applied to the Oldroyd-B constitutive equation and is tested intwo benchmark problems—flow in a planar channel and flow past a cylinder in achannel. Results show that (1) the mesh-convergence rate of GLS is comparable tothe DEVSS-TG/SUPG method; (2) the LS stabilization permits using equal-orderbasis functions for all fields; (3) GLS handles effectively the advective terms in theevolution equation of the conformation tensor; and (4) GLS yields accurate resultsat lower computational costs than DEVSS-type methods.

Keywords: Stabilized finite element method, Viscoelastic flow,Galerkin/Least-Squares, Oldroyd-B fluid, Flow past a cylinder in a channel

Preprint submitted to Elsevier Science 27 March 2006

1 Introduction

In the past decades, extensive research has been done on flows of liquids withmicro-macro structure (also known as complex fluids); these fluids are foundin several industrial and biological applications, e.g., polymer processing, coat-ing of polymer solutions, ink-jet printing, microfluidic devices, and human aswell as artificial organs (blood, synovial fluid). Usually these liquids displaya viscosity dependent on the rate of straining and the flow kinematics (shearversus extension); they also show elasticity on time scales that overlap withthe flow time scales.

Realistic models of flowing complex fluids are crucial for understanding andoptimizing flow processes. Two main classes of models have been proposed formodeling complex fluids: fine-grained models [1, 2], (e.g., bead-spring or bead-rod models of polymer solutions), where the microstructure is represented bymicromechanical objects governed by stochastic differential equations, andcoarse-grained ones, where the microstructure is modeled by means of one ormore continuum variables representing the expectation value of microscopicfeatures (e.g., the conformation tensor in models of polymer solutions) [3–6].Fine-grained models incorporate a richer degree of molecular details, but arestill limited to fairly simple flows because of computational cost [7–9].

Coarse-grained models represent the liquid microstructure in terms of one ormore conformation tensors; currently, these models are considered the mostappropriate for large-scale simulation of complex flows of complex fluids. Typ-ically, the conformation tensor obeys a hyperbolic partial differential trans-port equation. In polymer solutions and melts, this tensor represents the localexpectation value of the polymer stretch and orientation, e.g., gyration orbirefringence tensor. The elastic part of the stress is related to the confor-mation tensor through an algebraic equation [3–6]. Such models include most“classical” rate-type stress-based differential models (e.g., Oldroyd-B, PTT,Giesekus, etc.) [3–6].

Simulations of complex flows of complex fluids require solving simultaneouslythe hyperbolic transport equation of conformation (or rate-type equation forthe stress) together with the momentum and mass conservation equations; thisposes several numerical challenges. In particular, obtaining mesh-convergedsolutions in simple benchmark flows at high Weissenberg number (Wi, theproduct of characteristic strain rate and fluid relaxation time) is still consid-ered an open problem.

The Galerkin method is perhaps the most effective method for flows with

∗ Corresponding author, mp@rice.edu∗∗Corresponding author, behr@cats.rwth-aachen.de

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free surfaces and deformable boundaries. However, the Galerkin method isunstable in advection-dominated problems, and yields spurious oscillationsin the variable fields. Alternative methods have been developed to handleadvection-dominated as well as purely hyperbolic equations—e.g., Streamlineupwind/Petrov-Galerkin (SUPG) for high Reynolds number Newtonian flows[10] and viscoelastic flows [11], also Discontinuous Galerkin (DG) for viscoelas-tic flows [12].

When the Galerkin (or SUPG) method is applied to coupled partial differ-ential equations, the selection of the interpolating functions for the variousunknowns can be restricted by compatibility conditions—e.g., the Babuska-Brezzi condition in flows of incompressible Newtonian fluids [13, 14]. Somecompatibility conditions between the basis functions of velocity, pressure, ve-locity gradient, and conformation (or stress) must still be satisfied [15, 16]by current Galerkin-type methods for simulating viscoelastic flows—e.g., thestate-of-the-art DEVSS-TG/SUPG, which evolved from successive modifica-tions of the EVSS method [17–22] (see also reviews by Baaijens [23] and Owensand Phillips [24]).

These two key hurdles (handling advection-dominated problem and satisfyingcompatibility conditions) have been overcome in Newtonian flows by usingGalerkin/Least-Squares (GLS) methods [25–27]. Work on GLS methods ap-plied to Newtonian flows has shown that Streamline-upwind terms appearnaturally in the GLS form, that equal-order basis functions can be used forall fields (because the Least-Squares (LS) terms remove the compatibility con-dition), and that the resulting nonlinear algebraic equations yield a Jacobianmatrix that can be solved more easily with preconditioned Generalized Min-imum Residual Method (GMRES) (because the LS terms yield a positive-definite Jacobian component). Moreover, using equal-order basis functions forall fields allows “nodal” (rather than “elemental”) accounting, which speedsup greatly matrix operations on distributed memory parallel machines [28].

Weakly consistent forms of GLS method have been applied to viscoelasticflows. Behr [25] introduced a three-field (velocity-pressure-elastic stress) GLSmethod and studied the flow of an Oldroyd-B liquid in a 4-to-1 contraction.However, a detailed comparison between this method and other published re-sults was not performed, and the effect of the expression of the LS stabilizationcoefficient for the constitutive equation was not examined. This method hasbeen refined and extended more recently to improve consistency by recoveryof the velocity gradient as well as a more appropriate expression of the LSstabilization coefficient [29].

Fan et al. [30] independently introduced an incomplete GLS method for vis-coelastic flow and tested its performance in a flow between eccentric cylinders,flow around a sphere in a pipe, and flow around a cylinder in a channel. This

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method did not include terms due to the LS form of the momentum equation(because it degraded performance) and of the constitutive equation; therefore,the method of Fan et al. [30] is better characterized as a pressure-stabilizedSUPG method—see [31] for a description of pressure-stabilized methods forincompressible Newtonian flows.

This article presents a complete GLS method for computing flows of incom-pressible viscoelastic liquids modeled by conformation tensor or rate-typeequations. The flow equations are converted to a set of four first-order partialdifferential equations by representing explicitly the velocity gradient tensor(as in DEVSS-G). The GLS weighted residual equations include naturally theconsistent streamline upwinding for the advective terms in the conformationevolution equation (and in the momentum equation, although the presentationbelow is restricted to inertialess flows). The choice of basis functions for thefour unknown fields (velocity, pressure, velocity gradient, and conformation)is not restricted by compatibility conditions; here, the unknown fields are rep-resented by the simplest possible finite element basis functions—continuouspiecewise bilinear on quadrilateral elements. The method is termed GLS4 todistinguish it from the previous GLS3 [25, 29] method, in which the veloc-ity gradient was not represented explicitly. The accuracy and stability of themethod is demonstrated by using two benchmark problems—the flow in aplanar channel and the flow past a cylinder in a channel—for an Oldroyd-Bfluid.

It is worth noting that recent work [32–34] identified another source of insta-bility in low-order finite difference and finite element methods for computingviscoelastic flows—namely, the inability of low-order methods to capture expo-nentially growing profiles of conformation or elastic stress in regions of strongflow. Such instability can be avoided by using the logarithm of the conforma-tion tensor as field variable [32], which has the additional benefit of ensuringthat the conformation tensor is automatically positive definite everywhere inthe flow. The proposed GLS4 method does not address this source of insta-bility explicitly. However, as discussed in Ref. [32], the logarithmic change ofvariable is generally applicable to any finite element method (see, e.g., [34]);thus, it should be possible to combine the current GLS4 formulation with thelog-conformation method to improve the method further.

2 Governing Equations

The steady flow of an inertialess incompressible viscoelastic fluid, occupyinga spatial domain Ω, with boundary Γ is governed by the momentum andcontinuity equations,

4

∇ ·T=0 on Ω, (1)

∇ · v= 0 on Ω, (2)

where v is the liquid velocity, and T is the stress tensor, which can be de-composed into a constitutively undetermined isotropic contribution related toincompressibility, and viscous and elastic contributions,

T = −pI + τ + σ, (3)

respectively, where p is the pressure, I is the identity tensor, τ = 2ηs D is theviscous stress (usually due to solvent contribution), ηs is the solvent viscosity,and D is the rate-of-strain tensor, i.e., the symmetric part of the velocitygradient. In order to transform the equations of motion into a set of first-order partial differential equations (necessary for developing a consistent LSformulation for low-order elements), an additional variable L is introduced torepresent the velocity gradient,

L = ∇v− 1

tr I(∇ · v)I, (4)

where tr denotes trace.

The last term in Eq. (4) ensures that L remains traceless even in the finite-precision solution [22]; with this definition, D ≡ (L+LT )/2. In the Oldroyd-Bmodel, the elastic stress is related to the dimensionless conformation tensorM through a simple linear relationship σ = G(M − I), where G = ηp/λ isthe elastic modulus, ηp is the polymer contribution to the viscosity, and λis the relaxation time. The conformation tensor obeys a hyperbolic evolutionequation

λ5M + (M− I) = 0, (5)

where5M denotes an upper-convected derivative:

5M = v ·∇M− LT ·M−M · L. (6)

The equations governing the flow can be recast in dimensionless form as:

∇∗ ·T∗ =0, (7)

5

∇∗ · v∗ = 0, (8)

L∗ −∇∗v∗ +1

tr I(∇∗ · v∗)I=0, (9)

Wi5M + (M− I) =0, (10)

where v∗ = v/vc, p∗ = p/(ηvc/lc) and L∗ = L/(vc/lc) are dimensionless veloc-ity, pressure and interpolated traceless velocity gradient tensor, respectively.∇∗ = ∇ lc is the dimensionless gradient operator, vc is a characteristic veloc-ity and lc is a characteristic length. The dimensionless Weissenberg number isWi = λ(vc/lc). The dimensionless stress tensor T∗ is

T∗ = −p∗ I + β(L∗ + L∗ T ) +(1− β)

Wi(M− I), (11)

where β =ηs

ηs + ηp

is the viscosity ratio. Hereafter, all variables are dimen-

sionless and the (∗) is omitted for clarity.

Boundary conditions on the momentum equation are needed on the entireboundary Γ = Γg ∪ Γh. The essential and natural boundary conditions arerepresented as

v=g on Γg, (12)

n ·T=h on Γh, (13)

where g and h are given functions, and n is the outward unit vector normal tothe boundary. Because the equation of transport of conformation is hyperbolic,boundary conditions on the conformation tensor, represented by the tensor G,are imposed at inflow boundaries ΓG where v · n < 0,

M=G on ΓG . (14)

3 Four-Field Galerkin/Least-Squares Formulation (GLS4)

In this section, the GLS formulation of the governing equations (7)–(10) ispresented. The method is termed GLS4 because the equation set has fourbasic unknown fields—v, p, L and M. The basis (interpolation) and weightingfunction spaces are:

Shv = vh|vh ∈ [H1h(Ω)]nsd ,vh ≡ gh on Γg, (15)

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Vhv = vh|vh ∈ [H1h(Ω)]nsd ,vh ≡ 0 on Γg, (16)

Shp =Vh

p = ph|ph ∈ H1h(Ω), (17)

ShL =Vh

L = Lh|Lh ∈ [H1h(Ω)]n2sd, (18)

ShM = Mh|Mh ∈ [H1h(Ω)]ntc ,Mh ≡ G on ΓG, (19)

VhM = Mh|Mh ∈ [H1h(Ω)]ntc ,Mh ≡ 0 on ΓG, (20)

where H1h represents functions with square integrable first-order derivatives,nsd is the number of spatial dimensions and ntc = nsd(nsd +1)/2 is the numberof independent conformation tensor components. Bilinear piecewise continuousfunctions are used hereafter. The GLS4 formulation is: Find vh ∈ Sh

v, ph ∈ Shp ,

Lh ∈ ShL and Mh ∈ Sh

M such that:

∫

Ω

∇wh : ThdΩ +∫

Γh

wh · hhdΓ +

∫

Ω

τmom

∇qh − β∇ · (Eh + (Eh)T )− (1− β)

Wi∇ · Sh

︸ ︷︷ ︸A

·

[−∇ ·Th

]dΩ +

∫

Ω

qh(∇ · vh)dΩ +

∫

Ω

τcont(∇ ·wh)(∇ · vh)dΩ +

∫

Ω

Eh :[Lh −∇vh +

1

tr I(∇ · vh)I

]dΩ +

∫

Ω

τgradv

[Eh −∇wh +

1

tr I(∇ ·wh)I

]:[Lh −∇vh +

1

tr I(∇ · vh)I

]dΩ +

∫

Ω

Sh :[Wi(vh ·∇Mh − (Lh)T ·Mh −Mh · Lh) + (Mh − I)

]dΩ +

∫

Ω

τcons

[Wi (vh ·∇Sh − (Lh)T · Sh − Sh · Lh) + Sh

]:

[Wi (vh ·∇Mh − (Lh)T ·Mh −Mh · Lh) + (Mh − I)

]dΩ = 0,

∀qh ∈ Vhp , ∀wh ∈ Vh

v , ∀Eh ∈ VhL, ∀Sh ∈ Vh

M, (21)

where τmom, τcont, τgradv and τcons are the LS stabilization parameters for themomentum, continuity, interpolated traceless velocity gradient and constitu-tive equations, respectively. The underbraced term A is neglected at low Wibecause the (1/Wi) term grows large as Wi → 0, causing numerical problems.

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3.1 Design of the Stabilization Coefficients

The appropriate design of the four stabilization parameters—τmom, τcont, τgradv

and τcons—in Eq. (21) plays a crucial role in the performance of the method.

The τmom-term stabilizes the Galerkin form in advection-dominated flows, andalso removes the compatibility condition between velocity and pressure spaces.The parameter designed specifically for use with bilinear interpolations [31] isadapted here for the dimensionless system:

τmom =h2

4. (22)

where h is the dimensionless element length.

The τcont-term improves the convergence of non-linear solvers in advection-dominated problems. Hereafter, τcont = 0 because inertia is neglected.

The τgradv-term stabilizes Eq. (4); although the associated stabilization termis not strictly necessary, τgradv is taken here as τgradv = 1.

The τcons-term is introduced to stabilize the Galerkin form at high Wi, and tobypass the compatibility conditions between velocity and conformation spaces.No systematic derivation for τcons is available in the literature. However, thetransport equation of conformation can be viewed as an advection-generationequation, and considerable research has been done on stabilization param-eters for a simple advection-diffusion-generation equation [35–39]. Applyingthe definition proposed by Franca et al. [38], based on the convergence andstability analysis of advection-diffusion-generation equation, and extended byHauke [39], yields

τcons1 = 1, (23)

τcons2 =1

Wi‖Lh‖ , (24)

τcons3 =h

2Wi‖vh‖ . (25)

τcons1 and τcons2 are important in regions of the flow where generation is domi-nant, whereas τcons3 is important in advection-dominated regions. These threecontributions can be combined as:

τcons =

(1

τ rcons1

+1

τ rcons2

+1

τ rcons3

)−1/r

, (26)

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Hereafter, the switching parameter is set to r = 2 (see also Ref. [29]),

τcons =

1 + (Wi‖Lh‖)2 +

(2Wi‖vh‖

h

)2−1/2

. (27)

4 Numerical Results

The proposed GLS4 formulation is tested in flow in a planar channel and flowpast a cylinder in a channel. An analytical solution can be obtained in theformer case; in the latter, the numerical results from other state-of-the-artmethods are used for validation [20–22, 29, 34, 40]. The flow past a cylinderin a channel is a standard benchmark problem with desirable characteristicsof smooth boundaries, and poses several numerical challenges at high Wi dueto the formation of sharp boundary layers on the cylinder and in the wake.

4.1 Flow in a Planar Channel

Figure 1 shows a combination of Poiseuille flow (pushing liquid from left toright) and Couette flow (induced by the bottom wall dragging liquid from rightto left with velocity v0) in a planar channel of width w = 1 and length L = 4w.The flow of an Oldroyd-B fluid (β = 0.59) is simulated, and the results arecompared with the known analytical solution. The figure also shows velocityprofiles at the two open flow boundaries; both right and left ends of the chan-nel have respective inflow and outflow sections. A region ‘A’ (dotted area inFig. 1), which is 2w in length and centrally placed in the channel, is moni-tored for comparing numerical results with analytical solution; this sufficientlyeliminates the influences due to the boundary conditions. The problem setupclosely follows the numerical example employed by Xie and Pasquali [41]; theanalytical solution for velocity and conformation fields are:

vx =

[−∆p

2

w

L

[(y

w

)2

− y

w

]+

y

w− 1

]v0, vy = 0, (28)

Mxx = 1 + 2

(λ

dvx

dy

)2

, Mxy = λdvx

dy, Myy = 1, (29)

where ∆p = 50 is the differential pressure between the left and right bound-aries. Consequently, Wi = λ[∆p w/(2L) + 1](v0/w). The Dirichlet conditionsare imposed for velocity components on all boundaries, and the conformation

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tensor components are only specified at the corresponding inflows. The nu-merical results are obtained on four different uniform meshes—16×16, 24×24,32×32 and 64×64—followed by a node-by-node computation of the relativeerrors e = |(Numerical value - Analytical value)/(Analytical value) |×100%in region A. Figure 2 shows the maximum e in Myy (which has the highest eamong all unknown fields) versus the element size for Wi = 3, 5 and 7. Fromthe three curves the rate of mesh convergence is estimated to be 1.73, 1.63and 1.59, respectively. Because increase in Wi results in increased generation,subsequently forming steeper boundary layer close to the channel walls, therate of convergence is found to decrease. At Wi = 3, Xie and Pasquali [41]reported a rate of convergence of 1.89 using DEVSS-TG/SUPG method withbi-quadratic interpolation for velocity.

4.2 Flow Past a Cylinder in a Channel

The flow of an Oldroyd-B fluid past a cylinder in a rectangular channel hasbeen used as a standard benchmark problem to test several computationalmethods [20–22, 29, 34, 40]. For computational ease, the symmetry of theproblem is used and only half of the channel is simulated. Figure 3 shows theschematic of the problem, where Lu, Ld, Rc and w represent the upstreamlength, the downstream length, the cylinder radius, and the half channel width,respectively.

A no-slip boundary condition is imposed on the cylinder surface and chan-nel walls, and fully-developed flow conditions are assumed at the inflow andoutflow boundaries. Consequently:

vx = 1.5Q

w

(1− y2

w2

), vy = 0, (30)

Mxx =−3Q

wλ

y

w2, Mxy = Myx = 1− 2

(−3Qλy

w3

)2

, Myy = 1, (31)

where Q is the flow-rate. Whereas the velocity is imposed at both inflow andoutflow, the conformation tensor components are specified at the inflow only.At the symmetry line, n · T = 0 and vy = 0, where n is the unit vectornormal to the symmetry line. The computed drag on the cylinder fd has beentraditionally used to compare numerical methods,

fd = −2∫

S

e1n : T dS, (32)

where S represents the surface of the cylinder, n is the unit normal vector,

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and e1 is the unit vector in the x-direction.

4.2.1 Flow Past a Cylinder in a Channel: w/Rc = 2

In this case, w = 2, Rc = 1, Lu = −20, Ld = 20, Q = 2 and β = 0.59.Figure 4 shows the mesh M0 from which four systematically refined meshesare obtained; in these meshes, the elements are concentrated on the cylindersurface and in the wake along the symmetry line. The M1, M2, M3 and M4meshes are obtained by dividing every element side of M0 by 3, 4, 5 and 6,respectively. The number of elements and corresponding number of unknownsare listed in Table 1. This flow problem poses numerical challenges at high Wi;therefore, the maximum Wi up to which the numerical schemes converge hasbeen employed as a measure of robustness (but not necessarily accuracy). Forexample, using DEVSS-G/SUPG, Sun et al. [21] reported solutions up to Wi= 1.85, Fan et al. [30] using an incomplete GLS up to Wi = 1.05 and Hulsenet al. [34] using the log conformation up to Wi = 2.0; however, the accuracyof the solutions at Wi > 0.6 was not confirmed in these works.

Here, a sequence of flow states is computed by first-order arc-length contin-uation on Wi with automatic step control; the continuation terminates whenthe conformation tensor loses its positive-definiteness, which occurs at Wi ∼0.7. The positive-definiteness of M was not usually considered in past studies,with exception of the recent work of Hulsen et al. [34]. Figure 5 shows thedrag forces on the meshes M1, M2 and M3; a good agreement is found up toWi = 0.4 with the results reported by Hulsen et al. [34] and Sun et al. [21].Beyond that, the three methods show slight differences in the drag predictions,while following the same trend. Figures 6 and 7 show σxx versus s at Wi =0.6 and Wi = 0.7, respectively, where σxx = (ηp/λ)Mxx and s is defined as:0 < s < πRc on the cylinder and πRc < s < πRc + Ld −Rc in the wake alongthe symmetry line. At Wi < 0.6, a complete overlap is observed among theresults on the meshes M1, M2 and M3. In Fig. 6, σxx profiles computed withM1, M2 and M3 are overlapping in the wake; however, the result from M1shows underprediction on the cylinder, implying that refinement of M1 is notsufficient to capture the steep boundary layer on the cylinder. On the otherhand, in Fig. 7, differences are observed not only on the cylinder but also inthe wake flow. The figures also show the results reported by Hulsen et al. [34]at the corresponding Wi, and good agreement is found with results on M3.In previous works, the convergence of the stresses has been shown by com-paring stress profiles obtained on systematically refined meshes using p- andh-refinement [30, 34]. While overlap of the results demonstrates qualitativelymesh convergence, here, accuracy is measured more precisely by Richardsonextrapolation:

fd(h) = fd(0) + αhn, (33)

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where fd(0) is the drag for an infinitely refined mesh, n is the rate of meshconvergence and α is a constant. fd(0) is used to compute the relative errorse = |(fd(h) − fd(0))/fd(0)| × 100% in fd(h). Figure 8 shows fd predictionsfrom GLS4 and DEVSS-TG/SUPG along with the results presented by Hulsenet al. [34] at Wi = 0.6. The extrapolated values of fd for an infinitely refinedmesh are 117.979 and 117.778 for GSL4 and DEVSS-TG/SUPG, respectively.Thus, the GLS4 extrapolated results are within 0.2% of the values computedby high resolution finite volume (fd = 117.79 [40]), by pressure-stabilized finiteelements (fd = 117.78 [30]), by DEVSS-DG-Log conformation finite elements(fd = 117.77 [34]), and by DEVSS-TG/SUPG calculations. Figure 9 shows eversus h, and a mesh convergence rate of 2.39 is observed.

Similarly, Richardson extrapolation analysis is also performed for Mxx at apoint in the wake flow (x = 2; y = 0). The extrapolated value of Mxx is26.05 and the rate of mesh convergence is 1.74. Figure 10 shows e versus h forMxx. In all cases, results on M4 are also employed to obtain the extrapolatedvalues. Figure 11 shows the conformation contours at Wi = 0.7, in which, theformation of sharp boundary layers on the cylinder and along the symmetryline in the wake flow are observed. These boundary layers are difficult toresolve numerically, and the onset of oscillations in the conformation fields isobserved as the boundary layers grow at high Wi. The influence of the sharpboundary layer at high Wi is shown in Figs. 12–14, which plot M componentsalong line x = 2. At Wi = 0.5 and 0.6 a smooth profile for the M componentsis observed, whereas at Wi = 0.7 oscillations appear towards the symmetryline (y → 0). The maximum Wi attained in these simulations is slightly above0.7; beyond this, M loses its positive-definiteness.

A direct comparison of computational cost between GLS4 and DEVSS-TG/SU-PG is performed, while keeping the same number of degrees of freedom forconformation; the latter employs biquadratic interpolation functions for ve-locity, whereas bilinear for pressure, velocity gradient and conformation. Theresults at Wi = 0.6 on M2 are obtained from both methods and compara-ble accuracy is observed. Figure 15 shows σxx versus s along with the resultsof Hulsen et al. [34]. The GLS4 and DEVSS-TG/SUPG characteristics (num-ber of unknowns, time per Newton iteration and memory usage) are listed inTable 1.

Figure 16 shows a direct comparison of GLS4 and DEVSS-TG/SUPG withrespect to the number of degrees of freedom for conformation, and it can beobserved that GLS4 is ∼ 30% computationally faster and uses only 45% of thememory compared to DEVSS-TG/SUPG (for the same number of degrees offreedom for conformation).

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4.2.2 Flow Past a Cylinder in a Channel: w/Rc = 8

In this case, w = 8, Rc = 1, Lu = −40, Ld = 40, Q = 8 and β = 0.59.Following the same procedure as in the previous case, the M1, M2 and M3meshes are obtained by dividing every element side of the mesh M0 by 4, 5 and7, respectively. Figure 17 shows the mesh M0, and details of the subsequentmeshes are listed in Table 2. The drag on the cylinder from the three meshesare compared with the results reported by Sun et al. [21] in Fig. 18. For thiscase, a complete overlap of drag predictions from the three meshes is observed,and a good agreement with results of Sun et al. [21] is found up to Wi = 2.0.Moreover, the maximum Wi achieved in this simulation is ∼ 2.7. In Fig. 19,the contour plots for the M components are shown at Wi = 2.0. Figures 20and 21 show Mxx versus s (defined in 4.2.1) at Wi = 1.5 and 2.0, respectively.At Wi = 2.0, the streamwise normal conformation component Mxx has notyet converged in the wake. Figures 22–24 show the M components along theline x = 4 at Wi = 1.0, 1.5 and 2.0, respectively. It can be seen that significantoscillations appear towards the symmetry line (y → 0) at Wi = 2.0.

5 Conclusions

A complete four-field Galerkin/Least-Squares (GLS4) formulation to simulatethe flow of viscoelastic fluids is presented. The method successfully circum-vents the compatibility conditions associated with the multiple discrete un-known fields, thereby allowing equal-order polynomial interpolations for allvariables.

The formulation is presented for the equations governing the inertialess flow ofan Oldroyd-B (β = 0.59) fluid. The constitutive equation is written in terms ofthe conformation tensor, and can be easily extend to other constitutive models(e.g., Giesekus, FENE-P, FENE-CR, etc.). The set of governing equations—conservation of mass, momentum and the constitutive equation—are reducedto first-order by employing an interpolated traceless velocity gradient. Theequations are solved in a coupled way by using Newton’s method with analyt-ical Jacobian and a direct solver; the positive-definiteness of the conformationtensor is checked in all simulations. The method is evaluated for obtainingmesh-converged solutions in two benchmark problems.

The flow in a planar channel is computed on four meshes of increasing resolu-tion. The results are compared with the known analytical solution, and it isobserved that the GLS4 method is able to preserve the positive-definiteness ofM at high Wi. The mesh-convergence rate is also computed and found to becomparable to the state-of-the-art methods such as DEVSS.

13

The flow past a cylinder in a channel is computed on systematically refinedmeshes. Two different ratios of channel width to cylinder radius are used—2:1and 8:1. In both cases, at moderate Wi, the drag on the cylinder matcheswell with the state-of-the-art methods, and at high Wi, the results follow thesame trends. It is well known that in these problems, sharp boundary layersof M are formed on the cylinder and along the symmetry line in the wake,therefore mesh convergence for all components of M is analyzed. The onsetof the oscillations in the computed values of conformation are observed athigh Wi; this may be due to the non-optimal definition of the stabilizationparameters, or to the failure of the low order basis functions to capture ex-ponentially growing stress (conformation) profiles along streamlines in zonesof strong flow (see, e.g., Refs [33, 34]). On single processor machines, GLS4proves about 30% faster and 50% cheaper (memory-wise) than DEVSS whileproviding results of comparable accuracy. However, GLS4 is expected to scalebetter on distributed memory clusters because of nodal accounting of degreesof freedom and easier preconditioning of the GMRES solver [28].

In summary, this work demonstrates that GLS4 is on par with the state-of-the-art methods for solving viscoelastic fluid flows. The method is easy toimplement, because equal order polynomial interpolations can be used for allvariables. The method can further benefit from the latest developments in thisfield, e.g., from the logarithmic representation of the conformation tensor [34],which imposes the constraint of positive-definiteness. Other possibilities toimprove the performance of GLS4 may also be considered, including adjoint ofGLS or Variational Multiscale (VMS) [42] variant, and discontinuity captur-ing.

6 Acknowledgments

This work was supported by the National Science Foundation under awardCTS-ITR-0312764. Computational resources were provided by the Rice Com-putational Engineering Cluster, funded by NSF through an MRI award EIA-0116289, and by the Rice Terascale Cluster funded by NSF under grant EIA-0216467, Intel, and Hewlett-Packard.

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17

Table 1Flow past a cylinder in a channel, w/Rc = 2: Characteristics of the finite elementmeshes. εd is the % relative difference in the respective values from DEVSS andGLS4.Mesh Elements Unknowns Time per Newton Memory

iteration (s) usage (MB)

GLS4 DEVSS GLS4 DEVSS εd GLS4 DEVSS εd

M0 532 – – – – – – – –

M1 4788 49960 79102 220 322 31.7 608 1140 46.7

M2 8512 87890 139514 648 934 30.6 1410 2647 46.7

M3 13300 136460 216950 1460 2144 31.9 2720 5122 46.9

M4 19152 195670 311410 2914 – – 4650 – –

Table 2Flow past a cylinder in a channel, w/Rc = 8: Characteristics of the finite elementmeshes.Mesh Elements Unknowns Time per Newton Memory

iteration (s) usage (MB)

M0 250 – – –

M1 4000 41690 198 396

M2 6250 64610 440 863

M3 12250 125450 1505 1859

Fig. 1. Schematic of a flow in a planar channel with w/L = 1/4. The top wall is keptfixed, the bottom wall is moving from right to left at v0 and a differential pressureis applied between the left and right walls.

0.015 0.02 0.03 0.04 0.05 0.06

0.1

0.5

1.0

2.0

3.04.0

h

Max

imum

rel

ativ

e er

ror

for

Myy

(%)

log = 1.59 log + 2.39

log = 1.73 log + 2.01

log = 1.63 log + 2.23

Wi = 3

Wi = 5

Wi = 7

hWi = 3Wi = 3

h

hhe

e

e

Fig. 2. Mesh-convergence rate for a planar channel flow at different Wi. The slopeof the curves gives the rate of convergence with mesh refinement.

Channel wall

Symmetry line

Cylinder

OutflowInflow

(Lu,0) (Ld,0)

(Lu,w) (Ld,w)

Rc

x

y

Q

Fig. 3. Geometry of a flow past a cylinder in a half channel.

(a)

(b)

Fig. 4. Flow past a cylinder in a channel, w/Rc = 2: Finite element mesh M0 (a)complete domain (b) detail of the mesh from x = −2 to x = 2.

0 0.2 0.4 0.6 0.8 1

115

120

125

130

135

Wi

Dra

g fo

rce

0.5 0.6 0.7115

117

119

121

M1

M2

M3

Sun et al.

Hulsen et al.

M4

Fig. 5. Flow past a cylinder in a channel, w/Rc = 2: Drag force on the cylinder versusWi. The GLS4 results for the four meshes (M1, M2, M3 and M4) are compared withthe results presented by Sun et al. [21] and Hulsen et al. [34]. Inset: detail of thedrag force at high Wi. • represents the drag force on M4 at Wi = 0.6. At Wi =0.6, the extrapolated value of the drag force is 117.979, which is within 0.2% of thevalues reported in Refs. [30, 34, 40].

0 1 2 3 4 5 6−20

0

20

40

60

80

100

s

σ xx

M1M2M3Hulsen et al.

Fig. 6. Flow past a cylinder in a channel, w/Rc = 2: σxx on the cylinder and on thesymmetry line in the wake at Wi = 0.6. from Hulsen et al. [34].

0 1 2 3 4 5 6−20

0

20

40

60

80

100

120

s

σ xx

M1M2M3Hulsen et al.

Fig. 7. Flow past a cylinder in a channel, w/Rc = 2: σxx on the cylinder and on thesymmetry line in the wake at Wi = 0.7. from Hulsen et al. [34].

0.02 0.025 0.03 0.035 0.04 115

116

117

118

119

h

Dra

g fo

rce

GLS4DEVSS−TG/SUPG Hulsen et al.

Fig. 8. Flow past a cylinder in a channel, w/Rc = 2: Drag force at Wi = 0.6 forGLS4 and DEVSS-TG/SUPG for all meshes; dashed line represents the drag forcereported by Hulsen et al. [34] on their finest mesh.

0.02 0.025 0.03 0.035 0.04 0.045

0.2

0.4

0.6

0.8

1.0

h

Max

imum

rel

ativ

e er

ror

for

f d(%)

log = 2.39 log + 3.32e h

Fig. 9. Flow past a cylinder in a channel, w/Rc = 2: Mesh-convergence rate of thedrag force at Wi = 0.6.

0.02 0.025 0.03 0.035 0.04 0.045

2

3

4

5

6

7

8

h

Max

imum

rel

ativ

e er

ror

for

Mxx

(%)

log = 1.74 log + 2.7e h

Fig. 10. Flow past a cylinder in a channel, w/Rc = 2: Mesh-convergence rate of Mxx

at a point in the wake flow (x = 2; y = 0) at Wi = 0.6.

-4 -2 0 2 4X

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Y

Mxx127.3119.4111.4103.5

95.587.579.671.663.739.833.823.915.9

8.02.81.2

(a)

-4 -2 0 2 4X

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Y

Mxy41.337.834.330.727.220.216.7

9.66.12.60.6

-0.4-7.9

-18.5-22.0-29.1

(b)

-4 -2 0 2 4X

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Y

Myy33.431.329.227.125.022.920.816.612.410.3

3.91.81.31.00.9

(c)

Fig. 11. (color online). Flow past a cylinder in a channel, w/Rc = 2: (a) Mxx (b)Mxy and (c) Myy contours at Wi = 0.7 on mesh M2.

0 0.5 1 1.5 20

10

20

30

40

50

60

y

Mxx

0 0.1 0.20

20

40

60Wi = 0.5Wi = 0.6Wi = 0.7

Fig. 12. Flow past a cylinder in a channel, w/Rc = 2: Mxx along line x = 2 on meshM3. Inset: detail of Mxx near the centerline (y = 0).

0 0.5 1 1.5 2−4

−3

−2

−1

0

1

2

y

Mxy

0 0.1 0.2−0.5

0

0.5

1

1.5

Wi = 0.5Wi = 0.6Wi = 0.7

Fig. 13. Flow past a cylinder in a channel, w/Rc = 2: Mxy along line x = 2 on meshM3. Inset: detail of Mxy near the centerline (y = 0).

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

y

Myy

0 0.1 0.20.45

0.5

0.55

0.6

0.65

Wi = 0.5Wi = 0.6Wi = 0.7

Fig. 14. Flow past a cylinder in a channel, w/Rc = 2: Myy along line x = 2 for M3.Inset: detail of Myy near the centerline (y = 0).

0 1 2 3 4 5 6−20

0

20

40

60

80

100

s

σ xxGLS4 (M2)DVESS−TG/SUPG (M2)Hulsen et al.

Fig. 15. Flow past a cylinder in a channel, w/Rc = 2: σxx on the cylinder and on thesymmetry line at Wi = 0.6. The GLS4 and DEVSS-TG/SUPG results are obtainedfor M2. from Hulsen et al. [34].

5000 10000 15000 20000

500

1000

2000

3000

Tim

e pe

r N

ewto

n ite

r. (

s)

2000

4000

5000

Number of elements

1000

2000

4000

5000

Mem

ory

usag

e (M

B)

16000 30000 45000 60000

DEVSSGLS4

GLS4DEVSS

Degrees of freedom for conformation

Fig. 16. Direct comparison of GLS4 and DEVSS-TG/SUPG with respect to thenumber of elements (bottom axis) and to the number of degrees of freedom forconformation (top axis). The left and right axes represent the time per Newtoniteration (s) and memory usage (MB), respectively. A frontal solver is used in bothsimulations.

(a)

(b)

Fig. 17. Flow past a cylinder in a channel, w/Rc = 8: Finite element mesh M0 (a)complete domain (b) detail of the mesh from x = −4 to x = 4.

0.0 0.5 1.0 1.5 2.0 2.5 3.015.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

Wi

Dra

g fo

rce

1.8 2.0 2.2 2.4 2.616.5

17.0

17.5

18.0

18.5

19.0

M1M2M3Sun et. al

Fig. 18. Flow past a cylinder in a channel, w/Rc = 8: Drag force on the threemeshes. The dotted curve is obtained from Sun et al. [21]. Inset: detail of the dragforce at high Wi

-15 -10 -5 0 5 10 15X

0

1

2

3

4

5

6

7

8

9Y

Mxx 48.945.842.839.836.833.730.727.724.621.618.615.512.5

6.43.20.9

(a)

-15 -10 -5 0 5 10 15X

0

1

2

3

4

5

6

7

8

9

Y

Mxy 12.9

9.98.43.92.41.00.70.5

-0.1-0.5-2.0-3.5-5.0-6.5-8.0-9.5

(b)

-15 -10 -5 0 5 10 15X

0

1

2

3

4

5

6

7

8

9

Y

Myy26.825.121.820.218.515.213.611.9

8.67.05.33.62.01.01.00.8

(c)

Fig. 19. (color online). Flow past a cylinder in a channel, w/Rc = 8: (a) Mxx (b)Mxy and (c) Myy contours at Wi = 2.0 on mesh M2.

0 1 2 3 4 5 60

5

10

15

20

25

s

Mxx

M1M2M3

Fig. 20. Flow past a cylinder in a channel, w/Rc = 8: Mxx on the cylinder and alongthe symmetry line in the wake at Wi = 1.5.

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

s

Mxx

M1M2M3

Fig. 21. Flow past a cylinder in a channel, w/Rc = 8. Mxx on the cylinder and alongthe symmetry line in the wake at Wi = 2.0.

0 2 4 6 80

10

20

30

40

50

60

y

Mxx

Wi = 1.0Wi = 1.5Wi = 2.0

0 0.1 0.2 0.3 0.40

10

20

30

40

50

60

Fig. 22. Flow past a cylinder in a channel, w/Rc = 8: Mxx along line x = 4 on meshM3.

0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

y

Mxy

Wi = 1.0Wi = 1.5Wi = 2.0

0 0.2 0.4−1

−0.5

0

0.5

1

Fig. 23. Flow past a cylinder in a channel, w/Rc = 8: Mxy along line x = 4 on meshM3.

0 2 4 6 80.4

0.6

0.8

1

1.2

1.4

y

Myy

Wi = 1.0Wi = 1.5Wi = 2.0

0 0.2 0.40.5

0.55

0.6

0.65

0.7

0.75

Fig. 24. Flow past a cylinder in a channel, w/Rc = 8: Myy along line x = 4 on meshM3.