finite element analysis of planar 4:1 contraction flow with the tensor … · 2005-01-11 · finite...

Korea-Australia Rheology JournalVol. 16, No. 4, December 2004 pp. 183-191

Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmicformulation of differential constitutive equations

Youngdon Kwon*School of Applied Chemistry and Chemical Engineering, Sungkyunkwan University,

Suwon, Kyunggi-do 440-746, Korea(Received November 25, 2004; final revision received November 30, 2004)

Abstract

High Deborah or Weissenberg number problems in viscoelastic flow modeling have been known formidablydifficult even in the inertialess limit. There exists almost no result that shows satisfactory accuracy andproper mesh convergence at the same time. However recently, quite a breakthrough seems to have beenmade in this field of computational rheology. So called matrix-logarithm (here we name it tensor-logarithm)formulation of the viscoelastic constitutive equations originally written in terms of the conformation tensorhas been suggested by Fattal and Kupferman (2004) and its finite element implementation has been firstpresented by Hulsen (2004). Both the works have reported almost unbounded convergence limit in solvingtwo benchmark problems. This new formulation incorporates proper polynomial interpolations of the log-arithm for the variables that exhibit steep exponential dependence near stagnation points, and it also strictlypreserves the positive definiteness of the conformation tensor. In this study, we present an alternative pro-cedure for deriving the tensor-logarithmic representation of the differential constitutive equations and pro-vide a numerical example with the Leonov model in 4:1 planar contraction flows. Dramatic improvementof the computational algorithm with stable convergence has been demonstrated and it seems that there existsappropriate mesh convergence even though this conclusion requires further study. It is thought that this newformalism will work only for a few differential constitutive equations proven globally stable. Thus the math-ematical stability criteria perhaps play an important role on the choice and development of the suitable con-stitutive equations. In this respect, the Leonov viscoelastic model is quite feasible and becomes moreessential since it has been proven globally stable and it offers the simplest form in the tensor-logarithmicformulation.Keywords : high Deborah number, tensor-logarithm, constitutive equation, stability, Leonov model, con-

traction flow

1. Introduction

Numerical modeling of viscoelastic flow around a sharpcorner has posed great challenge in computational non-Newtonian fluid dynamics, because it is extremely difficultto obtain solutions in the case of high Deborah (or Weis-senberg) number flows. For viscoelastic fluid, the Deborahnumber defined as the ratio of liquid relaxation time tocharacteristic time of the flow expresses strength of elas-ticity accumulated in the flow. There are several reasonsconjectured for the failure of the numerical convergence athigh Deborah number in the planar 4:1 contraction flowmodeling, the representative benchmark flow problem withsingular geometry. It has been recognized that computa-tional algorithm certainly plays a significant role in deter-

mining the stability of the numerical scheme. However theapplied viscoelastic constitutive equation is thought to bealso responsible for the numerical degradation. Nowadaysone has come to a conclusion that selection of an appro-priate constitutive equation constitutes a very crucial stepalthough implementing a suitable numerical technique isstill important for successful discrete modeling of non-Newtonian flows (Leonov, 1995).

Irrespective of extensive stability analysis performed forthe constitutive equations as well as astonishing develop-ment of computational algorithm, it is indeed frustratingthat even the viscoelastic model proven globally stableshows quite unsatisfactory limitation of computational con-vergence whatever numerical algorithm may be applied.Recently a breakthrough in this field seems to have beenmade by Fattal and Kupferman (2004). They have sug-gested simple tensor-logarithmic transform of the confor-mation tensor in the differential viscoelastic constitutive*Corresponding author: [email protected]

© 2004 by The Korean Society of Rheology

Korea-Australia Rheology Journal December 2004 Vol. 16, No. 4 183

Youngdon Kwon

equations and proved existence of such representation fordifferential constitutive models. Their idea is based on thefollowing. The conformation tensor shows exponentialdependence in regions of high strain rate for high Deborahnumber flows, and the polynomial interpolation of thisbehavior is inappropriate. If one employs its logarithmicfunction instead of the conformation tensor, the multipli-cation is substituted by the addition operation, and thus thepolynomial interpolation of this logarithm function isthought to be proper and more effective in the spatial res-olution of steep stress gradients.

In addition, this new formulation may grant anotherimportant characteristic. It is well known that the positive-definiteness of the conformation tensor is crucial for well-posedness of its evolution equation (Dupret and Marchal,1986; Kwon and Leonov, 1995). In reality, even for theconstitutive equations proven Hadamard stable (Hadamardstability means the well-posedness of constitutive equa-tions under short and high frequency wave disturbance),violation of this positive-definiteness is frequentlyobserved (Dupret et al., 1985; Lee et al., 2004) probablydue to the spatial discretization error. However if we intro-duce this tensor-logarithm for the conformation tensor, thepositive-definiteness is strictly preserved in any compu-tational scheme and at any value of the Deborah number.Seemingly limitless convergence of the approximationscheme has already been reported in numerical compu-tation of benchmark viscoelastic flow problems such as lid-driven cavity flow (Fattal and Kupferman, 2004) and flowpast a cylinder in a straight pipe (Hulsen, 2004).

At this point, it is not certain that this new formalismbased on tensor-logarithm transform can successfully workfor every differential constitutive equation that can be writ-ten in the conformation tensor notation. Even if the exist-ence of the tensor-logarithm representation for differentialviscoelastic models has been proven (Fattal and Kupfer-man, 2004), it is not at all obvious that this sole refor-mulation stabilizes the computational scheme for any set ofequations. In the current author’s opinion, mathematicalstability analysis of constitutive equations may providequite feasible insight into optimal choice of viscoelasticmodels. Especially for Maxwell-type differential and time-strain separable single integral constitutive equations, onecan find extensive and detailed results on mathematical sta-bility (Joseph 1990; Kwon 2002; Kwon and Leonov 1995).

In this study we first present an alternative procedureobtaining the tensor-logarithmic formulation of the vis-coelastic equations originally suggested by Fattal and Kup-ferman (2004). Then this procedure is applied to the 2Ddescription of the Leonov model, which becomes moreimportant and also the simplest especially in this formal-ism. We consider the isothermal incompressible inertialess4:1 contraction flow problem to test this new represen-tation in high Deborah number flow modeling.

2. Tensor-logarithmic formulation of the consti-tutive equation

Most of differential viscoelastic constitutive equationscan be written into the following quite general form:

(1)

Here c is the conformation tensor that describes the mac-romolecular conformation under flow for the constitutivemodels based on molecular kinetic theory, v is the velocity,

is the material time derivative of c, is

the usual gradient operator in tensor calculus, is the upper convected time derivative, and θ is the

relaxation time. ψ is usually represented as the polynomialof c with coefficients possibly dependent on the invariantsof c or whose specific form has to begiven by the constitutive equation employed. For example,the upper convected Maxwell model can be rewritten asEq.(1) with ψ = c − δ and its extra-stress takes the form ofτ = G(c − δ) with G as the modulus. More examples can befound in Kwon and Leonov (1995) and Lee et al. (2004).The conformation tensor c reduces to the unit tensor δ inthe rest state and this condition also serves as the initialcondition in the start-up flow situation. In contrast, for theLeonov model that has been derived from the irreversiblethermodynamics and thus is not endowed with molecularconcept, the c-tensor explains the accumulated elasticstrain in the Finger measure during flow. In the asymptoticlimit of where the material exhibits purely elasticbehavior, it becomes the total Finger strain tensor.

The essential idea presented by Fattal and Kupferman(2004) in reformulating the constitutive equations is thetensor-logarithmic transformation of c as follows:

h = log c. (2)

Here the logarithm operates as the isotropic tensor func-tion, which implies the identical set of principal axes forboth c and h. In the case of the Leonov model, this hbecomes another measure of elastic strain, that is, twice theHencky elastic strain.

In this section, a simple alternative derivation of the ten-sor-logarithmic transform for Eq.(1) (formulation of theconstitutive equation in terms of h) originally given by Fat-tal and Kupferman (2004), is suggested. For this purposewe adopt following notations:ci : eigenvalues of c, hi : eigenvalues of h, ni : correspondingunit eigenvectors.

Here due to the isotropic function relation, c and h havethe same set of eigenvectors. The characteristic relation forc is written as

c · ni = cini (no sum). (3)

Differentiation and scalar product with another eigenvector

c· ∇ vT– c c– ∇ v⋅ ⋅ 1θ---ψ+ 0=

c· dcdt----- ∂c

∂t----- v ∇ c⋅+= = ∇

c· ∇ vT– c⋅c– ∇ v⋅

e ∇ v ∇ vT+( ) 2⁄ ,=

θ ∞→

184 Korea-Australia Rheology Journal

Finite element analysis of planar 4:1 contraction flow with the tensor-logarithmic formulation of differential constitutive equations

yield

(no sum). (4)

We may rewrite it as

(no sum). (5)

For h-tensor, an equivalent relation is readily obtained as

(no sum). (6)

Due to Eq.(2), one has and thus (no sum)

holds. With this and Eqs.(5), Eq.(6) represents the fol-lowing resultant relation for the time variation of h-tensor:

(no sum). (7)

When one denotes Q as the orthogonal matrix that trans-forms c or h into the corresponding diagonal form or in principal axes, the following simple relations hold:

(8)

The orthogonal matrix Q contains the eigenvectors ni as itscolumn vectors. In principle, all the components of ni andeigenvalues hi can be obtained by solving the characteristicequation for h, even though obtaining its analytic expressionmay become a formidable task in 3D flow domain. If we sub-stitute the constitutive equation (1) for in Eqs.(7) that doesnot contain the derivative of c any more and utilize the solu-tion of the characteristic equation for h, then Eqs.(7) even-tually represent the desired constitutive relation written interms of h. For (i, j) such as (1,1), (2,2), (3,3), (1,2), (2,3) and(1,3), Eqs.(7) yield exactly 6 differential equations for hij.

Above derivation holds in every flow situation exceptwhen there locally exists 2D or 3D isotropy. However inthis case, the relation yields simpler equation. Without lossof generality, we assume that ni and are the vec-tors perpendicular to the cylindrical symmetry axis (i.e. ni

and nj are contained in the plane of isotropy). Then ci = cj,hi = hj, and ni and nj are indeterminate except the fact thatthey consist in the isotropic plane. Since the asymptotic

relation is valid, Eqs.(7) combine to

Using the representation for ni in terms of hi

and hij and applying appropriate asymptotic relations, wepossibly obtain corresponding constitutive equations in h-tensor form. Since the original constitutive model (1) in thec formulation is known to be regular for ci = cj, such trans-formation into h-form certainly exists. In the followingsections we present illustrative example in simple planar2D flow with numerical results for the Leonov constitutiveequation.

3. Formulation in 2D flow

In the case of 2D planar flow, the direction orthogonal tothe computation domain automatically constitutes one fixedeigenvector. Here we designate the axes of flow domain asx1 and x2. When we write the first eigenvector as the fol-lowing, the second eigenvector is determined accordingly:

and then with (9)

Solving the characteristic equation of h, one obtains

(10)

In the rest state, all hij and the eigenvalues vanish, andtherefore the components of the eigenvector become inde-terminate as can be seen from the equations. In Eqs.(10),the two characteristic values may be interchanged, how-ever the resultant constitutive equations remain invariable.

In this simple 2D consideration, Eqs.(7) with (9) rep-resent 3 equations as

(11)

Substitution of the specific constitutive equations for with Eqs.(10) and transformation rule (8) eventually yieldsthe desired set of viscoelastic field equations in 2D rep-resented in h-tensor form.

From now on, only the Leonov constitutive equation(Leonov, 1976) is considered. Its one simple version hasthe following form for the evolution equation (1) and thestress relation:

nj c· ni⋅ ⋅ c· iδij ci cj–( )n· i nj⋅+=

i) c· i ni c· ni⋅ ⋅= when i j=

ii) n· i nj⋅ 1ci cj–------------= nj c· ni ⋅ ⋅ when i j≠

nj h·

ni⋅ ⋅ h·

iδij hi hj–( )n· i nj⋅+=

ci ehi= h

·i

1ci---c· i=

i) ni h·

ni⋅ ⋅ 1ci---c· i

1ci---ni c· ni⋅ ⋅= = when i j=

ii) ni h·

nj⋅ ⋅ hj hi–( )= n· j ni⋅hi hj–ci cj–------------= ni c· nj ⋅ ⋅ when i j≠

c h

c Q exp h( ) QT⋅ ⋅= exp h( ) ce

h1 0 0

0 eh2 0

0 0 eh3

.= =,

c·

nj i j≠( )

hi hj–ci cj–------------

hj hi→lim 1

ci---= ni h

·nj⋅ ⋅

1ci---ni c· nj.⋅ ⋅=

n1n1

n2

,= n2n– 2

n1

= n12 n2

2+ 1.=

h112--- h11 h22 h11 h22–( )2 4h12

2++ +[ ] ,=

h212--- h11 h22 h11 h22–( )2 4h12

2+–+[ ] ,=

n12 h12

2

h1 h11–( )2 h122+

------------------------------------,= n22 h1 h11–( )2

h1 h11–( )2 h122+

------------------------------------,=

n1n2h12 h1 h11–( )

h1 h11–( )2 h122+

------------------------------------.=

n12h

·11 2n1n2h

·12 n2

2h·

22+ + 1c1---- n1

2c·11 2n1n2c·12 n22c·22+ +( ),=

n22h

·11 2n1– n2h

·12 n1

2h·

22+ 1c2---- n2

2c·11 2n1– n2c·12 n12c·22+( ),=

n1– n2h·

11 n12 n2

2–( )h·

12 n1n2h·

22+ +

h1 h2–c1 c2–-------------- n1n2c·11– n1

2 n22–( )c·12 n1n2c·22+ +[ ] .=

c· ij


Youngdon Kwon

(12)

Here I1 = tr c and I2 = tr c−1 are basic invariants of c, andthey coincide in planar flows. In order to rigorously exam-ine the computational robustness of the formulation we donot include any retardation (Newtonian viscous) term thatbestows stabilizing effect on the numerical scheme by aug-menting the elliptic character in equations of motion. Theextra-stress tensor is obtained from the elastic potential Wbased on the Murnaghan’s relation. Since the extra-stress isinvariant under the addition of arbitrary isotropic terms,

when we present our numerical results we use ×

(c − δ) instead in order to set 0 for the stress in the reststate. In addition to the linear viscoelastic parameters, itcontains 2 nonlinear constants m and n, which can bedetermined from simple shear and uniaxial extensionalflow experiments. The value of the parameter m does nothave any effect on the flow characteristics in 2D situation,since two invariants are equal.

These Leonov equations contain one essential featurethat becomes especially important and practically useful incurrent tensor-logarithmic formulation as follows:

det c = 1. (13)

It can be seen as the first integral of the evolution equation(1), and it implies the liquid is incompressible (this relationis derived from the fact that volume change during the flowis completely recoverable). In 2D it reduces to = 1. Since the logarithmic function transforms the multi-plication into the operation of addition, this incompress-ibility relation (13) becomes

tr h = 0, (14)

which dramatically simplifies all the relations derivedabove. In addition, it gives another advantage in com-putation. For example, in 3D due to onecan eliminate one variable (and accordingly one equation)from the set of governing equations. In this 2D analysis,we remove h22 from the set, and thus the viscoelasticmodel contains only 2 additional unknowns such as h11

and h12. Based on our numerical scheme explained after-wards, the computation time has diminished to a half.

Now employing Eq.(14), one can get simplified versionof Eqs.(10) valid only for the Leonov model as

(15)

Then the evolution relation (11) for hij becomes

(16)

Subtraction of the first two equations yields

that merely illustrates equivalence of

these two equations (here we used and Thus we can

safely remove one equation from further consideration.When we insert all the necessary relations (15) and (1)

with (12) and then solve for and , following finalform of viscoelastic equations in h form results:

(17)

Here again and the initial condition ish = 0. Together with the equations of continuity andmotion, they constitute a complete set to describe iso-thermal incompressible viscoelastic flow in 2D. Howeverdue to the form presented in Eqs.(17), artificial numericaldifficulty may arise. Except for the case of rest state, dur-ing flow vanishing of the eigenvalue h (it means h = 0)may occur locally, e.g. along the centerline in the fullydeveloped Poiseuille flow through a straight pipe. Then the

coefficients of and hij become apparently indetermi-

nate. However proper introduction of asymptotic relationfor vanishing h results in

when (18)

When one employs a Newton’s method in solving the non-linear equations, partial derivatives of the coefficients areto be specified. For example, denoting the coefficient of

ψ 12--- I1

I2----

m

c2 I2 I1–3

------------c δ–+ ,= τ G

I1

3----

n

c,=

W 3G2 n 1+( )------------------ I1

3----

n 1+

1– .=

τ GI1

3----

n

=

c11c22 c122–

h11 h22 h33 0=+ +

h1 h≡ h112 h12

2+ ,= h2 h– ,=

n12 1

2--- 1 h11

h------+

= , n22 1

2--- 1 h11

h------–

= , n1n2h12

2h------.=

n12 n2

2–( )h·

11 2n1n2h·

12+ e h– n12c·11 2n1n2c·12 n2

2c·22+ +( ),=

n12 n2

2–( )h·

11 2n1n2h·

12+ eh n22c·11 2n1– n2c·12 n1

2c·22+( ),=

2n1n2h·

11 n12 n2

2–( )– h·

122h

eh e h––--------------- n1n2c·11 n1

2 n22–( )c·12 n1n2c·22– ] .–[=

0 c22c·11 2c12 ×–=

c·12 c11c·22+ ddt---- det c( )=

c11 ehn12 e h– n2

2+= ,c22 e h– n1

2 ehn22+= , c12 eh e h–+( )n1n2).=

h·

11 h·

12

∂h11

∂t---------- v1

∂h11

∂x1---------- v2

∂h11

∂x2---------- 2

h2----- h11

2 h122heh e h–+

eh e h––----------------+

∂v1

∂x1-------–+ +

h12h11

h2------ 1 heh e h–+

eh e h––----------------–

1+∂v1

∂x2-------– h12

h11

h2------ 1 heh e h–+

eh e h––----------------–

1–∂v2

∂x1-------–

1θ---eh e h––

2h---------------h11 0,=+

∂h12

∂t---------- v1

∂h12

∂x1---------- v2

∂h12

∂x2---------- 2h11h12

h2----------------- 1 h– eh e h–+

eh e h––----------------

∂v1

∂x1-------–+ +

1h2----- h12

2 h112+ heh e h–+

eh e h––----------------

h11–∂v1

∂x2-------–

1h2----- h12

2 h112+ heh e h–+

eh e h––----------------

h11+–∂v2

∂x1------- 1

θ---eh e h––

2h---------------h12 0.=+

h h112 h12

2+ ,=

∂vi

∂xj------

∂h11

∂t---------- v1

∂h11

∂x1---------- v2

∂h11

∂x2---------- 2

∂v1

∂x1-------– h12

∂v1

∂x2------- h12

∂v2

∂x1------- 1

θ---h11+ + 0,≈–+ +

∂h12

∂t---------- v1

∂h12

∂x1---------- v2

∂h12

∂x2---------- 1 h11–( )

∂v1

∂x2-------– 1 h11+( )

∂v2

∂x1------- 1

θ---h12+ 0,≈–+ +

h 0.≈



in the first of Eqs.(17) as φ(h11, h12) we have

and its limit behavior becomes when

We want to make a last additional remark on the h-tensorformulation of the Leonov model. Even in the simplified2D case, the constitutive relation appears to be quite awk-ward in its tensor-logarithmic form (17), however theLeonov equation suggests the simplest of all nonlinear vis-coelastic differential equations (and thus possibly the sim-plest of all viscoelastic constitutive models) due to theexistence of the first integral, Eq.(13) or (14). Hence in thecurrent author’s opinion, this viscoelastic model forms themost efficient and practical equations in modeling espe-cially 3D viscoelastic flows, where we may have to applyfor the eigenvalues the cubic formula, the analyticalexpression of the roots for a cubic polynomial.

4. Numerical scheme

We investigate planar 4:1 abrupt contraction flow with

centerline symmetry. The flow geometry and boundaryconditions are illustrated in Fig. 1. We apply no-slipboundary condition at the wall and specify symmetric nat-ural boundary on the centerline. To remove indeterminacyof pressure, we also set the pressure variable at the exitwall. Fully developed flow conditions are applied for thevelocity and h tensor at the inlet but only for the velocityat the outlet. When we denote the half width of the down-stream channel as H0, we set the length of the downstreamchannel as 15H0 and the length of the reservoir as 20H0.Even though the downstream channel length is rather shortto achieve fully developed flow, in order to alleviate com-putational burden we simply choose this flow geometry,and it seems that the length of the downstream channeldoes not have any effect on the stability of the compu-

∂v1

∂x1-------

∂φ∂h11---------- 2

h11h122

h4---------------- 2 heh e h–+

eh eh–----------------– 2h

eh e h––---------------

2– ,=

∂φ∂h11---------- 0≈ h 0.≈

: essential boundary condition for velocity (v = 0) : essential boundary condition for velocity : symmetric boundary condition : essential boundary condition for velocity and h

Fig. 1. Boundary conditions of the 4:1 contraction flow problem.

∂Ω1∂Ω2∂Ω3∂Ω4 Fig. 2. Partial view of the (a) coarse and (b) fine meshes

employed in this study.


Youngdon Kwon

tational procedure.With the standard Galerkin formulation adopted as basic

computational framework, either streamline-upwinding(SU) or streamline-upwind/Petrov-Galerkin (SUPG) methodas well as discrete elastic viscous stress splitting (DEVSS)algorithm is implemented in order to build relatively robustnumerical scheme at high Deborah number flows. Theupwinding algorithm developed by Gupta (1997) has beenapplied. The SU method additionally adds to the originalGalerkin formulation an inconsistent upwind term thatgives only the first order accuracy in spatial discretization.However the SUPG scheme is consistent and endows asecond order accuracy. Even though we report in this paperresults based on both upwinding algorithms, we are mainlyconcerned with results under the SUPG method. Two typesof meshes are employed for the computation, and they areillustrated in Fig. 2. The coarse mesh (Fig. 2a) has thesmallest corner element with the side length of 0.1H0,while the fine mesh (Fig. 2b) contains the smallest onewith the side length of 0.05H0. Corresponding mesh detailsare given in Table 1. As has been mentioned beforehand,in the h-formulation we have fewer unknowns.

Linear for pressure and strain rate and quadratic inter-polation for velocity and h-tensor are applied for spatialcontinuation of the variables. In this work, we only con-sider steady inertialess flow of the isothermal incompress-ible liquid. In order to mimic dimensionless formulation,we simply assign unit values for G and θ and adjust theDeborah number by the variation of the average flow rate.The Deborah number in this contraction flow is usuallydefined as

(19)

where U is the average downstream velocity. Also n = 0.1is set to guarantee the mathematical stability even in stresspredefined flow history (Kwon and Leonov, 1995) (e.g. inthe situation where one assigns traction boundary condi-tions at the inlet and outlet), whereas n = 0 corresponds tothe case of neo-Hookean potential in finite elasticity.

In order to solve the large nonlinear system of equationsintroduced, the Newton iteration is used in linearizing thesystem and the frontal elimination method is implementedfor solution update. As an estimation measure to determine

the solution convergence, the L∞ norm scaled with themaximum value in the computational domain is employed.Hence when the variation of each nodal variable in theNewton iteration does not exceed 10−4 of its value in theprevious iteration, the algorithm concludes that the con-verged solution is attained. For the viscoelastic variables,we examine the relative error in terms of the eigenvalue ofthe c-tensor. We have found that this convergence criterionimposes less stringent computational procedure, and itseems quite practical and appropriate since we mainlyobserve the results in terms of physically meaningful c-ten-sor or stress rather than h.

5. Results and discussion

The convergence limit in the scale of the Deborah num-ber is listed in Table 2 for the original c-formulation andthe logarithm-transformed h-formulation. With the SUPGmethod, in the conventional c-tensor representation theconvergence limits are 3.2 for the coarse mesh and as lowas 0.3 for the fine mesh. In addition to this severe lim-itation of flow rate in computation, one can observe steepdecrease of the limit Deborah number with the meshrefinement. On the contrary, when we switch to the h-ten-sor equations, we can examine huge increase of the limitDeborah number to 132 and even higher value of 193 infiner discretization. Even though the results under the SUscheme are not major concern in this work due to its lowerorder accuracy, they deserve a brief summary. Even thoughthe limit value is relatively high, it decreases as the meshbecomes finer under the c-formulation. However with thelogarithmic tensor formulation the limit values exceed 200for both meshes and we simply have stopped further com-putation due to no further interest.

In Fig. 3 the streamlines at the highest Deborah number

De UθH0-------,≡

Table 1. Total numbers of elements, nodes and variables for two meshes

No. ofelements

No. oflinear nodes

No. ofquadratic nodes

No. ofunknowns

Coarse meshc-formulation

3491 1920 733044330

h-formulation 37000

Fine meshc-formulation

7174 3835 1484389555

h-formulation 74712

Table 2. Limit of Deborah numbers achievable for each meshtypes under the SUPG method with each constitutiveformulation indicated (the limit value for the SU schemeis given inside the parenthesis)

Coarse mesh (Fig. 2a) Fine mesh (Fig. 2b)c-formulation 3.2 (142) 0.3 (65)h-formulation 132 (> 200) 193 (> 200)



of 193 for the fine mesh is depicted, where the size of thecorner vortex reaches almost twice the half width of thereservoir channel. Fig. 4 illustrates highly inhomogeneousdistribution of elastic potential and normal and shearstresses, respectively, all of which display extremely steepvariation near the corner. Near the corner these three rheo-logical variables achieve their maximum values such asWmax = 93.9, τyy,max= 189 and In Fig. 5 theextra-stress profiles are shown as a function of y (the flowdirection) at x = 1 (scaled with H0). Thus the region of

means the location at the downstream wall. Herewe do not observe fluctuation of stress variables along thewall, which have been frequently examined in many pub-lications. At this point, we do not intend to explain this dif-ference in results. This disappearance of numerical artifactsmay be due either to this appropriate formulation or to theconstitutive equations employed herein. At least, one mayobserve one proper tendency that the peaks of the stressvariables become sharper as the mesh becomes finer.

The important advantage of this new formulation in com-putational rheology consists not only in the stabilizingeffect of the whole numerical scheme demonstrated justnow, but also in appropriate mesh convergence possiblyachievable with this formalism since finer discretizationseems to yield better convergence. Even though Hulsen(2004) and Fattal and Kupferman (2004) have reportedalmost unlimited convergence with this h-formulation, inthis current flow geometry one may not be able to expectsuch unconditional numerical stability since there exists

corner singularity. In this type of computational domain,even the linear problem exhibits singular behavior, that is,the gradient of the solution function approaches infinity(actually integrable singularity) near the corner (e.g.Johnson, 1995). In this case, mesh refinement resolvessuch difficulty, in other words the solution becomes moreaccurate as the discretization refines. However until now,there has been no conclusive result in viscoelastic flowmodeling, and most probably finer mesh in finite elementanalysis deteriorates the numerical scheme more severely(Crochet et al., 1984; Baaijens, 1998).

With thorough examination of all the available resultsobtained until now (Hulsen, 2004; Fattal and Kupferman,2004) including the one in this manuscript, there is rel-atively high possibility of eventual resolution of the highDeborah number flow problem, even though more exten-sive study of mesh convergence, etc. in various flow geom-etries is certainly required. At least, one may conclude thatthis formalism seems quite promising and deserves inten-

τxy max 31.=

0 y 5≤ ≤

Fig. 3. Streamlines at De = 193 for the fine mesh (Fig. 2b) com-puted with the SUPG method.

Fig. 4. Contour lines of (a) elastic potential W, (b) normal stressin the flow direction and (c) shear stress at De = 193 forthe fine mesh (Fig. 2b) computed with the SUPG method.


Youngdon Kwon

sive further investigations.In the author’s opinion, understanding why this new for-

mulation completely equivalent to the original model equa-tions preserves better numerical characteristics andverifying whether it may work well for all other trans-formable differential constitutive equations, are extremelycrucial. First, as mentioned by Fattal and Kupferman(2004), the polynomial interpolation of the logarithm of cseems quite appropriate and efficient, since inside the vis-coelastic boundary layer or near the singular corner thesolution of c exhibits steep exponential variation and thislogarithmic formulation transforms exponential depen-dence into linear or polynomial one.

At this point, it is worthwhile to examine more carefullythe result given by Hulsen (2004), where the benchmarkproblem of flow around a cylinder inside a straight pipehas been considered. When implementing this h-formal-ism, he has observed almost limitless numerical conver-gence with the Giesekus model, whereas almost no

improvement has been attained with the Oldroyd-B equa-tion. The cause of this substantial difference is clearly evi-dent if one is well aware of the mathematical stabilitycharacteristics of these two equations. There is one the-orem on the positive definiteness of the conformation ten-sor c and its boundedness proven for the differentialconstitutive equations (Hulsen, 1990; Leonov, 1992;almost all available stability results are summarized inKwon and Leonov, 1995). The upper-convected Maxwelland thus the Oldroyd-B models violate this theorem, andtheir unbounded stress behavior in uniaxial extensionalflow in the strain rate exceeding the half of reciprocalrelaxation time is this well-known example. However theGiesekus model with the numerical parameter (written as αin Kwon and Leonov, 1995) not exceeding 1/2 is alwaysmathematically stable when we include the Newtonianterm or consider only pre-defined strain history. Similartrend may be seen in finite element modeling of contrac-tion flow in c-formalism (Lee et al., 2004).

It has been known for quite long time that the confor-mation tensor c has to remain strictly positive-definite inthe whole flow domain at all time, the violation of whichimmediately invokes the Hadamard instability (Dupret andMarchal, 1986; Kwon and Leonov, 1995). Even for theconstitutive equations proven Hadamard stable, numericalerror seems to cause the violation of the positive-defi-niteness (Dupret et al., 1985; Lee et al., 2004). The tensor-logarithmic transformation by Fattal and Kupferman(2004) rigorously preserves this essential requirement,which seems to be the most important among all featuresin this formulation.

Note that for some differential constitutive equationseven the positive-definiteness of the conformation tensor isnot guaranteed (Kwon and Leonov, 1995; Kwon and Cho,2001), for which the tensor-logarithmic transform itself isnot valid at all. Therefore it is quite certain that this h-ten-sor formulation will work in the high Deborah numberflow modeling only for very limited versions of consti-tutive equations. As a conclusion, the stability criteria sug-gested in Kwon and Leonov (1995) may play an importantrole on the choice of viscoelastic equations in flow mod-eling and even on formulating new constitutive relations.

6. Conclusions

We present an alternative procedure obtaining the tensor-logarithm representation of the differential constitutiveequations for the conformation tensor, which has been sug-gested originally by Fattal and Kupferman (2004). Con-crete example of the procedure is also given in 2D case andnumerical result for 4:1 planar contraction flow has beendemonstrated with the Leonov constitutive equation. Theresult shows better convergence with mesh refinement aswell as astonishing stabilization of the numerical scheme.

Fig. 5. Extra-stress profiles along the downstream wall (x = 1) atDe = 100 for the coarse and fine meshes (Fig. 2) com-puted with the SUPG method: (a) the normal stress τyy inthe flow direction, (b) shear stress τxy.



This new formalism exactly equivalent to the original con-stitutive model seems to work extremely well as long asproper viscoelastic field equations are employed. This ten-sor-logarithm formulation, if applied with careful consid-eration of mathematical stability of viscoelastic models,seems quite promising and possibly resolves the high Deb-orah number problems in computational rheology.

Acknowledgements

This study was supported by research grants from theKorea Science and Engineering Foundation (KOSEF)through the Applied Rheology Center (ARC), an officialKOSEF-created engineering research center (ERC) atKorea University, Seoul, Korea.

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