On the accuracy and stability properties of an hybrid �nite volume/element method for

viscoelastic ows

M. Aboubacar and M.F. Webster

Institute of Non-Newtonian Fluid Mechanics

Department of Computer Science

University of Wales, Swansea

Singleton Park, Swansea SA2 8PP, U.K.

Tel: (0)1792 295656, Fax: (0)1792 295708

email: m.f.webster@swansea.ac.uk

Abstract

A cell-vertex hybrid �nite volume/element method is investigated that is implemented on tri-

angles and applied to the numerical solution of Oldroyd model uids in contraction ows.

Particular attention is paid to establishing high-order accuracy, while retaining favourable sta-

bility properties in reaching high levels of elasticity. The main impact of this study reveals

that switching from quadratic to linear �nite volume stress representation with discontinuous

stress gradients, and incorporating a local reduced integration at the re-entrant corner, provide

enhance stability properties. Solution smoothness is achieved by adopting the non-conservative

ux form with area integration, by appealing to quadratic recovered velocity gradients, and

through consistency considerations in the treatment of the time term in the constitutive equa-

tion. In this manner, high-order accuracy is maintained, stability is ensured, and the �ner

features of the ow are con�rmed via mesh re�nement. Lip vortices are observed for We > 1,

and a trailing-edge vortex is also apparent. Entry pressure drop, loss of evolution, and solution

asymptotic behaviour towards the re-entrant corner are also discussed.

1 Introduction

This study considers the accuracy and stability properties of a new hybrid �nite volume/element

scheme in its application to the numerical solution of viscoelastic contraction ows. Previously,

a second-order scheme has been constructed that demonstrates its e�ciency over �nite element

counterparts on model problems. The current direction of investigation focuses on highly elastic

solutions, with retention of accuracy and favourable stability properties. Holding this goal in

mind, this article presents results for the benchmark problem of ow through a 4:1 planar

contraction with sharp re-entrant corners for an Oldroyd-B model. A comprehensive literature

review on this problem may be found in Matallah et al. [1] with further detail in Matallah [2].

Typically, incompressible viscoelastic ows may be classi�ed through mixed parabolic-

hyperbolic di�erential systems. The central ethos of the current approach is to apply �nite

element (fe) technology to the self-adjoint sections of the system (optimal), and �nite volume

(fv) schemes to the hyperbolic parts. Finite volume technology has advanced considerably,

over the last decade, in its treatment of pure advection equations, and has matured su�ciently

to attack the viscoelastic regime. In this regard, we may cite the important contributions of

Morton and co-workers [3, 4], Struijs et al. [5] and Tomaich and Roe [6], that have advocated

cell-vertex fv formulations for advection, Euler and compressible Navier-Stokes equations. In

contrast to their cell-centred counterparts (Berzins and Ware [7]), cell-vertex schemes are less

susceptible to spurious modes and maintain their accuracy for broader families of meshes, in-

cluding unstructured forms. In the viscoelastic context, the emergence of non-trivial source

terms, via the constitutive equations, gives rise to some open questions for �nite volume repre-

sentations. In fact, in shear ow or in no-slip boundary vicinities, ux terms vanish and source

terms dominate. Such stress source terms display dependence on stress and velocity gradients,

and their treatment is addressed comprehensively in this study.

Features of the current hybrid approach include a time-stepping procedure that combines a

�nite element discretisation (semi-implicit Taylor-Galerkin/Pressure-Correction) for continuity

and momentum equations, and a cell-vertex �nite volume scheme for the constitutive equation.

The combination is posed as a fractional-staged formulation based upon each time-step. The

positioning of �nite volume nodal values and control volumes, is such that four linear �nite

volume triangular cells are formed as embedded sub-cells of each parent quadratic �nite element

triangular cell (note association with superconvergence points, see recovery in Matallah et al.

[1]). Sub-cell reference has arisen earlier in the fe literature, see for example, the SU method

implementation of Marchal and Crochet [8]. Here, this is accomplished by connecting the mid-

side nodes of the parent �nite element triangular cell. An important aspect is that with stress

variables located at the vertices of the �nite volume cells, no interpolation is required to recover

the �nite element nodal stress values. The cell-vertex approach is naturally associated with

2

uctuation distribution, an upwinding technique that distributes control volume contributions

for each equation to provide nodal solution updates. The consideration of triangular cells o�ers

the exibility of either structured or unstructured meshing as equally viable options, without

further complication.

The literature on fv implementations for viscoelastic ow divides into two camps: those

dealing with a full system through fv and hybrid versions, see [9] for review. Most consider

established standard low-order discretisation, with cell-centred or staggered grid systems, and

SIMPLER-type algorithms for steady-state solutions on structured rectangular grids. Essen-

tially, this implies piecewise constant interpolation. There, principal attention has been directed

towards highly elastic solutions, often to the detriment of establishing accuracy in this compli-

cated scenario. Two recent papers adopting this approach for the abrupt 4:1 planar contraction

ow are those of Phillips and Williams [10] and Alves et al. [11]. Phillips and Williams used

a semi-Lagrangian method to simulate the ow of an Oldroyd-B uid, with and without the

inclusion of inertia. This is a �rst-order implementation that applies a semi-Lagrangian treat-

ment for convection terms and uses a staggered grid, that avoids the calculation of normal

stresses at the re-entrant corner. Only the shear stress variable is located at the corner sin-

gularity, though an averaging procedure is invoked, replacing the corner value by shear stress

values at cross-stream and upstream neighbouring nodes. These authors assessed accuracy by

demonstrating that the size of the salient corner vortex remained fairly constant with mesh

re�nement, and concurred with Matallah et al. [1] and Sato and Richardson [12], both for

creeping ow and with inertia (Re = 1). Phillips and Williams commented upon the absence of

lip vortex activity for coarse meshes, and on their �nest mesh, its appearance around We = 2

with growth as the level of uid elasticity increases up to a maximum value of We = 2:5. In

contrast, Alves et al. considered the same ow for an Upper-Convected Maxwell (UCM) uid,

comparing both �rst-order and second-order schemes, again using staggered grid systems. How-

ever, unlike with the study of Phillips and Williams, none of the stress components are located

at the re-entrant corner. A �rst-order implementation was able to converge up to We = 8

on a re�ned mesh (third in a series of four, consecutively re�ned meshes), but proved inaccu-

rate, re ecting large spurious vortex structure. An alternative second-order scheme failed to

converge at Weissenberg numbers above unity on the same re�ned mesh. Alves et al. related

this premature breakdown of numerical convergence (inhibiting advance in We) to oscillations

in the stress �eld, associated with the use of high-order upwinding in regions of high stress

gradients. Therefore, these authors invoked, in conjunction with their second-order scheme, a

limiting (min-mod) procedure applied to the stress convection (see also, Sato and Richardson

[12]). In this manner, Alves et al. were able to reach a Weissenberg number of �ve on their third

�nest mesh, and three on their fourth, most �ne mesh. Relevant issues that arise in reference

3

[11] can be summarised as follows: diminishing lip vortex with mesh re�nement as reported by

Matallah et al. [1], and Xue et al. [13]; lip vortex growth (see [1] and [10] to a lesser extent),

and diminishing salient corner vortex with increasing elasticity; agreement with the asymptotic

behaviour of velocities and stresses near the re-entrant corner, as predicted theoretically by

Hinch [14] for an Oldroyd-B uid (see also, Renardy [15, 16] for an UCM uid). The present

article develops these themes somewhat further.

The methodology proposed behind this cell-vertex �nite volume scheme is novel in this

domain and addresses both accuracy and stability behaviour, along with calculating stress

�elds in the presence of a singularity. Consistent treatment for stress ux and source terms

is maintained, reminiscent of fe Petrov-Galerkin variational weighting (supg). Likewise, such

considerations may be extended to embrace the transient terms involved. The hybrid scheme

of Sato and Richardson [12] deserves some mention, bearing resemblance to the present. In

contrast, this scheme employed a time-explicit fe method for momentum and time-implicit

fv for pressure and stress of cell-centred type. Higher-order upwinding was achieved through

application of a TVD ux-corrected transport scheme to the advection terms of the stress

equation. Clearly, some attempt must be made to address the complications of ux-source

interaction, and to accommodate for highly elastic convection, when this arises. In this respect,

the direction adopted by Tanner, Phan-Thien and co-workers [17, 18] was to incorporate a sta-

bilising arti�cial stress di�usion, reminiscent of the SU method of Marchal and Crochet [8]. The

fe studies of Baaijens [19, 20] consider various alternative Galerkin least-squares stabilisation

procedures, including strain-rate stabilisation. The discontinuous Galerkin aspects involved,

strive for low-order stress interpolation and gain stability accordingly. The correspondence to

fv weighting and the localised treatment of solution has similar characteristics to the present

approach. Under such discontinuous stress representation on a global scale, we also note the

work of Fortin and Fortin [21] with discontinuous quadratic interpolation and Basombr�io et al.

[22] with discontinuous linear interpolation.

Fluctuation splitting schemes, natural to the cell-vertex treatment, may be of linear or

non-linear type, and introduce such properties as positivity and linearity preservation. To date,

various forms have been considered and their properties analysed [9], recognising the importance

of linearity preservation to achieve high-order accuracy. Our prior studies established accuracy

properties against analytical solutions for smooth ows [9], before addressing stability issues

for complex smooth and non-smooth ows [23]. Here, we extend and revise our methodology

with complex ows and accuracy in mind.

We identify the relative strengths and weaknesses of various strategies for dealing with

ux and source terms. For the stress nodal update, we consider a uctuation distribution

contribution over the fv-triangle, and a uniform distribution over the median dual cell. The

4

latter is a unique non-overlapping region associated with each �nite volume node, a new concept

arising with cell-vertex schemes. Other issues covered concern, the use of conservative/non-

conservative stress ux treatment (area or line integrals), the choice of stress representation,

the order and continuity of stress gradient and velocity gradient representations, the treatment

of the solution in the re-entrant corner neighbourhood, and time consistency. The correspon-

ding properties of the schemes considered are adjudged through identifying solution behaviour

(particularly in stress pro�les), as a consequence of scheme implementation.

2 Governing equations

Incompressible viscoelastic ows are governed by conservation laws for mass and momentum

and a constitutive equation for stress. In non-dimensional form, the balance equations for

isothermal ows read

r � uuu = 0 (2.1)

Re@uuu

@t= �Reuuu � ruuu�rp+r �

�2�s

�ddd+ ���

�(2.2)

and the constitutive equation for an Oldroyd-B uid may be expressed as

We@���

@t= �Weuuu � r��� � ��� + 2

�e

�ddd+We (LLL � ��� + ��� �LLLT ) (2.3)

where uuu, p, ��� represent the uid velocity, the hydrodynamic pressure and the extra-stress tensor.

The total viscosity � is split into Newtonian (�s) and polymeric (�e) contributions, such that

� = �s + �e ; ddd = (LLL + LLLT )=2 represents the Euler rate-of-deformation tensor and LLLT = ruuu,the velocity gradient.

The Reynolds and Weissenberg numbers are de�ned according to convention as:

Re =�UL

�; We =

�U

L(2.4)

where �, � are the uid density and relaxation time and U , L are characteristic velocity (average

at outlet) and length scale (channel exit half-width) of the ow. Creeping ow is assumed

throughout and, as such, the momentum convection terms are discarded, whilst transient terms

are retained to provide the time-stepping facility, see Carew et al. [24].

5

3 Background to methodology

3.1 Numerical framework

We recall brie y the algorithmic structure upon which both the fe and hybrid fe/fv are articu-

lated. The reader is referred to Matallah et al. [1] and Wapperom and Webster [9] for detailed

discussion on these methodologies.

One may commence with the structure of the fe scheme, and point to the hybrid fe/fv

options in contrast. The general scheme follows a time-splitting semi-implicit formulation.

There are two distinct aspects to this approach: a Taylor-Galerkin scheme and a pressure-

correction scheme. The Taylor-Galerkin scheme is a two-step Lax-Wendro� time stepping

procedure, extracted via a Taylor series expansion in time [25, 26]. The pressure-correction

method accommodates the incompressibility constraint to ensure second-order accuracy in time

(see [27, 28]). This leads to a three-stage structure for each time step that can be stated in

discrete form as follows [23]:

Stage 1a: Au(Un+1=2 � Un) = bu(Pn;Un; T n;Dn);

2We

�tA� (T n+1=2 � T n) = b� (Un; T n;Dn);

Stage 1b: Au(U� � Un) = bu(Pn;Un;Un+1=2; T n+1=2;Dn+1=2);

2We

�tA� (T n+1 � T n) = b� (Un+1=2; T n+1=2;Dn+1=2); (3.1)

Stage 2:�t

2A2(Pn+1 � Pn) = b2(U�);

Stage 3:2

�tA3(Un+1 � U�) = b3(Pn;Pn+1);

where the superscript n denotes the time level, �t the time step and U ;U�;P; T ;D the nodal

values of velocity, non-solenoidal velocity, pressure, extra-stress and velocity gradient, respec-

tively. Au; A2; A3 are the standard velocity, sti�ness and mass matrices. The precise form of

A� depends on the implementation (fe or hybrid fe/fv), as discussed in the following sections.

The momentum equation in Stage 1, the pressure-correction in Stage 2 and the incompres-

sibility constraint in Stage 3 are discretised spatially via a Galerkin �nite element method.

For reasons of accuracy, the resulting Galerkin Mass matrix-vector equations at stages 1 and

3 are solved using an e�cient element-by-element Jacobi scheme, requiring only a handful of

iterations. A direct Choleski decomposition procedure is invoked to handle stage 2. Finally, the

di�usive terms in Eq. (2.2) are treated in a semi-implicit manner in order to enhance stability.

The parent fe mesh element is a triangular cell with three vertices and three mid-side nodes.

To solve for the momentum and continuity equations, the velocity is represented by quadratic

shape functions, using the six nodal values, and the pressure by linear functions from the

vertices alone, see �gure 1. All the above features are common to both fe and hybrid fe/fv

6

implementations. The fe and fv representations depart in their spatial discretisation of the

stress constitutive equation (see on). The fv sub-cell is an internal triangle of each fe triangle,

built on the mid-side node connections. Representation of stress upon each sub-cell may be

taken in a variety of ways, including linear on the sub-cells or quadratic from the parent fe cell.

These are crucial aspects expanded upon below.

3.2 Finite element method

In this instance with fe discretisation of stress, quadratic interpolation functions are used for

both velocity and stress, and the matrix A� is sparse. Therefore the resulting equation is similar

in structure to that of the momentum equation and is solved accordingly. Other features are

the incorporation of supg weighting and recovery of velocity gradients (as coe�cients of the

constitutive equation) that are found necessary to enhance stability of convergence [1, 23]. Here,

recovery of velocity gradients implies gathering continuous representations, that themselves

depend upon superconvergent recovered nodal values for these quantities. This implementation

(fe/supg) has proved successful in reaching highly elastic solutions (criticalWeissenberg number,

Wecrit = 4:1) for the present 4:1 contraction ow and compares favourably with the extensive

literature available on the subject, see Matallah et al. [1] for review and discussion thereon.

Nevertheless, the fe implementation su�ers a heavy computational overhead when compared to

its hybrid fe/fv counterpart, as reported in our previous communications [9, 23].

3.3 Hybrid �nite volume method

For clarity of the expos�e, we summarise the key features of our base hybrid fe/fv method, which

has been extensively documented in the precursor papers of Wapperom and Webster [9, 23]. In

this implementation, the matrix A� is the identity matrix, the need to resolve a matrix-vector

equation is avoided, and the right-hand side vector bu is straightforward to construct. These

two features have led to a hybrid fe/fv implementation that is up to �ve times more e�cient

than fe/supg for viscoelastic ow past a cylinder and the abrupt 4:1 contraction problem.

Cell-vertex uctuation distribution (FD) schemes, which are the foundation of the present

fv method, require only the vertices of a triangular cell to evaluate the solution of a given scalar

�eld. The theory associated with the variants of such FD-schemes is centred around appropriate

discretisation choices for pure advection equations. They feature such properties as:

� Conservation, that requires the distribution coe�cients �Tlto sum to unity for each

triangle T, the summation running over its three vertices l.

� Linearity preservation, that is associated with second-order accuracy at steady state.

� Positivity, that is related to accuracy in the transient development, prohibiting the

7

occurrence of new extrema in the solution. Particular attention must be paid to the

issue of dealing with sources, as these may lead to physical extrema, that should not be

suppressed arti�cially by imposing positivity.

Furthermore, FD schemes can be divided into linear and non-linear classes. Non-linear

schemes may be both linearity preserving and positive; alternatively, linear schemes can only

possess one such property. Since we are interested in viscoelastic ows where sources can be

dominant, from our previous experience we have selected the Low Di�usion B (LDB) scheme.

This is a linear scheme with linearity preserving properties, that has proven e�cient in dealing

with model problems, as well as some complex ows [9, 23]. The LDB distribution coe�cients

�i are de�ned per cell node i according to the angles 1 and 2 in the fv triangle, subtended on

both sides of the cell advection velocity aaa (an average vector per cell, see �gure 2). The LDB

coe�cients are de�ned as:

�i = (sin 1 cos 2)= sin( 1 + 2);

�j = (sin 2 cos 1)= sin( 1 + 2);

�k = 0:

(3.2)

We point out that the closer the advection velocity aaa is to being parallel to one of the cell boun-

daries, the larger the contribution to the downstream node at that boundary, hence minimising

the introduction of spurious numerical di�usion.

To avoid interpolation when recovering the stress nodal values for the momentum equation,

the �nite volume tessellation is constructed from the �nite element grid by connecting the mid-

side nodes. This generates four triangular fv sub-cells per fe parent triangle, as indicated in

�gure 1. Accordingly, the stresses are computed on the vertices of the fv cells as outlined below.

First, we recast the stress constitutive equation (2.3) in conservative form,

@���

@t= �r � R +QQQ (3.3)

R = uuu��� (3.4)

QQQ =1

We(2�e

�ddd� ���) +LLL � ��� + ��� �LLLT (3.5)

where R and QQQ are the ux and the source terms, respectively.

Next, we consider each scalar stress components, � , acting on an arbitrary volume . Its

variation is controlled through the uctuations of the ux vector RRR = uuu� and the scalar source

term Q

@

@t

Zl

� d =

I�

RRR �nnn d� +

Zl

Q d: (3.6)

The core of this cell-vertex uctuation distribution scheme is to evaluate these two variations

on each fv triangle, and to distribute them to the three vertices of the cell according to the

8

preferred strategy. The update of a given node l is obtained by summing the contributions from

its control volume, l, that is composed of all the fv triangles surrounding node l, see �gure 3.

When source terms are involved, the standard treatment advocated widely in the literature

consists in dealing separately with the ux and the source terms, say in the form

T

l

�n+1l

� �nl

�t= �T

lRT +Ql

MDCT(3.7)

for a particular triangle T. T

lis the area of the median dual cell (MDCT ) associated with node l

within the triangle T (�gure 3) and the subscripts T and MDCT indicate the control volume over

which the ux and the source terms are integrated. However, this method has proved itself

to be highly inaccurate, even for model viscoelastic problems with non-trivial source terms.

This is due to the incompatibility of control volumes between the two contributions and their

alternative discretisations [29]. Wapperom and Webster [9] developed an alternative scheme

that consistently distributes both ux and source terms over the fv triangle,

T

l

�n+1l

� �nl

�t= �T

l(RT +QT ): (3.8)

Note, by design, the LDB scheme does not impose positivity upon the source terms. To evaluate

the integrals RT and QT , the velocity and stress were approximated as piecewise linear from the

vertices of the fv triangle and the velocity gradients are recovered from the parent fe velocity

�eld (hence, are linear and continuous across the fv sub-cells). This approach, denoted FVL in

reference [9], achieved second-order accuracy at steady state for sink ows and Cartesian model

problem ows. However, it was necessary to have recourse to the quadratic representation from

the parent fe for the velocity and stress (FVQ form, also reference [9]), to maintain second-

order accuracy when the ow was nearly parallel to the boundaries (as would arise near no-slip

boundaries).

Switching attention from these mainly extensional model problems to pure shear and mixed

shear/extensional ows, required two additional strategies to retain stability; consistent stream-

line-wise upwinding and area integral \median dual cell" contributions. Hence, Wapperom and

Webster proposed the following generalised distribution scheme [23]:

T

l

�n+1l

� �nl

�t= �T�

T

l(RT +QT ) + �MDC(R

l

MDCT+Ql

MDCT): (3.9)

The parameters �T and �MDC are used to discriminate between various update strategies. With

�T = f(We; a; h) and �MDC = 1, the nodal update is similar to consistent streamline upwinding

as used in �nite elements. The function f is such that f = �=3 if j�j � 3 and 1 otherwise,

9

where � = We(a=h), with a the magnitude of the advection velocity per fv-cell and h the

square-root of the area of the fv-cell in question. In this manner, Wapperom and Webster [23]

generated solutions for a Wecrit of 1:5 for smooth ow past a cylinder and 1:3 for the abrupt

4:1 contraction problem, without imposing additional stabilisation techniques. This situation

was marginally improved upon in the latter instance with momentum strain-rate stabilisation,

then reaching a critical Weissenberg number of 1:5. Accordingly, we retain this method as the

foundation for our present hybrid fe/fv implementation and extended study.

The above deliberations have emphasised the importance of treating the contributions from

the ux and source terms in a consistent manner. Therefore, we proceed to extend such con-

siderations to the time terms, following Hubbard and Roe [30] for pure convection problems.

Equation (3.9) for a single triangle update, is labelled scheme CT0: a form that lacks time

consistency, due to area weighting. The complete nodal update corresponds to the sum of the

contributions from all triangles surrounding the node (see �gure 3), given by

�n+1l

� �nl

�t=

P8T

�T�T

l(RT +QT )

l

+

P8T(Rl

MDCT+Ql

MDCT)

l

(3.10)

where l is the area of the MDC associated with node l. To enforce consistency, the contribution

from the triangles (upon which the uctuation distribution scheme impinges) and that from

the MDC (distributed uniformly) are weighted with an appropriate area and then summed to

yield the nodal update, viz.

�n+1l

� �nl

�t=

P8T

�T�T

l(RT +QT )P

8T�T�

T

lT

+

P8T(Rl

MDCT+Ql

MDCT)

l

(3.11)

where T is the area of the triangle T. We introduce two possible options as consistent schemes,

dependent upon the area sumP

8T�T�

T

lT . Both schemes follow Eq. (3.11), with the exception

of when the area sum falls below a prede�ned threshold (say 10�14). Under such circumstances:

1. CT1 reverts to a pure MDC approach

�n+1l

� �nl

�t=

P8T(Rl

MDCT+Ql

MDCT)

l

: (3.12)

2. CT2 retains the uctuation-distributed contribution by recourse to CT0 (of Eq. (3.10)).

The motivation for such options lies in the fact that the area sum, dividing the contributions

from the triangles Eq. (3.11), can vanish near no-slip boundaries, where the average velocity

tends to zero. Indeed, in pure shear ow regions, the velocity is parallel to the walls, and it is

10

our experience that then, a pure median dual cell approach is a reasonable choice. Hence, we

expect CT1 to be more accurate than CT0 in the shear ow zones of the 4:1 contraction problem.

However, in the immediate vicinity of the corner, where the ow is mainly extensional, one needs

to retain the information from the uctuation distribution as well. This is the rationale upon

which we have constructed scheme CT2, which is superior to CT1, as we proceed to demonstrate

in section 5.3.

Other issues that in uence trends in solution quality are: the discretisation of the stress

ux integral (RT ), the order of the solution representation, the order and continuity of gradient

approximation, and the quadrature rules used to evaluate the integrals. The combination of

these speci�c options generates the fv schemes summarised in table 1, that are discussed in

detail in section 5.

4 Problem speci�cation

The benchmark problem of ow through an abrupt 4:1 planar contraction with a re-entrant

corner for an Oldroyd-B uid is known to be a formidable test problem, in terms of stability

at high Weissenberg number. It is equipped with an analytical representation for the corner

solution and is well-documented in the literature. Therefore, it is a natural choice in this nu-

merical study. In a subsequent article [31], we go further to address more robust and physically

representative constitutive models, and complex smooth ows such as contraction ow with a

rounded corner.

Figure 4 displays the set of structured triangular meshes employed, showing zoomed sections

around the re-entrant corner, following hierarchical re�nement based on zonal mesh density in

the contraction region. M1, M2 and M3 have been successfully utilised earlier by Matallah et

al. [1] to demonstrate solution convergence with mesh re�nement for the fe/supg scheme. This

o�ers a convenient reference point to qualify the various fv strategies outlined in the previous

section. Here, we re�ne further generating mesh the hybrid NM3 mesh. In contrast to M3,

NM3 mesh displays increased re�nement in the salient corner region, but also has one quarter

of the element size in the immediate corner vicinity and one half just beyond. This choice

allows us to discern vortex activity more clearly in this critical region. Table 2 displays the

mesh characteristics. As before, the inlet and outlet lengths are 27:5L and 49L to ensure fully

developed ows in these regions, L being the half-width of the downstream channel.

By convention, we apply no-slip boundary conditions at the wall and enforce symmetry at

the centerline. Velocity is imposed at the inlet and exit through Waters and King transient

boundary conditions [32]. The pressure is held �xed at a single exit point to remove the

indeterminacy of pressure level. We have appealed to a point-wise time-dependent initial-value

solution for inlet stresses, obtained by imposing vanishing stress ux and kinematics as above.

11

In this manner, we have reduced the CPU time consumption by 25%, without degradation in

the steady-state solution. The non-dimensional parameters are �e=� = 8=9, �s=� = 1=9 and

Re = 0 (creeping ow). Typical time steps are 0(10�3) and �ve Jacobi iterations are employed

for the fe stages. The time-stepping termination criteria is taken as 10�5, a relative solution

temporal increment within a least squares measure [1, 9]. To investigate the stability behaviour

of the various scheme alternatives summarised in table 1, we commence each simulation at

We = 0:1 from a quiescent initial state in all �eld variables and increment the Weissenberg

number by 0:1, till the scheme fails to converge (encountering either temporal oscillations or

numerical divergence).

5 Results for present fv scheme variants

5.1 Standard implementation

The baseline fe/fv scheme (FV0, see table 1), is that previously established in Wapperom and

Webster [23]. To be precise, this uses a consistent combination of uctuation distribution

and median dual cell contributions for stress ux and source terms, as outlined in section 3.3

(Eq. (3.10)). It appeals to a quadratic representation for stress and a linear recovered stress

gradient, with recourse to the parent �nite element tessellation. The same treatment is applied

to the velocity and its gradient, respectively. A line integral (conservative) representation is

used to evaluate the stress ux RT .

For the ow of an Oldroyd-B uid past a cylinder in an in�nite domain [23], FV0 attained

a critical Weissenberg number of 1.4 and displayed similar stress contour patterns to those re-

ported by Matallah et al. [1], Townsend [33] and Pilate and Crochet [34]. Streamline plots with

increasing elasticity showed no evidence of vortex enhancement and replicated the experimental

downstream shift away from the cylinder between Newtonian and elastic solutions [35, 36]. For

the ow of an Oldroyd-B uid through a 4:1 planar contraction, FV0 performed less well, reach-

ing onlyWecrit of 1.1 on mesh M2, as compared to 4.1 for the fe/supg [1] scheme. Nevertheless,

stress contour patterns were almost identical to those for the fe/supg scheme. A close exam-

ination reveals that, although FV0 solutions match the fe/supg counterparts almost exactly

in core ow, FV0 stresses su�er from oscillations, along the downstream wall just beyond the

re-entrant corner. This is apparent especially in �xx, as displayed in �gure 5 for We = 1. We

note, in passing, the �ndings of Renardy [37], i.e. ill-conditioned nature of the solution in this

region, and use of variable transformation. We focus on this issue and employ these oscillations

around the corner, as a signal of solution quality, both in scheme comparison and against the

literature, empirically and theoretically.

We compare such fv �xx-pro�les against the fe/supg scheme of Matallah et al. [1] that em-

ployed recovery. This was the approach adopted in our earlier published work [9, 23]. Results

12

are contrasted on mesh M2, exposing the steady-state stress component pro�les on a hori-

zontal line intersecting the re-entrant corner and passing along the downstream wall. Mesh

M2 provides an acceptable level of accuracy for the fe/supg scheme, see [1], but is su�ciently

coarse to stimulate discrepancies in solution smoothness across scheme variants. In this man-

ner, we are able to calibrate the various fv implementations against a fe/supg scheme, and to

identify optimality, before proceeding to assess solution accuracy with mesh re�nement. The

additional consideration of strain-rate stabilisation, as employed in Wapperom and Webster

[20, 23], is avoided here, so as not to over-complicate the comparisons. This issue is considered

in a proceeding article [31]. All critical Weissenberg numbers reported are subject to such

deliberations.

5.2 Linear stress representation

Noting the anomalies of scheme FV0, we revisit a linear representation of the stress, based

on the fv sub-cell with a recovered constant stress gradient (FV1). We note that recovery

implies a single-valued gradient across the control volume of a given node, �gure 3. Linear

interpolation of all primary variables was successfully utilised in the fe context by Basombr�io

et al. [22] to address spurious oscillations in the stress �eld near singularities. In our case, the

quadratic representation of the velocity from the parent fe is still retained. In this form, the

representation of the stress and velocity match the intrinsic order expected within each of the

fe and fv components of the system.

Figure 6 demonstrates the comparison between the two representations of the stress for

We = 1. The linear fe/fv representation of the stress reduces the amplitude of post-corner

oscillations along the downstream wall, shifting the solution towards the fe/supg representation.

We observe a major step forward in stability response, elevating Wecrit from 1:1 for scheme

FV0 to 1:6 with scheme FV1. However, the stress pro�le for scheme FV1 is slightly out of

phase with the fe/supg solution, displaying a conspicuous double peak immediately beyond the

corner. Note, the undershoot is far better represented here with scheme FV1 than with its

quadratic counterpart (FV0).

5.3 Transient consistency: time term treatment

The consistent treatment of time terms would be vital in a transient context, though may

prove to have some bearing upon accuracy in the steady state. Hence, at this point in the

investigation, we invoke the various treatments of the time term discussed above, namely the

default version CT0 (used thus far), and the two consistent alternatives CT1 and CT2, following

section 3.3. The introduction of CT1 (FV1.1) improved the smoothness of the solution in

the exit shear ow, but caused only minor adjustment to the corner neighbourhood solution.

This indicates the e�ect of appropriate area weightings in the transient and its in uence upon

13

solution accuracy in the steady state.

In Figure 7 with We = 1:5, we go somewhat further in time term treatments with the

introduction of CT2 option (FV1.2). This alternative scheme has all the features of CT1, but

retains the standard CT0 form when the area sum of Eq. (3.11) is unacceptably small. The CT2

form removes oscillations after the corner that are apparent with the CT1 (and CT0) version.

There is barely any change in the exit zone solution, where the di�erence between CT2 and CT1

solutions is no more than 0.15 percent. Consequently, scheme FV1.2 reaches a Wecrit of 2.0, as

compared to 1.7 for FV1.1: the clear preference for the CT2 option is established. AtWe = 1:5,

we note the non-monotonic behaviour identi�ed in �xx with the fe/supg scheme just beyond the

corner. This is a particular feature that emerges in the solution at and around We = 1:5. We

do not observe this non-monotonic response in rising up to We = 1:0, or subsequently beyond

We = 2:0, see below. Its occurrence seems to coincide with the onset of a lip vortex.

5.4 Conservative/non-conservative representation of stress ux

Here, we are concerned with the discrete treatment of the stress ux term over the fv-triangle

governed by uctuation distribution. The conservative form leads to line integrals (FV1.x),

where the integrand is a function of primary variables only. We note, in passing, that line

integral ux evaluation involves an averaging procedure for the convective terms around the

perimeter of a triangular cell. Alternatively, the non-conservative form has no such feature: it

leads to area integrals (FV2.x, FV3 and FV4.x) which require evaluation of stress gradients.

When using quadratic stress representation and area integration of RT , the limiting Weis-

senberg number, Wecrit, was merely 0:3 (scheme FV2). With the linear stress representation

(scheme FV2.1), Wecrit increases to 0.8. Both schemes, FV2 and FV2.1, incorporate CT2 time

treatment (note, table 1). This non-conservative area integral form, FV2.1, is most promising

in terms of solution quality, as demonstrated in �gure 8 at We = 0:5. Although this scheme

over-estimates the stress peak at the corner, particularly in �xx, it sharpens the post-corner

stress pro�le on the downstream wall, above and beyond both line integral (FV1.2) and fe/supg

alternatives. At the same time, it produces the most accurate fully-developed exit ow achieved

with a linear stress representation, both in smoothness and magnitude.

5.5 Stress and Velocity gradient representations

Thus far, we have identi�ed the steps necessary to capture solution smoothness in the hybrid

scheme. Stability is lacking, however, with restriction to a relatively modest level of Wecrit of

0.8. Our attention is now turned to resolve this issue.

It is generally acknowledged that the breakdown of convergence is accompanied with large

stress peaks within the corner neighbourhood, see for example �gure 8. Here, such large stress

peaks may well be a consequence of the recovery (smoothing) applied to the stress gradient

14

representation near the discontinuity. These approximations take e�ect within the discretisation

of the non-conservative form of the ux, RT . To test this hypothesis, we have implemented a

non-recovered constant stress gradient version, that is multi-valued from sub-cell to sub-cell.

In �gure 9 at We = 0:8, we compare the recovered (FV2.1) and non-recovered stress gradient

versions (FV3); this corresponds to the maximum attainable We-level for the recovered form.

The comparison indicates that the corner stress peak is lowered in the non-recovered version by

about a factor of one half, tallying with the fe/supg solution. With the recovered version, there

are oscillations present in the less dominant �xy and �yy components prior to the corner that

are now removed with the non-recovered implementation. This is yet another indication of the

superior solution quality attained with the non-recovered stress gradient, area integral, FV3

scheme over scheme FV2.1. Of major signi�cance is the impact on stability: by imbuing the

area integral with multi-valued stress gradients, Wecrit is raised from 0.8 to 3.0. We associate

this elevation with enhanced accuracy, leading to improved stability properties.

Still, there remains a post-corner phase lag between the FV3 and fe/supg solutions. This

we are able to address in an elegant manner by revisiting the velocity gradient representation.

>From a linear recovered instance (continuous), with scheme FV4 we adopt a quadratic recov-

ered form, mimicking that employed in the fe/supg implementation [1]. Except for the peak

value itself, the result is a close match between FV4 and fe/supg pro�les, as demonstrated in

�gure 10 at We = 1:0. The downside is a reduction in Wecrit from 3:0 to 2:4 for the FV4

scheme (see on).

5.6 Corner neighbourhood and local reduced integration

We have identi�ed the treatment of the corner solution as being associated with the levels of

attainment of Wecrit. This is manifest through the the onset of an underestimation of �yy

prior to the corner, falling to negative values as we increase elasticity (see section 6.4, with

e�ect on condition number S). The contraction ow displays discontinuous corner solutions

in stress and velocity gradient �elds, which we may incorporate into the fv approximation

through reduced quadrature for ux and source term integrals. In scheme FV4.1, we adjust

our fv quadrature rule, exact for cubics, to a linear form, within the control volume surrounding

the re-entrant corner. Reduced integration has the e�ect of sampling the integrand in such a

manner as to be equivalent to adopting a lower order representation of this quantity from cell to

cell. Such local considerations can be implemented within the fe/supg framework as well, since

both methods are variational in structure. Nevertheless, it is somewhat more straightforward

in the fv case, due to its local compactness properties. Figure 11 illustrates how this local

treatment adjusts the �xx and �yy FV4.1 pro�les, bringing them closer in line with those of the

fe/supg scheme (similar adjustment is noted in �xy). Notice, also, how scheme FV4.1 corrects

15

the underestimated value of �yy at the corner. This, in turn, has increased Wecrit from 2:4 for

scheme FV4 to 3:7 for scheme FV4.1, approximating that for the fe/supg scheme of 4:1 on mesh

M2.

At this juncture, we summarise the achievements thus far leading to our optimal hybrid

implementation, scheme FV4.1. First, a linear representation of the stress is preferred, based

on the fv sub-cell with a non-recovered (multi-valued) constant stress gradient. Second, velocity

gradients are taken of quadratic form over the parent �nite element. Third, a key issue has

been the strong in uence of non-conservative area integrals for RT , as opposed to conservative

line integrals (standard to fv-schemes). This is found to be all important in improving solution

quality. In this respect, it is e�ective to invoke multi-valued stress gradients. Lastly, we have

established the importance of a consistent treatment of the time term, and for the need of a local

treatment of the re-entrant corner solution gradients, to allow for some degree of discontinuity.

6 Mesh re�nement

We proceed to demonstrate that the fv method (scheme FV4 and its variant FV4.1) not only

compares favourably with the recent literature, but also highlights some fresh features.

6.1 Vortex behaviour

Figure 12 displays the streamlines on mesh M2 with increasing Weissenberg number, following

the format and conventions of Matallah et al. [1]. This is the standard mesh employed for the

scheme development work (section 5), and is less re�ned than that to follow. Both fv schemes

exhibit a lip vortex for We > 1 and indicate signi�cant lip vortex growth, as we increase uid

elasticity through We. For constant viscosity uids, such behaviour is widely reported in the

literature in the same range of Weissenberg number and at comparable levels of mesh size (see

references [1, 10, 11, 38, 39]). We note that this lip vortex growth is spurious (see below), at

least up to levels ofWe = 2. With regard to �gure 12 d we note that, Carew et al. [24] displayed

similar vortex structure for a linear PTT model at We = 10 and Re = 1, with equal-sized lip

and salient corner vortices.

Upon mesh re�nement, �gure 13 displays the streamlines produced by both fv schemes, FV4

and FV4.1, at the larger value We = 2 for the four meshes used in this study. Lip vortices

diminish and practically vanish at We = 2 with mesh re�nement with either fv scheme, re-

echoing the results of Matallah et al. [1] for Oldroyd-B and Alves et al. [11] for an UCM uid.

Moreover, in addition to a signi�cant improvement in solution smoothness, our �nest mesh

(NM3) demonstrates two important features: the �rst, is that there is a miniscule lip vortex at

this level of uid elasticity (We = 2); the second is, a closer examination reveals that FV4 and

FV4.1 streamlines present the same features, including a faint trailing-edge vortex. Scheme

16

FV4.1 points to this even on mesh M3. This is con�rmed in �gure 14, where we have plotted

these streamlines with increasing Weissenberg number on our �nest mesh, NM3. When the

mesh is su�ciently re�ned, we can see clearly that, with a discontinuity capturing technique

applied at the corner (scheme FV4.1), or without (scheme FV4), these fv schemes indicate the

persistence of the lip vortex beyond We = 2, say with We of O(3) (see table 3). This must

be regarded as a convincing trend, though one cannot be categoric on this point without still

further mesh re�nement. Unfortunately, further re�nement to the corner stimulates stronger

stability restrictions and these relatively high levels of elasticity are increasingly more di�cult

to access for the Oldroyd model. In addition, the existence of a small trailing-edge vortex is

con�rmed on the downstream wall, immediately after the re-entrant corner. Other authors may

well have overlooked such a feature, even with very re�ned meshes [10, 11], arguably because

of their staggered grid discretisation. Moreover, such a feature has been observed by Dennis

and Smith [40] in the viscous context, who showed that with considerable mesh re�nement, a

post-corner trailing-edge vortex was present in the Newtonian case for Reynolds numbers in

excess of 150 for ow in a 2:1 contraction geometry. In the viscoelastic regime and for the

present problem, Baloch et al. [41] hinted at the existence of such a trailing-edge vortex when

they incorporated a local discontinuity capturing technique into their Taylor-Galerkin fe/supg

implementation (see also Carew et al. [42]), following a Galerkin-Least-Squares approach. This

was observed at We = 5, for a pseudo-steady-state Oldroyd-B solution. We note, in passing,

the intensity of the lip vortex increases with We on each mesh and for both level four schemes,

as indicated in table 3. Generally, the �nest mesh NM3 provides lowest intensity levels for

each scheme. The strength of the salient corner vortex is charted in table 4, where the �nest

mesh values re ect a gradual decline with increasing We. These trends are in agreement with

Matallah et al. [1], and Alves et al. [11].

The cell size of the salient corner vortex (X), is de�ned as the length between the salient

corner and the intersecting point of the vortex separation line and the upstream wall. For this

problem, X has been reported widely to remain fairly constant (at around 1.4) with increasing

elasticity [1, 8, 10, 12, 38]. This we con�rm to be the case on the coarser mesh M2 with

scheme FV4.1, the identical mesh used to produce the fe/supg results of Matallah et al. [1].

All consistent methods agree with these �ndings on coarser meshing. Nevertheless, on our

�nest mesh NM3, table 5 suggests that the size of the salient corner vortex infact decreases

with increasing Weissenberg number. This is only apparent with su�cient mesh re�nement

as demonstrated on mesh NM3 at We = 2:8, with 18% reduction in X from the position at

We = 0:1. Indeed, Alves et al. [11] reported a signi�cant decrease in salient corner cell size for

an UCM uid (38%, asWe was raised from zero to three). By using consistent mesh re�nement

and applying Richardson's extrapolation, these authors showed that the uncertainty of their

17

results was below 2:5% for We � 3. Figure 16 demonstrates satisfactory agreement between

FV4.1 (or FV4) on mesh NM3, and the min-mod scheme of Alves et al. on their third and

fourth �ner meshes, both in trends and magnitude (see table 5 for relevant data).

6.2 Entry pressure drop

The entry pressure, as a function of uid elasticity or inertia, is an important quantity in con-

traction ows, that can be measured experimentally. Tanner [43] provides a review of available

measurements, empirical expressions, and numerical results. For shear-thinning uids, experi-

mental data demonstrate an increase of pressure drop with increasing We (or ow rate) [44].

This behaviour has been reproduced numerically, see for example Keunings and Crochet [45].

There, using a linear PTT uid in an axisymmetric contraction ow, these authors showed, with

increasing elasticity, an initial decrease in entry pressure drop (de�ned via Couette correction,

see on), followed with a monotonic increase. In contrast, Keunings and Crochet [45] reported

that, for an Oldroyd-B uid, the entry pressure drop always decreases, and this is in agreement

with results of Basombr�io et al. [22], Carew et al. [24], Yoo and Na [38] and Keiller [46].

The Couette correction is de�ned as

C =�pe � Lu ��pu � Ld ��pd

2�w; (6.1)

where �pe is the total pressure drop at the centerline of the contraction; �pu and �pd represent

the upstream and downstream fully-developed pressure gradients, respectively; Lu and Ld are

the upstream and downstream channel lengths, and �w is the exit ow fully-developed shear

stress at the wall. Figure 17 displays a plot of Couette correction against We for the series of

meshes employed and scheme FV4.1. Mesh convergence is apparent, and the Couette correction

drops to negative values as We rises [22, 38, 45, 46]. We note that C decreases linearly with

increasing We at a slope of ??. This trend is argued and reported by Keiller [46] in his study of

entry contraction sink ow, considering axisymmetric and planar geometries. In addition, our

numerical results correspond to the situation where We is elevated via the uid parameters at

a �xed owrate, not the reverse.

6.3 Stress patterns

Stress contour lines emphasise solution convergence with mesh re�nement for scheme FV4.1,

con�rming scheme convergence on the �nest mesh, NM3. In �gure 18, we contrast the three

stress components obtained on mesh M2 against those on NM3 for scheme FV4.1 at We = 1.

These is no discernible di�erence in the patterns in core ow away from the corner: solutions

on mesh NM3 are smoothed considerably in the corner region, as expected. We report similar

comparisons in �gure 19, but at the higher value of Weissenberg number, We = 2. The same

18

conclusions hold and there is a clear indication of the build up of boundary layers in �xx and �xy

with We increase, �nely resolved on mesh NM3. Here, we note the relevant work of Renardy

[47], Hagen and Renardy [48], with respect to width of stress boundary layers for various

models and ows (width of We�1 for Oldroyd-B and this ow). These �ndings are in close

correspondence with those of Phillips et al. [10]. On our �nest mesh NM3 at We = 2, and

with corner treatment introduced into scheme FV4 producing scheme FV4.1, almost identical

patterns are observed in all stress components on both schemes. Only minor modi�cation in

the corner region is apparent with FV4.1. This demonstrates how the local treatment of the

re-entrant corner solution is crucial in guaranteeing stability on coarser meshes, and essential

for accuracy if the �ner features of the ow are to be captured on the more re�ned meshes.

6.4 Loss of evolution

The ability of a numerical scheme to retain the evolutionary character of the underlying equation

system may be used as a stability indicator for that particular scheme [8, 10, 22, 24, 49, 50].

Dupret and Marchal [51] proposed a criterion to indicate loss of evolution for a Maxwell uid,

via the condition number S,

S = 2s1 � s2s12 + s22

; (6.2)

where s1 and s2, accordingly, are the non-trivial eigenvalues of the tensor TA = � + �e

WeI. With

frozen initial and boundary conditions applied, S should remain positive throughout the ow

domain to avoid the loss of evolution. Connected regions of such behaviour would be particularly

disastrous, that would consequently lead to numerical divergence. Strictly speaking, the theory

applies to the strong solution of a Maxwell model. Therefore, in the case of the Oldroyd model,

the condition number S is only an indicator of failure in scheme evolution, relying upon similar

discrete structure.

Figure 20 a displays �eld plots of negative values of the condition number S on mesh M2,

contrasting the introduction of multi-valued stress gradients (scheme FV3) against the single-

valued alternative (scheme FV2.1, We = 0:8), see also �gure 9 and section 5.5. Scheme FV3

has repaired the loss of evolution (Smin > 0), and this is re ected in its stability response, as

this scheme reaches a Wecrit value of three. The enhanced stability properties of the FV4.1

implementation, with a local discontinuity capturing technique at the reentrant corner, over

scheme FV4 (without same) is illustrated in �gure 20 b. At the higher levels of elasticity

(We = 2:4), scheme FV4.1 also shows signs of loss of evolution. Nevertheless, it does decrease

the minimum negative level of S by a factor of one half, and the number of nodes infected

accordingly, see also �gure 11 b.

19

Next, we proceed to investigate the relationship between the behaviour of factor S and the

appearance of lip and trailing-edge vortices. In �gure 21, we report the condition number on

our �nest mesh NM3, for schemes FV4 and FV4.1, with increasing We. We comment that

for We < 1, there is a complete lack of negative values of S. At We = 1, there is a single

negative value located at the corner. For higher We, isolated sporadic points occur that are

restricted to the downstream wall, just beyond the corner. This is a strong indication that the

appearance of the lip vortex (We > 1, see �gure 14) may trigger the local loss of evolution at

isolated points. The question arises, as to whether it is this resultant loss of evolution that gives

rise to the trailing-edge vortex. This is best discussed with recourse to the coarser mesh M2.

At We = 2, scheme FV4 exhibits a signi�cant lip vortex without a trailing-edge vortex, while

scheme FV4.1 displays both (�gure 12 b). However, �eld plots for S factor indicate similar

polluted zones for the two schemes (see �gure 22). Hence, the appearance of negative values in

S may be attributed to the onset of the lip vortex.

6.5 Asymptotic behaviour

A number of theoretical calculations have been conducted to predict the singular asymptotic

behaviour of the stress at the re-entrant corner [14, 15, 16, 52]. For the ow of an Oldroyd-B

uid past a sharp corner and for a given angle �, Hinch [14] established that velocities vanish

asymptotically, approaching the corner with form r5=9; likewise, stresses are singular as r�2=3.

Here, (r,�) represent a polar coordinate system of reference, centred at the corner. Hinch

assumed the absence of lip vortices, so that the ow is viscometric near the walls, away from

the corner. In a subsequent paper, Renardy [16] con�rmed the �ndings of Hinch, providing the

angle � is not too small, noting that Hinch's formulation violates the momentum equation near

the walls.

Here, we consider two such angles: � = �=2 and � = �. The discussion is restricted to

scheme FV4.1, since we have found no signi�cant di�erences with scheme FV4. All results

are plotted for the four meshes at We = 1 (prior to lip-vortex onset), and illustrate mesh

convergence. At the cross-stream angle (� = �=2), Ur re ects the theoretical decay (�gure

23 a). For stress, likewise, �rr and �r� reproduce the r�2=3 slope reasonably well (�gures 24 a

and b). U� and ��� deviate slightly from the predicted slopes, see �gures 23 b and 24 c. Alves

et al. [11] reported similar conclusions for the ow of an UCM uid through an abrupt planar

contraction. Upon considering the asymptotic response in the upstream direction (� = �,

�gures 25 and 26), we found satisfactory agreement in all quantities between the computed

solutions and the analytical slopes. Such agreement is held to be strong con�rmation of the

high-order of accuracy in the solutions thus generated.

20

7 Conclusions

We have fashioned a stable and accurate implementation of our novel cell-vertex hybrid �nite

volume/element scheme, for abrupt planar contraction ow of an Oldroyd-B uid. This has been

achieved through a systematic study of stress behaviour along a line intersecting the downstream

wall. The post-corner oscillations, that are commonly observed when stresses are calculated

at the re-entrant corner, are alleviated once a non-conservative form of the ux is invoked,

along with a consistent treatment of the transient stress update, and a recovery technique for

velocity gradients. Linear stress representation, with non-recovered stress gradients, proved

crucial to enhance stability. We have also highlighted the importance of incorporating a local

discontinuity capturing technique for the re-entrant corner solution, achieved here via reduced

integration. Accuracy has been established in a number of ways: �rst, we have demonstrated

convergence with mesh re�nement of salient corner vortex size, entrance pressure drop, and

stress contour plots; second, we have found close agreement with the theoretical predictions, in

asymptotic behaviour of velocity and stress, as one approaches the re-entrant corner. Finally,

we have shown that our results concur with the literature for the size of the salient corner vortex,

and re ect the linear decrease of the Couette correction with increasing elasticity. We con�rm

the existence of a lip vortex for this problem (for We > 1), that grows in size and intensity

with increase of We. Alternatively, the salient corner vortex decreases in size and strength

with increasing We, for such a constant viscosity model. Additionally, we have encountered

the existence of a trailing-edge vortex, accompanying the onset of a lip vortex. It is clear that

the appearance of negative values in the condition number S correlates with the presence of

a lip vortex. There is no evidence to suggest that these highly localised negative values are

associated with the trailing-edge vortex activity.

Here, the use of the Oldroyd-B model has allowed a measure of scheme performance in

terms of stability to be established. There is no doubt that the presence of a re-entrant corner

has been all important in providing the sharpness of solution to distinguish the independent

properties of the proposed scheme combinations. In a related article on contraction ows, we

proceed to consider three important steps: rounding the corner to produce smooth complex

ows, thus removing the uncertainties associated with the corner singularity; retaining the

Oldroyd-B model, to assess stability and to place current results in context against the abrupt

corner ow; and �nally, variations in rheology, with recourse to Phan-Thien-Tanner constitutive

models (linear and exponential), to consider alternative uid behaviour.

Promising future directions to advance and apply this fe/fv methodology include, conside-

ration of time-dependent and free-surface problems, three-dimensional ows and the extreme

conditions identi�ed in industrial processes. There, the attractive properties o�ered of e�-

ciency, accuracy and stability, identi�ed thus far, await to be further tested. One particularly

21

interesting possibility is to investigate time accurate solutions, via uctuation redistribution.

22

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24

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25

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26

Scheme uuu ruuu ��� r��� RRT CT C.R.I. Wecrit { M2

FV0 2 1 Rec. 2 1 Rec. � 0 � 1.1

FV1 2 1 Rec. 1 0 Rec. � 0 � 1.6

FV1.1 2 1 Rec. 1 0 Rec. � 1 � 1.7

FV1.2 2 1 Rec. 1 0 Rec. � 2 � 2.0

FV2 2 1 Rec. 2 1 Rec. � 2 � 0.3

FV2.1 2 1 Rec. 1 0 Rec. � 2 � 0.8

FV3 2 1 Rec. 1 0 Mul. � 2 � 3.0

FV4 2 2 Rec. 1 0 Mul. � 2 � 2.4

FV4.1 2 2 Rec. 1 0 Mul. � 2p

3.7

Table 1: Table of schemes; for uuu, ruuu, ��� , and r��� : 2, 1, and 0 indicate quadratic, linear and constant represen-

tation, respectively; Rec. and Mul. refer to recovered and multi-valued options; � and � indicate line and area

integral; and C.R.I. stands for corner reduced integration.

Meshes Elements Nodes Degrees of freedom Rmin Densities

||||||||{ |||{ (element/salient

uuu, p, ��� corner)

M1 980 2105 11088 0.024 28

M2 1140 2427 12779 0.023 63

M3 1542 3279 17264 0.019 135

NM3 2987 6220 32717 0.011 201

Table 2: Mesh characteristics.

27

We Scheme M1 M2 M3 NM3

1.0 FV4 � � � 0.011

FV4.1 � � � �1.5 FV4 0.385 0.577 0.697 0.085

FV4.1 0.145 0.267 0.538 0.020

2.0 FV4 1.066 1.272 1.482 0.126

FV4.1 0.334 0.880 1.053 0.079

2.5 FV4 1.578 (D) (O) 0.177

FV4.1 0.420 1.365 1.773 0.286

2.8 FV4 1.813 (D) (O) (D)

FV4.1 0.372 1.635 (D) 1.192

Table 3: Lip vortex intensity (�10�3); (D) indicates numerical divergence, and (O) temporal oscillations.

28

We Scheme M1 M2 M3 NM3

0.1 FV4 0.605 0.962 0.989 0.860

FV4.1 0.603 0.960 0.987 0.860

1.0 FV4 0.665 0.956 0.944 0.816

FV4.1 0.643 0.918 0.913 0.811

1.5 FV4 0.687 0.897 0.924 0.647

FV4.1 0.614 0.807 0.850 0.645

2.0 FV4 0.706 0.867 0.921 0.447

FV4.1 0.573 0.677 0.767 0.489

2.5 FV4 0.978 (D) (O) 0.368

FV4.1 0.557 0.622 0.801 0.377

2.8 FV4 1.256 (D) (O) (D)

FV4.1 0.581 0.610 (D) 0.350

Table 4: Salient corner vortex intensity (�10�3); (D) indicates numerical divergence, and (O) temporal oscilla-

tions.

We Scheme M1 M2 M3 NM3

0.1 FV4 1.500 1.400 1.600 1.412

FV4.1 1.500 1.400 1.600 1.412

1.0 FV4 1.500 1.400 1.600 1.358

FV4.1 1.500 1.400 1.600 1.358

2.0 FV4 1.500 1.400 1.500 1.203

FV4.1 1.500 1.400 1.500 1.203

2.8 FV4 1.500 (D) (O) (D)

FV4.1 1.500 1.400 (D) 1.154

Table 5: Salient corner vortex cell-size, X , across mesh with increasing elasticity, FV4 and FV4.1; (D) indicates

numerical divergence, and (O) temporal oscillations.

29

fe triangular elementfv triangular sub-cellsfe vertex nodes (p;uuu;���)fe midside nodes (uuu;���)fv vertex nodes (���)

Figure 1: Schematic diagram of a fe-cell with four fv sub-cells, and variable location.

i

j

k

0aaa

1

2

Figure 2: Graphical representation of LDB-scheme de�ning 1 and 2 in a fv triangular cell.

30

T1

T2

T3

T4

T5 T6

l

Figure 3: Control volume of a given node l with its unique associated median dual cell (MDC) area.

31

M1 M2

M3 NM3

Figure 4: Magni�cation of the set of meshes around the re-entrant corner.

32

0

5

10

15

20

25

30

17 18 19 20 21 22 23 24 25

Txx

x

We=1.0

FE/SUPGFV0

Figure 5: FE/SUPG versus Standard fv (FV0) { Line integral RT , We = 1:0.

0

5

10

15

20

25

30

17 18 19 20 21 22 23 24 25

Txx

x

We=1.0

FE/SUPGFV0 FV1

Figure 6: Quadratic (FV0) versus Linear Stress (FV1) { Line integral RT , We = 1:0.

33

0

5

10

15

20

25

30

35

40

17 18 19 20 21 22 23 24 25

Txx

x

We=1.5

FE/SUPGFV1.1 FV1.2

Figure 7: Consistent Time Term treatments, FE/SUPG and FV1.1 (CT1), FV1.2 (CT2) { Linear stress, Line

Integral RT , We = 1:5.

34

0

5

10

15

20

25

17 18 19 20 21 22 23 24 25

Txx

x

We=0.5

FE/SUPGFV1.2 FV2.1

Figure 8: FE/SUPG versus FV Line (FV1.2) and Area (FV2.1) Integration variants for RT Linear stress, CT2,

We = 0:5

.

0

5

10

15

20

25

30

35

40

17 18 19 20 21 22 23 24 25

Txx

x

We=0.8

FE/SUPGFV2.1 FV3

Figure 9: E�ect of the Non-recovered Stress gradient : FE/SUPG versus (FV2.1 and FV3) Linear Stress, Area

RT , CT2, Corner Region; Multi-valued stress gradient FV3.

35

0

5

10

15

20

25

30

17 18 19 20 21 22 23 24 25

Txx

x

We=1.0

FE/SUPGFV3 FV4

Figure 10: FE/SUPG vs FV3 (Linear ruuu) vs FV4 (Quadratic ruuu) { Stress Line Pro�les along y = 3:0

36

a) �xx

0

5

10

15

20

25

30

35

17 18 19 20 21 22 23 24 25

Txx

x

We=1.0

FE/SUPGFV4

FV4.1

b) �yy

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

17 18 19 20 21 22 23 24 25

Tyy

x

We=1.0

FE/SUPGFV4

FV4.1

Figure 11: FE/SUPG vs FV4 (without Corner Reduced Integration) vs FV4.1 (with C.R.I) { Stress Line Pro�les

along y = 3:0

37

FV4 FV4.1

a) We = 1:0

b) We = 2:0

c) We = 3:0

d) We = 3:7

Figure 12: Streamlines with increasing We. Mesh M2

38

FV4 FV4.1

a) M1

b) M2

c) M3

d) NM3

Figure 13: Streamlines with mesh re�nement. We = 2:0

39

FV4 FV4.1

a) We = 1:0

b) We = 2:0

c) We = 2:5

d) We = 2:8

Figure 14: Streamlines with increasing We. Mesh NM3

40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5 4

X

We

FV4.1 Mesh M2 Matallah et al. [1]

Marchal and Crochet [8]Phillips and Williams [10]

Sato and Richardson [12]Yoo and Na [38]

Figure 15: Size of the salient corner vortex with increasing We; FV4.1 Mesh M2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3

X

We

FV4.1 Mesh NM3 Alves et al. Mesh 3 [11]Alves et al. Mesh 4 [11]

Figure 16: Size of the salient corner vortex with increasing We; FV4.1 Mesh NM3 vs Alves et al. [11] (min-mod

scheme).

41

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

C

We

FV4.1

M1 M2 M3

NM3

Figure 17: Couette Correction vs We, with mesh re�nement, scheme FV4.1.

42

M2 NM3

a) �xx

0.0

1.02.0

3.0

0.0

1.02.0

3.0

b) �xy

0.0

0.5 1.0

0.0

0.5 1.0

c) �yy

0.0

0.0

0.4 0.8

0.0

0.0

0.4 0.8

Figure 18: Stress Contour lines , We = 1:0, FV4.1.

43

M2 NM3

a) �xx

0.0

1.0

2.03.0

0.0

1.0

2.03.0

b) �xy

0.0

0.5 1.0

0.0

0.5 1.0

c) �yy

0.0

0.0

0.40.8

0.0

0.0

0.40.8

Figure 19: Stress Contour lines , We = 2:0, FV4.1.

44

FV2.1, We = 0:8, Mesh M2. FV3 , We = 0:8, Mesh M2.a)

Smin = �0:4508 Smin = 0:0532

FV4 , We = 2:4, Mesh M2. FV4.1, We = 2:4, Mesh M2.b)

Smin = �0:9998 Smin = �0:4315

Figure 20: Steady-state negative values of the condition number S; 11 contours between

�1 and 0; a) Single-valued (FV2.1) vs Multi-valued (FV3) stress gradient; b) Recovered

velocity gradient wihtout Local corner treatment (FV4) and with Reduced integration at

the corner (FV4.1).

45

FV4 FV4.1

x

a) We = 1:0

Smin = �0:8620 Smin = �0:9913

x x

b) We = 2:0

Smin = �0:6109 Smin = �0:9712

x x

c) We = 2:5

Smin = �0:9957 Smin = �0:9817

d) We = 2:8

Smin = �0:8458

Figure 21: Condition number S with increasing We; Mesh NM3.

46

FV4 FV4.1

x x

Smin = �0:7031 Smin = �0:3298

Figure 22: Condition number S at We = 2; Mesh M2.

47

a) Ur

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Log(r)

NM3 M3 M2 M1

Slope 5/9

b) U�

-0.8

-0.6

-0.4

-0.2

0

0.2

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Log(r)

NM3 M3 M2 M1

Slope 5/9

Figure 23: Asymptotic behaviour of the velocity near the re-entrant corner, along the direction � = �=2 (cross-

stream). FV4.1, We = 1:0.

48

a) �rr

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Log(r)

NM3 M3 M2 M1

Slope -2/3

b) �r�

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Log(r)

NM3 M3 M2 M1

Slope -2/3

c) ���

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0Log(r)

NM3 M3 M2 M1

Slope -2/3

Figure 24: Asymptotic behaviour of the stress near the re-entrant corner, along the direction � = �=2 (cross-

stream). FV4.1, We = 1:0.

49

a) Ur

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

(Ur)

Log(r)

FV4.1

NM3 M3 M2 M1

Slope 5/9

b) U�

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

(Ut)

Log(r)

FV4.1

NM3 M3 M2 M1

Slope 5/9

Figure 25: Asymptotic behaviour of the velocity near the re-entrant corner, along the direction � = � (upstream).

FV4.1, We = 1:0.

50

a) �rr

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

(Trr

)

Log(r)

FV4.1

NM3 M3 M2 M1

Slope -2/3

b) �r�

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

(Trt

)

Log(r)

FV4.1

NM3 M3 M2 M1

Slope -2/3

c) ���

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Log

(Ttt

)

Log(r)

FV4.1

NM3 M3 M2 M1

Slope -2/3

Figure 26: Asymptotic behaviour of the stress near the re-entrant corner, along the direction � = � (upstream).

FV4.1, We = 1:0.

51