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NUMERICAL PR OCED UR E FOR THE C OMPUTATION OF FLUID FLOW WITH ARBITR ARY STR ESS-STRAIN RELATIONSHIPS Pau lo J. Oli v eira à Departam ento de Engenh aria Electromecanica, Un iv ersidade d a Beira Interior, 6200 Co v ilha,Ä Portugal
Fernando T. Pinho Centro de Estudos de Fenom  enos de Transporte, DEMEG I, Faculdade de Engenh aria, Univ ersidade do Porto, 4099 Porto Codex, Portugal A finite- v olume method is presented th at allows for general stress-strain constituti v e equ ations to be incorporated into a standard momentum ± pressure-correction procedure. The method is sequenti al and segregated in nature, the v arious equ ations for mass and momentum conser v ation and for the e v olution of the stress tensor are sol v ed following a predefined order, and one of its features is the use of nonstaggered, and generally nonorthogon al, computational meshes. Two types of constituti v e equations are used to test the method: the standard explicit and algebraic Newtonian model, and one of the simplest implicit differential equ ations, the upper-con v ected Maxwell model. In spite of its apparent simplicity, this latter model is known to pose the most se v ere numerical difficulties. Howe v er, the results in this article show the method to be effecti v e in sol v ing the equations for the flow of Newtonian and v iscoelastic fluids through abrupt plan ar contractions with an area reduction of 4 to 1, one typical benchm ark problem. The results are compared with a v ailable data and with solutions from a standard and v alidated code, and good agreement and consistency is found. A new formulation to e v alu ate stresses at cell faces is presented and shown to lead to improv ed results.
1 . INTR OD UC TION Most procedures to calculate fluid flow are based on Newtonian constitutive stress-strain relationships that are implicitly e mbedde d into the flow equations, resulting in the usual Navie r-Stoke s e quations. Howe ver, ve ry often real fluids e xhibit non-Ne wtonian viscoe lastic be havior, and in such case s it is ne ithe r advantageous nor always possible to find an explicit relation be twee n the stress tensor and ve locity gradie nt components to substitute into the momentum e quations. As a conseque nce, it is useful to deve lop nume rical procedures adequate for ge ne ral stress tensors, which are often relate d with the flow kine matics in a complex and implicit manne r via additional differential e quations, thus separating the stress calculation from the solution of the momentum e quations. The de velopme nt of such a method is the scope of the present article. The stress-strain relationships to Receive d 2 October 1998; accepte d 9 De ce mber 1998 The authors acknowledge the financial support of Junta Nacional de Inve stigacË ao Ä Cientõ  fica e s JNICT. under project PBIC r 1980. The authors are listed alphabetically. Te cnologica  Address correspondence to Dr. P. J. Oliveira, Unive rsidade da Beira Interior, Departame nto de Eng a E lectromecanica, à 6200 Covilha, Ä Portugal. E-mail:
[email protected] 295
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P. J. OLIVEIRA AND F. T. PINHO
NOMENC LATURE aP , aF bli B, B l i De f x, f y Ff H L1, L 2 N1 p, P Re S t Ti j u i, s u, v .
U xi s x, y. xR
coe fficients in the discretized e quations coe fficients in the discretized stress e quations are a, i component of are a of a ce ll surface aligned with direction j l De borah number s s l U r H . e xpansion r contraction factors to distribute cell spacing m ass flow rate across cell face f half-width of downstream ch anne l lengths of upstream and downstream channels primary normal-stress difference pressure, normalized pressure s s p r s m U r H .. Reynolds number s s r UH r m . source term in the discretized e quations time normalized stress components s s t i j r s m U r H .. Carte sian velocity components s streamwise and cross-stream components.
d
ij
d x, d y d t w D u xl l m
n
r
t
j
l ij
C
Su bscripts an d Su perscripts i, j, k l, m P, F f n, n q 1
Ä
average velocity in downstream channel Carte sian coordinates s stre amwise and cross-stream . re circulation length
identity tensor ce ll sizes s normalized as d x r H . time step difference between u values along direction l re laxation time viscosity coe fficient ce ll volume density general coordinates Cartesian components of the e xtra stress tensor stre am function value
9 *
Cartesian components s from 1 to 3. directions along ge neral coordinates s from 1 to 3 . generic control volume and neighbor s F from 1 to 6. ce ll face between ce lls P and F s varie s from 1 to 6 . denotes previous and prese nt time level, re spectively special cell-face interpolation linear interpolation divided by central coefficient a P intermediate value s s calculated implicitly.
be considered will eithe r follow the Newtonian viscosity law, but in this case with the stress compone nts e valuate d from inde pende nt equations and then incorporate d into the stress-diverge nce term of the momentum equations, or are give n by a line ar differential e quation for the e volution in time of the stress tensor. Future work will conside r the case of quasi-line ar and nonline ar stress-strain relations. The de ve lope d numerical procedure is of the finite -volum e type along the lines e xpose d by Patankar w 1 x , but incorporating modern technique s such as the use of nonstagge red and nonorthogonal meshe s which allows for gre ate r ve rsatility in terms of flow geometry w 2, 3 x . The important contribution that m ade possible the widespread use of the nonstagge red mesh arrange ment was the nove l interpolation scheme of Rhie and Chow w 4 x for de te rmining ve locity value s at cell faces, and a similar philosophy has be en recently advance d by O live ira e t al. w 5x for the proble m of de te rmining stress value s at cell faces. We shall use he re an improvement of this latte r method to inte rpolate the stress compone nts require d in the stressdive rge nce term of the line ar momentum e quation. The numerical procedure is then applie d to a typical proble m often used as a te st case in the non-Ne wtonian community w 6 ] 10 x : the flow of both Newtonian and viscoelastic fluids obe ying the
FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS
297
upper-convected Maxwe ll mode l s UCM. through a 4-to-1 planar contraction. This particularly simple ge ometry doe s not require the nonorthogonal capability of the procedure , a matte r left for future inve stigation. It pose s, howeve r, seve re numerical difficultie s, especially for the UCM fluid w 10, 11x , because of unbounde d stresses at the ree ntrant corner with a ve ry inte nse localize d growth rate for all stress compone nts; furthermore, for this fluid model the resulting flow fe atures are not ye t fully understood s see revie ws in w 11x and w 12x ..
2. GOVERNING EQUATIONS AND CONSTITUTIVE MODELS The basic e quations to be solve d are those expre ssing conse rvation of mass,
- r uj - xj
s 0
s 1.
and of line ar momentum,
- r ui - t
q
- r u jui - xj
s y
- p - xi
q
- t
ij
s 2.
- xj
In these e quations u i is the velocity component along the Carte sian axis xi , r is the fluid de nsity, and the total Eule rian stress te nsor s i j has be en de composed into an isotropic pre ssure te rm plus an e xtra stress tensor t i j , as s i j s y p d i j q t i j , whe re d i j is the identity te nsor. Here we will be concerned with incompre ssible and steady flow, and the main de pe ndent variable s to be solve d for are the ve locity components, the pre ssure, and the stress components. The ve locity compone nts result from the momentum e quation s 2 ., for incompressible flow the pressure is an arbitrary field required to constrain the ve locity fie ld so that it conforms to e quation s 1 ., and the stress components must be give n by a rheological constitutive e quation. In ge neral, the e quation for the stress tensor t i j is of the hype rbolic type w 13 x and is derived e ithe r from continuum mechanics or from kine tic theory w 14 x , whe re force balance s acting on simplified mode ls of molecular be havior are utilize d. The precise form of these equations varie s according to the fluid considered, but must respe ct some ge neral rules such as obje ctivity, realizability, etc., and must also provide physic ally realistic responses and preferably be supporte d by kine tic theory argume nts. In this article , two constitutive models are considered. First, and in order to compare the pre sent method with standard proce dures, we consider the well-known Newtonian mode l give n by the alge braic e xplicit stress-strain relationship,
t
ij
s m
t
- ui - uj q - xj - xi
/
y
2 3
m
- uk d - xk
ij
s 3.
whe re m is the constant dynamic viscosity of the fluid, and the ve locity dive rge nce s div u ’ - u k r - xk . in the last term vanishe s for incompressible flows but is
298
P. J. OLIVEIRA AND F. T. PINHO
ne verthele ss retaine d be cause it is not exactly zero in the numerical approxim ation whe n the nonstagge red mesh arrangement is utilized. Beside s, inclusion of this term le ads to more accurate results, as will be shown in Se ction 4. The second constitutive e quation considered for the e xtra stress compone nts t i j is the upper-convected Maxwe ll model s UCM. w 13 x ,
t
ij
q l
t
- t
ij
- t
q
- u kt - xk
ij
/ t s m
q l
- ui - xj
t t
- uj
q
- xi - uj
ik
- xk
y
q t
2 - uk 3 - xk
- ui jk
- xk
d
/
ij
/ s 4.
whe re m can still be seen as a constant viscosity coefficie nt and l is anothe r parame te r of the mode l with the dimensions of time and commonly calle d the relaxation time of the fluid. Equation s 4 . is one of the simple st mode ls to represe nt viscoe lastic fluid behavior, and it is note d that the Newtonian mode l is obtaine d as a spe cial case of the UCM model whe n l is set to zero. Howeve r, in the gene ral case of l / 0, the UCM e quations introduce conside rable complication into the proble m of de termining the motion of a fluid: six new implicit differential equations on the stresses have to be solve d in conjunction with the momentum and mass conservation equations. Furthermore , although Eq. s 4 . is line ar on the stresses t i j , when couple d with Eq. s 2 ., implicit nonline arities will arise through the terms with ve locity gradie nts.
3 . NUMER IC AL METHOD The diffe rential equations are integrated ove r control volum es in ge ne ral body-fitte d computational meshes s j l . with the help of standard discretization procedure s w 1, 2 x . The depe nde nt variable s are the three Cartesian ve locity compone nts, the pressure, and the six Cartesian stress components; all the se variable s are stored at the cente r of the control volume s, as implied by the nonstagge red mesh arrange ment adopte d he re. Spe cial procedures are thus require d to avoid pressure ] ve locity de coupling w 4 x and stress ] ve locity decoupling w 5 x . Details of the discretization procedure can be found in the latter reference and he re, for the sake of concisene ss, only the constitutive equation will be tre ate d with more de tail. Discretization of the continuity Eq. s 1 . give s 6
p
Ff s 0 f
with Ff s
p s
r B f j uÄ j . f
s 5.
j
e xpre ssing m ass conservation: the sum of the outgoing mass flow rate s s Ff . over the six faces de limiting any give n cell must vanish. The tilde over the ve locity is used to de note a spe cial Rhie-and-Chow type of inte rpolation; the pre cise form is as give n in w 15x . Here, and in the following, the i compone nt of the are a of any cell surface oriented along direction j l is de noted Bl i ; if this surface coincide s with an actual cell face with dire ction j f , the n we write B f i as the i component of that cell r face are a s the scalar are a is B f s s p j B f j B f j .1 2 ..
FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS
299
After discretization over any cell P, the momentum e quation can be written under the standard line arize d form: 6
p
aP s u i . P s
a F s u i . F q Su i
s 6.
F
whe re the coefficie nts a F are made up of conve ctive and diffusive contributions, whose precise form de pends on the differencing schemes adopte d, and the central coefficie nt is aP s
r n d t
q a0
with a 0 s
p
s 7.
aF
the sum being again over the six cell neighbors s F . surrounding the cell P under conside ration. Here d t is the time step used in the time-marching computations. The source term in Eq. s 6. results from all contributions in the momentum e quation that have not be en include d into the coefficients and, from inspe ction of Eq. s 2., these are Su i s Su i y p r e s q Su i y str e ss q Su i y d t q Su i y H O S q Su i y di ff
s 8.
for the pressure gradie nt, the stress dive rgence, the inertia term, a possible term arising from use of a high-orde r diffe rencing scheme s HOS. for convection, and an adde d diffusion term that cancels e xactly the diffusion contribution in the coefficients, since there is no e xplicit diffusion in the original equation w Eq. s 2 .x . The discretized stress equation s 4 . can be cast into a form similar to Eq. s 6 . as at P s t
6
. ij P s
p
at F s t
. q St
ij F
F
s 9.
ij
with the central coefficient now give n by at P s n q
l n d t
q at 0
s 10 .
and the other coefficie nts compose d only of conve ctive fluxe s due to the abse nce of diffusion in the constitutive equation s 4.. It is straightforward to arrive at the source te rm in the stress equation after transforming the Cartesian de rivative s in Eq. s 4 . into de rivative s with respe ct to the ge neral coordinate s, by making use of the rule - r - xi s Jy1b l i - r - j l , and re alizing that in the discretized e quations the Jacobian J be comes a cell volum e n and the metric coefficie nts b l i be come are a components Bl i w 16x ; that source term can the n be written as 3
St i j s
p l
q
t
bli w D u j x l q bl j w D u i x l y
l n d t
st
n. ij P
q St
i j y HO S
2 3
tp
/
m Bl k w D u k x l d k
ij
/
P
s 11 .
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P. J. OLIVEIRA AND F. T. PINHO
whe re the te rm multiplie d by 23 results from ke eping the div u term in the Newtonian part of the constitutive e quation and the b coefficients are give n by b l i s m Bl i q l
Bl kt
p
s 12 .
ik
k
O ne key fe ature in the nume rical procedure is the de termination of the stress components at cell faces s t Ä i j . required for the dive rge nce term in the momentum e quation, that is, 6
p
Su i y s tr e ss s
f
tp
B f jt Ä i j j
/
s 13 . f
In order to avoid stress-ve locity decoupling and along the lines of the procedure give n by w 5x , the face stresses are de fined as
s t Ä ij. f ’
Xt aF s t
p
. q
ij F
t
p l/ f
F
X X bli w D u j x l q bl j w D u i x l y
q bÄ Xf i w D u j x f q bÄ Xf j w D u i x f y
2
p 3
2
m BXl k w D u k x l d
p
3
ij
m BÄ Xf k w D u k x f d
ij
/
f
k
q
k
t / l n
9st
d t
n. ij P
q St X y H O S ij
s 14 . whe re the ove rbar here de notes line ar inte rpolation at the cell face position, and the prime de notes division by the central coefficient at P . It is simple r to use the following expression, obtaine d after dividing Eq. s 9 . by the central coefficie nt, applying line ar ave raging to it, and subtracting the resulting e quation from Eq. s 14.:
t /
s t Ä ij. f s t
ij f
t
y b Xf i w D u j x f q b Xf j w D u i x f y
t
q bÄ Xf i w D u j x f q bÄ Xf j w D u i x f y
2 3
2 3
m BXf k w D u k x f d
p k
m BÄ Xf k w D u k x f d
p
ij
k
/
ij
/ s 15 .
with
X bÄ f i ’
t
m Bf i q l
B f jt
p j
n fs aP r n t
P
.
ij
/
f
and
BÄ f i ’ X
Bf i
n f s at P r n
P
.
s 16 .
Expre ssion s 15. is e ssential to avoid the proble m of stress ] ve locity de coupling and is only slightly different from that proposed in our previous work w 5x , ye t it improve s the results, yie lding be tter stress inte rpolation whe n the mesh spacing change s abruptly, as will be shown be low.
FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS
301
Solution Procedure The sets of discretized e quations are solve d in a sequential manne r following a pse udo-time -marching approach de scribed be low, whe re ne w or inte rmediate value s are de note d with an aste risk, and value s from the pre vious time step with inde x n. 1. O btain cell-ce nte red stresses s t at P s t
U ij
U
. ij P s
. from the implicit e quation w Eq. s 9 .x , 6
at F s t
p
U . ij F
q
s
St
ij
.
n
s 17 .
F
and store the central coefficie nt at P . 2. Compute cell face stresses s t Ä i j . from Eq. s 15 ., base d on newly obtaine d U stresses t i j , the stored central coefficie nt at P , and velocity gradie nts at the previous time step. U 3. Solve for the Cartesian ve locity compone nts at cell cente rs, u i , from the discretized momentum conservation e quation w Eq. s 6 .x , U
aP s u i . P s
6
U
p
a F s u i . F q s Su i .
n
s 18 .
F
4. O btain cell face velocitie s s see w 15x . and form the corresponding mass flow rate s s Eq. 5., U Ff s
p s
r B f j uÄ Uj . f
j
5. Solve the pressure-correction e quation s following the SIMPLEC algorithm of w 17x ; see w 15x for the time-marching ve rsion . and correct the pressure UU fie ld, p*, the ve locity fie ld, u**, and the mass flow rate s, Ff , which will now satisfy the continuity constraint. 6. Check for conve rge nce to a ste ady state , whe n the norm of the residuals of all equations has falle n be low a prescribe d tole rance s 10 y4 .; otherwise, take the variable s as pe rtaining to a ne w time le ve l s n q 1 . and go back to step 1. Implicit solution of the line ar sets of e quations in steps 1, 3, and 5 is carried out by applic ation of standard pre conditione d conjugate gradie nt methods w 16 x .
4 . R ESULTS The numerical method described above was implemented into a computational code that we shall de note as the ``stress’’ code , and was the n applie d to a typical proble m: the flow of Newtonian and UCM fluids through a planar contraction with a cross-sectional are a ratio of 4 to 1. This particular flow has be en often used as a test case in the non-Ne wtonian community} see , for example , Marchal
302
P. J. OLIVEIRA AND F. T. PINHO
and Crochet w 6 x , Webste r and co-worke rs w 10, 11x , Xue e t al. w 12 x , to name only a few of the rele vant works. The flow is completely de scribe d by two nondime nsional parame te rs, the Reynolds and Deborah numbe rs. The former is define d he re with the ave rage ve locity U in the downstre am channe l of half-width H, Re s r UH r m , and the latte r give s an indication of the e lasticity of the UCM fluid, De s l U r H. Results are first presente d for the Newtonian case, De s 0, allowing comparison be twee n the present ``stress’’ code and a ``standard’ ’ Navie r-Stoke s solve r which has be en used in previous work and is by now sufficiently validate d s e .g., w 15, 16, 18 x .. This standard code solve s the momentum and continuity e quations in collocate d meshes and is essentially similar to that de velope d by Pe ric w 2 x . Agreement be twee n the results of the two code s will give an indication of the correct imple mentation of the method de velope d to handle ge neral stress e quations. Then, the flow of the UCM fluid with increasing degree of e lasticity will be conside red, mainly for Re s 1. Here we will concentrate on the streamline patterns and on the stress fie lds, which show interesting phe nome na that are not prese nt in the Newtonian case. Whe re possible , comparisons are made between our predictions and others from the lite rature.
Newtonian Fluid The contraction ge ometry has bee n mappe d with three successive ly refined computational meshe s, of which Figure 1 shows a zoomed portion of the fine st s mesh 3 .. The mesh is orthogonal but nonuniform , with incre ase d concentration of cells in the are a of the contraction, especially around the ree ntrant corner whe re the stress gradie nts are expected to be high. For the mesh of Figure 1, the minimum nondime nsional cell size is 0.01 in both the x and y directions w s d x r H . min and s d y r H . min x ; this value is double d for the medium mesh 2 and double d again for the coarse mesh 1. In e ach mesh the cell size varie s following a ge ometric progre ssion at constant ratio s f x ’ d xiq 1 r d xi . inside e ach subblock use d to gene rate the overall mesh. The f x and f y were carefully chosen to guarante e a smooth cell-size variation at boundarie s between the patche d subblocks. In orde r to go from mesh 1 s the coarsest . to mesh 2 s the medium ., and from mesh 2 to mesh 3 s the fine st ., the numbe r of cells along the x and y dire ctions inside e ach subblock was double d and the corresponding e xpansion r contraction ratios s f x and f y. were root-square d. With this proce dure a consistent refine ment is achie ved in these nonuniform meshe s s see w 19 x ., with all mesh spacings being approxim ately halve d at e ach step and e nabling e rror e stim ation through Richardson’ s e xtrapolation to the limit. Other details of the meshes are give n in Table 1, whe re NC is the total number of cells and where the inle t and outle t channe l le ngths were taken as L 1 s 20 H and L 2 s 50 H, respectively. These le ngths were found sufficient to achie ve fully deve loped conditions at the outle t in both the Newtonian and in all the e lastic flow cases. The results to be give n were obtaine d with the fine r mesh unless stated othe rwise. The e ffect of mesh refine ment can be asse ssed from the value s for the length of the recirculation zone in the salie nt corner s xR . and the amount of flow in the e ddy scaled with the inlet flow rate s C R . give n in Table 2. They were obtaine d using both the central diffe rencing scheme s CDS. and the upwind scheme
FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS
303
Figu re 1. Z oomed view of the finer me sh s mesh 3 ..
304
P. J. OLIVEIRA AND F. T. PINHO Table 1. Mesh ch aracteristics Mesh Mesh 1 Mesh 2 Mesh 3
NC
s d x r H . mi n
fx
fy
942 3598 14258
0.04 0.02 0.01
1.2179 1.1036 1.0505
0.8475 0.9206 0.9595
Note : f x and f y are for the subblock in the downstream channel.
Table 2. Eddy characte ristics with mesh re finement Me sh Mesh 1 Mesh 2 Mesh 3 Limit
C
R
= 10 y 4
4.83751 4.39882 4.37856 4.37975
xR r H
s N1m a x . C L
1.1838 1.2020 1.2072 1.2090
2.2672 2.2912 2.2896
C
R
= 10 y 4 s UDS. 5.29766 4.60744 4.48108
xR r H s UDS. 1.2073 1.2141 1.2134
s UDS. for the convective te rms in the momentum equations. There is little difference be tween the results of the two schemes s e spe cially for xR ., an e xpe cted outcome for this low-Reynolds-num ber flow s recall that Re s 1 and so the local cell Reynolds number is much lowe r still ., but the CDS being ide ally second-orde r accurate enable s be tter Richardson e xtrapolation. Based on the value s in Table 2, the error estimate for the simulation in the fine mesh is 0.15% and the medium mesh is able to give results for e ddy size and inte nsity to within 1.0% . Richardson’ s the ory give s the order of the numerical approxim ation as p s ln w s xR 2 y xR1 . r s xR 3 y xR2 .x r ln s 2. s 1.81, base d on the xR value s in the three meshes, in good agree ment with the the ore tical value of 2 for the central differencing scheme. Me sh 2 and mesh 3 also show good superposition of local quantitie s, as e xe mplified in the profile of the first normal stress difference w N1 s Tx x y Ty y, whe re Ti j are normalized stresses, Ti j s t i j r s m U r H .x along the centerline s y s 0 ., give n in Figure 2. The maxim a of the se profile s are give n in Table 2, use ful data for be nchmarking. Similar conve rge nce with mesh refinement has be e n found for other variable s, such as ve locity and stress compone nts, but this comparison is not shown he re for conciseness. For the viscoelastic fluid obeying the UCM constitutive e quation the re is a tendency for finer meshes to be required as the Deborah number is incre ase d, due to the resulting stee pe r stress gradie nts. However, at De s 2 the results with mesh 2 and mesh 3 are still not too diffe rent, as shown in Figure 2, although accuracy to the same leve l attaine d by the Newtonian runs m ay re quire fine r meshe s. The results for Newtonian fluids obtaine d with the ``stress’’ code are in e xcellent agre ement with those obtaine d with the standard code , which yie lds C R s 4.3869 = 10 y4 and xR r H s 1.2076 for mesh 3. It has also bee n checke d that profile s of local variable s at various locations agre ed ve ry close ly, and consequently we m ay conclude that the imple mentation of the stress method has bee n done correctly. In terms of conve rge nce rate , Figure 3 compares the history of the residuals for the two code s and for the combination of parame te rs Re s 1, De s 0,
FLUID FLOW WITH ARBITRARY STRESS-STRAIN RELATIONSHIPS
305
Figu re 2. Effect of mesh refinement on the primary normal stress difference along the centerline s De s 0 and 2 ..
Figu re 3. Decay of the norm of the residuals of the u momentum and continuity equations for the ``stress’’ and ``standard’’ procedures.
mesh 2, d t r s H r U . s 0.01, with a converged solution be ing assumed whe n the normalize d residuals of all equations fell be low a tole rance of 10 y4 . These residuals are define d as the l 1 norm of the alge braic e quations to be solve d, with all te rms shifted to one side of the equal sign, and should e ssentially tend to zero as the solution is approache d. Figure 3 shows that the new method follows an ide ntical convergence path to the standard one , and the number of time steps to the solution is only slightly highe r s 798 compared with 674 ., be cause the residuals of the stress equations s not shown in Figure 3 . are now also conside red in the
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P. J. OLIVEIRA AND F. T. PINHO
stopping criterion and te nd to lag be hind those for the momentum and mass conservation equations. Still, for an ide ntical stopping criterion, the CPU require d by the ne w method whe n applie d to a Newtonian fluid is only 18% large r than that of the standard proce dure, a re asonable computing time ove rhead to pay for the adde d advantage of choosing any gene ral constitutive relation. Naturally, the use of a different constitutive e quation will require additional computer time. It is inte resting and rele vant at this stage to corroborate the point made in Se ction 2, that inclusion of the y 23 div u te rm in the stress equation for the Newtonian fluid tends to give more accurate results and is also be tter at reducing the computing time. Figure 4 a compares the normalized pressure variation P ’ p r s m U r H . at the entrance to the smalle r channe l, just downstre am of the contraction plane , with and without the div u-term and using the medium-sized mesh. When the div u te rm is not include d to force a tracele ss de viatoric stress tensor, there are some perturbations in the pressure profile ne ar the wall s at y r H , 1.. Similar unphysical pe rturbations are obse rve d in the late ral compone nt of the ve locity v r U s Figure 4 b . and the norm al stress compone nt Ty y s Figure 4 c ., but the y are suppre ssed whe n the div u te rm is included in the stress te nsor. In gene ral, not including this te rm le ads to highe r maximum le ve ls of pre ssure and stress components w e.g., s Tx x . m ax incre ase s from y 5.1 to y 7.4 x , which typically occur close to the reentrant corner where the high gradie nts are localize d, and the computing time increases by 30% and 15% in the two tested case s s Re s 0.01 and 1, respe ctive ly..
Viscoelastic Fluid We turn now to the more complicated case of the implicit diffe rential UCM constitutive e quation w Eq. s 4 .x , which was solve d with the proce dure outline d in Se ction 3. Since the stress contribution to the momentum balance w the div t term give n by Eq. s 13 .x is treate d e xplicitly in the numerical method, so that it lags in time as the solution is re ache d through a sequential treatment of the various e quations, it is expe cted that convergence to a ste ady state will be slower than for the Newtonian fluid. This is inde ed the case as shown in Figure 5, which give s the history of the residuals for the slowest variable , for various runs with increasing value s of De. The runs for the Newtonian fluid s De s 0 . and for De s 1 have bee n started from an initial condition of uniform ve locitie s and vanishing stresses and pressure e verywhe re; the othe r runs have be en restarted from the corresponding solution at lower Deborah numbe r. This procedure is convenie nt to reduce computer time but was not found ne cessary to guarante e convergence. All runs in Figure 5 used the same time step s d t s 10 y2 H r U . in spite of increased elasticity, an indication of the robustne ss of the present numerical method. The conve rge nce rate , proportional to the inclination of the curve s, is reduced as De incre ase s, and so computations at high De will require more time steps for a ste ady solution to be re ache d. This situation is common to all methods reporte d in the non-Ne wtonian literature, most of them of the finite -e le ment type w 6, 8 ] 11 x , and with the drawback that those methods often could only obtain solutions at very low Deborah numbe rs s e .g., Yoo and Na w 7 x , De - 1.04; Carew e t al. w 8 x , De ( 2; Sato and Richardson w 9 x , De ( 2.. Marchal and Crochet w 6 x could attain De s 6, but with ve ry coarse meshes.
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Figure 4. E ffect of including the div u term in the constitutive equation: transve rse profiles at x s 0.01 H of s a. normalized pressure, s b . v r U, and s c . normal stress Ty y .
A comparison of some of the results of the literature and those of the pre sent stress method is carried out in Table 3. The exte nsion of the re circulating zone predicted by w 9 x , utilizing meshes with a minimum spacing of 0.025 and 0.05, compared well with the present pre dictions, and so did those of w 11 x for the low-De range. For the high-De numbe r range , howe ver, these latte r authors used a mesh that was too coarse to yie ld adequate predictions s minimum mesh spacing of 0.14 .,
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Figure 5. Convergence history for the Ne wtonian s De s 0. and various UCM fluid flows.
Table 3. V alues of xR r H: comparison with other calculations De 0 1 2 3 4 5 6 8
Sato a w 9 x , Re s 1
Ours, Re s 1
Matallah a w 11x , Re s 0.5
Ours b Re s 0.5
1.145 1.000 0.927
1.213 0.988 0.833 0.784 0.851 1.081 1.339 1.755
1.318 1.114 1.0 1.0 1.0
1.341 1.189 1.175 1.319 1.550 1.784
} } } } } a b
}
}
}
} }
Oldroyd-B model. With mesh 2.
and that m ay explain the fact that the ir pre dictions of xR tende d to a constant value for De 0 2. Some of the pote ntial of the present method for the study of viscoelastic fluid flow is illustrated by Figure 6, which shows the streamline patterns for increasing De s from 0 to 8. at constant Re s 1, and by Figure s 7 and 8, which show some of the corresponding contour plots of the nondime nsional normal-stress difference s N1 . and shear stress s Tx y . fie lds, respectively. Care ful analysis of the se figures and othe r data from the solution fie lds, a matte r outside the scope of the pre sent article , may explain some of the peculiar feature s observe d in e xpe riments with viscoe lastic fluids. The stre amline patte rns in Figure 6 show the appe arance of a lip vorte x at De f 1 and its growth at De f 2 ] 3, finge ring of the corner vortex toward
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309
Figu re 6. Streamline patterns of UCM fluids for increasing Deborah numbers, at constant Re s 1. s Iso-levels in the e ddies are e qually spaced, with d C = 10 3 s 0.05 for De s 0 ] 3; 0.5 for De s 4 ] 6; and 1.0 for De s 8.
the lip vorte x at De ( 4, the e ventual merging of the two vortice s at De , 4 ] 5, and the subse que nt growth of the single remaining vorte x. The flow patterns at low De are in good agre ement with the recent simulations for an Oldroyd-B fluid of Matallah e t al. w 11x s mainly so for De s 1 to 3 ., which also show the onse t of the lip vorte x that has bee n quite e lusive in previous works s e.g., Sato and Richardson w 9 x ; see also review in w 11 x .. We may also add that the flow feature s see n in Figure 6 are not just a peculiar effect resulting from the the ore tical UCM model and abse nt from the reality; in fact, similar vorte x-re late d phe nome na have be en obse rved e xpe rimentally in flow visualizations by Walte rs s in planar contractions . and Boge r
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Figure 7. Contour m aps of the nondimensional primary normal stress difference, N1 : s a. De s 0; s b . De s 2; s c . De s 5.
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Figure 8. Contour m aps of the nondimensional shear stress, Tx y : s a. De s 0; s b . De s 2; s c . De s 5.
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s in round contractions . w 20 x . The sketches of vorte x growth mechanisms give n by Boger e t al. w 21 x s see the ir Fig. 10 ., base d on visualizations of viscoe lastic e ntry flows, are closely emulate d by the succession of stre amline patterns in Figure 6. Lip vortice s, corne r vorte x e nhancement, and the othe r differences between Newtonian and viscoelastic flow patte rns see n in Figure 6 results from distinct stress structure, as can be infe rred from Figures 7 and 8. The main differences are the very high normal and she ar stresses that build up around the ree ntrant corner for viscoe lastic fluids, and the enhancement of the elongational flow along the centerline evide nt in Figure 9, and caused by a large and positive N1 at the e ntrance to the contraction. The stress field forces the viscoe lastic fluid to be more de flected toward the cente rline than the Newtonian or the le ss e lastic fluids, whe re the flow is further accelerate d by the positive Tx x gradie nts; as a conseque nce the axial velocity profile s of the viscoe lastic fluid at the contraction plane are more uniform, and the centerline velocity reaches highe r value s than those of the Newtonian case and may eve n go be yond the fully de velope d value in the downstream channe l, as shown in Figure 9. The centerline ve locity ove rshoot e ffect has be e n obse rved in other simulations s cf. w 6, 12 x . and also experimentally s e.g., w 22x ., and the increase relative to its fully de velope d value is in reasonable agree ment with othe r works where diffe rent fluids have bee n used s he re we obtain 2.8% , 7.1% , and 17.8% for De s 1, 2, and 5, respective ly; Xue e t al. w 12x got 4% for De s 1.6; Marchal and Crochet w 6x got 22% for De s 4.7 .. We e nd this work with a comparison betwee n the new formulation used to obtain stresses at cell faces give n by Eq. s 15. and that proposed in a previous work w 5 x . The main diffe rence be twee n the two formulations lies on the way inte rpolation is done, which affe cts the results when nonuniform meshe s are utilized. In the former formulation the forces acting on a surface across cell centers were interpolated directly s with arithme tic ave raging . to the cell faces, thus the b l i were give n by Blt i j n j w cf. Eq. s 16 .x ; howeve r, it see ms more correct to interpolate the stress components and the n obtain forces at cell faces via the usual te nsorial relationship, Tis Blt i jn j s where Bl and n i are now evaluate d directly at cell faces, i.e ., Bl is the scalar are a of the cell face along dire ction l and n i is its unit normal ve ctor., as
Figu re 9. V ariation of the longitudinal ve locity component along the centerline for various De borah number flows.
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implie d in the ne w formulation. Care should also be taken so that, in the limiting case of a uniform ve locity field in a nonuniform mesh, the resulting cell-face stresses do vanish, and Eqs. s 15 . ] s 16 . respect this premise. Figure 10 shows two transve rse profile s of the longitudinal ve locity compone nt across the large channe l, at 1 H and 2 H upstre am of the contraction plane ; some localized oscillations are visible at y r H f 0.5 H, where an inte rface be twee n two subblocks use d to gene rate the mesh creates a sudden nonuniform mesh-spacing change. With the pre sent formulation there are no oscillations, an indication of the be tter inte rpolation strategy achie ved with Eq. s 15 ..
5 . CONCLUSIONS A seque ntial se gre gate d approach base d on a ge neral finite-volume methodology in nonstagge red meshes is shown to be e ffective in solving the flow motion e quations in conjunction with arbitrary additional constitutive e quations e xpre ssing stress-strain relationships. The te st case chosen was the typical be nchmark flow through an abrupt 4:1 planar contraction for Newtonian and viscoelastic fluids obe ying the upper-convected Maxwe ll model. A new formulation to obtain stresses at cell faces shows improve ment for nonuniform meshes, and inclusion of the div u term in the constitutive equation improve s the stability and the conve rge nce rate of the method. Results are presente d for a range of Deborah numbe rs from 0 to 8, and for Reynolds numbe rs of 0.5 and 1.0. The streamline patte rns show the existence of corner and lip vortices at De , 1 to 4, and a vortex enhancement mechanism through lip vortex intensification followe d by finge ring of the corner vortex toward the lip, with subse que nt e nve loping and merging of the two, in agre ement with the literature. For the Newtonian mode l the results are identical to those obtaine d with a standard and well-validate d code , and the computing time is just about 18% highe r. For the UCM mode l the required computing time s and number of time steps .
Figu re 10. Influence of the ce ll-face stress formulation on the solution smoothness. Tranverse profiles of the streamwise ve locity in the upstream channel at x r H s y 2 and y 1, for De s 5.
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tends to incre ase with the e lasticity of the fluid. This was e xpe cted from the segre gate d nature of the method, with the stresses obtaine d from the constitutive e quation being e xplicitly inse rted into the momentum equations, and similar de te rioration of the convergence rate is found in most methods reported in the literature. Howe ve r, the present method shows good robustness, with solutions for the viscoe lastic cases achie ved for highe r De numbers than with other e xisting methods and where, furthermore , the time step utilize d in the computations was chosen based on the Newtonian case and was kept at a constant le vel for all runs.
REFERENC ES 1. S. V . Patankar, Num erical Heat Transfer an d Fluid Flow, Hemisphere, Washington, DC, 1980. 2. M. Peric, Â A Finite-Volume Method for the Prediction of Three -Dimensional Fluid Flow in Complex Ducts, Ph.D. thesis, Imperial College, University of London, 1985. 3. J. H. Ferziger and M. Peric, Â Com putation al Methods for Fluid Dyn am ics, Springer V erlag, Berlin, 1996. 4. C. M. Rhie and W. L. Chow, A Numerical Study of the Turbulent Flow Past an Airfoil with Trailing E dge Separation, AIAA J., vol. 21, pp. 1525 ] 1532, 1983. 5. P. J. O liveira, F. T. Pinho, and G. A. Pinto, Numerical Simulation of Non-linear Elastic Flows with a Ge neral Collocate d Finite-V olume Me thod, J. Non-Newtonian Fluid Mech., vol. 78, pp. 1 ] 43, 1998. 6. J. M. Marchal and M. J. Crochet, A Ne w Mixe d Finite Element Method for Calculating V iscoelastic Flow, J. Non -Newtonian Fluid Mech., vol. 26, pp. 77 ] 114, 1987. 7. J. Y. Yoo and Y. Na, A Numerical Study of the Planar Contraction Flow of a V iscoelastic Fluid Using the SIMPLER method, J. Non-Newtonian Fluid Mech., vol. 29, pp. 89 ] 106, 1991. 8. E. O. A. Carew, P. Townsend, and M. F. We bster, A Taylor-Petrov-Galerkin Algorithm for V iscoelastic Flow, J. Non-Newtonian Fluid Mech., vol. 50, pp. 253 ] 287, 1993. 9. T. Sato and S. M. Richardson, E xplicit Numerical Simulation of Time-De pendent V iscoelastic Flow Problems by a Finite E lement r Finite V olume Me thod, J. Non-Newton ian Fluid Mech., vol. 51, pp. 249 ] 275, 1994. 10. A. B aloch, P. Townsend, and M. F. Webster, On V ortex Deve lopment in V iscoelastic Expansion and Contraction Flows, J. Non-Newtonian Fluid Mech., vol. 65, pp. 133 ] 149, 1996. 11. H. Matallah, P. Townsend, and M. F. Webster, Recovery and Stress-Splitting Schemes for V iscoelastic Flows, J. Non-Newtonian Fluid Mech., vol. 75, pp. 139 ] 166, 1998. 12. S.-C. Xue, N. Phan-Thien, and R. I. Tanner, Three Dimensional Numerical Simulations of V iscoelastic Flows through Planar Contractions, J. Non-Newtonian Fluid Mech., vol. 74, pp. 195 ] 245, 1998. 13. R. B. Bird, R. Armstrong, and O . Hassage r, Dyn am ics of Polymeric Liquids, Vol. 1, Fluid Mechanics, 2d ed., Wiley, Ne w York, 1987. 14. R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dyn am ics of Polym eric Liquids, Vol. 2, Kinetic Theory, 2d e d., Wiley, New York, 1987. 15. R. I. Issa and P. J. Oliveira, Numerical Predictions of Phase Separation in Two-Phase Flow through T-Junctions, Compu t. Fluids, vol. 23, pp. 347 ] 372, 1994. 16. P. J. Oliveira, Computer Modelling of Multidimensional Multiphase Flow and Application to T-Junctions, Ph.D. thesis, Imperial College, University of London, 1992. 17. J. P. V an Doormaal and G. D. Raithby, Enhanceme nts of the SIMPLE Me thod for Predicting Incompressible Fluid Flows, Num er. Heat Transfer, vol. 7, pp. 147 ] 163, 1984.
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18. P. J. Oliveira and F. T. Pinho, Pressure Drop Coefficient of Laminar Newtonian Flow in Axisymmetric Sudden Expansions, Int. J. Heat Fluid Flow, vol. 18, pp. 518 ] 529, 1997. 19. F. Ferziger and M. Peric, Â Further Discussion of Numerical Errors in CFD, Int. J. Numer. Meth . Eng., vol. 23, pp. 1263 ] 1274, 1996. 20. D. V . Boger and K. Walte rs, Rheological Phenom ena in Focus, Rheology Series, V ol. 4, Elsevier, Amsterdam, 1993. 21. D. V. Boger, D. U. Hur, and R. J. Binnington, Further Observations of Elastic Effe cts in Tubular E ntry Flows, J. Non-Newtonian Fluid Mech., vol. 20, pp. 31 ] 49, 1986. 22. L. Q uinzani, R. C. Armstrong, and R. A. Brown, Birefringence and Laser-Doppler V e locimetry s LDV . Studies of V iscoelastic Flow through a Planar Contraction, J. Non-Newtonian Fluid Mech., vol. 52, pp. 1 ] 36, 1994.
