A computational method for flows of incompressible fluids based on a non conformal transformation of the physical domain

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Voi.13(5), 239-246, 1986. Printed in the USA. Copyright (c) 1987 Pergamon Journals Ltd.

A COMPUTATIONAL METHOD FOR FLOWS OF INCOMPRESSIBLE FLUIDS BASED ON A NON CONFORMAL TRANSFORMATION OF THE PHYSICAL DOMAIN

J.R. CLERMONT and M.E. DE LA LANDE I n s t i t u t de M~canique de Grenoble, B.P. n ° 68 - Domaine Universitaire 38402 - Saint Martin d'H~res C~dex, France

(Received 30 December 1985; accepted for print 31 March 1986) Introduction In general, numerical solutions of Newtonian or non-Newtonian f l u i d s in plane or axisymmetric flows are obtained by means of f i n i t e f i n i t e element techniques I I I ,

difference

Ill,

or, more widely

121, 131. The unknowns are the two components of the

v e l o c i t y and the pressure which are computed by considering the whole domain D of the flow, subjected to boundary conditions. The purpose of t h i s paper is to present r e s u l t s from a new formulation related to axisymmetric flows of Newtonian f l u i d s .

(The work can be extended to non-Newtonian

f l u i d s as w e l l ) . The general method has been presented elsewhere 141, t51, 161. At the difference of classical methods, the unknowns to be determined are the pres sure and a transformation between the physical domain D of the flow and a rectangular domain D1 where the transformed streamlines are p a r a l l e l

s t r a i g h t lines. The

present method l i e s on the concept of stream tube in r e l a t i o n with the incompressi bility

condition. We show here how i t is possible to compute the flow f i e l d in

domain D by considering successive sub-regions in domain DI. Formulation Consider the steady axisymmetric flow in a c y l i n d r i c a l ~

coordinate system : (I)

The domain under consideration is assumed to involve no secondary flows. Under the assumption of incompressibility, we can write the two components of the velocity in terms of a stream function

L~ = A_ " ~~

[~,~I )

V--

r

as : - ~ ~_H

f 239

(2)

240

J.R.

CLERMONT

For a p o i n t of the same s t r e a m l i n e Let us suppose t h a t t h e r e e x i s t s

and M.E. DE LA LANDE

(~),

the stream f u n c t i o n

a s e c t i o n - of r a d i u s

where the k i n e m a t i c s i s known. The stream f u n c t i o n

H(~,z)

is a constant

~o - of the f l o w at Z=Zo

f o r Z=Zo i s given by "

o According to 141, 151, 16L, we can define a one-to-one transformation between a streamline (L) of the region D and its transformed line in region D] (as shown in Fig.

l).

The domains D and Dl can be

p_

d i v i d e d i n t o subdomains b and B l i m i t e d by two s u c c e s s i v e s t r e a m l i n e s .

~Y~

~

(~,z)

~

~'

~)~)!~

We g e t "

F D

z

~

/7//I/~//}~

,,1

D1

.(m,z)

by u s i n g t h e r e l a t i o n s h i p s

:

= ~ (R,Z)

l

(4)

I

Z = Z From e q u a t i o n ( 4 ) , line

(~m)

/¢0

'

! i

is a straight

i n D1 and the t r a n s f o r m e d stream

FIG.1

f u n c t i o n H of H o n l y depends on R :

T r a n s f o r m a t i o n of the p h y s i c a l domain R =~,Zo=Zo

H(~,Zo)

The f u n c t i o n H and i t s and, from e q u a t i o n 9

= H(R) derivatives

From e q u a t i o n s ( 5 ) , =

_

H',

H", H'" . . . . .

( 4 ) , we o b t a i n the f i r s t

.4_ ~

u

(5)

o r d e r o p e r a t o r s as :

with

(6),

(7),

{~z " --' H {R.~

are known from e q u a t i o n (3)

-

(6

the components of the v e l o c i t y and

v

:

_-1

---~

,

are given by : .,

(8)

C o n s e r v a t i o n e q u a t i o n s - BQund.ary c o n d i t i o n s

1 ° The f l u i d

d e n s i t y mo i s assumed t o be a c o n s t a n t and the mass c o n s e r v a t i o n i s

NON CONFORMAL TRANSFORMATIONS

241

equivalent to the incompressibility condition. 2° The momentum conservation in steady flow leads to the d i f f e r e n t i a l dynamical e q u a t i o n s

form of the

171, Fv denotes the body f o r c e v e c t o r :

~ V. ~,~ ~" -__~oF~ - ~ where the stress tensor is :

k~':

-~

p ~-a~T,-"T.,,

(9)

~ ~T'.,.

(I0)

p is the isotropic pressure, T the stress tensor given by a rheological model. The transformed domain D1 is divided into successive stream tubes. The momentum conservation equations are p a r t i c u l a r l y convenient for the description of flows as i t is considered in the present paper. In steady flows, these equations are 171

" V" d"~ J~Fis the surface which limits the volume ~'-. In our case, the control surfaces are stream tubes limited by cross-sections of the tubes. By equations (9), we get for the dynamical equations a non-linear system 151, 161 as : (12)

These equations, of unknowns ~-and p are to be solved numerically according to boundary conditions on domain D.

Numerical procedure

The numerical procedure presented in the following concerns an axisymmetric flow the region of which involves the axis of symmetry (Fig. 2a).(A similar approach can be made for planar flows under the same conditions). The domain D is limited

S ..... $3

& FIG. 2a

FIG. 2b

242

J.R.

CLERMONT and M.E. DE LA LANDE

by two sections S1 and $3 where_boundary conditions are known for ~ and p, Using equation ( I I ) for S = ~ U %uS3 ' and neglecting i n e r t i a and body forces, (without any loss of generality) we get :

~.~s= which leads to

~T~'~s+~

~ ~ [T.n'&S ~ - I.

~'~%

According to equation (14), the quantity

~ s ÷

~'~-0

.o

/K ~.~'~%

d~ I

(13) (14)

which involves the functions

and P at the wall depends only on boundary conditions upstream and downstream the region D. In a similar way, by considering the momentum equation ( I I ) for the stream tube j ~ U

In this case,

~/~US,

we get :

~

~.n~ is s t i l l a function of boundary conditions upstream and q downstream the region D (~aCSl andfl3cS3 )" From equations (14) and (15), i t is possible to determine numerically ~ and P on successive stream tubes from the wall to the axis of the flow. It can be seen by considering the domain D' of Fig. 2b, t h a t t h i s procedure cannot be applied.

The flow domain D' is such that :

By equation (16) i t is clearly shown that the quantity / ~ z [ T . ~ s

on the wall $2

depends on boundary conditions for S1 and $3 and also on conditions at the inner wall $5' which are unknowns. These considerations lead to the following remarks " 1° For every stream tube between (~',o~) and ( ~ o ~ ) in DI, the conservation law of mass is veri f i ed from the present formulation which uses the stream functions 2 ° The computations of the unknowns ~ and p can be carried out by considering successively the bands ( ~

Computation of ~ .

,~),

(~,~)

.....

from the wall to the axis.

and p

Let us consider the f i r s t

band Bo between ( ~ o ) and ( ~ ) .

elements from Z=Zo to Z=z I (Fig. 3)

Bo is divided into N

NON CONFORMAL TRANSFORMATIONS

f

known ,

p

unknown ~

p known- ~ f known--I~ I Z=~ L

f known -~p unknown,~p/@R,~pA)Zknown

LZ---~,

•

f ,

2A3

p

unknown -/~ FIG. 3 Elements and boundary conditions in Bo

On every element, the unknowns

~- and p are approximated by second or t h i r d order

polynomials of R and Z : L

o~ L

•

g

L~-

3

~L L

~L

The subscript i is related to the element and the total number of unknowns is ION for second-order polynomials and 18N for third-order polynomials. On Fig. 3 are shown the boundary conditions for ~. and p. The dynamical equations at the nodes AL, BL, CL, DL and boundary conditionV equations lead to a non-linear set of equations which are solved by the Newton-Raphson method. Details concerning the equations have been presented in a previous paper 161.

Application to the flow of a newtonian fluid in a.conver~ent Let us consider the domain D of a converging flow limited by two Poiseuille flow s e c t i o n s at Zo and z 1 ( F i g . 4). The s t r e s s

t e n s o r S i s given by :

S -= 2 /' ~ ~

(19)

and, from relations (5) to (8), the equation (9) can be written as [5[, [61: ~ ~'< 'p ~" "'~T~"

+

,

o --

I~ t S'.,)TF

~-z

+ A s . LR!

.--

¢2oi

-

02 where AI , A2 , A3 , B1 , B2, B3 are functions of and the variable TT is given by : =

-~-

f'

f'R

,

f'

Z, ....

f"'R3 . . . . .

f"'Z3

(22)

244

J.R. CLERMONT and M.E. I)E LA LANDE

The boundary

conditions for the velocities are :

(23) At

z:z I

c-[o

:

and, since

(24) .//

Ro = ~Po , the stream

/ / / /

function and i t s derivatives are given by the equations •

H" (R)= ~ o [(~-(~,~(~-~

(26)

H"(R)= c < ~ t 3 ~ - ~ o z]

N

B~

(27)

From equations (23) and (24), the corresponding value at Z:z I is ~F=~,~ ,R is the initial value (Z=Zo)

c

FIG. 4

Apart boundary condition equations given by values of

and p and th e ir derivatives (see Fig. 3), we apply the momentum theorem to surfaces S2 and S4 (lines

(-~) and ( ~ )

for the z component. Since : (28)

and

(29)

I

we f i n a l l y obtain, with H'(Ro) = 0 (from equation {26)) :

p(B l) and p(CN) are the pressure values at the ends BI and CN of ( - ~ ) . The second boundary condition equation for the line (~i_) - o f radius Rl - i s written by considering the following equations (see Fig. 4) : . and

.~.~ as

~

~.fl~s

S

./S.~ ~.q~Js ~

(31) ~o'~s

-O

(32)

NON CONFORMAL TRANSFORMATIONS

245

Since

(33) (34)

It

then, we get :

-

(R,,-G).H (R~

+

_

"-I~'l('

-

r'-I~'J~ll

-

-

=

Equations (30) and (35) are to be used together with the set of non-linear dynamical equations and linear boundary condition equations.

Results We present on Fig. 5 the evolution of the computed values of f and p, for the f i r s t band Bo in Dl for a converging flow of angle or= 15°. As shown, we used 8 trapezoidal elements. The viscosity of the fluid wasv/~ = lO0 poise and the pressure drop for the upstream Poiseuille flow was 400 dynes/cm3. The pressure at the section Z=Zo was assumed to be 400 dynes/cm2.

I / I / I / I J--n°

R1 =O'85cml

I

i

I

• I,~" f -~' ~,_lcmL, ~ . . . . .

i ts

_

° I I _L I I ,~:o.e~ ~ - - - ~ --L__. L

:

I I

/ I

I i

IP,-'°'893 ~ I oo.~, o

~-~,~ ~ - - ~ - ~ ~ I ~ ; . ~ 5

30o ~ - -

~

__~o,I~4'

0

-100 .

Zo=

o.;.

o~6

'~ ...~, Cm

Zl= 0.8

Fig. 5 The mesh in domain DI, the computed values of ~ f o r the streamline ( ~ ) of the physical domain D, the pressure values for (~Jo) and ( ~ ) .

246

J.R.

CLERMONT and M.E. DE LA LANDE

I ne results presented in Fig. 5 are consistent with those obtained in the current literature (finite

difference or f i n i t e

elements), when using a resolution tech-

nique on the complete domain of the flow. Concerning the convergence behaviour, v a r i a t i o n s of the band size (at f i r s t

assumed to be 0.25) w i t h i n 60 % have modified

the downstream pressure in an order of less than 1%. When increasing the number of elements in a band, the r e l a t i v e v a r i a t i o n of the downstream pressure is found to be about 2 %. In all cases, the computed streamlines are p r a c t i c a l l y unaffected.

Conclusion In t h i s paper, a numerical simulation method for axisymmetric - or plane - flows of incompressible f l u i d s has been presented. A special procedure can be carried out when the axis of symmetry of the flow belongs to the physical domain. This method was developed for a converging flow. The approach leads to lower the computation time and l i m i t the storage. Moreover, in the computation domain of the unknowns and p, the transformed streamlines are parallel

s t r a i g h t lines and the formulation

is p a r t i c u l a r l y well-adapted to the problem of non-Newtonian flows in various geometries.

References II

Crochet M.J., Davies A.R. and Walters K., "Numerical simulation of non-Newtonian Flow", Elsevier, Amsterdam-Oxford-New-York-Tokyo,

21

1984.

Pearson J.R.A., Richardson S.M., "Computational Analysis of polymer processing" Applied Science Publishers, London and New-York, 1983.

3h

Chung T . J . , " F i n i t e Element Analysis in Fluid Dynamics", Mc Graw-Hill,

4n

Clermont J.R. "Sur la mod~lisation num6rique d'6coulements plans et m6ridiens de f l u i d e s incompressibles", C . R . A . S . t . 2 9 7 (S~rie I I . I )

5L

1978.

Paris, 1983.

Clermont J.R., De La Lande M.E., "A new numerical method for simulation of plane or axisymmetric flows of incompressible f l u i d s " ,

Proc. of the Numeta 85

Conference, Swansea, pp 155-160, January 1985. 61

Clermont J.R., De La Lande M.E., "A method for the simulation of plane or axisymmetric flows of incompressible f l u i d s based on the stream function concept", submitted to Engineering Computations, 1985.

7L

Batchelor G.K., "An introduction to Fluid dynamics", Cambridge U n i v e r s i t y Press,1967