di equazioni integrali di contorno per fIussi viscosi con su- perficie libera. Tale metodo ~ applicato allo studio della con- vezione termocapillare ed al processo di ...
MECCANICA
A BOUNDARY-INTEGRAL EQUATION METHOD FOR FREE SURFACE VISCOUS FLOWS*
19 (1984), 294-299
Shuyao Ma**l, Giorgio Graziani**, Renzo Piva**
SOMMARIO. Si propone un metodo basato sulla soluzione di equazioni integrali di contorno per fIussi viscosi con superficie libera. Tale metodo ~ applicato allo studio della convezione termocapillare ed al processo di formazione di una goccia, entrambi in condizioni di microgravit& La presenza dei termini non lineari nell'equazione di Navier-Stokes comporta un integrale di volume che viene approssimato mediante un processo di linearizzazione. Risultati numerici per flussi termocapillari con superficie libera sia fissa che mobile sono confrontati con altri ottenuti in precedenza con un metodo alle differenze finite. Si presentano inoltre alcuni risultati preliminari sul problema della formazione della goccia ed in particolare l'evoluzione nel tempo della configurazione geometrica della superficie libera. Nei due casi si analizzano solo campi bidimensionali. SUMMARY. A boundary integral equation method is proposed for the solution o f viscous recirculating flows with free surfaces. In particular the method is applied to thermocapillary convection and to drop formation, both in micro-gravity conditions, the latter to test its capability to handle real unsteady problems. The presence o f non linear terms in Navier-Stokes equations leads to a volume integral, which has to be approximated by a linearization procedure. Several numerical results for thermocapillary flows, both with fixed and moving free surface, are discussed in comparison with previously obtained finite difference solutions. Some preliminary results, and in particular the time evolution o f the free surface shape, are also presented for the drop formation problem. Only plane two dimensional fields are considered for both problems.
1. INTRODUCTION Free surface flow fields have been recently studied with a particular attention to the applications concerning thermocapillary convective motion and drop formation, which both have a fundamental importance in microgravity material process. Thermocapillary flows usually present small deformations of the free surface, and finite difference methods in curvili-
near coordinate have been shown [1,2] to be an efficient approach for their numerical simulation, in particular when steady state conditions are desired. Different is the case for drop formation which implies the analysis of large deformations of the free surface and the study of transient conditions to follow the time evolution of the drop. In this case difficulties arise for the grid generation and also for the numerical solution of the equations. In fact the grid stretching becomes excessive at large times, enhancing the negative effect of non orthogonal cell sides, and, as a consequence, the accuracy of the solution decreases [3]. Moreover the unsteady character of the problem requires all the conservation equations, mass included, to be satisfied at each time step. This is accomplished for primitive variable schemes (to be preferred when pressure boundary conditions are imposed) by an iterative procedure, which becomes very time consuming if a real transient behavior has to be described. Considering the drawbacks met for large deformations with F.D. domain integration schemes, we propose here a different approach based on boundary integral equations which seems more suitable for such problems, as free surface flow fields, greatly affected by the moving boundary configuration. This approach, that originates from the Green function methods in potential theory and from the classical relations of Betti and Somigliana in elastostatics, has recently revived for the capability shown in the frame of numerical solutions. At this purpose, very interesting numerical applications have been proposed in potential compressible flows, e.g. by Morino [4], or in elastostatics e.g. by Rizzo and Cruse [5, 6]. Many results in these two fields are reported in Brebbia [7]. Closer to the present problem is the application proposed by Bush and Tanner [8] to low Reynolds number flows as an extension of the solution for Stokes flows. The mathematical formulation of the flow field equations in a boundary integral form is described in section 2, while the solution procedure which is adopted to account for the volume integral of the non linear acceleration term, and for the dynamic and kinematic boundary conditions, is summarized in section 3. Several different applications, restricted to two dimensional flows, with growing influence of the surface deformation are discussed in section 4, in comparison with results obtained with F.D. schemes.
* Presented at the VII National Conference AIDAA, Naples, September 1983. *~ Dipartimento di Meccanica e Aeronautica, Universitg di Roma. t In leave of absence from Tianjin University, China. 294
2. INTEGRAL EQUATION FORMULATION The governing equations for steady incompressible viscous flows are, in tensor notation: MECCANICA
%k
conservation of momentum - -
=
p(a.
(1)
3x k where the acceleration a/is given by
Ouk
conservation of mass - -
Oxk Oxi ai = u k (•
+p* - ~x k
dV= (8)
k)
[",("
(2)
=0
~xi
Oxk
dV+
7
with the constitutive equation given by +
a/k = --p~/k + g, XOxk +
+
(3)
3xi
Substituting eq. 3 in (1) we obtain the momentum equation in terms of pressure and velocity
~)2uy I.t
~(~Uk]
-~Xk - 3xk + # -OXk X~x]/
3p 3x.J - p(a] --f])
(4)
Notice that we maintain the second term of (4) instead of interchanging the order of the derivatives to obtain the divergence of u, which can be canceled by substituting (2), as usually done with incompressible flows. Notice also that we wrote eq. (4) maintaining at the left hand side only the linear terms, while the convective term (typically non linear) is on the right hand side together with the external force f. We now consider for the system of equations (4), (2), the possibility to obtain, via a Green's function method, boundary integral equations, more appropriate for the problem under consideration. Let us recall the classical way [9, 10] of introducing the formal adjoint operator L* associated with the linear operator L, using integration by parts
f
gLfdV = fv V" P(g, f) dV + f fL*g dV "v "1I
(5)
(6)
where at the right hand side only boundary terms appear. If we choose g as the Green's function satisfying the equation
L'g= ~(P)
(7)
where 6(P) is the Dirac delta function, and we are able to find it, for the specific L* under consideration, eq. (6) greatly simplifies, giving the unknown f(P) in terms of boundary and volume (for non homogenous equations) integrals. To follow this line in our case, we have to consider that f, g are vectors and L is a matrix in order to give the system (4), (2). A general theory of Green's function method for the fluid dynamics system of equations extended to unsteady and compressible flow is described in details in Piva, Morino [ 11 ]. In the present specific case of incompressible and steady flows, integrating by parts, as in (5), and applying GreenGauss theorem, as in (6), we find 19 (1984)
i
.;
--
; .j T do
(9)
"a
The components ui* , p* of the Green's function satisfying eq. (7) with the expression of L* deduced from (8), apt to determine the velocity components u/, are given by
oh9* - -
Oxg ~x k
OuZ
that is
(gL f -fL*g) dV = foP(g, f) 9nda
where u.* and p* are the components of the Green's vector, I which will be determined later on. It's worth to be noted that L* is equal to L, which is therefore a self-adjoint operator. Recalling the constitutive equation (3) and introducing the traction t / = o/k n k the two surface integrals in (8) may be expressed as
- o
+u
~ Ou~ Op* -
3x k Ox]
kfi(P)
(lOa)
Oxi (10b)
bx k where ki are the components of a unit vector, which multiplies the Dirac delta function for vector equations. The components u7, p* of the Green vector apt to determine directly the variable p (if needed) could be given by an analogous system with the first equation homogeneous and a Dirac delta function at the right hand of conservation equation. We will neglect here this part which is not strictly required for incompressible flows. The second term of eq. (10a) may be cancelled now by interchanging the partial derivatives and accounting for eq. (10b). The unit vector k may assume any direction. In particular to obtain from (8) the velocity component in P along the (i) Cartesian axis, we consider in P a unit vector in the same (i) direction, that is k~0 - 8!0 I
(11)
where 5~i) is the Kronecker delta function. For a unit vector ~(0 in point P, we thus obtain a set of components u]*(i)(P, Q), p,(O(p, Q) of the corresponding Green's vector in point Q. The system (14) gives rise to the well known fundamental solution called > after Hancock [ 12] and used later by Hasimoto and Sano [13] and by Allen and Yao [14] to 295
study Stokes flows in the frame of singularity methods. More recently it has been used by Bush and Tanner [8] as weighting function in a Galerkin method which arrives with a different procedure to the same final set of equations presented here. The Stokeslet solution for the two-dimensional case is given by 1
uT(i)(P' Q)= 47r#
0r )
Oxi Oxi
Or 2rr r Oxi
1 1 . . .
p,(i)(p, Q) . t*(i)(P, Q) = 1
0r -- 6~i) l n r +
(12b)
-- --
7rr
(12a)
(12c)
. ~x.
i
1
where r is the distance between P and Q and n is the unit outward normal vector. If we substitute (2), (4), (9) and (10), considering also the expression (11) for the unit force together with the definition of the Dirac delta function, equation (8) assumes the form
the surface values of velocity and traction are known from the solution o f e q . (14). A numerical solution of these equations may be obtained by a discretization procedure and an iterative scheme, which are described in the following section.
3. NUMERICAL PROCEDURE To solve numerically the integral equation (14) we discretize the boundary into N geometrically linear boundary elements and the internal domain into M triangular cells. Furthermore we assume, as a first approximation, u / a n d t/ to be piecewiese constant along the boundary elements, while each constant value is located at the mid-point of the nth element [18]. Consistently with the above assumption, the volume integral is evaluated by taking constant acceleration values in each internal cell. Hence equations (13) and (14) may be written in the form N
Cu~(e) + ~
u/Q.)
tTr
(15)
Q) d~ =
n=l
ui(P) = fvul(i) (P, Q)p(f]-a])
dV+
(13)
= Z ~(Qn)
uT(i)(P'Q) do +
n=l
-- f u/t*"(i)(P' Q) d~ + ( u;(i)(P' Q) tl"
+ ~ p[f/(Qm)-ai(Qm)]
u;(i)(P,Q) dV
m=l
which gives the velocity field inside the considered volume of fluid. The integral containing behaves like a double layer potential giving rise to a discontinuity across the surface, while the integral containing behaves like a simple layer potential and is continuous across the boundary surface. If the interior point P is taken to approach from inside the bounding surface, equation (13) assumes the limiting
tT(i)
uT(i)
form [15, 16]
where C = 1 for internal points, C = 1/2 for boundary points, and are the values of velocity and traction for the boundary element and analogously and are the values of body force and acceleration for the m-th internal cell. If we now consider points P on the surface and we solve eq. (15) by an iterative procedure, computing the volume integral term using the known solution at each cycle, we may write
u](Qn)
ti(Qn) n-th al(Qm)
fi(Qm)
1
1
- ui(P) = 2
f u;'(~)(e,Q) p(fj - a )
dr-
(14)
"V
N
N
n=]
n=l
(16)
B(i)
where are the components of the known vector term and the coefficients @ 0 , G/(i) are given by
-- Iuitf(i)(P, Q)do + f uT(P, Q) tid~ provided the boundary is sufficiently smooth. The integrals appearing in eq. (14), which have to be interpreted in the sense of the Cauchy principal value, contain only weak singularities and therefore equation (14) may be analyzed as a Fredholm integral equation of the second kind [16, 17]. Equation (14) together with the boundary conditions (to be specified in the following sections) relates the surface values of velocity and traction with a body force term containing f] and the non linear acceleration a/ which is not known a priori. The acceleration term may be calculated from the velocity field which is obtained by eq. (13) after 296
It](O(n)=fntl*(O(P,Q)da;Gi(i)(n)=fnU~.(O(P,Q)do (17 a, b) Equation (16) is written in each boundary point P for the two (i) components. We have therefore 2N algebraic equations plus 2N boundary conditions to determine the set of 4N variables u F tF The boundary conditions in discretized form may be expressed as follows: for a solid wall u 1 = 0,
u2 = 0
(18)
for the free surface, defining ~ as the angle between the norMECCANICA
mal to the surface and the x 1 axis
R G~2)
t 1=
-- t n
where 2R is the length of the element under consideration. We can write eq. (16), where now all the coefficients are known, in the vector form
where t n and t t are given by
Hu -- Gt = B
do -- cos~ dx 1
(20)
o is the surface tension, which is assumed to be a linear function of x I and R is the radius of curvature. If the free surface changes its configuration, besides the above dynamic conditions, we have to impose the kinematic condition ah
ah
--
=
u 2 -- u 1
(21)
at
ax 1
where h is the displacement of the surface in the x 2 direction, which is used to find the new shape of the surface. The surface integrals/~/0 and G!] i) are evaluated for each boundary element by means of a four points Gaussian quadrature formula. Recalling the expressions of uT(i) ( 1 2 a ) a n d t/*(i) (12c) and the relation ~r ax i
- cos 0
f%
uT(i)(P, Q) dcr
(22)
=
1[
y-'co k k= 1
&x i (i) + _ _
-- In(r) ~
r
8]
Ax. ] 1
7
0
n
(23)
tT(i)(P, Q) du = n
~
Ax = D
(26)
where A is a completely populated matrix and it can be solved by using a Gaussian elimination technique. After the boundary values are found through the above computational procedure, equation ( 1 5 ) w i t h C = 1 is used to evaluate the velocity components at the internal points. In vector form it may be expressed as (27)
These field velocities are then used to update the acceleration term and a new iteration cycle may be performed.
COk
--
r
7/.
17 n
In particular for Q and P on the same element (arian) = O, hence integral (23) gives H{i) = 0. The integral (22), instead, 1 is evaluated analytically by letting the segment 2e across point P to shrink to a zero length R G~ 1) = - [(1 - - l n R ) + cos20] 2~r# R G(22) = - [(1 -- lnR) + sin 20 ] 2=p
The computational procedure, described in the previous section, has been applied first to a quite simple case, thermocapillary convection with fixed free surface configuration, to be considered as a test case for a preliminary evaluation of the procedure itself. In fact very well established finite difference solutions, are available in this case for comparison purpose. We assume, for the sake of simplicity, a linear temperature profile in the flow field in order to obtain a constant driving force, given by the surface tension gradient along the free surface
A particular attention has to be paid when the integration is performed on the same element where the point P is. In this case the integrals (22) and (23) contain singularities of the order ln(r) and 1/r respectively and they have to be evaluated in the sense of Cauchy principal value, as indicated before.
19 (1984)
We can also rearrange eq. (25) by setting the unknown boundary values vector x (either uj or t.) and the coefficient matrix A. The remaining known quantities, going to the right hand side form together with the components B (i) the known vector D. Thus in compact form we have
4. NUMERICAL RESULTS
4
k=l
where H is the coefficient matrix with elements H{i) and 1/2 I on the principal diagonal.
where H' has now elements H~/) also on the principal diagonal and u(Q), t(Q) are the surface values obtained from the solution of (26).
r
we can compute the integrals (17 a, b) as
i
(25)
Iu(P) = Gt(Q) - H'u(Q) + B
Ax.!
(24)
(19)
t 2 = t n cos a + t t sin a
tt =
sin 0 cos 0
G ~ 1) =
2rr#
sin a + t t cos a
a =-tn R
=
(24)
be
be
aT
aT
ax 1
-
aX 1
where
=
C
(2a)
ao/aT has a constant value.
Fig. 1 shows the numerical results obtained both with a ~, oo finite difference scheme (F.D.) and with the present boundary integral equation method (BIE) for R e = 1. The comparison seems very satisfactory. The u 2 velocity component of the free surface is evaluated better by BIE (where it is a real unknown of the problem) than by F.D. (where it is extrapolated from flow field values). The better agreement of F.D. with BIE with N = 20 rather than with BIE with N = 40 seems to confirm the above conclusion. The computer time (CPU seconds) is quite the same for F.D. (32 sec) and for BIE with N = 40 (31 see), while it decreases drastically for BIE with N = 20 (4.5 sec). In the 297
0 ,/1
Fig.
trace
Fig. 3.
U1
I .
.
.
.
.
.
velocity profiles and surface configuration, comparison for
Re = 100, moving surface.
Fig. 2. u 1 velocity profiles, comparison for Re = 100, fixed surface.
Fig. 4. Drop formation: surface shape evolution in time.
former BIE solution the coefficient matrix to be solved is
rent approximating technique, which is under investigation, for the volume integral, which contain the non linear terms. F o r the same thermocapillary flow problem, we consider now a moving free surface configuration, which then assumes, at steady state, a curvilinear shape. In these conditions, F.D. method need to be reset completely in general curvilinear coordinate [1, 2], while BIE may be applied without any major change with respect to the previous case. In a linearized procedure, the geometrical variations of the surface are assumed to be small between two subsequent steps and the dynamic boundary conditions are evaluated from the known surface shape. The resulting surface velocity compo-
80 x 80 (two unknows for each element) while in the latter is decreased to 40 x 40. The still good results obtained with the fewer elements solution, together with the small computing time required, give a first sign of the good qualities and capabilities o f BIE method. A good agreement is maintained for larger Reynolds numbers, as shown in fig. 2 for Re = 100. In this case the convergence rate was poor for the larger values of the non linear convective terms which require a larger number of iterations to reach the steady state conditions. At this purpose an improvement should be obtained, at large Re, by using a diffe298
MECCANICA
nents are then used to find the new configuration according to the kinematic condition (21), which was not required in the previous case. The convergence of the above procedure is strongly dependent on the time step. An overextimation of the surface deformation may easily bring to the divergence of the solution if a particular care is not taken in the discretization of equation (21). The velocity profiles and the surface shapes for R e = 100 are shown in fig. 3 in comparison with results previously obtained by the authors [1,2] with a F,D. method in generalized curvilinear coordinates. The agreement of resuits is still satisfactory in this case, but the computer time
use is now much in favour of BIE method. A second free surface application considers the problem of drop formation in microgravity conditions. The real transient character of this problem, introduces further difficulties in the numerical procedure, which is still under investigation for this case. The time evaluation of the drop is approximated by a sequence of steady states, for which we have to know both flow field and surface configuration, as shown in fig. 4. The internal domain is divided into triangular elements whose area is evaluated at each time step according to the surface deformation. No comparison is available for these preliminary results. Received: December 14, 1983; in revisedform: April 19, 1984.
REFERENCES [ 1] STRANIM., PWA R., Surface tension driven flows in micro-gravity conditions, Int. Jour. for Numerical Methods in Fluids, vol. 2, 367 - 386, 1982. [2] STRANIM., PWA R., GRAZlANIG., Thermocapillary convection in a rectangular cavity: asymptotic theory and numerical simulation, J. Fluid Mech. vol. 130, 347, 1983. [3] PIVA R., DI CARLOA., FAV1NIB., and GUJ G., Adaptive Curvilinear Grids for Large Reynolds Number Viscous Flows, Lecture Notes in Physics n. 170, 414 - 420, 1982. [4] MORINO L. and KUO C.C., Subsonic Potential Aerodynamics for Complex Configurations: a General Theory, AIAA J., vol. 12, n. 2, t91 - 197, 1974. [5] Rlzzo F., An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics, Q. Applied Math., vol. 25, 83, 1967. [6] CRUSET.A., Numerical Solutions in Three Dimensional Elastostatics, Int. J. Solids Struet., vok 5, 1295, 1969. [7] BREBBIAC.A., The Boundary Element Method for Engineers, Pentech Press, London, 1978. [8] BUSH M.B., TANNER R.I., Numerical Solution of Viscous flows using lntegral Equation Method, Int. J. Numer, Meth. Fluids, vol. 3, 71, 1983. 19 (1984)
[9] MORSE P.M,, FESHBACH H., Methods of theoretical Physics, McGraw-Hill, 1953. [10] GREENBERGM.D., Application of Green's Functions in Seiem:e and Engineering, Prentice Hall, 1971. [11] P1VAR., MORINOL, Green's Vector Method for Unsteady Navier Stokes Equations, (to appear), 1984. [12] HANCOCKG.J., The self propulsion of Microscopic Organisms through Liquids, Proc. Roy. Soc. A. 217, 96, 1953~ [13] HASIMOTOH., SANOO., Stokestet and Eddies in CreepingFlow, Ann. Rev. Fluid Mech., vol. 12, 335, 1980. [14] CHWANG A.T., WU T.Y., Hydromechanics of low Reynolds number flow. Part 2. Singularity method for Stokes flows, J. Fluid Mech., vol. 67, 787, 1985. [15] LADYZHENSKAYAO.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Beach, New York, 1963. [16] MIKHLIN S.G., Integral Equations, Pergamon Press, London, t957. [17] ZABREYKOP.P., et al., Integral Equations. a Reference Text, Noordrhoff, t975. [18] MA S.Y, The Boundary Elements Applied to Steady Heat Conduction Problems, Proc. Third Int. Conf. Num. Meth. in Thermal Problems, Seattle Aug. 1983. 299
