Invited Paper Time-Dependent Stokes Flow for Interior and Exterior

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Invited Paper

Time-dependent Stokes flow for interior and exterior domains via dual reciprocity boundary element

method H. Power", P.W. Partridge*

Wessex Institute of Technology, University of Portsmouth,

Ashurst Lodge, Ashurst, Southampton S04 2AA, UK " On leave from the Instituto de Mecdnica de Fluidos,

Universidad Central de Venezuela, Venezuela * On leave from the Department of Civil Engineering,

University of Brasilia, Brazil

ABSTRACT:

This paper presents a boundary element formulation for the transient Stokes equations for interior and exterior domains, in which the well known closed form fundamental solution to the steady Stokes equations is employed and the time derivative is taken to the boundary with Dual Reciprocity Method. In order to satisfy the re-

quired asymptotic behavior at infinity in the case of an unbounded domain, special considerations have to be taken into account in the formulation of the Dual Reci- procity Method. This approach has the advantage of simplicity of formulation and implementation in relation to alternative boundary element schemes. Results are presented for three-dimensional internal and external problems showing the accuracy of the method.

1) INTRODUCTION:

The boundary element method will be applied to the non-permanent Stokes system of equations:

here, u is the flow velocity, p is the fluid density, p is the pressure, p, is the dynamic viscosity, and 12 is a bounded or an unbounded three-dimensional domain with respect to the contour surface S, that we will consider to be of Lyapunov type. In general, the flow field can satisfy the following mixing boundary condition:

for all

332 Boundary Elements and

^(^P)^K) = /*(() ^r all where S\ U 5o = 5, and ffij(u,p) = -pkj + ^(du^/dxj + duj/dxi) is the stress tensor corresponding to the flow field (u,p); and the following initial condition

Ui(x,Q)=U?(x) for all x € ft

The above system of equations is a first approximation to the complete Navier- Stokes system of equations for incompressible flow, when the flow phenomena occur over a time scale of the order O(L*/v), where L is a characteristic length and is = n/p is the kinematic viscosity, and the Reynolds' number, Re — U L/v, is small, then the acceleration term Oui/dt is of the same order as the viscous terms, but the non-linear term will be 0(Re) smaller than the remaining terms. The fundamental solution of the above system of partial differential equations, ), is given by (see Ladyzhenskaya [1]):

where B(x,y,t) = (4xi>t)~3/-exp(-r- /4vt) is the fundamental solution of the heat equation, 6*% is the Kronecker Delta, 6(t) is the Dirac Delta function and r =| x-y \ . The integral representation formulae based upon this fundamental solution leads to a time-boundary integral equation, see for instance Gavze [2j. The principal difficulty encountered with this approach, is that the above fundamental solution cannot be evaluated explicitly, requiring the use of numerical integration. Another possibility to obtain an integral representation formulae that can be used as a basis for the boundary element numerical solution of the above system of partial differential equations, is to transform the non-permanent Stokes system of equations to the frequency domain as was done by Pozrikidis [3]. A simple closed form fundamental solution exists for the transformed equation. The main disadvantage of this approach is that it is necessary to carry out the inverse transform on the solution. An alternative approach, to be employed here, is to use the fundamental solution to the permanent Stokes equations, and express the time derivative as a domain integral. This fundamental solution has a simple closed form, however the domain integral will introduce extra numerical computations. Several methods have been developed to take domain integrals to the boundary in order to eliminate the need for internal cells. One of the most effective to date being the Dual Reciprocity Method (DRM). This method was introduced by Nardini and Brebbia [4], the method is general and straightforward to apply. In this work the DRM will be employed in order to take the corresponding domain integral to the boundary. In this way a boundary integral equation will be obtained without domain terms, using a closed form fundamental solution and without the need to recourse to inverse transforms. This approach has been suggested by Jin and Brown [5] for the interior two-dimensional case, however no numerical implementation was carried out.

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2) DUAL RECIPROCITY METHOD: Consider the unsteady Stokes equation written as a non-homogeneous steady

form, with the original non-permanent term as the forcing term in the new system of equation:

Let us consider first the case of a bounded domain, in this case the velocity integral representation or Green's formulae for a point x e!7 is given by Ladyzhenskaya [11 as:

(x) = I Kkj(x,y)*j(y)dSy- I U*(x,y)

where f/* is the velocity field of the fundamental' solution of the Stokes' equation known as a Stokeslet, with a corresponding pressure q* :

and

(3,c)

In order to express the boundary integral in equation (2) in terms of equivalent domain integrals, a dual reciprocity approximation is introduced (see Partridge et al. [6]). The basic idea is to expand the time-derivative Oui/dt in the form,

gt(x) = P^=J2 /{x. lT )ai"(0*« (4) m=l

The above series involves a set of known functions f(x, y™) which are dependent only on geometry, a set of unknown vector coefficients a™ which are time-dependent only, and y™ , with m = 1,2, • • -p, been p fixed collocation points chosen to be in the closed domain where the function is approximated (also called nodal points). With this approximation, the domain integral becomes

(5)

334 Boundary Elements

To reduce the last domain integral in equation (5) to an equivalent boundary integral, let us define a new auxiliary non-homogeneous Stokes' flow field (£//(#, y™)ei, ;3'(o:,7/™)), for each collocation point, t/"\ in the following way:

Applying Green's formulae to the non-homogeneous Stokes' flow field (U-(x, y™) ei,p*(x,y™)), we obtain

"/ m / ^"^^ ) = y^

J^ f kf m Jn or equivalently

r k m ~/ m / " / m J^ kj , j y

f k "I ~/

y^ Substituting the last equation into equation (5), the domain integral can be recast in the form

y,/")<#:,

+ /^,^(0'(^-),,,^,-.-))»W«5) "" A ^ " '" ^ ^J and using the resulting expression in equation (2), one finally arrives at a boundary only integral representation formula for the velocity field at point xeO, in the form:

4- 2^ #r i &\(3\^™) - / Kkj(x,y)Uj(y,z™) dSy (9) m=l ^ J $ { k "/ m ~l A ' ' U % )- M i- J x €

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Evaluating the above equation at points £ on the contour surface 5, and using the known jump property of the double layer potential, we obtain the following system of Fredholm's integral equations:

- %(()%,(() - / /W(, Z/) %,(%/) <#% 7s , ^)- / ;w(,y)^j(2/, m=l •'S

where (7%;(42/ Cij(£) = 0(f)<$t;, and 0^-(() is the solid angle at the point f G 5. It is usual in DRM to choose the interpolation nodal points as the surface source

points used in the boundary element method plus additional internal points, where the velocity in the interior nodal points is a unknown to be determined. Internal nodes are needed in the formulation in order to be able to consider the initial condition of the problem and to advance in time with the numerical algorithm. For simplicity, equation (9) evaluated at the internal nodal points and equation

(10) will be written as a single equation

f Js

(11, a)

In the above equation Atj(f) is equal to Ckj(£), and the vector u is equal to the

boundary velocity when the evaluation point £ is at the contour surface, and they are equal to zero and equal to the interior velocity, respectively, when the evaluation point is one of the internal nodal points, i.e., £ G ft. In the case of unbounded domain, an integral relation like the above can be found as long as the following asymptotic behaviour at infinity is satisfied

Ui(x), U\(x] -> 0(R~*) and p(x),p\x) -» O(R~*) where R = |x|,

and given by:

y,z^)dSy (11,6) S

- f U Js

336 Boundary Elements where the difference in sign between the expressions for the interior and exterior cases, comes from the relation between the direction of the normal vector and the location of the fluid domain in each case.

3) CHOICE OF FUNCTION / :

Previous work on dual reciprocity has shown (see Partridge et.al. [6]), that although a variety of functions can in principle be use as a basic approximation function, best results are normally obtained with simple expansions, the most popular of which is / = 1 + R, where R is the distance between pre-specified fixed collocation points, y™, and a field point x where the function is approximated, i.e.. R =| x - y™~ \ . In the reference given, as well as in previous DRM literature, the choice is based on experience rather than formal mathematical analysis. However, recent mathematical work, related to the theory of mathematical interpolation, and unrelated to the integral equations literature, based on the so called radial functions, has partially sustained these numerical findings, the approximation function, / = 1 4- R, defined previously, in general use in DRM is just one such radial function. This mathematical work provides general convergence criteria for the interpolation series based on radial functions. The convergence is derived from the fact that the behaviour of these interpolation series is local, i.e.. increasing the number of fixed nodal points makes the interpolation more localized. It is interesting to note that although the basic functions are defined globally, the interpolation series have local behaviour. For the case of radial functions defined by the above *'/" function in one dimensional space, these new theories have shown that the resulting interpolation series happens to be a natural spline interpolation. Much of this mathematical work has addressed the properties of these approximations when the data points, collocation points y™, form an infinite regular grid, because this structure allows the order of accuracy to be derived when an interpolation or quasi-interpolation procedure is applied to a smooth underlying function. Unfortunately, the subject of non-regular grids has insufficient theoretical support, and is an on-going topic of mathematical research. In practical application of the DRM, the grid for the interpolation series is usually non-regular (for more detail on the theory of radial basis function approximation see Powell [7] or Powell [8]). By evaluating expression (4) at all nodal points and inverting, one arrives at

Micchelli [9] has proved that for the case when the nodal points are all distinct, the matrix (F')~\ resulting from the basic function defined previously, for all positive integers p, (the dimension of the matrix), and n, (the dimension of the space, #",) is always non-singular, i.e.. the matrix is invertible. Therefore, as long as the function g(x) is regular, then, the above vector coefficients a™, that are used in the dual reciprocity schemes, are well defined.

In order to find the corresponding particular solution of the non-homogeneous Stokes system of equations (6,a,b), we will use an approach suggested by Happel and Brenner [10] to find the fundamental solution of the Stokes homogeneous equations, defining the second rank tensor Ul(x,y™) in terms of an auxiliary potential 0, as follows:

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*•'-•-'

By differentiating the above equation, we find that, for any choice of the potential V>, the continuity equation (6,b)is automatically satisfied. Substituting expression (13) into equation (6, a), we obtain

If we now assume that ^ satisfies the non-homogeneous biharmonic equation

the solution of which is given by

lh/ - -!/#( < 1 R* .), ^ //120 360'' then, the pressure field has to satisfy the following equation:

+,% -—' = .0 n(16m)

The biharmonic potential 0, thus obtained is substituted into equations (13) and (17), in order to obtain the following expression for the non-homogeneous Stokes flow field (t//(x,2T)e;,P'(z, ))

and

with the stress tensor cr^(£7/(?/, ^™)e,-,p'(y, z™)) given by:

1 O = ( (x, -

f )^

338 Boundary Elements

As can be observed, the above auxiliary non-homogeneous Stokes flow field is only appropriated for the case of bounded domain, since it does not satisfied the required asymptotic behaviour at infinity for the unbounded case. For the unbounded case, it can be proved that a function

tf=±(l + #)'", (20)

solution of equation (15) with a non-homogeneous term equal to

which is also a radial function, will lead to the following auxiliary non-homogeneous Stokes flow field

and

Therefore

The asymptotic value at infinity of the above flow field is just the required and 0(R~~) for the velocity and shear stress field, respectively.

4) NUMERICAL SOLUTION:

For the numerical solution of the interior problem, equation (11-a) is written in a discretized form where the boundary integrals are approximated by a summation of integrals over individual boundary elements, i.e.,

r E / f/f(f,%)4%K% 4- ;=1./AS. N r

- Y] / Kkj(t,y)uj(y)dSy = (25)

f € S

Equation (25) can be written in matrix form

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G - Hu = a(G^ - HU*) (26)

In the above system, G and H are square matrices, the coefficients of which are calculated through integrating products of U* and A'**' by the interpolation function, respectively, over each boundary element. In the examples considered here, quadratic eight node elements were used. Details of standard boundary element techniques may be found in, for instance, (Brebbia, Telles and Wrobel, [11]). By substituting equation (12) into (26) we obtain

G<£-Hu = Su (27) where the dot denotes temporal derivative, and

S = F-\G

Equation (27) can be integrated in time using standard finite difference time- stepping techniques between time levels m and (m + 1), in such a way that the values of U® will be the initial conditions for the problem.

Ui = ^(«r"-C) (29)

Substituting equation (29) into (27) one obtains

Go™+i - (H + A)u«"+* = (H - -|-)U™ (30) L\T> L\L

It should be noted that the coefficients of the matrices H, G and S depend on geometry only, thus they can be computed once only and stored. Finally the above linear algebraic system for the unknown surface density and internal velocity at the advanced time-step is solved using a standard Gauss procedure. Letting Urn+i become Um values for a new time-step can be computed until the desired total time is reached. In the case of bounded domain the integral surface was discretized using 64 continuous elements with 8 source points each, in the series (4) we used 277 nodal points, of which, 194 were the same source points of the surface elements and the remaining 83 corresponded to internal nodes, and a time-step of 6t — 0.01 was employed.

Results where obtained for the problem of starting flow in a pipe, where the fluid contained in a long pipe of circular cross-section is initially at rest, and is set in motion by a difference between the pressures at the two ends of the pipe suddenly imposed. This interior problem can be solved evaluating equation (30) with the following boundary and initial conditions:

Uj = 0. i — 1.2,3 for every

u-2 = (63 = 0 for every

340 Boundary Elements where S*, is the surface of the pipe's wall, So is a control surface at the pipe's inlet, Si is a control surface at the pipe's outlet, and we have chosen the or-coordinate in the axial direction of the pipe. It can be observed that the assumption of zero tangential velocity at the inlet and outlet control surfaces is equivalent to the assumption of uniform flow there, since the condition u

(H - -p-)U™+i - G

In this case, results where obtained for the problem of a constant accelerating sphere moving in an otherwise quiescent fluid, and starting from the state of constant velocity u% = 1 for t < 0. The general analytical solution of this problem was found by Basset^] (see Landau and Lifshitz^' for the derivation) and is given by the following expression:

F(t) =

and in particular in the case of constant acceleration, i.e dU/dt = 1, the total force as a function of time is equal to:

F(t) =

In figure 2, we present the comparison between our numerical results and the corresponding analytical solution for the above sphere problem. In the numerical solution the integral surface was discretized using 64 continuous elements with 8 source points each, in the series (4) we used 204 nodal points, of which, 194 were the same source points of the surface elements and the remaining 10 corresponded to external nodes, and a time-step of 6t = 0.01 was employed.

CONCLUSIONS:

In this paper a dual reciprocity boundary element scheme has been used to solve the unsteady Stokes equations for internal and external problems. This scheme uses the closed form steady Stokes fundamental solution which is very well known and the numerical integration of which is well established (Youngren and Acrivos [15]). The scheme has the advantage of simplicity of formulation and implementation in relation to the alternative boundary element approaches, discussed in the introduction, as numerical integration of the fundamental solution and inverse transforms are not required. The accuracy of the method is shown by the results presented.

Boundary Elements 341

REFERENCES

[1] Ladyzhenskaya, U.A. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, (1963). [2] Gavze, E. The Accelerated Motion of Rigid Bodies in Non-Steady Stokes Flow.

Int. J. Multiphase Flow Vol. 16 No.l (1990). [3] Pozrikidis, C. A Study of Linearized Oscillatory Flow Past Particles by the Boundary-Integral Method. J. Fluid Mech. Vol. 202(1989). [4] Nardini, D. and Brebbia, C.A. A New Approach to Free Vibration Analysis using Boundary Elements, in Boundary Element Methods in Engineering, Computational Mechanics Publications, Southampton and Springer-Verlag, Berlin and New York,

(1982). [5] Jin, X. and Brown, O.K. Dual Reciprocity BIE in Lagrangian Form for In- compressible Unsteady Viscous flow, in Boundary Element Technology VI, Compu- tational Mechanics Publications, Southampton and Elsevier Applied Science London New York (1991). [6] Partridge, P.W., Brebbia, C.A. and Wrobel, L. The Dual Reciprocity Boundary

Element Method. Computational Mechanics Publications, Southampton and Elsevier Applied Science London New York (1992). [7] Powell, M.J.D. Radial Basic Functions for Multivariable Interpolation: a Re- view, in Algorithms for Approximation, eds. J.C. Mason and M.G. Cox, Oxford University Press, Oxford (1987). [8] Powell, M.J.D. The Theory of Radial Function Approximation in 1990, presented at the Numerical Analysis Summer School, Lancaster (1990). [9] Micchelli, C.A. Interpolation of Scattered Data: Distance Matrice and Condi- tionally Positive Definite Functions, Constr. Approx., Vol.2, (1986). [10] Happel, J. and Brenner, H. Low Reynolds Number Hydrodynamics. Martinus

Nijhoff, The Hague, (1983). [11] Brebbia, C.A., Telles, J.C.F. and Wrobel, L. Boundary Element Techniques, Springer-Verlarg, Berlin and New York (1984). [12] Batchelor, G.K. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, (1967). [13] Basset, A.B. .4 Treatise on Hydrodynamics, Vol. 2, Deighton, Bell and Co.,

Cambridge, (1888). [14] Landau, L.D. and Lifshitz, E.M. Fluid Mechanics, Pergamon Press, New York, (1959). [15] Youngren, G.K. and Acrivos, A. Stokes Flow Past a Particle of Arbitrary Shape: A Numerical Method of Solution, J. Fluid Mech. Vol. 69 (1975).

Figure 1 Longitudinal velocity in the problem of starting flow in a pipe for dimensionless time t = 0.1,0.2, 0.4 and 0.8.

analytical solution, ••• numerical solution.

Figure 2 Dimensionless drag force in the problem of a moving sphere at constant acceleration as a function of time. analytical solution, ••• numerical solution.