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Invited Paper Time-Dependent Stokes Flow for Interior and Exterior

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Invited Paper Time-Dependent Stokes Flow for Interior and Exterior Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 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 fun- damental 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 pre- sented 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 Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 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. Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 333 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: dp 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)<Tij(u(y),p(y}}nj Js Js (2) + I U?(x,y)gi(y)dy Jn 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) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 334 Boundary Elements To reduce the last domain integral in equation (5) to an equivalent boundary in- tegral, 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: 7s 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 € Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 335 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 4- 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 Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 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 al- though 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.
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