Boundary Element Method Of AnalysingNon-linear Phenomena On Gas-liquidTwo-phase Free Surface At Middle And HigherReynolds Numbers

Abstract

A boundary element method for analysing non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers has been developed. Employing this method we analyze two kinds of non-linear phenomena on gas-liquid two-phase free surface (l) Solitary wave on free falling thin liquid film. (2) Surface tension drives convective flows in open-boat type crystal-growth techniques in microgravity environment. INTRODUCTION Although some new schemes of discretizing convective terms are developed as QUICK and QUICK ER in finite difference methods[FDM] and finite element methods[FEM], but it is very difficult to avoid numerical diffusion introduced by convective discretizations, especially, at high Reynolds number. In [1], they developed a boundary element iterative scheme(BEIS) to calculate convective terms and compared the advantages of using upwind or central difference schemes and BEIS. As shown as numerical results, BEIS of the convective terms was the most accurate of the two approaches. However, computation was complex. The main objective of this research is to develop a boundary element method for analysing some non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers. On the basis of the boundary element formulation with the pressure penalty function, the divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation. Consequently, velocity gradients are eliminated, the complete formulation is written in terms of velocity. This avoids the difficulty of convective discretizations, and provides considerable reducing storage and computational requirement while improving accuracy. In order to calculate free surface, we construct shooting iterative method of free surface on the basis of normal stress balance condition Project supported by the National Natural Science Foundation of China Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 168 Free and Moving Boundary Problems similar to [2] [3]. The method is applied to analyze two kinds of free surface problems. (1) Solitary wave on free falling thin liquid film. This problem was researched by Wasden and Kheshgi etc. with finite difference method [4] and finite element method [5]. However, in finite difference method[4], one applied an adjustment procedure for wave velocity in order to obtain reasonable flow fields, in finite element methods[5] numerical solution was limited to low flow rates. Moreover, the upwind schemes of discretizing convective momentum terms common to FDM and FEM introduce numerical diffusion or oscillatory behavior of the wave peak region, more importantly, the upwind scheme lacks sensitivity to cross stream diffusion and source terms which are especially important in the case of a thin film. Therefore it is valuable to research a non-traditional numerical method such as BEM. (2) Surface tension drives convective flows in open-boat type crystal growth techniques in microgravity environment. The analytical or numerical study of free surface problem in crystal-growth techniques is very difficult. We shall restrict ourselves to model a problem of open-boat type crystal growth technology, and we only study the melt-gas free boundary, assume the melt-crystal interface shape is flat. The free surface problem of open-boat type is simplified as the free surface problem in open rectangular cavity. The left vertical boundary of the cavity can be interpreted as the crucible wall. The right vertical boundary of the cavity can be interpreted as the melt-crystal interface. The temperatures equal #/ and 0,., respectively. Some important numerical results are presented in paper. BASIC EQUATIONS We consider a system of steady incompressible viscous liquid in a open rectangular cavity with differentially heated lateral walls (the temperature difference is A0*). There is a gradient of surface tension on the free surface since the distribution of temperature is non-homogeneous. The gradient of surface tension drives the convection of liquid. On the basis of the basic law of fluid mechanics and Boussinesq approximation, the governing equations of these problems can be described by a tensor notation and a non-dimensional form as follow: continuity equation «,-„• = o (i) momentum equations energy equation V,-^,=0 (3) where, #,„% is the Kronecker delta symbol, m = x for solitary wave, m = y for thermocapillary convection. The lengths, velocities are scaled with respect to H, Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X