Thermocapillary-Flow Simulation And Linear Stability Analysis Based On Spectral Element Method

The thermocapillary flow,driven by the non-uniform surface tension along free surface,appears often in industrial processing.Numerical simulation on thermocapillary flow,coupling with the dynamic deformation of free surface,is a long-term challenge in computational fluid dynamics.Meanwhile,the investigation on thermocapillary flow and its instability is important in many engineering application.In this paper,a Galerkin spectral element method(GSEM)and linear stability analysis based on GSEM were developed,and then thermocapillary flow and its instability were investigated.My main works are summaried briefly as:(1)In accordance with the spectral approximation theory of the Fourier/Legendre orthogonal function systems,the Legendre-Legendre spectral element method and the Fourier-Legendre spectral element method for the Poisson-type equations were developed,then a comparison of the computational accuracy between the two spectral element methods(SEM)was performed.(2)In solving the unsteady Navier-Stokes equations,the time splitting method(TSM),including the classical time splitting method(CSM)and the normal time splitting method(NSM),is widely applied in literature.In this paper,an improved TSM,so-called the present time splitting method(PSM)was proposed.By modifying the pressure boundary condition,the proposed PSM is superior to CSM in accuracy and better than NSM in stability.(3)In solving the steady Navier-Stokes equations,the PN×PN-2 method was modified by introduing the SIMPLE algorithm,decoupling velocity and pressure in iteration was achieved,and the matrix dimension in computation was reduced.In addition,the results indicate that the PN×PN-2 method is better than TSM in accuracy,but it is worse than TSM in the algorithm stability.(4)The SEM incorporating with the dynamic mesh technique was developed,the shape control equation and the boundary condition for the dynamic deformation of free surface were derived from the stress balance on the free surface.Then,the thermocapillary flow with considering the dynamic deformation of free surface was simulated numerically,and the effects of several dimensionless parameters on the deformation of free surface were invesitigated.(5)The linear stability analysis based on SEM was developed successfully.The numerical method was verified by the thermocapillary flow in liquid bridge model.Then,thermocapillary flow in the shallow annular pool model with a slowly rotation,was invesitigated,and a new instability phenomenon was observed,the instability mechanism was investigated based on the perturbation-energy analysis.