This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.
The instability of forced flow in a rotating cylindrical pool with a differentially rotating disk on the free surface is investigated through a series of unsteady three-dimensional numerical simulations.The results show that the basic flow state of this system is axisymmetric and steady,but has rich structures at the meridian plane.However,when the rotation Reynolds number exceeds a critical value,the flow will undergo a transition to three-dimensional oscillatory flow,characterized by the velocity fluctuation waves traveling in the azimuthal direction.The main characteristics of the flow patterns are presented,including the propagating direction,velocity,amplitude and wave number,which depend on the rotation rates and directions of the disk and the cylindrical pool,and the critical conditions for the onset of oscillatory flow are also determined.For the case of disk-only rotation,the centrifugal instability is responsible for the flow transition,and when the disk isoand counter-rotates with the cylindrical pool,the mechanisms for the transition are elliptic and of circular shear instabilities,respectively.
WU ChunMei & LI YouRong College of Power Engineering,Chongqing University,Chongqing 400044,China
The steady laminar two-dimensional thermocapillary convection of two superposed horizontal liquid layers in a shallow annular cavity was investigated using asymptotical analysis.The liquids were supposed to be immiscible with a nondeformable interface.The cavity was heated from the outer cylindrical wall and cooled at the inner wall.Bottom and top surfaces were rigid and adiabatic.Asymptotic solutions were obtained in the core region in the limit as the aspect ratio,which was defined as the ratio of the lower layer thickness to the gap width,and trended to zero.The numerical experiments were also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection.It is found that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.
LI YouRong,WANG ShuangCheng,WU ShuangYing & SHI WanYuan College of Power Engineering,Chongqing University,Chongqing 400044,China
