THERMAL-CONVECTION IN VERTICAL CYLINDERS - A METHOD BASED ON POTENTIALS OF VELOCITY

Any solenoidal vector field can be written in terms of two scalar fiel ds. Solving the Navier-Stokes equation for these two so-called potenti als is a widely used method, although difficulties have been reported in using this method to describe non-Cartesian confined flows, in part icular cylindrical geometries. Difficulties arise because the potentia ls are coupled at the boundaries but also because of possible numerica l instabilities associated with the high order of the equations. This paper shows how all these difficulties can be circumvented: firstly by properly deriving the required boundary conditions, but also through an adequate discretization, which is the final requirement. The use of spectral methods together with a reduction of order for the axisymmet ric modes is sufficient for this purpose. In this paper, as a test, a velocity potentials method is used to solve the stability problem for the onset of convection in a fluid confined in a cylindrical container .