Linear Stability Analysis of Axisymmetric Convective Flows

Abstract

Knowledge of bifurcation behavior is often very important for understanding of heat transfer properties and material processing properties. Linear stability analysis (LSA) of convective flows is a powerful tool for fluid flow and heat transfer predictions, providing quite fast and reliable results. Also, it allows to avoid long time 3-D numerical simulations for idealized flows to reveal bifurcation points. An experience of development and application of linear stability analysis for axisymmetric convective flows is present. Axisymmetric convective flows of forced, thermogravitational and thermocapillary nature are studied on the basis of NavierStokes equations in Boussinesq approximation. Boundaries might be solid walls or fixed shape, free-shear-stress boundaries. The last case includes the kinematic and dynamic boundary conditions in the problem formulation. LSA is perfomed by generalized eigen-value problem solution. The numerical procedure includes basic flow determination by iterative matrix Newton-Raphson method and eigen-values (-functions) calculation by inverse iterations. Finite volumes are employed for problem discretization. Sparse linear systems are solved by BiCGStab with ILU-preconditioning. Efficiency, reliability and accuracy are checked for wide-area standard modern fluid flow problems. Examples from crystal growth processes are performed. An experimental two-branch marginal curve for thermocapillary flow in high-Prandtl liquid bridge is reproduced by application of LSA. The IMA-2 benchmark on low-Prandtl liquid bridge is another example of successful application of developed LSA technique. Thermocapillary convective flow in a rotating shallow annular pool is also studied by LSA. Destabilizing effect of pool rotation at low rotation rates is demonstrated. Efficient possible ways for performing LSA for 3-D flows are discussed.