Stability analysis of uniform and non-uniform annular passages conducting incompressible laminar flows for small and large amplitude oscillatory motions of the outer cylinder

Abstract

A computational method is developed involving the simultaneous integration of the Navier–Stokes and structural equations for the purpose of studying the stability of concentric annular passages conducting incompressible laminar flows. It is assumed that one side of the annulus, i.e. the centre-body, is fixed and the outer cylindrical duct is flexibly supported. The outer cylinder is displaced from its equilibrium position and is then released. In this situation, the fluid part of the problem is solved by an accurate method using a three-point backward implicit scheme, followed by a pseudo-time iteration using an artificial compressibility factor. The fluid equations are discretized in space based on a finite-difference formulation and primitive variables, for which stretched staggered grids are used. The resulting equations are cast in delta form and are solved using an ADI scheme. The fluid forces acting on the vibrating cylinder are calculated from the integration of the unsteady pressure and shear stresses resulting from the unsteady primitive variables calculated. The equations of motion of the structure, subjected to the calculated fluid forces are solved using the Runge–Kutta scheme to obtain the displacement of the moving cylinder. The problem is solved: (a) for small-amplitude motions, by means of the so-called mean position (MP) analysis and (b) for large amplitude oscillations of the outer cylinder for which a time-dependent coordinate transformation (TDCT) is used to fix the computational domain. Both these approaches (MP and TDCT) are applied to uniform and non-uniform (backstep-shaped) annuli for translational motion of the cylinder. The problem is also solved for (i) rocking motion and uniform annuli and (ii) translational motion for diffuser-shaped annuli, with only MP analysis. When the MP analysis is considered, it is shown that, for translational motion of the outer cylinder, the most stable configuration is that of a uniform geometry and the least stable one is the backward step geometry. For rocking motion in uniform annular geometry, for some system parameters and high enough flow velocity, the outer cylinder can develop flutter (limit-cycle oscillation). The comparison between the results of MP and TDCT analyses for uniform and backstep geometries for translational motion of the cylinder indicates that the outer cylinder is less stable when TDCT is used and the coupled frequency of oscillation is also changed considerably.