Axisymmetric annular pipe flow over a backward-facing step on the inner cylinder and its global instabilities

Abstract

The two-dimensional axisymmetric incompressible Newtonian flow in an annular pipe with a backward-facing step in the radial direction is considered. The flow is driven by a pressure gradient and the radius of the inner cylinder decreases suddenly whereas the radius of the outer cylinder remains constant. In addition, the case in which the flow in an annular pipe expands into a simple pipe is also considered. The inlet and outlet lengths, upstream and downstream of the step, respectively, are selected long enough not to influence the flow. Therefore, the flow is characterized by the Reynolds number, the outlet radius ratio (inner-to-outer radius ratio at the outlet cross section) and expansion ratio (ratio of the step height to the outlet gap). The variation of all three these parameters is investigated. The axisymmetric steady basic flow is computed discretizing the Navier-Stokes equations by a second-order finite volume method on a staggered grid. The resulting system of algebraic equations is solved using Newton-Raphson method and polynomial line-search strategy is used to globalize the convergence of this method. Reducing the outlet radius ratio, the Reynolds number at which the flow separates from the outer cylinder decreases. The recirculation zones on the outer cylinder and inner outlet cylinder correlate with each other. The growth of the recirculation zone on the inner outlet cylinder with Reynolds number is strongly reduced, as soon as the recirculation zone on the outer cylinder appears. It is seen that by increasing the expansion ratio, the original separation zone on the outer cylinder shrinks in axial direction while it grows radially. As a result, a strong annular jet between two separation zones appears. In comparison with the flow over a backward-facing step in plane channels, the flow separation on the outer cylinder is stronger and by increasing the Reynolds number this separation point moves toward the upstream which leads to arise a strong annular jet on the inner cylinder. A global temporal linear stability analysis considering three-dimensional perturbations in form of normal modes is performed. An implicitly restarted Arnoldi algorithm and a Cayley transformation are used. Depending on the outlet radius ratio and the expansion ratio, stationary and oscillatory instabilities with different critical azimuthal wave numbers $m$ have been found. We found a pronounced clustering of neutral modes for small radius ratios. In general, deceleration of the flow and lift-up mechanism which arise when a high shear gradient exists were the most important mechanism to destabilize the flow. For small outlet radius ratios, deceleration of the basic flow is found to be the most dominant instability mechanism. Comparing the obtained results with the instability of the flow over a backward-facing step in a plane channel, it can be seen that the cylindrical effect introduced by the annular geometry leads to a much thinner wall jet downstream of the separation zone on the inner cylinder. As a consequence, deceleration and acceleration effects are much more significant in axisymmetric geometries. On the other hand, by increasing the outlet radius ratio, lift-up effects dominate and the instability of the flow over backward-facing step in an annular pipe become more similar to the instability of the plane channel flow over a backward-facing step.