On the transient behaviour of a laminar rotor–stator cavity

Abstract

The unsteady laminar flow between two large rotating disks when one of them is impulsively started is described using the von Kármán similarity analysis to reduce the solution of the Navier–Stokes equations to a set of ordinary differential equations. It is found that the transient phenomenon consists of three distinct phases. Firstly, the development of the Ekman boundary layer, where a quasi-steady Stewartson-type of flow is created. Secondly, angular momentum is initially diffused in the axial direction until the inward radial convection of angular momentum becomes dominating. Thirdly, a Batchelor flow is created once the Bödewadt boundary layer is developed and the entrainment of disk and stator boundary layers are balanced. The dependences of the characteristic dimensionless times on the Reynolds number based on the cavity gap for the second and third stages have been identified numerically and analytically. It is found that the diffusive part of the transient is bypassed if the flow, initially at rest, is perturbed with a small circumferential velocity. The flow and heat transfer transient during a ramp of finite duration has been computed numerically. The integral angular momentum and energy balances of the cavity have been performed in order to understand the energy and momentum budget of the cavity. It is concluded that the spin-up and the spin-down process of a rotor–stator cavity using a quasi-stationary approximation of the fluid, where the time derivatives are neglected, is questionable for realistic gas turbine applications. Finally, the self-similar solution is compared against two-dimensional axisymmetric, steady and unsteady, laminar simulations to assess the limitations and validity of the self-similar analysis. It has been identified that in a closed squared cavity the outer shroud modifies the physics of the transient, but the general conclusions drawn from the one-dimensional model are still valid.