UNSFLO : a numerical method for the calculation of unsteady flow in turbomachinery

Abstract

Introduction: There are four principal sources of unsteadiness in a single stage of a turbomachine in which there is one row of stationary blades (stators) and one row of moving blades (rotors). As shown in Fig. 1.1, wake/rotor interaction causes unsteadiness because the stator wakes, which one can consider to be approximately steady in the stator frame of reference, are unsteady in the rotor frame of reference since the rotor is moving through the wakes and chopping them into pieces. This causes unsteady forces on the rotor blades and generates unsteady pressure waves. Although the stator wakes are generated by viscosity, the subsequent interaction with the rotor blades is primarily an inviscid process and so can be modelled by the inviscid equations of motion. This allows two different approaches in numerical modelling. The first is to perform a full unsteady Navier-Stokes calculation of the stator and rotor blades.

The second is to perform an unsteady inviscid calculation for just the rotor blade row, with the wakes being somehow specified as unsteady inflow boundary conditions. This latter approach is computationally much more efficient, but assumes that one is not concerned about the unsteady heat transfer and other viscous effects on the rotor blades. Potential stator/rotor interaction causes unsteadiness due to the fact that the pressure in the region between the stator and rotor blade rows can be decomposed approximately into a part that is steady and uniform, a part that is non-uniform but steady in the rotor frame (due to the lift on the rotor blades) and a part that is non-uniform but steady in the stator frame (due to the lift on the stator blades).

As the rotor blades move, the stator trailing edges experience an unsteady pressure due to the non-uniform part that is locked to the rotors, and the rotor leading edges experience an unsteady pressure due to the non-uniform part that is locked to the stators. This is a purely inviscid interaction which is why it is labelled a "potential" interaction. There are again two approaches to modelling this interaction. The first is an unsteady, inviscid calculation of the stator and rotor blade rows. The second is an unsteady, inviscid calculation of just one of the blade rows, either the stator or the rotor, with the unsteady pressure being specified as a boundary condition. The latter approach is more efficient, but unfortunately the situation in which the potential stator/rotor interaction becomes important is when the spacing between the stator and rotor rows is extremely small, and/or there are shock waves moving in the region between them.

Consequently, one does-not usually know what values to specify as unsteady boundary conditions. The first two sources of unsteadiness were both due to the relative motion of the stator and rotor rows. The remaining two sources are not. The viscous flow past a blunt turbine trailing edge results in vortex shedding, very similar to the Karman vortex street shed behind a cylinder. In fact real wakes lie somewhere between the two idealized limits of a Karman vortex street and a turbulent wake with steady mean velocity profile. It is believed that provided the integrated loss is identical the choice of model does not affect the subsequent interaction with the downstream rotor blade row. However, this is an assumption which needs to be investigated sometime in the future. The importance of vortex shedding lies in the calculation of the average pressure around the blunt trailing edge, which determines the base pressure loss, a significant component of the overall loss.

There is also experimental evidence to suggest that the vortex shedding can be greatly amplified under some conditions by the potential stator/rotor interaction. Finally, there can be unsteadiness due to the motion of the stator or rotor blades. The primary concern here is the avoidance of flutter. This is a condition in which a small oscillation of the blade produces an unsteady force and moment on the blade which due to its phase relationship to the motion does work on the blade and so increases the amplitude of the blade's unsteady motion. This can rapidly lead to very large amplitude blade vibrations, and ultimately blade failure.