Second-Order Effects of Unsteadiness on the Performance of Turbomachines

Abstract

A 2D, linearized approach is used to investigate second-order effects of unsteadiness on the time-mean efficiency of turbomachinery. The objective is to quantify unsteady losses with a nonzero time-mean and to examine numerical simulations with respect to the modeling of unsteady flow fields and loss mechanisms. Results of simulations constitute the input to the analytical models employed. Two unsteady loss mechanisms, one of inviscid and the other of viscous nature, are considered. The unsteady circulation losses, i.e., the transfer of kinetic energy into the unsteady part of the flow field through vorticity shed at the trailing edge of a blade, was first considered by Keller (1935) and later by Kemp and Sears (1956). The vorticity is shed in response to an unsteady blade circulation and determined from Kelvin’s circulation theorem which is valid in compressible homentropic flow. Use of a numerical simulation to obtain circulation amplitudes avoids the limitations of thin-airfoil theory and yields results more realistic for modern turbomachinery. For the unsteady viscous loss mechanism, i.e., the dissipation in an unsteady boundary layer on the blade surface, Lighthill’s high-reduced-frequency limit (1955) is used to obtain the local velocity distribution in the laminar sublayer and the corresponding time-mean unsteady dissipation. The input to the model is the time-harmonics of the pressure gradient along a blade surface obtained from a simulation. A numerical study of the errors introduced by a departure from the high-reduced-frequency limit is presented. Losses from both sources are found to be small.