The acoustic impedance of three-dimensional turbines

Abstract

Thermoacoustic stability assessment of gas turbine combustors requires an acoustic impedance boundary condition for the downstream turbomachinery. Actuator disk type analytical methods offer rapid predictions of acoustic impedance in an iterative design process; previous studies show that this class of models works well for cascades consistent with a two-dimensional assumption. Real turbines, however, are three-dimensional with multiple stages, coolant flows, and leakage flows. The first part of this paper validates a cambered semi-actuator disk model for the acoustic impedance of a realistic multi-stage turbine using non-linear time-marching computations: the two methods agree to within 9% for incident pressure waves and 14% for incident entropy waves. Simulations of the multi-stage turbine with different inlet conditions confirm that, to a close approximation, inlet corrected flow and hence Mach numbers and acoustic impedance are constant during off-design operation. The second part of the paper then applies both analytical and computational approaches to families of parametrically generated turbine stages to quantify three-dimensional design effects. The results show that the influence of hub-to-tip ratio on acoustic impedance is weak, and the two-dimensional analytical model is accurate even for high aspect ratio stages. Front-loaded camber lines increase axial Mach number within blade passages, raising acoustic impedance by up to 51% compared to a datum quadratic camber line. Varying the stator–rotor axial gap changes the relative phase of reflections from vanes and blades, causing the total impedance to either increase or decrease, at different frequencies, by up to 11%. The cambered semi-actuator disk method consistently captures the correct trends, showing that the physical basis of the model is sufficient to produce a broadly applicable design tool for rapid assessment of turbine impedance boundary conditions.