Abstract
T HE computation of unsteady ows in high-speed airbreathing inlets requires a compressor-faceboundarycondition (CFBC). The physical basis for the analytical or numerical formulation of a realistic CFBC is the acoustic re ection coef cient of the operating compressor. In Ref. 1 an approximate, one-dimensionalmethod is offered to calculate this information for a single blade-row compressor, when the total approach velocity (vector sum of axial and tangentialcomponents, relative to the rotor blades) is subsonic.The present Note extends the calculation to the practically important case where the total approach velocity is supersonic and the axial component is subsonic. (In this Note, the term “supersonic” refers to this limited range.)The background,motivation,physicalmodel, method of analysis, and nomenclature described in Ref. 1 apply to thisNote without change and are not repeatedhere. Familiaritywith Ref. 1 is essential to the understandingof this Note. In supersonic cascade ows expansion and compression waves can propagate upstream and can modify the approach ow, which makes them considerablydifferent from subsonic cases. It has been well established that a steady, uniform, supersonic ow upstream of an in nite, linear cascade of blades can exist only for a unique incidence angle. The existence of such an incidence implies that an upstreammoving acoustic disturbance (which might be initiated by a downstream-movingdisturbancearrivingto thebladerow) cancels the initial disturbance,restoring the undisturbedupstream ow. The unique incidence angle can be determined in the knowledge of the blade geometry and the upstream Mach number (methods given in Ref. 2). The unique incidence angle is generally small, several degrees only. In the following,the exitMach number is assumed to be subsonic, which is a practically common situation. This assumption also de nes a unique steady ow when using the simplemean owmethod of Ref. 1. The choice of subsonic out ow implies the presence of shocks (and, hence, a total pressure loss) in the blade passage. The effect of total pressure losses on the re ection coef cient is demonstrably small and a reasonableestimate is suf cient. Shock lossmay be estimated as that associatedwith a normal shock at the upstream Mach numberMu . The estimationof viscous lossesmay be made on the basisof empiricalinformationvalid for similar bladegeometries. The analysis deals with a transient initiated by the arrival of an acoustic step change to the blade row, the goal being the prediction of the magnitude of the re ected wave, which is also a step change. The equations used in the present analysis are the same as those of Ref. 1, with one exception. In the subsonic case, it was assumed that the direction of the exit ow, after completion of the transient, is the same as the direction of the undisturbed exit ow [Eq. (25)
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