Time-Averaged Entropy and Total Pressure Ratios Across Unsteady Inlet Shock Waves

Abstract

Unsteady shock-wave motion can have a significant effect on the flow in a supersonic inlet, or in a compressor or turbine passage. One particular consequence of unsteadiness is that the time average of moving shock-wave properties can be noticeably different than the corresponding values for a steady shock. Of special interest to propulsion applications, shock unsteadiness can either increase or decrease the time-averaged total pressure ratio in the shock reference frame and the time average of frame-independent entropy jump, depending on the shocknormal Mach number. However, unsteadiness will always lead to a decrease in time-averaged total pressure ratio in the absolute frame and thus downstream of the unsteady shock. An analysis is presented, which includes time averaging of the sinusoidally perturbed, quasi-steady Rankine-Hugoniot equations, from which the time-dependent responses are calculated using both a differential and integral approach. Functions are derived that permit the direct calculation of time-averaged unsteady losses in total pressure and increase in entropy, assuming sinusoidal perturbation. Both computational and analytical studies have been performed to quantify the conditions under which the shock entropy jump will either increase or decrease as a result of periodic unsteadiness. Nomenclature M = Mach number P = pressure, Pa R = gas constant for air, 287 J/kg-K s = entropy, J/kg-K T = temperature, K β = shock-wave angle γ = ratio of specific heats = shock jump value ε = ratio of perturbed value to steady state θ = oblique surface angle µ = Mach number perturbation amplitude ρ = density, kg/m 3 φ = pressure perturbation amplitude Subscripts 0 = total conditions 1 = upstream value 2 = downstream value ∞ = freestream conditions