Multi-valued solutions of steady-state supersonic flow. Part 1. Linear analysis

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

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“…Numerous studies have been performed on the stability of oblique shocks, 7−10 as well as their oscillations characteristics. 8,12 …”

Section: Application To Oblique Shock Wavesmentioning

confidence: 99%

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

Lewis

Smith

2005

Journal of Propulsion and Power

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“…This process was considered by Carrier 4 and Van Dyke 5 with particular regard to the problem of supersonic flow past a wedge performing small amplitude oscillations, the shock remaining attached to the wedge tip. These problems raise questions regarding the stability of the shock, and this aspect has been considered in the twodimensional context ͑associated with wedge flows, the shock remaining attached to the wedge tip at all times͒ by Levinson, 6 Carrier, 7 Henderson and Atkinson, 8 Rusanov and Sharakshannae 9 and Salas and Morgan. 10 The overall conclusion is that if the flow downstream of the shock is subsonic ͑loosely classified as the strong shock solution͒, then the shock is unstable, in so far as disturbances grow downstream.…”

Section: Introductionmentioning

confidence: 99%

The effect of three-dimensional freestream disturbances on the supersonic flow past a wedge

1997

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The interaction between a shock wave ͑attached to a wedge͒ and small amplitude, three-dimensional disturbances of a uniform, supersonic, freestream flow are investigated. The paper extends the two-dimensional study of Duck et al. ͓P W. Duck, D. G. Lasseigne, and M. Y. Hussaini, ''On the interaction between the shock wave attached to a wedge and freestream disturbances,'' Theor. Comput. Fluid Dyn. 7, 119 ͑1995͒ ͑also ICASE Report No. 93-61͔͒ through the use of vector potentials, which render the problem tractable by the same techniques as in the two-dimensional case, in particular by expansion of the solution by means of a Fourier-Bessel series, in appropriately chosen coordinates. Results are presented for specific classes of freestream disturbances, and the study shows conclusively that the shock is stable to all classes of disturbances ͑i.e., time periodic perturbations to the shock do not grow downstream͒, provided the flow downstream of the shock is supersonic ͑loosely corresponding to the weak shock solution͒. This is shown from our numerical results and also by asymptotic analysis of the Fourier-Bessel series, valid far downstream of the shock.

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“…At the same time, non-uniqueness of some two-dimensional steady Riemann problems is a well-known feature (e.g. Henderson & Atkinson 1976), the most relevant example being the parameter domain where both regular and Mach reflections are possible.…”

mentioning

confidence: 99%

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

2020

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Multi-valued solutions of steady-state supersonic flow. Part 1. Linear analysis

Cited by 9 publications

References 13 publications

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

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

The effect of three-dimensional freestream disturbances on the supersonic flow past a wedge

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

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