JOURNAL OF PROPULSION AND POWER Vol. 21, No. 3, May–June 2005 Interactions Between Shock and Acoustic Waves in a Supersonic Inlet Diffuser Jong Y. Oh, ∗ Fuhua Ma, † Shih-Yang Hsieh, ‡ and Vigor Yang § Pennsylvania State University, University Park, Pennsylvania 16802 The interactions between shock and acoustic waves in a supersonic inlet diffuser are investigated numerically. The model treats the viscous flowfield in an axisymmetric, mixed-compression inlet operating under supercritical conditions. It is solved by means of a finite-volume approach using a four-stage Runge–Kutta scheme for tempo- ral derivatives and the Harten–Yee upwind total-variation-diminishing scheme for spatial terms. Various distinct flow structures, including shock/boundary-layer and shock/shock interactions, are studied under the effects of externally imposed pressure oscillations at the diffuser exit over a wide range of forcing frequencies and ampli- tudes. As a result of the terminal shock oscillation induced by the impressed disturbances and the cyclic variation of the oblique/normal shock intersection, large vorticity fluctuations are produced in the radial direction. The characteristics of the shock/boundary-layer interactions (such as the size of the separation bubble, the terminal shock configuration, and the vorticity intensity) are also greatly influenced by the acoustic-driven shock oscillation. The overall response of the inlet aerodynamics to acoustic waves can be characterized by the mass-transfer and acoustic-admittance functions at the diffuser exit. Their magnitudes decrease with increasing frequency. A super- sonic inlet acts as an effective acoustic damper, absorbing disturbances arising downstream. Severe flow distortion, however, may arise from shock oscillation and subsequently degrade the combustor performance. Nomenclature A = cross-sectional area A d = acoustic admittance function a = speed of sound c p = constant-pressure specific heat e t = specific total energy f = fluctuation frequency i = imaginary unit M = Mach number Ms = Mach number immediately in front of terminal shock ˙ m = mass flow rate p = pressure p b = back pressure (pressure at inlet exit under steady-state calculation) p 0 = total pressure R = gas constant R c = radius of cowl lip R e = radius of inlet exit R m = mass response function, identical to ( ˙ m ′ / ¯ ˙ m)/( p ′ / ¯ p) r = radial coordinate s = entropy T = temperature t = time u = axial velocity v = radial velocity x = axial coordinate Received 29 March 2004; revision received 13 October 2004; accepted for publication 13 October 2004. Copyright c  2004 by the authors. Pub- lished by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0748-4658/05 $10.00 in correspondence with the CCC. ∗ Graduate Research Assistant, Department of Mechanical Engineering; currently Agency for Defense Development, Republic of Korea. † Postdoctoral Research Associate, Department of Mechanical Engineer- ing; mafuhua@psu.edu. ‡ Research Associate, Department of Mechanical Engineering; currently General Electric Aircraft Engines, Cincinnati, OH. § Distinguished Professor, Department of Mechanical Engineering. Fellow AIAA. x s = axial position of terminal shock β = acoustic reflection coefficient γ = specific heat ratio x s = shock-displacement amplitude ε = relative amplitude of imposed pressure fluctuation ρ = density  = dimensionless frequency, defined by Eq. (19) ω = radian frequency, 2π f Subscripts e = flow properties at inlet exit 1 = flow properties immediately upstream of shock 2 = flow properties immediately downstream of shock Superscripts ′ = fluctuating property ¯ = mean property I. Introduction A N inlet and its interaction with a combustor represent a crucial aspect in the development of ramjets and other supersonic air- breathing engines. The inlet is designed to capture and supply stable airflow at a rate demanded by the combustor and to maintain high pressure recovery and an appropriate stability margin under various engine operating conditions. The overall vehicle performance de- pends greatly on the energy level and flow quality of the incoming air. A small loss in inlet efficiency translates to a substantial penalty in engine thrust. Furthermore, any change in the inlet flow structure may modify the downstream combustion characteristics and sub- sequently lead to undesirable behavior, such as flame blowoff and flashback. Thus, matching inlet flow properties to engine require- ments is of fundamental importance to designers. 1,2 The oscillatory behavior of an inlet diffuser flow caused by longi- tudinal combustion instabilities has often plagued the development of ramjet engines. 3 As a result of unsteady combustion processes, acoustic waves are produced in the combustor and propagate up- stream to interact with the shock waves in the inlet. The resultant flow oscillations in the inlet diffuser then either propagate down- stream in the form of acoustic waves or are convected downstream with the mean flow in the form of vorticity and entropy waves and 486