The linear stability theory is used to investigate the emergence, in supersonic free shear flows such as mixing layers and fully expanded plane jets, of supersonically travelling instability waves, which do not vanish in the ambient space. It is shown that, at supersonic convective Mach numbers, the slow and fast supersonic modes in the mixing layer as well as the sinuous supersonic mode in the plane jet should lead to Mach wave radiation. Direct numerical simulations are further used to study nonlinear stages of instability development in high-speed mixing layers and jets. They have shown that the formation of oblique shock waves attached to large-scale structures is observed in free shear flows forced by modes with supersonic phase speeds. The relevance of this phenomenon to the noise generation by high-speed jets is discussed.
References
1.
GoldsteinM.E.1976. Aeroacoustics. New York: McGraw-Hill. 293 pp.
2.
MuninA.G.KuznetsovV.M., and Leont'evE.A.1981. Aerodynamic Noise Sources. Moscow: Machine-Building. 248 pp. (in Russian).
3.
LilleyG.M.1974. On the noise from jets. Noise mechanisms. AGARD CP-131, pp. 13.1–13.12.
4.
BetchovR., and CriminaleW.R.1967. Stability of Parallel Flows. New York: Academic Press. 330 pp.
5.
BrownG.L., and RoshkoA.1974. On density effects and large structure in turbulent mixing layers. J. Fluid Mech., Vol. 64, pp. 775–816.
6.
GoldsteinM.E.1984. Aeroacoustics of turbulent shear flows. Ann. Rev. Fluid Mech., Vol. 16, pp. 263–285.
7.
CrightonD.G.1981. Acoustics as a branch of fluid mechanics. J. Fluid Mech., Vol. 106, pp. 261–298.
8.
TamC.K.W., and ChenK.C.1979. A statistical model of turbulence in two-dimensional mixing layers. J. Fluid Mech., Vol. 93, pp. 303–326.
9.
TamC.K.W.1998. Jet noise: Since 1952. Theoret. Comput. Fluid Dynamics, Vol. 10, pp. 393–405.
10.
JacksonT.L., and GroschC.E.1989. Inviscid spatial stability of a compressible mixing layer. J. Fluid Mech., Vol. 208, pp. 609–637.
11.
RagabS.A., and WuJ.L.1989. Linear instabilities in two-dimensional compressible mixing layers. Phys. Fluids A, Vol. 1, pp. 957–966.
12.
KudryavtsevA.N., and SolovievA.S.1989. Shear layer stability in compressible fluid, Prikl. Mekh. i Tekhnich. Fiz., No. 6, pp. 119–127 (in Russian, English translation in J. Appl. Mech. Tech. Phys.)
13.
TamC.K.W., and MorrisP.J.1980. The radiation of sound by the instability waves of a compressible plane turbulent shear layer. J. Fluid Mech., Vol. 98, pp. 349–381.
14.
TamC.K.W., and BurtonD.E.1979. Sound generated by instability waves of supersonic flows. Part 1. Two-dimensional mixing layers. J. Fluid Mech., Vol. 138, pp. 249–271.
15.
SeinerJ.M.BhatT.R.S., and PontonM.K.1994. Mach wave emission from a high-temperature supersonic jet. AIAA Journal, Vol. 32, pp. 2345–2350.
16.
RagabS.A., and SheenS.1992. The nonlinear development of supersonic instability waves in a mixing layers. Phys. Fluids A, Vol. 4, pp. 553–566.
17.
BassetG.M., and WoodwardP.R.1995. Numerical simulation of nonlinear kink instabilities on supersonic shear layers, J. Fluid Mech., Vol. 284, pp. 323–340.
18.
MitchellB.E.LeleS.K., and MoinP.1997. Direct computation of Mach wave radiation in an axisymmetric supersonic jet. AIAA Journal, Vol. 35, pp. 1574–1580.
19.
KudryavtsevA.N., and KhotyanovskyD.V.1998. Numerical simulation of nonlinear interactions in subsonic and supersonic free shear layersProc. of IUTAM Symposium on Dynamics of Slender Vortices (Aachen, Germany, 31 August – 3 September 1997). Dordrecht: Kluwer, pp. 133–142.
20.
AvitalE.J.SandhamN.D., and LuoK.H.1998. Mach wave radiation by mixing layers. Analysis of the sound field. Theoret. Comput. Fluid Dynamics, Vol. 12, Part I, pp. 73–90; Part II, pp. 91–108.
21.
MohseniK.ColoniusT., and FreundJ.B.2002. An evaluation of linear instability waves as sources of sound in a supersonic turbulent jet. Phys. Fluids, Vol. 14, pp. 3593–3600.
22.
KourtaA., and SauvageR.2002. Computation of supersonic mixing layers. Phys. Fluids, Vol. 14, pp. 3790–3797.
23.
ShuC.-W.1999. High-order ENO and WENO schemes for computational fluid dynamics. High-Order Methods for Computational Physics. Ed. by BarthT.J. and DeconinckH.New York: Springer, pp. 439–582.
24.
JiangG.S., and ShuC.-W.1996. Efficient implementation of weighted ENO schemes. J. Comput. Phys., Vol. 126, pp. 202–228.
25.
BalsaT.F., and GoldsteinM.E.1990. On the instabilities of supersonic mixing layers: A high-Mach-number asymptotic theory. J. Fluid Mech., Vol. 216, pp. 585–611.
26.
GillA.E.1965. Instabilities of “top-hat” jets and wakes in compressible fluids. Phys. Fluids, Vol. 8, pp. 1428–1430.
27.
WatanabeD., and MaekawaH.2002. Transition of supersonic plane jet due to symmetric/antisymmetric unstable modes. Journal of Turbulence, Vol. 3, pp. 1–17.
28.
ThompsonK.W.1987. Time-dependent boundary conditions for hyperbolic systems. J. Comput. Phys., Vol. 68, pp. 1–24.
29.
HairerE.NørsettS.P., and WannerG.1993. Solving Ordinary Differential Equations. I. Nonstiff Problems (2nd edition). New York: Springer, 528 p.
