×

Insight into the two-source structure of the jet noise spectrum using a generalized shell model of turbulence. (English) Zbl 1258.76137

Summary: There is a large body of experimental evidence which shows that the jet noise spectrum is composed of two sources. Our aim here is to prove, mathematically, that the two-source paradigm can be derived using a minimum number of self-consistent approximations based on our current knowledge of jet turbulence in cold flows. The starting point of the paper is M. E. Goldstein’s [J. Fluid Mech. 488, 315–333 (2003; Zbl 1063.76630)] exact re-arrangement of the Navier-Stokes equations, which shows that turbulence enters the acoustic spectrum formula through the Reynolds stress auto-covariance tensor. We extend the shell model of turbulence using a more general symmetry approximation that amounts to assuming that the Reynolds stress auto-covariance is, firstly, axi-symmetric, and secondly is equivalent to the same tensor only after it has been averaged (point-wise) over the azimuthal separation.
As a consequence of these two assumptions, the space-ime Fourier transform of the Reynolds stress auto-covariance (which we refer to as the spectral tensor) depends on the transverse wave vector only through the square of its magnitude and, moreover, is also an axi-symmetric tensor. This defines the generalized shell model (or GSM) and we apply it to the jet noise problem. The final result shows that the acoustic spectrum can be written as the sum of two groups of terms, one of which corresponds to the peak jet noise in the weakly non-parallel flow limit.

MSC:

76Q05 Hydro- and aero-acoustics
76F10 Shear flows and turbulence

Citations:

Zbl 1063.76630
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lighthill, M. J., On sound generated aerodynamically: I. General theory, Proc. Roy. Soc. A, 222, 564-587 (1952) · Zbl 0049.25905
[2] Lush, P. A., Measurements of sub-sonic jet noise and comparison with theory, J. Fluid Mech., 46, 477-500 (1971)
[3] Ahuja, K. K., Correlation and prediction of jet noise, J. Sound Vib., 29, 155-168 (1973)
[4] C.K.W. Tam, M. Golebioski, J.M. Seiner, Two components of turbulent mixing noise from supersonic jets, AIAA Paper, 96-1716, 1996.; C.K.W. Tam, M. Golebioski, J.M. Seiner, Two components of turbulent mixing noise from supersonic jets, AIAA Paper, 96-1716, 1996.
[5] Tam, C. K.W., Influence of nozzle geometry on noise of high speed jets, AIAA J., 36, 1396-1400 (1998)
[6] Tam, C. K.W.; Zaman, K. M.B. Q., Subsonic jet noise from axi-symmetric and tabbed nozzles, AIAA J., 38, 592-599 (2000)
[7] Dahl, M. D.; Papamoschou, D., Analytical predictions and measurements of the noise radiated from supersonic coaxial jets, AIAA J., 38, 584-591 (2000)
[8] Viswanathan, K., Analysis of the two similarity components of turbulent mixing noise, AIAA J., 40, 1735-1744 (2002)
[9] Viswanathan, K., Aeroacoustics of hot jets, J. Fluid Mech., 516, 39-82 (2004) · Zbl 1060.76518
[10] Tam, C. K.W.; Viswanathan, K.; Ahuja, K.; Panda, J., The sources of jet noise: Experimental evidence, J. Fluid Mech., 615, 253-292 (2008) · Zbl 1175.76026
[11] Morris, P. J., A note on noise generation by large scale turbulent structures in subsonic and supersonic jets, Int. J. Aero-ac., 8, 301-316 (2009)
[12] ERCOFTAC, Symposium on sound source mechanisms in turbulent shear flows. University of Poiters, France, See http://ercoftac-ssmtsf.ceat.univ-poitiers.fr/; ERCOFTAC, Symposium on sound source mechanisms in turbulent shear flows. University of Poiters, France, See http://ercoftac-ssmtsf.ceat.univ-poitiers.fr/
[13] Ribner, H. S., Quadrupole correlations governing the pattern of jet noise, J. Fluid Mech., 38, 1-8 (1969) · Zbl 0179.56606
[14] Goldstein, M. E.; Rosenbaum, B., Effect of anisotropic turbulence on aerodynamic noise, J. Sound Vib., 54, 630-645 (1973)
[15] Bishop, K. A.; Ffowc-Williams, J. E.; Smith, W., On the sources of the unsuppressed high-speed jet, J. Fluid Mech., 50, 21-31 (1971)
[16] M. Harper-Bourne, Jet noise turbulence measurements, AIAA Paper, 2003-3214, 2003.; M. Harper-Bourne, Jet noise turbulence measurements, AIAA Paper, 2003-3214, 2003.
[17] S.A. Karabasov, M.Z. Afsar, T.P. Hynes, A.P. Dowling, W.A. McMullen, C.D. Pokora, G.J. Page, J.J. McGuirk, Using large eddy simulation within an acoustic analogy approach for jet noise modeling, AIAA Paper, 2008-2985, 2008.; S.A. Karabasov, M.Z. Afsar, T.P. Hynes, A.P. Dowling, W.A. McMullen, C.D. Pokora, G.J. Page, J.J. McGuirk, Using large eddy simulation within an acoustic analogy approach for jet noise modeling, AIAA Paper, 2008-2985, 2008.
[18] W.A. McMullen, C.D. Pokora, G.J. Page, J.J. McGuirk, Large eddy simulation of a high Reynolds number subsonic turbulent jet for acoustic source capture, AIAA Paper, 2974-3768, 2008.; W.A. McMullen, C.D. Pokora, G.J. Page, J.J. McGuirk, Large eddy simulation of a high Reynolds number subsonic turbulent jet for acoustic source capture, AIAA Paper, 2974-3768, 2008.
[19] C.D. Pokora, J.J. McGuirk, Spatio-temporal turbulence correlation using a high speed PIV in an axi-symmetric jet, AIAA Paper, 3028-3687, 2008.; C.D. Pokora, J.J. McGuirk, Spatio-temporal turbulence correlation using a high speed PIV in an axi-symmetric jet, AIAA Paper, 3028-3687, 2008.
[20] Goldstein, M. E., A generalized acoustic analogy, J. Fluid Mech., 488, 315-333 (2003) · Zbl 1063.76630
[21] Afsar, M. Z., Solution of the parallel shear layer Green’s function using conservation equations, Int. J. Aero-ac., 8, 585-602 (2009)
[22] Goldstein, M. E.; Leib, S. J., The aero-acoustics of slowly diverging supersonic jets, J. Fluid Mech., 600, 291-337 (2008) · Zbl 1151.76573
[23] Isserlis, L., On a formula for the product moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika, 12, 134-139 (1918)
[24] Frisch, U., Turbulence: The Legacy of A.N. Kolmogorov (1996), Cambridge University Press
[25] Batchelor, G. K., The Theory of Homogeneous Turbulence (1953), Cambridge University Press · Zbl 0053.14404
[26] Frenkiel, F. N.; Klebanoff, P. S., Higher-order correlations in a turbulent field, Phys. Fluids, 10, 507-520 (1967)
[27] Van Atta, C. W.; Chen, W. Y., Correlation measurements in grid turbulence using digital harmonic analysis, J. Fluid Mech., 34, 497-515 (1968)
[28] Van Atta, C. W.; Yeh, T. T., Some measurements of multi-point time correlations in grid turbulence, J. Fluid Mech., 41, 169-178 (1970)
[29] Monin, A. S.; Yaglom, A. M., Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. II (2007), Dover Publications · Zbl 1140.76004
[30] P.J. Morris, K.M.B.Q. Zaman, Velocity measurements in jets with application to noise source modeling, AIAA Paper, 2009-2017, 2009.; P.J. Morris, K.M.B.Q. Zaman, Velocity measurements in jets with application to noise source modeling, AIAA Paper, 2009-2017, 2009.
[31] Lesiuer, M., Turbulence in Fluids (1990), Kluwer Academic Publishers
[32] McComb, W. D., The Physics of Fluid Turbulence (2003), Oxford University Press · Zbl 0748.76005
[33] Orszag, S. A., Analytical theories of turbulence, J. Fluid Mech., 41, 363-386 (1970) · Zbl 0191.25601
[34] Proudman, I.; Reid, W. H., On the decay of a normally distributed and homogeneous turbulent velocity field, Proc. Roy. Soc. A, 247, 163-189 (1954) · Zbl 0056.19901
[35] Kraichnan, R. H., Relation of the fourth order to the second order moments in stationary isotropic turbulence, Phys. Rev., 107, 1485-1490 (1957) · Zbl 0078.17801
[36] Ogura, Y., A consequence of the zero-fourth-cummalant approximation in the decay of isotropic turbulence, J. Fluid Mech., 16, 38-40 (1963)
[37] O’ Brien, E. E.; Francis, G. C., A consequence of the zero fourth cummalant approximation, J. Fluid Mech., 13, 369-382 (1963)
[38] A.P. Mirabel, An example of the application of the Millionshchikov hypothesis to the problem of isotropic turbulence, Izv. Akad. Nauk, SSSR, Ser. Mech. Zhidk. i Gaza, 1969, 171-175 (cited in Monin and Yaglom [29]; A.P. Mirabel, An example of the application of the Millionshchikov hypothesis to the problem of isotropic turbulence, Izv. Akad. Nauk, SSSR, Ser. Mech. Zhidk. i Gaza, 1969, 171-175 (cited in Monin and Yaglom [29]
[39] Arad, I.; Dhruva, B.; Kurien, S.; L’vov, V. S.; Procaccia, I.; Sreenivasan, K. R., Extraction of the anisotropic contribution in turbulent flows, Phys. Rev. Lett., 81, 5330 (1998)
[40] Kurien, S.; Sreenivasan, K. R., Anisotropic scaling contributions to high-order structure functions in high Reynolds number turbulence, Phys. Rev. E, 62, 2206 (2000)
[41] Arad, I.; L’vov, V. S.; Procaccia, I., Correlation functions in isotropic and anisotropic turbulence: The role of the symmetry group, Phys. Rev. E, 59, 6753 (1999)
[42] Gaite, J.; Hochberg, D.; Molina-Paris, C., Large-scale features of rotating forced turbulence, Phys. Rev. E, 67, 0206304 (2003)
[43] Gaite, J., Anisotropy in homogeneous rotating turbulence, Phys. Rev. E, 68, 1063 (2005)
[44] Saguat, P.; Cambon, C., Homogeneous Turbulence Dynamics (2008), Cambridge University Press: Cambridge University Press UK
[45] Ditlevsen, P. D., Turbulence and Shell Models (2011), Cambridge University Press: Cambridge University Press UK · Zbl 1229.76001
[46] M.Z. Afsar, M.E. Goldstein, A. Fagan, The influence of ‘enthalpy flux-momentum flux’ on the acoustic spectrum of heated jets. Presented at the 49th AIAA Aerospace Sciences Meeting. 4-7th January 2011, Orlando Florida. AIAA 2011-1033.; M.Z. Afsar, M.E. Goldstein, A. Fagan, The influence of ‘enthalpy flux-momentum flux’ on the acoustic spectrum of heated jets. Presented at the 49th AIAA Aerospace Sciences Meeting. 4-7th January 2011, Orlando Florida. AIAA 2011-1033.
[47] Lilley, G. M., The sound radiated by isotropic turbulence with application to the theory of jet noise, J. Sound Vib., 190, 463-476 (1996)
[48] Morfey, C. L.; Szewczyk, V. M.; Tester, B. J., New Scaling laws for hot and cold jet mixing noise based on a geometric acoustics model, J. Sound Vib., 61, 255-292 (1978)
[49] P.J. Morris, K.M.B.Q. Zaman, Two component velocity correlations in jets and noise source modeling, AIAA Paper 2010-3781, 2010.; P.J. Morris, K.M.B.Q. Zaman, Two component velocity correlations in jets and noise source modeling, AIAA Paper 2010-3781, 2010.
[50] A. Khavaran, Private communication, 2009.; A. Khavaran, Private communication, 2009.
[51] Gilmore, R., Lie Groups, Lie Algebras and Some of their Applications (2002), Dover: Dover New York, USA
[52] Hamermesh, M., Group Theory and its Application to Physical Problems (1962), Dover: Dover New York, USA · Zbl 0151.34101
[53] Morse, P. M.; Feshbach, H., Methods of Theoretical Physics (1953), McGraw Hill: McGraw Hill USA · Zbl 0051.40603
[54] Bowman, F., Introduction to Bessel Functions (1958), Dover Publications: Dover Publications USA · Zbl 0020.11205
[55] Oboukhov, A. M., On some general characteristics of the equations of the dynamics of the atmosphere, Izv. Akad. Nauk SSSR. Fiz. Atmos. Okeana, 7, 695-704 (1971)
[56] Gledzer, E. B., System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl. SSSR., 18, 216-217 (1973) · Zbl 0292.76038
[57] Yamada, M.; Okhitani, K., Lyapunov spectrum of a model of two-dimensional turbulence, Phys. Rev. Lett., 60, 983-986 (1988)
[58] Yamada, M.; Okhitani, K., The inertial sub-range and non-positive Lyapunov exponents in fully-developed turbulence, Progr. Theoret. Phys., 79, 1265-1268 (1988)
[59] L’vov, V. S.; Podivilov, E.; Pomyalov, A.; Procaccia, I.; Vandembroueq, D., Improved shell model of turbulence, Phys. Rev. E, 58, 1811-1822 (1998)
[60] M.Z. Afsar, J.J. McGuirk, C.D. Pokora, Statistical axi-symmetry of the two-point time-delayed Reynolds stress auto-covariance tensor for jet noise modeling, in: 19th Polish Fluid Mechanics Conference, Poznan, Poland. 2010.; M.Z. Afsar, J.J. McGuirk, C.D. Pokora, Statistical axi-symmetry of the two-point time-delayed Reynolds stress auto-covariance tensor for jet noise modeling, in: 19th Polish Fluid Mechanics Conference, Poznan, Poland. 2010.
[61] Afsar, M. Z., Asymptotic properties of the overall sound pressure level of subsonic jet flows using isotropy as a paradigm, J. Fluid Mech., 664, 510-539 (2010) · Zbl 1221.76178
[62] Robertson, H. P., Theory of isotropic turbulence, Proc. Cam. Phil. Soc., 36, 209-223 (1939) · Zbl 0023.42604
[63] Chandrasekhar, S., The theory of axisymmetric turbulence, Proc. Roy. Soc. A, 242, 557-577 (1950) · Zbl 0037.40602
[64] Weyl, H., The Classical Groups: Their Invariants and Representations (1997), Princeton University Press: Princeton University Press New Jersey, USA · Zbl 1024.20501
[65] S. Kurien, K.R. Sreenivasan, Measures of anisotropy and the universal properties of turbulence, in: New Trends in Turbulence, NATO Advanced Study Institute, 2001, pp. 55-111.; S. Kurien, K.R. Sreenivasan, Measures of anisotropy and the universal properties of turbulence, in: New Trends in Turbulence, NATO Advanced Study Institute, 2001, pp. 55-111. · Zbl 1309.76098
[66] C.G. Mouron, The first fundamental theorem of invariant theory for the unimodular and orthogonal groups, MSc Thesis. Texas Tech University, USA, 1996.; C.G. Mouron, The first fundamental theorem of invariant theory for the unimodular and orthogonal groups, MSc Thesis. Texas Tech University, USA, 1996.
[67] Rajan, C. S., Unique decomposition of tensor products of irreducible representations of simple algebraic groups, Ann. Math., 160, 2, 683-704 (2004) · Zbl 1084.17005
[68] Butcher, P. N.; Cotter, D., The Elements of Non-linear Optics (1990), Cambridge University Press: Cambridge University Press UK
[69] Cartan, E., The Theory of Spinors (1981), Dover: Dover New York, USA · Zbl 0147.40101
[70] Stillwell, J., Naive Lie Theory (2008), Springer Publications: Springer Publications New York, USA · Zbl 1143.22001
[71] Bishop, R. L.; Goldberg, S. I., Tensor Analysis on Manifolds (1980), Dover publications: Dover publications USA · Zbl 0218.53021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.