Heating effects on the structure of noise sources of high-speed jets

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA 2009-291 47th Aerospace Sciences Meeting and Exhibit 5 8 January 2009, Reno, NV Heating effects on the structure of noise sources of high-speed jets Daniel J. Bodony Department of Aerospace Engineering University of Illinois at Urbana-Champaign Urbana, IL 61801 The noise from hot, high-speed jets is not yet fully understood. Predictive models, such as JeNo from NASA Glenn, face difficulties in constructing an effective closure strategy that correctly accounts for changes in the sound sources with heating. By utilizing two databases from a large-eddy simulation of an acoustic Mach number 1.47 jet the Lighthill sound sources are analyzed in detail with respect to changes with heating. It is shown that the individual components of the Lighthill stress tensor are tightly coupled, implying they are difficult to model separately. An alternative form of Lighthill s stress tensor which attempts to more usefully isolate the various effects is presented, but is shown to be only partially successful. A form of the acoustic analogy developed by Ffowcs Williams is also applied and it is found that the noise radiated to the 30 degree aft angle is dominated by the term [ u r / ] ρ/, which is linear in the density. Introduction It is known that the sound emitted by a turbulent jet is a function of both its Mach number and of its temperature. For low speed jets with fixed acoustic Mach number M a = U j /a < 0.7, heating increases the radiated sound while, for higher speed jets, heating decreases the sound. 1, 2 Several theoretical explanations for these observations exist but none are universally accepted. 2 4 In the context of Lighthill s acoustic analogy 5 the heating effects on the noise source are often associated with the so-called entropy term, [p p a 2 (ρ ρ )]δ i j. Lilley 6 rearranged this term into an alternative form γ 1 [ ( )] 2 ρu ku k + a 2 h h s ρu k dt δ i j } {{ } x k h } {{ } term I term II where the enthalpy fluctuations are more closely tied the source structure. In high-speed heated jets it has been shown 7, 8 that, indeed, the enthalpy source term does increase with heating. However, those investigations also suggest that the effects of heating are not solely tied to the enthalpy fluctuations but, instead, alter the spatial and temporal structure of the source. The objective of this paper is, thus, to examine the changes which occur in the noise sources due to heating, while keeping the jet velocity constant. To do this the LES database of Ref. 9 will be used and post-processed using the methods described below. As there is not yet one accepted aeroacoustic theory, initially we choose to consider the source described by Lighthill. 5 In particular the roles ofρ and T, the density and temperature fluctuations, will be highlighted as much as feasible. I. LES database of Bodony & Lele Large-eddy simulations were carried out in cylindrical coordinates for the filtered, compressible equations of motion using Reynolds-averaged variables in a 31 diameters in the axial direction and 25 diameters in the radial AIAA member.bodony@illinois.edu Copyright c 2009 by D. J. Bodony. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Copyright 2009 by Daniel J. Bodony. Published by the American American Institute Instituteof ofaeronautics Aeronauticsand andastronautics, AstronauticsInc., Paper with 2009-0291 permission. 1 of 5

direction. The dynamic Smagorinsky model 10 was used to close the subgrid scale stresses. Sixth order optimized compact finite difference schemes were used in the radial and axial directions; Fourier-spectral differencing was used in the azimuthal direction. Time integration used a low-dispersion, low-dissipation Runge-Kutta scheme. 11 Forcing and absorbing sponges 12 provide boundary conditions on the computational boundaries. For all boundaries the sponges absorb, without reflection, the outgoing vortical, entropic, and acoustic waves. At the inflow boundary the sponge also induces jet unsteadiness by forcing disturbances formed by a normal-mode solution of the linearized stability equations for a spatially-growing disturbance, on the inflow mean flow profile. Azimuthal mode number combinations, including n = ±1,..., ±4, are random walked in time to provide approximate broadband forcing without generating unphysical noise; the axisymmetric mode was not explicitly forced. The forcing amplitude, when summed over all modes, was u rms /U j = 0.03. Consequences of this type of inflow condition are discussed in Bodony ( & Lele. [ 13 { }]) The initial mean flow profile, specified at x/r 0 = 0, was of the form U/U j = 1 2 1 tanh 1 r 4θ 0 r 0 r 0 r. whereθ0, the initial momentum thickness, is a parameter. In all calculationsθ 0 /D j = 0.045. Assuming constant static pressure, fixed stagnation temperature and known jet centerline temperature the density was found from the equation of state of an ideal gas. The reference solution used in the sponge zones was found from Reynolds-averaged Navier-okes solutions of the parabolized Navier-okes equations using the v 2 - f turbulence model. A Kirchhoff surface was used to extrapolate the sound field to the far-field. II. Modified Lighthill s analogy results It was shown in Ref. 13 that interpreting the Lighthill stress tensor, T i j =ρu i u j + [(p p ) a 2 (ρ ρ )]δ i j (1) as the sum of two independent sources was not feasible for high-speed jets, regardless of being heated. For, in the far-field, the individual spectra ofρu i u j and of the so-called entropy term did not resemble that of the overall far-field pressure. As a consequence there was significant cancellation between the two, such that they could not be considered independent. Part of the reason was the increased role ofρ andρ, the time-averaged density, and the near sonic speed of the jets which allowedρ u 2 x a 2 ρ. An attempt to make more independent the terms of T i j was made based on previous work 6, 14, 15 on separatingρ from T. The result is an equivalent form of T i j given by (assuming an ideal gas) { T i j =ρ u i u j γ 1 } ( 2 u ku k δ i j (ρ ρ ) a 2 δ i j + a2 ρ T 0 ρ ) T δ i j (2) γt ρ where T 0 = T[1+{(γ 1)/2}M 2 ] is the temperature and T is the ambient fluid temperature. In this form bothρ andρt 0 are more isolated, but not completely. ThatρT 0 changes with heating is shown in Fig. 1. Note that for the hot jetρ and T 0 are almost completely out of phase. The far-field predictions using Eq. (2) are shown in Fig. 2 for observers at 30 and rees. In the figure the labels, term B, and refer to the first, second, and third bracketed expression in Eq. (2), respectively. At both angles terms B and C are of the same amplitude over the available frequency range of 0.2 1.0, with an amount of phase difference that decreases with frequency indicating that, at low frequencies, terms B and C are strongly dependent. For both angles, is of the same amplitude as the combined term B+C, but with a phase difference that depends on angle. At 30 degrees there is an increasing amount of phase cancellation with increasing frequency, again suggesting that the terms cannot be treated independently, but at rees they are more independent, with their sum being slightly less than their individual contributions. For the 30 degree observer note that has a spectral shape that is unlike the, as was found earlier by Bodony & Lele 13 for the conventional expression for T i j. As argued in Ref. 13 this is a consequence of containing the productρu i u j and the selection by the free space Green s function for the componentρu x u x to contribute essentially to this observer. The result is that for high-speed jets, where compressibility effects on turbulence is important, theρ u 2 x contribution is non-negligible and, since u x a over a large region of the jet, the temporal spectral shape of the overall sound is found by significant cancellation betweenρu i u j and the other terms. This conclusion also holds for the far-field form of analogy using the second time derivative of T rr, the component of T i j in the direction of the observer. 2 of 5

6 5 Lipline 6 5 Centerline Phase of ρt0 4 3 2 1 Hot Cold Phase of ρt0 4 3 2 1 Hot Cold 0 0 1.4 0 0 1.4 Figure 1: Phase ofρ-t 0 correlation taken at the maximum velocity fluctuation point. 180 30 deg term B term B+ Kirchhoff 0.2 0.4 0.6 0.8 1 term B Kirchhoff term B+C 100 Figure 2: Far-field sound predictions using alternative form of T i j, given in Eq. (2). 3 of 5

III. Ffowcs Williams s analogy results The apparent necessity of strong inter-term dependency for T i j due to the presence ofρu i u j suggests that if one is interested in modeling the noise sources more independently, one cannot choose sources with that kind of nonlinearity. An alternative form of T i j which avoids this issue was derived by Ffowcs Williams in the context of Mach waves emitted from shear layers 16 to highlight the amplifying role the velocity gradient tensor. The result was an exact rearrangement of T i j in the far-field, with the expression for the density fluctuations given by {ρ ρ 0 }(x, t)= 1 4πa 2 0 V {[ (a 0 u r ) ρ ρ u r (ρu r) ]} dy (3) where a subscript r denotes in the direction of the observer. It will be observed thatρu i u j does not appear in Eq. (3). Indeed only a quadratic non-linearity appears in the last term, the first two terms being linear in the density. Application of Eq. (3) is shown in Fig. 3 for the cold jet and in Fig. 4 for the hot jet. In these figures we observe that terms A and C, which are first and last terms in Eq. (3), are very nearly identical over the entire range of frequencies, with a significant amount of cancelation. In contrast, the middle term proportional to [ u r / ] ρ/, which is strictly linear in the density, is more closely related to the sound pressure spectrum for three of the four observation points; at rees for the hot jet it is not dominant. There is some amount of cancellation between this term on the sum of terms A and C but, evidently, the velocity gradient-weight density is important and a useful indicator of the far-field spectrum for these jets. 180 30 deg Kirchhoff surface + Kirchhoff surface Figure 3: Ffowcs Williams analogy applied to the cold jet. The direct connection between the near-field density and the far-field pressure is, of course, not a new idea. Panda, 15 for example, found similar correlation betweenρ measurements in the jet and far-field measurements of p. However, what is useful is the fact that a linear source was, in some cases, dominant. It is not yet clear what this observation means in detail, however we can note that, from the continuity equation one can derive ρ = ρ u u ρ { 1 = ρ a 2 [ 1 2 u u2 + 1 2 u 2 1 ρ ]} p u ρ+o(re 1 D ) which relates the linear time derivative of the density to convective non-linearities. (4) IV. Conclusions Observations of the far-field spectra for two high-speed jets at the same velocities but at different temperatures suggested that the inter-dependence of the terms within T i j was a consequence of compressibility and not due to flowsound interaction. The dependence stemmed from theρu i u j term and an alternative form of T i j, which included this 4 of 5

180 30 deg + Kirchhoff surface 100 Figure 4: Ffowcs Williams analogy applied to the hot jet. term but better isolated the density, also exhibit a strong inter-term dependence. When the traditional form of T i j was replaced by an equivalent (in the far-field) expression due to Ffowcs Williams 16 which had terms linear inρamore useful source decomposition resulted in which cross-term dependancy was reduced. Acknowledgements This work is supported by NASA grant number NNX07AC86A, Cliff Brown and James Bridges program managers. Additional support by the University of Illinois Center for Simulation of Advanced Rockets research program, supported by the US Department of Energy through the University of California under subcontract B523819, is gratefully acknowledged. References 1 Tanna, H. K., An Experimental udy of Jet Noise Part I: Turbulent Mixing Noise, J. Sound Vib., Vol. 50, No. 3, 1977, pp. 405 428. 2 Viswanathan, K., Aeroacoustics of Hot Jets, J. Fluid Mech., Vol. 516, 2004, pp. 39 82. 3 Morfey, C. L., Szewczyk, V. M., and Tester, B. J., New scaling laws for hot and cold jet mixing noise based on a geometric acoustics model, J. Sound Vib., Vol. 61, No. 2, 1978, pp. 255 292. 4 Lilley, G. M., The radiated noise from isotropic turbulence with applications to the theory of jet noise, J. Sound Vib., Vol. 190, No. 3, 1996, pp. 463 476. 5 Lighthill, M. J., On sound generated aerodynamically I. General theory, Proc. R. Soc. London A, Vol. 211, 1952, pp. 564 587. 6 Lilley, G. M., On the noise from jets, Tech. Rep. AGARD CP-131, March 1974. 7 Bodony, D. J. and Lele, S. K., Generation of Low Frequency Sound in Turbulent Jets, AIAA Paper 2005-3041, Presented at the 11th AIAA/CEAS Aeroacoustics Conference and Exhibit, Monterey, CA, 2005. 8 Bodony, D. J. and Lele, S. K., Low Frequency Sound Sources in High-Speed Turbulent Jets, J. Fluid Mech., Vol. 617, 2008, pp. 231 253. 9 Bodony, D. J. and Lele, S. K., On Using Large-Eddy Simulation for the Prediction of Noise from Cold and Heated Turbulent Jets, Phys. Fluids, Vol. 17, No. 085103, 2005. 10 Germano, M., Piomelli, U., Moin, P., and Cabot, W. H., A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A, Vol. 3, No. 7, 1991, pp. 1760 1765. 11 anescu, D. and Habashi, W. G., 2N-orage low dissipation and dispersion Runge-Kutta schemes for computational aeroacoustics, J. Comp. Phys., Vol. 143, 1998, pp. 674 681. 12 Bodony, D. J., Analysis of Sponge Zones for Computational Fluid Mechanics, J. Comp. Phys., Vol. 212, 2006, pp. 681 702. 13 Bodony, D. J. and Lele, S. K., Current status of jet noise predictions using large-eddy simulation, AIAA J., Vol. 46, No. 2, 2008. 14 Antonia, R. A. and Van Atta, C. W., On the correlation between temperature and velocity dissipation fields in a heated turbulent jet, J. Fluid Mech., Vol. 67, No. 2, 1975, pp. 273 288. 15 Panda, J., Experimental investigation of turbulent density fluctuations and noise generation from heated jets, J. Fluid Mech., Vol. 591, 2007, pp. 73 96. 16 Ffowcs Williams, J. E. and Maidanik, G., The mach wave field radiated by supersonic turbulent shear flows, J. Fluid Mech., Vol. 21, No. 4, 1965, pp. 641 657. 5 of 5