Prediction of Near-Field Jet Cross Spectra

Abstract

A prediction method is developed based on the acoustic analogy for the cross-power spectral density in the convecting near field of compressible fluid turbulence. Equivalent source near-field, midfield, and far-field terms within the model integrand create corresponding near-field, midfield, and far-field radiating waves. These equivalent sources are modeled with a single equation for the two-point cross correlation of the Lighthill stress tensor that is dependent on the jet operating conditions. An alternative equivalent source model based on steady Reynoldsaveraged Navier-Stokes solutions is proposed. The cross-power spectral density model automatically reduces to a traditional autopower spectral density model when observers are at the same location. Predictions of radiation intensity and coherence compare favorably with measurements in the near field, midfield, and far field for a wide range of jet Mach numbers and temperature ratios.fully expanded jet diameter e s = number of ensembles F t = far-field integrand terms f = frequency G = cross-power spectral density g = Green's function k = turbulent kinetic energy l si = turbulent length scale M = Mach number M c = convective Mach number coefficient M d = design Mach number M j = fully expanded Mach number M t = midfield integrand terms M ∞;i = component of the vector freestream Mach number N t = near-field integrand terms P f = prefactor Pr = Prandtl number Pr t = turbulent Prandtl number p = pressure R = magnitude of x R g = gas constant R ijlm = two-point cross correlation of T ij r = source vector S = spectral density S ij = source tensor S y = equivalent spatial source distribution St = Strouhal number T ij = Lighthill stress tensor T j = fully expanded temperature t = time u = velocity vector u j = fully expanded jet velocity x = observer spatial coordinate y = source spatial coordinate y c = core length α = constant length scale coefficient β s= ratio of specific heats δ = Dirac delta function δ ij = Kronecker delta function ϵ = dissipation of turbulent kinetic energy ηξ; η; ζ = vector within source region ρ = density σ = anisotropic dependence parameter σ f = anisotropic amplification exponent τ = retarded time τ s = turbulent time scale ϕ = azimuthal observer angle Ψ = inlet observer angle ω = radial frequency

Spatiotemporal-Correlation Analysis of Jet Noise from a High-Performance Military Aircraft

“…For example, both two-point space-time pressure correlation functions [2][3][4][5] and the shape of single-point autocorrelation functions have been used to distinguish between the fine-vs large-scale nature of the jet-noise radiation [1,6]. These measurements have also been used to provide spatiotemporal length scales, either broadband [1] or band limited [7], which are useful in validating and improving jet-noise models [8][9][10]. Consequently, the correlation results presented in this paper for full-scale tactical engine noise extend the growing number of laboratory and computational jet studies that use auto-(single-point) and cross-(two-point) correlation functions of the acoustic field to obtain not only valuable information of the spatial structure of the noise field, but also insights into the noise sources found within the turbulent jet plume.…”

Theoretical investigation of alteration and radiation of large-scale structures due to jet impingement

“…Integrations of the double divergence of T ij are shown in Appendix 1. We adopt the length scale models of Miller, 39 where l x ¼ ð1:07=D j Þ 0:1028M j þ 0:0654 ð Þ y 1 D j and l r ¼ 0:33l x . We model the convection velocity as, u c % 0:7 u, where…”

“…Integrations of the double divergence of scriptTij are shown in Appendix 1. We adopt the length scale models of Miller, 39 where lx=(1.07/Dj)true(0.1028Mj+0.0654true)y1Dj and lr=0.33lx. We model the convection velocity as, trueu¯c≈0.7u¯, where and y c is the jet core length.…”

Semi-Empirical Prediction of Noise from Non-Zero Pressure Gradient Turbulent Boundary Layers

“…of turbulence, and the recent work of Naka et al [8] supports this viewpoint. In a series of philosophical papers, Ffowcs Williams [9,10] and Ffowcs Williams and Purshouse [11] Recently, Miller [18] developed the cross-spectral 102 acoustic analogy (CSAA), which is capable of predict-103 ing the near-field cross-spectra of acoustic pressure 104 from an arbitrary turbulent field in motion. When 105 both observers are placed at the same spatial location 106 the model predicts the auto-spectra of acoustic pres-107 sure.…”