Subsonic Jet Without Considering Convection
For subsonic jets, as shown schematically in Fig. 4.1 for a 2-dimensional or axisymmetric jet emanating parallel to the axis, different regions can be described.
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Subsonic Jet Without Considering Convection
4.1 Dimensional Analysis by Lighthill For subsonic jets, as shown schematically in Fig. 4.1 for a 2-dimensional or axisymmetric jet emanating parallel to the axis, different regions can be described. It is found that for moderate gas velocities and on the axis of the jet, the velocity remains constant up to x/Do = 4, in which x is the coordinate beginning from the exit plane and Do is the exit jet height or the exit diameter. The velocity on the axis of the jet decreases rapidly beyond x/Do > 4, and there are similar velocity profiles beyond x/Do > 6, whereas y/Do increases continuously. In addition, from another experiment with three different gases, Lassiter and Hubbard [56] found that larger jets generate noise in which the low-frequency component is somewhat higher than in smaller jets. Furthermore, a higher frequency emanates from a point just outside the jet, whereas the low-frequency components come from x/Do = 3 to 5. In his first treatise on aerodynamic noise, Lighthill did not take into account the variation of the frequency in the jet, which was later included by Ribner [96] and Powell [90]. The equation for the production of the turbulent jet noise is written from (3.28) in which, for the source term, the appropriate quadrupole term is replaced by (3.24). Thus, one gets B(x, θ ∗ = 0) = (ρ − ρo)2 =
1 16π 2c4o
(xi − yi )(x j − y j )(xk − yk )(xl − yl ) c4o |x − y|4
T¨i j (y,t)T¨kl (y + δ ,t + θ )dΩ (ydΩ ∗ (δ ),
(4.1)
and the intensity of sound radiation is I(x) =
c3o B(x, θ ∗ = 0) . ρo
T. Bose, Aerodynamic Noise: An Introduction for Physicists and Engineers, Springer Aerospace Technology 7, DOI 10.1007/978-1-4614-5019-1 4, © Springer Science+Business Media New York 2013
(4.2)
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4 Subsonic Jet Without Considering Convection
Fig. 4.1 Different regions of a jet. (I) Mixing region, and (II) Transition region followed by fully-developed region
For a constant radian frequency, ω , there will be a phase shift between fluctuations emanating at two different places if measurements are taken simultaneously, and because there is an additional time difference θ , the following fluctuations for Ti j and Tkl are assumed: Ti j = A1 (y) expjω t and Tkl = A2 (δ ) expjωδ δ expjω (t+θ ) .
(4.3)
Thus, it can be shown that
∂ 4 Ti jkl ∂ 2 Ti j ∂ 2 Tkl (y, δ , θ ) = , 4 ∂θ ∂ t2 ∂ t2
(4.4)
Ti jkl = Ti j (y,t)Tkl (δ ,t + θ )
(4.5)
where and θ is computed from (3.31) for θ ∗ = 0, that is, θ = δ · x/|x|. For the quadrupole moment Ti j is approximated from (3.13) by the relation Ti j ρo ui u j . With |x − y| = r, L3 ∼dΩ ∗ , where L is a characteristic dimension of the correlation volume Ω ∗ , the velocities are proportional to a characteristic velocity U, and time θ is inversely proportional to the characteristic frequency of the acoustic radiation; we can write from (4.1) and (4.2) that dB ∼
ρo2U 4 ν 4 L3 ρoU 4 ν 4 L3 c3 dΩ and dI ∼ o dB ∼ dΩ 8 2 co r ρo c5o r2
(4.6)
and the acoustic power dP = 4π r2 dI(r) ∼ r2 dI ≈
ρoU 4 ν 4 L3 dΩ . c5o
(4.7)
For th
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