ARTICLE IN PRESS182P. Testud et al. / Journal of Fluids <strong>and</strong> Structures 23 (2007) 163–18910 0 f (<strong>in</strong> Hz)10 -1S p +2 /(ρ w c) (<strong>in</strong> W/Hz)10 -210 -310 -410 -510 -6σ=0.74, c=1280 m/sσ=0.41, c=660 m/sσ=0.25, c=1430 m/s10 1 10 2 10 3Fig. 20. Acoustical power spectra <strong>in</strong> developed cavitation (<strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice).10 -1 f (<strong>in</strong> Hz)10 -2p+ 2 S/(ρ c) (<strong>in</strong> W/Hz)10 -310 -410 -510 -610 -7σ=0.74, c=1420 m/sσ=0.45, c=1420 m/sσ=0.28, c=1425 m/sσ=0.17, c=1425 m/s10 1 10 2 10 3Fig. 21. Acoustical power spectra <strong>in</strong> developed cavitation (<strong>multi</strong>-<strong>hole</strong> orifice).taken <strong>in</strong>to account to choose the scal<strong>in</strong>g variables. This assumption of predom<strong>in</strong>ance of cavitation noise is globallyvalid, but seems to fail at low frequencies (<strong>in</strong> this work, below 200–300 Hz approximately). Also, it is assumed thatwhistl<strong>in</strong>g does not alter cavitation noise, generaliz<strong>in</strong>g the hypothesis that broadb<strong>and</strong> noise is not affected <strong>by</strong> whistl<strong>in</strong>g[as shown <strong>in</strong> Verge (1995) for a flue organ pipe].Follow<strong>in</strong>g Blake (1986), the amplitude of noise produced <strong>by</strong> cavitation should be made dimensionless <strong>by</strong> divid<strong>in</strong>gwith the downstream pressure, <strong>and</strong> not the pressure drop, when us<strong>in</strong>g a Rayleigh–Plesset bubble dynamic model for aspherical isolated free bubble. However, r<strong>in</strong>g vortices <strong>generated</strong> <strong>by</strong> an orifice are not isolated bubbles <strong>in</strong> free space, so
ARTICLE IN PRESSP. Testud et al. / Journal of Fluids <strong>and</strong> Structures 23 (2007) 163–189 183Fig. 22. Choice of the variables d <strong>and</strong> U d for the scal<strong>in</strong>g of the noise spectra <strong>in</strong> the developed cavitation regime for the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong>orifice.that this scal<strong>in</strong>g should not hold <strong>in</strong> the case of the present study, as it also does not for sheet cavitation on airfoils(Keller, 1994).As we lack a precise model, the scal<strong>in</strong>g data are chosen for the sake of simplicity: the basic idea, as shown <strong>in</strong> Fig. 22,relies on the fact that, <strong>in</strong> the developed cavitation regime, the bubbles are created <strong>in</strong> the mix<strong>in</strong>g high-shear region of thejet.For the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice experiments, the velocity U d <strong>and</strong> the orifice diameter d are representative of the conditions<strong>in</strong> this region, hence those quantities are used <strong>in</strong> order to scale the noise spectra <strong>in</strong> this regime. Furthermore, the scal<strong>in</strong>gpressure is def<strong>in</strong>ed as the pressure drop DP across the orifice, which is a measure of the k<strong>in</strong>etic energy density <strong>in</strong> the jet.Hence for the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice <strong>in</strong> developed cavitation, fd=U d is the nondimensional frequency <strong>and</strong> p þ2 U d =ðDP 2 dÞ isthe nondimensional magnitude.For the <strong>multi</strong>-<strong>hole</strong> orifice experiments, we assume the noise issu<strong>in</strong>g from <strong>in</strong>coherent N <strong>hole</strong>s sources.Each source represents the radiation of one <strong>hole</strong>. It radiates on a characteristic surface of S=N <strong>hole</strong>s . The strength ofeach source is assumed to be <strong>in</strong>dependent of the environment of the source. This assumption is natural, as we havepreviously supposed (see previous section) that the noise <strong>generated</strong> <strong>by</strong> the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice does not depend on thediameter of the pipe. However, it should be po<strong>in</strong>ted out that this assumption is wrong when whistl<strong>in</strong>g occurs.In this model, the total acoustic power P measured downstream is a summation of the acoustic power P each sourceemitted <strong>by</strong> each source [the key element is that sources are supposed to be <strong>in</strong>coherent between each other, see Pierce(1981)]:P ¼ N <strong>hole</strong>s P each source . (15)The acoustic power of each source P each source is, <strong>by</strong> def<strong>in</strong>ition, the total acoustic <strong>in</strong>tensity flux I <strong>multi</strong>plied the surface ofthis source:P each source ¼ IS=N <strong>hole</strong>s . (16)Hence the total acoustic power isP ¼ SI. (17)The total acoustic power is consequently <strong>in</strong>dependent of the number of <strong>hole</strong>s. As previously, we ignore any downstreamreflections, so that I ¼ p þ2 =ðrcÞ.This argumentation based on energy considerations can also be conducted <strong>in</strong> terms of forces: if the source isrepresented as a force act<strong>in</strong>g on the orifice, tak<strong>in</strong>g the form Sp þ , the total force imposed on the <strong>multi</strong>-<strong>hole</strong> orifice is dueto the contribution of the forces imposed <strong>by</strong> N <strong>hole</strong>s equivalent <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> <strong>orifices</strong> with open surface S=N <strong>hole</strong>s . Thoseforces are supposed to be uncorrelated with each other. Thus, the total force squared equals N <strong>hole</strong>s times the forcesquared due to one <strong>hole</strong>. As the force squared of one <strong>hole</strong> is ðp þ S=N <strong>hole</strong>s Þ 2 , the total force squared is expressed as:N <strong>hole</strong>s ðp þ S=N <strong>hole</strong>s Þ 2 , or <strong>by</strong> simplify<strong>in</strong>g: ðp þ SÞ 2 =N <strong>hole</strong>s .The scal<strong>in</strong>g of the acoustic power is based on each source of surface S=N <strong>hole</strong>s . The scal<strong>in</strong>g velocity is the velocity atthe orifice, which is taken equal to the velocity of the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice U d , as the open surface of the two <strong>orifices</strong> arevery similar, <strong>and</strong> the scal<strong>in</strong>g length is the diameter of one <strong>hole</strong> d <strong>multi</strong> . In conclusion for the <strong>multi</strong>-<strong>hole</strong> orifice <strong>in</strong>developed cavitation: fd <strong>multi</strong> =U d is the nondimensional frequency, <strong>and</strong> p þ2 U d =ðDP 2 d <strong>multi</strong> Þ is the nondimensionalmagnitude.4.3.2. Nondimensional noise spectra <strong>generated</strong> downstreamThe dimensionless acoustical power spectra obta<strong>in</strong>ed downstream of the orifice for the developed cavitation regimeare given <strong>in</strong> Fig. 23 for the <strong>s<strong>in</strong>gle</strong>-<strong>hole</strong> orifice <strong>and</strong> <strong>in</strong> Fig. 24 for the <strong>multi</strong>-<strong>hole</strong> orifice. The follow<strong>in</strong>g observations areworth not<strong>in</strong>g.
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