Scaling Laws and Experimental Validations
6.1 Scaling Laws and Asymptotic Trends
Scaling laws in aeroacoustics are often used as empirical prediction means. They can be obtained either from experimental data bases or from theoretical arguments. In the first case, the point is that the laws may fail when applied to configurations not covered by the original data base. The theoretical background is more reliable. Scaling laws are discussed in this section for turbulence-interaction noise and trailing-edge noise, from the aforementioned analytical models. They are based on the very important assumption that the flows are self-similar. This is very often so in practice because unsteady flows exhibit higher frequencies and levels with increasing mean-flow speeds. Typically the frequency is proportional to the mean flow speed U0 and can be made dimensionless by introducing a Strouhal number St = fl/Uo based on a relevant length scale l. Depending on the configuration, the fluctuating velocity amplitudes can be also proportional to the mean flow speed, or not (especially for three-dimensional flows, highly Reynolds-number dependent flows…). Therefore the PSD of the acoustic pressure is divided by U0 where the exponent n is to be determined. Aforementioned analytical models state that the acoustic intensity is proportional to the mean square value of the forcing disturbance, w1rms or P2ms, to its span-wise correlation length £y and to the span-wise extent of the edge L. The derivations assume that the ratio £y/L is small. Furthermore the incident disturbances must have the properties of homogeneous and stationary random processes. Examples will be given in the next section.
The asymptotic high-frequency trend for turbulence-impingement noise is derived by only retaining the first term I of the radiation integral, since the trailing-edge correction gets smaller and smaller. For kc = 2 Mp Щ ^ ж and accounting for the developments of Fresnel integrals (Abramowitz & Stegun (1970)), some algebra leads to
Spp _ pi Up L Mo cos2(ee/2)
_ nR2 (1 + Mo cos ве)3
with S0 = Re (1+Mo cos 9e). The result is expressed in emission coordinates with respect to the surrounding flow for the sake of a better physical insight, and 9e is the observation angle from the streamwise direction (see chapter 2 for definitions). In the limit of small Mach numbers M2 ^ 0, a cardioid directivity pattern is found. The sound goes to zero upstream at small angles close to the airfoil plane and is maximum downstream. For the low- frequency limit, Amiet’s formalism must be replaced by Sears’ theory and the radiation integral reads I = S(k/e2)/в. Small values of kc and reasonable values of Mo make the function get close to 1, so that now the ratio becomes
The radiation is that of a compact dipole in motion, zero in the plane of the airfoil and maximum in the normal direction. Except that it is not specified in the mid-span plane, the general case illustrated by the sample results of Figs. 7 and 8 is between the two asymptotic regimes. Apart from the change in the directivity, the exponent of convective amplification is also found to go from 4 to 3 from low to high frequencies because of increased non-compactness.
A similar analysis for trailing-edge noise only makes sense for the high- frequency limit. In this case, introducing the convection Mach number Mc = Uc/c0 yields the result
Spp _ LMc sin2 (9e/2)_________ 1 – (Mp – Mc)
Фрр ty n2 R2 (1 + Mp cos 9e) [1 + (Mp – Mc) cos 9e]2
in which the last factor has a secondary importance, especially for low-Mach number applications. The opposite cardioid trend to leading-edge noise is found, no sound being radiated downstream and a maximum radiation upstream. Both results are in accordance with what is expected from the half-plane Green’s function for a distant observer and a source located close
to the edge. In the limit of high frequencies, what happens at one edge radiates as if the complementary edge was removed to infinity. Trailing-edge noise radiates preferentially upstream and turbulence-impingement noise preferentially downstream.
Even though different in essence, both vortex-shedding noise and trailing – edge noise have common features because the underlying vortex dynamics responsible for sound radiation takes place in the very vicinity of the trailing edge in terms of emitted wavelengths. As a result if the radiation is interpreted as quadrupole sound according to Lighthill’s analogy it tends to obey the asymptotic half-plane Green’s function and produce a cardioid pattern. The present statements using Schwarzschild’s technique adopt Ffowcs Williams & Hawkings’ interpretation and directly address the equivalent distributed dipoles the radiation of which is defined by the free-space Green’s function. Again it is verified that both interpretations provide the same result.