# Dimensional Analysis

A rapid analysis is proposed in this section to assess both parts of the loading-noise term for a compact segment of lifting surface, keeping in mind that similar principles could be applied to other source terms. The kinematics of interest is that of a blade segment rotating at constant speed U at some radius R0, so that the characteristic velocity scale is U0 = U R0. The corresponding centripetal acceleration is Гі = Uq/R0. The chord length is c and the spanwise extent is assumed of the same order of magnitude. Steadyloading noise involves the steady-state aerodynamic force which scales according to F0 ^ CL c2 p0 Uq /2 (see chapter 5 for justifications). In contrast the fluctuations around the mean value F0 arise from time variations of the relative flow velocity on the segment; the velocity disturbances of amplitude w are in most cases proportional to the mean-flow speed, leading to some disturbance rate w/U0 which is assumed constant in the present analysis. According to simple unsteady aerodynamic arguments, typically Sears’ theory outlined in chapter 5, the fluctuating force amplitude is evaluated as F ^ n p0 c2 wU0 and its time derivative as

Ж npo c2 wUg2/Rg,

where n stands for a multiple of the rotational frequency (R0/U0 is taken as the relevant time scale). Furthermore dMr/dt’ <x U2/(c0 R0). Finally, introducing the acoustic far-field intensities IS and Iu associated with steadyloading noise and unsteady-loading noise respectively leads to the dimensional evaluations

MO = Ug/cg being the characteristic Mach number and R the distance to the observer. This is made irrespective of possibly different constant scaling factors. Unsteady-loading noise intensity is found to scale with the sixth

power of a characteristic flow speed and to include the fourth power of the Doppler factor in the denominator. This is typical of dipoles in translating motion and is generally retained as the scaling law of dipole sources in aeroacoustics. In contrast steady-loading noise scales like the eighth power of flow speeds with the sixth power of the Doppler factor in the denominator, similarly to what is obtained for translating quadrupoles (see section 3.4). Comparing both contributions is achieved by forming the ratio

Is _! f 4s Y f Mo Y

Iu ^ n2 w ) Y – Mo)

to be considered with the rate of unsteadiness w/40 as parameter. It appears that the flow Mach number strongly determines which contribution dominates. At low to moderately subsonic Mach numbers and for a significantly disturbed oncoming flow, steady-loading noise can be neglected. The fluctuating forces on the moving bodies are the most efficient sound sources. This holds especially at higher frequencies. Conversely, at high Mach numbers, and more precisely at lower frequencies, steady-loading noise is the dominant mechanism, even with effective unsteadiness in the flow. But unsteady-loading noise must be considered anyway.

For constant rotating motion thus constant centripetal acceleration, the dipole part of thickness noise in eq. (27) would be found to behave like steady-loading noise. Without going further into the details, the same simple developments also suggest that quadrupole sources associated with flows around a body become efficient at transonic speeds.

Finally the dimensional analysis justifies that preferred attention is paid to unsteady loading noise in most applications. This is not only true for the broadband noise of bodies in translating motion but also for subsonic rotating blade segments. As a consequence all predictions methods developed in chapter 5 will hold for low-speed fans as well, even though the rotating motion will not be explicitly taken into account. Now once recognized as the primary source of sound, the unsteady loads on moving surfaces must be known as a first step to enable any prediction of the radiated sound field based on the acoustic analogy. This makes aeroacoustic predictions generally more challenging than classical aerodynamic predictions for engineering applications. Simulating steady-state aerodynamics is clearly affordable with modern computational tools; but it is more specifically related to the performances and mechanical efficiency of a system (fan, propeller, turbofan engine…). Acoustic predictions require the description of all kinds of unsteadiness in the flow, which is a much more demanding task in terms of resources. Moreover flow unsteadiness is often not intrinsic to the system

but rather depending on the environment or the installation of the system.

6 Concluding Remarks

The acoustic analogy has been presented in this chapter as a theoretical background for the analysis of aerodynamically generated sound. Once the equations of gas dynamics are reformulated as a linear wave equation, all flow features responsible for sound production are grouped in equivalent source terms. This mathematical statement makes the radiated sound field easily expressed from the sources, provided that the latter are previously determined accepting some simplifications. In that sense the analogy is an indirect approach. The flow must be first described assuming that it develops independently of its acoustic signature, and the sound calculation appears as a post-processing step. The equivalent sources are moving with respect to the observer and/or the surrounding medium. Therefore attention has been paid to emphasize all effects of motion in the formal solution, classically expressed by the Green’s function technique. The most crucial point for practical sound predictions is the description of the sources. Applications to lifting surfaces are presented in chapter 5.

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