THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

388 14. Machinery Noise ControlNoise is generated from gas jets through the creation of fluctuating pressures fromthe turbulence and shearing stresses, as the high-velocity gas impacts with theambient gas. In establishing the theory of aeroacoustics, the nonlinear effects ofthe momentum flux ρu i u j (i.e., the rate of transport of any momentum ρu i acrossa unit area by any velocity component u j ) cannot be neglected as they were forlinear acoustics. The momentum flux acts as a stress, since the rate of changeof momentum constitutes a force. This momentum flux ρu i u j generates soundas a distribution of time-varying stresses. The forces between the airflow andits boundary radiate sound as distributed dipoles, and the stresses (which act onfluid elements with equal and opposite dipole-type forces) radiate as distributedquadrupoles.The nature of the noise from jets cannot be accurately predicted, owing to thecomplex nature of the jet itself and the uncertainties associated with turbulence,nozzle configuration, temperature vacillations, and so on. However, first-orderestimates can be derived from empirical data obtained for the most part fromexperimentation in the aviation industry. The earliest measurements of jet noisedemonstrated that intensity and noise power varied very closely with the eighthpower of the jet exit velocity (Lighthill’s eighth power law), and it is now generallyagreed that the overall sound power P can be expressed asP = Kρ 0U 8 D 2c 5 0(14.41)where K is a constant, with a value 3–4; D is the jet diameter (in meters); U isthe jet flow velocity (m/s); ρ 0 is the density of ambient air (kg/m 3 ); and c 0 is theambient speed of sound (m/s). The factor ρ 0 U 8 D 2 c −50is often called Lighthill’sparameter. Because the kinetic power of a jet is proportional to 1 2 ρ 0U 2 · UD 2 , thefraction of the power converted into noise is the noise-generating efficiency η,η ∝ M 5 (14.42)where M = U/c 0 , the Mach number of the flow referenced to the ambient speedof sound.Aerodynamic noise can be modeled as monopoles, dipoles, and quadrupoles.A jet pulse through a nozzle or discharge from HVAC ducts can be modeled as amonopole. In fans and compressors, the turbulent flow generally encounters rotoror stator blades, grids, and baffles; this type of flow can be modeled as a dipole.Quadrupole modeling applies to noise occurring from turbulent mixing in jetswhere there is no interaction with confining surfaces.The velocity term U in Equation (14.41) is the fluctuating velocity which variesthroughout the jet stream. Consequently, U is not easily measured nor amenableto analytical treatment. But we can consider the average velocity V and assumethat the size of the energy-bearing eddies are of the same order of magnitude as thejet diameter, and the total radiated acoustic power P is proportional to the kineticenergy of the jet flow. The total radiated power is simply a fraction of the total