Advances in Aeroacoustics

Advances in Aeroacoustics Outline ƒ What is Aeroacoustics ƒ Direct approach for analyzing aeroacoustic phenomenon ƒ Integral methods Free-space Green’s function Monopole, dipole, quadrupole Lighthill’s analogy ƒ Extensions of the analogy Ffowcs-Williams and Hawkings Eq./Curle’s eq. Kirchhoff method Howe’s analogy ƒ Acoustically compact sources 1 What is aeroacoustics? Sound produced by or in the presence of a fluid flow Free-space problems -- turbulence Free-space problems with solid surfaces -- wings etc. Bounded problems -- piping systems Direct approach Governing equations of fluid motion : Continuity NavierStokes Energy Definitions: Sounds of interest : 10-130 dB 6.3X10-5 - 63 Pa Atmospheric pressure at sea level 1X105 Pa 2 Direct approach: LEE Linearized Euler equations, mean flow denoted with 0 subscript Neglected H20, air viscous effects : µ; 10-3, 10-5 thermal effects : κ; 10-6, 10-5 For inviscid, nonheat-conducting, uniform mean flow: sound generated due to initial or boundary conditions. Acoustic/vortical splitting: Helmholtz Decomposition Split unsteady velocity field into solenoidal (vortical) & irrotational (acoustic) parts Constant mean flow equations: =0 Vorticity is purely convected Couples to the acoustic velocity only at solid boundaries Unsteady pressure IS the acoustic pressure. No pressure associated with the vorticity. 3 Acoustic/vortical splitting: Comments • Method has been used to compute interaction sound see notes for references • Popular applications: airfoil/gust, cascade/gust Vortical, acoustic, and entropic waves are decoupled in this approach NOT true when shocks occur or in swirling flow settings Acoustic pressure associated with the irrotational portion of the flow which is driven by a coupling to the vortical and entropic portions of the field at solid surfaces. Thus: VORTEX sound is a topic of great interest Integral Methods 4 Free-space Green’s function observer source time of travel retarded time How to use free-space Green’s function to find a solution to: P Integrate wrt time Forced wave equation Add a volume source (q) to the continuity equation and an external force (F) to the momentum equation, combine to form a wave equation… more details to come monopole dipole quadrupole Monopole For point monopole at the origin: concentric circles origin 5 Source types cont. Dipole One extra step needed before integrating wrt to time: integration by parts using For point dipole at the origin: Superposition of monopoles: _ + * θ x As compared to harmonic source: l Source types cont. Quadrupole Group the quadrupole terms into: Superposition of pair of dipoles: tij = lihj dq/dt _ + h As compared to harmonic source: Superposition of monopoles Longitudinal quadrupole (i=j) _ l _ + + h l _ Lateral quadrupole + Four leaf clover… _ l + h l 6 Far-field expansion: rules 1) 2) Integration by parts to change independent variable in differential 3) Far-field expansion 4) Space differential to time differential (for far-field expansion) Lighthill’s Equation creation of sound generation of vorticity refraction, convection, attenuation, known a priori Lighthill stress tensor mean speed of sound mean density excess momentum transfer attenuation of sound wave amplitude nonlinearity mean density variations 7 Forms of solution to Lighthill’s Eq. Quadrupole like source! Direct application of Green’s function Far-field expansion, integration with respect to retarded time What we can learn from far-field form For low Mach number, M << 1 (Crow) If the source is oscillates at a given frequency The far-field approx to the source in Lighthill’s equation can be written as Therefore the solution becomes Scalings: velocity -- U, length -- L, f of disturbance -- U/L Acoustic wavelength/source length >> 1 Acoustic field pressure fourth power of velocity Acoustic Power eighth power of velocity 8 Explicit dependence on vorticity Low Mach number,high Reynolds number flow, Lighthill’s stress tensor dominated by The double derivative of this term can be related to the vorticity Howe/Powell source term The solution to the wave equation with this source term becomes dipole like term must degenerate to quadrupole in the far field quadrupole term Dipole like term can cause problems numerically for flows in free-space Further comments… ¾ Analogy is based on the fact that one never knows the fluctuating fluid flow very accurately ¾ Get the equivalent sources that give the same effect ¾ Insensitivity of the ear as a detector of sound obviates the need for highly accurate predictions ¾ Just use good flow estimates… ¾ Alternative wave operators that include some of the refraction etc. effects that can occur due to flow nonuniformity near the source have been derived: Phillips’ eq. , Lilley’s eq. ¾ Using these is getting close to the direct calculation of sound. 9