University:
Boston University
Course:
CAS PY 105 | Physics 1
Academic year:

2021
Views:

284

Pages:

12
Author:

ciliegelfs5

Extensions of the analogy FWH Equation Need a more general solution when: Turbulence is moving Two distinct regions of fluid flow Solid boundaries in the flow Define a surface S by f = 0 that encloses sources and boundaries (or separates regions of interest) Surface moves with velocity V Heavy side function of f : H(f) Rule: 10 FWH Equation cont. Multiply continuity and Navier Stokes equation by H Rearrange terms, add and subtract appropriate quantities Take the time derivative of the continuity and combine it with the divergence of the NS equation Differential form of the FWH Eq. Dipole type term i i Quadrupole Monopole type term FWH Equation cont. unsteady surface pressure Reynolds stress i rate of mass transfer across the surface viscous stresses i Only nonzero on the surface 11 FWH Equation cont. Integral form of the FWH Eq. Quadrupole Dipole Monopole Square brackets indicate evaluation at the retarded time If S shrinks to the body dipole = fluctuating surface forces monopole = aspiration through the surface Curle’s Equation When the surface is stationary the equation reduces to Curle’s Equation 12 Example: 3D BVI acoustics validation We have a BEM for calculating the near field (surface forces) ?? Is the acoustic calculation correct?? (contributed by Trevor Wood -- not in the notes) Example: 3D BVI acoustics validation BEM computes unsteady pressure on the wing surface Curle’s Eq. 13 Example: 3D BVI acoustics validation (cont.) far-field expansion integration of pressure is lift interchange space and time derivatives Acoustic pressure in non-dimensional form Example: 3D BVI acoustics validation results our CL vs. t Curle acoustic calc using our CL analytic result (in appendix) our acoustic calc using our CL Curle acoustic calc using our CL Purely analytic acoustic calc (based on analytic CL 14 FWH Eq. Moving Coordinate Frame Introduce new Lagrangian coordinate Inside integral, the δ function depends on τ now and Where the additional factor that appears in the denominator is = angle between flow direction and R because = unit vector in the direction of R FWH Eq. Moving Coordinate Frame (cont) Volume element may change as moves through space density at τ = τ0 Volume element affected by Jacobian of the transformation When control surface moves with the coordinate system … becomes ratio of the area elements of the surface S in the two spaces Retarded time is calculated from 15 FWH Eq. Moving Coordinate Frame (cont) When f is rigid: uj = Vj When body moves at speed of fluid: Vj = vj Square brackets indicate evaluation at the retarded time τe Doppler shift > 1 for approaching subsonic source < 1 for receding subsonic source accounts for frequency shift heard when vehicles pass Comparison of turbulent noise sources Stationary turbulence (low M) Moving turbulence (high M) 16 Stationary turbulence (low M) Far-field form From before…. Pressure goes as fourth power of velocity and power as eighth power of velocity Moving turbulence (high M) Pressure goes as scaled fourth power of velocity Power goes as eighth power scaled by 17 Comments on Kirchhoff method • Solve the homogeneous wave equation using the free-space Green’s function approach • All sources of sound and nonuniform flow regions must be inside the surface of integration • FWH is same if the surface is chosen as it is for the Kirchhoff method • FWH superior • Based on the governing equation of motion (not wave equation) • Valid in the nonlinear region Howe’s acoustic analogy Howe formulated an analogy based on the total enthalpy The wave equation that is formulated : In the far field, away from sources of sound: Good for thermal sources such as temperature fluctuations on a surface 18 Example of usefulness of explicit ω dependence Spinning vortex pair rate of travel position *a *a vorticity associated with each one velocity associated with each one source term expanded about s Sound from spinning vortex pair The governing equation The source term The general solution using the free-space Green’s function Perform the integration: 1) 2) Note that *a 19 Solution for spinning vortex pair (cont.) 3) Compute the integral using make a change of variables assume the observer distance r is much larger than the acoustic wavelength Acoustic pressure from spinning vortex pair Spinning vortex pair discussion Acoustic pressure from spinning vortex pair Dependence on distance Power dependence on velocity 2D : 7th power When one uses the Lighthill form : not explicit with ω Source term for incompressible flow becomes Oseen correction needed for computations 20 Comparison of calculation methods Familiar spiral pattern Calculated vs. analytical Calculated: • Analytical source with Oseen correction • Second order finite difference in space and time • First order characteristic type radiation boundary conditions Sound power scalings Far-field behavior …. dependence on velocity Turbulence in low Mach number uniform flow --- eighth power Turbulence in low Mach number variable density flow --- sixth power Turbulence in high Mach number flow --- third power Turbulent fluctuations in a 2D low Mach number uniform flow --- seventh power Simple source --- fourth power Simple source in 2D--- third power Dipole in low Mach number uniform flow --- sixth power Dipole in 2D --- fifth power Directivity in notes 21

Advances in Aeroacoustics Extensions of the Analogy

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