University:
Boston University
Course:
ENG ME 302 | Engineering Mechanics II
Academic year:

2020
Views:

486

Pages:

20
Author:

lleferwchguld

Time cont. Spurious numerical solutions exist because ω~ is not unique based on ω. Optimization range has been selected based on the behavior of the approximated frequency. In particular, ω~Δt is well behaved for values less than 0.6, and the optimization ranges from -0.5 to 0.5. For stability, the entire computation must be restricted to the range of ωΔt from -0.6 to 0.6. Time step must be chosen to ensure numerical stability: M is the mean flow Mach number, c0 is the mean flow speed of sound, 0.4 is the value under which all of the roots of ω are damped. Numerical damping due to the small imaginary part of the approximate frequency. This gives a more stringent requirement on the numerator above. (Details in Tam) What we can learn from the far-field form For low Mach number, M << 1 (Crow) If the source is oscillating 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 FWH Eq. 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: Curle Eq. When the surface is stationary the equation reduces to Curle’s Equation Comments on analogy 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. FWH 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 Example of application of FWH/Curle We have a BEM for calculating the near field (surface forces) (Wood, Grace) 3D BVI (rotor-type problem) BEM computes unsteady pressure on the wing surface Curle’s Eq. Ex. cont. far-field expansion integration of pressure is lift interchange space and time derivatives Acoustic pressure in non-dimensional form Ex. cont. our CL vs. t Curle acoustic calc using our CL our acoustic calc using our CL Curle acoustic calc using our CL Purely analytic acoustic calc (based on analytic CL Additional info… FWH moving frame FWH moving 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 = because angle between flow direction and R = unit vector in the direction of R FWH moving 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 FWH moving 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 Turbulent noise sources Stationary turbulence (low M) Moving turbulence (high M) 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 Pressure goes as scaled fourth power of velocity Power goes as eighth power scaled by Example Method based on : Unsteady CFD -> forced wave equation solved numerically Goal : Make use of existing CFD through a hybrid method for computational aeroacoustics (i.e. no integral formulation for the acoustics) viscous acoustic Results for U = 33.1m/s, L/D = 8.0, Re=8100 Unsteady non-dimensional source term Acoustic pressure field t=0.5168 s t=0.5326 s t=0.5247 s t=0.5405 s