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

2021
Views:

133

Pages:

15
Author:

Cevariog7jk

Perfectly matched layer PML Computational domain Following Hu, J. of Comp. Phys. 173:455-480, 2001. • Add absorbing layer at end of computational domain where the absorption quantities are based on the plane wave solutions of the linearized Euler equations. • Create the absorption quantities such that all three wave types are absorbed in the appropriate area. • PML differs from the regular buffer zone technique in that the equations used in the added region will not cause ANY reflection when entering the region at any frequency and angle of incidence. Hu uses the nondimensional form of LEE, but follows the same process of forming the plane wave solutions (based on the dispersion relations) discussed previously. One forms but in nondimensional quantities, ρ is 1. and then find the eigenvalues of A which when set to zero give the dispersion relations. PML cont. Combined acoustic eigenvalues give y wave angle Eigenvalue for the entropy and vortical modes gives entropy vortical A single Fourier/Laplace component of U is still formed from where X_j are the eigenvectors PML cont. Absorption coefficients are then introduced using a splitting method absorption coefficients Recombining the two equations, and considering a single frequency gives Clear denominator of σ and reformulate to space-time Introduction of scaled spatial parameters, allows one to write the acoustic mode for a given Fourier/Laplace component in such a way that the damping is clear. There are similar expressions for the entropy and vortical modes PML cont. It was shown that there is an instability arising due to convective acoustic waves that have a positive group velocity but a negative phase velocity in the x-direction. Another spatial transformation is used to overcome this instability, and the final equation that one uses in the PML becomes 0 in vertical layer q is only introduced in the PML domain. 0 in horizontal layer vertical layer σx=0 PML cont. The absorption coefficients can be varied gradually, for example (given in Hu’s paper): σm Δx = 2 (Δx is the grid size) β=2 location where PML domain starts Term to allow absorption rates to be the same in the x and y directions width of PML domain Outer edge of the PML can use characteristic, asymptotic, or even very simple reflective type boundary conditions. yl D xl Solution methods using the integral approach Acoustic analogy Ffowcs-Williams and Hawkings Kirchhoff Flow field quantities are known in a region near the source, use the integral approaches to find the propagation of the acoustics to the farfield. Lighthill’s Eq. 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 Solution to Lighthill’s Eq. Quadrupole like source! Direct application of Green’s function Far-field expansion, integration with respect to retarded time Some example applications of Lighthill’s analogy ¾ Uzun et al. AIAA Paper No. 2004-0517. Applied to jet flow, coupled to an LES. Source term for acoustics propagating in a specific direction (figure) ¾Colonius, Freund, AIAA J. 38(2):368-370. Applied to jet flow, coupled to a DNS. ¾Oberai et al. (AIAA J., 40(11):2206-2216, 2002) - airfoil selfnoise, coupled to FEM LES Turbulence is moving Two distinct regions of fluid flow Solid boundaries in the flow FWH Differential form of the FWH Eq. unsteady surface pressure Reynolds stress i rate of mass transfer across the surface viscous stresses i Only nonzero on the surface FWH 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 Comments on FWH method • There is a formulation for moving surfaces (some discussion included as an appendix of this presentation) • There is a formulation for permeable (or porous) surfaces • developed for use with CFD where the surface has to be placed quite close to the body, but not on the body • Paper to appear in AIAA J (in near future) - A. Morgans et al. CFD + permeable surface FWH for transonic helicopter noise • Brentner, Farrassat, Progress in Aerospace Sciences 39:83-120, 2003 • Great review/overview of use of FWH in rotor noise studies • Gloerfelt et al., JSV 266:119-146, 2003, FWH and porous FWH for 2D cavity problem coupled to DNS •Acoustic part computed in the frequency domain (no need for retarded time variable this way.) • Kim at colleagues at FLUENT, AIAA Paper No. 2003-3202, couples FWH to their LES solver. Comment 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. Integration surface must be placed in the linear region of the flow. • 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 • Both methods when used in the linear regime, may capture a lower maximum frequency (CFD method may use grid-stretching). Kirchhoff and FWH cont. • Brentner, Farrassat AIAA J. 36(8):1379-1386, 1998 • Compare FWH and Kirchhoff • FWH does separate contributions to the noise (if surface is placed closed to or on body) • Patrick(Grace) et al.ASME FED 147:41-46, 1993, used the Kirchhoff method coupled to the splitting method for fluid/airfoil interaction noise • Gloerfelt et al. also compares the two methods (for the cavity problem) • Lyrintzis gives a great review of coupling CFD to FWH and Kirchhoff in Int. J. of Aeroacoustics 2(2):95-128, 2003 • Uzun et al., AIAA Paper No. 2004-0517, shows LES coupled to open FWH and Kirchhoff surfaces for jet (meaning jet outflow not enclosed by the surface - see figure above) • Rahier et al. Aero. Sci. & Tech. 8:453-467,2004, open vs. closed surfaces for jet analysis. Open surface makes more sense. Alternative couplings • Freund, J. of Comp. Phys. 157:796-800, 2000. Solve the linearized Euler equations (with an additional term) using near-field DNS or LES as boundary information. Additional term drives the density towards the Navier-Stokes value. Applied to jet M = 0.9, JFM 438:277-305,2001) • Grace, Curtis. ASME NCAD, 1999. Low M applications, compute solution to wave equation in appropriate region using CFD input