Computational acoustics
7.2 Acoustic far-field (FAST c )
As the name itself indicates, this acoustic tool is based in the Formulation 1A of Farassat; it was de- veloped in C language and is capable of predicting two important types of noise: the loading and the thickness noise. Since it was developed to predict the rotor noise generated by helicopters, where the blades rotate perpendicularly to a central axis, the code of the programme as well as the input files had to be properly studied in order to find the best way to use the programme in the concerning case of a centrifugal fan.
7.2.1 Theoretical Background
As almost all actual rotor noise prediction tools, FASTc is based on the Ffows Williams-Hawkings (FW- H) equation [22]. This equation is a time-domain formulation based on the Lighthill’s acoustic analogy and involves enclosing the sound sources with a moving surface that is mathematically represented by a function, f(x, t) = 0, where (x, t) are the observer space-time variables andf > 0 stands for the exterior whilef <0for the interior of the control surface. The FW-H acoustic analogy is essentially an inhomogeneous wave equation that can be derived manipulating the mass and moment Navier-Stokes equations. Note that these continuity equations change in the presence of the artificial surface f = 0 and the use of generalized functions is therefore required (see [57] for a more detailed explanation about generalized functions). Below, is presented the final FW–H wave equation governing the generation and propagation of sound
∂¯2
∂t2 −c2
∂¯2
∂xi2
(ρ−ρ0) =
∂¯
∂t[ρ0vnδ(f)]− ∂¯
∂xi[pijujδ(f)] +
∂¯2
∂xi∂xjTij, (7.1) whereTijis the Lighthill’s stress tensor defined as
Tij =ρuiuj+pij−c2(ρ−ρ0)δij, (7.2)
outside any surfaces and is zero within them and
• ∂¯is the generalized mixed partial derivative operator;
• ρandρ0are respectively the actual and the mean fluid densities;
• uiare the components of local fluid velocity;
• vnis the local normal velocity of the surface;
• δstants for the Dirac delta function;
• δij is the Kronecker delta;
• pij =p0δij is the compressive stress tensor andp0 is the acoustic pressure.
The first two source terms in the governing wave equation (Eq. (7.1)) are monopole (thickness) and dipole (loading) sources, respectively, based on their mathematical structure. The monopole source term models the noise generated by the displacement of fluid as the body passes; the dipole or loading source term models the noise that results from the unsteady motion of the force distribution on the body surface. Both of these sources are surface sources: i.e. they act only on the surfacef = 0as indicated by the Dirac delta functionδ(f). The third source term is a quadrupole source term that acts throughout the volume that is exterior to the data surface.
The wave equation can be integrated analytically under the assumptions of the free-space flow and the absence of obstacles between the sound sources and the receivers. The complete solution consists of surface integrals and volume integrals. However, since the contribution of the quadrupole source in
the region outside the source surface is small when the flow is subsonic, the volume integrals were neglected. Moreover, this term requires volume integration and an accurate prediction of the flow field, which involves large computational demands. The solution to Eq. (7.1) can be obtained using the free-space Green’s function:
G(x, t;y, τ) =
δ(τ−t+r/c)/4πr, τ≤t
0, τ > t
, (7.3)
wherer=|x−y|and(y, τ)are the source space-time variables.
Applying the Green function (Eq. (7.3)) to the monopole and the dipole terms in Eq. (7.1) finally leads to the Formulation 1A of Farassat, which is basically an integral representation of the solution to the FW-H:
p0(x, t) =p0T(x, t) +p0L(x, t), (7.4) In the equation above,p0T(x, t)andp0L(x, t)are the thickness and loading terms, respectively, given by:
4πp0T(x, t) = Z
f=0
"
ρ0v˙n r(1−Mr)2
#
ret
dS+ Z
f=0
"
ρ0vn r2(1−Mr)3
hrM˙r+ Mr−M2i
#
ret
dS (7.5)
and
4πp0L(x, t) = 1 c Z
f=0
" l˙r r(1−Mr)2
#
ret
dR+1 c
Z
f=0
lr
rM˙r+cMr−cM2 r2(1−Mr)3
ret
dR+
Z
f=0
"
lr−lM r2(1−Mr)2
#
ret
dR, (7.6)
where
• the subscriptretstands for the retarded time;
• Ris the spanwise integration;
• lstands for the section loading, inN/m;
• li=pijnjis the local force intensity andlr=lirˆiis the local force intensity in the radiation direction;
• M is the Mach number andMris the Mach number in the radiation direction.
7.2.2 Assumptions
The radial component of the velocity vector varies with the azimuth exhibiting a minimum after the volute tongue and a maxima somewhere in the direction of the outlet volute. However, since SolidWorksR does not have any tool to extract data except on real surfaces - where the velocity vectors are zero due to the non-slip condition - it was not possible to find out the radial component of the velocity at each point of the azimuth. Therefore, it was not possible to determine the radial component of the velocity along the azimuthal direction.
Since the fan in study rotates at velocities well below subsonic, the acoustic radiation efficiency of the thickness noise is low and does not play an important role on the noise prediction of these type of fans [28]. Thus, the radial component of the velocity is going to be neglected and it is assumed that the blades only have tangential speed.
In addition, given that SolidWorksR cannot give the distribution of the forces along the blades span, it is considered that the pressure distribution is constant in the spanwise direction and equal to the mean force applied on the blade divided by the span.