The scattering matrix of the diaphragm(s) when there is no flow is calculated using the two-port method similar to the experimental case investigated in Chapter 4. Considering the case shown in Figure 4.12 with zero flow velocity, the scattering behavior of both the single and tandem diaphragms are expected to be symmetric along the duct axis. The reflection and transmission
Figure 6.5:Comparison of the acoustic responses obtained from numerical solver with the tai-lored Green’s function. The vertical dashed line indicates the first cut-off frequency.
matrices,Ra,b andTa,b reduce toRandT, respectively. Eq. (4.4) then becomes;
·p+a p+b
¸
=
·R T T R
¸ ·p−a p−b
¸
. (6.31)
The matricesRandTare calculated up to the 2ndazimuthal mode, which required 10 different load cases for[pa pb]>to be invertible. Different load cases are obtained by randomly placing a test source within the source zone shown in Figure 6.7, and calculating the acoustic response at the listeners on both sides of the diaphragm(s). 15 listeners are located at each of the listener zones, and the calculation is repeated for 15 different source positions, yielding an
overdeter-Figure 6.6:Comparison of the acoustic responses obtained from numerical solver with the tai-lored Green’s function for tandem diaphragm case with the test source located in between (top) and downstream (bottom) the diaphragms. The vertical dashed line indicates the first cut-off frequency.
Figure 6.7:Source and listener zones defined for the system identification of the diaphragm(s).
mined system by a factor of 1.5. The source strength is defined to be constant over the frequency range with the value1+i1kg/ms2. The acoustic response calculations are performed using the tailored Green’s function for the single and tandem diaphragms. The resulting reflection and transmission coefficients up to the 1st azimuthal mode are plotted in Figures 6.8 and 6.9. The off-diagonal elements are observed to be zero as a result of the axisymmetry, and thereby are not shown in the plots.
Figure 6.8:The reflection/transmission coefficient for the single (left) and the tandem (right) diaphragms form=0.
Figure 6.9:The reflection/transmission coefficient for the single (left) and the tandem (right) diaphragms form=1
prediction
7.1 Noise prediction using compressible flow data
7.1.1 Characteristic Based Filtering method
To extract the acoustic field in the plane wave region from the LES calculations, a DNC method called Characteristic Based Filtering (CBF) is used. The method was proposed by Kopitz et al. [41] to provide non-reflecting boundary conditions in plane wave region for LES, and it is based on the order of magnitude difference in the correlation lengths and speed of propagation of the acoustic and the turbulent fields. The details of the method are given as follows.
In case of a left-to-right going flow with a mean velocity,U; the acoustic information is carried with the characteristic waves at a convection speed,cf =U+c0for the right-going character-istic wave, f, and cg =U−c0 for the left-going characteristic wave, g. Considering the flow field obtained from the numerical simulation, the unsteady part which is easily computed by subtracting the mean can be written as the sum of the turbulent and the acoustic fields:
p=pt+p0, (7.1)
u=ut+u0, (7.2)
where p and u correspond to the unsteady pressure and velocity fields, respectively. The subindex ‘t’ and the ‘prime’ denote the turbulent and the acoustic perturbations, respectively.
The relations between the characteristic waves, and the acoustic perturbations are given by:
u0=f −g, (7.3)
p0=ρc0(f +g), (7.4)
where ρ andc0 are density and the speed of sound, respectively. The above equations can be reformulated to obtain the characteristic waves, f andg:
f =1 2
µ p0 ρc0+u0
¶
, (7.5)
g=1 2
µ p0 ρc0−u0
¶
. (7.6)
These waves are monitored at two different sections with axial positions,x1andx2=x1+d (see
f, c0+ U
Figure 7.1:Characteristic waves traveling inside a duct with mean flow.
Figure 7.1), to have:
At this point, two auxiliary variables in terms of the unsteady pressure, p and velocity, u are introduced in the form of the characteristic wave equation:
f∗(p,u)=1
Please note that the auxiliary variables f∗(p,u)andg∗(p,u)can directly be obtained from the numerical flow field as they are defined in terms of the unsteady flow data. Eq. (7.1) and (7.2) implies that these auxiliary variables can also be written in terms of the acoustic and turbulent components of the unsteady flow data as f∗(p0,u0,pt,ut)andg∗(p0,u0,pt,ut). A time-shifted averaging of these auxiliary variables is performed over the two sections mentioned above to have〈f∗(x1|t),f∗(x2|t+τf)〉and〈g∗(x1|t),g∗(x2|t+τg)〉. Since the characteristic wave equa-tion is linear, this averaging can be applied to each of the arguments of f∗ andg∗, separately.
When the axial separation,d is sufficiently large, the turbulent components pt andut become uncorrelated between the two sections. According to the statistics theory, averaging two uncor-related signals yields a zero mean. Then, the time-shifted averaging of the auxiliary variables eliminates the turbulent contribution in the unsteady flow data, yielding approximate character-istic wave equations:
f ≈ 〈f∗(x1|t),f∗(x2|t+τf)〉, (7.11) g≈ 〈g∗(x1|t),g∗(x2|t+τg)〉. (7.12) Once the characteristic waves are computed, the acoustic pressure and velocity can be retrieved using Eq. (7.3) and (7.4). It should be noted that the CBF method is only applicable in the plane wave region since the wave phase velocity becomes frequency dependent for the higher order modes, which prevents filtering the characteristic waves using time domain data. In the present analysis, the CBF method has been implemented locating four sections at each side
of the diaphragm(s) which are separated from each other by 0.5D. The distance between the section planes has been selected to be greater than the integral time scale multiplied by the mean convection velocity for the turbulent structures to be uncorrelated. For the time-shifted averaging approach to be valid, the flow field enclosing the selected cross-sections is to be source-free. To ensure a source-free region, the distances between the diaphragm(s) and the nearest cross-sections at the upstream and downstream sections have been selected to be 3D and5D, respectively.