In this section a finite element simulation method is described, adapting the technique proposed by Lefebvre & Scavone [91, 92]. The main difference of the current simulation setup and the one presented in [91] is that here rectangular tuning slots are used instead of cylindrical bores.
The technique to emulate free field conditions of the open tonehole is also different from the one applied in [91] and [92].
Following [91], the tuning slot is represented as a two-port, characterized by its frequency dependent transmission matrix (TM, see Section 3.3.2). Withpin,vin andpout,vout denoting in-put and outin-put pressure and particle velocity, respectively, the transmission matrixTis written similar to (3.52) as
Using (7.11) the TM of a cylindrical tube with lengthLreads as (see also eq. 3.57) Tcyl(kL) = withZP0=ρ0c/(πa2)denoting the acoustic plane wave impedance of the tube, the transimssion matrix of the slotTslotcan be written as
TM of the
the tuning slot can only be examined as a part of a tube section, in order to get its TM, the slot is placed into a symmetrical tube of length2Lcyl, similar to the methodology of [91]
and [92]. The simulation arrangement is shown in Figure 7.5. To be able to emulate free field radiation conditions near the tuning slot opening, the finite element model is extended with an exterior part, which is indicated by the transparent green box in the figure. This extension is
3There is a mistake in [91] as they give the acoustic plane wave impedance, however they use the specific impedance in the deduction. Therefore the specific impedances are used herein.
7.4. SIMULATION METHOD 89
Tuning slot
Input plane (z= 0) (pin, vin)
Domain extension (PML on surface) Plane A (x= 0)
Plane B (z=Lcyl)
Ladd
x z y
Figure 7.5.Arrangement for numerical characterization of tuning slots
needed to provide a transition into the absorbing layer and for visualizing the radiated field near the slot. There are no explicit requirements on the size of this extensionLadd[30]. This exterior simulation domain is bounded by absorbing elements, provided by the Perfectly Matched Layer (PML) technique. The PML formulation was implemented using unbounded absorbing functions following [30] (see also Section 4.4.2 and eq. 4.49).
Since the arrangement is symmetric, only one quarter of the whole domain needs to be sim- symmetry ulated, which allows better resolution (more degrees of freedom) of the geometry discretization.
The separated quarters are illustrated by planeAandBin Figure 7.5. We define the two symme-try planes as follows. PlaneAis they−zplane located atx= 0and planeB is thex−yplane located atz=Lcyl. Because of the symmetry, zero particle velocity can be assumed on planeAin case of plane wave excitation at the input plane (z= 0).
The transmission matrix of the whole simulated objectTobjcan be written as
Tobj=TcylTslotTcyl, (7.14)
thus, the TM of the tuning slot is expressed as
Tslot=T−1cylTobjT−1cyl. (7.15) To be able to extract the TM of the whole objectTobjfrom the FE calculations, the simulation
is performed using two different sets of boundary conditions. boundary conditions On the input plane the pressure
is always defined as constant. A symmetric and an antisymmetric case is set up, by defining zero particle velocity (Neumann) and zero pressure (Dirichlet) condition on symmetry planeB, respectively. On the output plane, the pressure and particle velocity values can be given as
p(s)out=p(s)in v(s)out=−v(s)in,
p(a)out =−p(a)in v(a)out=vin(a). (7.16a) The upper indices ·(s) and ·(a) denote the symmetric and antisymmetric cases, respectively. It should be noted that in the simulation arrangement the cylindrical sections and the domain ex-tension are only needed to be able to attain the matrixTslot. The results are not influenced by the
Figure 7.6. Typical shunt (left plot) and series (right plot) impedancesZsandZafor a tuning slot. (a = 17.5 mm,hS= 20 mm,wS= 12 mmandt= 1.5 mm)
sizesLcylandLaddif the latter are chosen properly. To obtain the values in the TM, the following relation is used:
After equation (7.17) is solved, the parametersZ¯aandZ¯scan be calculated as Z¯a= 2(T11−1)/T21 and
Z¯s= 1/T21. (7.18)
It is worth discussing the choice of the parameterLcyl. choice of
tube length
Theoretically, for an arbitrary value of Lcyl,Tcylcan be evaluated analytically using (7.12). If the propagation is assumed to be lossless—
as this is the case in the present FE simulations—the resultingTcylis always invertible and well-conditioned, and (7.15) can be applied to calculateTslot fromTobj and Tcyl. However, from a numerical viewpoint, choosing a too high value forLcylis not recommended. This is explained by the inevitable appearance of the numerical dispersion error.
A lower limit onLcylcan be imposed by the requirement that the transfer matrix description of the object and the boundary conditions (7.16) are only valid when the wavefront at the input and output planes can be regarded as planar. Since at the tuning slot this assumption does not hold,Lcylhas to be large enough that the perturbations caused by the slot vanish. It was found by trying a few different setups that choosingLcylas 3–4 times the inner diameter of the tube is sufficient.4
Figure 7.6 displays the typical behavior typical
behavior
of the shunt and series impedancesZsandZa eval-uated from the finite element model by means of equation (7.18). In the low frequency regime (ka <1) the equivalent length approximations are suitable for both impedances, since a linear fit provides good match in this region. At low frequencies, the real part of the impedances is negli-gible compared to the imaginary part. At higher frequencies, however, the shunt impedanceZs
shows a resonant behavior with a remarkably strong peak atka≈1.65. Another peak atka≈1.9 is also observable in both impedance curves. This resonant peak has been observed at approxi-mately the same frequency with different slot sizes in the same tube. By looking at the simulated pressure field inside the tube, it is found that this second resonant peak is due to the inevitable excitation of the first transversal mode of the tube. This phenomenon is illustrated in Figure 7.7.
As it can be seen in the left panel, there is a strong transversal component in the simulated pres-sure field already atka= 1.53. At higher frequencies, as shown in the right panel forka= 1.91, longitudinal and transversal oscillations are both present in the whole tube.
4There is no information given on the value ofLcylin Refs. [91, 92] by Lefebvre & Scavone.