An other problem occuring in case of open conical pipe ends is the calculation of the radation impedance. There are no analytical formulations known to the author that could explicitly eval-uate the radiation impedance from a flaring or tapering pipe end. In the corresponding literature different approximations are applied for the characterization of the radiation and wave propaga-tion properties of conical pipe ends and juncpropaga-tions, see e.g. references [39, 40, 59, 70]. In order to overcome the limitations of analytical approximations of the radiation impedance, finite element (FE) modeling is applied in this study. These simulations are discussed in the sequel.
In the finite element simulation finite
element model
, the air column enclosed by a conical resonator section was meshed using hexahedral elements. Models of conical air columns with different opening angles αwere constructed. Free field radiation conditions were taken into account by means ofinfinite elements(see Section 4.3). To be able to apply the infinite element method, the simulation domain had to be extended by a circumscribing cylinder, onto which infinite elements were attached.
Since the model had two planes of symmetry in the cartesian coordinate system, only1/4of the whole geometry had to be built allowing better spatial resolution at the same computational cost.
The simulation arrangement is depicted in Figure 8.3.
The opening angleαof the cone models was varied from−60◦ to+60◦ in2◦steps resulting in a total of61different models. All meshes were created automatically utilizing the mesh cre-ation and manipulcre-ation tools ofNiHu(see Appendix D). The parameters of the generated finite
8.3. SIMULATION OF THE RADIATION IMPEDANCE 105
Figure 8.3. Arrangement of the finite/infinite element simulation of the radiation impedance from a conical pipe end
Parameter Symbol Value [mm]
Radius at open end rL 25.0
Maximal cone length Lmax 75.0 Minimal cone length Lmin 12.0
Wall width w 2.00 Table 8.1. Geometry and model parameters of the finite / infinite element models applied in radiation impedance simulations
element models are summarized in Table 8.1. In order to keep the number of elements and nodes at a computationally affordable quantity, the conditionsr0,min≤r0≤r0,maxandLmin≤L≤Lmax were always enforced. Infinite elements of polynomial orderP = 6were applied, with the poly-nomials defined in accordance with equations (4.26) and (4.27). The latter choice resulted in very small reflection from the inner surface of the infinite elements and a reasonable computational ef-fort at the same time. The element size was chosen to provide at least12elements per wavelength at the largest test frequency, which resulted in165 000to276 000degrees of freedom.
The excitationwas given as a point source located at the apex of the cone, except the case of excitation the cylindrical pipe (i.e.α = 0◦), when a plane wave source was assumed. The amplitude of
the source was always normalized to give unit pressure at the center point of the bottom (input) plane of the conical section. The excitation was incorporated into the simulation by imposing Dirichlet boundary conditions on the input plane of the conical section.
The simulations were run using the FE/IE model assembler tools of theNiHutoolbox and the built-in BiCGSTAB iterative solver of Matlab. The test frequencies covered the range of nondi-mensional frequencieskrLfrom0.001to3.82, divided into256equal steps. The simulation was run in six parallel threads on one node of a computational grid with Intel Xeon X5680 cores run-ning at3.33 GHzand took approximately a total of120hours to finish for all61models.
From the FE/IE simulation
postpro-cessing the pressure fieldp(x, k)ˆ was obtained for the whole simulation
domain, given by the finite element interpolation functions and the corresponding nodal weights.
The radiation impedance was calculated using the formula
radiation
with S denoting the cross section surface at the open end of the cone and vn(x, k)being the normal component of the particle velocity. The latter was obtained making use of the linearized Euler equation (3.12). The integral in equation (8.11) was calculated as the weighted sum of the nodal values with the weighting factors being proportional to the size of the corresponding elements. By means of equation (8.11) the finite element results of the61models are transformed
Figure 8.4.Comparison of the radiation impedance from an unflanged circular pipe end obtained by finite element simulations and the formulation of Levine & Schwinger [93]
into a single equivalent concentrated parameter, represented by the bivariate functionZR(k, α), which can readily be inserted into the one-dimensional acoustic model shown in Figure 8.2.
Simulation results were compared to the theoretical values calculated for a cylindrical tube by Levine & Schwinger [93], see also equation (3.65). The comparison
comparison with Levine &
Schwinger
of the resulting radia-tion impedance for the unflanged cylindrical (α = 0◦) tube is shown in Figure 8.4. The mag-nitude of the impedance |ZR(k,0)| is shown in units normalized by the acoustic plane wave impedanceZ0 =ρ0c/S. The finite element method gives a greater magnitude for the radiation impedance, with a maximal difference of≈ 20% from the analytical approximation, observed aroundkrL ≈2.0. The phase of the impedanceargZR(k,0)is very similar for the two models, only small deviations of7–8◦are observed. One reason of the deviation between the two models can be the finite thickness of the walls, which is inherently taken into account in the FEM, but disregarded by the analytical approximation.
Figure 8.5 presents the results of the finite element model for the full range of simulated opening angles (−60◦ ≤ α ≤ +60◦). The amplitude and the phase of the complex quantity ZR(sim)/ZR(L&S) are displayed, withZR(sim) andZR(L&S) denoting the simulated and analytically cal-culated values, respectively. In the frequency range above the cutoff frequency, no results are displayed. (Note: the cutoff frequency is smaller thankrL = 3.82in case of tapering sections, since the input diameter is greater in these cases.) The results shown in Figure 8.4 are the same profiles as the ones found atα= 0◦in Figure 8.5.
From the top plot of Figure 8.5 it can be seen that the difference in the magnitudes is sig-nificant in case of highly flaring or tapering cones. For example, in case of a flaring tube with α = −30◦ angle of opening, the estimation for the absolute value of the radiation impedance differs as much as30%even in the low frequency range. The phase difference shown in the bot-tom plot of Figure 8.5 has a typical range of−10◦to+10◦. Greater differences up to−25◦ are only observed in case of quickly tapering cones at high frequencies. With the phase differences being relatively small, the deviations in the magnitude of the radiation impedance can readily be interpreted as discrepancies of the corresponding length corrections. Since
expecta-tions
the length correc-tion affects the effective length of the resonator, better prediccorrec-tion of the radiacorrec-tion impedance can result in the more accurate approximation of the input admittance function and frequencies of natural resonance of the whole resonator. These expectations are tested later in Section 8.5.