Figure 7.7. Resonant behavior observed in FE simulations of tuning slots (a = 17.5 mm,hS = 20 mm, wS= 12 mmandt= 1.5 mm). Symmetric boundary conditions. Left:ka= 1.53. Right:ka= 1.91.
Case #1 #2
a) b) c) a) b)
a 10 mm 10 mm
δ 0.70 1.00
t/b 1.30 1.01
hS/wS 1.0 2.0 3.0 2.0 4.0
Lcyl 80 mm
Ladd 15 mm
LPML 5 mm
NPML 6
DOF ≈74 000 ≈83 000
(ka)max 6.37 6.32
Table 7.1. Physical and model
parame-ters of validation cases Figure 7.8.Effect of numerical dispersion in validation cases
7.5 Comparison of tonehole and tuning slot models
7.5.1 Validation
validation cases In order to validate the method and the numerical model, two test cases of references [45]
and [91] have been examined. In the original test cases cylindrical toneholes were used, which were replaced by rectangular ones, keeping the effective radiib = p
hSwS/π of the holes un-changed. In order to examine the effect of the oblongness of the slot, models with differenthS/wS ratios were created. The parameters of the validation models are listed in Table 7.1.
Parameters Lcyl, Ladd andLPML refer to the length of the cylindrical section, the additional length of the domain extension, and the thickness of the PML, respectively. NPMLdenotes the number of elements in the PML along the layer thickness. The maximal frequency(ka)maxhas been calculated from the generated mesh as a maximal frequency, where the element edge length is smaller thanλ/12withλdenoting the wavelength. The reason for using such a low tolerance (12 elements per wavelength instead of the usual 6 to 8) is to minimize the effect of numerical dispersion errors.
To be able to identify the magnitude of numerical errors, the error of the TM of the cylindrical
Figure 7.9.Open tonehole shunt length correction – validation case #1
tube section is evaluated as numerical
error e(FEM)= kT(ana)cyl −T(FEM)cyl k
kT(ana)cyl k , (7.19)
with the upper indices (ana) and (FEM) denoting matrices calculated from the analytical and FE models, respectively. The notationk·kdenotes the 2-norm.
The numerical error is displayed in Figure 7.8. As it can be seen, the numerical error scales approximately linearly with the frequency with some small oscillations. To be able to obtain the TM of the tuning slot the inverse ofTcylis needed, therefore these errors should be kept as small as possible. Using the parameters given in Table 7.1 the numerical error remained< 5%in the whole frequency range of interest (ka <1) for the two validation cases. Furthermore, the effect of numerical dispersion errors on the resultingTslotcan remarkably be reduced by usingT(FEM)cyl in (7.15). This approach was followed in the evaluation.
To be able to compare the results of different models, the length correction parameters are derived after the impedances are obtained from eq. (7.18) as
ta=Re
Z¯aδ2 jk
and ts=Re
Z¯sδ2 jk
. (7.20)
The factorδ2arises from the ratio of the tube and hole plane wave impedancesZP0/ZH0=δ2. The results for the open tonehole shunt length correctiontsare displayed in Figure 7.9. Beside the analytical results, the data points of Figure 2 of Ref. [91] are also shown in the diagram.
validation case #1
For case #1 with hS/wS = 1the data shows good match in the low and middle frequency range. As the ratiohS/wSis increased, the low frequency shunt length correction decreases. At higher frequencies (ka > 0.5) some deviations are observed between the simulation results and the analytical model. Our FE simulation provides higher maximaltsvalues than the analytical model and the other FE model. At the same time the location of the maximum position is shifted a bit downwards (ka ≈ 0.85instead ofka ≈0.90). The maximal deviation from the FE model of Lefebvre & Scavone is around 20% withhS/wS = 1. By increasinghS/wS
effect of oblongness
, the maximal value of the shunt length correction increases and the frequency of the maximum is slightly shifted upwards, as it can be seen in the figure.
The results for the series length corrections are displayed in Table 7.2.5 Data of numerical sim-ulations and the analytical model show a good match, for test case #1 withhS/wS = 1, however, when the slot is more oblong, the resulting values oftabecome larger. This effect leads to more significant deviations for test case #2 because of the larger value ofδ.
5The series length correctiontahas a negative value (see eq. 7.8), therefore the positive values reported in Table 3 of Ref. [91] have been corrected. Also the results of Dalmontet al.[45] have been rescaled with the correctδ4factor.
7.5. COMPARISON OF TONEHOLE AND TUNING SLOT MODELS 93
Case δ t/b hS/wS Model ta[mm]
Case #1 0.7 1.30
N/A Dalmontet al.[45] −0.47 N/A Lefebvre & Scavone[91] −0.50 1.0 Present thesis, FEM −0.50 2.0 Present thesis, FEM −0.75 3.0 Present thesis, FEM −0.97
Case #2 1.0 1.01
N/A Dalmontet al.[45] −2.90 N/A Lefebvre & Scavone[91] −2.80 2.0 Present thesis, FEM −4.20 4.0 Present thesis, FEM −6.60
Table 7.2.Comparison of series length correctionstain the two validation cases using different models
Figure 7.10.Comparison of results from different formulations of the series length correctiontato numerical simulations of tuning slot setups withδandhS/wSvarying over a great range. “tacorrected” is calculated using eq. (7.22).
In the following the background of the discrepancies observed between the numerical and the analytical slot models is briefly reviewed. The examination is performed by comparing the predictions of length correction coefficients of both models.
7.5.2 Evaluation of the series length correction
Figure 7.10 displays the comparison of different approximations for the series length correction ta. The diagram also displays the results obtained by Lefebvre & Scavone [92], given as
ta=bδ2[−0.35 + 0.06 tanh(2.7t/b)]. (7.21) As it is seen, the discrepancy between the numerical tuning slot simulation and tonehole models from Dalmontet al.[45] and Lefebvre & Scavone [92] are higher in case of larger values ofδandhS/wS. In these cases both tonehole models fail to foretell the series length correction provided by the FEM. This observation is in good correspondence with the results to be presented in Section 7.6, as the underestimation of the series length correction leads to the underestimation of the fundamental frequency.
Nevertheless, with a small modification of the formula of Dalmontet al.for the series length correction a good fit to the simulation results can be achieved. The proposed correction takes the oblongness of the slot into account simply as multiplying the original length correction value in equation (7.8) by the ratio ofhS/wSas
modified series length correction ta=−0.28δ4bhS
wS. (7.22)
Figure 7.11. Frequency dependent behavior of the shunt length corrections given by different models. Ex-perimental pipe #2,hS= 20 mm,wS= 12 mm.
This simple correction form provides very good agreement for the series length correctiontain the whole range of the simulated parameters, as it can be seen in Figure 7.10 indicated by the triangular markers.
7.5.3 Evaluation of the shunt length correction
From Figure 7.6 it can immediately be seen, that the equivalent length approximation ofZsonly works in the low frequency regime since the impedance curve becomes non-linear at higher fre-quencies. In the models of Dalmontet al.[45] and Lefebvre & Scavone [92] this behavior is treated by regarding the shunt length correctiontsas dependent on the frequency as it was indicated e.g.
in Figure 7.9.
Figure 7.11 displays the evaluated shunt length correction for the tuning slot of pipe #2 with hS = 20 mmand wS = 12 mmprovided by different models as a function ofka. A significant dissimilarity
discrepan-cies
is observed between the FEM and the other two models. In the very low frequency range (ka <0.1) similar values fortsare provided by all three models, however, strong depen-dence on the frequency is only given by the FE model. The dissimilarities are significant where ka >0.6.
Dalmontet al.[45] treat the inner length correctionti—which is already incorporated into the shunt length correction—as independent of frequency, and hence, the frequency dependence of ts is a result of that of the radiation impedanceZrand the finite wall thicknesst. The latter is too small in case of tuning slots to show frequency dependent behavior in this frequency range.
Lefebvre & Scavone gave a formula for the approximation of the frequency dependence oftiused for toneholes of shorter heights (see equation (32) of reference [92]). However, this formulation only means a slight deviation from the model of Dalmontet al.in this case.
Due acoustic
coupling
to the thin walls (t/b 1), the coupling between the inner and outer acoustic fields is remarkably stronger in case of tuning slots than for woodwind toneholes, which prevents the separation of inner and outer length corrections [83, 91]. It can be stated that the limitations of traditional tonehole models relying on the equivalent T-circuit low frequency length corrections are exceeded at such low tonehole heights.
From the validation results presented above, it can be assessed that the proposed numerical treatment is capable of predicting the equivalent parameters of toneholes and tuning slots with sufficient accuracy. Furthermore, it can also be assessed that the rectangular geometry can have a remarkable effect on the corresponding length correction parameters, especially in case of oblong slots. In the following sections the equivalent length correction parameters of real tuning slots are calculated and the results are compared to measurement data.
7.6. COMPARISON OF SIMULATIONS AND MEASUREMENTS 95