Errors expected in aeroacoustics predictions

The previous section noted that the evaluation of a similarity metric requires a choice of which errors are important in aeroacoustic predictions and which are

not. The results of the far-field predictions in the validation problem of § 4.4.4.3 (particularly figure 4.20) give an example of the some of the types of errors that can arise in prediction schemes in aeroacoustics problems. Figure 4.20shows that the occurrences of the peaks and troughs in the prediction correspond well to the reference solution even though there is an≈6 dB discrepancy in magnitude for the statistical comparisons. This is an example ofamplitude error: although the features of the signal are well represented, the absolute level of the fluctuations are not captured. Arguably, this is the most important type of error since the magnitude of a signal across a wide range of frequencies is precisely what jet noise prediction for regulatory purposes requires. However, this information is already available directly from statistical measures, and no real benefit is gained from incorporating this error again in a time-domain metric. To reiterate, the perceived advantage of a time-domain metric is to better understand what part of flow physics is either captured or not captured as opposed to determining absolute levels for a final design decision. In fact, due to scaling in the SSIM procedure used here, it will be seen that the SSIM is completely insensitive to such amplitude errors. But due to the fact that this type of error is not particularly informative in the type of applications imagined for the SSIM in aeroacoustics, this is not considered a drawback but rather an advantage of the technique. Ultimately, a successful similarity metric will not simply replace simple statistical measures but complement them, leading to a more effective comparison technique.

A second group of errors, which will be the focus of the current investigation, arises from uncertainties in the source and propagation mechanics of jet noise emissions:

Random-noise error This error is apparent particularly infigure 4.20(c), where the prediction captures the behaviour of the reference signal very well except there are some small random fluctuations about the reference signal. These fluctua- tions are not particularly important in noise prediction if they only make up a very small portion of the signal energy. This is especially true if the fluctuations occur at frequencies far from the spectral peaks of the signal. An ideal metric would be relatively insensitive to this type of error as long as the noise is at reasonable levels.

Time-lag error Uncertainty in the exact propagation time from a source to an ob- server is a common issue in jet noise predictions. Uncertainty in the exact source location is the first contributor to this issue, but there are others. In a quiescent fluid, since the propagation path is direct, the propagation time can be determined directly from τ = |xo−xs|/c0. However, in the presence of a non-zero mean flow, the effect of ray refraction frequently causes the propagation

5.1. INTRODUCTION

path—even on average—to be unknown. Unsteady fluctuations in the jet also have an effect on the exact propagation time. The result of these factors is an ambiguity that is particularly important in beamforming and correlation-based source localization techniques, which assume a particular propagation time. The monotonic departure from the optimal metric value is especially important in this case, because the shape of the autocorrelation, which will be seen in § 5.3.2to affect certain types of metrics, can create local maxima and minima far from the optimum time lag. If the departure is monotonic, than this allows a more robust determination of the optimum lag by incorporating a metric into a cross-correlation-like routine.

Frequency-shift error This is most likely to arise from the Doppler effect when the source of the noise is convecting in the flow. If the observation location is stationary and the source is moving toward it, the observed frequency will be higher than the source frequency. An ideal metric would be able to distinguish if a frequency shift is present in the same way that the an optimal time lag is found in a correlation. The signal could then be systematically Doppler shifted and the shift at which the most optimal metric value is determined would estimate the frequency shift. Again, this requires a monotonic departure from the optimal metric value.

Jittery-phase (Random-cancellation) error Unsteady fluctuations in the jet can cause variations in propagation paths allowing for multiple paths of arrival for the emissions from a single in-jet sound source. When the propagation paths are different lengths, the signal phases at arrival will be different, leading to self- interference of the sound source’s emissions. If the fluctuations are random and normally distributed around the mean path length, the effect will most likely be decreased strength at the observer as well as a broadening of the signal. It is difficult to prescribe how an ideal metric should respond to such an error beyond a monotonic departure from the optimal value.

Non-linear errors Non-linear interactions can lead to far-field fluctuations that are related to a near-field source but will not correlate well. Traditional tools for detecting these types of interactions include the bispectrum and the bicoherence. Again, prescribing a behaviour for an ideal metric beyond the monotonic depar- ture from the optimal value is difficult. In this work, the non-linear effects will be simulated by quadratic and cubic interactions.

Figure 5.1: Process model for current study