# A systematic approach to modelling

Analysis of aeroacoustic systems is, like that of most of complex fluid systems, largely an exercise in system reduction. We are interested in discerning the essential aspects of the fluid system with regard to the quantity (observable) that interests us (the radiated sound in the present case), our end objective being to come up with a simplified model of the flow (both kinematically and dynamically). And, of course, it is a prerequisite that this simplified model provide as accurate as possible a prediction of the radiated sound field: how best to model the flow turbulence as a sound source. The acoustic analogy can be useful as an aid, but, as we saw in the previous section, used in isolation it is not sufficient.

The information neglected in a simplified model of an aeroacoustic system can be seen as an error, and the success or failure of that model will be reflected by the degree to which the acoustic analogy considered is sensitive to that error. Note, however, that such errors can arise, or be perceived, in two quite different contexts. The errors might be due to there being incomplete flow information available to us. Or, alternatively, the ‘error’ might be something that we intentionally introduce, through the removal of flow information that we consider non-essential where the sound production problem is concerned. In the latter case, the missing information is something that we are required to consider and choose carefully. An analysis methodology is outlined in this section, concerned with such a considered

removal of non-essential information: we intentionally introduce considered and calculated ‘errors’.

The sensitivity issue has been studied in an ad hoc manner by Samanta et al. (2006) with the former idea in mind: how sensitive are acoustic analogies to unwanted errors? The authors considered a DNS of a twodimensional mixing layer, which they used in conjunction with a number of acoustic analogy formulations (Lighthill-like and Lilley-like formulations were assessed); the sound fields computed by all analogies showed good agreement with the DNS, consistent with the results of the model problem considered in the previous section. The full solution of the DNS was then artificially modified so as to introduce an error, which we here denote Js(q). This error was produced through a manipulation of the coefficients of the POD modes [11] of the full solution. The sound field was then recomputed, by means of the different acoustic analogies, using the contaminated flow data, and the error in the sound field so computed was assessed in each case.

Different kinds of source error were explored: effects analogous to low – pass filtering, and the reduction of energy in narrow frequency bands, are two examples. In many cases the resultant error in the sound field was found to be similar for all of the acoustic analogies considered. For one particular case, however, where the error corresponds to a division of the first POD mode coefficient by 2 (this amounts to a significant reduction of the low frequency fluctuation energy of the flow), the Lighthill-like formulation showed greater sensitivity than the other formulations.

The problem can be thought about as follows. Consider an acoustic analogy, written in the general form Lp = s(q). The parameter space of the source, s(q), can be expressed in terms of an orthonormal basis, to which there corresponds an inner product; such is the case, for instance, for the POD basis of Samanta et al. (2006). If we now consider the eventual impact of the introduction of a small disturbance (which simulates a modelling error) to the source, Js(q) (as per Samanta et al. (2006)), we are interested in the impact that this will have on the acoustic field, i. e. 5p. The problem comes down to the following situation: if Js(q) || VL then the sound field will be sensitive to small perturbations in the source, Js(q). Js(q) is in this case aligned with the direction of maximum sensitivity of the propagation operator L in the parameter space considered. If, on the other hand, Js(q) T VL, then changes in s(q) will have no impact on the sound field, p. [12]

This way of viewing the aeroacoustic problem means that the modelling

problem can be formulated in the following way: beginning with full flow information q, from a numerical simulation for example, we are required to find the directions (in a suitably chosen parameter space) of the flow solution that can be eliminated without adversely affecting the quality of sound prediction. We must identify the ‘errors’ dq, such that we obtain a simplified flow field, q = q — dq; the source computed from this simplified flow field, s(q), has an associated error, and this error must be such that the component of s(q) aligned with the propagation operator is unaffected.

The following analysis methodology, based on the above reasoning, is intended as a guide for the analysis of complex aeroacoustic systems, from the point of view of source mechanism identification and the design of simplified models (from both kinematic and dynamic standpoints).

Analysis methodology

1.Obtain full or partial information associated with the complete flow solution, q (whose dynamic law we know: the Navier-Stokes operator, N(q) = 0); this data could be provided by experimental measurements or from a numerical simulation;

2. Identify and extract, from q, the observable of interest: the radiated sound in our case, q^;

3. Construct an observable-based filter, FqA, which, applied to the full solution removes information not associated with sound production, and thereby provides a reduced-complexity sound-producing flow skeleton (kinematics), q_p = FqA (q);

4. Analyse qD with a view to postulating a simplified ansatz for the source, s(qD);

5. Using an acoustic analogy, compute q^ = L-1s(qD), and verify that min || q^ — q^||;

6. Determine a reduced-complexity dynamic law, N(<d) = 0, that governs the evolution of qd.

Let us consider step 3 for a moment, as the observable-based filter, FqA, can be defined with varying degrees of rigour. The following are some possible scenarios. (i) In some situations the application of FqA might be quite heuristic, e. g. no more than the simple observation of the flow—we see with relative ease that this structure interacted with that to produce this aspect of the sound field, whence we propose a model. (ii) Alternatively,

VLi and VL2 (where the subscripts 1 and 2 indicate the two analogies) have different directions in the parameter space, one will always be able to find a perturbation that causes one operator to appear less robust than the other.

it could comprise a more sophisticated flow visualisation, or perhaps a series of measurements giving quantitative access to the flow solution, from which a simplified model might be proposed, provided the essential mechanisms show themselves clearly in this data. However, in the context of high Reynolds number turbulent flows, it is frequently necessary to approach the design of FqA in a more rigorous, methodological and objective, manner. Two further avenues can be pursued in this regard: (iii) it may be possible, using a purely theoretical deduction, to identify flow (or source) information that can be safely removed (examples are provided in what follows); and, (iv) signal processing tools can be used to decompose the complex system into more easily manageable ‘building blocks’, whose relative importance for sound production can then be tested.

Early analysis in aeroacoustics (1950s-1980s) was largely undertaken in contexts (i) and (iii), due to the limited capabilities of measurement and signal-processing. With the progressive improvement of the two latter disciplines, analysis in contexts (ii) and (iv) has become more common. In what follows we will show how a complete analysis will generally involve a combination of (i)-(iv).

In the following, we provide a short historical sketch (contexts (i) and (iii) are preponderant) outlining how the complexity of the turbulent jet was observed, considered, discerned and finally modelled with respect to both its internal turbulence mechanisms and the associated sound sources.

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