- Email: [email protected]
On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers
Journal of Sound and Vibration 332 (2013) 2808–2815
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers Philippe R. Spalart Boeing Commercial Airplanes, Seattle, WA 98124, USA
a r t i c l e in f o
abstract
Article history: Received 24 April 2012 Received in revised form 20 December 2012 Accepted 26 December 2012 Handling Editor: J. Astley Available online 30 January 2013
We seek a clear statement of the scaling which may be expected with rigour for transportation or other noise at low Mach numbers M, based on Lighthill’s and Curle’s theories of 1952 and 1955. In the presence of compact solid bodies, the leading term in the acoustic intensity is of order M6. Contrary to the belief held since that time that it is of order M8, the contribution of quadrupoles, in the presence of dipoles, is of order only M7. Retarded-time-difference effects are also of order M7. Curle’s widely used approximation based on unsteady forces neglects both effects. Its order of accuracy is thus lower than was thought, and the common estimates of the value of M below which it applies appear precarious. The M6 leading term is modified by powers up to the fourth of ð1M r Þ, where Mr is the relative Mach number between source and observer; at speeds of interest the effect is several dB. However, this is only one of the corrections of order M7, which makes its value debatable. The same applies to the difference between emission distance and reception distance. The scaling with M6 is theoretically correct to leading order, but this prediction may be so convincing, like the M8 scaling for jet noise, that some authors rush to confirm it when their measurements are in conflict with it. We survey experimental studies of landing-gear noise, and argue that the observed power of M is often well below 6. We also object to comparisons across Mach numbers at fixed frequency; they should be made at fixed Strouhal number St instead. Finally, the compact-source argument does not only require M 5 1; it requires MSt 5 1. This is more restrictive if the relevant St is well above 1, a situation which can be caused by interference with a boundary or by wake impingement, among other effects. The best length scales to define St for this purpose are discussed. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Simple theoretical–empirical laws such as the dependence of various types of noise on powers of the Mach number are very convenient, and widely used in industry both to predict transportation noise across a speed range, and to identify putative sources of noise in a complex environment. In a complex system, simply by varying Mach number, one may thus ideally separate jet noise, with M8 dependence, from dipole airframe noise with M6, and even turbulent-boundary-layer/ trailing-edge noise with M5. Overwhelmingly, the powers of M were suggested by the Acoustic Analogy of Lighthill and extensions of it, notably Curle’s [1] and then supported by experiments and now simulations; this gave them a high stature. Nevertheless, theoretical challenges to the Acoustic Analogy have been numerous over the years [2–4]; the subject remains controversial, although
E-mail address: [email protected] 0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.12.029
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2809
specific charges that the Acoustic Analogy gives erroneous results have been refuted mathematically [5,7]. The experimental confirmation is far from perfect. Our attitude is to examine and possibly correct the interpretation and uses of the Acoustic Analogy, but certainly not to suggest that it is ‘‘wrong’’ and should be discarded. There have been qualifications to Curle’s theory also [8–10]. Attractive as the M6 law for aerodynamic noise is, abuses are possible, and we believe we have witnessed simplistic applications, as well as applications outside the true range of validity of the theory. This becomes more relevant as the progress of simulations and instrumentation stokes the desire for quantitative predictions with error bars of very few dB, which is indispensable if the predictions are to have value in practice. This motivated a communication that contains little new theory except for explicit statements which appear to be new of the order of accuracy of the approximations in terms of M, but may stimulate discussion and serve as a warning against too broad an application of Curle’s approximation. The discussion distinguishes the certain predictions of dimensional analysis and far-field decay laws, in Section 2, from the stronger but tentative predictions which arise from the Acoustic Analogy and asymptotic theories, in Section 3. It is reminiscent of one by Tam for jet noise [11]. Then Section 4 presents a survey of landing-gear noise measurements and their interpretation, and Section 5 our conclusions. 2. Dimensional analysis Consider the flow past an object with characteristic dimension D, moving at speed U 1 through a uniform medium at rest. The fluid is modelled as an ideal gas, with specific heat ratio g (omitted from here on), with the freestream speed of sound c1 , density r1 , and viscosity n1 . Let M U 1 =c1 be the Mach number and Re DU 1 =n1 be the Reynolds number. Consider the unsteady pressure at an observation point defined by its spherical coordinates ðr, y, fÞ, with y ¼ 0 in the direction of flight. The relative Mach number between the model and a point at rest is thus M r ¼ M cos y. By dimensional analysis, the pressures in similar solutions of the Navier–Stokes equations with different values of D, c1 , and so on, are related by a scaling of the type: r tU 1 , y, f, ; M,Re : (1) p0 ðr, y, f,tÞ ¼ p1 F D D Here and throughout the paper the calligraphic F ð. . .Þ, sometimes with indices as in F ij or F 1 , means ‘‘a non-dimensional function of a list of non-dimensional arguments.’’ This function F depends only on the geometry of the model, normalized by D. The list of arguments separates the four spatio-temporal coordinates (here, spherical) from the two parameters M and Re with a semi-colon. This first equation is valid everywhere in the field, but others may be valid only in some regions, usually the far-field. Naturally, there are many choices besides p1 to normalize F , such as the dynamic pressure q r1 U 21 =2. However, the content of the ‘‘F assertion’’ in (1) is unchanged (since q ¼ ðgM2 =2Þp1 , and M is retained in the arguments of F ). The definitions of the y angle and distance r are also non-unique and dependent on M (emission versus reception versus open wind tunnel), but as M is in the list of parameters, it does not need to be dealt with yet. There is much interest in spectra, and we use Tam’s notation S for the ‘‘narrow-band’’ spectral density: Z 1 p02 ðr, y, fÞ ¼ Sðr, y, f,f Þ df , (2) 0
with f being the frequency, in which case (1) implies Sðr, y, f,f Þ ¼
p21 D r , y, f,St; M,Re , F U1 D
(3)
where St fD=U 1 is the Strouhal number. Dimensional analysis makes no further predictions; in particular, any statement (a few of which are found in the literature) that a behaviour such as proportionality to M6 or M8 is a matter of dimensional analysis is not sustainable. 3. Scaling predictions 3.1. Background The dependence on Reynolds number will be neglected. Either it is weak, for instance a logarithmic dependence of the skin-friction coefficient on Re, or it is strong, for instance the drag crisis of a cylinder, and then it is simply out of reach of simple scaling laws. The plausible conditions for such laws to succeed are to have both a high Reynolds number, of the order of 106, and a repeatable presence of turbulence over a range of speeds, say within a factor of 2 on either side of the initial speed. This presence is sometimes ensured by boundary-layer trips. Curle presented his scaling laws ‘‘times a function of Re.’’ For tests with a given physical model and viscosity, as is very often the case, Re is proportional to M and such a statement is empty, in the following sense. Suppose a quantity has been determined to be only a function of M and Re, that is, F ðM,ReÞ. Curle’s words mean that F ðM,ReÞ ¼ F 1 ðMÞF 2 ðReÞ, and he gives only F 1 . In a wind-tunnel test of a given model, we have Re ¼ ðDc1 =nÞM so that the M dependence can be written F 1 ðMÞF 2 ððDc1 =nÞMÞ. If the F 1 function is specified but the F 2 function is unspecified, then the M dependence is still
2810
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
unknown. The implication must be ‘‘the scaling law F 1 in terms of M, times a weak function of Re.’’ This is also our position, and the Reynolds number will be omitted from here on. We move on to the far-field behaviour of the sound, which is proportional to 1=r for p0 . This is asymptotic, for r b D, but un-controversial. As a result, the scaling law (3) (with Re also omitted) becomes Sðr, y, f,f Þ ¼
p21 D3 F ðy, f,St; MÞ: U 1 r2
(4)
From here on, all the equations apply in the limit as r=D-1. There is great theoretical and industrial interest in finding a simple dependence on the remaining variables and/or parameters, and this is where controversies begin. It appears that the chances that the dependence on St is analytical (for instance, that the spectrum would follow a power of St over most of its range) are very remote, so that M is the prime candidate, and ðy, fÞ are secondary ones. The literature contains several candidates for a simple Mach scaling beyond (4), which are usually of the form S p21
D2 D F 1 ðMÞF 2 ðy, f,StÞ, r2 U1
(5)
and in which generally the function of M is a power: S p21
D2 D M n F ðy, f,StÞ: r2 U 1
(6)
Inside the turbulence, n would be 4 (but not with the far-field 1=r dependence, of course), i.e., pressure fluctuations scale with the dynamic pressure, but for noise the exponent n if any exists is higher than 4 and non-trivial. The factor D=U 1 ¼ dSt=df in (6) leads to an integration versus St when (2) is executed. However, a scaling at fixed frequency rather than fixed Strouhal number is implied here and there in the literature, as discussed at length in Section 4, and takes the form S p21
D2 D n M F ðy, f,MStÞ, r2 c1
(7)
since f ¼ StU 1 =D ¼ MStc1 =D. The factor D=c1 ¼ dðMStÞ=df (instead of dSt=df ) here leads to the integration versus MSt; this gives n the same role in the two formulas (6) and (7) when p02 is calculated. In both cases (6) and (7), M having been removed from the list of individual arguments of F placed after the semi-colon, (2) can be converted to an integral versus either St or MSt, and the far-field acoustic mean squared pressure follows as p02 p21
D2 n M F ðy, fÞ: r2
(8)
Much literature has been devoted for determining both the value of n and the nature of the scaling of the spectrum, between (6) and (7), combinations of the two, or other ideas.
3.2. Theory We now draw on theory to propose a scaling law. The simplest one (and to our knowledge the only simple one) arises if, following Curle, we neglect both the quadrupoles and the differences in retarded times in the Ffowcs-Williams–Hawkings (FWH) calculation (performed with the solid surface and all the fluid quadrupoles). Both assumptions flow from the low Mach number. Within this approximation, the sound is attributed to the time-derivatives df =dt of the forces on the model, where t is the emission time [1]. We further assume that in units of r1 , D, and U 1 , these forces do not depend on M. This amounts to q and St scaling for the energy-containing portion of the turbulence, as M-0, which is un-controversial. The density fluctuations in the turbulence are of order r1 M 2 . The time-derivative of the force fi in the i direction has the form df i =dt ¼ qU 1 D½dC i =dðtU 1 =DÞ, where Ci is the time-dependent force coefficient and the quantity in square brackets is independent of M, to order M2. Using the far-field approximation of the FWH equation [12], stripped of the quadrupole term and of the differences in retarded times, the acoustic pressure perturbation is 9x9 1 xi df i t þ p1 OðM 4 Þ: (9) p0 ðx,tÞ ¼ c 4p91Mr 99x9c1 9x9 dt In this formula, 9x9 is the emission distance, and the observer is fixed in the atmosphere, thus giving a Doppler shift relative to the internal frequency of the source. From here on we do not use the symbol as in Section 3.1, but ¼ signs with specific remainders of the type OðM m Þ. The fact that the approximation in (9) is valid to order M4 (and for instance (11) below to order M7) is of some importance, and is covered in Section 3.3.
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
The emission and reception times are related by dt ¼ ð1M r ðtÞÞ dt so that 9x9 1 xi df i þp1 OðM4 Þ: t p0 ðx,tÞ ¼ c 4pð1M r Þ2 9x9c1 9x9 dt
2811
(10)
For an observer moving at the speed of the model (e.g., a fixed microphone in a wind tunnel), the pressure has the same amplitude but without a Doppler shift. Squaring (10) and using the definition of force coefficients Ci yields for the spectrum " # xi xj g2 D2 D M6 2 7 St C ðStÞ þOðM Þ , (11) Sðr, y, f,f Þ ¼ p21 ij 2 16 r 2 U 1 ð1M r Þ4 9x9 R where Cij is the cross-spectral density between force coefficients in the i and j directions (i.e., C i C j ¼ C ij ðStÞ dSt) Note that xi xj 2 depends only on (y,f). 9x9 The scaling produced by the theory is indeed of the type postulated in (5). At first sight, it is not quite of the type proposed in (6), but rather of the type " # 2 D M6 7 2 D Sðr, y, f,f Þ ¼ p1 2 F ðy, f,StÞ þ OðM Þ : (12) r U 1 ð1M r Þ4 The first term in M6 =ð1M r Þ4 is well-known, and for instance implicit in the ANOPP empirical noise-prediction code [13]. The meaning of the ð1M r Þ4 factor is discussed below. In (11) the angular dependence is also analytical, and thus much more specific than that in (5): " # 2 xi xj D M6 7 2 D F ij ðStÞ 2 þ OðM Þ : (13) Sðr, y, f,f Þ ¼ p1 2 r U 1 ð1M r Þ4 9x9 The tensor F ij is symmetric and positive-definite, proportional to the tensor Cij of cross-correlations of the force time-derivatives, and therefore due to (11) S is always positive. This angular dependence is actually not compatible with the empirical ones assumed in ANOPP, which make F ðy, f,StÞ in (12) become F 1 ðy, fÞF 2 ðStÞ. In any case, the theoretical predictions are as specific regarding directivity, which was described as a ‘‘secondary candidate’’ in Section 3.1, as they are regarding Mach number. However, the M trends are far easier to test (one parameter M versus six in the F ij tensor) and have received most of the attention. The total mean-squared pressure then obeys " # xi xj D2 M6 7 p02 ðr, y, fÞ ¼ p21 2 F ðÞ þ OðM Þ (14) r ð1M r Þ4 ij 9x92 (where F ij ðÞ is the integral versus St of F ij from (13) and has no arguments left), rather than (8). The directional dependence is analytical, as inherited from (13). Again, r and y are emission coordinates. They are related to reception qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coordinates by r r ¼ r e 1 þ M 2 2M r and r e sin ye ¼ r r sin yr , with M r ¼ M cos ye . This deduction from theory suggests that (6) is too simple to be adequate at values of M such as 0.2, since the factors of ð1M r Þ in (14) make a difference of 4 dB in the forward direction, y ¼ 0. This is shown in Fig. 1, in which each vertical division represents 10 dB, and the Mach-number range covers up to high-speed trains. Thus, the correction is far from negligible for industrial practice in air.
Fig. 1. Acoustic intensity factors. —, M6; - - -, M 6 =ð1 þ MÞ4 ; - — -, M6 =ð1MÞ4 .
2812
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
Perspective on this correction is however gained by pointing out that it is of order M7, and the quadrupole and retardedtime effects lumped in the OðM7 Þ term (Section 3.3) have no deep reason to be weaker than the ð1M r Þ correction. In other words, (13) and " # xi xj D2 D M 6 F ij ðStÞ 2 þ OðM7 Þ (15) Sðr, y, f,f Þ ¼ p21 2 r U1 9x9 are mathematically equivalent (simply because M r r M and M 6 =ð1MÞ4 ¼ M6 þ OðM7 Þ as M-0), and there is no rigorous difference between (6) and the first term of (12) after all. The figure makes the point that the correction by ð1M r Þ4 is not negligible, but quantitatively, it must be viewed with caution. It is merely plausible that it alters the results in the right direction, relative to the pure M6 formula. In the same vein, this shows that, for a smooth spectrum, the approximation is too low-order to predict the Doppler frequency shift, because that is also an effect of relative order M.
3.3. Order of magnitude of the quadrupole term Curle in an article vetted by Lighthill and his following base the neglect of quadrupoles in his Eq. (3.7), which states that ‘‘IQ =ID ¼ OðM 2 Þ times a function of Re,’’ where IQ and ID are the intensities due to quadrupoles and dipoles. We challenge this. The FWH and Curle’s equivalent equation show that the pressure p0 can be written as p0 ðtÞ ¼ p0D ðtÞ þ p0Q ðtÞ with obvious notation, and p0D =p1 ¼ OðM 3 Þ while p0Q =p1 ¼ OðM 4 Þ (as confirmed by Phillips [14] and many others). Then p02 ¼ p0D2 þ2p0D p0Q þ p0Q2 ,
(16)
and the leading correction due to quadrupoles, 2p0D p0Q , is the cross-term of order M7. This applies instantaneously, as written above, and to the time average p02 which gives the sustained sound intensity. Thus, in fact the extra contribution of quadrupoles, in the presence of dipoles, is given by IQ =ID ¼ OðMÞ, where ID refers to the dipoles radiating alone. It is only a region of quadrupoles by itself, as in Lighthill’s jets, that gives an intensity of order M8. This error has survived countless readings (many by the present author). The difference in order between IQ and ID is thus less favourable than was thought, which lowers the value of M below which the approximation can be expected to hold. The 2p0D p0Q term will be zero if the two factors are uncorrelated, but that amounts to expecting the turbulent eddies and their pressure footprint on the surface to be uncorrelated. This is most unlikely for eddies that are close to the surface, compared with their own size. The equations in Section 3.2 will thus be stated more precisely by adding that the frequency range of the OðM 7 Þ remainder is the same as that of the leading term, so that they are most probably correlated. In the FWH integral, the differences in retarded times are of the order of D=c1 . At a frequency f, the phase differences are therefore of the order of fD=c1 ¼ MSt, leading to a correction of order M7 to the leading term of order M6; to be specific, p0D is modified by a term we would call p0RT of order M4 [14], altering p02 by OðM 7 Þ through (16). Recall that the Cij function in (11) depends on M only to OðM 2 Þ, so that to leading order, the dominant St does not depend on M. This is a second reason for the order of accuracy given in Section 3.2. Finally, the distinction between emission and reception distances is also of relative order M; therefore, strictly to the order of the approximation, these two distances are equivalent. The same applies to the emission and reception angles.
3.4. Definition of the Strouhal number Because of the neglect of retarded-time differences, the force-based scaling only extends to the part of the spectrum that makes the source compact, that is, frequencies f such that fD 5c1 ; this corresponds to MSt 51 as discussed in Section 3.3. Curle himself offered no guess as to how low M needs to be for the approximation to be accurate; that could be 0.1, or 0.01; air, or water. In fact, Ffowcs Williams clearly considered that 0.01 was safe, but aeronautical and now railroad values of M were not [12]. The condition MSt 5 1 is required to neglect differences in retarded times, and arrive at the formula based on unsteady forces (9), a point made by Crighton [9] in his Section 6 among others. In asymptotic terms, any length scale of the model is adequate to define St and the theory rests on M tending to 0, but when the approximation is applied in practice, values of St very far from 1 matter. The safe definition of St is based on the total extent L of the system. Landing-gear studies have used the diameter D of the wheels. However, if the flow has a plane of symmetry, this drives L up, to twice the distance from the model to the plane of symmetry (the mirror image model being included in the system). For landing-gear research, it is common to have such a plane, roughly in the position the wing would have. The differences in retarded times cause strong interference effects when StL fL=U 1 is of order 0.3–3, as clearly seen for the Rudimentary Landing Gear research model which has a plane of symmetry about 2D away from the model so that L 4D, and M values in the 0.1–0.2 range ([15], used in Fig. 2). As could be expected, a scaling by (12) did not apply well at all. Now the symmetry is somewhat artificial (it would of course not apply between the left and right landing-gear of a real airplane), but landing-gear trucks especially with six wheels have plausible values of L of around 5D to 8D independently of any reflections.
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2813
Fig. 2. Spectra at two Mach numbers, using simulation results [15]. Lower curve, M ¼ 0.115; upper curve, M ¼0.23. Solid arrow, comparison at the same frequency, proportional to MSt; dashed arrow, comparison at the same Strouhal number St.
Another mechanism for strong correlation beyond a reach of order D, so that L is appreciably larger than D, exists in cases of strong vortex shedding impinging on a second solid body. A test case of much interest has been that of tandem cylinders, with L 4D; rod–airfoil configurations have also been prominent. Conversely, it is clear that the combined noise of small independent sources, say of size d, will satisfy Curle’s approximation at a frequency f once the product of M and Std fd=U 1 is much smaller than 1, even if they are considered together in a complex of total size L. This is because these sources are uncorrelated, in contrast with the model and its mirror image. For such sources, it is highly plausible that Std rather than StL is the relevant Strouhal number. In this context, Curle stated that ‘‘o will in general be of order U 1 =D,’’ where o ¼ 2pf and D was again a ‘‘typical dimension of the solid bodies.’’ In other words, he stated that StD is near 1 (or rather near 1=2p, which actually happens to be more typical of a circular cylinder, for example). He did not distinguish D, L and d, which now appears quite necessary for systems such as airplane landing gear, or trains with their running gear and pantographs. 4. Application to flight and wind-tunnel tests We use data from papers of Siller and Drescher [16], Michel and Qiao [17], Dobrzynski and Buchholz [18], Dobrzynski et al. [19], Guo et al. [20]), and Manoha et al. [21]. They all concern landing-gear noise, which is the prominent example of noise created by solid bodies and is much easier to isolate than high-lift-system noise. The noise of train and road-vehicle components will be of great interest also. In most of the studies, the frequency range did not extend to low enough values to calculate the SPL and test equation (14). This would have been the simplest test. The insufficient frequency range is evident in the figures since the spectral densities do not fall deeply, if at all, at the low-frequency ends. Our comments will therefore concern the upper frequency range, even while noting that the condition MSt 51 is often violated for the diameter Strouhal number. Results that include both sides of the spectral peak would be more instructive, because where the spectral slope is close to constant, it is impossible to assert which of (6) and (7) fits the facts better (assuming n is left free). We look at angles near 901, and neglect the ð1M r Þ factors. In each case, we provide estimates of the different Strouhal numbers discussed in Section 3.4. We first clarify the options of comparing spectra from different Mach numbers with the same object at fixed frequency or at fixed Strouhal number, as illustrated in Fig. 2, using a simulation dataset [15]. Comparison at fixed frequency as per the vertical arrow is common, but there is nothing in the theory to recommend it; recall (11), which is compatible with a collapse of spectra at fixed frequency only if the spectra follow a power law. In our opinion, apparent successes of fixedfrequency comparisons have been brought about by two effects among others. First, there can be interference effects due to reflections on a plate (plane of symmetry in a simulation), as seen over the approximate range ½0:7,1:7 in the figure. Interference effects result from the model and its mirror image being too distant to constitute a compact source together, so that the theory simply does not apply. Second, limitations in the instrumentation such as the transfer function of a microphone tend to corrupt the results at the same frequency even as M is varied, which can easily cause a fall-off that is mistaken for physics. We of course note that simulations have similar limitations, although the dependence of their errors on frequency and Strouhal number is more obscure. Inasmuch as we trust the simulations used in the figure, visually translating the two arrows makes it clear that only the comparison at fixed Strouhal number has any chance of fitting both 10 the region with a weak sustained slope on the left, and the region with a steep slope (approximate f power law) on the right.
2814
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
We begin with two sets of measurements in flight, and follow with the larger supply of wind-tunnel studies. Siller and Drescher’s Fig. 11, for the nose gear of an airliner, displays the spectral density versus frequency over the range of 1000–6000 Hz, giving roughly MStD between 3 and 20, and MStL between 6 and 40 (for a two-wheel nose landing gear, L ¼ 2D is plausible). With these values, at first sight, agreement with the theory would be fortuitous. However, the geometry had many small components, with d one and two orders of magnitude smaller than D, which may well dominate the noise in this frequency range. We proceed with the theory in any case. In the framework of fixed St, 3000 Hz at 78 m/s corresponds with 4614 Hz at 120 m/s. The difference in S is about 5.5 dB, in the graph. We neglect Mr although the angle is near 601, and apply (12) with an unknown n instead of 6 as the power of M, i.e., Sðf 2 ,M2 Þ=Sðf 1 ,M 1 Þ ¼ ðM 2 =M 1 Þn1 with f 2 =f 1 ¼ M 2 =M1 . This gives only n ¼3.9, in sharp contrast with Siller and Drescher’s own analysis at fixed frequency, which arrives at n 6. The same analysis between frequencies 1000 Hz and 1538 Hz at the two speeds leads to n 5:2. Siller and Drescher’s Fig. 10, for the noise of the entire aircraft instead of only its nose gear, leads to values between 3 and 4.5, depending on St. Therefore, a safe deduction is that the power n is both well below 6, and not uniform across the spectrum. Note that at the lower speed, the aircraft had a higher angle of attack, and therefore the ratio of local velocity near the landing gear to flight velocity was slightly lower. Therefore, the ratio of local velocities between the two flights was slightly larger than 120/78, so that the value of n corrected for this angle-of-attack effect would be even lower than the values calculated above. Michel and Qiao also measured noise under airliners, focussing on the nose landing gear using a phased array, and included lower frequencies, nominally down to 280 Hz. This still has MStL above 1. They declare the trends are ‘‘roughly compatible’’ with n¼5, but over most of the directions their data in Fig. 7 at three speeds are far from consistent with a power dependence on M (i.e., the points would not fall on a straight line in a log–log plot). The SPL ratios for the two most distant speeds lead to n 4. These are A-weighted SPL levels. Since a lot of the acoustic intensity is below 1 kHz, the higher-speed cases are less penalized by A-weighting; therefore, un-weighted SPL levels would place n even lower. In fact, the fixed high-pass limit may be more severe than the A-weighting. From their wind-tunnel study of full-size landing gear, Dobrzynski et al. present spectra versus St which appear to be in 1/3-octave bands, which is equivalent to weighting the spectral density by f, and SPL. They cover StD from 1 to 100, and M from 0.15 to 0.23. The total size L is of the order of 4D. The log of the velocity ratio between runs, multiplied by 10, is 1.9, so that the results over this speed range are discriminating on a dB scale, assuming of course no Reynolds-number influence. These are full-size and complete models, with Reynolds numbers in the 4 106 range for the wheel diameter, and a variety of smaller values for the other components. They find very good agreement for scaling with n ¼6, for both spectra and SPLs. The agreement is for fixed Strouhal number, as predicted by theory, but the peak near St ¼ 15 that illustrates this fact has MStD values of 2–3.5, which are a priori well outside the range of the theory, if the wheel diameter D is the appropriate length scale. In addition, in the older study of Dobrzynski and Buchholz in the same facility, some spectra have powerful fixed-frequency bumps. Guo et al. had MStD between about 0.15 and 100. They support n¼6 at low frequencies, but find a very good fit with n ¼7 at high frequencies (MSt reaching 100). However, they plotted 1/3-octave energies versus frequencies, not Strouhal 1:4 numbers. This fit is in a region in which the spectra drop roughly like f . Therefore a comparison at fixed St based on (13) would give n 5:6, rather than n ¼7. We conclude that this dataset truly points at values between 5 and 6, and disagree with their conclusion that the spectrum scales like M7, even locally, which would lead to M8 scaling for the SPL (even as the idea of an increasing role for quadrupoles at high frequencies would be seductive). Manoha et al. in their Fig. 16 have SPL results slightly over n ¼6, over the very wide M range from 0.1 to 0.28, but no comparison of spectra. However, we presume that the SPL was calculated over a fixed frequency range; therefore, measurements with a fixed St range or a lower cutoff than 100 Hz would yield a lower value for n. A worrisome pattern appears, with n values between 5 and 6 (usually close to 6) in wind tunnels, and below 5 if not below 4 in flight tests. Flight tests involve ‘‘real’’ physics, but are difficult, and require separating the contribution of distinct regions of the airframe using phased arrays. Wind-tunnel tests in contrast only require checking the backgroundnoise level, and applying mild shear-layer corrections, but the Reynolds numbers are much less safe from transition phenomena. This concern is added to the fact that the condition MSt 5 1 is violated so strongly if St is based on wheel diameter that claiming agreement with theory is debatable. In addition, it can be difficult to place the microphones in the true far-field for all frequencies of interest (L. Cattafesta, Personal communication, 2012). A bias towards theory does not appear to be recent in the literature. In his 1956 study of the Aeolian noise radiated by circular cylinders [14], Phillips presented the M6 formula, and then a relevant but not definitive critique of earlier experimental findings which led to an M4 (or possibly M5) dependence. The critique of that ‘‘anomaly’’ was obscured, first by the usual mention ‘‘times a function of Reynolds number,’’ and second by the length effects for a nominally twodimensional body. The lateral two-point correlations are always uncertain. Phillips then supported the M6 theory with new measurements, thought to be in ‘‘remarkably good’’ agreement with it, but that was not deserved because the actual trend in these measurements placed the exponent n much closer to 7 than to 6. 5. Conclusions The principal points of the study are Eqs. (13) and (14) which are not new, except of course for the OðM 7 Þ accuracy of the approximation, which appears to be new as an explicit statement. The argument in Eq. (16) bringing out an error of
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2815
order M7 instead of M8 surprised the author and others, but seems inescapable. The other point made is the limitation of simple scaling to combinations of Mach number and frequencies which satisfy MSt 5 1, where the Strouhal number is based on the ‘‘appropriate’’ length scale on the model, which proved delicate and probably frequency-dependent. Finally, comparisons across Mach numbers must be made at the same Strouhal number St rather than the same Helmholtz number (scaled frequency) MSt. The M6 law should be applied with care. A subtlety pointed to by a reviewer is that diffraction phenomena, predictable at high frequencies, bring in other powers of M when the geometry has sharp edges: M5 for a sharp plate [22], or M 16=3 for a right angle [9]. These principles were applied to a re-interpretation of some experimental results for landing-gear noise, with mixed results and low n values especially for flight tests, and in several cases disagreement with the authors’ own analyses. There is the suggestion of fortuitous agreement, or else the compact-source argument is too narrow, and the theory can be improved until it covers a much wider frequency range.
Acknowledgments This work draws on a long collaboration with Drs. M. Strelets and M. Shur, who made substantial comments about the manuscript. Prof. P. Bradshaw, Dr. J. Larssen and Dr. D. Wetzel also contributed. A very early version was an appendix in AIAA paper 2012-1174. I dedicate this article to Dr. Feri Farassat, who was a guide in this domain. References [1] [2] [3] [4]
N. Curle, The influence of solid boundaries upon aerodynamic sound, Proceedings of the Royal Society A 231 (1955) 505–514. S.C. Crow, Aerodynamic sound as a singular perturbation problem, Studies in Applied Mathematics XLIX (1) (1970) 21–44. A.T. Fedorchenko, On some fundamental flaws in present aeroacoustic theory, Journal of Sound and Vibration 232 (4) (2000) 719–782. C.K.W. Tam, Computational aeroacoustics examples showing the failure of the acoustic analogy theory to identify the correct noise sources, Journal of Computational Acoustics 10 (4) (2002) 387–405. [5] N. Peake, A note on ‘‘Computational aeroacoustics examples showing the failure of the acoustic analogy theory to identify the correct noise sources’’ by C.K.W. Tam, Journal of Computational Acoustics 12 (2004) 631–634. [7] P.R. Spalart, P.R. 2007, Application of the full and simplified acoustic analogies to an elementary problem, Journal of Fluid Mechanics 578 (2007) 113–118. [8] A. Powell, Aerodynamic noise and the plane boundary, Journal of the Acoustical Society of America 32 (8) (1960) 982–990. [9] D.G. Crighton, Basic principles of aerodynamic noise generation, Progress in Aerospace Sciences 16 (1) (1975) 31–96. [10] J.E. Ffowcs-Williams, On the role of quadrupole source terms generated by moving bodies, AIAA Paper, AIAA-79-576. [11] C.K.W. Tam, Dimensional analysis of jet-noise data, AIAA Journal 44 (3) (2006) 512–522. [12] D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl, F.G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes, Springer-Verlag, Berlin, Heidelberg, 1992. [13] C.L. Burley, T.F. Brooks, W. Humphreys, J.W. Rawls Jr., ANOPP landing gear noise prediction comparisons to model-scale data, AIAA Paper, AIAA2007-3459. [14] O.M. Phillips, The intensity of Aeolian tones, Journal of Fluid Mechanics 1 (1956) 607–624. [15] P.R. Spalart, M.L. Shur, M.Kh. Strelets, A.K. Travin, Initial noise predictions for rudimentary landing gear, Journal of Sound and Vibration 330 (17) (2011) 4083–4097. [16] H.A. Siller, M. Drescher, Investigation of engine tones in flight, AIAA Paper, AIAA-2011-2946. [17] U. Michel, W. Qiao, Directivity of landing-gear noise based on flyover measurements, AIAA Paper, AIAA-99-1956. [18] W. Dobrzynski, H. Buchholz, Full-scale noise testing on Airbus landing gears in the German Dutch wind tunnel, AIAA Paper, AIAA-97-1597-CP. [19] W. Dobrzynski, L.C. Chow, M. Smith, A. Boillot, N. Molin, Experimental assessment of low noise landing gear component design, AIAA Paper, AIAA2009-3276. [20] Y.P. Guo, K.J. Yamamoto, R.W. Stoker, Experimental study on aircraft landing gear noise, Journal of Aircraft 43 (2) (2006) 306–317. [21] E. Manoha, J. Bulte´, V. Ciobaca, B. Caruelle, LAGOON: further analysis of aerodynamic experiments and early aeroacoustics results, AIAA Paper, AIAA2009-3277. [22] M.S. Howe, Acoustics of Fluid–Structure Interactions. Monographs on Mechanics, Cambridge University Press, 1998.
Contents lists available at SciVerse ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
On the precise implications of acoustic analogies for aerodynamic noise at low Mach numbers Philippe R. Spalart Boeing Commercial Airplanes, Seattle, WA 98124, USA
a r t i c l e in f o
abstract
Article history: Received 24 April 2012 Received in revised form 20 December 2012 Accepted 26 December 2012 Handling Editor: J. Astley Available online 30 January 2013
We seek a clear statement of the scaling which may be expected with rigour for transportation or other noise at low Mach numbers M, based on Lighthill’s and Curle’s theories of 1952 and 1955. In the presence of compact solid bodies, the leading term in the acoustic intensity is of order M6. Contrary to the belief held since that time that it is of order M8, the contribution of quadrupoles, in the presence of dipoles, is of order only M7. Retarded-time-difference effects are also of order M7. Curle’s widely used approximation based on unsteady forces neglects both effects. Its order of accuracy is thus lower than was thought, and the common estimates of the value of M below which it applies appear precarious. The M6 leading term is modified by powers up to the fourth of ð1M r Þ, where Mr is the relative Mach number between source and observer; at speeds of interest the effect is several dB. However, this is only one of the corrections of order M7, which makes its value debatable. The same applies to the difference between emission distance and reception distance. The scaling with M6 is theoretically correct to leading order, but this prediction may be so convincing, like the M8 scaling for jet noise, that some authors rush to confirm it when their measurements are in conflict with it. We survey experimental studies of landing-gear noise, and argue that the observed power of M is often well below 6. We also object to comparisons across Mach numbers at fixed frequency; they should be made at fixed Strouhal number St instead. Finally, the compact-source argument does not only require M 5 1; it requires MSt 5 1. This is more restrictive if the relevant St is well above 1, a situation which can be caused by interference with a boundary or by wake impingement, among other effects. The best length scales to define St for this purpose are discussed. & 2013 Elsevier Ltd. All rights reserved.
1. Introduction Simple theoretical–empirical laws such as the dependence of various types of noise on powers of the Mach number are very convenient, and widely used in industry both to predict transportation noise across a speed range, and to identify putative sources of noise in a complex environment. In a complex system, simply by varying Mach number, one may thus ideally separate jet noise, with M8 dependence, from dipole airframe noise with M6, and even turbulent-boundary-layer/ trailing-edge noise with M5. Overwhelmingly, the powers of M were suggested by the Acoustic Analogy of Lighthill and extensions of it, notably Curle’s [1] and then supported by experiments and now simulations; this gave them a high stature. Nevertheless, theoretical challenges to the Acoustic Analogy have been numerous over the years [2–4]; the subject remains controversial, although
E-mail address: [email protected] 0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.12.029
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2809
specific charges that the Acoustic Analogy gives erroneous results have been refuted mathematically [5,7]. The experimental confirmation is far from perfect. Our attitude is to examine and possibly correct the interpretation and uses of the Acoustic Analogy, but certainly not to suggest that it is ‘‘wrong’’ and should be discarded. There have been qualifications to Curle’s theory also [8–10]. Attractive as the M6 law for aerodynamic noise is, abuses are possible, and we believe we have witnessed simplistic applications, as well as applications outside the true range of validity of the theory. This becomes more relevant as the progress of simulations and instrumentation stokes the desire for quantitative predictions with error bars of very few dB, which is indispensable if the predictions are to have value in practice. This motivated a communication that contains little new theory except for explicit statements which appear to be new of the order of accuracy of the approximations in terms of M, but may stimulate discussion and serve as a warning against too broad an application of Curle’s approximation. The discussion distinguishes the certain predictions of dimensional analysis and far-field decay laws, in Section 2, from the stronger but tentative predictions which arise from the Acoustic Analogy and asymptotic theories, in Section 3. It is reminiscent of one by Tam for jet noise [11]. Then Section 4 presents a survey of landing-gear noise measurements and their interpretation, and Section 5 our conclusions. 2. Dimensional analysis Consider the flow past an object with characteristic dimension D, moving at speed U 1 through a uniform medium at rest. The fluid is modelled as an ideal gas, with specific heat ratio g (omitted from here on), with the freestream speed of sound c1 , density r1 , and viscosity n1 . Let M U 1 =c1 be the Mach number and Re DU 1 =n1 be the Reynolds number. Consider the unsteady pressure at an observation point defined by its spherical coordinates ðr, y, fÞ, with y ¼ 0 in the direction of flight. The relative Mach number between the model and a point at rest is thus M r ¼ M cos y. By dimensional analysis, the pressures in similar solutions of the Navier–Stokes equations with different values of D, c1 , and so on, are related by a scaling of the type: r tU 1 , y, f, ; M,Re : (1) p0 ðr, y, f,tÞ ¼ p1 F D D Here and throughout the paper the calligraphic F ð. . .Þ, sometimes with indices as in F ij or F 1 , means ‘‘a non-dimensional function of a list of non-dimensional arguments.’’ This function F depends only on the geometry of the model, normalized by D. The list of arguments separates the four spatio-temporal coordinates (here, spherical) from the two parameters M and Re with a semi-colon. This first equation is valid everywhere in the field, but others may be valid only in some regions, usually the far-field. Naturally, there are many choices besides p1 to normalize F , such as the dynamic pressure q r1 U 21 =2. However, the content of the ‘‘F assertion’’ in (1) is unchanged (since q ¼ ðgM2 =2Þp1 , and M is retained in the arguments of F ). The definitions of the y angle and distance r are also non-unique and dependent on M (emission versus reception versus open wind tunnel), but as M is in the list of parameters, it does not need to be dealt with yet. There is much interest in spectra, and we use Tam’s notation S for the ‘‘narrow-band’’ spectral density: Z 1 p02 ðr, y, fÞ ¼ Sðr, y, f,f Þ df , (2) 0
with f being the frequency, in which case (1) implies Sðr, y, f,f Þ ¼
p21 D r , y, f,St; M,Re , F U1 D
(3)
where St fD=U 1 is the Strouhal number. Dimensional analysis makes no further predictions; in particular, any statement (a few of which are found in the literature) that a behaviour such as proportionality to M6 or M8 is a matter of dimensional analysis is not sustainable. 3. Scaling predictions 3.1. Background The dependence on Reynolds number will be neglected. Either it is weak, for instance a logarithmic dependence of the skin-friction coefficient on Re, or it is strong, for instance the drag crisis of a cylinder, and then it is simply out of reach of simple scaling laws. The plausible conditions for such laws to succeed are to have both a high Reynolds number, of the order of 106, and a repeatable presence of turbulence over a range of speeds, say within a factor of 2 on either side of the initial speed. This presence is sometimes ensured by boundary-layer trips. Curle presented his scaling laws ‘‘times a function of Re.’’ For tests with a given physical model and viscosity, as is very often the case, Re is proportional to M and such a statement is empty, in the following sense. Suppose a quantity has been determined to be only a function of M and Re, that is, F ðM,ReÞ. Curle’s words mean that F ðM,ReÞ ¼ F 1 ðMÞF 2 ðReÞ, and he gives only F 1 . In a wind-tunnel test of a given model, we have Re ¼ ðDc1 =nÞM so that the M dependence can be written F 1 ðMÞF 2 ððDc1 =nÞMÞ. If the F 1 function is specified but the F 2 function is unspecified, then the M dependence is still
2810
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
unknown. The implication must be ‘‘the scaling law F 1 in terms of M, times a weak function of Re.’’ This is also our position, and the Reynolds number will be omitted from here on. We move on to the far-field behaviour of the sound, which is proportional to 1=r for p0 . This is asymptotic, for r b D, but un-controversial. As a result, the scaling law (3) (with Re also omitted) becomes Sðr, y, f,f Þ ¼
p21 D3 F ðy, f,St; MÞ: U 1 r2
(4)
From here on, all the equations apply in the limit as r=D-1. There is great theoretical and industrial interest in finding a simple dependence on the remaining variables and/or parameters, and this is where controversies begin. It appears that the chances that the dependence on St is analytical (for instance, that the spectrum would follow a power of St over most of its range) are very remote, so that M is the prime candidate, and ðy, fÞ are secondary ones. The literature contains several candidates for a simple Mach scaling beyond (4), which are usually of the form S p21
D2 D F 1 ðMÞF 2 ðy, f,StÞ, r2 U1
(5)
and in which generally the function of M is a power: S p21
D2 D M n F ðy, f,StÞ: r2 U 1
(6)
Inside the turbulence, n would be 4 (but not with the far-field 1=r dependence, of course), i.e., pressure fluctuations scale with the dynamic pressure, but for noise the exponent n if any exists is higher than 4 and non-trivial. The factor D=U 1 ¼ dSt=df in (6) leads to an integration versus St when (2) is executed. However, a scaling at fixed frequency rather than fixed Strouhal number is implied here and there in the literature, as discussed at length in Section 4, and takes the form S p21
D2 D n M F ðy, f,MStÞ, r2 c1
(7)
since f ¼ StU 1 =D ¼ MStc1 =D. The factor D=c1 ¼ dðMStÞ=df (instead of dSt=df ) here leads to the integration versus MSt; this gives n the same role in the two formulas (6) and (7) when p02 is calculated. In both cases (6) and (7), M having been removed from the list of individual arguments of F placed after the semi-colon, (2) can be converted to an integral versus either St or MSt, and the far-field acoustic mean squared pressure follows as p02 p21
D2 n M F ðy, fÞ: r2
(8)
Much literature has been devoted for determining both the value of n and the nature of the scaling of the spectrum, between (6) and (7), combinations of the two, or other ideas.
3.2. Theory We now draw on theory to propose a scaling law. The simplest one (and to our knowledge the only simple one) arises if, following Curle, we neglect both the quadrupoles and the differences in retarded times in the Ffowcs-Williams–Hawkings (FWH) calculation (performed with the solid surface and all the fluid quadrupoles). Both assumptions flow from the low Mach number. Within this approximation, the sound is attributed to the time-derivatives df =dt of the forces on the model, where t is the emission time [1]. We further assume that in units of r1 , D, and U 1 , these forces do not depend on M. This amounts to q and St scaling for the energy-containing portion of the turbulence, as M-0, which is un-controversial. The density fluctuations in the turbulence are of order r1 M 2 . The time-derivative of the force fi in the i direction has the form df i =dt ¼ qU 1 D½dC i =dðtU 1 =DÞ, where Ci is the time-dependent force coefficient and the quantity in square brackets is independent of M, to order M2. Using the far-field approximation of the FWH equation [12], stripped of the quadrupole term and of the differences in retarded times, the acoustic pressure perturbation is 9x9 1 xi df i t þ p1 OðM 4 Þ: (9) p0 ðx,tÞ ¼ c 4p91Mr 99x9c1 9x9 dt In this formula, 9x9 is the emission distance, and the observer is fixed in the atmosphere, thus giving a Doppler shift relative to the internal frequency of the source. From here on we do not use the symbol as in Section 3.1, but ¼ signs with specific remainders of the type OðM m Þ. The fact that the approximation in (9) is valid to order M4 (and for instance (11) below to order M7) is of some importance, and is covered in Section 3.3.
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
The emission and reception times are related by dt ¼ ð1M r ðtÞÞ dt so that 9x9 1 xi df i þp1 OðM4 Þ: t p0 ðx,tÞ ¼ c 4pð1M r Þ2 9x9c1 9x9 dt
2811
(10)
For an observer moving at the speed of the model (e.g., a fixed microphone in a wind tunnel), the pressure has the same amplitude but without a Doppler shift. Squaring (10) and using the definition of force coefficients Ci yields for the spectrum " # xi xj g2 D2 D M6 2 7 St C ðStÞ þOðM Þ , (11) Sðr, y, f,f Þ ¼ p21 ij 2 16 r 2 U 1 ð1M r Þ4 9x9 R where Cij is the cross-spectral density between force coefficients in the i and j directions (i.e., C i C j ¼ C ij ðStÞ dSt) Note that xi xj 2 depends only on (y,f). 9x9 The scaling produced by the theory is indeed of the type postulated in (5). At first sight, it is not quite of the type proposed in (6), but rather of the type " # 2 D M6 7 2 D Sðr, y, f,f Þ ¼ p1 2 F ðy, f,StÞ þ OðM Þ : (12) r U 1 ð1M r Þ4 The first term in M6 =ð1M r Þ4 is well-known, and for instance implicit in the ANOPP empirical noise-prediction code [13]. The meaning of the ð1M r Þ4 factor is discussed below. In (11) the angular dependence is also analytical, and thus much more specific than that in (5): " # 2 xi xj D M6 7 2 D F ij ðStÞ 2 þ OðM Þ : (13) Sðr, y, f,f Þ ¼ p1 2 r U 1 ð1M r Þ4 9x9 The tensor F ij is symmetric and positive-definite, proportional to the tensor Cij of cross-correlations of the force time-derivatives, and therefore due to (11) S is always positive. This angular dependence is actually not compatible with the empirical ones assumed in ANOPP, which make F ðy, f,StÞ in (12) become F 1 ðy, fÞF 2 ðStÞ. In any case, the theoretical predictions are as specific regarding directivity, which was described as a ‘‘secondary candidate’’ in Section 3.1, as they are regarding Mach number. However, the M trends are far easier to test (one parameter M versus six in the F ij tensor) and have received most of the attention. The total mean-squared pressure then obeys " # xi xj D2 M6 7 p02 ðr, y, fÞ ¼ p21 2 F ðÞ þ OðM Þ (14) r ð1M r Þ4 ij 9x92 (where F ij ðÞ is the integral versus St of F ij from (13) and has no arguments left), rather than (8). The directional dependence is analytical, as inherited from (13). Again, r and y are emission coordinates. They are related to reception qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coordinates by r r ¼ r e 1 þ M 2 2M r and r e sin ye ¼ r r sin yr , with M r ¼ M cos ye . This deduction from theory suggests that (6) is too simple to be adequate at values of M such as 0.2, since the factors of ð1M r Þ in (14) make a difference of 4 dB in the forward direction, y ¼ 0. This is shown in Fig. 1, in which each vertical division represents 10 dB, and the Mach-number range covers up to high-speed trains. Thus, the correction is far from negligible for industrial practice in air.
Fig. 1. Acoustic intensity factors. —, M6; - - -, M 6 =ð1 þ MÞ4 ; - — -, M6 =ð1MÞ4 .
2812
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
Perspective on this correction is however gained by pointing out that it is of order M7, and the quadrupole and retardedtime effects lumped in the OðM7 Þ term (Section 3.3) have no deep reason to be weaker than the ð1M r Þ correction. In other words, (13) and " # xi xj D2 D M 6 F ij ðStÞ 2 þ OðM7 Þ (15) Sðr, y, f,f Þ ¼ p21 2 r U1 9x9 are mathematically equivalent (simply because M r r M and M 6 =ð1MÞ4 ¼ M6 þ OðM7 Þ as M-0), and there is no rigorous difference between (6) and the first term of (12) after all. The figure makes the point that the correction by ð1M r Þ4 is not negligible, but quantitatively, it must be viewed with caution. It is merely plausible that it alters the results in the right direction, relative to the pure M6 formula. In the same vein, this shows that, for a smooth spectrum, the approximation is too low-order to predict the Doppler frequency shift, because that is also an effect of relative order M.
3.3. Order of magnitude of the quadrupole term Curle in an article vetted by Lighthill and his following base the neglect of quadrupoles in his Eq. (3.7), which states that ‘‘IQ =ID ¼ OðM 2 Þ times a function of Re,’’ where IQ and ID are the intensities due to quadrupoles and dipoles. We challenge this. The FWH and Curle’s equivalent equation show that the pressure p0 can be written as p0 ðtÞ ¼ p0D ðtÞ þ p0Q ðtÞ with obvious notation, and p0D =p1 ¼ OðM 3 Þ while p0Q =p1 ¼ OðM 4 Þ (as confirmed by Phillips [14] and many others). Then p02 ¼ p0D2 þ2p0D p0Q þ p0Q2 ,
(16)
and the leading correction due to quadrupoles, 2p0D p0Q , is the cross-term of order M7. This applies instantaneously, as written above, and to the time average p02 which gives the sustained sound intensity. Thus, in fact the extra contribution of quadrupoles, in the presence of dipoles, is given by IQ =ID ¼ OðMÞ, where ID refers to the dipoles radiating alone. It is only a region of quadrupoles by itself, as in Lighthill’s jets, that gives an intensity of order M8. This error has survived countless readings (many by the present author). The difference in order between IQ and ID is thus less favourable than was thought, which lowers the value of M below which the approximation can be expected to hold. The 2p0D p0Q term will be zero if the two factors are uncorrelated, but that amounts to expecting the turbulent eddies and their pressure footprint on the surface to be uncorrelated. This is most unlikely for eddies that are close to the surface, compared with their own size. The equations in Section 3.2 will thus be stated more precisely by adding that the frequency range of the OðM 7 Þ remainder is the same as that of the leading term, so that they are most probably correlated. In the FWH integral, the differences in retarded times are of the order of D=c1 . At a frequency f, the phase differences are therefore of the order of fD=c1 ¼ MSt, leading to a correction of order M7 to the leading term of order M6; to be specific, p0D is modified by a term we would call p0RT of order M4 [14], altering p02 by OðM 7 Þ through (16). Recall that the Cij function in (11) depends on M only to OðM 2 Þ, so that to leading order, the dominant St does not depend on M. This is a second reason for the order of accuracy given in Section 3.2. Finally, the distinction between emission and reception distances is also of relative order M; therefore, strictly to the order of the approximation, these two distances are equivalent. The same applies to the emission and reception angles.
3.4. Definition of the Strouhal number Because of the neglect of retarded-time differences, the force-based scaling only extends to the part of the spectrum that makes the source compact, that is, frequencies f such that fD 5c1 ; this corresponds to MSt 51 as discussed in Section 3.3. Curle himself offered no guess as to how low M needs to be for the approximation to be accurate; that could be 0.1, or 0.01; air, or water. In fact, Ffowcs Williams clearly considered that 0.01 was safe, but aeronautical and now railroad values of M were not [12]. The condition MSt 5 1 is required to neglect differences in retarded times, and arrive at the formula based on unsteady forces (9), a point made by Crighton [9] in his Section 6 among others. In asymptotic terms, any length scale of the model is adequate to define St and the theory rests on M tending to 0, but when the approximation is applied in practice, values of St very far from 1 matter. The safe definition of St is based on the total extent L of the system. Landing-gear studies have used the diameter D of the wheels. However, if the flow has a plane of symmetry, this drives L up, to twice the distance from the model to the plane of symmetry (the mirror image model being included in the system). For landing-gear research, it is common to have such a plane, roughly in the position the wing would have. The differences in retarded times cause strong interference effects when StL fL=U 1 is of order 0.3–3, as clearly seen for the Rudimentary Landing Gear research model which has a plane of symmetry about 2D away from the model so that L 4D, and M values in the 0.1–0.2 range ([15], used in Fig. 2). As could be expected, a scaling by (12) did not apply well at all. Now the symmetry is somewhat artificial (it would of course not apply between the left and right landing-gear of a real airplane), but landing-gear trucks especially with six wheels have plausible values of L of around 5D to 8D independently of any reflections.
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2813
Fig. 2. Spectra at two Mach numbers, using simulation results [15]. Lower curve, M ¼ 0.115; upper curve, M ¼0.23. Solid arrow, comparison at the same frequency, proportional to MSt; dashed arrow, comparison at the same Strouhal number St.
Another mechanism for strong correlation beyond a reach of order D, so that L is appreciably larger than D, exists in cases of strong vortex shedding impinging on a second solid body. A test case of much interest has been that of tandem cylinders, with L 4D; rod–airfoil configurations have also been prominent. Conversely, it is clear that the combined noise of small independent sources, say of size d, will satisfy Curle’s approximation at a frequency f once the product of M and Std fd=U 1 is much smaller than 1, even if they are considered together in a complex of total size L. This is because these sources are uncorrelated, in contrast with the model and its mirror image. For such sources, it is highly plausible that Std rather than StL is the relevant Strouhal number. In this context, Curle stated that ‘‘o will in general be of order U 1 =D,’’ where o ¼ 2pf and D was again a ‘‘typical dimension of the solid bodies.’’ In other words, he stated that StD is near 1 (or rather near 1=2p, which actually happens to be more typical of a circular cylinder, for example). He did not distinguish D, L and d, which now appears quite necessary for systems such as airplane landing gear, or trains with their running gear and pantographs. 4. Application to flight and wind-tunnel tests We use data from papers of Siller and Drescher [16], Michel and Qiao [17], Dobrzynski and Buchholz [18], Dobrzynski et al. [19], Guo et al. [20]), and Manoha et al. [21]. They all concern landing-gear noise, which is the prominent example of noise created by solid bodies and is much easier to isolate than high-lift-system noise. The noise of train and road-vehicle components will be of great interest also. In most of the studies, the frequency range did not extend to low enough values to calculate the SPL and test equation (14). This would have been the simplest test. The insufficient frequency range is evident in the figures since the spectral densities do not fall deeply, if at all, at the low-frequency ends. Our comments will therefore concern the upper frequency range, even while noting that the condition MSt 51 is often violated for the diameter Strouhal number. Results that include both sides of the spectral peak would be more instructive, because where the spectral slope is close to constant, it is impossible to assert which of (6) and (7) fits the facts better (assuming n is left free). We look at angles near 901, and neglect the ð1M r Þ factors. In each case, we provide estimates of the different Strouhal numbers discussed in Section 3.4. We first clarify the options of comparing spectra from different Mach numbers with the same object at fixed frequency or at fixed Strouhal number, as illustrated in Fig. 2, using a simulation dataset [15]. Comparison at fixed frequency as per the vertical arrow is common, but there is nothing in the theory to recommend it; recall (11), which is compatible with a collapse of spectra at fixed frequency only if the spectra follow a power law. In our opinion, apparent successes of fixedfrequency comparisons have been brought about by two effects among others. First, there can be interference effects due to reflections on a plate (plane of symmetry in a simulation), as seen over the approximate range ½0:7,1:7 in the figure. Interference effects result from the model and its mirror image being too distant to constitute a compact source together, so that the theory simply does not apply. Second, limitations in the instrumentation such as the transfer function of a microphone tend to corrupt the results at the same frequency even as M is varied, which can easily cause a fall-off that is mistaken for physics. We of course note that simulations have similar limitations, although the dependence of their errors on frequency and Strouhal number is more obscure. Inasmuch as we trust the simulations used in the figure, visually translating the two arrows makes it clear that only the comparison at fixed Strouhal number has any chance of fitting both 10 the region with a weak sustained slope on the left, and the region with a steep slope (approximate f power law) on the right.
2814
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
We begin with two sets of measurements in flight, and follow with the larger supply of wind-tunnel studies. Siller and Drescher’s Fig. 11, for the nose gear of an airliner, displays the spectral density versus frequency over the range of 1000–6000 Hz, giving roughly MStD between 3 and 20, and MStL between 6 and 40 (for a two-wheel nose landing gear, L ¼ 2D is plausible). With these values, at first sight, agreement with the theory would be fortuitous. However, the geometry had many small components, with d one and two orders of magnitude smaller than D, which may well dominate the noise in this frequency range. We proceed with the theory in any case. In the framework of fixed St, 3000 Hz at 78 m/s corresponds with 4614 Hz at 120 m/s. The difference in S is about 5.5 dB, in the graph. We neglect Mr although the angle is near 601, and apply (12) with an unknown n instead of 6 as the power of M, i.e., Sðf 2 ,M2 Þ=Sðf 1 ,M 1 Þ ¼ ðM 2 =M 1 Þn1 with f 2 =f 1 ¼ M 2 =M1 . This gives only n ¼3.9, in sharp contrast with Siller and Drescher’s own analysis at fixed frequency, which arrives at n 6. The same analysis between frequencies 1000 Hz and 1538 Hz at the two speeds leads to n 5:2. Siller and Drescher’s Fig. 10, for the noise of the entire aircraft instead of only its nose gear, leads to values between 3 and 4.5, depending on St. Therefore, a safe deduction is that the power n is both well below 6, and not uniform across the spectrum. Note that at the lower speed, the aircraft had a higher angle of attack, and therefore the ratio of local velocity near the landing gear to flight velocity was slightly lower. Therefore, the ratio of local velocities between the two flights was slightly larger than 120/78, so that the value of n corrected for this angle-of-attack effect would be even lower than the values calculated above. Michel and Qiao also measured noise under airliners, focussing on the nose landing gear using a phased array, and included lower frequencies, nominally down to 280 Hz. This still has MStL above 1. They declare the trends are ‘‘roughly compatible’’ with n¼5, but over most of the directions their data in Fig. 7 at three speeds are far from consistent with a power dependence on M (i.e., the points would not fall on a straight line in a log–log plot). The SPL ratios for the two most distant speeds lead to n 4. These are A-weighted SPL levels. Since a lot of the acoustic intensity is below 1 kHz, the higher-speed cases are less penalized by A-weighting; therefore, un-weighted SPL levels would place n even lower. In fact, the fixed high-pass limit may be more severe than the A-weighting. From their wind-tunnel study of full-size landing gear, Dobrzynski et al. present spectra versus St which appear to be in 1/3-octave bands, which is equivalent to weighting the spectral density by f, and SPL. They cover StD from 1 to 100, and M from 0.15 to 0.23. The total size L is of the order of 4D. The log of the velocity ratio between runs, multiplied by 10, is 1.9, so that the results over this speed range are discriminating on a dB scale, assuming of course no Reynolds-number influence. These are full-size and complete models, with Reynolds numbers in the 4 106 range for the wheel diameter, and a variety of smaller values for the other components. They find very good agreement for scaling with n ¼6, for both spectra and SPLs. The agreement is for fixed Strouhal number, as predicted by theory, but the peak near St ¼ 15 that illustrates this fact has MStD values of 2–3.5, which are a priori well outside the range of the theory, if the wheel diameter D is the appropriate length scale. In addition, in the older study of Dobrzynski and Buchholz in the same facility, some spectra have powerful fixed-frequency bumps. Guo et al. had MStD between about 0.15 and 100. They support n¼6 at low frequencies, but find a very good fit with n ¼7 at high frequencies (MSt reaching 100). However, they plotted 1/3-octave energies versus frequencies, not Strouhal 1:4 numbers. This fit is in a region in which the spectra drop roughly like f . Therefore a comparison at fixed St based on (13) would give n 5:6, rather than n ¼7. We conclude that this dataset truly points at values between 5 and 6, and disagree with their conclusion that the spectrum scales like M7, even locally, which would lead to M8 scaling for the SPL (even as the idea of an increasing role for quadrupoles at high frequencies would be seductive). Manoha et al. in their Fig. 16 have SPL results slightly over n ¼6, over the very wide M range from 0.1 to 0.28, but no comparison of spectra. However, we presume that the SPL was calculated over a fixed frequency range; therefore, measurements with a fixed St range or a lower cutoff than 100 Hz would yield a lower value for n. A worrisome pattern appears, with n values between 5 and 6 (usually close to 6) in wind tunnels, and below 5 if not below 4 in flight tests. Flight tests involve ‘‘real’’ physics, but are difficult, and require separating the contribution of distinct regions of the airframe using phased arrays. Wind-tunnel tests in contrast only require checking the backgroundnoise level, and applying mild shear-layer corrections, but the Reynolds numbers are much less safe from transition phenomena. This concern is added to the fact that the condition MSt 5 1 is violated so strongly if St is based on wheel diameter that claiming agreement with theory is debatable. In addition, it can be difficult to place the microphones in the true far-field for all frequencies of interest (L. Cattafesta, Personal communication, 2012). A bias towards theory does not appear to be recent in the literature. In his 1956 study of the Aeolian noise radiated by circular cylinders [14], Phillips presented the M6 formula, and then a relevant but not definitive critique of earlier experimental findings which led to an M4 (or possibly M5) dependence. The critique of that ‘‘anomaly’’ was obscured, first by the usual mention ‘‘times a function of Reynolds number,’’ and second by the length effects for a nominally twodimensional body. The lateral two-point correlations are always uncertain. Phillips then supported the M6 theory with new measurements, thought to be in ‘‘remarkably good’’ agreement with it, but that was not deserved because the actual trend in these measurements placed the exponent n much closer to 7 than to 6. 5. Conclusions The principal points of the study are Eqs. (13) and (14) which are not new, except of course for the OðM 7 Þ accuracy of the approximation, which appears to be new as an explicit statement. The argument in Eq. (16) bringing out an error of
P.R. Spalart / Journal of Sound and Vibration 332 (2013) 2808–2815
2815
order M7 instead of M8 surprised the author and others, but seems inescapable. The other point made is the limitation of simple scaling to combinations of Mach number and frequencies which satisfy MSt 5 1, where the Strouhal number is based on the ‘‘appropriate’’ length scale on the model, which proved delicate and probably frequency-dependent. Finally, comparisons across Mach numbers must be made at the same Strouhal number St rather than the same Helmholtz number (scaled frequency) MSt. The M6 law should be applied with care. A subtlety pointed to by a reviewer is that diffraction phenomena, predictable at high frequencies, bring in other powers of M when the geometry has sharp edges: M5 for a sharp plate [22], or M 16=3 for a right angle [9]. These principles were applied to a re-interpretation of some experimental results for landing-gear noise, with mixed results and low n values especially for flight tests, and in several cases disagreement with the authors’ own analyses. There is the suggestion of fortuitous agreement, or else the compact-source argument is too narrow, and the theory can be improved until it covers a much wider frequency range.
Acknowledgments This work draws on a long collaboration with Drs. M. Strelets and M. Shur, who made substantial comments about the manuscript. Prof. P. Bradshaw, Dr. J. Larssen and Dr. D. Wetzel also contributed. A very early version was an appendix in AIAA paper 2012-1174. I dedicate this article to Dr. Feri Farassat, who was a guide in this domain. References [1] [2] [3] [4]
N. Curle, The influence of solid boundaries upon aerodynamic sound, Proceedings of the Royal Society A 231 (1955) 505–514. S.C. Crow, Aerodynamic sound as a singular perturbation problem, Studies in Applied Mathematics XLIX (1) (1970) 21–44. A.T. Fedorchenko, On some fundamental flaws in present aeroacoustic theory, Journal of Sound and Vibration 232 (4) (2000) 719–782. C.K.W. Tam, Computational aeroacoustics examples showing the failure of the acoustic analogy theory to identify the correct noise sources, Journal of Computational Acoustics 10 (4) (2002) 387–405. [5] N. Peake, A note on ‘‘Computational aeroacoustics examples showing the failure of the acoustic analogy theory to identify the correct noise sources’’ by C.K.W. Tam, Journal of Computational Acoustics 12 (2004) 631–634. [7] P.R. Spalart, P.R. 2007, Application of the full and simplified acoustic analogies to an elementary problem, Journal of Fluid Mechanics 578 (2007) 113–118. [8] A. Powell, Aerodynamic noise and the plane boundary, Journal of the Acoustical Society of America 32 (8) (1960) 982–990. [9] D.G. Crighton, Basic principles of aerodynamic noise generation, Progress in Aerospace Sciences 16 (1) (1975) 31–96. [10] J.E. Ffowcs-Williams, On the role of quadrupole source terms generated by moving bodies, AIAA Paper, AIAA-79-576. [11] C.K.W. Tam, Dimensional analysis of jet-noise data, AIAA Journal 44 (3) (2006) 512–522. [12] D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl, F.G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes, Springer-Verlag, Berlin, Heidelberg, 1992. [13] C.L. Burley, T.F. Brooks, W. Humphreys, J.W. Rawls Jr., ANOPP landing gear noise prediction comparisons to model-scale data, AIAA Paper, AIAA2007-3459. [14] O.M. Phillips, The intensity of Aeolian tones, Journal of Fluid Mechanics 1 (1956) 607–624. [15] P.R. Spalart, M.L. Shur, M.Kh. Strelets, A.K. Travin, Initial noise predictions for rudimentary landing gear, Journal of Sound and Vibration 330 (17) (2011) 4083–4097. [16] H.A. Siller, M. Drescher, Investigation of engine tones in flight, AIAA Paper, AIAA-2011-2946. [17] U. Michel, W. Qiao, Directivity of landing-gear noise based on flyover measurements, AIAA Paper, AIAA-99-1956. [18] W. Dobrzynski, H. Buchholz, Full-scale noise testing on Airbus landing gears in the German Dutch wind tunnel, AIAA Paper, AIAA-97-1597-CP. [19] W. Dobrzynski, L.C. Chow, M. Smith, A. Boillot, N. Molin, Experimental assessment of low noise landing gear component design, AIAA Paper, AIAA2009-3276. [20] Y.P. Guo, K.J. Yamamoto, R.W. Stoker, Experimental study on aircraft landing gear noise, Journal of Aircraft 43 (2) (2006) 306–317. [21] E. Manoha, J. Bulte´, V. Ciobaca, B. Caruelle, LAGOON: further analysis of aerodynamic experiments and early aeroacoustics results, AIAA Paper, AIAA2009-3277. [22] M.S. Howe, Acoustics of Fluid–Structure Interactions. Monographs on Mechanics, Cambridge University Press, 1998.