19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 007 Numerical simulation of low level sonic boom propagation through random inhomogeneous sound speed fields PACS: 43.8.Mw Ollivier, Sébastien 1 ; Blanc-Benon, Philippe 1 Université de Lyon, Université Lyon 1, LMFA UMR CNRS 5509; 36 Avenue Guy de Collongue, Ecully, Ecully cedex, France; sebastien.ollivier@ec-lyon.fr Université de Lyon, Ecole Centrale de Lyon, LMFA UMR CNRS 5509; 36 Avenue Guy de Collongue, Ecully, Ecully cedex, France; philippe.blanc-benon@ec-lyon.fr ABSTRACT Atmosphere random inhomogeneities are responsible for random distortion of initial N-wave sonic boom pressure waveforms. In some cases waveforms are rounded, in some others spikes can occur on the waveforms, and sometimes the peak pressure level is increased and the rise time is shortened, leading to considerably increased perceived levels. At focus, perceived levels could be unacceptable. In order to evaluate the influence of these inhomogeneities on initial low level sonic booms, numerical simulations based on KZK equation have been performed for propagation distances of 5 km through random inhomogeneous sound speed fields. Inhomogeneous fields have been modelled by using von Karman distributions, larger scales being of the order of 100 m. Statistical results on sonic boom random distortion will be presented for different configurations. This work is a part of HISAC European research program on the feasibility of an Environmentally Friendly High Speed Aircraft (project no AIP ). INTRODUCTION As soon as sonic boom measurements have been done, it has been reported that atmosphere random inhomogeneities are responsible for random distortion of initial N-wave shape sonic booms [1-4]. These distortions can be explained by focusing and defocusing of waves by random sound speed inhomogeneities. In some cases waveforms are rounded, in some others spikes can occur on the waveforms, and sometimes the peak pressure level is increased and the rise time is shortened, leading to considerably increased perceived levels. At focus, perceived levels could be unacceptable, even in the case of an initial low boom pressure wave. In order to evaluate the influence of these inhomogeneities on initial low level sonic booms, numerical simulations based on the solution of the KZK equation have been performed for propagation distances of 3 km through random inhomogeneous sound speed fields. Inhomogeneous fields have been modelled by using von Karman distributions, larger scales being of the order of 100 m. SIMPLIFIED CONFIGURATION The structure and behaviour of the atmosphere close to the ground are very complex, and vary greatly during one day. Simulation of sonic boom through a realistic atmosphere is not possible within the present knowledge, and would still be limited due to of computer limitations. As a first step, simulations are limited a simplified configuration. The configuration is limited to two dimensions in space. The initial wave front is assumed to be plane. The mean sound speed and temperature are assumed to be constant (i.e. no temperature gradient, no wind). Inhomogeneities of the atmosphere are limited to (scalar) deviations of sound speed c (x,y) from the mean reference sound speed c o, i.e. only sound speed variations in the propagation direction are taken into account: c' = c' temp + c' // is the sum of the local sound speed fluctuation c' temp= c0. T ' T0 due to random deviation of the temperature T from the mean temperature T 0, and the random velocity fluctuation of sound speed in the propagation direction c '//. Contrary to

2 real sonic boom propagation, turbulence characteristics (length scales, energy ) are assumed to be independent of the altitude (i.e. independent of propagation distance). Propagation time of the sonic boom through turbulence is much smaller than the characteristic evolution time of turbulent structures. Concerning the geometry, it is assumed that the atmosphere is homogeneous for x < 0, i.e. above the planetary boundary layer, and that the atmosphere is inhomogeneous for x > 0, into the planetary boundary layer where the initial pressure N-wave is distorted during propagation due to random focusing and scattering by inhomogeneities. The propagation distance of sonic boom through the PBL under trace depend on flight speed, meteorological conditions, location on earth, period of the year, and period of the day. In most cases this distance is of the order of 1 to 3 km before ground impact. Direct-reflected wave interference is not modelled; ground reflection is taken into account by the amplification of the pressure by a factor two. The simplified configuration differs on several points from the real atmosphere. Consequently the results from these simulations are not yet accurate predictions, but they can give indications on the variations of sonic boom waveforms and sound level when propagation parameters (temperature, humidity ) or initial boom waveform vary. This technique appears to be a wellsuited tool to investigate the influence of atmosphere inhomogeneities on sonic boom, and in particular it could be helpful to investigate the sensitivity of shaped sonic booms. plane waves y 0 PBL x Figure 1- Definition of the x- and y-axis. Initial pressure wave front at x = 0 is a plane N-wave. Atmosphere is inhomogeneous starting from x = 0. Wave front and pressure waves p(x,y,t) are distorted by inhomogeneities for x > 0. (Sonic booms are taken from Lee & Downing) SONIC BOOM PROPAGATION MODEL Sonic boom propagation is computed by assuming that at the initial position of the calculation (x=0) the pressure field has a plane wave front normal to the main propagation path. The field of scalar sound speed inhomogeneities c (x,y) is known. The properties of air (temperature, humidity, pressure, etc.) are also known. Then, the pressure wave propagation is computed in time domain as it propagates along the x-axis, normal to the initial plane wave front. For this purpose, we use a modified KZK equation, which accounts for nonlinear effects, linear dissipation, relaxation, diffraction, and sound speed inhomogeneities [5, 6]. The propagation equation for the pressure p(x,y,t) is written as follows : p c0 = x t' δ pdt" + 3 c p β + t' ρ c p p + t' 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID c ' t' p e t" c' dt" + c ν ( t' t")/ tν 3 ν c0 0 where the five right-hand terms account respectively for diffraction, linear thermoviscous attenuation, nonlinear effects due to high pressure level, molecular relaxation processes (ν = O p t'

3 and N ), and scalar sound speed inhomogeneities due to temperature and wind field. The main parameters which depend on atmospheric pressure, temperature, humidity and chemical composition of air are sound speed c o, air density ρ o, linear absorption δ, molecular vibration relaxation times t ν and corresponding sound speed increments c ν, where ν = O, N. The solution of the KZK propagation equation is based on finite difference schemes in time domain. The input wave at x = 0, P 0 (x=0, y, t) and the inhomogeneous sound speed field are firstly defined, then the wave propagation is computed step by step along the x-axis. Atmosphere modelling Two kinds of parameters have to be defined to model the atmosphere. Firsts are non fluctuating parameters like mean temperature, humidity, pressure etc. These can be deduced from meteorological data and models [7]. Seconds are random fluctuating parameters (T and c '// ), which have to be computed from their respective statistical distribution. The difficulty is that available meteorological data in the PBL concerns long term evolution of the atmosphere, and thus large scales, since data are usually measured every 6 hours. Sound propagation through the PBL is mostly influenced by scales of the order of the wavelength. Consequently, the values of the random type parameters (T and c // ) have to be computed from a model. ' Non-fluctuating parameters Partners of HISAC Task.3.1 defined two typical cases: arid atmosphere and temperate atmosphere. Corresponding typical meteorological conditions have been chosen. Actually, arid and temperate atmosphere differ mainly in the relaxation time of oxygen and nitrogen, which could have a significant influence on sonic boom rise time, thus on the perceived loudness. The arid case is the over flight of Lancaster (California, USA) near the Mojave desert. The temperate case is the over flight of Paris, representative of maritime temperate zone. Actually, atmosphere characteristics vary with the altitude, but as a first step, constant parameters have been chosen. Random sound speed fields Random fluctuations of sound speed are computed by using Fourier mode synthesis [8]. The phase and direction of 00 turbulent wave numbers, which correspond to 00 length scales, are chosen randomly, and the energy of these modes follows a von Karman energy spectrum distribution. This choice allows generating sound speed fields with large to small scale random fluctuations. Figures -a show an example of a spatial distribution of sound speed inhomogeneities obtained by using this technique. The 3 figures (a-1, a-, a-3) differ only in turbulence level: c RMS /c 0 = , , The order of magnitude of scales and sound speed fluctuation c are of the order of magnitude of the effect of inhomogeneities in the PBL. Temperature and velocity fluctuations relative influence is not discussed, therefore inhomogeneities are here defined by the RMS value of c and by the outer and inner length scales (L 0, l 0 ) without distinction between temperature and velocity contributions. Input pressure waves Input pressure waves have been chosen to be representative of medium and low boom, i.e. the corresponding peak pressure levels (if turbulence, relaxation and dissipation is neglected) are respectively of the order of ~30 Pa and ~15 Pa with account for ground pressure doubling. Initial waveforms have been calculated by F. Coulouvrat for flights over Paris (temperate case) and Lancaster (arid case). The propagation of these input waves is computed by solving the KZK equation, and finally the resulting pressure wave is multiplied by in order to account for pressure doubling due to ground reflection. Differences in shock structures are mainly due to humidity, which affects mainly molecular relaxation frequency and absorption, the worst case being propagation over Paris. RESULTS Hereafter we give results for the case of an initial low boom (peak pressure ~ 6 Pa, duration ~154 µs at x =0), that propagates through moderate turbulence (outer length scale 100 m, c RMS /c 0 = , ,.10 - ), in an arid atmosphere. 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 3

4 Figure.- case 1 (a,b,c): c RMS /c 0 = , case : c RMS /c 0 = , case 3 : c RMS /c 0 =.10 - a-1, a-, a-3: c fields, colour scale in m/s b-1, b-, b-3: peak pressure variation 100.[P max (x,y) P 0 (x,y)] / P 0 (x,y) [ in % ] c-1, c-, c-3: rise time variation 100.[τ (x,y) τ 0 (x,y)] / τ 0 (x,y) [ in % ] d-1, d-, d-3: shock front steepness variation 100.[S(x,y) S 0 (x,y)] / S 0 (x,y) [ in % ] black dots: «microphones» positions On figure, maps a-1, a-, and a-3 are the spatial fields of sound speed inhomogeneities for the 3 turbulence levels c RMS /c 0 = , , Maps b-1, b-, and b-3 show the relative peak pressure fluctuation [P max (x,y) P 0 (x,y)]/p 0 (x,y) due to sound speed inhomogeneities c (x,y), where P max (x,y) is the peak pressure at the position (x,y), and P 0 (x,y) is value of the peak pressure at the same position in the case where there is no sound speed fluctuations (i.e. c =0 and c=c 0 x, y). These maps show that the peak pressure can vary by +/-10% even for weak inhomogeneities (b-1: c RMS /c 0 = ), and vary by +/- 60% for moderate level of sound speed inhomogeneities (b-3: c RMS /c 0 =.10 - ). Some simulations done the case of strong turbulence and medium or high booms showed that peak pressure can increase up to 300% at focusing points. Maps b- also outline that the area where the peak pressure is enhanced is smaller than the area where the peak pressure is decreased. Concerning sonic boom, it means that in most cases, the perceived level would be lower than in the case of the propagation into a homogeneous atmosphere. However, the area where the pressure level is increased is not negligible thus, in future regulations, it could be necessary to take into account the probability to expose people to louder boom by defining a tolerance on pressure level. 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 4

5 Maps c-1, c-, and c-3 show the relative increase of the shock front rise time [τ (x,y) τ 0 (x,y)]/τ 0 (x,y), where τ (x,y) is the rise time at the position (x,y), and τ 0 (x,y) is the rise time at the same position in the case where there is no sound speed fluctuations. These maps show that in most cases the rise time is of the same order with or without sound speed inhomogeneities, and that it can be significantly higher due to sound speed inhomogeneities, but there is not evidence that it could be significantly lower. Detailed analysis of the rise time decreases only at focusing area. The rise time increase and the peak pressure decrease are thus correlated. In order to outline this observation, we have plotted the relative fluctuation S(x,y) S 0 (x,y)]/s 0 (x,y) of the slope of the shock front defined as S(x,y)=P max (x,y)/τ (x,y). The reference is S 0 (x,y)=p 0 (x,y)/τ 0 (x,y). The colour scale of maps d-1, d- and d-3, is the same as for maps b-1, b- and b-3. It appears that the contrast of maps d- is higher than for maps b-. This is due to the fact that areas where the peak pressure level increases also correspond to areas where the rise time is shortened, and areas where the peak pressure level decreases also correspond to areas where the rise time increases. This could also be observed on waveforms. This behaviour is of major importance concerning sonic boom because the increase of the peak pressure combined with the shortening of rise time induces a significant increase of the perceived level. At the location of the black dots plotted on maps, waveforms have been recorded and plotted on figure 3. For each case (a,b,c), the pressure waves at the upper boundary of the PBL is given (i.e. at x = 0.06 km) and for four typical propagation distances through the PBL (x = 1., 1.5, 1.8,.1 km) the waveforms recorded by the height microphones of each vertical line are plotted. The horizontal dotted lines show the peak pressure level recorded at the same position in the case of the homogeneous atmosphere. This figure outlines that: (i) the boom waveform is not much distorted in the case of weak inhomogeneities, (ii) distortion of the waveforms increases with the propagation distance an with the level of inhomogeneities, (iii) the enhancement or attenuation of the peak pressures by sound speed fluctuations increases with the propagation distance and with the level of sound speed inhomogeneities c RMS. CONCLUSIONS Numerical simulations of low level sonic boom propagation through a simplified inhomogeneous atmosphere have been done by solving the KZK equation. The results discussed here outline that, even in the case of moderate turbulence levels and low level booms, atmospheric wind and temperature inhomogeneities can have a significant influence on the perceived level (increase up to 60% of the ratio peak pressure / rise time). Results obtained for different atmospheric conditions and sonic boom levels will be discussed at the congress. 19 th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 5

6 Figure 3.- Pressure waveforms recorded for 5 propagation distances in the PBL (x=0.06, 1., 1.5, 1.8,.1 km ). For each propagation distance, the waveforms recorded by 8 microphones are plotted (see figure ). References: [1] R.A. Lee, J.M. Downing, Sonic Boom produced by United states Navy aircraft: measured data, AL-TR , Biodynamic Environment Branch, Biodynamics and Bioengenineering Division, Armstrong Laboratory,Wright-Patterson Air Force Base, Ohio (1991). [] A.D. Pierce, Spikes on sonic boom pressure waveforms, J. Acoust. Soc. Am. 44(4) (1968), [3] A.D. Pierce, D.S. Maglieri, Effect of atmospheric irregularities on sonic-boom propagation, J. Acoust. Soc. Am. 51 (197) [4] H.E. Bass, R. Raspet, J.P. Chambers, M. Kelly, Modification of sonic boom wave forms during propagation from the source to the ground, J. Acoust. Soc. Am. 111(1) (00), [5] R.O. Cleveland, M.F. Hamilton, D.T. Blackstock, 1996, Time domain modelling of finite amplitude sound in relaxing fluids, J. Acoust. Soc. Am. 99 (1996), [6] Ph. Blanc-Benon, B. Lipkens, L. Dallois, M.F. Hamilton, D.T. Blackstock, Propagation of finite amplitude sound through turbulence: modelling with geometrical acoustics and the parabolic approximation, J. Acoust. Soc. Am., 111(1) Pt (00), [7] L.C. Sutherland, N.H. Bass, Atmospheric absorption in the atmosphere up to 160 km, J. Acoust. Soc. Am. 115(3) (006), [8] M. Karweit, Ph. Blanc-Benon, D. Juvé, G. Comte-Bellot, J. Acoust. Soc. Am. 89(1) (1991), th INTERNATIONAL CONGRESS ON ACOUSTICS ICA007MADRID 6