Numerical study of sonic boom propagation through atmospheric turbulence using open data of weather research and forecasting Shinya Koganezawa 1 Japan Aerospace Exploration Agency Mitaka, Tokyo, 181-0015, Japan Yusuke Naka 2 Japan Aerospace Exploration Agency Mitaka, Tokyo, 181-0015, Japan Hiroaki Ishikawa 3 Japan Aerospace Exploration Agency Mitaka, Tokyo, 181-0015, Japan Ryo Shimada 4 ASIRI inc. Chiyoda, Tokyo, 101-0047, Japan

ABSTRACT Sonic boom noise is a major issue to be addressed for the realization of supersonic transport. In particular, the deformation of waveforms and changes in noise levels due to atmospheric turbulence have been studied from the viewpoint of assessing annoyance and certification standards. Although many studies on the effects of atmospheric turbulence on sonic booms have been conducted, few studies have investigated realistic trends of atmospheric turbulence. In this study, numerical analysis of sonic boom propagation considering atmospheric turbulence is performed for C609, an early type of NASA’s X-59 Low-Boom Flight Demonstrator. The model equation for propagation is heterogeneous one-way approximation for the resolution of diffraction (HOWARD) and the Ostashev and Wilson model is chosen to define the atmospheric turbulence spectra. The parameters characterizing the atmospheric turbulence are set based on publicly available datasets of weather research and forecasting (WRF) simulation. The Edwards Air Force Base, where the first flight of X-59 is planned, was selected as the data acquisition point. The sonic boom waveforms obtained by the numerical simulations and their noise level variations are statistically processed to evaluate seasonal and temporal trends. A series of numerical analyses demonstrate the usefulness of WRF-based atmospheric data.

1 koganezawa.shinya@jaxa.jp 2 naka.yusuke@jaxa.jp 3 ishikawa.hiroaki2@jaxa.jp 4 r-shimada@asiri.co.jp

a Shea mar ce 21-24 AUGUST SCOTTISH BENT caso

1. INTRODUCTION

Recently, significant research efforts have focused on the development of supersonic transport in many institutes and industrial companies, including startups. Sonic boom noise is a major issue to be addressed for the realization of supersonic transport and has therefore attracted considerable attention among stakeholders as well as researchers. In particular, the deformation of waveforms and changes in noise levels owing to atmospheric turbulence have been studied from the viewpoint of assessing annoyance and certification standards.

NASA conducted a sonic boom in atmospheric turbulence (SonicBAT) program [1] to investigate the effects of atmospheric turbulence on the propagation of sonic booms. Distorted waveforms due to atmospheric turbulence were measured, and variations in the perceived level [2] (PL) were quantitatively evaluated. Numerical simulation using the Khokhlov–Zabolotskaya– Kuznetzov (KZK) propagation equation and the Ostashev and Wilson model [3,4] (the O-W model) was also performed, and the results showed good agreement with the measurement data [5]. This shows that the numerical simulation is sufficiently useful for predicting the PL variances and annoyance of sonic booms through atmospheric turbulence.

Atmospheric turbulence conditions vary depending on environmental factors (e.g., region, season, and time); therefore, it is necessary to set turbulence parameters that are consistent with realistic trends. However, measuring atmospheric turbulence requires large-scale equipment, such as lidars and sodars, which are costly and difficult to investigate globally and comprehensively.

In recent years, as a mesoscale numerical weather prediction model, the WRF model has been applied to various fields. High-resolution rapid refresh response (HRRR) [6] is a WRF-based model developed by the National Oceanic and Atmospheric Administration (NOAA) that provides U.S. weather data. The HRRR is a 3-km resolution, cloud-resolving, convection-allowing atmospheric model that is updated hourly. The output data were archived by MesoWest of the University of Utah and are publicly available [7].

In this study, a numerical analysis of sonic boom propagation considering atmospheric turbulence was performed for C609. The atmospheric turbulence spectrum is defined by four turbulence parameters: atmospheric boundary layer (ABL) thickness z i , friction velocity u * , friction temperature T * , and mixed-layer velocity scale w * . These parameters were set based on publicly available datasets of the HRRR. Mean values calculated from the past three years near the Edwards Air Force Base, where the first flight of X-59 is planned, were used as atmospheric conditions. The sonic boom waveforms obtained by the numerical simulations and their noise level variations were statistically processed and evaluated. 2. NUMERICAL CONDITIONS

A numerical analysis of sonic boom propagation considering atmospheric turbulence was performed for C609. The near-field ( R / L =3) pressure signature data distributed for the SBPW3 [8,9] were used as input for the simulations. The numerical analysis was divided into two processes. First, propagation analysis from the near field of the aircraft to the upper layer of the ABL was performed using Xnoise [10]. Subsequently, the propagation analysis in the ABL was performed using SPnoise for a sonic boom [11]. These codes were developed at JAXA to estimate sonic booms in stratified atmospheres and atmospheric turbulence, respectively. The waveforms obtained on the ground and their noise-level variations were statistically processed and evaluated. The PL was used as the evaluation metric. The PL metric has been shown to estimate annoyance caused by sonic booms well [12] and is used in SonicBAT [1]. The governing equation for propagation in a stratified atmosphere is the augmented Burgers equation. Xnoise [10] solves the equation by separating it into the following effects: nonlinearity, geometrical spreading, stratification of atmosphere, thermoviscous attenuation, and molecular vibration relaxation. The model equation used in SPnoise for propagation in the ABL is a HOWARD, and the O-W model was chosen to define the atmospheric turbulence spectra. The O-W model provides the variances and length scales of the temperature and velocity fluctuations as a function of altitude h as follows:

𝜎𝜎 𝑇𝑇 2 (ℎ) = 4.0𝑇 𝑇 ∗ 2

𝐿 𝐿 𝑚 𝑚 𝑚 𝑚 ቁቅ 2 3 ⁄ , (1)

ቄ1+10ቀ− ℎ

2 = 3.0𝑢𝑢 ∗

𝜎𝜎 𝑆𝑆

2 , (2)

2 = 0.35𝑤 𝑤 ∗

𝜎𝜎 𝐵𝐵

2 , (3)

𝐿𝐿 𝑇𝑇 (ℎ) = 2ℎ 1 + 7 ቀ− ℎ

𝐿 𝐿 𝑚 𝑚 𝑚 𝑚 ቁ

, (4)

1 + 10 ቀ− ℎ

𝐿 𝐿 𝑚 𝑚 𝑚 𝑚 ቁ

𝐿𝐿 𝑆𝑆 (ℎ) = 1.8ℎ , (5)

𝐿𝐿 𝐵𝐵 = 0. 23𝑧 𝑧 𝑖 𝑖 , (6)

𝐿𝐿 𝑚𝑚𝑚𝑚 = − 𝑧 𝑧 𝑖 𝑖 𝑢 𝑢 ∗

3

κ𝑤 𝑤 ∗ 3 (7)

where σ represents the variance of the fluctuations; L represents the length scale of the fluctuations; the subscripts T , S , and B represent the temperature, shear, and buoyancy terms, respectively. L mo is the Monin-Obukov length scale and κ(=0.4) is the von Karman constant. Finally, the thermal and kinetic turbulence energy spectra are given by:

𝐺𝐺( 𝑘 𝑘 , ℎ) =

Γ(11 6 ⁄ )

8𝑘 𝑘 2

𝜎 𝜎 𝑇 𝑇 2 𝐿 𝐿 𝑇 𝑇 3

Γ(1 3 ⁄ )

√𝜋

11 6 ⁄ , (8)

൫1+𝑘 𝑘 2 𝐿 𝐿 𝑇 𝑇 2 ൯

𝐸𝐸(𝑘𝑘, ℎ) = 55𝑘 𝑘 4

Γ(1 3 ⁄ ) ቊ 𝜎 𝜎 𝑆 𝑆 2 𝐿 𝐿 𝑆 𝑆 5

൫1+𝑘 𝑘 2 𝐿 𝐿 𝑆 𝑆 2 ൯ 17 6 ⁄ + 𝜎 𝜎 𝐵 𝐵 2 𝐿 𝐿 𝐵

൫1+𝑘 𝑘 2 𝐿 𝐿 𝐵 𝐵 2 ൯ 17 6 ⁄ ቋ (9)

5

Γ(5 6 ⁄ )

9√𝜋

where k is the wavenumber. The above equations show that in the O-W model, atmospheric turbulence is characterized by four parameters: ABL thickness z i , friction temperature T * , friction velocity u * , and mixed-layer velocity scale w * . In this study, these parameters were set based on the HRRR datasets [6,7] (refer to Section 3.2). Other conditions and parameters are listed in Table 1.

Table 1: Parameter values used in numerical simulations.

Parameters Value Unit

Aircraft C609 -

Mach Number 1.4 -

Cruise Altitude 54000 (16459.2) ft (m)

Ground Altitude 2264 (690) ft (m)

Propagation start distance, R 270 (82.296) ft (m)

R / L 3.0 -

Azimuth angle 0 (under-track) degrees

Propagation Step Size 0.20 m

Nonlinearity Coefficient 1.2 -

Inner length scale 0.1 mm

Sampling rate 50000 Hz

Computational domain Ly ± 2000 m

# of grid Ny 4096 -

# of Fourier mode 1000 -

# of turbulence realizations 10 -

Reflection factor 1.9 -

3. ATMOSPHERIC CONDITIONS

In this study, the Edwards Air Force Base (lat=34.8852, lon=-117.8323) was selected as the data acquisition point and corresponding atmospheric data were acquired from the HRRR. The profiles of temperature and relative humidity were obtained for stratified atmospheric conditions, and the ABL thickness z i , heat flux H , friction velocity u *, and temperature and pressure at the surface layer T 0 and p 0 were obtained to calculate the atmospheric turbulence parameters. The following equations were used to calculate the two atmospheric turbulence parameters other than z i and u * , which were directly obtained from the archive [7].

20000 17500 15000 m 12500 3 gs s s 7500 Altitude 5000 2500 Time(UCT)=8 800 220 240 260 280 300 320 Temperature, K

𝑇𝑇 ∗ =

𝜌 𝜌 0 𝐶 𝐶 𝑝 𝑝 𝑢 𝑢 ∗ , (10)

𝐻

1 3

𝑤𝑤 ∗ = ቆ 𝑔 𝑔 𝑔 𝑔 𝑧 𝑧 𝑖 𝜌 𝜌 0 𝐶 𝐶 𝑝 𝑝 𝑇 𝑇 0

ቇ

, (11)

𝜌𝜌 0 = 𝑝 𝑝 0

𝑅 𝑅 𝑇 𝑇 0

(12)

where ρ 0 is the density at the surface, g is the gravitational acceleration, C p is the specific heat at constant pressure, and R is the gas constant. The mean value was calculated from the data obtained for the past three years (2019–2021) and used as the calculation condition.

Time(UCT)=20 20000 — Tan 17500 Feb — Mar 15000 —— Apr — May & 12500 ¥ _ — hn s 10000 — ml 2 —— Aug 7500 — sep 5000 sei — Nov 2500 — Dee 800 220 240 260 280 300 320 Temperature, K.

3.1. Stratified atmosphere

Two representative results of the temperature and relative humidity profiles are shown in Figures 1 and 2, respectively. In these figures, the horizontal dashed line indicates the ground surface altitude.

Figure 1: Temperature profiles.

Altitude, m Time(UCT)=8 60 Relative humidity, %

Time(UCT)=20 20000 — hn 17500 Feb — Mar 15000 —— Apr — Ma & 12500 Y * — in 3s J 10000 — hl £ —— Aug 7500 — sep 5000 Ort — Nov 2500 — Dee 0 0 20 40 60 30 100 Relative humidity, %

Figure 2: Relative humidity profiles.

Note that the local time zone PST is UTC-8, and these results correspond to midnight and noon, respectively. Both the temperature and relative humidity profiles show that noticeable difference between the midnight and noon was found only near the ground surface. These temporal changes near the surface are due to the formation of an inversion layer, which is seen up to approximately 300-500 m from the ground surface. However, seasonal trends can be observed above the surface layer, although there is little temporal change. Note that the relative humidity is low in September and October and high in December and January over a wide range from the surface layer to approximately 12000 m. Wind effects were not considered in this study.

Z;,m 2500 2000 1500 1000 8 12 Time(UTC) 16 20

3.2. Atmospheric turbulence The variations in the four atmospheric turbulence parameters for each month are shown in Figure 3.

(a)

(b)

0.0 0 4 8 12 16 20 Time(UTC) Sep Oct

16 12 Time(UTC)

(c)

(d)

Figure 3: Time variation of turbulence parameters. (a) ABL thickness z i (b) friction velocity u * (c) friction temperature T * (d) Mixed-layer velocity scale w * The ABL thickness tended to be low at night and gradually developed around sunrise. A seasonal trend can also be observed, with low values in January and December and high values in July and

August. The other three parameters also showed obvious correlations with ABL development. This suggests that turbulence parameters should not be set independently but should be set in consideration of their correlations. From this perspective, using the results of weather forecast models, such as the WRF, seems to be a promising method. The seasonal and temporal variations in the atmospheric turbulence parameters and their correlations are expected to depend on climatic classification. While this is an important and interesting topic, it is beyond the scope of the current study and left for our future plan. 4. RESULTS

The results are presented in the following subsections: showing the results assuming a stratified atmosphere (i.e., no turbulence), and showing the results of the analysis considering turbulence. Atmospheric conditions were based on the HRRR data presented in Section 3, and 288 cases (12 months × 24 h) were simulated with and without turbulence, respectively.

4.1. Without turbulence cases

Figure 4 shows the variations in PL without atmospheric turbulence.

Time(UTC)

PL, dB 14 nD 70 68 teetittittte teeter tt itt 00 ol 02 03 0s 06 07 08 09 10 u 12 B 14 15 16 7 18 19 20 a 22 23

Figure 4: Seasonal variation of PL without atmospheric turbulence.

There is a seasonal periodicity in the PL; higher in December and January and lower in September and October. It is known that relative humidity has a significant effect on PL. Lower humidity attenuates sonic boom noise more due to the change of relaxation effect [13]. The seasonality observed can be explained by the difference in the relative humidity profiles shown in Figure 2. Conversely, temporal changes in PL were smaller than seasonal changes. As mentioned in Section 3, this is thought to be because the stratified atmospheric profile changes only in a thin region of a few hundred meters from the ground surface over time. To confirm the effect of changes in the stratified atmospheric profile on the waveforms, Figure 5 shows two representative waveforms and enlarged views of front and aft shocks.

Figure 5: Representative two waveforms (red: Jan, blue: Oct) and their enlarged views.

These are waveforms showing relatively high and low PL during the year. The sonic boom attenuation due to the relaxation effect has been shown to affect peaky waveforms significantly but rounded ones slightly [13]. Comparing the two waveforms, a noticeable difference is observed in the negative peaks in the rear portion. This suggests that these differences are due to humidity variation.

4.2. With turbulence cases The atmospheric turbulence is essentially chaotic phenomenon; therefore, the waveforms affected by it, and thus the PL, should be statistically processed and evaluated. The statistics for four representative months (January, April, July, and October) are shown in Figure 6. The vertical axis shows the PL difference with and without turbulence ( ∆ PL), defined in Equation 13, to purely discuss the atmospheric turbulence effect.

“5 8 12 Time(UTC) 16 20

∆𝑃𝑃𝑃𝑃= 𝑃𝑃𝑃𝑃 w turb. ⁄ −𝑃𝑃𝑃𝑃 w o turb. ⁄ (13)

“5 Apr Mean - Max. = Min. 8 12 Time(UTC) 16 20

“5 Jul @ Mean - Max. = Min. 8 12 Time(UTC) 16 20

-5 12 Time(UTC) 16

Figure 6: Statistics for four representative cases.

Time(UCT)=0 Jan —— Oct 180 160 80 z 6 2 92 ¢ @ 2a oso Ss 5S G F gsd ‘omssodioAQ Time, msec

Here, we focus on the mean, standard deviation, and maximum and minimum values, which were calculated from more than 25000 waveforms in each case. Note that ∆ PL is negative for all months and times; that is, in statistical terms, noise is reduced by atmospheric turbulence. However, the maximum value is positive; that is, it is louder than the PL without turbulence, indicating the importance of properly assessing the effects of atmospheric turbulence. The standard deviations and maximum-minimum differences appear to be smaller at night and larger during the day, respectively. This trend is similar to the time variation of the atmospheric turbulence parameters shown in Figure 3, which indicates that the higher the turbulence strength, the larger the waveform distortion, and as a result, the larger the PL variances. The results presented herein are consistent with those reported for SonicBAT [1]. The contribution of each of the four atmospheric turbulence parameters to PL distribution is currently under investigation. The probability density histograms are shown in Figure 7 for a better understanding of the PL distribution.

=8 Time(UTC): ME Jan Apr ee Jul mE Oct 25° 00° «#25 50 75 100 PL, dB =5.0 133 4 “4 2 2 © gy e¢5 4 4S 3266 Isuop Arpiqeqorg

Figure 7: Probability density histograms of PL.

The results are shown here for the time of small (8:00 UTC) and large variances (20:00 UTC). It can be observed more visually how the PL distribution widens owing to the atmospheric turbulence. From the perspective of evaluating annoyance and certification standards, it is necessary to discuss probabilistically whether noise is amplified or attenuated by atmospheric turbulence. Therefore, the ratio of positive ∆ PL ( R p ), that is, the probability that the noise is amplified by atmospheric turbulence, is shown in Figure 8.

© © a ES Probability density 0.1 Time(UTC)=20

Figure 8: The ratio of positive ∆ PL. At night, when the turbulence strength is low, R p is less than 10%, whereas, during the day, when it is high, R p is approximately 30%. As shown on the right in Figure 7, the PL distribution is different in the four cases; however, R p at that time (20:00 UTC) is almost the same, which is a notable result and suggests the possibility of saturation. Figure 9 shows the waveforms that have the maximum and minimum PL at 20:00 UTC for each representative month, along with those without turbulence.

100 —* Jan —e— Apr ee jul oe Oct 8 12 16 20 Time(UTC)

Overpressure, psf Jan 06 — wioturb. w/ turb.(max) 04 — w/turb(min) 02 0.0 02 0.4, 40 80 120 160 Time, msec 200 240

Figure 9: Waveforms showing maximum and minimum PL in each representative case. In all the cases, remarkable deformations were observed in the waveforms. In particular, the rear side, which was originally divided into several steps, exhibited a marked change, becoming a smooth curve in the minimum PL cases. However, in the maximum PL cases, the overall pressure is larger than the original waveform; in particular, the amplification at the peak portion is noticeable. The deformation of these waveforms is qualitatively reasonable given that the high-frequency component contributes significantly to the PL. 5. SUMMARY

Overpressure, psf Apr 0.6 — wioturb. —— w/turb.(max) 04 — w/turb.(min) 02 0.0 0.2 0.4, 40 30 120 160 Time, msec 200

Numerical analysis of sonic boom propagation through atmospheric turbulence was performed for C609, an early type of NASA X-59. The O-W model [3,4] was used to define the atmospheric turbulence spectra. The parameters characterizing the atmospheric turbulence were set using a WRF-based public dataset [6,7]. The average trend of the stratified atmospheric profile and turbulence parameters for the past three years at the Edwards Air Force Base, where the first flight of X-59 is planned, was presented. Based on the atmospheric data obtained, 288 cases (12 months × 24 h) were simulated with and without turbulence. The sonic-boom waveforms and their PL variations were evaluated, and seasonal and temporal trends were presented. Statistical processing was employed for the results with turbulence, and the mean, standard deviation, and maximum and minimum PLs were discussed. The mean PL was lower than that without turbulence, indicating that the sonic boom is attenuated by the turbulence effect. This is the same result as that shown for N- waves in SonicBAT [1]. The probability density of the PL was shown, and it was found that the higher the turbulence strength, the greater the PL variance. A new evaluation index, amplification probability R p was introduced to quantify the probability of the PL being amplified by turbulence. R p was found to increase by up to approximately 30% as the effect of turbulence became stronger.

Overpressure, psf 06 — wlo tub w/ turb.(max) 04 — w/turb(min) 0.2 0.0 02 04, Jul 40 80 120 160 Time, msec

A series of numerical analyses demonstrated the usefulness of WRF-based atmospheric data. In the future, the analysis will be extended to other regions to clarify the climatic zone dependence of the atmospheric turbulence effects.

Overpressure, psf 06 — wlotub. — w/turb.(max) 0.4 — w/turb.(min) 0.2 0.0 02 04, 40 80 120 160 Time, msec 200

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