Assessment of Common Practice for Sonic Boom Calculation

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2011, Orlando, Florida
AIAA 2011-1279
Assessment of common practice for sonic boom calculation
Angelo L. Scandaliato 1
Ohio Aerospace Institute, Cleveland, OH, 44142
Meng-Sing Liou 2
NASA Glenn Research Center, Cleveland, OH, 44135
In this paper we investigate the issues encountered in the state-of-the-art practice for
sonic boom calculation, specifically the near-field flow fidelity, the axial-symmetry criteria
for wave-form propagation, and the sensitivity of perceived loudness to wave-form. This
work has been carried out with the intent for inclusion in a shape optimization framework
for aerodynamic design and analysis for low boom supersonic transport. Using an adaptive
Cartesian meshing method for solving the Euler equations, Cart3D1, we analyze the effects
of both grid refinement and pressure sampling distance away from an aircraft on the
extrapolated sonic boom signature and perceived loudness. In this study, two delta-wing
models and the NASA experimental F-15-Active aircraft are used to test the sonic boom
propagation procedure and loudness calculation. Convergence of the ground pressure
signature with respect to the near-field sampling distance from the aircraft is achieved
rather quickly. The perceived loudness (PLdB) is tested for its sensitivity to changes in
signature shape; our study reveals a surprising insensitivity to the near-field sampling
distance, axial-symmetry condition, and mesh size. Finally, to gain further insight into the
link between the ground signature and loudness, we construct several nominal signature
models and assess the effectiveness of the controlling parameters.
Nomenclature
PLdB
H
L
𝜂𝑖
𝐶𝐿
𝐶𝐴
=
=
=
=
=
=
perceived loudness in decibels
radial distance from aircraft centerline
model aircraft length
measure of axial-symmetry
lift coefficient
axial force coefficient
I. Introduction
S
UPERSONIC flights can generate rapidly changing off-body pressure disturbances. These disturbances over
long distances can coalesce to form strong shocks or spread to produce a considerable pressure decrease,
followed by a sudden compression/shock to accommodate for the pressure at the rear of the vehicle. Observers
and/or materials that encounter such pressure distributions will experience what is commonly referred to as the sonic
boom. The sonic boom has the potential to be damaging to human hearing and structures. One major outcome of
the recent NASA and collaborators’ supersonics project is the development of a low boom Supersonic Business Jet
(SSBJ). It has been demonstrated through in-flight experimentation and computational simulation that a reduction
1
Principal Researcher, RT, NASA Glenn Research Center 21000 Brookpark Road/Mail Stop 5-11; currently
Graduate student, Department of Mathematic, University of California, San Diego. Member, AIAA
2
Senior Technologist, RT, NASA Glenn Research Center 21000 Brookpark Road/Mail Stop 5-11. Associate
Fellow, AIAA
1
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This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
in sonic boom strength could be realized by modifying both aircraft shape and component positioning2, 3, 4, 5. Hence,
it is necessary to determine the issues that most effectively minimize the sonic boom and properly incorporate them
into a multidisciplinary analysis optimization (MDAO) framework. This work is focused on the quantitative
evaluation of all issues taken into account in the current practice for computing the sonic boom carpet and loudness
level at the ground.
The distance between observer and aircraft is very large compared to the scales needed to accurately resolve the
aircraft geometry. Even with moderately sized computational domains, complex aircraft geometries require
immense volume meshes, pushing the limits of the world’s most powerful supercomputers. Hence, the disparity in
scales makes it impractical to do a full computational fluid dynamics (CFD) analysis of the physical domain. In
order to tackle this issue the problem is generally broken into parts – either two or three domains6. The one we
adopt here involves breaking the problem into two domains. The first domain is considered the near to mid-field
surrounding the aircraft and is solved by a high-fidelity CFD analysis. The second domain, the far-field, is the
remaining distance to the observer and is solved by propagating the pressure distribution extracted from sampling
the near-field.
The available linear sonic boom extrapolation algorithms7, 8 are well established, essentially a standard approach,
and can quickly give a solution at far distances. However, the methods do not account for non-linear off radial flow
effects, which may influence the disturbance, but their effects are not quantified in the literature. As an underlying
condition for using this extrapolation approach, the complicated 3D near field must have propagated away from the
body to a sufficiently long distance such that axial-symmetry is nearly obtained. Therefore, an appropriate 2D
cylindrical cut revolving around the aircraft is defined and the pressure disturbance on the cylindrical surface,
especially the portion underneath the body, extracted. The flow field is further generalized to allow a 1D line cut of
the 2D cylinder to be used as input into the extrapolation algorithms. This line source is known as the sonic boom
signature or signal.
Previous studies for sonic boom reduction concentrate on altering aircraft shape and flight conditions that have a
direct effect on the pressure disturbance’s maximum overpressure, shape, or decibel level. Decibel levels are a
logarithmic scaling of pressure distributions for noise comparisons and are generally tailored to give increased
weighting on frequencies considered to have a greater impact on human hearing. Three particular decibel metrics,
AdB, CdB, and PLdB, have been previously used as possible loudness measures9,10,11. Of the three, the perceived
loudness PLdB has been cited as having the closest relation to the human auditory response as well as being
calculated over the entire pressure signature. We investigate the PLdB sensitivity to changes in the ground signature
and its potential to be used as a fitness value for the low boom objective.
In section 2, the Euler equation solver, Cart3D1, is adopted for the calculation of the near-field flow region. The
amount of grid refinement necessary for Cart3D to accurately resolve off-body pressure signatures is investigated
for a delta-wing model. In section 3, a measure of axial-symmetry of the near-field pressure distribution is defined.
Section 4 studies the measure of axial-symmetry on two delta-wing models. Section 5 discusses two available linear
sonic boom extrapolation codes, NFBOOM and PCBOOM, and the calculation of perceived loudness. The nearfield signatures from the delta-wing and F-15-Active configurations are propagated to the ground; a grid refinement
study for the F-15-Active is also performed. Section 6 investigates the metric of PLdB sensitivity to changes in
ground signature shapes.
II. Near-Field Flow Solution and Grid Refinement
In the present work we choose Cart3D to calculate the non-linear compressible Euler equations of a perfect gas
in the near-field. Cart3D is a second order solver, which automatically generates adaptive Cartesian multi-grids
with an adjoint output-based procedure12. It has been extensively tested and is known to provide high quality
solutions where viscous effects are not dominant. Cart3D is chosen primarily because of its hands off ability in
creating reliable volume grids, which is particularly important for automation in multidisciplinary studies, and its
robust solution adaptation capability, allowing a very efficient use of mesh for problems requiring a considerably
large amount of mesh points. The technique of Cart3D’s use particularly for sonic boom simulation, as outlined in1,
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is followed here. This includes appropriate procedures for rotation of the grid to better align with the Mach cone,
grid stretching in the direction of shock propagation, and both on-body aerodynamic force and off-body pressure
sensors to drive mesh growth (see Figure 3 below).
As a first example, the classical Model4 delta-wing geometry13 is used and denoted as config1, see Figure 1a.
Cart3D has been shown to produce a near-field pressure signature reasonably comparable with wind-tunnel data1, 14.
We have chosen to do recomputation of a particular test case for the config1 model. Figure 2 is the comparison of
the computed pressure distribution at H/L=3.6 body lengths below config1 with the experimental data, where the
computation does not include the attached sting. The large discrepancy seen at the tail section is due to the missing
sting.
a)
b)
Figure 1: Model4 delta-wing; a) config1 with 𝜽 = 𝟏𝟐. 𝟑𝟑𝟒𝟗° , b) config2 with 𝜽 = 𝟏𝟔. 𝟑𝟐𝟕𝟖° .
Figure 2: Over-pressure distribution on config1 after 13 adaptation cycles and 5.6 million hexes.
(Mach=1.68, 𝑪𝑳 = 𝟎. 𝟎 , H/L=3.6 )
Determining whether the flow has become axial-symmetric is particularly difficult because even though our
measurement 𝜂𝑖 defined in equation (1) may be small, indicating good axial-symmetry, it does not necessarily mean
the sampled signatures are accurately resolved. In fact, as mentioned earlier, the problem of an under developed or
under resolved flow solution may artificially give a small 𝜂𝑖 due to smearing. This leads to the need that each
sampled signature’s accuracy must be verified by a grid refinement study. That is, if finer grids do not change the
signature’s general characteristic shape, then we can be assured all features have been captured and converged.
The config1 geometry, with sensors at azimuth angle (defined in Figure 5 below) of 0 degrees, is used for the
grid refinement study. The flight conditions are specified with a Mach number of 1.4 and angle of attack (aoa) of 0
degrees. The radial distance H from the aircraft centerline is expressed in terms of the model aircraft length L. Line
sensors are equally spaced from H/L=1 to 10 body lengths beneath the geometry and grids are build over half the
body; see Figure 3. The starting location of each sensor is placed just in front of the theoretical Mach cone. The
functional used to drive Cart3D’s grid adaptation is specified as a weighted sum of the off-body pressure sensor at
H/L=10 and the axial force coefficient 𝐶𝐴 in the body frame. The goal is to find the coarsest grid allowable while
still capturing all features in the signatures. All later runs for the axial-symmetry test will be adapted up to this level
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of refinement. In the plots below, L1, L2, … refer to the signatures taken from H/L=1 body length away, H/L=2
body lengths away, and so on. It is found, see Figure 4, that the shock due to the end fuselage (at sensor coordinate
around 24 units) is not captured in the farther distant line sensors (say H/L>5) in adaptation cycles 16 and 17. By
adaptation cycle 18, all of the pressure signature characteristics could be seen up to the farthest sensor H/L=10. This
level of grid refinement is used for all subsequent runs concerning the Model4 geometry.
The jobs were preformed on NASA’s Columbia supercomputer which is comprised of an Altix 4700 architecture
with Itanium2 9040 and 9150M 1.6 GHz processors’ with 4 cores per node, and approximately 2 GB of memory per
core. The wall-clock time to complete all 18 adaptation cycles using 144 cores was approximately 3 hours.
Figure 3: Pressure contours with line sensors positioned at various distances from config1.
a)
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b)
c)
Figure 4: Grid and corresponding overpressure signatures for config1; 0 degree azimuth, Mach=1.4, aoa=0.0 degree.
a) adapt cycle 16 (7.675 million hexes); b) adapt cycle 17 (19.021 million hexes); c) adapt cycle 18 (47.275 million hexes).
III. Axial-Symmetry Criteria
Two available propagation codes, NFBOOM and PCBOOM7, use ray tracing as a means to evolve a pressure
signature extracted from the near-field region. NFBOOM is a packaged collection of sonic boom extrapolation and
sound-level prediction codes maintained by Donald Durston of NASA Ames. Both NFBOOM and PCBOOM work
on the assumption that all non-linear flow effects have already diminished sufficiently before the pressure sample is
to be taken. As a means to guarantee the underlying signal convergence, the sampling distance must be chosen at a
location off the aircraft body where the flow field is locally axial-symmetric about the centerline of the aircraft.
Further evolution of the signature is regarded as being influenced entirely by the flight atmospheric conditions and
the imprint of the particular shape of the aircraft is already contained in the input signature. As cited in6, 15, 14, based
on the linear sonic boom generation and propagation theory, the flow field is considered converged and axialsymmetric when the pressure signature decreases proportionally to (H/L)−1⁄2 .
For computational efficiency it is desirable to sample at radial distances as close to the aircraft as possible for
reducing the number of fine mesh points employed for resolving the near body nonlinear flow field, but ensuring a
“reasonable” flow convergence to axial-symmetry. Any distance closer would be inappropriate for input into the
propagation code and any further would incur unnecessary computational expense. The hope is that a minimum
radial distance could be reliably estimated before engaging a CFD analysis for any given set of flight conditions and
aircraft geometry. Previous works have determined this computational boundary to be the minimum radial distance
H at which the pressure decays at the rate of (H/L)−1⁄2 ; this process typically requires a posteriori confirmation and
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it is unclear where (e.g., which azimuth angle) the pressure should be sampled. In this work we use a metric
measuring the closeness to axial-symmetry and then choose a sampling distance that meets a desired level of that
measure.
For each H/L, three sensors at azimuth angles 0, 45, and 90 degrees, see Figure 5, are chosen for pressure
signature sampling by line sensors parallel with the direction of the flow. A measure of the difference between the
signatures is constructed at each point 𝑖 = 1, ⋯ , 𝑁 along the sensors.
2
where Δ𝑝j,𝑖 =
𝑝j,𝑖 −𝑝∞
𝑝∞
2
𝜂𝑖 = ��Δ𝑝1,𝑖 − Δ𝑝2,𝑖 � + �Δ𝑝1,𝑖 − Δ𝑝3,𝑖 � + �Δ𝑝2,𝑖 − Δ𝑝3,𝑖 �
2
(1)
is the overpressure and j=1, 2, 3 represents data sets taken from the 0, 45, and 90 degree
azimuth angles respectively. Once a well refined volume grid is determined for a given test case the signatures are
compared across all the azimuth angles for axial-symmetry. Thus, axial-symmetry is reached as 𝜂𝑖 → 0 ∀ 𝑖 =
1, ⋯ , 𝑁.
Figure 5: Convention of sensor locations in terms of azimuth angle, 0 degrees is directly beneath the body centerplane.
IV. Axial-Symmetry Example
In this section the axial-symmetry test described in section 3 is applied to the original Model4, config1, and a
slightly modified version named config2. Config2, as shown in Figure 1b, has an increased nose to wing-tip angle
and is created by translating the wing toward the nose along the body centerline. The purpose of including this
variation is to study the effects of rate of longitudinal area changes while keeping the maximum area unchanged.
Keeping the flight conditions the same in both configurations as in the previous case, we generate six grids with the
same adaptation level as Figure 4c, and calculate the pressure signatures along azimuth angles of 0, 45, and 90
degrees; see Figure 6. In order to keep the grid aligned with the direction of shock propagation, the model geometry
is rotated by the azithum angle, keeping the grid fixed. Next, equation (1) is calculated for each configuration and
plotted point wise in Figure 7. These plots indicate as expected, the axial-symmetry as measured by 𝜂𝑖 is largest for
H/L=1, and a minimum distance of H/L= 4 body lengths is necessary to guarantee a difference from axial-symmetry
of no more than 𝜂𝑖 = 0.02. The L-infinity norm for each signature is given in Table 1, showing that the maximum
magnitude in the distribution of 𝜂𝑖 is decreasing as H/L increases, except after L7, which may indicate that the grid
resolution is sufficient in the far outer region or the numerical accuracy has been lost there.
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a)
b)
c)
Figure 6: Near-field signatures for config1 (left column) and config2 (right column).
a) 0 degree azimuth, b) 45 degree azimuth, c) 90 degree azimuth
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a)
b)
Figure 7: Axial-symmetry measure 𝜼𝒊 for config1(left column) and config2 (right column).
a) full, b) zoomed signatures 4-10.
Model
config1
config2
L1
0.05211
0.05121
L2
0.03191
0.02927
L3
0.02315
0.02272
L4
0.01858
0.02021
𝐿∞ (𝜂𝑖 )
L5
L6
0.01582 0.01256
0.01591 0.01238
L7
0.01188
0.0117
L8
0.01169
0.0136
Table 1: The L-infinity norm of the point wise residual shown in Figure 7.
L9
0.00982
0.01313
L10
0.0122
0.0138
V. Signature Propagation and Loudness
The propagation algorithm used to extrapolate near field pressure signature is the Thomas waveform parameter
method8. This method completely describes the signal with three parameters and a system of ordinary differential
equations describing each parameter’s time rate of change is solved by finite differencing. In comparison to the Ffunction method, no area-balancing rule is needed for locating shocks. For a brief overview of the method, see15,
pages 578-581. Here Alauzet and Loseille observe how the chosen near-field sampling distance affects the
propagated signature shape for three supersonic business jet (SSBJ) configurations. They found that a minimum
pressure sampling distance of at least H/L=10 body lengths was necessary to give a converged ground sonic boom
signature.
In addition to signal extrapolation over the far field domain, NFBOOM has built in subroutines to calculate three
types of loudness metrics, AdB, CdB, and PLdB, in decibels associated with the extrapolated signature. Each
loudness metric has its own method of weighting the frequency spectrum in order to better represent the
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psychoacoustic perception of human hearing. Some advantages and the comparison of these metrics are given in11, 9,
. We choose the PLdB metric as a measure of sound-level prediction16, 17. This metric takes into account the entire
signature and is calculated using Stevens’ Mark VII equal loudness procedure, as described below.
10
For any given pressure versus time signature 𝑝(𝑡) of the ground sonic boom, we would like to associate an
appropriate scalar loudness value. To do this the signature must be transformed into a sound pressure level versus
frequency spectrum in one-third-octave bands. As outlined in10, by taking the Fourier transform of 𝑝(𝑡)
𝑇2
𝐹(𝜔) = � 𝑝(𝑡)𝑒 −𝑖𝜔𝑡 𝑑𝑡
(2)
𝑇1
we effectively acquire the energy spectral density versus frequency. The energy in each one-third-octave band is
then given by
𝜔2
𝐸(𝜔𝑖 ) = � |𝐹(𝜔)|2 𝑑𝜔
(3)
𝜔1
where 𝜔1 and 𝜔2 are the lower and upper band frequencies of the 𝑖 𝑡ℎ band (units 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 2 × 𝑇𝑖𝑚𝑒). The band
energy levels 𝐸(𝜔𝑖 ) are then converted into band sound pressure levels by
𝐸(𝜔𝑖 )
𝑆𝑃𝐿(𝜔𝑖 ) = 10 log10 �
(4)
2 � − 3 𝑑𝐵
𝑡𝑐 ∙ 𝑃𝑟𝑒𝑓
where 𝑡𝑐 ≈ .07 𝑠𝑒𝑐 is the approximate critical time of the auditory system (the time necessary to evoke a full
auditory response) , and 𝑃𝑟𝑒𝑓 ≈ 20 𝜇𝑃𝑎 is the pressure threshold of human hearing. Notice that the band sound
pressure levels 𝑆𝑃𝐿(𝜔𝑖 ) in equation (4) are reduced by 3 decibels. The 3-decibel reduction is a correction for the
energy bands being calculated over the entire pressure signature, which typically will contain two distinct sounds
separated by a time larger than the critical time 𝑡𝑐 . This creates a doubling in power which corresponds to a 3dB
increase.
To calculate the perceived loudness level PLdB, each sound pressure level 𝑆𝑃𝐿(𝜔𝑖 ) is assigned a loudness index
𝑆𝑖 in sones. The sone indexes 𝑆𝑖 are determined by experimentally verified equal loudness contours, as a function of
the band center frequency and SPL. The specification of the 𝑆𝑖 ’s incorporates our perspective weighting over the
frequency spectrum. A total perceived level 𝑆𝑡 in sones is then calculated by summing all the band loudness indices
𝑆𝑖 together with a summation formula given by
𝑆𝑡 = 𝑆𝑚 + 𝐹�(∑𝑁
𝑖=1 𝑆𝑖 ) − 𝑆𝑚 � sones
(5)
where 𝑆𝑚 is the largest loudness index of all the bands, N is the number of bands, and 𝐹 < 1 is a weighting factor
determined by 𝑆𝑚 .
This procedure has two scaling mechanisms. By following the equal loudness contours to assign each band
loudness index 𝑆𝑖 , the first scaling determines the effect each frequency has on the human auditory system. Then by
multiplying all indices less than the largest index, 𝑖 ≠ 𝑚, by the factor 𝐹 < 1, the second scaling incorporates the
masking effect the largest band SPL, 𝑖 = 𝑚, has on the total perceived level in sones. The final perceived level 𝑆𝑡 in
sones could then be converted to phons by
(6)
𝑃𝑡 = 40 + 33.3 log10 (𝑆𝑡 ) phons
or decibels
𝑃𝐿 = 32 + 9 log 2 (𝑆𝑡 ) PLdB
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(7)
A. Model4 Example
The near-field signals of the Model4 configurations shown in Figure 6 are propagated through a U.S. standard
atmosphere to produce the pressure distribution on the ground, see Figure 8. It is of particular interest that the
sampling distance H/L does not have much effect on the propagated signatures shape, with the exception of H/L=1
in config2. This may indicate nonconvergence since one body length away may be of significant departure from
axial-symmetry, this effect becomes more pronounced when the area changes more rapidly, as is the case of config2.
The associated PLdB measurements are shown in Table 2 and Table 3 in which config2 with a sharper increase in
area (volume) than config1 gives a lower PLdB by up to 3dB; this result at first glance seems counter intuitive.
Config2 gives rise to a milder expansion and two consecutive shocks, instead of a strong one as in config1; also the
two front shocks are close to each other with the second one being much weaker than that in config1.
a)
b)
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c)
Figure 8: config1 (left column) and config2 (right column) propagation through NFBOOM with U.S. standard
atmosphere (no wind), a) Signatures from 0 degree azimuth, b) 45 degree azimuth, c) 90 degree azimuth.
Azimuth
angle
0
45
90
Azimuth
angle
0
45
90
L1
L2
103.95
104.32
104.69
104.08
104.43
104.69
L1
L2
101.01
101.15
101.59
101.03
101.15
101.68
PLdB (NFBOOM) U.S. Standard Atmosphere
L3
L4
L5
L6
L7
104.07
104.44
104.72
104.05
104.46
104.68
104.17
104.47
104.66
104.17
104.43
104.72
104.13
104.48
104.60
Table 2: NFBOOM decibel predictions for config1
PLdB (NFBOOM) U.S. Standard Atmosphere
L3
L4
L5
L6
L7
101.08
101.09
101.68
101.06
101.24
101.65
100.97
101.25
101.56
101.06
101.32
101.58
101.04
101.39
101.41
Table 3: NFBOOM decibel predictions for config2
L8
L9
L10
104.14
104.50
104.82
104.16
104.56
104.91
103.95
104.62
104.75
L8
L9
L10
101.05
101.43
101.40
101.19
101.46
101.48
101.21
101.45
101.49
B. F-15-Active Example
This next example simulates NASA’s experimental F-15-Active military jet. The F-15-Active has been used as
a test bed for the Lift and Nozzle Change Effects on Tail Shock (LaNCETS) project 3,4. It has been modified with
adjustable forward mounted canards, and multi-axis thrust vectoring nozzles. By changing canard bias, horizontal
stabilizer bias, nozzle area ratio, and thrust vectoring, the resulting in-flight shock structure is altered and measured
by a probing aircraft.
We have chosen one of the first test flights, flight number 228, of the LaNCETS project for simulation, with
Mach number 1.4, angle of attack 1.587 degrees, and at an altitude of 40,000 feet. Static power boundary conditions
are specified at fan face and nozzle throat locations using engine specific operating conditions for the given flight.
The first test, denoted test1, adapts up to an H/L=5 and the second, denoted test2, up to H/L=10, with a mesh cell
aspect ratio of 4 and grid rotation of 48.58° (Mach angle + 3° for sonic glitch), see Figure 9. The computational
domain contains the full body geometry with domain boundaries of H/L=15 body lengths away from the body in all
directions except in the direction of propagation which is H/L=20 away. The adjoint functional is a weighted sum of
the line sensor, a point sensor located 5 feet behind the aircraft in the plum, and the axial force coefficient 𝐶𝐴 in the
body frame.
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The final mesh contains 86.3 million hexes after 14 adaptation cycles, for test1, and 87.58 million hexes after 15
adaptation cycles, for test2. Since practically the same amount of hexes was used in both tests, while test2 adapts
twice as far away compared to test1, it is expected that test1 will have better resolution of the near-field signatures.
These jobs were preformed on NASA’s Columbia supercomputer, and with 64 cores test2 a complete case took
approximately 29 hours. Pressure contours are shown in Figure 10a and Mach contours in Figure 10b and 10c for
test1. Cut planes are chosen through the center line of the body and center line of one of the nozzles.
The pressure sensor located at H/L=1, denoted L1, from test1 is compared with the experimentally captured
flight data from flight 228 signature 3, formally denoted as f2s3c, in Figure 11. As shown in Figure 11a, the
computational data extracted at H/L=1 and the flight path of the data-collecting aircraft is given by the dashed line,
clearly suggesting that this discrepancy must be taken into account. Figure 11b highlights that the extracted
computational signature along line sensor L1 does capture the overall shape of the experimental signature. Two
major discrepancies are seen. The first discrepancy is in the magnitudes of the sharp peaks which could be due to
numerical dissipation of the grid quality and Euler equation solver. The second discrepancy is the shifted shock
location of the second major shock due to the front of the canards. It is assumed this is due to the difference of the
actual flight path of the sensing aircraft in flight and the ideal line sensor L1 in computation. Also included in the
figure is the interpolated CFD result from test1 onto the actual flight path of the sensing aircraft. The interpolated
signal now is in better agreement with the actual flight data.
The near-field and propagated far-field pressure signatures for test1 are shown in Figure 12. Both the uniform
and U.S. standard atmosphere models are used for propagation. The far-field signatures seem to be converging as
the radial sampling distance increases. The corresponding PLdB measurements are shown in Table 4. At most only
a 1 dB variation is obtained across the sampling locations, H/L=1 to 5, even though the near-field distributions are
vastly different. More striking is the extent of attenuation of acoustic waves by the standard atmosphere, a whopping
decrease of nearly 10 dB.
Near-field pressure signatures for adaptation cycles 11 to 14 are shown in Figure 13 for test2. Adaptation from
cycle 11 to 12 is performed for grid propagation, while the remaining two are grid refinement cycles. In the grid
refinement cycles the majority of cells are usually refined at the geometry surface and therefore the off-body
pressure signatures do not change much. We have found that a combination of using a large amount of grid
refinement levels in the initial mesh and then designating many propagation cycles works efficiently. The signatures
from all four adaptation cycles are propagated through the uniform and U.S. standard atmospheres, see Figure 14.
The associated PLdB measurements are shown in Table 5. Again, the variation in PLdB is practically negligible.
This indicates that finer grids do not necessarily make any significant difference in terms of loudness predictions.
This then further implies that coarser grid solution may be sufficient, and indeed this is a significant relief on
computational resources from the design optimization perspective.
a)
b)
Figure 9: F-15 Active (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and throat) a) test1 final mesh
with 86.3 million hexes, b) test2 final mesh with 87.58 million hexes.
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a)
b)
c)
Figure 10: F-15 test1 with 86.3 million hexes (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and
throat) a) Zoomed view pressure, c) Zoomed view Mach body-center-line, d) Zoomed view Mach nozzle-center-line.
a)
b)
Figure 11: F-15 Active test1 with 86.3 million hexes (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and
throat). a) Signature L1 and actual flight path positioning, b) –- Signature L1 from adapt14 at 0 degrees azimuth, –experimentally sampled flight data near-field pressure signal, and –- CFD interpolated valued to experimental flight path.
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a)
b)
c)
Figure 12: F-15 Active test1 with 86292857 hexes (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and
nozzle throat) a) Signatures from adapt14 at 0 degree azimuth, b) propagation through NFBOOM with uniform pressure
atmosphere, c) NFBOOM with U.S. standard atmosphere (no wind).
Atmosphere
Uniform
Standard
PLdB (NFBOOM)
L1
L2
L3
108.80 109.37 109.56
99.04
99.62
99.78
L4
109.65
99.91
L5
109.62
99.95
Table 4: F-15 Active test1 NFBOOM decibel predictions.
a)
b)
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c)
d)
Figure 13: F-15 test2 (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and throat) Near-field Signatures
at 0 degree azimuth. a) adapt11 with 20.636 million hexes, b) adapt12 with 29.084 million hexes, c) adapt13 with 40.383
million hexes, d) adapt14 with 55.286 million hexes.
a)
b)
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c)
d)
Figure 14: F-15 test2 (Mach=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and throat) (Left column is
propagation through NFBOOM with uniform pressure atmosphere, and right column is propagation through NFBOOM
with U.S. standard atmosphere (no wind). a) 20.636 million hexes, b) 29.084 million hexes, c) 40.383 million hexes, d)
adapt14 with 55.286 million hexes.
Adapt Cycle
11
12
13
14
L1
99.64
99.95
99.75
99.63
PLdB (NFBOOM) U.S. Standard Atmosphere
L2
L3
L4
L5
L6
L7
99.75
99.84
99.89
99.86
99.93
99.93
100.23 100.17 100.22 100.21 100.28 100.32
100.14 100.18 100.18 100.20 100.24 100.29
100.14 100.17 100.09 100.13 100.17 100.21
L8
99.93
100.31
100.29
100.22
L9
100.00
100.32
100.23
100.23
L10
99.80
100.18
100.18
100.11
Table 5: F-15 test2 (M=1.4, aoa=1.587, Altitude=40,000 ft, PowerBC’s at fan face and throat) PLdB comparison chart.
VI. PLdB Sensitivity
Based on the loudness prediction shown in the previous section, we are baffled by its insensitivity to the nearfield pressure distribution. Hence, a more in-depth investigation into the cause of this insensitivity should be
pursued. In this section we define generalized shaped sonic boom signatures to investigate the PLdB metrics
sensitivity and determine signature properties that have the greatest influence on perceived loudness. The most
common sonic boom signature shape is the N-wave, a wave characterized by an initial positive jump in overpressure
followed by steady expansion to negative overpressure and then a jump back to the ambient pressure. For our first
test, we choose a two-parameter N-wave signature where the span from peak to peak is fixed at 0.25 seconds long
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and the maximum overpressure, dPmax, varies from 0.001 psf to 10.0 psf, see Figure 15a. The corresponding PLdB
as shown in Figure 15b increases with the maximum overpressure dPmax, following a strict logarithmic
relationship.
a)
b)
Figure 15: a) Family of N-waves 0.25 seconds long with varying dPmax=dPmin. b) Loudness vs. dPmax for the family of
N-waves.
Next, four different general signature shapes are tested by appending an initial shock structure 0.05 seconds long
to the front of a constant N-wave with maximum overpressure equal to 1.0 psf and duration 0.2 seconds. Each
initial disturbance is characterized by a specific shape, either a single peak, ramp, flat top, or double peaks; they are
designed to be completely determined by the initial shock strength, which is varied from 0.0 psf to 1.5 psf. These
families of signatures are shown in Figure 16, with the red arrow indicating the direction of the variable signature
shape. For the single and double peaks initial shock structure family of signatures, signals 1 and 4, all expansions
are equal in slope to that of the following N-wave.
The initial shock overpressure versus loudness in PLdB for each signature family is shown in Figure 17. From
this plot, we can draw several conclusions. First, changes in overpressure across all shock locations sum up together
to give the final PLdB reading. Second, signatures with the largest change in overpressure at any given location
have higher PLdB readings. Signals 1, 3, and 4 all experience their lowest PLdB readings when the change in
overpressure at time 0.05 seconds and 0.10 seconds are equal. Third, shock rise time has a significant impact on the
perceived loudness. Fourth, signal 2 has the lowest PLdB readings and experiences its minimum when the initial
shock structure is reduced to a constant overpressure rise, effectively increasing the initial shock rise time. Signal 2
is the only one exhibiting a monotonic relationship with the initial shock strength (dP), while the other three find its
own minimum loudness, in fact producing high loudness as the initial dP diminishes, reducing to a single front
shock. Thus, we conclude from this family that it is beneficial to have a rise time in the front pressure distribution.
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a)
b)
c)
d)
Figure 16: Four signals with varying initial shock dP followed by constant expansion.
a) signal1 (single peak), b) signal2 (slope-top), c) signal3 (flat-top), d) signal4 (double peak).
Figure 17: Four signals with varying initial shock dP followed by constant expansion. Initial Shock dP versus Loudness
PLdB.
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In order to test the effect of the rise time of the initial shock on PLdB, a 5th member of the previous family of
signatures is constructed, see Figure 18a. For this signature, the initial shock overpressure is a constant flat top at
1.0 psf. We vary the rise time of the initial shock from 0.00001 seconds to 0.05 seconds. The loudness versus initial
shock rise time is shown in Figure 18b. This plot confirms our suspicion that a more gradual increase in
overpressure, that is an increase in rise time, decreases the perceived loudness.
a)
b)
Figure 18: a) Signal5 flat-top with varying rise time followed by constant expansion, b) Initial Shock rise time versus
Loudness PLdB.
Next, an N-wave of 0.25 seconds wide and with maximum overpressure 1.0 psf is used to test how PLdB varies
as the signature is symmetrically deformed. Two deformations of the N-wave are investigated as rise time on both
the initial and final shocks increase. Signal 6, see Figure 19a, keeps the maximum overpressure constant as rise time
increases, which gradually increases the slope of expansion and eventually evolves into a single shock. In Signal 7,
see Figure 19b, the maximum overpressure decreases with increasing rise time, which keeps the slope of expansion
constant and eventually evolves into a flat-line signal of 0 dP.
a)
b)
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c)
Figure 19: a) Signal 6 N-wave with equally varying rise time on both initial and final shocks, b) Signal 7 N-wave with
equally varying rise time and maximum and minimum overpressure on both initial and final shocks, c) Initial and Final
Shock rise time versus Loudness PLdB.
VII. Conclusion
In conclusion, we have attempted to investigate the procedures that are involved and commonly used for
predicting sonic booms. First, we split the computational domain into two regions, in which the near-field is solved
by Cart3D and the far-field solved with NFBOOM or PCBOOM. The issue concerning the need for attaining axial
symmetry before extracting the near-field pressure signature for the subsequent wave propagation has been
addressed. We also examined the effect (sensitivity) of sensor distance away from the body on the ground pressure
signature and perceived loudness. Our result suggested that the calculated perceived loudness in PLdB was rather
insensitive to the condition of axial-symmetry and the sensor distance. This in turn implies that an increased
precision by obtaining higher accuracy or fidelity of the off-body signatures might not give rise to a significant
difference in ground PLdB.
Finally, a simplified ground signature modeling was initiated in order to gain a further insight into the most
effective parameters in trimming the PLdB. By examining the perceived loudness PLdB sensitivity to changes in
signature shape, it has been determined that the largest change in pressure amongst all the shocks and the shock rise
time are the dominant contributors to the loudness metric.
Acknowledgements
We thank Michael J. Aftosmis, Marian Nemec, and Mathias Wintzer of NASA Ames for their help in the proper
use of Cart3D and the grid adaptation module. We also thank Sudheer Nayani of NASA Langley and Edward A.
Haering Jr. of NASA Edwards for providing us with the F-15-Active geometry and experimental flight data. Donald
A. Durston of NASA Ames provided NFBOOM source code.
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