12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

TURBULENCE STRUCTURES IN AN AXISYMMETRIC
ADVERSE-PRESSURE-GRADIENT BOUNDARY LAYER AND THEIRROTOR AEROACOUSTIC RESPONSE

Di Zhou, Kan Wang and Meng Wang
Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN 46556, USAdzhou@nd.edu, kwang2@nd.edu, m.wang@nd.edu

ABSTRACT

ing the axisymmetric TBL of a BOR (Hickling et al., 2019;
Hickling, 2020; Balantrapu et al., 2021). The evolution of thetail-cone boundary layer was examined through measurementsof single-point velocity statistics and two-point velocity correlations at the rotor inlet plane, and their unsteady pressure
footprint on the tail-cone surface was also measured. Acoustic measurements, conducted at different rotor advance ratios,
showed broadband noise and haystacking peaks at frequencies
7–12% higher than the BPF and its harmonics. In the present
work, the BOR boundary layer and its interaction with the rotor as in the VT experiment are simulated numerically, and theresults are validated by the experimental data. To gain a detailed understanding of the acoustic source mechanisms, a systematic analysis is conducted on the evolution of velocity twopoint and spatiotemporal correlations in the tail-cone TBL,
acoustic dipole source distributions on rotor blades, and bladeto-blade correlations and coherence of the dipole sources.

The flow over an axisymmetric body of revolution (BOR)
is investigated numerically at Reynolds number of 1.9 × 106
based on the BOR length, with a focus on the decelerating
turbulent boundary layer in the tail-cone section and its acoustic radiation when ingested by a rotor. The boundary layeron the nose and midsection of the BOR is computed using
wall-modeled LES whereas that in the acoustically important
tail-cone section is wall resolved. The radiated acoustic fieldis calculated based on the Ffowcs Williams-Hawkings equation. The computed flow statistics and sound pressure spectra agree well with the experimental measurements at Virginia
Tech. Correlation and spectral analyses demonstrate rapidly
growing turbulence structures in the tail-cone boundary layer,
whose interaction with successive rotor blades generates coherent unsteady-loading dipole sources on neighboring blades
and thereby haystacking peaks in the broadband sound pressure spectra. The spatial and frequency characteristics of blade
acoustic dipole sources and their correlations are examined.

COMPUTATIONAL APPROACH
The BOR geometry, shown in figure 1a, consists of acylindrical midsection of diameter D and equal length, an ellipsoidal nose of length D, and a tail cone with a 20◦ half apexangle that is connected to a thin cylindrical support pole ofdiameter 0.15D. The tail-cone angle was selected to achieve
the thickest possible boundary layer at the tail end without flow
separation. The Reynolds number based on the free-stream velocity U∞ and the length of the body, L = 3.17D, is 1.9 × 106 .
Transition to turbulence is triggered by a circumferential trip
ring located at the end of the nose. Large-eddy simulation(LES) is conducted under the VT experimental conditions withand without a five-bladed rotor installed at the tail-cone end of
the BOR. The rotor, whose geometry is shown in figure 1b,
has a diameter of 0.5D and is at the advance ratio of J = 1.44,
which nominally corresponds to zero thrust.
A finite volume, unstructured-mesh LES code for incompressible flow developed at Stanford University (You et al.,2008) was adapted and enhanced to carry out the flow simulations. It employs energy-conserving and low-dissipative
spatial discretization, a fully implicit fractional-step method
for time advancement, the Generalized Product Bi-Conjugate
Gradient with safety convergence (GPBiCGSafe) method forthe pressure Poisson equation, and the dynamic subgrid-scalemodel. The BOR-rotor flow is simulated in the rotor frame ofreference using the absolute velocity formulation. To reducethe computational cost, wall modeling is employed in the noseand midsection regions of the BOR while the acoustically im-

INTRODUCTION
The noise produced by a rotor ingesting a turbulent
boundary layer (TBL) and/or wake is of considerable interest in many aeronautical, marine, automotive and wind-energy
applications. The noise level and spectral characteristics arestrongly dependent on the turbulent inflow, which is in general spatially inhomogeneous and often subjected to complex
pressure-gradient and surface-curvature effects. Accurate prediction and modeling of rotor turbulence-ingestion noise require knowledge of not only low-order statistics but also structures of the inflow turbulence.
This study is aimed at numerically predicting the noise ofrotor interaction with a thick turbulent boundary layer at therear of an axisymmetric body of revolution (BOR) and elucidating the acoustic source mechanisms. The BOR is at zeroangle of attack and thus the TBL is also axisymmetric statistically. Of particular interest is the generation of spectral humps,
known as haystacking peaks, in the broadband sound pressure
spectra near the blade passing frequency (BPF) and its harmonics, and their relation to the turbulence structures in theboundary layer. The BOR-rotor configuration is of practicalsignificance as it captures the key physics of rotor interaction
with fuselage (hull) boundary layers on flight (marine) vehicles.
Researcher at Virginia Tech (VT) recently conducted experiments to study the sound from a five-bladed rotor ingest-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

axial locations as well as the magnitude and frequency spectraof pressure fluctuations on the tail-cone surface.
The evolution of turbulence structures in the tail-cone
TBL is investigated through spectral and correlation analyses. Figure 3 depicts the two-point correlations of the fluctuating axial velocity in a center plane through the BOR axis
for seven anchor positions. These anchor points are along amean streamline that passes through (x/D, r/D) = (2.17, 0.18),
which is in the vicinity of peak turbulence intensity in the rotor
inlet plane. It can be seen that the spatial scales of flow structures increase rapidly in the downstream direction with growing TBL thickness. The corresponding correlations from the
LES without a rotor (not shown) are virtually the same except
at the last station, where the effect of the rotor becomes noticeable. The two-point correlations of the fluctuating pressure
on the tail-cone surface have also been examined, and they exhibit even more drastic increases in length scales in the downstream direction (Zhou et al., 2020). Two-point correlations of
the fluctuating axial velocity in the y-z plane at the tail-coneend with and without the rotor are compared in figure 4. The
correlation lengths in both radial and azimuthal directions are
increased by the rotor, although the growth of the azimuthal
correlation length is small. Two negative correlation regions
are observed when the rotor is present.
Figure 5 shows the radial spatial-temporal correlations of
the fluctuating axial velocity at the tail-cone end with the anchor position at (x/D, r/D) = (2.17, 0.18). Results from simulations with and without the rotor are both illustrated. The
correlation contours for the flow undisturbed by a rotor show
a typical convective pattern with a small negative convection
velocity in the radial direction. In the presence of the rotor,
multiple strips of relatively low-correlation levels appear due
to fluid motions induced by the rotating blades. The main correlation contours are distorted and show interaction of coherent structure with successive blades. As will be discussed in
the next section, such interaction leads to blade-to-blade correlations and coherence between the unsteady-loading dipole
sources, and thereby gives rise to haystacking in the soundpressure spectra.

portant tail-cone section is wall-resolved.
Simulations are conducted in a cylindrical domain withthe same blockage ratio as in the VT wind tunnel. It is 11.2Dlong and 2.39D in radius. Structured-mesh blocks are usedaround the entire BOR, rotor blades, the rotor hub and the support pole, and unstructured mesh blocks are employed elsewhere. Approximately 5 × 108 mesh cells are used for the
BOR block. The wall-resolved tail-cone section has 1200
streamwise cells and 1600 azimuthal cells, both uniformly
spaced, and 130 cells across the thickness of the boundary
layer at the end of the cone. Grid convergence for flow overthe BOR has been established through simulations without therotor on coarser and finer meshes (Zhou et al., 2020). In therotor block, two meshes with 5 × 107 and 1.2 × 108 cells are
employed to evaluate the grid convergence of the rotor acoustic response.
Noise calculations are based on the Ffowcs WilliamsHawkings (FW-H) equation (Ffowcs Williams and Hawkings,1969). At low Mach numbers sound generation is dominatedby the unsteady forces on rotor blades, and the sound pressure
at an acoustic far-field location x and time t can be computedfrom (Brentner and Farassat, 2003; Wang et al., 2021)
1 ∂
p(x,t) ≈4π c∞ ∂ tZ Srdi pi j n j
dS
rd2 |1 − Mr | τ ∗(1)where c∞ is the free-stream speed of sound, pi j is the compressive stress tensor dominated by pressure, n j are components of
the unit normal vector of the blade surface S pointing into thefluid, rd = |x − y(τ )| is the distance between the observer andthe source coordinates, rdi = xi − yi , and Mr = (rdi /rd )Mi is
the Mach number of the source in the radiation direction with
Mi being components of the source Mach number vector. The
integrand is evaluated at the retarded time τ ∗ , which is the rootof the equation τ = t − |x − y(τ )|/c∞ . Acoustic scattering by
the BOR, significant only at high frequencies and shallow observer angles relative to the flow direction, is ignored in thecalculation. Accuracy of this computational approach has been
demonstrated by Wang et al. (2021) for rotor noise generation
in a turbulent wake by comparison with experimental results.

ACOUSTICS AND SOURCE ANALYSIS
TAIL-CONE BOUNDARY LAYER

Figure 6 shows the sound pressure levels (SPLs) predictednumerically using equation (1) with surface pressure obtained
from the LES along with the integrated spectra measured by
Hickling et al. (2019) with a 64-channel microphone array.
The free-stream Mach number is 0.059, and the center of the
microphone array is at a distance of 3.86D from the rotor center and 106◦ counter-clockwise from the downstream direction. The computed spectra are obtained by averaging thespectra from all 64 microphone positions in the array. Twosets of LES data, from the coarse and fine meshes in the rotor
block, are presented. The SPLs are in dB per Hz with reference to 2 × 10−5 Pa, and the frequency is normalized by the
BPF.
It can be observed that the sound pressure spectra arebroadband with four distinct haystacking peaks. These peaks
occur at frequencies 8–12% higher than (i.e., right-shifted) the
BPF and its first three harmonics. The numerical predictions
from the two meshes compare well for frequencies up to tentimes the BPF, demonstrating grid convergence. They are also
in reasonable agreement with the experimental spectrum overthe frequency range of the experimental data. In particular,
the simulations capture the haystacking peaks and their frequencies, although the peak levels are underpredicted by 2–

Figure 1a shows an instantaneous flow field in terms of
isocontours of the axial velocity in a plane through the BOR
axis. The flow accelerates around the nose and on approachto the sharp corner between the midsection and the tail cone.
Downstream of the corner, the flow decelerates under the influence of strong adverse pressure gradient (APG), and theboundary layer thickens rapidly. A comparison of the computed velocity statistics with the PIV measurements at VT(Hickling, 2020; Balantrapu, 2021) is shown in figure 2 for
x/D = 2.17, which is at the end of the BOR tail-cone. Thislocation is only 0.055D upstream of the rotor leading edge andthus considered the rotor inlet plane in the following analysis.
The agreement between the LES and experimental profiles isreasonable for both the mean axial velocity and the root-meansquare (r.m.s.) values of axial and azimuthal velocity fluctuations. Although there are small discrepancies in the velocity fluctuations, the overall distributions and locations of peakfluctuations agree well with the experiment. More extensive
validation against experimental measurements has been conducted for BOR flow without a rotor under otherwise the same
conditions (Zhou et al., 2020), and good agreements have beenobtained for velocity statistics and energy spectra at various

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

blade-to-blade correlations of the sectional dipole sources (notshown) exhibit the same trend as the sectional forces but thecorrelation levels are lower since the time derivative places
more weights on smaller scales (Wang et al., 2021). Nonetheless, they are sufficient to generate haystacking in the radiatedsound pressure spectra.
The statistical dependence among blades in the frequency
domain can be determined by examining the coherence of
blade sectional dipole sources, which is the same as the coherence of sectional forces for stationary turbulence (Yang andWang, 2013). Figure 9 shows the coherence between the sectional dipole source at r/D = 0.18 on a blade and those on afront neighbor, itself, and two back neighbors at varying radial
positions. On the same blade (figures 9b), the coherence length
decreases with frequency and becomes very small at high frequencies. Regions of pronounced coherence are found on thetwo neighboring blades in the forward (figure 9a) and backward (figure 9c) directions around the frequencies of haystacking peaks and valleys between these peaks in the sound pressure spectra (cf. figure 6). As the frequency increases, the coherence decays gradually and becomes insignificant beyond
3.5 times the BPF. The value of the coherence with the second
neighboring blade at the back (figure 9d) is smaller than 0.2 at
low frequencies and practically negligible at higher frequencies. As in the correlation figure, the region of large coherence shifts downward on succeeding blades as a result of thedownward convection of the turbulence structures. The significant source coherence between neighboring blades leads
to strong interference among radiated acoustic waves at these
discrete frequencies. At the haystacking frequencies, the interference is constructive, resulting in the peaks in the rotor soundpressure spectra. At the frequencies between two neighboring
haystacking peaks, destructive interference causes the valleys
in the spectra. Additionally, figure 9 indicates that the same coherent structure is cut through by only two successive blades,
which is consistent with figure 5a and apparently sufficient togenerate the haystacking effect.

4 dB. The broadband component of the sound is predicted accurately.
To reveal the acoustic source characteristics and themechanisms for generating the haystacking phenomenon, it
isilluminating to examine the unsteady sectional force, fi =R
pi j n j dΓ where Γ is along the blade circumference at given
radial position r, and its time derivative ∂ fi /∂ t. The latter isrepresentative of the sectional acoustic dipoles associated withrotor blades. If the blade chord is acoustically compact, whichis the case at low and intermediate frequencie, equation (1) canbe rewritten asZ
1 ∂ Nb Rtip h rdi
fi inp(x,t) ≈dr(2)∑
4π c∞ ∂ t n=1 Rhub rd2 |1 − Mr | τ ∗where Nb is the number of blades, and Rhub and Rtip are theradii of the blade hub and tip, respectively. Sectional force anddipole properties are particularly useful for theoretical modeling of rotor noise (e.g., Glegg et al., 2015).
Figure 7 shows the radial distributions of the frequency
spectra of the unsteady sectional force and dipole source atfive discrete frequencies corresponding to the four haystacking peaks and a higher frequency. Only the component perpendicular to the chord, denoted by L, is considered since it
is significantly larger in magnitude than the chordwise component. The level of the sectional-force spectra decreases withincreasing frequency and grows in the radial direction initially
until the blade outer region. The location of the spectral peakdepends on the frequency, but in terms of the overall magnitude, i.e., the r.m.s. of L, the maximum value is reached atr/D ≈ 0.18, which coincides with the location of the maximum chord length and turbulence intensity. For the spectra of
the sectional dipole, DL = ∂ L/∂ t, the radial variations are similar to those of the sectional-force spectra but their frequency
variations are much weaker and non-monotonic. Among thefive frequencies, the second haystacking peak at f / fBP ≈ 2.2
has the highest dipole spectral level overall. It should be mentioned that the weak oscillations in the spectral curves seen in
the figure are caused by limited statistical sampling period, andthey do not affect the above conclusions.
Figure 8 shows the correlations of the unsteady sectional(m,n)
forces, CLL (r, ∆r, ∆t), between one blade (blade m) at the location of maximum chord length, r/D = 0.18, and those on
several blades (blade n) including itself at varying radial positions. Correlations are strong with the neighboring blade at
the front (figure 8a), the same blade (figure 8b), and the neighboring blade at the back (figure 8c). Correlations with the second neighboring blade at the back (figure 8d) are significantly
weaker. The location of maximum correlation moves downward from above the anchor point on the front neighbor to theanchor point on the same blade, and then to below the anchor
point on the back neighbor. This downward shift is related tothe downward convection of the coherent turbulence structuresin the tail-cone boundary layer, as can be seen in the two-point
velocity correlations in figure 3. Blade-to-blade correlations of
the sectional forces are caused by successive blades interactingwith the same coherent structure (cf. figure 5a). As the structure is convected downstream and through the rotor, succeeding blades cut through the same structure at lower positions,leading to the downward shift of the maximum correlation location on the blades. It can be noticed that the temporal separations corresponding to maximum correlations are slightly lessthan the corresponding blade-passage times. This is related to
the right-shift of haystacking frequencies due to the diagonal
track taken by the rotating blades through the convecting turbulence structures (Murray et al., 2018; Martinez,1996). The

CONCLUSIONS
The development of an axisymmetric turbulent boundary layer in the tail-cone section of a BOR and its interaction with a five-bladed rotor have been investigated numerically to predict the rotor turbulence-ingestion noise and elucidate the noise source mechanisms. The results presented inthis paper demonstrate accurate prediction of the boundarylayer turbulence structures by LES and, in conjunction with
the FW-H equation, accurate prediction of the radiated acoustic field under experimental conditions. The two-point correlations of velocity fluctuations show significant growth of turbulence length scales in the downstream direction under strong
APG. The computed sound pressure spectra are broadband
with haystacks caused by successive blades cutting throughthe same turbulence structure. Acoustic sources are analyzed
in terms of the blade sectional unsteady-loading dipole, whosestrength increases with the local chord length, inflow turbulence intensity and rotational speed, and is largest in the outerregion near the location of maximum chord length. As anticipated from the velocity two-point and space-time correlations,large correlations are found between sectional dipole sourceson neighboring blades, and the corresponding blade-to-blade
coherence is significant around the frequencies of haystacking
peaks and valleys. The acoustic interference among correlated
blade dipoles gives rise to the haystacks in the rotor noise spectra.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

ACKNOWLEDGEMENTS

Hickling, C. 2020 Inhomogeneous, anisotropic turbulence ingestion noise in two open rotor configurations. PhD
Thesis, Virginia Tech, Blacksburg, VA.
Hickling, C., Balantrapu, N. A., Alexander, W. N. A.,
Millican, J., Devenport, W. J. & Glegg, S. A. 2019 Turbulence ingestion into a rotor at the rear of an axisymmetric body.
AIAA Paper 2019-2571, 25th AIAA/CEAS Aeroacoustics Conference, 20–23 May, 2019, Delft, The Netherlands.
Martinez, R. 1996 Asymptotic theory of broadband rotor
thrust, Part II: Analysis of the right frequency shift of the maximum response. Journal of Applied Mechanics 63, 143–148.
Murray, H. H., Devenport, W. J., Alexander, W. N.,
Glegg, S. A. L. & Wisda, D. 2018 Aeroacoustics of a rotor
ingesting a planar boundary layer at high thrust. Journal of
Fluid Mechanics 850, 212–245.
Wang, J., Wang, K. & Wang, M. 2021 Computational prediction and analysis of rotor noise generation in a turbulent
wake. Journal of Fluid Mechanics 908, A19.
Yang, Q. & Wang, M. 2013 Boundary-layer noise induced
by arrays of roughness elements. Journal of Fluid Mechanics
727, 282–317.
You, D., Ham, F. & Moin, P. 2008 Discrete conservation
principles in large-eddy simulation with application to separation control over an airfoil. Physics of Fluids 20, 101515.
Zhou, D., Wang, K. & Wang, M. 2020 Large-eddy simulation of an axisymmetric boundary layer on a body of revolution. AIAA Paper 2020-2989, AIAA Aviation Forum,15–19
June, 2020.

This work was supported by the U.S. Office of Naval
Research under Grants N00014-17-1-2493 and N00014-20-12688, with Drs. Ki-Han Kim and Yin Lu Young as Program
Officers. Computer time was provided by the U.S. Department of Defense High Performance Computing Modernization Program. The authors acknowledge Drs. William Devenport, Nathan Alexander, N. Agastya Balantrapu and Christopher Hickling for providing experimental data.

REFERENCES
Balantrapu, N. A. 2021 The space-time structure of an axisymmetric turbulent boundary layer ingested by a rotor. PhD
Thesis, Virginia Tech, Blacksburg, VA.
Balantrapu, N. A., Hickling, C., Alexander, W. N. & Devenport, W. 2021 The structure of a highly decelerated axisymmetric turbulent boundary layer. Journal of Fluid Mechanics
929, A9.
Brentner, K. S. & Farassat, F. 2003 Modeling aerodynamically generated sound of helicopter rotors. Progress in
Aerospace Sciences 39, 83–120.
Ffowcs Williams, J. & Hawkings, D. 1969 Sound generation by turbulence and surfaces in arbitrary motion. Philosophical Transactions of the Royal Society 264, 321–342.
Glegg, S. A. L., Devenport, W. J. & Alexander, W. N.
2015 Broadband rotor noise predictions using a time domain
approach. Journal of Sound and Vibration 335, 115–124.

(b)

(a)

Figure 1: (a) Instantaneous axial velocity ux /U∞ in the x-y plane through the BOR axis; (b) rotor geometry.

0.3

0.3

0.25

0.25

0.25

0.2

0.2

0.2

r/D

r/D

LES
VT Exp.

0.15

0.15

0.15

0.1

0.1

0.1

0.05

0.05
0

0.2

0.4

0.6

0.8

1

0.05
0

Ux /U∞
(a)

r/D

0.3

0.02

0.04

0.06

u′x,rms /U∞

(b)

0.08

0

0.02

0.04

0.06

uθ′ ,rms /U∞

0.08

(c)

Figure 2: Velocity statistics at x/D = 2.17 compared with VT experimental data (Hickling, 2020; Balantrapu, 2021): (a)mean axial velocity; (b) axial turbulence intensity; (c) azimuthal turbulence intensity.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

Figure 3: Two-point correlations of axial velocity fluctuations at seven locations along a mean streamline in the tail-cone

TBL.

(a) With rotor(b) Without rotor

Figure 4: Two-point correlations of axial velocity fluctuations at x/D = 2.17, r/D = 0.18 in a cross-flow plane, with andwithout the rotor.

(a) With rotor(b) Without rotor

Figure 5: Radial space-time correlations of axial velocity fluctuations at x/D = 2.17, r/D = 0.18 with and without therotor.
40

SPL (dB/Hz)

30
20
10
LES
LES, coarse mesh
VT Exp.

0
-10

100

101

f / fBP
Figure 6: Computed sound pressure spectra compared with the microphone array measurement from the VT experiment(Hickling et al., 2019).
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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19–22, 2022

0.25

0.25
f/fBP≈1.1
f/fBP≈2.2
f/fBP≈3.3
f/fBP≈4.4
f/fBP=8

0.2

r/D

r/D

0.2

0.15

0.15

0.1

0.1
10

f/fBP≈1.1
f/fBP≈2.2
f/fBP≈3.3
f/fBP≈4.4
f/fBP=8

-17

10

-16

10

-15

10

-14

10

-13

10

-12

10

-11

10

10

-10

10

-9

10

-8

ΦDL DL D/(ρ∞2 U∞5 )ΦLL /(ρ∞2 U∞3 D)

(a)

-11

(b)

Figure 7: Radial distributions of the frequency spectra of (a) unsteady sectional force, and (b) sectional dipole source atfive discrete frequencies.

(a)

(b)

(c)

(d)
(m,n)

Figure 8: Space–time correlation coefficient of the unsteady sectional forces on blades m and n, CLL(r/D = 0.18, ∆r, ∆t):

(1,5)
(1,1)
(1,2)
(1,3)
(a) CLL ; (b) CLL ; (c) CLL ; (d) CLL .

(a)

(b)

(c)

(d)
2(m,n)

2(1,5)

Figure 9: Coherence between the sectional dipole sources on blades m and n, γDL DL (r/D = 0.18, ∆r, ∆t): (a) γDL DL ; (b)2(1,1)

2(1,2)

2(1,3)γDL DL ; (c) γDL DL ; (d) γDL DL .
6