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

LES ANALYSIS OF A SUPERSONIC AIR INLET
EXPERIENCING BUZZ PHENOMENON

Riwan Hammachi, Romain Gojon, Maxime Fiore, Jérémie Gressier and Stéphane Jamme∗
ISAE-SUPAERO, University of Toulouse, France∗ email: stephane.jamme@isae-supaero.fr

ABSTRACT

challenge today because of the complex unsteady flows thatneed to be captured in off-design conditions, with large effectsof turbulence, shocks and acoustic waves that interact witheach other. A possible path to study these flow configurations
is the use of high fidelity numerical simulation. The large-eddy
simulation (LES) approach can be used since it can cope withthe above-mentioned flow features at a high but nowadays affordable computational cost. The objective of the present work
is to analyze the flow in a supersonic air intake with different
back-pressure levels using LES. Its purpose is to characterize
the unsteady features of supersonic air inlets and the underlying mechanisms driving the flow when a back-pressure occurs
and triggers the buzz phenomenon.

This paper investigates the flow in a supersonic mixed
compression air intake using numerical simulation. The configuration has been experimentally tested for different backpressures imposed by a moving plug set at the exit of the internal intake (throttled) or without plug (unthrottled) for whichdata are available in the literature and to which the simulations
can be compared. The unthrottled and three different throttled
positions are investigated using large-eddy simulation (LES).
The purpose of the study is to analyze the shock structures andtheir interaction with the boundary layers developing on thesurfaces of the air intake for these different cases. Also, because the experiments reported potential unsteady positions of
the shock structures for large throttling ratios and, more generally, to analyze the unsteady phenomena, the spectral proper
orthogonal decomposition (SPOD) algorithm has been used.

CONFIGURATION/NUMERICAL SETUP
The flow configuration selected in the present work follows the experiments from Chen et al. (2018). The geometryof the air intake can be seen in Fig. 1. In the experiments, theupstream flow is characterized by a Mach number of M = 2.41
(corresponding to an overspeed mode for this air intake designed for a shock-on-lip Mach number of 2), a total pressureof p0 = 100, 900 Pa, and a total temperature of T0 = 294.1 K.
The Reynolds number, based on the upstream quantities andthe height h = 47.5 mm of the final straight portion of the air
inlet duct, equals 494, 640.
Several cases have been investigated in the experiments.
The cases differ in the back-pressure imposed at the exit inorder to mimic a blockage due to the presence of a downstream engine. This is achieved by placing a plug at the exitof the air inlet duct characterized by a half-top angle of 15.5◦ .
The plug reduces the mass flow and leads to an increase of
the back-pressure which modifies the operating point of the
air inlet. This procedure allows to investigate different flow
regimes of the air inlet, ranging from steady configurations forthe mean flow to fully unsteady behaviors featuring the buzz
phenomenon. In order to quantify the downstream blockage, athrottling ratio (TR) is defined as TR = 1−Ath,plug /Aexit , whereAth,plug is the throat area of the plug and Aexit is the crosssectional area of the duct exit without plug (see Fig. 1). In
the numerical simulations, in addition to the case without plug
(T R = 0), the latter has been set to different positions to reachthrottle ratios of T R = 0.35, 0.45 and 0.6, in order to evaluate
how the flow progressively deviates from a steady state. These
different cases will be respectively denoted TR0, TR35, TR45
and TR60.
In the present work, the Mach number is kept constant
between the different simulations and is the same as the oneused in the reference experiments. However, in order to keep areasonable computational time, the Reynolds number has been

INTRODUCTION
Aerobic propulsion systems require sufficient airflow
through the engine to operate at nominal conditions. The design of the air intake then plays a key role in the performanceof the propulsion system. This is particularly true in the supersonic flight regime where the incoming flow must be decelerated before entering into the core of the engine. The supersonicflight regime involves compressible phenomena: shock waves
and expansions from the ramp compression devices or fromthe inlet cowl lips. The latter can impact the boundary layers
developing on the opposite walls. These shock wave/boundary
layer interactions (SBLI), which vary according to the flight
conditions, have direct consequences on the performance andoperation of the supersonic air inlet (Chen et al. (2018)). The
strong adverse pressure gradient induced by a shock wave on
a boundary layer may indeed cause a separation of this lowspeed region and lead to the development of a separation bubble. The mass flow rate is then reduced, which is detrimental
to the propulsion system.
One major well-known problem of these configurations is
the supersonic inlet buzz (Oswatitsch (1980)) which can be agreat threat to air-breathing supersonic vehicles. Usually, it istriggered by an accidental downstream pressure- or thermaldriven flow blockage, which can throw the inlet into the undesirable subcritical mode featuring the expected terminal shock
standing upstream of the inlet entrance. Once the buzz occurs,
self-excited streamwise normal-shock oscillations are generated along with periodic duct pressure fluctuations. This promotes a sharp drop in captured air mass flow and, as a consequence, an engine thrust penalty. The inlet buzz with intense
fluid unsteadiness should thus be avoided as much as possible.
Ensuring the efficiency of supersonic air intakes is still a1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
halved to 247, 320 in our simulations by changing the viscosityof the flow. The spanwise extent of the computational domain
has been set to Lz = 3δ0 where δ0 = 2.4 mm is an expected estimate of the thickness of the turbulent boundary layer (basedon a 1/7th power law) developing with no pressure gradient on
the bottom ramp of the inlet, up to the point where the bow
shock generated by the cowl lip would impact the bottom wallin the absence of a boundary layer on the ramp (i.e. using a
direct extension of the shock angle, see location IV in Fig. 2).
This results in a ratio of δ0 /h = 0.05 for the thickness of
the ramp boundary layer at the first interaction with the cowlinduced shock wave.
All variables of the supersonic incoming flow are specified at the inlet boundary of the domain, while they are all
extrapolated at outlets. This is consistent with the supersonicflow condition in either the unthrottled or throttled cases. The
ramp and cowl walls of the air inlet are set as adiabatic no-slip
walls, while the plug walls are set to adiabatic slip walls whenset (TR 6= 0). Symmetry is imposed at the upper and lower
boundaries, and periodic boundary conditions are used in the
spanwise direction (see Fig. 1). The computational grid is refined close to the ramp and cowl walls, as well as in the coreregion in the vicinity of the entrance of the air intake. At the
no-slip walls, the size of the cells have been chosen such that∆x+ = 20, ∆y+ = 1 and ∆z+ = 15. This results in meshes ofaround 160 × 106 cells.
The compressible LES presented herein are performed using the in-house IC3 flow solver, which solves the spatiallyfiltered compressible Navier-Stokes equations for conserved
quantities using a finite volume formulation on unstructured
meshes. An explicit third-order Runge-Kutta (RK3) scheme
is used for time advancement. 20,000 time steps are necessary to resolve one convective time period ∆T = h/uin of
the flow, uin being the longitudinal velocity at the inlet boundary of the domain. The solver relies on the Vreman subgridscale model (Vreman (2004)) to represent the influence of unresolved small-scale fluid motions. It also features a solutionadaptive methodology which combines a low-dissipative centered numerical scheme and a first order upwind scheme inregions of the flow where discontinuities are present. For
this purpose, a dilatation-vorticity-pressure gradient (DVPG)
shock sensor based on the values of dilatation β , vorticity ωand pressure gradient ∆p, is used to identify the discontinuities
in the flow. In the cells where the ratio DVPG= ωβ ∆pp exceeds
a defined threshold value (set to unity in the present simulations), the first order scheme is applied while in the other cells,
the low-dissipative scheme is kept. Fifty-five convective time
periods h/uin have been used to converge the flow field andan additional fifty-five ∆T have been simulated to extract thestatistical data. The total computational cost of a single simulation represents 2 MCPUh on AMD ROME processors.
In order to analyze the phenomena occurring in the air
intake for the different throttling ratios and in particular theunsteady phenomena that are expected to occur, the spectral
proper orthogonal decomposition (SPOD) algorithm is used.
The SPOD method relies on discrete Fourier transform (DFT)and proper orthogonal methods to extract the main energetic
structures and their characteristic frequency in the flow field.
We use in this work the mathematical basis of the SPOD algorithm in its current and more widely used version relying on the Welch formulation, as introduced by Towne et al.
(2018) and Schmidt & Colonius (2020). This algorithm isapplied to two-dimensional (x,y) planes stored in time at themid position in the spanwise direction. We use 4,000 2Dsnapshots of the full two-dimensional simulation domain for

Aexit

y
z
h

11 h
(0,0)

Ath,plugyz
13 h

No-slip wall

Inlet

Slip wall

OutletSymmetry
Figure 1: Geometry of the air inlet and associated computational domain in the (x,y) plane, featuring a plug at the exit of
the inlet duct. The unthrottled case (TR0) does not have any
plug.

each LES case with an extraction frequency f corresponding
to St = f h/uin = 70. The dataset is partitioned into smaller
50%-overlapping blocks, each of which represents an ensemble realization of the flow. A temporal DFT of each block isthen calculated. To reduce spectral leakage, a Hanning window is used to smooth the discontinuities with a hypotheticalnext period. The frequency resolution corresponds to around∆St = 0.05.

COMPARISON WITH EXPERIMENTAL DATA
As a validation step for our LES approach, a comparison
with experiments is first performed for a steady case where thesystem of shocks in the air intake is stationary in time. This operating point corresponds to the unthrottled case TR0. Figure 2
shows the corresponding global structure of the flow obtainedin the experiments (schlieren) and in the present LES (density
gradient). In the experiments, because of installation requirements, the flow data are not available over a box downstreamof the cowl lip marked in black in Fig. 2 (a) and the corresponding shock structures have been manually reconstructed aposteriori (red lines). The compression of the incoming supersonic flow is achieved across an attached ramp shock (I), followed by an isentropic compression wave fan (II) and a bow
shock (III) forming in front of the upper cowl lip. This bow
shock impinges on the boundary layer developing on the ramp
surface and triggers a flow separation (IV) similar to the one
occurring in canonical SBLI. One can notice that the intersection between the ramp shock and the cowl-induced shock
occurs inside the inlet entrance. It takes place below the cowl
lip because the design Mach number of this air inlet equals
2, which is lower than the freestream Mach number of 2.41
studied in this work. On the upper side of the duct, a subsonic region appears downstream of the bow shock (locally anormal shock wave close to the cowl lip), and full separationfrom the bottom cowl surface happens due to the decelerationof the flow. Additional shock reflections on the ramp and cowl
surfaces then occur further downstream inside the duct. The
waves pattern obtained with LES is in good accordance withthe experimental schlieren.
A more quantitative comparison with experiments is pre-2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022of the pressure is under-predicted at x/h ≈ 1.6 but the othermeasurement locations match well. On the cowl side (Fig. 3
bottom), the positions of the shock structures characterized by
the increase/decrease of pressure obtained numerically correspond well to the experiments. In terms of amplitude, the increase of pressure at x/h = 1.9 is slightly over-predicted by
the LES. Given the difference in Reynolds number between
the present LES and Chen’s experiments, a perfect agreement
between numerical and experimental data should not be expected. The first SBLIs occurring on the ramp near the inlet
section of the intake duct are indeed affected by this Reynolds
number discrepancy, which in turn influences the behavior of
the subsequent flow. We therefore consider that our LES results can be validated in view of their overall qualitative andquantitative agreement with the reference experiment.

(a)
IV
V
I

VI

II
IV

(b)
SPOD ANALYSISThe focus of the analysis will now be dedicated to thecharacterization of the unsteadiness level of the flow in the
different considered cases (TR0 to TR60). To this purpose,
we rely on the SPOD approach previously evoked.

Figure 2: Schlieren image of the unthrottled air inlet (i.e. without plug, TR0 case) from Chen et al. (2018) (a) ; Contours ofthe magnitude of the mean density gradients from LES for thesame flow configuration (b).

Unthrottled case TR0
The TR0 case is first studied, because, as stated in thecomparison against experiments, this case has a shock cellstructure steady in time. The source of unsteadiness is thus expected to be related to turbulence at relatively high frequency
compared to buzz phenomena that are expected to induce fluctuations of the flow in the low frequency range. The eigenvalues associated with the two (out of the three) SPOD modes
maximizing the fluctuating compressible energy in the domain
for the TR0 case are shown in Fig. 4. Each eigenvalue has
been normalized by the sum of all eigenvalues for each SPOD
mode and over all frequencies (total fluctuating compressibleenergy in the domain) to provide the relative contribution tothis balance of each SPOD mode at each frequency. The analysis showed that the spanwise velocity fluctuations u02
z had loweigenvalue magnitudes (low disturbance energy), and that themodes are homogeneous along the span. This is why only thevariables ρ, ux , uy , T are considered in the energy disturbancebalance. Mode 1 (the most energetic POD mode over all thefrequency range denoted SPOD1) exhibits a few main peaks
but no large amplitude ratio can be observed between the low
frequency range and the other part of the spectrum, which corroborates the steady nature of the shock structures. Moreover,
we note that the second most energetic mode SPOD2 has amagnitude two to three times smaller than the mode SPOD1,
so that only the mode SPOD1 will be investigated. The spatial structures of density, longitudinal and vertical velocity andtemperature fluctuations of the first and second most energetic
peaks of SPOD1 at St = 0.623 and St = 2.909 (denoted 1 and
2 in Fig. 4 and sorted from low to large frequency) are shownin Fig. 5. For the first peak corresponding to St = 0.623, thespatial organization is characterized by large scale structuresof density and temperature fluctuations developing inside theinlet duct on the cowl side, as well as more localized longitudinal and vertical velocity structures close to the cowl. An
instantaneous snapshot of the actual flow field is shown in
Fig. 6. The analysis of the snapshots confirms that the shock
cell structures is steady as observed in the experiments. It also
shows that the unsteady phenomenon revealed by the SPOD
analysis at St = 0.631 can be associated to the shear layer developing at the edge of the flow separation region observed
below the cowl. At a higher frequency St = 2.909, the sec-

4
3
p
p0 2
[-]
1

Experiments
LES

0
0

2

4

6

8

10

x/h [-]
6

4
p
p0
[-] 2

Experiments
LES

0
0

2

4

6

8

10

x/h [-]
Figure 3: Wall distributions of mean pressure for the TR0 case:ramp side (top) ; cowl side (bottom).

sented in Fig. 3 for the TR0 case where mean pressure distributions along the ramp and the cowl are shown. On the ramp
surface (Fig. 3 top), the axial positions of increase/decreaseof static pressure obtained with the LES match well the experimental data indicating that the shock structures that cause
these pressure variations are well predicted. From the positionx/h ≈ 3, an axial shift occurs between the LES and experiments indicating that the shock structures do not match anymore from this position. In terms of amplitude, the decrease3

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

SPOD1
SPOD2
1

10−2

n

1.2
0.8
y/h [-]
0.4

2

Eigenvalue

n

λk,l

dft
modes
λk,l
∑i=0
∑l=0

[-]

10−1

10−3

z/h [-] 0
0.4

1

0

1

2

3

4

5

0

ux /uin
0.35 0.7

1.05

x/h [-]

10−4

0

4

3

2

Figure 6: Iso Q-criterion colored by the longitudinal velocity
and contours of M = 1 characterizing the shock structures (thesimulation domain has been replicated three times in the spanwise direction), TR0 case.

St [-]
Figure 4: SPOD eigenvalues as a function of frequency (St)for the two most energetic SPOD modes (SPOD1 and SPOD2)
based on the fluctuating compressible energy, TR0 case.

Throttled cases TR35, TR45 and TR60
1

We now examine the cases with a non-zero throttling ratio where unsteady effects should be promoted. We present in
Fig. 8 instantaneous snapshots of density for the all the cases
considered. One can clearly see that, when the back-pressure
is increased, the flow inside the duct is strongly affected compared to the TR0 case. If this influence is limited to the downstream part of the duct for TR35, where the waves pattern issimilar to the unthrottled case near the entrance of the air inlet,
this is no more true for the higher throttling ratios where flow
disturbances move further upstream and contaminate the entire
duct. The imprint of this behavior on the pressure distribution
on the ramp is shown in Fig. 9 where the influence of the backpressure is striking, with a change in regime for TR60 where
the external upstream part of the ramp wall is also contaminated. For TR45 the intersection between the ramp and bow
shock is not regular anymore and a Mach stem forms below
the cowl lip. Finally, as expected from the pressure distribution, the foot of the bow shock is completely expelled from theduct in the TR60 case, displaying a flow structure characteristic of a strong buzz phenomenon.
The inception of more and more unsteadiness when thethrottling ratio increases is also visible in Fig.7. For TR35, themaximum mass flow rate fluctuation represent 11.5% of themean mass flow rate, this value being greater than the unthrottled TR0 case where the maximum fluctuation reached 3.8%.
For the operating point with maximum back pressure (TR60),

2

(a)

(b)

(c)
y/h

1
0
0

1.25

x/h

(d)

Figure 5: Most energetic spatial structures of density (a), longitudinal (b) vertical (c) velocity and temperature (d) fluctuations related to the peak 1 (left) and 2 (right) of SPOD1, TR0case. The spatial structures of the modes are scaled between -1
and 1, the energy being held by the corresponding eigenvalue
and provided in Fig. 4.

ond peak exhibits structures developing close to the ramp wallaround x/h = 2.5, downstream of the throat. Based on the instantaneous snapshot shown in Fig. 6, theses structures can be
related to the ramp boundary layer separation due to the impingement of the bow shock developing at the cowl lip. For
this operating point, the main unsteadiness is thus related toboundary layer separation at the cowl lip and further downstream on the ramp side due to SBLI. This process related toturbulence spreads the corresponding energy over a broad frequency range, leading to the low amplitude levels observed forthe two main SPOD modes in this case.

0.8
(c)
x
ṁ 0.6(a) (b)
ρin uin hLz xx[-] 0.4
(d)
x
0.2

TR0
TR35
TR45
TR60

0
50

The unsteadiness related to separation, downstream of thelip on the cowl side and inside the diffuser on the ramp side,
can also be observed on the time evolution of the mass flowrate at the throat shown in Fig. 7. The maximum mass flowrate fluctuations represent however no more than 3.8% of themean mass flow rate for this unthrottled case.

60

70

80
90
tuin
[-]
h

100

110

Figure 7: Mass flow rate at the throat as a function of time forthe different cases studied. The letters on the TR60 curve markthe instants represented in Fig. 12.

4120

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
the mass flow rate does not reach the mean value obtained forother operating points. Also, the maximum mass flow rate
fluctuation reaches 83.3% of the mean mass flow rate, withan instantaneous mass flow rate that can become very low forsome flow events (see for example time tuin /h = 58 in Fig.7).

ρ
ρ0

12
10
8
p
p0 6
[-] 4

3.4

TR0
TR35
TR45
TR60

2
0
0

0.34

2

4

6

8

10

x/h [-]

ρ
ρ0 5.1Figure 9: Wall distributions of mean pressure on the ramp sideof the duct for the different LES cases.

[-]

1

n

10−2

Eigenvalue

n

λk,l

ρ
ρ0 4.5

SPOD1
SPOD2

dft
modes
λk,l
∑i=0
∑l=0

0.34

0.38
ρ
ρ0 4.9

10−4

10−6

0

1

2

3

4

5

St [-]
Figure 10: SPOD eigenvalues as a function of frequency (St)for the two most energetic SPOD modes (SPOD1 and SPOD2)
based on the fluctuating compressible energy, TR60 case.

0.23
Figure 8: Instantaneous contour of density for the different
LES cases. From top to bottom: TR0, TR35, TR45 and TR60.

velope of angles comprised between 41 and 111◦ with respect
the x axis and centered at the nose of the ramp. Some spots
are also visible on the vertical velocity fluctuations inside theduct of the air intake. This information can be correlated witha global oscillation of the flow during which the instability of
the ramp-side flow separation plays the dominant role. To that
purpose, instantaneous snapshots of the actual flow field describing a period of the phenomenon are shown in Fig. 12,
based on isosurfaces of Q-criterion colored by the longitudinal velocity. The cycle presented in this figure starts at an instant (a) when the ramp-side flow is massively separated due tothe strong adverse pressure introduced by the presence of theplug, with the corresponding separation shock at the ramp tip
inclined at an angle close to the higher limit of the previously
identified envelope on SPOD1 mode. This huge separation
leads to a serious blockage of the duct entrance that results ina significant reduction of the inlet mass flow rate in comparison with smaller TR cases (see Fig. 7). The separation bubble
first expands and then retracts between instants (a) and (b) sothat the minimum mass flow is found between these two snapshots. The bubble subsequently breaks down into a large-scale
clockwise-rotating structure that is advected into the diffuser(c), explaining the activity observed on the vertical velocity
structure of SPOD1 mode inside the duct, see Fig. 11 (c). Afterwards, the shrinked separation region on the ramp starts toincrease again (c), and the inclination of the ramp separation
shock wave is at its lower limit of the angle envelope (NoteThe SPOD analysis of the throttled cases reveals that alarge part of the energy disturbance is held at low frequency, incontrast with what has been observed for the TR0 case. This is
illustrated in Fig. 10 which displays the eigenvalues associatedwith the two (out of the three) SPOD modes maximizing thefluctuating compressible energy in the domain for the TR60
case (to be contrasted with its TR0 counterpart in Fig. 4). In
particular, the mode SPOD1 at the frequency St = 0.05 holds
38.4% of the total energy disturbance for TR60 (25.8% forTR35) while the dominant peaks of SPOD1 in the TR0 case
did not exceed 2%. Thus, for the throttled cases, the main
source of unsteadiness does not correspond to the same mechanism as in the TR0 case.
The mechanisms responsible for this unsteady behaviorof the flow can be identified by looking at the spatial structureof the corresponding energetic low-frequency SPOD modes.
This can be done for all the throttled cases, but for sake ofconciseness, we present herein only the TR60 case that corresponds to the most intense one. Fig. 11 displays the spatial structures of density, longitudinal and vertical velocity andtemperature fluctuations of the most energetic peak of mode
SPOD1 at St = 0.05. These spatial structures show that themain mechanisms underlying the activity of the flow at thislow frequency are mostly taking place above the external ramp,
close to the ramp wall and within a zone delimited by an en-512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

1.2
0.8
y/h [-]
0.4(a)z/h [-] 00.4
01x/h [-]

4

3

2
0

ux /uin
0.35 0.7

1.05

3

4

(a) t = 0
1.2
(b)

0.8
y/h [-]
0.4
z/h [-] 0
0.4
0

(c)

1

2
0

ux /uin
0.35 0.7

1.05

x/h [-](b) t = T /4

1
y/h

1.2
0.8
y/h [-]
0.4

0
0

1.25

x/h

(d)

z/h [-] 0
0.4

Figure 11: Most energetic spatial structures of density (a), longitudinal (b) vertical (c) velocity and temperature (d) fluctuations related to the dominant peak of SPOD1 (St = 0.05), TR60
case.

0

1

3

2
0

4

ux /uin
0.35 0.7

1.05

x/h [-](c) t = 2 × T /4

1.2

also that the maximum mass flow rate is obtained close to instant (c) when the separated region on the ramp is small). This
process keeps on going (d) and the progressive increase of theextent of the separation region gradually provokes a steepening of the separation shock wave and a return to the beginningof the cycle. This oscillating behavior of the flow operatingin the undesirable sub-critical regime of the intake is typicallycorresponding to the so-called big buzz phenomenon which
is here occurring at a frequency around 900 Hz (St = 0.076),
close to the resolution of our SPOD analysis given the current
total duration of our LES simulation.

0.8
y/h [-]
0.4
z/h [-] 0
0.4
0

1

3

2
0

ux /uin
0.35 0.74
1.05

x/h [-](d) t = 3 × T /4

Figure 12: Isosurfaces of Q-criterion colored by the longitudinal velocity at four consecutive instants during a buzz cycleof period T (the simulation domain has been replicated threetimes in the spanwise direction), TR60 case.

ACKNOWLEDGEMENTS
This research was supported by the French Ministry of
Defense through the financial support of the DGA (grant#2019-650068004707501). This work was performed using
resources from GENCI [CCRT-CINES-IDRIS] (Grant 2021[A0102A07178]) and CALMIP (Project 1425-21A).

Schmidt, O. T. & Colonius, T. 2020 Guide to spectral proper
orthogonal decomposition. AIAA Journal 55 (12), 4013–
4041.
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral
proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journalof Fluid Mechanics 847 (1), 821–867.
Vreman, A. W. 2004 An eddy-viscosity subgrid-scale modelfor turbulent shear flow: Algebraic theory and applications.
Physics of Fluids 16 (10), 132–144.

REFERENCES
Chen, Hao, Tan, Hui Jun, Zhang, Qi Fan & Zhang, Yue 2018
Throttling process and buzz mechanism of a supersonic inlet at overspeed mode. AIAA Journal 55 (5), 1953–1964.
Oswatitsch, Klaus 1980 Pressure Recovery for Missiles with
Reaction Propulsion at High Supersonic Speeds (The Efficiency of Shock Diffusers). In Contributions to the Development of Gasdynamics.

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