The 24th Computational Fluid Dynamics Symposium
Paper No.

The Effects of Turbulent Inflow Condition on Feedback-loop Mechanism inSupersonic Cavity Flowso

Weipeng Li, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, li@flab.isas.jaxa.jp
Taku Nonomura, ISAS/JAXA, 3-1-1, Yoshinodai, Sagamihara, Kanagawa, 252-5210
Kozo Fujii, ISAS/JAXA, 3-1-1, Yoshinodai, Sagamihara, Kanagawa, 252-5210

The effects of turbulent inflow condition on feedback-loop mechanism are numerically studied by implicit
large-eddy simulations of supersonic flows over a cavity under two turbulent-boundary-layer-profiled inflow
conditions with fluctuations (TurBL-flu case) and without fluctuations (TurBL-ave case). Results show thathigher-speed recirculation flows and high pressure distributions are observed in the TurBL-flu case. The
behaviors of the shear-layer in these two cases are much different. Two-dimensional shedding vortices inspanwise are clearly observed in the shear-layer of the TurBL-ave case, while plenty of small-scale
three-dimensional vortex structures are present in the shear-layer of the TurBL-flu case and bridge the cavity
from the leading edge to trailing edge. In the case of TurBL-ave, intense Mach waves radiate from theshear-layer and generate feedback compression waves, while no intense Mach wave and no intense
feedback compression wave are generated in the TurBL-flu case. The dominant cavity tone in the TurBL-flu
case has higher amplitude but exists in lower frequency than that of TurBL-ave case.
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1. Introduction
Feedback-loop mechanism is the driving factor ofself-sustained oscillations between shear-layer instability andacoustic forcing in supersonic open cavity flows. It could be
described as three steps: (1) shear-layer is excited by the acoustic
forcing of the feedback compression wave, which leads toshedding vortex and large-scale coherent structures; (2) shedding
vortex convects downstream and impinges on the solid wall; and(3) a new feedback compression wave is generated, and it willpropagate upstream and excite the shear-layer. Although the
feedback-loop mechanism itself is well-established and accepted,
1-3some experimental phenomenon has not been well-understood ,especially for high-speed compressible cavity flows. Questions
underlying the basic physical phenomenon still remain and further
investigations are needed.
4
In our previous work , feedback-loop mechanism is clearly
verified using the visualization of one feedback loop. To be
remarkable, we found that the feedback compression wave isgenerated by the reflection of Mach wave at the rear wall under
the laminar-profiled inflow conditions. However, when fully
turbulent inflows with presentence of random fluctuations andsmaller-scaled vortex structures pass over the cavity, the behaviorof the shear-layer and compression wave radiation may differ fromthose with laminar-profiled inflows. Understanding the behavior of
the turbulent inflows could lead to significant contributions to the
feedback-loop mechanism. The aim of this work is to clarify theeffects caused by different turbulent inflows and to put insight into
the physics of supersonic cavity flows.
2. Numerical Methods(1) Flow Configures
Table 1 Parameters of the two cases
Cases
Ma ReD L/D W/D θ
Inflow
Turbulent
Profilefluctuation
5
0.02D Turbulent No
TurBL-ave 2.0 102
0.6
TurBL-flu
YesSupersonic flows over a cavity of L/D=2, W/D=0.6 arenumerically studied, where L is the length of cavity, W is width ofcavity, and D is the depth of cavity. Freestream Mach number is
set to 2.0 and Reynolds number based on the cavity depth is set
1

to 10 . The momentum boundary-layer thickness (θ) at the leading
lip is 0.02D. Two cases are conducted as shown below.
(2) Governing Equations and Numerical Algorithms
The governing equations are three-dimensional compressible
Navier-Strokes equations in conservative form. Implicit large eddy
simulations (ILES) are conducted, which rely on the numericaldissipation to dissipate high frequency turbulent energy. In order tomeet the requirements of low dispersion and dissipation forcomputational
aeroacoustics(CAA)
andhigh-resolution
simulations of turbulence flows, seventh-order weighted nonlinear
5,6
compact scheme (WCNS)
is employed for the spatial
derivatives. This numerical scheme had been validated to be7
effective for high-speed jet noise simulations. The numericalfluxes are evaluated by the simple high-resolution upwind scheme
8(SHUS) which is a family of advection-upstream-splitting-method(AUSM) type schemes. Viscous terms are evaluated by thesixth-order difference scheme. Alternate directional implicit
9
symmetric Gauss-Seidel (ADI-SGS) scheme is applied for time
integration and Newton-like three sub-iterations are used tomaintain second order temporal accuracy. The CFL number isapproximately equal to 1.2.
(3) Grids and Boundary Conditions
The computational domain is shown in Fig. 1. The
computational domain consists of inside the cavity region andupper cavity region. Structured grids are adopted and grid points
are 213x211x150 in the inside cavity region, 488x177x150 in theupper cavity region, respectively. The total number of grid points isapproximate 19.7 millions. The length from inflow boundary to thecavity leading lip is equal to 1.5D, and 4D is extended from trailing
lip to outflow boundary. A distance of 4D is extended in the verticaldirection and no buffer region is set because of supersonicfreestream. In the spanwise direction, 150 points are equallydistributed within the total length of 0.6D. The grids are refined at+++near wall region. The values of Δx , Δy , Δz are 4.0, 0.8 and
8.0 respectively at the cavity leading lip. Inside the cavity, the++largest grid spacing is constrained by Δy max=24, Δx max=24,+
Δz max=8. Fine grids are used for the upstream region and inside
cavity region for accurate simulation of the resolution of oncoming
turbulent boundary-layer and the instable shear-layer
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The 24th Computational Fluid Dynamics Symposium
Paper No.
0.5
TurBL-ave
TurBL-flu

y/D

0

-0.5

Fig. 1 Computational grids (shown every other five points)
No-slip adiabatic wall boundary condition is imposed at all thewall boundaries. Zero-gradient pressure condition is employed at
the outflow, and periodical boundary condition is imposed in the10
spanwise direction. Recycling and rescaling technique isadopted for the simulation of zero-pressure-gradient flat plate in
the purpose of generation of a fully turbulent inflow. The results arevalidated in Fig.2 and Fig. 3.
The mean streamwise velocity profile in a semi-logarithmic plotusing Van Driest transformation is shown in Fig. 2, which indicatesthe profile agrees well with theoretical formulation. Threecomponents of velocity fluctuations in the turbulent boundary layer
are compared with direct numerical simulation(DNS) conducted
11
by Pirozzoli et al , in which Mach number equals to 2 and Reσis
44
set to1.74x10 . While in our simulation the Reσ equals 2.0x10
which is slightly different from the DNS. The comparison shows
the results of ILES simulation agree well with DNS and theturbulent inflow is adoptable for the simulation of supersonic cavity
flow.

-1

0

0.5

1

1.5

2

2.5

x/D

Fig. 4 Mean streamwise velocity profiles at the internal cavity
The mean streamwise velocity profiles at the internal cavity
appear in Fig. 4. The profiles near the bottom wall indicate that therecirculation flow of the TurBl-flu case has higher speed than thatof the TurBL-ave case. In the shear-layer region the streamwise
velocity profiles of the TurBL-ave case are steeper than the othercase, especially in the region near leading edge.
The contours of turbulence kinetic energy appear in Fig. 6. Highvalues are found near the leading edge for the TurBL-flu casewhich implies rapid turbulent mixing at this region. Nonzero values
are present in the oncoming flow since fully turbulent inflow
condition is imposed at upstream. For the TurBL-ave case, highvalues are observed near the trailing edge and almost zero values
are present in oncoming upstream.

25
LES-recyclingu+ = y+log law (1/0.41)*log(y+)+5.0

20

Uvd

15

(a ) TurBL-ave case(b) TurBL-flu caseFig. 5 Contours of turbulence kinetic energy

10

5

0

0

1

10

2

10

0.25

3

10

10

y+

Cp

Fig. 2 Mean streamwise velocity profile of turbulent boundary
layer of zero-pressure-gradient flat plate
3

0.2

0

4

0.15

1

3

TurBL-ave
TurBL-flu

0.1(u')rms /u

Velocity fluction

2.5

0.05

(v')rms /u(w')rms /u20

DNS by Pirozzoli et al.
1.5

-0.05

0

0.5

1

1.5

2

2.5

3

3.5

4

1

S/D

0.5

Fig. 6 Mean pressure coefficient distributions
The mean pressure coefficient distributions of the two cases
are shown in Fig. 6. The mean flowfield of the TurBL-flu case haslarger pressure coefficients at the right bottom corner andtrailing-edge lip, which is supposed to be caused by higher speed
flow injection flow from trainging edge and impingement at righbottom corner, as illustrated in Fig. 4.

0

0

0.2

0.4

0.6

0.8

1

y/

Fig. 3 Velocity fluctuations in one boundary layer thickness ofzero-pressure-gradient flat plate
3. Results and Discussions
In the present study, the analysis is mainly focused on three
aspects: 1) mean flowfields; 2) the behavior of the shear-layer andcompression wave radiation, and 3) pressure spectrum of cavity
tones.
(1) Mean Flowfields
2

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The 24th Computational Fluid Dynamics Symposium
Paper No.

(a ) TurBL-ave case(b) TurBL-flu caseFig. 7 Contours of pressure fluctuation
The distributions of pressure fluctuation appear in Fig. 7, in
which the contours are much different from each other. The
TurBL-ave case has larger and higher values of pressure
fluctuation near the trailing edge of the cavity, and has smaller andlower values near the leading edge. The TurBL-ave case has
several peaks in the region from leading edge to trailing edge,
while the TurBL-flu case only has one larger high pressure
fluctuation region approximately at the mid of the cavity. These
differences are caused by the much different behavior ofshear-layer in the two simulation cases, which is discussed later.
(2) The Behavior of Shear-layer and Compression Wave
Radiation
The behavior of shear-layer in cavity flows is an important clue
for revealing the physical mechanism of self-sustained oscillations.
In the two cases, the behavior of the shear-layer significantly
differs from each other due to the difference in fluctuation
components. The isosurface of the second invariant of velocity
gradient tensors and the spanwise-averaged vorticity magnitudeof an instantaneous flowfield is shown in Fig. 8 and Fig. 9,
respectively.
Since the inflow of the TurBL-ave case does not containturbulent fluctuations, the oncoming wall boundary-layer isdominated by two-dimensional characteristic without the presenceof small-scaled vortex structures before it passes over the cavity.
After the shear-layer separates from the leading edge,
three-dimensional characteristic grows owning to K-H instability. In
Fig. 8(a) a bulk of three-dimensional vortices is observed in theshear-layer near mid of the cavity, and a larger one exists near the
trailing edge of the cavity. Although the vortices in the shear-layer
are three-dimensional, highly two-dimensional characteristic isobserved in spanwise direction by the visualization of
spanwise-averaged vorticity magnitude in Fig. 9(a). Two shedding
vortices are clearly observed. The scales of the vortices arecomparable to the thickness of the shear-layer.

(a ) TurBL-ave case(b) TurBL-flu caseFig. 9 Contours of vorticity magnitude of spanwise-averaged
flowfields
However, the behavior of the shear-layer in the TurBL-flu caseis much different. As shown in Fig. 8(b), plenty of small-scalevortex structures are present not only in the oncoming stream but
also in the shear-layer. The small-scale vortex structures consist of
the shear-layer and bridge the cavity from the leading edge to thetrailing edge. No clear large-scale coherent structure is observedin the shear-layer. In Fig. 9(b) contours of spanwise-averaged
vorticity of the TurBL-flu case are shown, which indicates that the
spanwise two-dimensional characteristic is weak. One large-scalevortex structure is observed near the trailing edge as marked by
the white dot line. Many small-scale vortex structures are present
near the right rear wall for the both cases because massively
vortices inject into the cavity and become parts of recirculation
flows.
In order to investigate the spreading rate of the shear-layer,
vorticity boundary-layer thickness is adopted. The distribution ofvorticity boundary-layer thickness is shown in Fig. 10 from leadingedge to the trailing edge of the cavity. It indicates that theshear-layer of the TurBL-flu case grows rapidly from the leadingedge to the region 0.5D from leading edge, and then the
spreading rate slightly decreases but also at high value. However,the shear-layer of the TurBL-ave case has a small growth ratefrom the leading edge to the region approximately 0.3D fromleading edge, and then grows with a high spreading rate. The
vorticity boundary-layer thickness decreases near the trailing edge
because of the impingement of shear-layer on the rear wall of thecavity.
0.5
TurBL-ave
TurBL-flu

0.4



0.3

0.2

0.1

0

0

0.5

1

1.5

2

x/D

Fig. 10 The resolution of vorticity boundary layer thickness fromleading edge to the trailing edge
The compression waves are captured by the absolute value of
density gradient, which is shown as contour surface in Fig 11
coupling with visualization of positive values of Q2nd as black
contour lines. The category of compression waves in supersonic1
cavity had been summarized in our previous work. As shown in+
Fig. 11(a), compression waves III and compression waves III are(a ) TurBL-ave case(b) TurBL-flu caseFig. 8 Isosurface of the second invariant of velocity gradienttensors (Q2nd=(U∞/σ) )2

3

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The 24th Computational Fluid Dynamics Symposium
Paper No.
wall are shown in Fig. 12. First, the amplitudes of dominant cavity
tone are different. The TurBL-flu case has larger amplitude fordominant cavity tone than that of the other case. The reason isprobably related to more and higher-speed massively injection. As
illustrated in Fig. 4 and Fig. 10, the TurBL-flu case hashigher-speed recirculation flows at the internal cavity, and theshear-layer has larger spreading rate from the leading edge to thetrailing edge, which lead to more mass flows inject into the cavity
from the trailing edge and impinge on the right bottom corner.
Second, the frequencies of the dominant cavity tone are different.
The second mode at lower frequency is the dominant cavity tonein the TurBL-flu case, while the forth one in the TurBL-ave case.
The cavity tones are greatly related to the behavior of theshear-layer.

Mach waves in the upper cavity region and internal cavity region,
respectively. Compression wave I and II are shock waves
generated at the leading edge and trailing edge, and compression
waves IV is feedback compression waves traveling toward front
wall. Compression wave V and compression wave VI are
generated by the reflection of feedback compression wave at thefront wall and bottom wall, respectively. Detailed discussions about
1
their features could refer to our previous work.

III

I

IV

III+

+

IIIV

III+
I

III+

II

II
IIIIV

VI
TurBL-ave
TurBL-flu
Exp. Zhuang et al.

170(a ) TurBL-ave case(b) TurBL-flu caseFig. 11 Compression wave radiation of a instantaneous flow
(background contour surfaces are absolute values of density
gradient; black contour lines are the second invariant of velocity
gradient tensors)
Here we mainly focused on the discussion of Mach wave
radiation, which is one of distinguishing features in supersoniccavity flows. As we known, Mach wave radiation occurs when the
difference between the freestream velocity and the shear-layer
convective velocity exceeds the local sound speed. Both sides of1
shear-layer could radiate Mach wave. In our previous work , weproposed that the feedback compression wave is generated by
the reflection of Mach wave at the rear wall when thelaminar-profiled inflow condition is imposed. In the present study,
whether Mach wave reflection plays a crucial role in supersoniccavity flows is discussed.
In Fig. 11(a), Mach wave radiation in both sides of shear-layer isobserved to be traveling with the shedding vortex. As time goes,the Mach wave inside the cavity reaches at the rear wall, and aclear reflection is observed which leads to a new feedback
compression wave. This is similar to our previous simulation forlaminar-profiled cases. However, no intense Mach wave radiates
at the region inside the cavity from the shear-layer in thesimulation of the TurBL-flu case. Instead, many weak Machwaves periodically radiate from the shear-layer but no clear
reflection is observed near the rear wall. Moreover no intense
feedback compression wave is observed inside the cavity, but thecavity tones exist in this case as in the pressure spectra shown
later.
The different characteristic in Mach wave radiation is related tothe different behavior of shear-layer as discussed above. If thesupersonic convective shear-layer consists of plenty of small-scalevortex structures and has highly three-dimensional characteristic,
the Mach waves radiated from the shear-layer may become
weaker. The simulation of the TurBL-flu case demonstrates that
Mach wave does not play a critical role to feedback-loop
mechanism when the shear-layer is fully turbulent. But
self-sustained oscillations are observed and the deriving
mechanism on generation of feedback compression remains
unclear.
(4) Cavity Tones
The spectra of pressure fluctuations at the mid-point of the front

SPL(dB)

160
150
140
130
120

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

St

Fig. 12 Pressure spectra at the mid-point of the cavity front wall
The spectra are compared with the experiment conducted by
12
Zhuang et al. on Mach 2 flows over a cavity with length-to-depth(L/D) ratio of 2. The Reynolds number based on the depth of the5cavity and the momentum boundary-layer thickness is 5.4*10
4
and 1.8*10 , respectively. The peak values of the cavity tones inthe experiment are plot in dash-blue line in Fig. 12, and thesecond mode is the dominant cavity tone. The comparison showsthat the simulation of the TurBL-flu case matches well withexperimental results.
4. Conclusion
The effects of turbulent inflow condition on feedback-loop
mechanism are numerically studied by implicit large-eddy
simulations of supersonic flows over a cavity under two
turbulent-boundary-layer-profiled inflow conditions with fluctuations(TurBL-flu case) and without fluctuations (TurBL-ave case).
Results show that higher-speed recirculation flows and highpressure distributions are observed in the TurBL-flu case. The
behaviors of the shear-layer in these two cases are much different.
Two-dimensional shedding vortices in spanwise are clearlyobserved in the shear-layer of the TurBL-ave case, while plenty ofsmall-scale three-dimensional vortex structures are present in theshear-layer of the TurBL-flu case and bridge the cavity from theleading edge to trailing edge. In the case of TurBL-ave, intense
Mach waves radiate from the shear-layer and generate feedback
compression waves, while no intense Mach wave and no intense
feedback compression wave are generated in the TurBL-flu case.
The dominant cavity tone in the TurBL-flu case has higher
amplitude but exists in lower frequency than that of TurBL-ave
case.
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The 24th Computational Fluid Dynamics Symposium
Paper No.
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