Journal of Sound and Vibration (1999) 221(1), 23–47

Article No. jsvi.1998.2012, available online at http://www.idealibrary.com.on

ATTENUATION OF CAVITY FLOW

OSCILLATION THROUGH LEADING EDGE
FLOW CONTROL
X. Z, X. X. C  A. R
Department of Aeronautics and Astronautics, University of Southampton,
Southampton SO17 1BJ, England

J. A. E
Weapons Systems Sector, DERA, Fort Halstead, Sevenoaks, Kent TN14 7BP,
England

(Received 28 July 1998)

The unsteady flows over a shallow rectangular cavity at Mach 1·5 and 2·5 are
modified at the leading edge by using compression ramps, expansion surfaces, and
mass injection. The study is performed through solutions of Short-time
Reynolds-Averaged Navier–Stokes equations (TRANS) with turbulence modelled
by a two-equation k–v model. When a compression ramp is introduced, two types
of responses are observed: at Mach 1·5, a strong flapping motion leads to small
changes in the frequency and sound pressure level in the cavity compared with
the baseline case of rectangular geometry. The roll-up of the shear layer produces
convective vortices, leading to enhanced pressure fluctuations on the downstream
surface; At Mach 2·5, a weak shear layer instability produces a reduction in the
sound pressure level, and the increased distance between the leading edge and the
trailing edge produces a reduction in frequency. An increase in the mean pressure
drag coefficient is produced due to the high pressure on the ramp. When an
expansion surface is employed, the mean pressure drag coefficient is also increased
slightly. When the flow is attached to the surface, the major flow physics are
similar to the baseline case. A reduction of the sound pressure level is observed
in the cavity with the surface height. When a shock induced separation occurs on
the surface, a steady flow is established in the cavity. When the mass injection is
introduced, a passive pressure response is observed at the leading edge, producing
local vorticity and vortex shedding. The flow mechanism remains the same at both
Mach numbers, with a weak sitting vortex near the rear corner. An optimal mass
injection pressure ratio is identified.
7 1999 Academic Press

1. INTRODUCTION
Flow inside a shallow cavity driven by a shear layer is known to be unsteady under
a wide range of flow and geometric conditions [1]. Experiments [2, 3] have
highlighted the presence of convected vortical structures in the driven shear layer,

0022–460X/99/110023 + 25 $30.00/0 7 1999 Academic Press

24 .   .
shear layer impingement onto the downstream face and pressure disturbances
inside the enclosure, which lead to a self-sustained fluid flow oscillation. This
phenomenon is responsible for wall pressure fluctuations, pressure drag increase,
and far-field aerodynamic noise radiation.
A substantial amount of research has been directed towards devising means of
effective unsteady pressure field control. The objective could be (a) to reduce the
pressure drag, (b) to attenuate the pressure fluctuations on aerodynamic surfaces,
or (c) to alter the far field noise radiation. Methods have been proposed to modify
the flow near the leading edge of the cavity, the shear layer impingement near the
trailing edge, and the pressure field inside the cavity. Devices used to modify the
pressure field inside the cavity include passive venting of the floor [4] and mass
injection near the trailing edge [5]; devices for leading edge flow modifications
include spoilers [6, 7], cowls [8], vortex generators [7], and mass injection [9];
Methods for trailing edge shear layer impingement modifications include
introducing a ramp [10, 11] and a round-nose [11].
The development/deployment of effective unsteady flow control devices depends
on a clear understanding of major flow physics. There are areas of flow physics
and their interpretation which deserve further attention. For example, modifying
the trailing edge geometry is generally believed to be effective in reducing the
pressure fluctuation [10]. Recent model tests [11], however, suggest an enhanced
oscillation with certain edge geometries. Devices employed to modify the shear
layer generally lead to changes in the dominant mode of oscillation–large vortical
structures in the shear layer. This will alter or attenuate narrowband tones
observed in the far field. However, as the shear layer is normally turbulent the
effects on the likely rise in the broadband noise have not been investigated in
depth. Pressure oscillation attenuation mechanisms proposed for some devices
such as the leading edge mass injection require further clarification to determine
whether the observed reduction in the pressure fluctuation is due to ‘‘active
control’’ of disturbances in shear layer [9], or in fact an alteration of shear layer
impingement. Questions also exist on the effectiveness of a compression ramp and
an expansion surface. Would a ‘‘ski-jump’’ type of ramp lift the shear layer and
modify the downstream reattachment? If so, will the oscillation be attenuated? Will
an expansion surface induce a stronger shear layer impingement onto the
downstream face, leading to higher pressure fluctuation?
In this study, methods including leading edge compression ramps, expansion
surfaces [12], and mass injection are investigated, which follows an earlier study
of pressure waves around a cavity [13]. It is hoped that the study will help clarify
confusion in the interpretation of model test results. Past experiments suggest that
near the leading edge it is possible that unsteady pressure disturbances could
selectively excite shear layer instability modes which characterize the
time-dependent velocity and vorticity fields. The initial amplitude of such
exponentially growing modes provides possibilities for exploring means of
reducing or suppressing their onset/growth by altering the flow passing the leading
edge. It is also possible that the shear layer reattachment position and
characteristics will be altered, leading to changes in the pressure inside the cavity,
thus reducing the pressure oscillation. The flows are investigated using
     25
computational fluid dynamics modelling. The calculation solves for the Short-time
Reynolds-averaged Navier–Stokes equations (TRANS) with turbulence modelled
by a two-equation k–v model. This follows an earlier study [14] where the
numerical model was validated. The authors believe this represents a valid
approach as the flow fields concerned are characterized by an unsteady pressure
field and are highly susceptible to external disturbances such as a pressure probe
in a model test.

2. FLOW CONDITIONS
2.1.  
The numerical algorithms and model are validated against a baseline model test
case: an open two-dimensional cavity flow [2]. The flow is driven by a shear layer
from separation of the oncoming boundary layer at the leading edge of the cavity.
The length to depth ratio of the cavity is 3 and depth of the cavity is fixed at
15 mm. Freestream Mach numbers are 1·5 and 2·5. The Reynolds number based
on the depth of the cavity is 4·5 × 105 at both Mach numbers. In the model tests,
a turbulent boundary layer approaches the enclosure upstream edge. At Mach 1·5,
the oncoming boundary layer had a thickness d of 0·287 D a displacement
thickness d* of 0·06 D, and a momentum thickness u of 0·0273 D. At Mach 2·5,
the respective values were 0·28 D, 0·0867 D, and 0·0227 D. A schematic of the
geometry is shown in Figure 1(a).

2.2.     

The approaching surface upstream of the leading edge of the cavity is replaced
by compression ramps and expansion surfaces of a fixed length (1D). The height
of the ramp/surface varies between 0·1 D, 0·2 D and 0·4 D. The ramps and the
surfaces are formed by a circular arc of a constant radius (see Figures 1(b) and
(c)).

2.3.  

In the mass injection test (see Figure 1(d)), the approaching surface upstream
of the leading edge of the cavity is replaced by a porous surface extending to 1 D
upstream. The porosity factor s is 0·607 (see section 3). Different back wall
pressures are selected. For the Mach 1·5 flow the back wall pressure pi ranges from
1·5 pa to 2·5 pa ; for the Mach 2·5 flow the back wall pressure pi ranges from
1·25 pa to 3·0 pa .

3. NUMERICAL MODEL
A numerical model of the unsteady flow is obtained through solutions of the
discretized short-time averaged Navier–Stokes equations [14] with Wilcox’s k–v
turbulence model [15], by which the time dependent predictions are obtained of
the large-scale structures characterizing the unsteady flow. The flow field is
discretized by using a multi-block structured grid. A second order Roe flux
26 .   .
difference split approximate Riemman solver estimates the inviscid fluxes which
are integrated in space with the turbulent fluxes by using a finite volume technique.
An explicit multi-step Runge–Kutta scheme with optimized coefficients [16]
advances the flow prediction in time. The method is formally second order time
and space accurate.
For the majority of the test cases, the computational domain covers an area
extending from x = −3 D to 9 D in the streamwise direction, and from y = −1 D
to 4 D in the transverse direction. The domain above the cavity is covered by 320
by 400 cells and the cavity by 80 by 80 cells. The cell sizes have been found to
be adequate for the flow following a cell size test and are therefore retained in this
study (for details, see reference [14]). For some cases, the upstream boundary is
extended to x = −6 D as the upstream influence of introducing a ramp and mass
injection needs to be taken into account. A fixed turbulent boundary layer is

y
U∞ Trailing edge
x

Leading edge (a)

D

Rear corner
L

= 2 arc tg(h/D)

D (b)

h (c)

= 2 arc tg(h/D)

SPL measurement positions

D (d)

0.33 D
0.4 D
2.33 D
3.6 D
5.6 D
7.6 D

Figure 1. Test geometries: (a) baseline; (b) compression ramp; (c) expansion surface; (d) mass
injection.
     27
T 1
Effect of leading edge compression ramp
Mach 1·5 Mach 2·5
ZXXXXXXCXXXXXXV ZXXXXXXCXXXXXXV
h/D Baseline 0·1 0·2 0·4 Baseline 0·1 0·2 0·4
Cd 0·0672 0·1069 0·1511 0·2785 0·0217 0·0345 0·0609 0·1314
St 0·0925 0·0945 0·0952 0·0976 0·0843 0·0839 0·0816 0·0787
SPL=(−0·4,0) 150·3 149·5 163·8 145·2 73·87 113·2 129·5 109·0
SPL=(0·33,−1) 171·1 170·1 168·7 169·9 160·3 157·9 156·1 156·5
SPL=(2·33, − 1) 176·0 176·4 176·2 176·3 164·0 162·5 161·6 160·6
SPL=(3·6,0) 168·3 170·3 171·9 174·9 161·3 160·7 158·9 157·7
SPL=(5·6,0) 164·6 165·7 166·8 168·7 155·9 154·9 151·4 149·4
SPL=(7·6,0) 162·8 164·2 166·0 169·8 152·5 151·0 144·9 140·3

defined at the inflow boundary; no-slip conditions are imposed on the solid walls;
extrapolated and non-reflecting conditions apply at the outflow (x = 6 D) and
upper boundary (y = 4 D), respectively. When the mass injection is considered, a
porous surface of constant porosity is introduced. A constant back pressure is
maintained below the surface area [17]. According to the linear form of the Darcy
pressure–velocity law, this gives

vw = (s/ra ua )(pi − pw ), (1)

where s is the geometric porosity [17] and pi is the back wall pressure (a list of
nomenclature is given in the Appendix).

4. RESULTS AND DISCUSSION

4.1.  
The changes in the flow field are summarized in Table 1. In terms of Strouhal
number (St) and sound pressure level (SPL), the response of the flow to the
introduction of a ramp differs at Mach 1·5 and 2·5. At Mach 1·5, there is only
a slight rise in St from 0·0925 for the baseline flow to 0·0945 at h/D = 0·1, 0·0952
at h/D = 0·2 and 0·0976 at h/D = 0·4. This compares with a value of 0·093 in the
model test. While the SPL on the floor of the cavity experiences negligible changes
(Q1·4% at x = 0·33 D and 2·33 D), its value downstream of the cavity rises
sharply with h/D, the rise in SPL (compared with the baseline flow) being 2·0 dB
at h/D = 0·1, 3·6 dB at h/D = 0·2 and 6·6 dB at h/D = 0·4 being observed at
x = 3·6 D. A different response with the ramp height is observed at Mach 2·5. Here
St drops from 0·0843 for the baseline flow to 0·0839 at h/D = 0·1, 0·0816 at
h/D = 0·2 and 0·0787 at h/D = 0·4, which compares with a value of 0·088 from
the model test. SPL on the floor as well as downstream of the cavity all experience
a reduction in value. The largest reduction occurs near the leading edge, a
28 .   .
reduction of 2·4 dB at h/D = 0·2, 4·2 dB at h/D = 0·2 and 3·8 dB at h/D = 0·4
being observed at x = 0·33 D.
A study of the pressure field suggests that the difference in the flow response
at the two Mach numbers could not be explained simply by the changes in the
mean pressure field. In fact, the influence of the ramp on the mean pressure field
is rather similar at the two Mach numbers (see Figure 2). The approaching flow
is compressed and a quasi-stationary shock forms in front of the ramp (for a
discussion of the pressure waves around a shallow cavity, see references [13, 18]),
which is followed by a compression region and then a rapid expansion around the
edge of the ramp as the flow is turned around. This leads to a generally low mean
pressure environment in the cavity which is indicated by the low mean pressure
level on the upstream face of the cavity (see Figure 2). The reduction in the mean
pressure level increases with the height of the ramp (h/D). The h/D = 0·4 ramp
produces the lowest mean pressure level at both Mach numbers. This trend
continues on the floor of the cavity, while the mean surface pressure rises sharply
approaching the rear corner of the cavity due to flow compression as a result of
shear layer impingement. The surface mean pressure distribution on the
downstream face suggests that the flow is still dominated by a large sitting vortex

2.4
(a) (b) (c) (d)
2.2
2.0 Mach 1.5

1.8
p/p ∞

1.6
1.4
1.2
1.0
0.8
0.6
–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D

2.4
(a) (b) (c) (d)
2.2
Mach 2.5
2.0
1.8
1.6
p/p ∞

1.4
1.2
1.0
0.8
0.6
–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D
Figure 2. Surface mean pressure with compression ramps: (a) approaching surface; (b) upstream
face; (c) floor; (d) downstream face. q, Test; ––, baseline; - - - , h/D=0·1; - · - · - , h/D = 0·2; – – – ,
h/D = 0·4.
     29
in the region near the rear corner [2]. In presenting the results, we define a mean
pressure drag coefficient, Cd , taking into account the pressure on the upstream
face, the downstream face, the ramp and the expansion surface. Thus, for the
baseline geometry

g
0
pdown − pup
Cd = dy; (2)
−D
q

for the expansion surface geometry,

g g g
0 −h 0
pdown pup pexp
Cd = dy − dy − dy; (3)
−D
q −D
q −h
q

for the ramp geometry,

g g g
0 h h
pdown pup pramp
Cd = dy − dy + dy. (4)
−D
q −D
q 0
q

It can be seen from Table 1 that the mean pressure drag coefficient is increased
sharply with h/D. At Mach 1·5, Cd rises to 1·59 times that of the baseline flow at
h/D = 0·1, 2·25 times at h/D = 0·2, and 4·14 times at h/D = 0·4. At Mach 2·5, the
corresponding values are 1·59, 2·81 and 6·06. This is mainly due to the presence
of the ramp. In fact, the characteristics of the surface mean pressure in the cavity
are quite similar.
Changes in the unsteady flow field (see Table 1 and Figure 3) involve variations
in both frequency and magnitude. These behave differently at Mach 1·5 and 2·5,
suggesting a difference in the physics. In Figure 3 only the baseline case and
h/D = 0·2 case are presented. The characteristics of the h/D = 0·1 and 0·4 flows
are rather similar to those at h/D = 0·2 and are therefore not presented. To
understand the observed flow features, a clear understanding of the basic flow
physics is necessary. We believe that one important aspect of the flow is a coupled
motion of the shear layer [13], the results of which can be seen in the velocity
vectors in Figures 4 and 5, where a sequence of the flow motion is presented and
the corresponding surface pressure variations are given in Figure 6. In the
transverse direction, the shear layer experiences a flapping motion due to the shear
layer instability. In the streamwise direction, there is a vortex convection due to
the non-linear propagation effects leading to significant wave steepening with
convection. These two motions are strongly coupled. At Mach 1·5, the instability
of the shear layer is rather strong and the non-linear propagation effects are such
that the wave steepening with convection produces large convective vortices
(Figure 4), leading to the transient pressure in the cavity (Figure 6). The flapping
motion of the shear layer is thus quite pronounced and causes the large vortex near
the rear corner to experience large motion in the transverse direction. As a result
of this coupled motion, although the unsteady mode is in the longitudinal
direction, the induced pressure fluctuation is characterized by the flapping motion
of the shear layer near the trailing edge. The (vertical) flapping motion of the shear
30 .   .
(a)
0.5
0.4

p/(2q ∞ )
0.3
0.2
0.1
0.0
150 175 200 225
(b)
0.5
0.4
p/(2q ∞ )

0.3
0.2
0.1
0.0
50 75 100 125 150
(c)
0.2
p/(2q ∞ )

0.1

0.0
100 110 120 130 140 150 160 170 180 190 200
(d)
0.2
p/(2q ∞ )

0.1

0.0
100 110 120 130 140 150 160 170 180 190 200
Figure 3. Surface pressure with compression ramps at x/D = 2·33: (a) baseline at Mach 1·5; (b)
h/D = 0·2 at Mach 1·5; (c) baseline at Mach 2·5; (d) h/D = 0·2 at Mach 2·5.

layer near the trailing edge is geometrically limited to the depth of of the cavity.
The length scale of this flapping motion is therefore primarily determined by a
flow-independent geometric restriction and so remains constant at value near 1 D.
Consequently the observed St value does not change significantly with h/D. As
the trailing edge is now lower than the leading edge with a ramp, the strongly
amplified disturbances/vortex structures are allowed to be convected downstream
of trailing edge (see Table 1).
At Mach 2·5, the instability of the shear layer is weak (the limiting Mach number
for a vortex sheet is 2z2 above which the vortex sheet is stable [19]). The
non-linear effect is times rather weak, producing small vortices convecting in the
longitudinal direction. The large vortex near the trailing edge (see Figure 5)
experiences a relatively smaller motion in the transverse direction than that of the
Mach 1·5 flow and the unsteady motion/pressure field is then dominated by the
convected disturbances/vortices in the longitudinal direction. As a result of this,
St drops with the ramp height h/D as the distance between the edge of the ramp
     31
and the trailing edge increases with h/D. The level of oscillation in terms of SPL
is lower than that of the baseline flow at Mach 2·5.

4.2.  

When a leading edge expansion surface is introduced, the pressure field in the
cavity is affected and the presence of the expansion surface acts as a limiting factor
in defining the pressure level in the cavity. This feature can be observed in Figure
7 where the surface mean pressure is given. There are two important features of
the mean flow: (i) unlike the ramp case, the mean pressures on the upstream face

U∞

(a)

U∞

(b)
U∞

(c)
U∞

(d)

Figure 4. Velocity vectors with a h/D = 0·2 compression ramp at Mach 1·5 over one period T:
(a) t = 0; (b) t = 0·25 T; (c) t = 0·5 T; (d) t = 0·75 T.
32 .   .
U∞

(a)

U∞

(b)
U∞

(c)
U∞

(d)

Figure 5. Velocity vectors with a h/D = 0·2 compression ramp at Mach 2·5 over one period T:
(a) t = 0; (b) t = 0·25 T; (c) t = 0·5 T; (d) t = 0·75 T.

(see Figure 7(b)) are nearly the same at the three heights; (ii) the sitting vortex near
the rear corner (see Figure 7(d)) is weaker than that of the baseline case, suggesting
a weak shear layer impingement. The second feature is rather surprising as it is
expected the expansion surface would lead to a stronger shear layer impingement
which will then produce high surface mean pressure on the downstream trailing
edge, and more importantly stronger pressure feedback and fluctuation in the
cavity. The fact that a weaker vortex is produced suggests that the separated shear
layer from the leading edge possesses different features from that of the baseline
     33
case. When the oncoming flow approaches the leading edge, the expansion surface
allows the flow to accelerate along the surface, hence the reduction in the surface
mean pressure (see Figure 7(a)) and associated expansion pressure waves (see
density contours in late figures). However, this process is terminated on the surface
by an oblique shock wave. The leading edge oblique shock wave is rather similar
to the type 2 wave described in reference [18] and type 1 wave of reference [13].
The appearance of the wave depends on the height of the expansion surface. The
higher the surface the earlier the appearance of the terminating shock wave, and
the stronger the wave. The surface mean pressure is at its lowest at h/D = 0·4
under the present flow conditions. Downstream of the oblique wave the pressure
recovers. The extent of the pressure recovery, though, depends on the flow
following the shock wave. For the h/D = 0·1 and 0·2 cases, the pressure recovery
processes are rather similar, leading to a surface mean pressure level close to that
of the baseline case at x = 0 (see Figure 7(a)). At h/D = 0·4, the recovery process

0.40
Mach 1.5 (i)
p/(2q ∞)

0.30

0.20
200 205

0.40
(ii)
(a) (c)
p/(2q ∞ )

0.30

0.20
(b) (d)
0.10
200 205
tU ∞/D
0.12
Mach 2.5 (i)
0.11
p/(2q ∞ )

0.10
0.09
0.08
160 165

0.14 (ii)
(a) (c)
0.12
p/(2q ∞ )

0.10
0.08
(b) (d)
0.06
160 165
tU ∞/D

Figure 6. Surface pressure over one period with a h/D = 0·2 compression ramp: (i) x/D = 0·33;
(ii) x/D = 2·33.
34 .   .
2.0
(a) (b) (c) (d)
1.8
1.6
1.4
1.2
p/p ∞ 1.0
0.8
0.6
Mach 1.5
0.4
0.2
–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D
2.0
(a) (b) (c) (d)
1.8
1.6
1.4
1.2
p/p ∞

1.0
0.8
0.6
Mach 2.5
0.4
0.2
–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D
Figure 7. Surface mean pressure with expansion surfaces: (a) approaching surface; (b) upstream
face; (c) floor; (d) downstream face. q, Model test; ––, baseline; - - - , h/D=0·1; - · - · - , h/D = 0·2;
– – – , h/D = 0·4.

is slightly different at both Mach numbers. There is an area of relatively high

surface pressure approaching x = 0 (leading edge) at both Mach 1·5 and 2·5. In
fact, the surface mean pressures at Mach 1·5 and 2·5 possess the same
characteristics. It will be shown (later with the velocity vectors) that a shock
induced flow separation occurs at h/D = 0·4 at both Mach numbers. The resulting
recircuiting flow from the flow separation is associated with nearly stable flow
environment at Mach 1·5 and stable flow at Mach 2·5. From Table 2, it can be
seen that the pressure drag coefficient is increased with the leading edge expansion
surfaces. The increase though is not as big as that with a compression ramp.
The accelerated flow along the expansion surface alters the stability
characteristics of the shear layer from the flow separation at the leading edge at
h/D = 0·1 and 0·2 at both Mach numbers. The pressure fluctuations are reduced
significantly by the introduction of the surfaces (see Figures 8 and 9, and Table
2). In Figure 8, the pressure fluctuations at h/D = 0·2 show a big reduction
compared with that of a baseline flow (see Figure 3). At h/D = 0·4, the flow is
nearly stable and the oscillation is very weak, SPL being about 35 dB lower than
that for the baseline flow. At Mach 2·5 (see Figure 9), the flow is completely stable
after an initial period of the starting process of the numerical computation [14].
The characteristics of the flows at Mach 1·5 and 2·5 are the same. The reduction
     35
T 2
Effect of leading edge expansion surface
Mach 1·5 Mach 2·5
ZXXXXXXCXXXXXXV ZXXXXXXCXXXXXXV
h/D Baseline 0·1 0·2 0·4 Baseline 0·1 0·2 0·4
Cd 0·0672 0·0645 0·0786 0·0787 0·0217 0·0267 0·0245 0·0306
St 0·0925 0·0893 0·0859 0·0823 0·0843 0·0804 0·0786 n/a
SPL=(−0·4,0) 150·3 154·0 162·0 142·0 73·87 105·7 127·3 n/a
SPL=(0·33,−1) 171·1 169·8 169·5 136·0 160·3 158·1 151·1 n/a
SPL=(2·33, − 1) 176·0 174·2 171·3 139·0 164·0 164·3 152·9 n/a
SPL=(3·6,0) 168·3 166·0 161·8 131·0 161·3 158·8 147·2 n/a
SPL=(5·6,0) 164·6 162·6 159·3 126·4 155·9 152·9 139·4 n/a
SPL=(7·6,0) 162·8 160·2 157·4 125·5 152·5 148·8 132·7 n/a

in the pressure oscillation occurs not only in the cavity but downstream after the
reattachment of the shear layer (see Table 2). A typical example is Mach 2·5 flow
at h/D = 0·2, where the SPL experiences a reduction of around 10 dB in the cavity
and a bigger reduction downstream.
The difference between the stable flow and the unsteady flow is shown in Figures
10 and 11. In Figure 10, the attenuated and nearly stable flow at Mach 1·5 is shown
with the velocity vectors and the density contours. Figure 11 gives examples of the

0.6
(a)
0.5
p/(2q ∞)

0.4
0.3
0.2
0.1
0 50 100 150
0.6
(b)
0.5
p/(2q ∞ )

0.4
0.3
0.2
0.1
0 50 100 150
0.6
(c)
0.5
p/(2q ∞ )

0.4
0.3
0.2
0.1
0 50 100 150
tU ∞/D

Figure 8. Surface pressure with expansion surfaces at x/D = 2·33 and Mach 1·5. h/D values: (a)
0·1; (b) 0·2; (c) 0·4.
36 .   .
0.20
(a)
0.15

p/(2q ∞ )
0.10
0.05
0.00
0 50 100 150
0.20
(b)
0.15
p/(2q ∞ )

0.10
0.05
0.00
0 50 100 150
0.20
(c)
0.15
p/(2q ∞ )

0.10
0.05
0.00
0 50 100 150
tU ∞/D

Figure 9. Surface pressure with expansion surfaces at x/D = 2·33 and Mach 2·5. h/D values; (a)
0·1; (b) 0·2; (c) 0·4.

unsteady flow at Mach 1·5 and h/D = 0·2. As the unsteady flow physics are rather
similar with an expansion surface at Mach 1·5 and 2·5, only the Mach 1·5 results
are discussed here. At h/D = 0·1 and 0·2, the observed flow physics such as the
sitting vortex and the pressure waves are similar. Therefore, only h/D = 0·2 results
are given in Figure 11, In Figure 11, the flow on the expansion surface is shown
to stay attached during the whole period of flow oscillation. In Figure 11(a), the
flow is associated with local low pressure near the leading edge and a downward
deflection of the shear layer into the cavity. The position of the large sitting vortex
near the trailing edge suggests a mass ejection process near the trailing edge [14].
In Figure 11(b), the upward deflection of the shear layer at the leading edge is
associated with local high pressure and the creation of a convecting vortex near
(but not at) the leading edge. The sitting vortex near the rear corner suggests a
mass entrainment process at the trailing edge [14].
In Figure 10, the density contours show the wave patterns above the cavity, in
particular the expansion–shock wave system on the expansion surface (see Figure
10(b)). In viewing the velocity vectors at h/D = 0·4 where the flow is nearly stable,
it is interesting to note the induced flow separation on the expansion surface
following the limiting oblique shock wave. The reversed flow indicates an induced
separation and the resulting vortex/vorticity distribution is also clear. This
vortex/vorticity distribution differs from those observed in Figures 4 and 5, and
Figure 11. When the flow is unstable (e.g., with a compression ramp), the shear
layer from the leading edge flow separation is unstable. The amplification of the
instability waves leads to significant wave steepening and convecting vortices.
     37
U∞

(a)

8
00 0.9

08
1. 08
1. 0

1.
1.0

0.76

0 .7 8 (b)

76
0.
Figure 10. Steady flow at Mach 1·5 with a h/D = 0·4 expansion surface: (a) velocity vectors; (b)
r/ra contours.

These vortices are not produced at the leading edge but some distance
downstream. In the current flow at h/D = 0·4, the vorticity produced by the flow
separation on the expansion surface is located on the surface and is not convected

U∞

(a)

U∞

(b)

Figure 11. Unsteady flow at Mach 1·5 with a h/D = 0·2 expansion surface: (a) downward
deflection of shear layer at the leading edge; (b) upward deflection.
38 .   .
T 3
Mach 1·5 flow with leading edge mass injection
pi /pa Baseline 1·5 2·00 2·25 2·50
rv n/a 0·0293 0·1018 0·1582 0·2058
Cd 0·0672 0·0388 0·0150 0·0139 0·0147
St 0·0925 0·0892 0·0805 0·0730 0·0756
SPL=(−0·4,0) 150·3 155·4 153·3 154·7 155·5
SPL=(0·33,−1) 171·1 168·4 160·8 160·0 162·3
SPL=(2·33,−1) 176·0 173·3 165·0 163·6 165·7
SPL=(3·6,0) 168·3 168·1 159·1 159·1 161·0
SPL=(5·6,0) 164·6 164·0 150·6 152·4 155·4
SPL=(7·6,0) 162·8 163·0 144·8 151·9 154·1

and no convected vortex forms. This changes the shear layer stability
characteristics and leads to a stable flow. The dominant feature of the flow at
h/D = 0·4 is thus the sitting vortex near the trailing edge/downstream face. We
have earlier noticed from the surface mean pressure (see Figure 7(d)) that the
sitting vortex is weaker than that of the baseline flow. It should be mentioned here
that the complex flow physics described above, in particular the shock/boundary
layer interaction on the expansion surface, is simulated with a two-equation k–v
turbulence model. Questions remain as to the applicability of the turbulence model
to this flow, which could be answered with a model experiment.

4.3.  

For the mass injection study, the calculated values of Strouhal number, sound
pressure level and mean pressure drag coefficient are given at different pressure
ratios and mean mass flow rates in Tables 3 and 4. The mass injection generally
introduces a significant drop in the sound pressure level. At Mach 1·5, the largest

T 4
Mach 2·5 flow with leading edge mass injection
pi /pa Baseline 1·25 1·50 2·00 2·25 2·50 2·75 3·00
rv n/a n/a 0·0085 0·0223 0·0319 0·440 0·0568 0·0587
Cd 0·0217 0·0193 0·0146 0·0052 0·0044 0·0047 0·0066 0·0065
St 0·0843 0·0829 0·0813 0·0758 0·0724 0·0691 0·0657 0·0263
SPL=(−0·4,0) 73·88 108·9 134·6 142·1 140·4 141·1 143·0 145·1
SPL=(0·33,−1) 160·3 158·6 152·1 152·3 149·2 148·3 148·7 149·9
SPL=(2·33,−1) 164·0 163·2 162·6 158·3 155·5 154·5 154·8 156·0
SPL=(3·6,0) 161·3 160·7 159·8 154·1 148·9 144·5 143·7 146·5
SPL=(5·6,0) 155·9 155·1 153·7 145·1 141·2 139·4 138·9 142·4
SPL=(7·6,0) 152·5 151·6 149·9 140·8 135·0 139·4 141·3 141·8
     39
1.8
(a) (b) (c) (d)
Mach 1.5
1.6

1.4
p/p ∞
1.2

1.0

0.8
–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D

(a) (b) (c) (d)

1.8

1.6
p/p ∞

1.4

1.2
Mach 2.5
1.0

–5 –4 –3 –2 –1 0 0 –1 0 1 2 3 –1 0
x/D y/D x/D y/D
Figure 12. Surface mean pressure with mass injection: (a) approaching surface; (b) upstream face;
(c) floor; (d) downstream face. q, Test; ––, baseline; - - - - , pi /pa=1·5; - · - · - · , pi /pa = 2·0; – – –,
pi /pa = 2·25; – · · – · · – , pi /pa = 2·5.

reduction in the cavity is observed at pi /pa = 2·25, being 11·1 dB at x = 0·33 D

and 12·4 dB at x = 2·33 D. Increasing pi /pa above 2·25 will not produce a further
reduction in SPL. At Mach 2·5, the largest reduction in SPL on the floor is
observed at pi /pa = 2·5, where a drop of 12 dB is observed at x = 0·33 D. Again,
a further increase in the mass flow rate of pi /pa will not lead to a further drop
in SPL. In Tables 3 and 4, the mean pressure drag coefficient is seen to experience
a significant drop in value with pi /pa . A nearly five-fold reduction is observed at
both Mach numbers.
The mean surface pressure distribution is given in Figure 12. Unlike the
compression ramp test cases, the mean pressure distribution does exhibit some
different features from the baseline cases, notably on the downstream face. The
mean pressure distribution on the downstream face, however, does point to the
existence of a large sitting vortex and the surface mean pressure distribution is that
of a typical vortex flow induced one. With the mass injection, it can be seen that
the difference in the mean pressure is reduced, suggesting a weakened sitting vortex
near the rear corner. On the mean flow field, there is again a quasi-stationary
oblique shock in front of the mass injection region, followed by a compression
region and then an expansion region near the leading edge. As a result, the mean
40 .   .
pressure field in the cavity is generally lower than the corresponding one for the
baseline flow. An interesting feature of the flow is that a high pi /pa or mean rv
does not necessarily lead to a low pressure drag coefficient. The lowest pressure
drag coefficient is observed at pi /pa = 2·5 at both Mach numbers. At Mach 2·5,
the mean pressure distribution shows little change between pi /pa = 2·0 and 3·0,
despite the increase in the mean mass flow rate rv. In explaining the behaviors
of the mean pressure drag coefficient and SPL with the mass flow rate, it is
interesting to note the rise in the surface mean pressure upstream of the cavity (see
Figure 12). The rise in the mean pressure above pi /pa = 2·0 is quite marginal and,
in the case of the Mach 1·5 flow, small. It follows that the expansion of the flow
around the leading edge remains similar for further increase of pi /pa . An optimal
pressure rate for both mean form drag coefficient and SPL reduction could then
be possible, and indeed is identified in the study at 2·5 based on the mean pressure
drag coefficient.
The reduction in SPL can be seen clearly in Figures 13 and 14, where the
unsteady surface pressure variations are presented. At both Mach numbers, the
Strouhal number is reduced substantially with the mass injection. At Mach 1·5,
St drops from 0·0925 for the baseline flow to 0·073 at pi /pa = 2·25. At Mach 2·5,
St drops from 0·843 for the baseline flow to 0·0724 at pi /pa = 2·25. The reduction

0.5
(a)
0.4
p/(2q ∞ )

0.3
0.2
0.1
0.0
50 75 100 125 150
0.5
(b)
0.4
p/(2q ∞ )

0.3
0.2
0.1
0.0
100 110 120 130 140 150 160 170 180 190 200
0.5
(c)
0.4
p/(2q ∞ )

0.3
0.2
0.1
0.0
100 110 120 130 140 150 160 170 180 190 200
0.5
(d)
0.4
p/(2q ∞)

0.3
0.2
0.1
0.0
200 210 220 230 240 250 260 270 280 290 300
tU ∞ /D

Figure 13. Surface pressure injection at x/D = 2·33 and Mach 1·5. pi /pa values: (a) 1·5; (b) 2·0;
(c) 2·25; (d) 2·5.
     41
0.2
(a)

p/(2q ∞)
0.1

0.0
50 75 100 125 150
0.2
(b)
p/(2q ∞)

0.1

0.0
100 125 150 175 200
0.2
(c)
p/(2q ∞)

0.1

0.0
100 125 150 175 200
0.2
(d)
p/(2q ∞ )

0.1

0.0
300 325 350 375 400
tU ∞ /D

Figure 14. Surface pressure with mass injection at x/D = 2·33 and Mach 2·5. pi /pa values: (a) 1·5;
(b) 2·0; (c) 2·5; (d) 3·0.

in the Strouhal number at both Mach numbers can be explained by the low
convection speed of the vortices in the shear layer following the mass injection.
In Figures 15 and 16, velocity vectors at four consecutive times in one oscillation
period are given for a typical pressure ratio. The monitored mass flow rate and
surface pressures upstream and on the floor of the cavity are presented for the
same period in Figures 17 and 18. It can be seen that rv responds passively to
the unsteady pressure at the leading edge. A high local pressure leads to a low mass
flow rate, while a high mass flow rate corresponds to a low local pressure. As a
result of the local passive pressure response, a local transient velocity/vorticity field
near the leading edge is generated. A vortex is produced immediately after the
leading edge but its strength is rather weak in comparison with that observed in
the ramp test cases under similar flow conditions (see Figure 4). We have seen that
due to this particular physics the flow oscillation mechanism remains the same at
both Mach numbers (see Figures 13 and 14). In Figures 15 and 16, the sitting
vortex near the trailing edge is weak in comparison with that in the baseline cases
and the ramp cases. Although not presented in the paper, the pressure wave
patterns at Mach 1·5 and 2·5 are rather similar to those reported by Zhang et al.
[13]. Apart from the unsteady waves around the trailing edge due to the shear layer
42 .   .
U∞

(a)

U∞

(b)

U∞

(c)

U∞

(d)

Figure 15. Velocity vectors at Mach 1·5 and pi /pa = 2·0 over one period T: (a) t = 0; (b)
t = 0·25 T; (c) t = 0·5 T; (d) t = 0·75 T.

flapping, the convective vortices also induce an associated wave which is convected
downstream.

5. CONCLUDING REMARKS
The effects of leading edge compression ramps, expansion surfaces and mass
injection on supersonic shallow cavity flow oscillations are investigated, through
     43
U∞

(a)

U∞

(b)

U∞

(c)

U∞

(d)

Figure 16. Velocity vectors at Mach 2·5 and pi /pa = 2·0 over one period T: (a) t = 0; (b)
t = 0·25 T; (c) t = 0·5 T; (d) t = 0·75 T.

solutions of Short-time Reynolds-Averaged Navier–Stokes equations with the

effect of turbulence modelled by a two-equation k–v model. This study has
provided insight into major flow physics and clarified confusion in the
interpretation of the effectiveness of different means for the unsteady flow control.
The findings are as follows.
1. The shear layer driven oscillation is characterized by a coupled motion of
shear layer flapping in the transverse direction due to the shear layer instability,
44 .   .
and vortex convection in the streamwise direction due to the non-linear
propagation effects leading to significant wave steepening with convection. This
mechanism explains different responses to the introduction of a leading edge
compression ramp at Mach 1·5 and 2·5.
2. At Mach 1·5, the shear layer is dominated by large non-linear roll-ups of
convective vortices which lead to a strong flapping motion of shear layer near the
trailing edge. The limiting length scale of the oscillation is the depth of the cavity.
The Strouhal number of the oscillation and SPL experience small changes with
h/D. With the introduction of a ramp, the large vortical structures are convected
downstream beyond the trailing edge of the cavity and enhanced pressure
fluctuations are observed downstream of the cavity as compared with the baseline
flow.
3. At Mach 2·5, a reduction in both the Strouhal number and SPL in and
around the cavity is observed. The non-linear effect is small at this Mach number
and the oscillation is in the longitudinal direction, with smaller shear layer flapping
in the transverse direction near the trailing edge. The increased length between the
edge of the ramp and the trailing edge thus leads to the reduction in St.
4. When an expansion surface is introduced, an expansion wave is formed on
the surface which is terminated by an oblique shock. At a high surface height of
h/D = 0·4, a shock induced flow separation occurs on the surface, inducing a
sitting vortex or vorticity distribution which is not convected downstream. A
nearly stable flow is established at Mach 1·5 and stable flow at Mach 2·5.
5. When mass injection is introduced on the surface upstream of the leading
edge, a passive pressure response is generated which modulates the mass flow rate.

0.023
(i)
(a) (c)
v

(b) (d)
0.022
260 265
0.180
(ii)
0.175
p/(2q ∞ )

0.170
0.165
0.160
260 265
0.130
(iii)
0.120
p/(2q ∞ )

0.110
0.100
0.090
260 265
tU ∞/D

Figure 17. Variation of mass flow rate and surface pressure over one period at Mach 1·5 and
pi /pa = 2·0: (i) rv; (ii) x/D = −0·4; (iii) x/D = 2·33.
     45
0.023
(i)
(a) (c)

v
(b) (d)
0.022
260 265
0.180
(ii)
0.175
p/(2q ∞)

0.170
0.165
0.160
260 265
0.130
(iii)
0.120
p/(2q ∞ )

0.110
0.100
0.090
260 265
tU ∞ /D

Figure 18. Variation of mass flow rate and surface pressure over one period at Mach 2·5 and
pi /pa = 2·0: (i) rv; (ii) x/D = −0·4; (iii) x/D = 2·33.

Local vorticity is produced immediately after the leading edge, leading to vortex
convection in the longitudinal direction. The flow physics remains the same at both
Mach numbers. Rsults suggest a reduction in St, SPL, and mean pressure drag
coefficient. The reduction in SPL could be as much as 10 dB.
6. The biggest reduction in SPL and mean pressure drag coefficient occurs
between pi /pa = 2·25 and 2·5. Further rise in the mass flow rate will not produce
a reduction in SPL and mean pressure drag coefficient.

ACKNOWLEDGMENTS
The authors wish to thank D. G. Mabey for his suggestion on the ramp study.
This work is supported by DERA under contract number WSF/U1961.

REFERENCES
1. H. H, D. G. H, E. E. C 1971 Journal of Sound and Vibration 18,
545–553. Flow-induced pressure oscillations in shallow cavities.
2. X. Z and J. A. E The Aeronautical Journal 94, 355–364. An investigation
of supersonic oscillatory cavity flow driven by thick shear layers.
3. K. K. A and J. M 1995 NASA CR-4653. Effects of cavity dimensions,
boundary layer, and temperature on cavity noise with emphasis on benchmark data
to validate computational aeroacoustics.
4. F. W, J., AIAA Journal 26, 374–376. Passive venting system for modifying
cavity flowfields at supersonic speeds.
46 .   .
5. V. S and P. F. M 1977 Journal of Aircraft 14, 833–877. Control of cavity
noise.
6. L. S 1982 AIAA Paper 82-0329. Full scale flight evaluation of suppression
concepts for flow-induced fluctuating pressure in cavities.
7. H. H. H and D. B 1975 AIAA Paper 75–491. The physical mechanisms of
flow-induced pressure fluctuations in cavities and concepts for their suppression.
8. E. R 1964 ARC R&M No. 3438. Wind tunnel experiments on the flow over
rectangular cavities at subsonic and transonic speeds.
9. A. V and C. G 1991 AIAA Paper 91-1645. Control of cavity flow by
upstream mass injection.
10. M. E. F and D. L. C 1975 AIAA Paper 75-0492. Effect of geometry on open
cavity flow induced pressure oscillation.
11. J. C. F. P and J. M. S 1994 AIAA Journal 32, 1737–1740. Influence of
impingement edge geometry on cavity flow oscillation.
12. D. G. M 1995 Private communication.
13. X. Z, A. R and J. A. E 1998 Journal of Sound and Vibration (to
appear). An observation of pressure waves around a shallow cavity.
14. X. Z 1995 AIAA Journal 33, 1404–1411. Compressible cavity flow oscillation due
to shear layer instabilities and pressure feedback.
15. D. C. W 1988 AIAA Journal 26, 1299–1310. Reassessment of the scale
determining equation for advanced turbulence models.
16. M. M 1992 Ph.D. Thesis, Université Catholique de Louvain. A three dimensional
high resolution compressible flow solver.
17. Y. N. J and U. J. P 1995 Journal of Aircraft 32, 363–369. Numerical study
of a supersonic open cavity flow and pressure oscillation control.
18. H. H and J. D 1996 Journal of Sound and Vibration 196, 248–252. Cavity
pressure oscillations: the generating mechanism visualized.
19. D. S. J 1975 Journal of Institute of Mathematics and its Applications 15, 33–57.
The scattering of sound by a vortex sheet.

APPENDIX: NOMENCLATURE
Cd mean pressure drag coefficient
D depth of cavity
f frequency of dominant mode of oscillation
h height of leading edge ramp and surface
M Mach number
p pressure
q ra u a
2
/2
Re Reynolds number ra ua D/ma
SPL sound pressure level in dB, 20 log (Prms /10 mPa)
St Strouhal number, fD/ua
u streamwise velocity
v transverse velocity
x, y Cartesian co-ordinates
d thickness of boundary layer
m molecular viscosity
v specific dissipation rate
r density
s geometric porosity
     47
rv mass flow rate, f rv/(ra ua ) dx/D
0
−D
( )down downstream face of cavity
( )exp expansion surface
( )i back wall condition
( )ramp ramp
( )up upstream face of cavity
( )w wall condition
( )a free stream condition