J. Fluid Mech. (2008), vol. 599, pp. 309–339.
c
2008 Cambridge University Press
doi:10.1017/S0022112007009925 Printed in the United Kingdom
309
Three-dimensional instabilities in compressible
flow over open cavities
GUILLAUME A. BR
`
ES AND TIM COLONIUS
Department of Mechanical Engineering, California Institute of Technology,
1200 E. California Blvd., Pasadena, CA 91125, USA
(Received 26 March 2007 and in revised form 30 October 2007)
Direct numerical simulations are performed to investigate the three-dimensional
stability of compressible flow over open cavities. A linear stability analysis is conducted
to search for three-dimensional global instabilities of the two-dimensional mean flow
for cavities that are homogeneous in the spanwise direction. The presence of such
instabilities is reported for a range of flow conditions and cavity aspect ratios. For
cavities of aspect ratio (length to depth) of 2 and 4, the three-dimensional mode has a
spanwise wavelength of approximately one cavity depth and oscillates with a frequency
about one order of magnitude lower than two-dimensional Rossiter (flow/acoustics)
instabilities. A steady mode of smaller spanwise wavelength is also identified for
square cavities. The linear results indicate that the instability is hydrodynamic
(rather than acoustic) in nature and arises from a generic centrifugal instability
mechanism associated with the mean recirculating vortical flow in the downstream
part of the cavity. These three-dimensional instabilities are related to centrifugal
instabilities previously reported in flows over backward-facing steps, lid-driven cavity
flows and Couette flows. Results from three-dimensional simulations of the nonlinear
compressible Navier–Stokes equations are also reported. The formation of oscillating
(and, in some cases, steady) spanwise structures is observed inside the cavity. The
spanwise wavelength and oscillation frequency of these structures agree with the linear
analysis predictions. When present, the shear-layer (Rossiter) oscillations experience
a low-frequency modulation that arises from nonlinear interactions with the three-
dimensional mode. The results are consistent with observations of low-frequency
modulations and spanwise structures in previous experimental and numerical studies
on open cavity flows.
1. Introduction
From the canonical rectangular cut-out to more complicated shapes with internal
structures, resonant cavity instabilities are endemic to a number of aircraft compo-
nents, including weapon bays, landing gear wells and instrumentation cavities. Self-
sustained oscillations and intense acoustic loading inside the cavity can lead to
structural damage, optical distortion and store separation problems. In compressible
flow, cavity oscillations are typically described as a flow–acoustic resonance
mechanism: small instabilities in the shear layer interact with the downstream corner
of the cavity and generate acoustic waves, which propagate upstream and create new
disturbances in the shear layer. The feedback process leads to reinforcement, and
ultimately saturation, at one or more resonant frequencies. This type of instability
is referred to as shear-layer (or Rossiter) mode, from the early work of Rossiter
310 G. A. Br
`
es and T. Colonius
(1964) who developed the now classic semi-empirical formula to predict the resonant
frequencies:
St =
fL
U
=
n − α
M +1/κ
, (1.1)
where n =1, 2,...leads to the Strouhal number St of mode I, II, etc. The empirical
constants κ and α correspond to the average convection speed of the vortical
disturbances in the shear layer, and a phase delay, respectively.
Over the past few decades, two-dimensional cavity flows have received significant
attention (see for instance review articles by Rockwell & Naudascher 1978;
Colonius 2001; Rowley & Williams 2006). Aside from numerical benchmarking,
the main motivations for studying cavity flow are noise reduction and flow control.
Fundamental research has been conducted recently to examine how active (open-
and closed-loop) flow control can be use to replace traditional passive devices such
as spoilers, ramps and rakes (e.g. Cattafesta et al. 1999; Alvarez, Kerschen & Tumin
2004; Rowley et al. 2006; Rowley & Williams 2006). Model-based closed-loop control,
in particular, promises efficient (low energy input) tone suppression, while passive
devices may be more effective for broadband noise reductions. However, past efforts
in this regard have typically ignored non-parallel and three-dimensional effects that
may, on one hand, reduce the effectiveness of model-based control, or on the other
hand disregard important three-dimensional mechanisms that could be exploited in
passive ways to reduce broadband noise.
Recently, some aspects of the three-dimensional cavity flow have been investigated
using large-eddy simulation (LES) (Rizzetta & Visbal 2003; Larchev
ˆ
eque et al. 2004;
Chang, Constantinescu & Park 2006) and proper orthogonal decomposition (POD)
(Podvin et al. 2006). These studies have been focused on the frequencies of oscillation
and coherence of the (two-dimensional) Rossiter modes, and the extent to which
there is agreement with experimental measurements of mean flow and spectra. Some
observations regarding the three-dimensionality of the large-scale turbulent structures
are also reported. Likewise, experiments by Maull & East (1963), Rockwell & Knisely
(1980), Neary & Stephanoff (1987), Forestier, Jacquin & Geffroy (2003) and Faure
et al. (2007) all reported three-dimensional structures in the flow. The origin of these
three-dimensional features has not yet been understood.
Therefore, the focus of the present work is to characterize the basic instabilities
of three-dimensional open cavity flows. Because the basic (steady or time-averaged)
cavity flow is complex and non-parallel, our stability analysis is focused on extracting
global instabilities from direct numerical simulations (DNS) of the full and linearized
compressible Navier–Stokes equations. We consider two- and three-dimensional
instabilities to basic cavity flows that are homogeneous in the spanwise direction.
The numerical methods used are presented in § 2.
The remainder of the paper is organized as follows. First, the onset of two-
dimensional cavity instability is characterized as a function of Reynolds number, Mach
number, cavity aspect ratio, and incident shear-layer thickness. The two-dimensional
modes are consistent, both in terms of oscillation frequency and eigenfunction
structure, with the typical Rossiter flow/acoustic resonant modes that have been
observed in many cavity experiments and flight tests. For basic cavity flows that are
two-dimensionally stable, we then search for three-dimensional instabilities of the
steady base flow, and identify, for the first time, the presence of such instabilities
(§ 3). For cavity length-to-depth ratios of 1, 2 and 4 considered here, the instability
appears to arise from a generic centrifugal instability mechanism associated with
Three-dimensional cavity flows 311
the internal recirculation vortical flow that occupies the downstream part of the
cavity (§ 4). Next, a few selected three-dimensional numerical simulations of the full
compressible Navier–Stokes equations are performed. The results (§ 5) exhibit three-
dimensional structures in good agreement with the linear analysis predictions. Finally,
the connections between the three-dimensional modes we report here and the flow
structures observed in previous numerical studies and experiments are discussed in § 6.
2. Numerical methods and stability analysis
2.1. Direct numerical simulations
Following previous work of Rowley, Colonius & Basu (2002) on cavity flows, we
developed a DNS code to solve the full compressible Navier–Stokes (NS) equations
and study the flow over three-dimensional open cavities. The equations are solved
directly, meaning that no turbulence model is used and all the scales of the flow
are resolved. A linearized version of the equations was also implemented, and the
existing DNS code can solve linear or nonlinear NS equations for both two- and
three-dimensional flows.
The flow equations are solved on a structured mesh, using a sixth-order compact
finite-difference scheme for spatial discretization in the x-andy-direction (Lele 1992)
and a fourth-order Runge–Kutta algorithm for time-marching. The cavity is assumed
homogeneous (periodic) in the spanwise direction (z-direction) and the derivatives
are computed using fast Fourier transform (FFT) with subroutines provided by the
FFTW library (e.g. Frigo & Johnson 2007). Each spanwise wavenumber is discretized
on a stretched Cartesian grid, with clustering of points near the walls and the shear
layer in the cavity. The boundary conditions are non-reflective for the inflow and
outflow, no slip and constant temperature (T = T
∞
) at the walls (Thompson 1990;
Poinsot & Lele 1992). In addition, a buffer zone is implemented at the inflow, outflow
and normal computational boundaries to reduce acoustic reflections (Colonius, Lele &
Moin 1993; Freund 1997). Unless stated otherwise, the simulations are initiated with
a Blasius flat-plate boundary layer spanning the cavity and zero flow within the
cavity. The code can handle any type of block geometry and is fully parallelized using
message-passing interface (MPI).
2.2. Simulation parameters and flow conditions
The cavity configuration and flow conditions are controlled by the following
parameters: the cavity aspect ratio L/D and spanwise extent Λ/D, the ratio of
the cavity length to the initial boundary layer momentum thickness at the leading
edge of the cavity L/θ
0
, the Reynolds number Re
θ
= Uθ
0
/ν and the free-stream Mach
number M = U/a
∞
(see figure 1). Typical grid sizes ranged from a few hundred
thousand to several million grid points for two- and three-dimensional simulations,
respectively. In each case, the computational domain extends several cavity depths
upstream, downstream and above the cavity.
Our study focuses on cavity flows at subsonic speed (0.1 6 M 6 0.8) and low
Reynolds number (35 6 Re
θ
6 400), with laminar upstream boundary layers. For
the shallow cavities considered here (L/D = 1, 2 and 4), the main two-dimensional
instability mechanism is the Rossiter aeroacoustic coupling, rather than the acoustic
modes of the cavity, even at low subsonic speed. The onset of a Rossiter mode as a
function of the parameters is summarized qualitatively as follows (Rowley et al. 2002):
there is a critical value of M, Re
θ
,andL/θ
0
beyond which oscillations occur. For low
Reynolds number and Mach number, the flow is subcritical and ultimately reaches
312 G. A. Br
`
es and T. Colonius
Λ
θ
D
U
Outflow
Buffer zone
Inflow
Outflow
L
y
x
z
Figure 1. Schematic diagram of the computational domain.
a steady state. As these parameters, or L/θ
0
, are increased, the two-dimensional flow
becomes supercritical and oscillates in shear-layer (Rossiter) mode.
2.3. Linear stability theory
To take into account non-parallel effects and possible coupling of the shear layer with
the acoustic scattering and recirculating flow inside the cavity, the so-called bi-global
linear stability theory has been used to study the stability of cavity flow (Theofilis &
Colonius 2003; Theofilis, Duck & Owen 2004). In this approach, the transient solution
of the equations of motion q =[ρu,ρv,ρw,ρ,e]
T
is decomposed into
q(x,y,z, t)=
¯
q(x,y)+q
(x,y, z, t), (2.1)
where
¯
q(x,y) is the unknown steady two-dimensional basic flow and q
(x,y, z, t)
an unsteady three-dimensional perturbation with ||q
|| ||
¯
q||.Asthedomainis
homogeneous in the spanwise direction, a general perturbation can be decomposed
into Fourier modes with spanwise wavenumbers β. At linear order, modes with
different wavenumber are decoupled and the following eigenmode Ansatz can be
introduced:
q
(x,y, z, t)=
n
ˆ
q
n
(x,y) exp[i(βz − Ω
n
t)] + c.c., (2.2)
where the parameter β is taken to be a real and prescribed spanwise wavenumber,
related to a spanwise wavelength in the cavity by λ =2
π/β,
ˆ
q
n
and Ω
n
= ω
n
+iσ
n
are the unknown complex eigenmodes and corresponding complex eigenvalues, both
dependent on β. Complex conjugation is required in (2.2) since q
is real. The frequency
and the growth/damping rate of the mode are given by ω
n
and σ
n
, respectively. The
long-time behaviour of the linear solution will be dictated by the mode with eigenvalue
Ω = ω +iσ of largest imaginary part. The flow is said to be subcritical (stable) if σ
is strictly negative, neutrally stable if σ = 0, and supercritical (unstable) if σ>0.
The determination of the least damped (or most unstable) modes for a given
wavelength β amounts to finding the eigenvalue Ω and corresponding eigenvector, by
integrating the governing equations directly in the time domain. More details on the
linear stability analysis and the residual algorithm used to determine the dominant
eigenmode are presented in Br
`
es & Colonius (2007).
2.4. Validation
The implementation of the numerical methods, the grid convergence, and boundary
treatments were successfully validated following the procedures described by Rowley
Three-dimensional cavity flows 313
3D unstable
2D unstable
2D & 3D stable
1000
1250
1500
1750
2000
0.1 0.2 0.3 0.4 0.5 0.6
M
Re
D
Figure 2. Schematic of the neutral stability curve for the cavity run series 2M (L/D =2,
L/θ
0
=52.8). Results from two-dimensional nonlinear simulations: two-dimensional stable (䊐)
and unstable (
䊏); Results from three-dimensional linear analysis: three-dimensional stable (◦)
and unstable (
䊉). The different shaded regions indicate the approximate stability transitions.
The critical conditions are estimated by linear interpolation between stable and unstable
conditions. Details of each run are given in the Appendix.
et al. (2002). For brevity, we do not report the results here. The reader is referred to
Br
`
es (2007) for details. To validate the linear stability results, a direct approach was
also considered (e.g. Br
`
es 2007), where the eigenmodes are directly searched for using
an Arnoldi method developed in the ARPACK software (e.g. Lehoucq et al. 2007),
rather than isolated through long-time integration. In practice, the use of ARPACK
was significantly limited by the size and complexity of our problem. The software was
therefore only used to validate our time-domain methods. As expected, the dominant
eigenmode and corresponding eigenvalue computed with ARPACK for the same test
case were in excellent agreement with the results of the linear stability analysis.
3. Linear stability results
As described in the previous section, the three-dimensional linear stability analysis
relies on the existence of subcritical conditions with a steady two-dimensional basic
flow
¯
q(x,y), an exact solution of the two-dimensional NS equations. However, for
most experiments and realistic flight conditions, the flow parameters would be such
that Rossiter modes do occur and eventually saturate into a periodically oscillating
flow. Also, it must be acknowledged that the presence of three-dimensional instabilities
is likely to alter the two-dimensional basic flow on which the present linear analysis
is based. With this in mind, our approach here is to investigate the three-dimensional
linear stability of a given base flow, regardless of potential interactions. Such an
approach has been widely use to predict the stability and growth rate of boundary
layers, for instance. As discussed in § 5and§ 6, the features observed in the linear
results are in fact relevant to full nonlinear simulations and experiments.
The stability analysis is conducted as follows: given a cavity configuration and
different flow conditions, several two-dimensional simulations are performed to
construct the estimated neutral stability curve for the two-dimensional instabilities of
the basic cavity flow (e.g. figure 2). The neutral stability curves are the starting point of
the three-dimensional stability analysis: the goal here is to investigate whether three-
dimensional instability takes place before the onset of two-dimensional instabilities.
For subcritical cases, the two-dimensional steady flow
¯
q is then extracted from the
