  The stability properties of two- (2D) and three-dimensional (3D) compressible
flows over a rectangular cavity with length-to-depth ratio of $L/D=6$ is
analyzed at a free stream Mach number of $M_\infty=0.6$ and depth-based
Reynolds number of $Re_D=502$. In this study, we closely examine the influence
of three-dimensionality on the wake-mode that has been reported to exhibit
high-amplitude fluctuations from the formation and ejection of large-scale
spanwise vortices. Direct numerical simulation (DNS) and bi-global stability
analysis are utilized to study the instability characteristics of the
wake-mode. Using the bi-global stability analysis with the time-average flow as
the base state, we capture the global stability properties of the wake-mode at
a spanwise wavenumber of $\beta=0$. To uncover spanwise effects on the 2D
wake-mode, 3D DNS are performed with cavity width-to-depth ratio of $W/D=1$ and
$2$. We find that the 2D wake-mode is not present in the 3D cavity flow for a
wider spanwise setting with $W/D=2$, in which spanwise structures are observed
near the rear region of the cavity. These 3D instabilities are further
investigated via bi-global stability analysis for spanwise wavelengths of
$\lambda/D=0.5-2.0$ to reveal the eigenspectra of the 3D eigenmodes. Based on
the findings of 2D and 3D global stability analysis, we conclude that the
absence of the wake-mode in 3D rectangular cavity flows is due to the release
of kinetic energy from the spanwise vortices to the streamwise vortical
structures that develops from the spanwise instabilities.
