  The flow field in a two-dimensional three-ramp hypersonic mixed-compression
inlet in a freestream Mach number of $M_\infty=5$ is numerically solved to
understand the unsteady throttling dynamics. Throttling conditions are
simulated by varying the exit area of the isolator in the form of plug insets.
Different throttling ratios between $0\leq \zeta \leq 0.7$ in steps of 0.1 are
considered. No unsteadiness is observed for $\zeta\leq 0.2$ and severe
unsteadiness is found for $0.3 \leq \zeta \leq 0.7$. The frequency of
unsteadiness ($f$) increases rapidly with $\zeta$. As $\zeta$ increases, the
amount of reversed mass inside the isolator scales with the frequency and the
exit mass flow rate. A general framework is attempted to scale the unsteady
events based on the gathered knowledge from the numerical study. The
inlet-isolator flow is modeled as an oscillating flow through a duct with known
upstream design conditions like the freestream Mach number ($M_\infty$) and the
isolator inlet Mach number ($M_i$). Factors like the mass occupied by the duct
volume, the characteristic unsteady frequency, throttling ratio, and the exit
mass flow rate through the duct are used to form a non-dimensional parameter
$\beta$, which scales with the upstream design parameter $\xi=M_i/M_\infty$.
The scaling parameters are further exploited to formulate a semi-empirical
relation using the existing experimental results at different throttling ratios
from the open literature. The unsteady frequencies from the present
two-dimensional numerical exercise are also shown to agree with the proposed
scaling and the resulting semi-empirical relation.
