  Linear instability of high-speed boundary layers is routinely examined
assuming quiescent edge conditions, without reference to the internal structure
of shocks or to instabilities potentially generated in them. Our recent work
has shown that the kinetically modeled internal nonequilibrium zone of straight
shocks away from solid boundaries exhibits low-frequency molecular
fluctuations. The presence of the dominant low frequencies observed using the
Direct Simulation Monte Carlo (DSMC) method has been explained as a consequence
of the well-known bimodal probability density function (PDF) of the energy of
particles inside a shock. Here, PDFs of particle energies are derived in the
upstream and downstream equilibrium regions, as well as inside shocks, and it
is shown for the first time that they have the form of the non-central
chi-squared (NCCS) distributions. A linear correlation is proposed to relate
the change in the shape of the analytical PDFs as a function of Mach number,
within the range $3 \le M \le 10$, with the DSMC-derived average characteristic
low-frequency of shocks, as computed in our earlier work. At a given Mach
number $M=7.2$, varying the input translational temperature in the range $89
\le T_{tr,1}/(K) \le 1420$, it is shown that the variation in DSMC-derived
low-frequencies is correlated with the change in most-probable-speed inside
shocks at the location of maximum bulk velocity gradient. Using the proposed
linear functions, average low-frequencies are estimated within the examined
ranges of Mach number and input temperature and a semi-empirical relationship
is derived to predict low-frequency oscillations in shocks. Our model can be
used to provide realistic physics-based boundary conditions in receptivity and
linear stability analysis studies of laminar-turbulent transition in high-speed
flows.
