Velocity structure functions in multiphase turbulence: interpreting
  kinematics of H$\alpha$ filaments in cool core clusters
  The central regions of cool-core galaxy clusters harbour multiphase gas, with
gas temperatures ranging from $10$ $\mathrm{K}$--$10^7$$\mathrm{K}$. Feedback
from active galactic nuclei (AGNs) jets prevents the gas from undergoing a
catastrophic cooling flow. However, the exact mechanism of this feedback energy
input is unknown, mainly due to the lack of velocity measurements of the hot
phase gas. However, recent observations have measured the velocity structure
functions ($\mathrm{VSF}$s) of the cooler molecular ($\sim10$$\mathrm{K}$) and
H$\alpha$ filaments ($\sim10^4$$\mathrm{K}$) and used them to indirectly
estimate the motions of the hot phase. In the first part of this study, we
conduct high-resolution ($384^3$--$1536^3$ resolution elements) simulations of
homogeneous isotropic subsonic turbulence, without radiative cooling. We
analyse the second-order velocity structure functions ($\mathrm{VSF}_2$) in
these simulations and study the effects of varying spatial resolution, the
introduction of magnetic fields, and the effect of projection along the line of
sight (LOS) on it. In the second part of the study, we analyse high-resolution
($768^3$ resolution elements) idealised simulations of multiphase turbulence in
the intracluster medium (ICM) from Mohapatra et al 2021. We compare the
$\mathrm{VSF}_2$ for both the hot ($T\sim10^7$$\mathrm{K}$) and cold
($T\sim10^4$$\mathrm{K}$) phases and find that their amplitude depends on the
density contrast between the phases. They have similar scaling with separation,
but introducing magnetic fields steepens the $\mathrm{VSF}_2$ of only the cold
phase. We also find that projection along the LOS steepens the $\mathrm{VSF}_2$
for the hot phase and mostly flattens it for the cold phase.
