  We perform as the first time hydrodynamic simulations to study the properties
of hot accretion flow (HAF) around a neutron star (NS). The energy carried by
the HAF will eventually be radiated out at the surface of the NS. The emitted
photons can propagate inside the HAF and cool the HAF via Comptonization. We
find that the Compton cooling can affect the properties of HAF around a NS
significantly. We define the Eddington accretion rate as $\dot M_{\rm
Edd}=10L_{\rm Edd}/c^2$, with $L_{\rm Edd}$ and $c$ being the Eddington
luminosity and the speed of light, respectively. We define $\dot m$ as the mass
accretion rate at the NS surface in unit of $\dot M_{\rm Edd}$. When $\dot m >
10^{-4}$, Compton cooling can effectively cool the HAF and suppress wind.
Therefore, the mass accretion rate is almost a constant with radius. The
density profile is $\rho \propto r^{-1.4}$. When $\dot m < 10^{-4}$, the
Compton cooling effects become weaker, wind becomes stronger, accretion rate is
proportional to $r^{0.3-0.5}$. Consequently, the density profile becomes
flatter, $\rho \propto r^{-1 \sim -0.8}$. When $\dot m < 10^{-6}$, the Compton
cooling effects can be neglected. We find that with a same accretion rate, the
temperature of HAF around a NS is significantly lower than that of HAF around a
black hole (BH). Also, the Compton $y-$parameter of HAF around a NS is
significantly smaller than that of HAF around a BH. This result predicts that
HAF around a NS will produce a softer spectrum compared to HAF around a BH,
which is consistent with observations.
