  Understanding the physical structures of the accreated matter very close to
the black hole in quasars and active galactic nucleus (AGNs) is an important
milestone to constrain the activities occurring in their centers. In this
paper, we numerically investigate the effects of the asymptotic velocities on
the physical structures of the accretion disk around the Kerr and
Einstein-Gauss-Bonnet (EGB) rapidly rotating black holes. The Bondi-Hoyle
accretion is considered with a falling gas towards the black hole in upstream
region of the computational domain. The shock cones are naturally produced in
the downstream part of the flow around both black holes. It is found that the
structure of the cones and the amount of the accreated matter depend on
asymptotic velocity $V_{\infty}$ (Mach number) and the types of the gravities
(Kerr or EGB). Increasing the Mach number of the inflowing matter in the
supersonic region causes the shock opening angle and accretion rates getting
smaller because of the rapidly falling gas towards the black hole. The EGB
gravity leads to an increase in the shock opening angle of the shock cones
while the mass accretion rates $\dot{M}$ are decreasing in EGB gravity with a
Gauss-Bonnet (GB) coupling constant $\alpha$. It is also confirmed that
accretion rates and drag forces are significantly altered in the EGB gravity.
Our numerical simulation results could be used to identify the accreation
mechanism and physical properties of the accretion disk and black hole in the
observed $X-$ rays such as NGC $1313$ $X-1$ and $1313$ $X-2$ and MAXI
$J1803-298$.
