  We numerically study the primordial black hole (PBH) formation by an
isocurvature perturbation of a massless scalar field on super Hubble scales in
the radiation-dominated universe. As a first step we perform simulations of
spherically symmetric configurations. For the initial condition, we employ the
spatial gradient expansion and provide the general form of the growing mode
solutions valid up through the second order in this expansion. The initial
scalar field profile is assumed to be Gaussian with a characteristic comoving
wavenumber $k$; $\sim\exp(-k^2R^2)$, where $R$ is the radial coordinate. We
find that a PBH is formed for a sufficiently large amplitude of the scalar
field profile. Nevertheless, we find that the late time behavior of the
gravitational collapse is dominated by the dynamics of the fluid but not by the
scalar field, which is analogous to the PBH formation from an adiabatic
perturbation in the radiation-dominated universe.
