  Protoplasmic streaming in plant cells is directly visible in the cases of
\textit{Chara corallina} and \textit{Nitella flexilis}, and this streaming is
understood to play a role in the transport of biological materials. For this
reason, related studies have focused on molecular transportation from a fluid
mechanics viewpoint. However, the experimentally observed distribution of the
velocity along the flow direction $x$, which exhibits two peaks at $V_x\!=\!0$
and at a finite $V_x(\not=\!0)$, remains to be studied. In this paper, we
numerically study whether this behavior of the flow field can be simulated by a
2D stochastic Navier-Stokes (NS) equation for Couette flow, in which random
Brownian force is assumed. We present the first numerical evidence that these
peaks are reproduced by the stochastic NS equation, which implies that the
Brownian motion of the fluid particles plays an essential role in the emergence
of these peaks in the velocity distribution. We also find that the position of
the peak at $V_x(\not=\!0)$ moves with the variation in the strength $D$ of the
random Brownian force, which also changes depending on physical parameters such
as the kinematic viscosity, boundary velocity and diameter of the plant cells.
