  In this paper, we report the numerical results obtained using the Langevin
Navier-Stokes (LNS) simulation of the velocity distribution of
three-dimensional (3D) protoplasmic streaming in plant cells, such as those of
{\it Nitella flexilis}. The LNS simulations are performed on 3D cylinders
discretized by regular cubes in which fluid velocities are activated by
boundary velocities parallel and nonparallel to the longitudinal direction and
a random Brownian force with strength $D$. We find that, for a finite $D$, the
velocity distribution $h(V), V\!=\!|\vec{V}|$, has two different peaks at a
small non-zero $V$ and a finite $V$, and the distribution $h(V_z)$ for $|V_z|$
along the longitudinal direction also has a peak at finite $V_z$. These results
are in good agreement with the reported velocity distributions observed using
laser Doppler velocimetry. Moreover, we study the effects of the Brownian force
on biological material mixing and find that mixing along the $\vec{V}$
direction enhanced by the nonparallel circular motion is further improved by
the Brownian force in the experimentally relevant region of $D$. In addition,
the experimentally relevant $D$ is found to be consistent with the expectation
from the fluctuation dissipation relation between the random stress and
viscosity in the LNS equation of Landau and Lifschitz for incompressible
fluids.
