Are water molecules orientation absolutely independent of the flow direction? Or is there a bias toward a specific angle in regards to the direction of the current?
 A: These types of questions are tempting to ask in a yes-no way, and you currently have an answer that says "yes" and and answer that says "no."
The physicist's approach is to ask how big the biggest effect might be; let's try that.
The current answer which proposes an aligning effect suggests an interaction between the electric dipole moment of the water molecule, $p \approx 6\times10^{-30}\,\mathrm{C\,m} \approx 0.4\,e\,Å$, and the Earths' magnetic field, via the Lorentz force, $\vec F = q \vec v \times \vec B$.  The energy associated with this interaction is what you get if the force interacts over the length scale of the molecule, which has a bond length of about $1\,Å$.  The typical thermal velocities obey $kT \approx mv^2$, or
\begin{align}
v^2 \sim \frac{kT}{m} = \frac{25\rm\,meV}{18\,\mathrm{GeV}/c^2}
&\approx \frac 43\times10^{-12}\ c^2
\\
v &\sim 10^{-6}\ c \approx 300 \rm\,m/s
\end{align}
So a typical Lorentz-force polarization energy would be
\begin{align}
U &\approx | p v B |
%\\ &= 6\times10^{-30} \mathrm{C\,m}\cdot 3\times10^2\mathrm{m/s} \cdot \frac12\times10^{-4}\mathrm T
%\\ &\approx 9\times10^{-32}\,\mathrm J
%\times\frac{1\rm\,eV}{1.6\times10^{-19}\rm\,J}
\\ &\approx \frac 58\times10^{-13} \rm\,eV
\approx 60 \rm\,feV
\end{align}
Those are femto-eV.
But the water molecule's rotational degree of freedom also has milli-eV energy sloshing around.  The ratio of the aligned and un-aligned populations will go like the Boltzmann factor for this energy difference,
$$
e^{\Delta E/kT} = e^{\text{femto/milli}} = 1 + 10^{-12},
$$
that is, a part-per-trillion difference.  I've been involved in several experiments looking for part-per-billion asymmetries; each one took ten years.  A few parts per trillion is a small effect, even if you go back through my arithmetic and futz around with some missing factors of two.
What's more, the preferred direction $\vec v \times \vec B$ is only well-defined if most of the water molecules are moving in generally the same direction.  That only happens if the rate of flow is much faster than the typical thermal velocity --- which doesn't really happen unless the flow approaches the speed of sound in water.
If you tried to enhance the effect --- by, say, shooting a hypersonic jet of water through the bore of ten-tesla magnet, to bring the asymmetry up to the part-per-million range --- you'd probably just learn something sneaky about hydrogen bonding.
A: No, there isn't a bias. Water molecules jiggle around so much in the liquid that any long-range order (like dipoles getting all lined up) has no chance to form, and if it did form for any reason, it would be very quickly erased.
A: Because water molecule has a dipole moment, in the earth magnetic field it is subjected to the Lorentz force.
