Are water molecules orientation absolutely independent of the flow direction?

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, $p approx 6times10^{-30},mathrm{C,m} approx 0.4,e,Å$, 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{25rm,meV}{18,mathrm{GeV}/c^2} &approx frac 43times10^{-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 | % &= 6times10^{-30} mathrm{C,m}cdot 3times10^2mathrm{m/s} cdot frac12times10^{-4}mathrm T % &approx 9times10^{-32},mathrm J %timesfrac{1rm,eV}{1.6times10^{-19}rm,J} &approx frac 58times10^{-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.