PeterDonis said:
An observer sits in a chair that is bolted to a disc of radius ##R## that rotates at ##\omega##. He is thus constrained to face the rotation axis. His four velocity expressed in non-rotating cylindrical polars centered on the rotation axis is $$\begin{eqnarray*}v^a&=&(v^t,v^r,v^\phi,v^z)\\&=&(\gamma_\omega,0,\gamma_\omega\omega ,0)\end{eqnarray*}$$where ##\gamma_\omega## is the Lorentz gamma factor associated with speed ##\omega R## and the metric is ##\mathrm{diag}(1,-1,-r^2,-1)##. At any event he can define three unit spacelike vectors, #m$$\begin{eqnarray*}r'^a&=&(0,1,0,0)\\
\phi'^a&=&(\gamma_\omega \omega R,0,\gamma_\omega/R,0)\\
z'^a&=&(0,0,0,1)\end{eqnarray*}$$using the same coordinate system. These are mutually orthogonal and also orthogonal to his four velocity, so these four four vectors form a tetrad at the event.
If there is a mark painted somewhere on the disc the observer can point to it (if very, very strong), or even point in the direction where he sees it to be (that's a different direction, but I don't care here). If he keeps pointing at it, I think this action matches the intuitive notion of "pointing in the same direction" in this case. The observer could always express the direction he's pointing as ##Ar'^a+B\phi'^a##, and we can define a one-parameter family of geodesics with these tangent vectors extending from the observer at all events he passes through. These geodesics will all pass through the same point (i.e. Langevin worldline) on the rim of the disc. The segments of geodesic from observer to point will all have the same length, and any pair of geodesics separated at the observer by Langevin proper time ##\Delta \tau## will also be separated by Langevin proper time ##\Delta\tau## at their other ends.
Note that the direction is not only expressible as a sum of those two tetrad vectors, the coefficients ##A## and ##B## are constants. Thus the tetrad field ##v^a, r'^a,\phi'^a,z'^a## is what we want to transport along the Langevin worldlines. This is not parallel transport (which would keep the tetrad fixed in an inertial frame) nor Fermi-Walker transport (which would precess with respect to these). As I said, I don't know if it has a name.
A couple of observations. First, many tetrads are possible. But the observer's four acceleration is parallel to ##r'^a## and he will feel a torque about ##z'^a## since he is not Thomas precessing, so this is a physically motivated choice. Second, given one tetrad, the rest of the field can be generated by the rotational and time-translational symmetries of this scenario.
Finally, I've been careful to not call my observer a Langevin observer. He is following a Langevin worldline but he is not Fermi-Walker transporting his tetrad, and I am not sure if Langevin observers are formally points or tetrads.