  We propose a novel framework of generalised Petrov-Galerkin Dynamical Low
Rank Approximations (DLR) in the context of random PDEs. It builds on the
standard Dynamical Low Rank Approximations in their Dynamically Orthogonal
formulation. It allows to seamlessly build-in many standard and well-studied
stabilisation techniques that can be framed as either generalised Galerkin
methods, or Petrov-Galerkin methods. The framework is subsequently applied to
the case of Streamine Upwind/Petrov Galerkin (SUPG) stabilisation of
advection-dominated problems with small stochastic perturbations of the
transport field. The norm-stability properties of two time discretisations are
analysed. Numerical experiments confirm that the stabilising properties of the
SUPG method naturally carry over to the DLR framework.
