  Autonomous marine vessels are expected to avoid inter-vessel collisions and
comply with the international regulations for safe voyages. This paper presents
a stepwise path planning method using stream functions. The dynamic flow of
fluids is used as a guidance model, where the collision avoidance in static
environments is achieved by applying the circular theorem in the sink flow. We
extend this method to dynamic environments by adding vortex flows in the flow
field. The stream function is recursively updated to enable on the fly waypoint
decisions. The vessel avoids collisions and also complies with several rules of
the Convention on the International Regulations for Preventing Collisions at
Sea. The method is conceptually and computationally simple and convenient to
tune, and yet versatile to handle complex and dense marine traffic with
multiple dynamic obstacles. The ship dynamics are taken into account, by using
B\'{e}zier curves to generate a sufficiently smooth path with feasible
curvature. Numerical simulations are conducted to verify the proposed method.
