A type of network analysis successfully demonstrated to quantify growth and development in ecosystems is applied to the purely physical phenomenon of fluid flow. Simple two-dimensional models of fluid flow are created and represented as networks of nodes and transfers or flows between nodes. Modeling flow data as a network enables the calculation of indicators or indices that quantify the activity and organization of the represented flow field. The method of cellular automata is used to create three flow field examples, two of which introduce obstacles in the flow field to disrupt the otherwise uniform flow. Four well understood examples from fluid dynamics are described analytically and then analyzed as networks. These conceptual examples of fluid flow demonstrate the utility of network analysis as a method of quantitatively characterizing complex patterns of fluid flow.