Many turbulent astrophysical, geophysical, and engineering flows are dominated by robust coherent structures on the largest scales, such as convection rolls in the atmosphere. One of the challenges in understanding such flows is that these structures have different shapes and dynamics depending on boundary conditions, so they defy universal statistical models of bulk turbulence. On the other hand, the organization imposed onto the turbulent flow by these structures holds some promise for the development of solvable models. I will present an approach to solvable universal models based on using empirically-known flow structures as approximate solutions to the Navier-Stokes equations, which leads to low dimensional dynamical systems models. As an example, I will present results of Rayleigh-Benard convection experiments, in which a container is filled with water and heated from below. Buoyancy drives a flow which organizes into a roll-shaped circulation which spontaneously breaks the symmetry of the system. As a consequence, this roll exhibits a wide range of dynamics including erratic meandering, spontaneous flow cessations, and several oscillation modes. A simple model consisting of stochastic ordinary differential equations quantitatively reproduces these observed flow dynamics. The effects of boundary geometry and different forcings are physically represented by different model terms. These results suggest that we may be able to develop general and relatively easy to solve models for a wide range of turbulent flows with potential applications to climate, weather, and other turbulent flow problems.