Unsteady simulations of horizontal ribbon growth of silicon were performed that included both Marangoni and buoyancy effects. An hp-finite element method (FEM) was used with an arbitrary Lagrangian-Eulerian moving mesh method to track the solidification interface. The hp-FEM was fifth-order accurate in space, third-order accurate in time and used a mesh adaptation scheme. A chaotic flow was observed that was dominated by Marangoni-driven jets forming near the points of minimum temperature on the free surface. Although buoyancy effects were not dominant, they were essential to observe an unstable chaotic flow. This chaotic flow led to oscillations of the leading edge of the sheet. These fluctuations resulted in the formation of corrugations on the top surface of the sheet and temporal variations in growth rate. Also, nonuniformities with larger amplitude and wavelength developed on the bottom of the sheet. These nonuniformities resulted in a sheet of varying thickness. All of these phenomena agreed with observations from experiments. To reduce the flow instabilities, numerical studies were performed by modifying the design parameters including the depth of the crucible, and the parameters determining the convective cooling flux.