Flow Down a Wavy Inclined Plane

Abstract

Under certain conditions, flow down an inclined plane destabilizes and a persistent series of interfacial waves develop. An interest in determining under what conditions a flow becomes unstable and how the interface develops has motivated researchers to derive several models for analyzing this problem.

The first part of this thesis compares three models for flow down a wavy, inclined plane with the goal of determining which best predicts features of the flow. These models are the shallow-water model (SWM), the integral-boundary-layer (IBL) model, and the weighted residual model (WRM). The model predictions for the critical Reynolds number for flow over an even bottom are compared to the theoretical value, and the WRM is found to match the theoretical value exactly. The neutral stability curves predicted by the three models are compared to two sets of experimental data, and again the WRM most closely matches the experimental data. Numerical solutions of the IBL model and the WRM are compared to numerical solutions of the full Navier-Stokes equations; both models compare well, although the WRM matches slightly better. Finally, the critical Reynolds numbers for the IBL model and the WRM for flow over a wavy incline are compared to experimental data. Both models give results close to the data and perform equally well. These comparisons indicate that the WRM most accurately models the flow.

In the second part of the thesis, the WRM is extended to include the effects of bottom heating and permeability. The model is used to predict the effect of heating and permeability on the stability of the flow, and the results are compared to theoretical predictions from the Benney equation and to a perturbation solution of the Orr-Sommerfeld equation from the literature. The results indicate that the model does faithfully predict the theoretical critical Reynolds number with heating and permeability, and both effects destabilize the flow. Finally, numerical simulations of the model equations are compared to full numerical solutions of the Navier-Stokes equations for the case with bottom permeability. The results are found to agree, which indicates that the WRM remains appropriate when permeability is included.