Conservation laws models in networks and multiscale flow optimization.
Jean Medard Techoukouegno. Ngnotchouye · 2011
The flow of fluids in a network is of practical importance in gas, oil and water transport for industrial and domestic use. When the flow dynamics are understood, one may be interested in the control of the flow formulated as follows: given some fluid properties at a final time, can one determine the initial flow properties that lead to the desired flow properties? In this thesis, we first consider the flow of a multiphase gas, described by the driftflux model, in a network of pipes and that of water, modeled by the shallow water equations, in a network of rivers. These two models are systems of partial differential equations of first order generally referred to as systems of conservation laws. In particular, our contribution in this regard can be summed up as follows: For the drift-flux model, we consider the flow in a network of pipes seen mathematically as an oriented graph. We solve the standard Riemann problem and prove a well posedness result for the Riemann problem at a junction. This result is obtained using coupling conditions that describe the dynamics at the intersection of the pipes. Moreover, we present numerical results for standard pipes junctions. The numerical results and the analytical results are in agreement. This is an extension for multiphase flows of some known results for single phase flows. Thereafter, the shallow water equations are considered as a model for the flow of water in a network of canals. We analyze coupling conditions at the confluence of rivers, precisely the conservation of mass and the equality of water height at the intersection, and implement these results for some classical river confluences. We also consider the case of pooled stepped chutes, a geometry frequently utilized by dams to spill floodwater. Here we consider an