Shallow Water Wave Equations

Dr Steve Roberts

Description of Project

Depth averaged models of shallow water flow over discontinuous terrain would provide for the efficient simulation of many practical physical situations. Steps are used in spillways to provide an efficient method for dissipating energy. Rivers which breach their banks can be modelled as a flow over discontinuous terrain. In estuaries, tidal flows play an important environmental role providing nutrients to plants and organisms in the tidal channels and on the mud flats. Flows from the well defined channels that incise the flat and wide estuary can be considered as flow over a discontinuity. The standard shallow water wave equation

    h uh     t  +      uh u2h+ 1 2 gh2      x  =      0 − gh   dz dx       

(0.1)

is a conservation law for fluid mass and horizontal momentum, together with a momentum dissipation term dependent on the slope of the terrain. This equation is obtained as a depth averaged approximation of the full two or three dimensional Euler equation. Flow over discontinuous terrain can be obtained as a limit of infinite slope. On the other hand, it has been reported by Alcrudo and Benkhaldoun that an alternative form of the shallow water wave equation

    h u/g     t  +      uh h+ 1 2 u2/g+z      x  =     0 0    

(0.2)

in which energy is the conserved quantity, provides a good approximation to the flow over discontinuous terrain. The alternative forms of the equations lead to different prospects for the numerical solution of the shallow water equation over discontinuous terrain.

Project

Investigate the use of depth averaged equations used to model shallow water wave equations. Investigate the derivation of the equations. Consider numerical methods. Investigate the use of Momentum and Energy Conserving Equations when approximating flow over a step.