| Abstract: |
| Modelling of surface wave motion in a fluid is normally based on classical
systems which are obtained in the framework of irrotational flow. In such a
context, the influence of vorticity is completely disregarded in the formulation
of the governing equations. Although this consideration is justified in many
circumstances, there are observed cases in the field of meteorology and
hydrology in which this approach is unsuitable due to the presence of currents.
The natural action of currents invariably create shear at the bottom of the
channel and in the realm of shallow-water theory, the shear effect can become
a dominant feature in the wave dynamics.
Recently, the classical shallow water equations have been modified to
incorporate the action of vorticity in the flow dynamics leading to the system
\begin{equation*}\label{SWSystem}
\begin{split}
\partial_tH\, &+\, \partial_x\left(\frac{\Gamma}{2}H^2 + uH\right) = 0,\
\partial_t\left(\frac{\Gamma}{2}H^2 + uH\right) &+\, \partial_x\left(\frac{\Gamma^2}{3}H^3 + \Gamma uH^2 + u^2H + \frac{1}{2}gH^2\right) =0.
\end{split}
\end{equation*}
A steady state solution of a flow with horizontal shear current is
investigated. In particular, we
analyse a stationary jump of a linear shear flow by using the Froude number
which is defined in terms of depth-averaging integral in other to account for
the average flow velocity over the entire fluid depth. It will be shown that in this way, the flow is
uniquely determined by three parameters, namely, the total fluid depth, the
Froude number and a non-dimensional measurable quantity which determines the
strength of the vorticity. |
|