5.2 Solution Scheme
The scheme for modelling 1D open channels is based on a numerical solution of the 1D unsteady St Venant fluid flow equations (momentum and continuity) including the inertia terms. The 1D solution uses an explicit finite difference, second-order, Runge-Kutta solution technique (Morrison & Smith, 1978) for the 1D SWE of continuity and momentum as given by the equations below. The equations contain the essential terms for modelling periodic long waves in estuaries and rivers, that is: wave propagation; advection of momentum (inertia terms) and bed friction (Manning’s equation).
1D Continuity:
\[\begin{equation} w \frac{\partial h}{\partial t} + \frac{\partial (A u)}{\partial x} = S \tag{5.1} \end{equation}\]
1D Momentum:
\[\begin{equation} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + g \frac{\partial (z+h)}{\partial x} + g \frac{n^2}{R^\frac{4}{3}}|u|u + \frac{k}{2 \mathrm{\Delta}x}|u|u = S_u \tag{5.2} \end{equation}\]
Where
- \(w\) is the channel width at the water surface
- \(h\) is the water depth in the channel
- \(x\) and \(t\) are channel flow dimension and time respectively
- \(z\) is the channel bed elevation
- \(A\) is the cross-sectional flow area (up to the water surface)
- \(u\) is the cross-sectional flow area averaged water velocity
- \(n\) is the Manning coefficient for bed friction (units time per cube-root (unit length))
- \(R\) is the hydraulic radius
- \(S\) is the lineal water source term (volume per unit time per unit length), usually lateral inflow/outflow
- \(S_u\) is the lineal momentum source term (force per unit volume per unit fluid density)
- \(k\) is an energy loss coefficient, used for local energy losses
- \(\mathrm{\Delta}x\) is the channel length
Note: For 1D structures, velocities are computed based on empirical equations instead of using the 1D momentum equation (see Section 5.7 for more details). Also note that as bed-friction and energy losses are explicitly stated, the additional momemtum source \(S_u\) is typically not used.