Saint Venant Equations¶

Solving the problem of an arbitrary flow of water through three dimensional canals is computationally impractical. Fortunately, the geometry of pipes and canals allows for a set of approximations, which naturally fit the problem space. The approximations rely on the following assumptions:

Water is incompressible and homogeneous, i.e., it has negligible variations in density
The bottom slope of pipes and canals is small, so that the cosine of the angle it makes with the horizontal may be taken as 1.
Wave lengths are large compared to the water depth. This ensures that the flow everywhere can be regarded as having a direction parallel to the bottom, i.e. vertical accelerations can be neglected and a hydrostatic pressure variation along the vertical can be assumed.
There is no acceleration of water perpendicular to the flow direction of a reach.

With these assumptions, two simplified equations can be derived from the Navier-Stokes equations, which are usually called the Saint-Venant equations, or Shallow Water equations:

\[ \frac{\partial q}{\partial x} + \frac{\partial A_{fl}}{\partial t} = q_{in} \]

Equation: The mass, or continuity, equation.

\[ \frac{\partial q}{\partial t} + \frac{\partial\left( \alpha \dfrac{q^{2}}{A_{fl}} \right)}{\partial x} + gA_{fl}\frac{\partial h}{\partial x} + gA_{fl}I_{f} = \frac{f}{\rho_{w}} \]

Equation: The momentum equation.

where,

\(q\) — discharge
\(A_{fl}\) — flow area, also called cross section area
\(q_{in}\) — lateral inflow (per length unit)
\(h\) — water level
\(\alpha\) — momentum distribution coefficient
\(I_{f}\) — flow resistance term, in the form of a friction slope
\(f\) — momentum forcing (per length unit)
\(\rho_{w}\) — density of water

The continuity equation is called the mass equation or continuity equation and expresses conservation of mass; the volume of water which is added, \(\partial q\), in reach section of length \(\partial x\), is balanced by an increase in cross section area \(\partial A_{fl}\) (volume).

The momentum equation is called the momentum equation and expresses conservation of momentum; the first two terms on the left side of the momentum equation represent the inertia forces (local and convective acceleration), the third term represents pressure forces due to gravity, and the last term represent resistance/friction. The right-hand side represents an external momentum forcing.

There are different formulations of the flow resistance term in the momentum equation, which are described in Resistance.

The derivation of the continuity and momentum equations is given in numerous text books on hydrodynamics. A version is also provided in Deriving the Saint-Venant Equations.

The Saint-Venant equations are only valid for free surface flows. Since pressurised flow in closed reaches is an important part of the 1D problem space, an approximate treatment of pressurised flow has been adopted. The approach is described in Pressurised Flow.

To solve the Saint-Venant equations, a finite difference scheme is implemented. An overview of this scheme is given in Numerical Schemes, and a more detailed description is provided in Conservation of momentum. This solution scheme is only accurate for sub-critical flows (Froude number less than 1). For super-critical flows (Froude number larger than 1), a simpler approximation where non-linear terms are suppressed provides an accurate flow description. Super Critical Flow describes how the transition to super-critical flow is handled.

Finally, it should be noted that the solution scheme does not include a detailed description of hydraulic jumps. However, flow conditions both upstream and downstream of a hydraulic jump are accurately described.