Shallow water equations

The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface).

[1] The shallow-water equations in unidirectional form are also called (de) Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below).

Shallow-water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height.

[3] Expanding the derivatives in the above using the product rule, the non-conservative form of the shallow-water equations is obtained.

Since velocities are not subject to a fundamental conservation equation, the non-conservative forms do not hold across a shock or hydraulic jump.

Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density):

This is called geostrophic balance, and is equivalent to saying that the Rossby number is small.

The 1-D Saint-Venant equations contain to a certain extent the main characteristics of the channel cross-sectional shape.

The 1-D equations are used extensively in computer models such as TUFLOW, Mascaret (EDF), SIC (Irstea), HEC-RAS,[5] SWMM5, InfoWorks,[5] Flood Modeller, SOBEK 1DFlow, MIKE 11,[5] and MIKE SHE because they are significantly easier to solve than the full shallow-water equations.

Common applications of the 1-D Saint-Venant equations include flood routing along rivers (including evaluation of measures to reduce the risks of flooding), dam break analysis, storm pulses in an open channel, as well as storm runoff in overland flow.

The system of partial differential equations which describe the 1-D incompressible flow in an open channel of arbitrary cross section – as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is:[6] and where x is the space coordinate along the channel axis, t denotes time, A(x,t) is the cross-sectional area of the flow at location x, u(x,t) is the flow velocity, ζ(x,t) is the free surface elevation and τ(x,t) is the wall shear stress along the wetted perimeter P(x,t) of the cross section at x.

Closure of the hyperbolic system of equations (1)–(2) is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional area A and the surface elevation ζ at each position x.

For non-moving channel walls the cross-sectional area A in equation (1) can be written as:

Here σ is the height above the lowest point in the cross section at location x, see the cross-section figure.

So σ is the height above the bed level zb(x) (of the lowest point in the cross section):

For a rectangular and prismatic channel of constant width B, i.e. with A = B h and c = √gh, the Riemann invariants are:[9]

The Riemann invariants and method of characteristics for a prismatic channel of arbitrary cross-section are described by Didenkulova & Pelinovsky (2011).

[12] The characteristics and Riemann invariants provide important information on the behavior of the flow, as well as that they may be used in the process of obtaining (analytical or numerical) solutions.

[13][14][15][16] In case there is no friction and the channel has a rectangular prismatic cross section, the Saint-Venant equations have a Hamiltonian structure.

It is numerically challenging to solve, but is valid for all channel flow scenarios.

The dynamic wave is used for modeling transient storms in modeling programs including Mascaret (EDF), SIC (Irstea), HEC-RAS,[18] InfoWorks_ICM Archived 2016-10-25 at the Wayback Machine,[19] MIKE 11,[20] Wash 123d[21] and SWMM5.

Models that use the diffusive wave assumption include MIKE SHE[22] and LISFLOOD-FP.

The kinematic wave is valid when the change in wave height over distance and velocity over distance and time is negligible relative to the bed slope, e.g. for shallow flows over steep slopes.

The x-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x-direction – can be written as:

The local acceleration (a) can also be thought of as the "unsteady term" as this describes some change in velocity over time.

Somewhat smaller wavelengths can be handled by extending the shallow-water equations using the Boussinesq approximation to incorporate dispersion effects.

[29] Shallow-water equations are especially suitable to model tides which have very large length scales (over hundred of kilometers).

An advantage of this, over Quasi-geostrophic equations, is that it allows solutions like gravity waves, while also conserving energy and potential vorticity.

One alternative is to modify the "pressure term" in the momentum equation, but it results in a complicated expression for kinetic energy.

[31] Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for kinetic energy, avoids shock formation, but conserves only linearized potential vorticity.