Deriving the Saint-Venant Equations

The 1D engine approximates the vertically integrated equations of conservation of volume and momentum (the ‘Saint-Venant equations), which are derived on the basis of the following assumptions:

• the water is incompressible and homogeneous, i.e. without significant variations in density
• the bottom slope is small
• the 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 acceleration can be neglected and a hydrostatic pressure variation along the vertical can be assumed
• the flow is sub-critical

For a rectangular cross section with a horizontal bottom and a constant width, the conservation of mass and momentum can be expressed as follows (in the first instance neglecting friction and lateral inflows): Conservation of mass:.

\[ \frac{\partial \left(\rho Hb\right)}{\partial t} = -\frac{\partial \left(\rho Hb\overline{u}\right)}{\partial x} \]

Equation: Conservation of mass equation for shallow water flow relating temporal density change to spatial momentum flux.

Conservation of momentum:.

\[ \frac{\partial(\rho H b\bar{u})}{\partial t} = -\frac{\partial\left(\alpha' \rho H b \bar{u}^2 + \frac{1}{2}\rho g b H^2\right)}{\partial x} \]

Equation: Conservation of momentum accounting for pressure and convective acceleration in open channel flow.

where r is the density, H the depth, b the width, \(\bar{u}\) the average velocity along the vertical and a’ the vertical velocity distribution coefficient.

Introducing the bottom slope, I_b, and allowing for the channel width to vary will give rise to two more terms in the momentum equation. These terms describe the projections in the flow direction of the reactions of the bottom and side walls to the hydrostatic pressure.

The momentum equation now becomes:

\[ \frac{\partial \left(\rho H b \overline{u}\right)}{\partial t} = -\frac{\partial \left(\alpha' \rho H b \overline{u}^{2} + \frac{1}{2}\rho g b H^{2}\right)}{\partial x} + \frac{\partial b}{\partial x}\frac{\rho g H^{2}}{2} - \rho g H b I_{b} = -\frac{\partial \left(\alpha' \rho H b \overline{u}^{2}\right)}{\partial x} - b\frac{\partial \left(\frac{1}{2}\rho g H^{2}\right)}{\partial x} - \rho g H b I_{b} \]

Equation: The momentum equation with the bottom slope and varying channel width terms included.

When the water level, h, is introduced into the relationship instead of water depth:

\[ \frac{\partial h}{\partial x}=~I_{b}+\frac{\partial H}{\partial x} \]

Equation: Water level gradient decomposed into bottom slope and surface slope components.

and the equations are divided by r, the conservation laws of mass and momentum become:

\[ \frac{\partial \left(\rho Hb\right)}{\partial t} = -\frac{\partial \left(\rho Hb\overline{u}\right)}{\partial x} \]

Equation: Continuity equation expressing conservation of mass in a reach.

\[ \frac{\partial(Hb\bar{u})}{\partial t} = -\frac{\partial(\alpha' Hb\bar{u}^2)}{\partial x} - Hbg\frac{\partial h}{\partial x} \]

Equation: Momentum conservation equation with vertical velocity distribution coefficient.

These equations can be integrated to describe the flow through cross sections of any shape when divided up into a series of rectangular cross sections as shown in the cross section divided into a series of rectangular channels:

Figure: Cross section divided into a series of rectangular channels.

According to the previous assumptions, \(\frac{\partial h}{\partial x}\) is constant across the channel and no exchange of momentum occurs between the sub-channels. If the integrated cross section area is called A and the integrated discharge q, and B is the full width of the channel, then:

\[ A = \int_0^B H \, db \]

Equation: Cross-sectional area computed by integrating water depth across the channel width.

\[ q=\int_{0}^{B}H\overline{u}{\mathrm{d}}b~=~\overline{u}A \]

Equation: Discharge as the integrated product of cross-sectional velocity and flow area.

Integrating the mass and momentum conservation equations and introducing Equations the cross-sectional area computed by integrating water depth across the channel and the discharge as the integrated product of cross-sectional velocity and flow area yields:

\[ \frac{\partial q}{\partial x}+\frac{\partial A}{\partial t}\quad0 \]

Equation: Continuity equation relating spatial discharge gradient to temporal cross-sectional area change.

\[ \frac{\partial q}{\partial t}+\frac{\partial\!\left(\alpha\frac{q^{2}}{A}\right)}{\widehat{\sigma}x}+g A\frac{\partial h}{\widehat{\sigma}x}=~0 \]

Equation: Simplified momentum equation with friction slope, pressure gradient, and convective inertia terms.

Including the flow resistance term in the form as a friction slope, a lateral inflow \(q_{in}\),and an external momentum forcing f into these equations leads to the basic equations used in the 1D engine:

\[ \frac{\partial Q}{\partial x}+\frac{\partial A}{\partial t}~=~q_{i n} \]

Equation: Continuity equation for discharge and cross-sectional area with lateral inflow.

\[ \frac{\partial q}{\partial t} + \frac{\partial\left(\alpha \frac{q^2}{A}\right)}{\partial x} + gA\frac{\partial h}{\partial x} + gAI_f = f \]

Equation: Momentum equation with flow resistance expressed as friction slope and external forcing.

Pressurised Flow, Generalisation of Saint-Venant Equations

The idea of introducing a fictitious slot was first presented by Preissmann and Cunge, 1961 /2/, and has since been used by Cunge and Wegner, 1964 /1/.

To show that a top slot can be included without modifying the Saint-Venant equations, we start by rewriting the continuity equation as:

\[ -\frac{\partial(\rho Q)}{\partial x} dx dt = \frac{\partial(\rho A)}{\partial t} dx dt \]

Equation: Continuity equation expressing conservation of mass flux with constant density.

assuming the density of water r is constant over the cross section.

By partial differentiation we find:

\[ \frac{\partial Q}{\partial x} + \frac{Q}{\rho}\frac{\partial \rho}{\partial x} + \frac{\partial A}{\partial t} + \frac{A}{\rho}\frac{\partial \rho}{\partial t} = 0 \]

Equation: Continuity equation including effects of spatial and temporal density variations.

For a circular closed reach, it can be shown that the density of the water can be approximated as:

\[ \rho\approx\!\rho_{0}\!\left(1+\frac{g(y-D)}{{{a}_{0}}^{2}}\right) \]

Equation: Water density variation with depth in a pressurised circular closed reach.

where:

• r_o — the density of water for a free surface flow [kgm^-3]
• a_o — the speed of sound in water [ms^-1]
• y — the water depth [m]
• D — the reach diameter [m]. Furthermore, it can be shown that the cross section area in the case of the excess pressure rg(y-D) is approximately equal to:

\[ A\approx A_{0}\bigg(1+\frac{g(y-D)}{{a_{r}}^{2}}\bigg) \]

Equation: Cross-sectional area as a function of excess pressure accounting for reach compressibility.

where

A_o = the area without excess pressure and a_r is given as:

\[ {a}_{r}~=~\sqrt{\frac{{E}_{r}\cdot{e}}{{\rho}_{0}\cdot{D}}} \]

Equation: Wave celerity in compressible flow including elastic effects of pipe walls.

and

E_r = the Young`s modulus of elasticity, [Nm^-2] e = the reach wall thickness, [m].

The a_r has the dimension ms^-1 and is in the order of 1400 ms^-1 for most concrete pipes.

Combining these equations yields:

\[ \frac{\partial A}{\partial t} + \frac{A}{\rho} \cdot \frac{\partial \rho}{\partial t} = g \cdot A\frac{\partial y}{\partial t}\left(\frac{1}{a_r^2} + \frac{1}{a_0^2}\right) = \frac{gA_0}{a^2} \cdot \frac{\partial y}{\partial t} \]

Equation: Rate of change of flow area accounting for water and pipe compressibility in pressurized flow.

where:

\[ a~=~\frac{a_{0}}{\sqrt{1a_{0}^{2}/a_{r}^{2}}} \]

Equation: Effective wave speed incorporating water compressibility and reach wall deformation.

represents the speed of sound in water considering the compressibility of water and the deformation of the reach wall.

Equation the continuity equation including effects of spatial and temporal density variations can now be written as:

\[ \frac{\partial Q}{\partial x} + \frac{Q}{\rho} \cdot \frac{\partial \rho}{\partial x} + \frac{gA_0}{a^2} \cdot \frac{\partial y}{\partial t} = 0 \]

Equation: Wave equation including discharge, density variation, and elastic storage due to compressibility.

If we further assume that the density of water varies only slowly along the reach so that

\[ \frac{\partial \rho}{\partial x} = 0 \]

Equation: Assumption that water density does not vary significantly along the reach.

and use that the cross section area for water level in the slot varies as:

\[ \frac{\partial A}{\partial t} = b_{slot} \frac{\partial y}{\partial t} \]

Equation: Cross-sectional area change per unit time related to slot width and water level change.

then the continuity equation can be recovered by choosing the slot width so that

\[ b_{s l o t}=\frac{g A_{0}}{a^{2}} \]

Equation: Slot width for the Preissmann slot method to recover the continuity equation in pressurized flow.

a is in the order of 1000 [ms^-1] for most concrete pipes.