Conservative form

The shallow-water equations are derived from equations of conservation of mass and conservation of linear momentum (the Navier–Stokes equations), which hold even when the assumptions of shallow-water break down, such as across a hydraulic jump. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are:

$${\displaystyle {\begin{aligned}{\frac {\partial (\rho \eta )}{\partial t}}&+{\frac {\partial (\rho \eta u)}{\partial x}}+{\frac {\partial (\rho \eta v)}{\partial y}}=0,\\[3pt]{\frac {\partial (\rho \eta u)}{\partial t}}&+{\frac {\partial }{\partial x}}\left(\rho \eta u^{2}+{\frac {1}{2}}\rho g\eta ^{2}\right)+{\frac {\partial (\rho \eta uv)}{\partial y}}=0,\\[3pt]{\frac {\partial (\rho \eta v)}{\partial t}}&+{\frac {\partial }{\partial y}}\left(\rho \eta v^{2}+{\frac {1}{2}}\rho g\eta ^{2}\right)+{\frac {\partial (\rho \eta uv)}{\partial x}}=0.\end{aligned}}}$$

Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. Further g is acceleration due to gravity and ρ is the fluid density. The first equation is derived from mass conservation, the second two from momentum conservation.^[3]

Non-conservative form

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):

$${\displaystyle {\begin{aligned}{\frac {\partial h}{\partial t}}&+{\frac {\partial }{\partial x}}{\Bigl (}(H+h)u{\Bigr )}+{\frac {\partial }{\partial y}}{\Bigl (}(H+h)v{\Bigr )}=0,\\[3pt]{\frac {\partial u}{\partial t}}&+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}-fv=-g{\frac {\partial h}{\partial x}}-ku+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right),\\[3pt]{\frac {\partial v}{\partial t}}&+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}+fu=-g{\frac {\partial h}{\partial y}}-kv+\nu \left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right),\end{aligned}}}$$

where

It is often the case that the terms quadratic in u and v, which represent the effect of bulk advection, are small compared to the other terms. This is called geostrophic balance, and is equivalent to saying that the Rossby number is small. Assuming also that the wave height is very small compared to the mean height (h ≪ H), we have (without lateral viscous forces):

$${\displaystyle {\begin{aligned}{\frac {\partial h}{\partial t}}&+H\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0,\\[3pt]{\frac {\partial u}{\partial t}}&-fv=-g{\frac {\partial h}{\partial x}}-ku,\\[3pt]{\frac {\partial v}{\partial t}}&+fu=-g{\frac {\partial h}{\partial y}}-kv.\end{aligned}}}$$

Equations

Cross section of the open channel.

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]

$${\displaystyle {\frac {\partial A}{\partial t}}+{\frac {\partial \left(Au\right)}{\partial x}}=0}$$

(1)

and

$${\displaystyle {\frac {\partial u}{\partial t}}+u\,{\frac {\partial u}{\partial x}}+g\,{\frac {\partial \zeta }{\partial x}}=-{\frac {P}{A}}\,{\frac {\tau }{\rho }},}$$

(2)

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. Further ρ is the (constant) fluid density and g is the gravitational acceleration.

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 example, for a rectangular cross section, with constant channel width B and channel bed elevation z_b, the cross sectional area is: A = B (ζ − z_b) = B h. The instantaneous water depth is h(x,t) = ζ(x,t) − z_b(x), with z_b(x) the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). For non-moving channel walls the cross-sectional area A in equation (1) can be written as:

$${\displaystyle A(x,t)=\int _{0}^{h(x,t)}b(x,h')\,dh',}$$

with b(x,h) the effective width of the channel cross section at location x when the fluid depth is h – so b(x, h) = B(x) for rectangular channels.^[7]

The wall shear stress τ is dependent on the flow velocity u, they can be related by using e.g. the Darcy–Weisbach equation, Manning formula or Chézy formula.

Further, equation (1) is the continuity equation, expressing conservation of water volume for this incompressible homogeneous fluid. Equation (2) is the momentum equation, giving the balance between forces and momentum change rates.

The bed slope S(x), friction slope S_f(x, t) and hydraulic radius R(x, t) are defined as:

$${\displaystyle S=-{\frac {\mathrm {d} z_{\mathrm {b} }}{\mathrm {d} x}},}$$

$${\displaystyle S_{\mathrm {f} }={\frac {\tau }{\rho gR}}}$$

and

$${\displaystyle R={\frac {A}{P}}.}$$

Consequently, the momentum equation (2) can be written as:^[7]

$${\displaystyle {\frac {\partial u}{\partial t}}+u\,{\frac {\partial u}{\partial x}}+g\,{\frac {\partial h}{\partial x}}+g\,\left(S_{\mathrm {f} }-S\right)=0.}$$

(3)

Conservation of momentum

The momentum equation (3) can also be cast in the so-called conservation form, through some algebraic manipulations on the Saint-Venant equations, (1) and (3). In terms of the discharge Q = Au:^[8]

$${\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {Q^{2}}{A}}+g\,I_{1}\right)+g\,A\,\left(S_{f}-S\right)-g\,I_{2}=0,}$$

(4)

where A, I₁ and I₂ are functions of the channel geometry, described in the terms of the channel width B(σ,x). 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 z_b(x) (of the lowest point in the cross section):

$${\displaystyle {\begin{aligned}A(\sigma ,x)&=\int _{0}^{\sigma }B(\sigma ',x)\;\mathrm {d} \sigma ',\\I_{1}(\sigma ,x)&=\int _{0}^{\sigma }(\sigma -\sigma ')\,B(\sigma ^{\prime },x)\;\mathrm {d} \sigma '\qquad {\text{and}}\\I_{2}(\sigma ,x)&=\int _{0}^{\sigma }(\sigma -\sigma ')\,{\frac {\partial B(\sigma ',x)}{\partial x}}\;\mathrm {d} \sigma '.\end{aligned}}}$$

Above – in the momentum equation (4) in conservation form – A, I₁ and I₂ are evaluated at σ = h(x,t). The term g I₁ describes the hydrostatic force in a certain cross section. And, for a non-prismatic channel, g I₂ gives the effects of geometry variations along the channel axis x.

In applications, depending on the problem at hand, there often is a preference for using either the momentum equation in non-conservation form, (2) or (3), or the conservation form (4). For instance in case of the description of hydraulic jumps, the conservation form is preferred since the momentum flux is continuous across the jump.

Characteristics

Characteristics, domain of dependence and region of influence, associated with location P = (x_P,t_P) in space x and time t.

The Saint-Venant equations (1)–(2) can be analysed using the method of characteristics.^{[9][10][11][12]} The two celerities dx/dt on the characteristic curves are:^[8]

$${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=u\pm c,}$$

with

$${\displaystyle c={\sqrt {\frac {gA}{B}}}.}$$

The Froude number Fr = |u| / c determines whether the flow is subcritical (Fr < 1) or supercritical (Fr > 1).

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

$${\displaystyle r_{+}=u+2{\sqrt {gh}}}$$

and

$${\displaystyle r_{-}=u-2{\sqrt {gh}},}$$

so the equations in characteristic form are:^[9]

$${\displaystyle {\begin{aligned}&{\frac {\mathrm {d} }{\mathrm {d} t}}\left(u+2{\sqrt {gh}}\right)=g\left(S-S_{f}\right)&&{\text{along}}\quad {\frac {\mathrm {d} x}{\mathrm {d} t}}=u+{\sqrt {gh}}\quad {\text{and}}\\&{\frac {\mathrm {d} }{\mathrm {d} t}}\left(u-2{\sqrt {gh}}\right)=g\left(S-S_{f}\right)&&{\text{along}}\quad {\frac {\mathrm {d} x}{\mathrm {d} t}}=u-{\sqrt {gh}}.\end{aligned}}}$$

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]}

Hamiltonian structure for frictionless flow

In case there is no friction and the channel has a rectangular prismatic cross section, the Saint-Venant equations have a Hamiltonian structure.^[17] The Hamiltonian H is equal to the energy of the free-surface flow:

$${\displaystyle H=\rho \int \left({\frac {1}{2}}Au^{2}+{\frac {1}{2}}gB\zeta ^{2}\right)\mathrm {d} x,}$$

with constant B the channel width and ρ the constant fluid density. Hamilton's equations then are:

$${\displaystyle {\begin{aligned}&\rho B{\frac {\partial \zeta }{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {\partial H}{\partial u}}\right)=\rho \left(B{\frac {\partial \zeta }{\partial t}}+{\frac {\partial (Au)}{\partial x}}\right)=\rho \left({\frac {\partial A}{\partial t}}+{\frac {\partial (Au)}{\partial x}}\right)=0,\\&\rho B{\frac {\partial u}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {\partial H}{\partial \zeta }}\right)=\rho B\left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+g{\frac {\partial \zeta }{\partial x}}\right)=0,\end{aligned}}}$$

since ∂A/∂ζ = B).

Derived modelling

Diffusive wave

For the diffusive wave it is assumed that the inertial terms are less than the gravity, friction, and pressure terms. The diffusive wave can therefore be more accurately described as a non-inertia wave, and is written as:

$${\displaystyle g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0.}$$

The diffusive wave is valid when the inertial acceleration is much smaller than all other forms of acceleration, or in other words when there is primarily subcritical flow, with low Froude values. Models that use the diffusive wave assumption include MIKE SHE^[22] and LISFLOOD-FP.^[23] In the SIC (Irstea) software this options is also available, since the 2 inertia terms (or any of them) can be removed in option from the interface.

Kinematic wave

For the kinematic wave it is assumed that the flow is uniform, and that the friction slope is approximately equal to the slope of the channel. This simplifies the full Saint-Venant equation to the kinematic wave:

$${\displaystyle S_{f}-S=0.}$$

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.^[24] The kinematic wave is used in HEC-HMS.^[25]

Derivation from Navier–Stokes equations

The 1-D Saint-Venant momentum equation can be derived from the Navier–Stokes equations that describe fluid motion. The x-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x-direction – can be written as:

$${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=-{\frac {\partial p}{\partial x}}{\frac {1}{\rho }}+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)+f_{x},}$$

where u is the velocity in the x-direction, v is the velocity in the y-direction, w is the velocity in the z-direction, t is time, p is the pressure, ρ is the density of water, ν is the kinematic viscosity, and f_x is the body force in the x-direction.

1. If it is assumed that friction is taken into account as a body force, then ${\displaystyle \nu }$ can be assumed as zero so:
   $${\displaystyle \nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)=0.}$$

   
2. Assuming one-dimensional flow in the x-direction it follows that:^[26]
   $${\displaystyle v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=0}$$

   
3. Assuming also that the pressure distribution is approximately hydrostatic it follows that:^[26]
   $${\displaystyle p=\rho gh}$$

   
   or in differential form:
   $${\displaystyle \partial p=\rho g(\partial h).}$$

   
   And when these assumptions are applied to the x-component of the Navier–Stokes equations:
   $${\displaystyle -{\frac {\partial p}{\partial x}}{\frac {1}{\rho }}=-{\frac {1}{\rho }}{\frac {\rho g\left(\partial h\right)}{\partial x}}=-g{\frac {\partial h}{\partial x}}.}$$

   
4. There are 2 body forces acting on the channel fluid, namely, gravity and friction:
   $${\displaystyle f_{x}=f_{x,g}+f_{x,f}}$$

   
   where f_x,g is the body force due to gravity and f_x,f is the body force due to friction.
5. f_x,g can be calculated using basic physics and trigonometry:^[27]
   $${\displaystyle F_{g}=\sin(\theta )gM}$$

   
   where F_g is the force of gravity in the x-direction, θ is the angle, and M is the mass.
   

   Figure 1: Diagram of block moving down an inclined plane.
   The expression for sin θ can be simplified using trigonometry as:
   $${\displaystyle \sin \theta ={\frac {\text{opp}}{\text{hyp}}}.}$$

   
   For small θ (reasonable for almost all streams) it can be assumed that:
   $${\displaystyle \sin \theta =\tan \theta ={\frac {\text{opp}}{\text{adj}}}=S}$$

   
   and given that f_x represents a force per unit mass, the expression becomes:
   $${\displaystyle f_{x,g}=gS.}$$

   
6. Assuming the energy grade line is not the same as the channel slope, and for a reach of consistent slope there is a consistent friction loss, it follows that:^[28]
   $${\displaystyle f_{x,f}=S_{f}g.}$$

   
7. All of these assumptions combined arrives at the 1-dimensional Saint-Venant equation in the x-direction:
   $${\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0,}$$

   
   $${\displaystyle (a)\quad \ \ (b)\quad \ \ \ (c)\qquad \ \ \ (d)\quad (e)\ }$$

   
   where (a) is the local acceleration term, (b) is the convective acceleration term, (c) is the pressure gradient term, (d) is the friction term, and (e) is the gravity term.

Terms

The local acceleration (a) can also be thought of as the "unsteady term" as this describes some change in velocity over time. The convective acceleration (b) is an acceleration caused by some change in velocity over position, for example the speeding up or slowing down of a fluid entering a constriction or an opening, respectively. Both these terms make up the inertia terms of the 1-dimensional Saint-Venant equation.

The pressure gradient term (c) describes how pressure changes with position, and since the pressure is assumed hydrostatic, this is the change in head over position. The friction term (d) accounts for losses in energy due to friction, while the gravity term (e) is the acceleration due to bed slope.