Theshallow-water equations (SWE) are a set ofhyperbolic 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, afree surface).^[1] The shallow-water equations in unidirectional form are also called(de) Saint-Venant equations, afterAdhémar Jean Claude Barré de Saint-Venant (see therelated section below).

The equations are derived^[2] from depth-integrating theNavier–Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity scale of the fluid is small compared to the horizontal velocity scale. It can be shown from the momentum equation that vertical pressure gradients are nearlyhydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow-water equations are thus derived.

While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow-water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation.

Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. They are used withCoriolis forces in atmospheric and oceanic modeling, as a simplification of theprimitive equations of atmospheric flow.

Shallow-water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow-water equations can describe the state.

The shallow-water equations are derived from equations ofconservation of mass andconservation of linear momentum (theNavier–Stokes equations), which hold even when the assumptions of shallow-water break down, such as across ahydraulic jump. In the case of a horizontalbed, with negligibleCoriolis forces,frictional andviscous forces, the shallow-water equations are:

Hereη is the total fluid column height (instantaneous fluid depth as a function ofx,y andt), and the 2D vector (u,v) is the fluid's horizontalflow velocity, averaged across the vertical column. Furtherg is acceleration due to gravity and ρ is the fluiddensity. The first equation is derived from mass conservation, the second two from momentum conservation.^[3]

Expanding the derivatives in the above using theproduct 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 orhydraulic jump. Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density):

where

It is often the case that the terms quadratic inu andv, which represent the effect of bulkadvection, are small compared to the other terms. This is calledgeostrophic balance, and is equivalent to saying that theRossby 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):

Theone-dimensional (1-D) Saint-Venant equations were derived byAdhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transientopen-channel flow andsurface runoff. They can be viewed as a contraction of the two-dimensional (2-D) shallow-water equations, which are also known as the two-dimensional Saint-Venant equations. The 1-D Saint-Venant equations contain to a certain extent the main characteristics of the channelcross-sectional shape.

The 1-D equations are used extensively incomputer models such asTUFLOW,Mascaret (EDF),SIC (Irstea),HEC-RAS,^[5] SWMM5, InfoWorks,^[5]Flood Modeller, SOBEK 1DFlow,MIKE 11,^[5] andMIKE SHE because they are significantly easier to solve than the full shallow-water equations. Common applications of the 1-D Saint-Venant equations includeflood 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 ofpartial differential equations which describe the 1-Dincompressible flow in anopen channel of arbitrarycross section – as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is:^[6]

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

1

and

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

2

wherex is the space coordinate along the channel axis,t denotes time,A(x,t) is the cross-sectionalarea of the flow at locationx,u(x,t) is theflow velocity,ζ(x,t) is thefree surface elevation and τ(x,t) is the wallshear stress along thewetted perimeterP(x,t) of the cross section atx. Further ρ is the (constant) fluiddensity andg is thegravitational acceleration.

Closure of thehyperbolic system of equations (1)–(2) is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional areaA and the surface elevation ζ at each positionx. For example, for a rectangular cross section, with constant channel widthB and channel bed elevationz_b, the cross sectional area is:A =B (ζ −z_b) =Bh. The instantaneous water depth ish(x,t) = ζ(x,t) −z_b(x), withz_b(x) the bed level (i.e. elevation of the lowest point in the bed abovedatum, see thecross-section figure). For non-moving channel walls the cross-sectional areaA in equation (1) can be written as:$${\displaystyle A(x,t)={\int }_0^{h(x,t)}b(x,h^{\prime })d{h}^{\prime },}$$withb(x,h) the effective width of the channel cross section at locationx when the fluid depth ish – sob(x,h) =B(x) for rectangular channels.^[7]

The wall shear stressτ is dependent on the flow velocityu, they can be related by using e.g. theDarcy–Weisbach equation,Manning formula orChézy formula.

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

The bed slopeS(x), friction slopeS_f(x,t) andhydraulic radiusR(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]

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

3

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

$${\displaystyle \frac{\mathrm{\partial }Q}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\frac{Q^2}{A}+g{I}_1\right)+gA\left(S_f-S\right)-g{I}_2=0,}$$

4

whereA,I₁ andI₂ are functions of the channel geometry, described in the terms of the channel widthB(σ,x). Here σ is the height above the lowest point in the cross section at locationx, see thecross-section figure. So σ is the height above the bed levelz_b(x) (of the lowest point in the cross section):

Above – in the momentum equation (4) in conservation form –A,I₁ andI₂ are evaluated atσ =h(x,t). The termgI₁ describes thehydrostatic force in a certain cross section. And, for anon-prismatic channel,gI₂ gives the effects of geometry variations along the channel axisx.

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 ofhydraulic jumps, the conservation form is preferred since themomentum flux is continuous across the jump.

The Saint-Venant equations (1)–(2) can be analysed using themethod of characteristics.^{[9][10][11][12]} The twocelerities dx/dt on the characteristic curves are:^[8]$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}=u\pm c,}$$ with$${\displaystyle c=\sqrt{\frac{gA}{B}}.}$$

TheFroude numberFr = |u| /c determines whether the flow issubcritical (Fr < 1) orsupercritical (Fr > 1).

For a rectangular and prismatic channel of constant widthB, i.e. withA =B h andc =√gh, theRiemann invariants are:^[9]$${\displaystyle r_{+}=u+2\sqrt{gh}}$$ and$${\displaystyle r_{-}=u-2\sqrt{gh},}$$so the equations in characteristic form 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 rectangularprismatic cross section,^[17] the Saint-Venant equations have aHamiltonian structure.^[18] The HamiltonianH is equal to the energy of the free-surface flow:$${\displaystyle H=\rho \int \left(\frac{1}{2}A{u}^2+\frac{1}{2}gB{\zeta }^2\right)\mathrm{d}x,}$$with constantB the channel width andρ the constant fluiddensity. Hamilton's equations then are:since∂A/∂ζ =B).

The dynamic wave is the full one-dimensional Saint-Venant equation. 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 includingMascaret (EDF),SIC (Irstea),HEC-RAS,^[19]Infoworks ICM^[20]MIKE 11,^[21] Wash 123d^[22] andSWMM5.

In the order of increasing simplifications, by removing some terms of the full 1D Saint-Venant equations (aka Dynamic wave equation), we get the also classical Diffusive wave equation and Kinematic wave equation.

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:$${\displaystyle g\frac{\mathrm{\partial }h}{\mathrm{\partial }x}+g(S_f-S)=0.}$$

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 includeMIKE SHE^[23] and LISFLOOD-FP.^[24] In theSIC (Irstea) software this options is also available, since the 2 inertia terms (or any of them) can be removed in option from the interface.

For thekinematic 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.^[25] The kinematic wave is used inHEC-HMS.^[26]

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The 1-D Saint-Venant momentum equation can be derived from theNavier–Stokes equations that describefluid motion. Thex-component of the Navier–Stokes equations – when expressed inCartesian coordinates in thex-direction – can be written as:$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }u}{\mathrm{\partial }z}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\frac{1}{\rho }+\nu \left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)+f_x,}$$

whereu is the velocity in thex-direction,v is the velocity in they-direction,w is the velocity in thez-direction,t is time,p is the pressure, ρ is the density of water, ν is the kinematic viscosity, andf_x is the body force in thex-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:$${\displaystyle \nu \left(\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2}\right)=0.}$$
2. Assuming one-dimensional flow in thex-direction it follows that:^[27]$${\displaystyle v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }u}{\mathrm{\partial }z}=0}$$
3. Assuming also that the pressure distribution is approximately hydrostatic it follows that:^[27]$${\displaystyle p=\rho gh}$$ or in differential form:$${\displaystyle \mathrm{\partial }p=\rho g(\mathrm{\partial }h).}$$ And when these assumptions are applied to thex-component of the Navier–Stokes equations:$${\displaystyle -\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\frac{1}{\rho }=-\frac{1}{\rho }\frac{\rho g\left(\mathrm{\partial }h\right)}{\mathrm{\partial }x}=-g\frac{\mathrm{\partial }h}{\mathrm{\partial }x}.}$$
4. There are 2 body forces acting on the channel fluid, namely, gravity and friction:$${\displaystyle f_x=f_{x,g}+f_{x,f}}$$ wheref_x,g is the body force due to gravity andf_x,f is the body force due to friction.
5. f_x,g can be calculated using basic physics and trigonometry:^[28]$${\displaystyle F_g=\sin (\theta )gM}$$whereF_g is the force of gravity in thex-direction,θ is the angle, andM is the mass.

   

   The expression for sin θ can be simplified using trigonometry as:$${\displaystyle \sin \theta =\frac{\text{opp}}{\text{hyp}}.}$$ For smallθ (reasonable for almost all streams) it can be assumed that:$${\displaystyle \sin \theta =\tan \theta =\frac{\text{opp}}{\text{adj}}=S}$$ and given thatf_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:^[29]$${\displaystyle f_{x,f}=S_{f}g.}$$
7. All of these assumptions combined arrives at the 1-dimensional Saint-Venant equation in thex-direction:$${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+g\frac{\mathrm{\partial }h}{\mathrm{\partial }x}+g(S_f-S)=0,}$$$${\displaystyle (a)(b)(c)(d)(e)}$$ 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 theinertia 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.

Shallow-water equations can be used to modelRossby andKelvin waves in the atmosphere, rivers, lakes and oceans as well asgravity waves in a smaller domain (e.g. surface waves in a bath). In order for shallow-water equations to be valid, thewavelength of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. Somewhat smaller wavelengths can be handled by extending the shallow-water equations using theBoussinesq approximation to incorporatedispersion effects.^[30] Shallow-water equations are especially suitable to model tides which have very large length scales (over hundreds of kilometers). For tidal motion, even a very deep ocean may be considered as shallow as its depth will always be much smaller than the tidal wavelength.

Shallow-water equations, in its non-linear form, is an obvious candidate for modellingturbulence in the atmosphere and oceans, i.e. geophysicalturbulence. An advantage of this, overQuasi-geostrophic equations, is that it allows solutions likegravity waves, while also conservingenergy andpotential vorticity. However, there are also some disadvantages as far as geophysical applications are concerned - it has a non-quadratic expression for total energy and a tendency for waves to becomeshock waves.^[31] Some alternate models have been proposed which prevent shock formation. One alternative is to modify the "pressure term" in the momentum equation, but it results in a complicated expression forkinetic energy.^[32] Another option is to modify the non-linear terms in all equations, which gives a quadratic expression forkinetic energy, avoids shock formation, but conserves only linearizedpotential vorticity.^[33]

• Waves and shallow water

• Derivation of the shallow-water equations from first principles (instead of simplifying theNavier–Stokes equations, some analytical solutions)

• Equations of fluid dynamics
• Partial differential equations
• Physical oceanography
• Water waves

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