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). The shallow-water equations in unidirectional form are also called Saint-Venant equations, after
Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant (23 August 1797 – 6 January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as ...
(see the
related section below).
The equations are derived
from depth-integrating the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Georg ...
, 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 nearly
hydrostatic
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
, 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 with
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
s in atmospheric and oceanic modeling, as a simplification of the
primitive equations
The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:
# A ''continuity equ ...
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.
Equations
Conservative form
The shallow-water equations are derived from equations of
conservation of mass and
conservation of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
(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,
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
al and
viscous forces, the shallow-water equations are:
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
. The first equation is derived from mass conservation, the second two from momentum conservation.
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):
where
It is often the case that the terms quadratic in ''u'' and ''v'', which represent the effect of bulk
advection
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
, 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 (), we have (without lateral viscous forces):
One-dimensional Saint-Venant equations
The one-dimensional (1-D) Saint-Venant equations were derived by
Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant (23 August 1797 – 6 January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as ...
, and are commonly used to model transient
open-channel flow and
surface runoff
Surface runoff (also known as overland flow) is the flow of water occurring on the ground surface when excess rainwater, stormwater, meltwater, or other sources, can no longer sufficiently rapidly infiltrate in the soil. This can occur when th ...
. 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 channel
cross-sectional shape.
The 1-D equations are used extensively in
computer models such as
TUFLOW,
Mascaret (EDF)
SIC (Irstea) HEC-RAS,
SWMM5, ISIS,
InfoWorks,
Flood Modeller, SOBEK 1DFlow,
MIKE 11
MIKE 11 is a computer program that simulates flow and water level, water quality and sediment transport in rivers, flood plains, irrigation canals, reservoirs and other inland water bodies. MIKE 11 is a 1-dimensional river model. It was develope ...
,
and
MIKE SHE
MIKE SHE is an integrated hydrological modelling system for building and Computer simulation, simulating surface water flow and groundwater flow. MIKE SHE can simulate the entire land phase of the hydrologic cycle and allows components to be used ...
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.
Equations
![Open Channel Section (alt)](https://upload.wikimedia.org/wikipedia/commons/6/6e/Open_Channel_Section_%28alt%29.svg)
The system of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s which describe the 1-D
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
in an
open channel
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
of arbitrary
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
– as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is:
and
where ''x'' is the space coordinate along the channel axis, ''t'' denotes time, ''A''(''x'',''t'') is the cross-sectional
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
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
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
along the
wetted perimeter
The wetted perimeter is the perimeter of the cross sectional area that is "wet". The length of line of the intersection of channel wetted surface with a cross sectional plane normal to the flow direction. The term wetted perimeter is common in ci ...
''P''(''x'',''t'') of the cross section at ''x''. Further ρ is the (constant) fluid
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and ''g'' is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
.
Closure of the
hyperbolic system of equations ()–() 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: . The instantaneous water depth is , with ''z''
b(''x'') the bed level (i.e. elevation of the lowest point in the bed above
datum
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
, see the
cross-section figure). For non-moving channel walls the cross-sectional area ''A'' in equation () can be written as:
with ''b''(''x'',''h'') the effective width of the channel cross section at location ''x'' when the fluid depth is ''h'' – so for rectangular channels.
[
The wall shear stress ''τ'' is dependent on the flow velocity ''u'', they can be related by using e.g. the ]Darcy–Weisbach equation
In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation ...
, Manning formula or Chézy formula
The Chézy formula is an semi-empirical resistance equation which estimates mean flow velocity in Open-channel flow, open channel conduits. The relationship was realized and developed in 1768 by French physicist and engineer Antoine de Chézy (1718 ...
.
Further, equation () is the continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
, expressing conservation of water volume for this incompressible homogeneous fluid. Equation () is the momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
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:
and
Consequently, the momentum equation () can be written as:[ Chow, Ven Te (1959), ''Open-channel hydraulics'', McGraw-Hill, , §18-1 & §18-2.]
Conservation of momentum
The momentum equation () can also be cast in the so-called conservation form, through some algebraic manipulations on the Saint-Venant equations, () and (). In terms of the discharge
Discharge may refer to
Expel or let go
* Discharge, the act of firing a gun
* Discharge, or termination of employment, the end of an employee's duration with an employer
* Military discharge, the release of a member of the armed forces from serv ...
:[Cunge, J. A., F. M. Holly Jr. and A. Verwey (1980), ''Practical aspects of computational river hydraulics'', Pitman Publishing, , §§2.1 & 2.2]
where ''A'', ''I''1 and ''I''2 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):
Above – in the momentum equation () in conservation form – ''A'', ''I''1 and ''I''2 are evaluated at . The term describes the hydrostatic
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
force in a certain cross section. And, for a non-prismatic channel, 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, () or (), or the conservation form (). For instance in case of the description of hydraulic jumps, the conservation form is preferred since the momentum flux
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechan ...
is continuous across the jump.
Characteristics
The Saint-Venant equations ()–() can be analysed using the method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial d ...
. The two celerities d''x''/d''t'' on the characteristic curves are:[
with
The ]Froude number
In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on t ...
determines whether the flow is subcritical
In nuclear engineering, a critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction. The critical mass of a fissionable material depends upon its nuclear properties (specifically, its nuclear fissio ...
() or supercritical ().
For a rectangular and prismatic channel of constant width ''B'', i.e. with and , the Riemann invariant Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obta ...
s are:[
and
so the equations in characteristic form are:][ Whitham, G. B. (1974) ''Linear and Nonlinear Waves'', §§5.2 & 13.10, Wiley, ]
The Riemann invariants and method of characteristics for a prismatic channel of arbitrary cross-section are described by Didenkulova & Pelinovsky (2011).[
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.
]
Derived modelling
Dynamic wave
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 including Mascaret (EDF)
SIC (Irstea)
HEC-RAS
InfoWorks_ICM
MIKE 11
MIKE 11 is a computer program that simulates flow and water level, water quality and sediment transport in rivers, flood plains, irrigation canals, reservoirs and other inland water bodies. MIKE 11 is a 1-dimensional river model. It was develope ...
, Wash 123d and SWMM5.
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.
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:
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
MIKE SHE is an integrated hydrological modelling system for building and Computer simulation, simulating surface water flow and groundwater flow. MIKE SHE can simulate the entire land phase of the hydrologic cycle and allows components to be used ...
and LISFLOOD-FP. In th
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
In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associ ...
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:
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 kinematic wave is used in HEC-HMS The Hydrologic Modeling System (HEC-HMS) is designed to simulate the precipitation-runoff processes of dendritic drainage basins. It is designed to be applicable in a wide range of geographic areas for solving the widest possible range of problems. ...
.
Derivation from Navier–Stokes equations
The 1-D Saint-Venant momentum equation can be derived from the Navier–Stokes equations that describe fluid motion
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. The ''x''-component of the Navier–Stokes equations – when expressed in Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
in the ''x''-direction – can be written as:
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.
# If it is assumed that friction is taken into account as a body force, then can be assumed as zero so:
# Assuming one-dimensional flow in the ''x''-direction it follows that:
# Assuming also that the pressure distribution is approximately hydrostatic it follows that: or in differential form: And when these assumptions are applied to the ''x''-component of the Navier–Stokes equations:
# There are 2 body forces acting on the channel fluid, namely, gravity and friction: where ''f''x,g is the body force due to gravity and ''f''x,f is the body force due to friction.
# ''f''''x'',''g'' can be calculated using basic physics and trigonometry: where ''F''g is the force of gravity in the ''x''-direction, ''θ'' is the angle, and ''M'' is the mass.
The expression for sin θ can be simplified using trigonometry as: For small ''θ'' (reasonable for almost all streams) it can be assumed that: and given that ''f''''x'' represents a force per unit mass, the expression becomes:
# 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:
# All of these assumptions combined arrives at the 1-dimensional Saint-Venant equation in the ''x''-direction: 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.
Wave modelling by shallow-water equations
Shallow-water equations can be used to model Rossby and Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
waves in the atmosphere, rivers, lakes and oceans as well as gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s in a smaller domain (e.g. surface waves in a bath). In order for shallow-water equations to be valid, the wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
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 the Boussinesq approximation to incorporate dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
effects. Shallow-water equations are especially suitable to model tides which have very large length scales (over hundred 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.
Turbulence modelling using non-linear shallow-water equations
Shallow-water equations, in its non-linear form, is an obvious candidate for modelling turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
in the atmosphere and oceans, i.e. geophysical turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
. An advantage of this, over Quasi-geostrophic equations, is that it allows solutions like gravity wave
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
s, while also conserving energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
and potential vorticity
In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It i ...
. 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 become shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
s. 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 for kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
. Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
, avoids shock formation, but conserves only linearized potential vorticity
In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It i ...
.
See also
*Waves and shallow water
When ocean surface wave, waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free Trochoidal wave, orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their ...
Notes
Further reading
*
*
*
External links
Derivation of the shallow-water equations from first principles
(instead of simplifying the Navier–Stokes equations, some analytical solutions)
{{physical oceanography
Equations of fluid dynamics
Partial differential equations
Physical oceanography
Water waves