The shallow water equations 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 (see the related section below).
The equations are derived from depth-integrating the Navier–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 nearly hydrostatic, 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 forces in atmospheric and oceanic modeling, as a simplification of the primitive 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 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, no Coriolis forces, frictional or 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. The first equation is derived from mass conservation, the second two from momentum conservation.
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):
|u||is the velocity in the x direction, or zonal velocity|
|v||is the velocity in the y direction, or meridional velocity|
|h||is the height deviation of the horizontal pressure surface from its mean height H: η = H + h|
|H||is the mean height of the horizontal pressure surface|
|g||is the acceleration due to gravity|
|f||is the Coriolis coefficient associated with the Coriolis force. On Earth, f is equal to 2Ω sin(φ), where Ω is the angular rotation rate of the Earth (π/12 radians/hour), and φ is the latitude|
|b||is the viscous drag coefficient|
|ν||is the kinematic viscosity|
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):
The one-dimensional (1-D) Saint-Venant equations were derived by Adhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transient open-channel flow and surface 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 channel cross-sectional shape.
The 1-D equations are used extensively in computer models such as Mascaret (EDF), SIC (Irstea), HEC-RAS, SWMM5, ISIS, InfoWorks, Flood Modeller, SOBEK 1DFlow, MIKE 11, 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:
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 zb, the cross sectional area is: A = B (ζ − zb) = B h. The instantaneous water depth is h(x,t) = ζ(x,t) − zb(x), with zb(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:
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.
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 Sf(x,t) and hydraulic radius R(x,t) are defined as:
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 zb(x) (of the lowest point in the cross section):
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.
so the equations in characteristic form are:
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.
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, 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.
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 and LISFLOOD-FP.. 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.
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:
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 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:
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 fx is the body force in the x-direction.
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.
Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain (e.g. surface waves in a bath). In order for shallow water equations to be valid, the wavelength 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 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.
Shallow water equations, in its non-linear form, is an obvious candidate for modelling turbulence in the atmosphere and oceans, i.e. geophysical turbulence. An advantage of this, over Quasi-geostrophic equations, is that it allows solutions like gravity waves, while also conserving energy and potential 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 become shock waves. 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. 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.
The ADCIRC model is a high-performance, cross-platform numerical ocean circulation model popular in simulating storm surge, tides, and coastal circulation problems.
Originally developed by Drs. Rick Luettich and Joannes Westerink,
the model is developed and maintained by a combination of academic, governmental, and corporate partners, including the University of North Carolina at Chapel Hill, the University of Notre Dame, and the US Army Corps of Engineers.
The ADCIRC system includes an independent multi-algorithmic wind forecast model and also has advanced coupling capabilities, allowing it to integrate effects from sediment transport, ice, waves, surface runoff, and baroclinicity.Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant (23 August 1797, Villiers-en-Bière, Seine-et-Marne – 6 January 1886, Saint-Ouen, Loir-et-Cher) 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 the Saint-Venant equations that are a fundamental set of equations used in modern hydraulic engineering. The one-dimensional Saint-Venant equation is a commonly used simplification of the shallow water equations. Although his full surname was Barré de Saint-Venant in mathematical literature other than French he is known as Saint-Venant. His name is also associated with Saint-Venant's principle of statically equivalent systems of load, Saint-Venant's theorem and for Saint-Venant's compatibility condition, the integrability conditions for a symmetric tensor field to be a strain.
In 1843 he published the correct derivation of the Navier–Stokes equations for a viscous flow and was the first to "properly identify the coefficient of viscosity and its role as a multiplying factor for the velocity gradients in the flow". Even though he published before Stokes, the equations do not bear his name.
Barré de Saint-Venant developed a version of vector calculus similar to that of Grassmann (now understood as exterior differential forms) which he published in 1845. A dispute arose between Saint-Venant and Grassmann over priority for this invention. Grassmann had published his results in 1844, but Barré de Saint-Venant claimed he had developed the method in 1832.
Barré de Saint-Venant was born at the château de Fortoiseau, Villiers-en-Bière, Seine-et-Marne, France.
His father was Jean Barré de Saint-Venant, (1737–1810), a colonial officer of the Isle of Saint-Domingue (later Haiti). His mother was Marie-Thérèse Josèphe Laborie (born Saint-Domingue, 1769). He entered the École Polytechnique, in 1813 at sixteen years old. where he studied under Gay-Lussac. Graduating in 1816 he worked for the next 27 years as an engineer, initially his passion for chemistry led him a post as a élève-commissaire (student commissioner) for the Service des Poudres et Salpêtres (Powders and Saltpeter Service) and then as a civil engineer at the Corps des Ponts et Chaussées. He married in 1837, Rohaut Fleury from Paris. Following a disagreement on an issue of road with the Municipal Administration of Paris, he was suddenly retired as "Chief Engineer, second class", on 1 April 1848. In 1850 Saint-Venant won a contest to be appointed the chair of Agricultural Engineering at the Agronomic Institute of Versailles, a post he occupied two years.He went on to teach mathematics at the École des Ponts et Chaussées (National school of Bridges and Roads) where he succeeded Coriolis.
In 1868, at 71 years old, he was elected to succeed Poncelet in the mechanics section of the Académie des Sciences, and continued research work for a further 18 years. He died in January 1886 at Saint-Ouen, Loir-et-Cher. Sources differ on his date of death: gives 6 January whereas 22 January. In 1869 he was given the title 'Count' (comte) by Pope Pius IX.André Robert
Dr. André Robert (April 28, 1929 – November 18, 1993) was a Canadian meteorologist who pioneered the modelling the Earth's atmospheric circulation.Boussinesq approximation (water waves)
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours.
While the Boussinesq approximation is applicable to fairly long waves – that is, when the wavelength is large compared to the water depth – the Stokes expansion is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).Geodesic grid
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron.Gravity current
In fluid dynamics, a gravity current or density current is a primarily horizontal flow in a gravitational field that is driven by a density difference in a fluid or fluids and is constrained to flow horizontally by, for instance, a ceiling. Typically, the density difference is small enough for the Boussinesq approximation to be valid. Gravity currents can be thought of as either finite in volume, such as the pyroclastic flow from a volcano eruption, or continuously supplied from a source, such as warm air leaving the open doorway of a house in winter.
Other examples include dust storms, turbidity currents, avalanches, discharge from wastewater or industrial processes into rivers, or river discharge into the ocean.Gravity currents are typically much longer than they are tall. Flows that are primarily vertical are known as plumes. As a result, it can be shown (using dimensional analysis) that vertical velocities are generally much smaller than horizontal velocities in the current; the pressure distribution is thus approximately hydrostatic, apart from near the leading edge. Gravity currents may be simulated by the shallow water equations, with special dispensation for the leading edge which behaves as a discontinuity. When a gravity current propagates along a plane of neutral buoyancy within a stratified ambient fluid, it is known as a gravity current intrusion.Hundred-year wave
A hundred-year wave is a statistically projected water wave, the height of which, on average, is met or exceeded once in a hundred years for a given location. The likelihood of this wave height being attained at least once in the hundred-year period is 63%. As a projection of the most extreme wave which can be expected to occur in a given body of water, the hundred-year wave is a factor commonly taken into consideration by designers of oil platforms and other offshore structures. Periods of time other than a hundred years may also be taken into account, resulting in, for instance, a fifty-year wave.Various methods are employed to predict the possible steepness and period of these waves, in addition to their height.Nambu mechanics
In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu-Poisson manifolds with more than one Hamiltonian.Radial basis function interpolation
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured data, possibly in high-dimensional spaces. The interpolant takes the form of a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate and stable for large numbers of nodes even in high dimensions.
Many interpolation methods can be used as the theoretical foundation of algorithms for approximating linear operators, and RBF interpolation is no exception. RBF interpolation has been used to approximate differential operators, integral operators, and surface differential operators. These algorithms have been used to find highly accurate solutions of many differential equations including Navier–Stokes equations, Cahn–Hilliard equation, and the shallow water equations.Riemann solver
A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.SPECfp
SPECfp is a computer benchmark designed to test the floating point performance of a computer. It is managed by the Standard Performance Evaluation Corporation. SPECfp is the floating point performance testing component of the SPEC CPU testing suit. The first standard SPECfp was released in 1989 as SPECfp89. Later it was replaced by SPECfp92, then SPECfp95, then SPECfp2000, then SPECfp2006, and finally SPECfp2017.SWE
SWE may refer to:
Samberigi Airport in Papua New Guinea by IATA airport code
Sensor Web Enablement, an Open Geospatial Consortium framework for defining a Sensor Web
Shallow water equations, a set of hyperbolic partial differential equations
Snow water equivalent
Society of Women Engineers, a non-profit engineering organization
Society of Wood Engravers, a British printmakers' group
Staebler–Wronski effect, light-induced changes in the properties of silicon
Standard written English
Sweden, the country's ISO 3166-1 alpha-3-code
Swedish language, the language's ISO 639-2 and ISO 639-3 language codeShallow water
Shallow water may refer to:
Shallow water blackout
Waves and shallow water
Shallow water equations
Boussinesq equations (water waves)
Shallow Water, Kansas, unincorporated community, United States
Shallow Water (album)TELEMAC
In computational fluid dynamics, TELEMAC is short for the open TELEMAC-MASCARET system, or a suite of finite element computer program owned by the Laboratoire National d'Hydraulique et Environnement (LNHE), part of the R&D group of Électricité de France. After many years of commercial distribution, a Consortium (the TELEMAC-MASCARET Consortium) was officially created in January 2010 to organize the open source distribution of the open TELEMAC-MASCARET system now available under GPLv3.Vance Faber
Vance Faber (born December 1, 1944 in Buffalo, New York) is a mathematician, known for his work in combinatorics, applied linear algebra and image processing.
Faber received his Ph.D. in 1971 from Washington University in Saint Louis. His advisor was Franklin Tepper Haimo.Faber was a professor at University of Colorado at Denver during the 1970s. He spent parts of 3 years at the National Center for Atmospheric Research in Boulder on a NASA postdoctoral fellowship where he wrote a second thesis on the numerical solution of the Shallow Water Equations under the direction of numerical analyst Paul Swarztrauber. In the 1980s and 1990s he was on the staff of the Computer Research and Applications Group at Los Alamos National Laboratory. He was Group Leader from 1990 to 1995.
From 1998 to 2003 Faber was CTO and Head of Research for three different small companies building imaging software: LizardTech, Mapping Science and Cytoprint. He is currently a consultant.
In 1981, Gene Golub offered a US$500 prize for "the construction of a 3-term conjugate gradient like descent method for non-symmetric real matrices or a proof that there can be no such method". Faber and his co-author Thomas A. Manteuffel won this prize for their 1984 paper, in which they gave conditions for the existence of such a method and showed that, in general, there can be no such method.Wave base
The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.Wave shoaling
In fluid dynamics, wave shoaling is the effect by which surface waves entering shallower water change in wave height. It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in energy density in order to maintain a constant energy flux. Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.
In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water. This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.Wave tank
A wave tank is a laboratory setup for observing the behavior of surface waves. The typical wave tank is a box filled with liquid, usually water, leaving open or air-filled space on top. At one end of the tank an actuator generates waves; the other end usually has a wave-absorbing surface. A similar device is the ripple tank, which is flat and shallow and used for observing patterns of surface waves from above.Waves and shallow water
When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies.