One-dimensional Saint-Venant equations
Contenido
Las ecuaciones unidimensionales (1-D) de Saint-Venant fueron derivadas por Adhémar Jean Claude Barré de Saint-Venant, y se utilizan comúnmente para modelar el flujo en canal abierto transitorio y la escorrentía superficial. Pueden considerarse como una contracción de las ecuaciones bidimensionales (2-D) de aguas poco profundas, que también se conocen como ecuaciones bidimensionales de Saint-Venant. Las ecuaciones de Saint-Venant 1-D contienen hasta cierto punto las principales características de la forma de la sección transversal "Sección (geometría)") del canal.
Las ecuaciones 1-D se utilizan ampliamente en modelos informáticos como TUFLOW"), Mascaret&action=edit&redlink=1 "MASCARET (Software) (aún no redactado)") (EDF), lang=es SIC (Irstea), HEC-RAS,[5] SWMM5, ISIS,[5] InfoWorks,[5] Flood Modeller, SOBEK 1DFlow MIKE 11"),[5] y MIKE SHE") porque son significativamente más fáciles de resolver que las ecuaciones completas de aguas poco profundas. Las aplicaciones comunes de las ecuaciones 1-D de Saint-Venant incluyen enrutamiento de inundaciones a lo largo de los ríos (incluyendo la evaluación de las medidas para reducir los riesgos de inundación), el análisis de la rotura de presas, los pulsos de tormenta en un canal abierto, así como la escorrentía de la tormenta en el flujo terrestre.
Equations
The system of partial differential equations that describe the 1-D incompressible flow in an open channel of arbitrary cross section - as derived and proposed by Saint-Venant in his 1871 article (equations 19 and 20) - is:[6].
and.
where x is the spatial coordinate along the channel axis, t denotes the 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 of the wetted perimeter P(x,t) of the cross section at x. Furthermore, ρ is the density of the fluid (constant) and g is the gravitational acceleration.
The solution of the hyperbolic system of equations (1)–(2) is obtained from the geometry of the cross sections, providing a functional relationship between the cross-sectional area A and the surface elevation ζ at each position x. For example,For example, for a rectangular cross section, with constant channel width B and channel bed elevation z, the cross section area is: . The instantaneous water depth is with z(x) the bed level (i.e. the elevation of the lowest point of the bed above the date, see cross-sectional figure). For stationary 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 for rectangular channels.[7].
The wall shear stress τ depends on the flow velocity u, they can be related using, for example, the Darcy-Weisbach equation, the Manning formula or the Chézy formula.
Furthermore, equation (1) is the continuity equation, which expresses the conservation of the volume of water for this incompressible homogeneous fluid. Equation (2) is the momentum equation, which gives the balance between forces and rates of change of momentum.
The bed slope S(x), the friction slope S(x, t) and the hydraulic radius R(x, t) are defined as:
and.
Consequently, the momentum equation (2) can be written as:[7].
Conservation of moment
The momentum equation (3) can also be stated in the so-called conservation form"), or Eulerian form, through some algebraic manipulations on the Saint-Venant equations, (1) and (3). In terms of the flow rate "Flow (fluid)"):[8].
where A, I and I are functions of the channel geometry, described in terms of the channel width B(σ,x). Here, σ is the height above the lowest point of the cross section at location x. Then σ is the height above the bed level z(x) (of the lowest point in the cross section):.
Above - in the momentum equation (4) in conservation form -A, I and I is evaluated as . The term describes the hydrostatic force at a given cross section. And, for a non-prismatic channel "Prism (geometry)"), gives the effects of geometry variations along the x axis of the channel.
In applications, depending on the problem at hand, it is often preferred to use the momentum equation in the non-conservative form, (2) or (3), or the conservative form (4). For example, in the case of the description of hydraulic jumps, the conservation form is preferred since the momentum flow is continuous across the jump.
Characteristics
The Saint-Venant equations (1)–(2) can be analyzed by the method of characteristics").[9][10][11][12] The two accelerations dx/dt in the characteristic curves are:[8]
with.
The Froude Number Fr= |u|/c determines whether the flow is subcritical () or supercritical ().
For a rectangular and prismatic channel of constant width B, that is, with and the Riemann invariants") are[9] and
so the equations in characteristic form are::[9]
The Riemann invariants and the method of characteristics for a prismatic channel of arbitrary cross section are described by Didenkulova and Pelinovsky (2011).[12].
The characteristics and Riemann invariants provide important information about the behavior of the flow, and can also be used in the process of obtaining solutions (analytical or numerical).[13][14][15][16].
Derivative modeling
The dynamic wave is the complete one-dimensional Saint-Venant equation. It is numerically difficult to solve, but is valid for all channel flow scenarios. The dynamic wave is used to model transient storms in modeling programs such as Mascaret&action=edit&redlink=1 "MASCARET (Software) (not yet written)") (EDF), SIC (Irstea), HEC-RAS,[17] InfoWorks_ICM Archived October 25, 2016 at the Wayback Machine.,[18] MIKE 11"),[19] Wash 123d[20] and SWMM5").
In the order of increasing simplifications, by eliminating some terms from the complete 1D Saint-Venant equations (also known as the dynamic wave equation), we obtain the also classical diffusive wave equation and the kinematic wave equation.
For the diffusive wave it is assumed that the inertia terms are smaller than the gravity, friction and pressure terms. Therefore, the diffusive wave can be more accurately described as a non-inertial 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 mainly subcritical flow, with low Froude values. Models using the diffusive wave hypothesis include MIKE SHE")[21] and LISFLOOD-FP.[22] In the SIC software (Irstea) these options are also available, as the 2 inertia terms (or either of them) can be optionally removed 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 complete 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, for example, for shallow flows over steep slopes.[23] The kinematic wave is used in HEC-HMS").[24].