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Description: |
In the 80’s, numerical models based on the Saint-Venant equations, or shallow water equations, were frequently used in practical applications. However, as has been widely demonstrated, in shallow water conditions and for some types of waves, models based on a non-dispersive theory, of which the Saint-Venant model is an example, are limited and are not usually able to compute satisfactory results over long periods of analysis.
The theory shows and the practice confirms that only models of order σ^2 (σ=hλ, where h and λ represent, respectively, depth and wavelength characteristics) or greater, of the Boussinesq and Serre types, are able to reproduce effects other than the dispersive effects, including the non-linearities resulting from wave-wave and wave-current interactions.
Especially since the early 90’s, not only our theoretical knowledge of the phenomena involved has improved greatly, but also the numerical methods have been used more efficiently. The great advances in computer technology, improving the processing of information and enabling large amounts of data to be stored, have made possible the use of mathematical models of greater complexity and with fewer restrictions.
The general shallow water wave theory is here used to develop different mathematical approaches, which are nowadays the basis of the most sophisticated models in hydrodynamics. For this purpose, we start from the fundamental equations of the Fluid Mechanics, written in Euler’s variables, relating to a three-dimensional and quasi-irrotational flow of a perfect fluid [Euler equations, or Navier-Stokes equations with the assumptions of non-compressibility (dρ dt = div v = 0), irrotationality (rot v = 0) and perfect fluid (dynamic viscosity μ = 0).
Considering the dimensionless quantities ε = a/h, in which a is a characteristic wave amplitude, and σ, as defined above, and proceeding by suitable dimensionless variables, the fundamental equations and the boundary conditions allows to obtain the continuity equation.
Following, accepting the fundamental hypothesis of the shallow water wave theory, σ <<1, and developing the dependent variables in power series of the small parameter σ^2, in second approach (order 2 in σ^2), the momentum equation is obtained after some mathematical manipulations. The resulting equations are known as Serre (or Green and Naghdi) equations.
Assuming additionally a relative elevation of the surface due to the wave (ε) having a value close to the square of the relative depth (σ^2), i.e., O(ε)=O(σ^2), another equation is obtained, known as Boussinesq approach. Further simplifying these equations, retaining only terms up to order 1 in σ, i.e., neglecting all terms of dispersive origin, a new system of equations is obtained, known as Saint-Venant approach, or shallow water equations.
The classical Serre (or Green and Naghdi) equations are fully-nonlinear and weakly dispersive. The standard Boussinesq equations only incorporate weak dispersion and weak non-linearity, and are valid only for long waves in shallow waters. Such as for the standard Boussinesq models, also Serre’s equations are valid only for shallow water conditions.
To allow applications in a greater range of h/λ and Boussinesq type equations, with additional terms of dispersive origin, have been developed in recent years. Several theoretical approaches solved by efficient methods of finite-element and finite-difference, as well as numerical results in the field of Hydraulic Engineering, will be presented and discussed.
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Date: |
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| Start Time: |
14:30 |
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Speaker: |
José S. Antunes do Carmo (Marine and Environmental Sciences Centre/UC)
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Institution: |
MARE/University of Coimbra, Department of Civil Engineering
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| Research Groups: |
-Numerical Analysis and Optimization
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