Project topics and materials
Get Complete Project Material File(s) Now! »
Physical and geometrical set-up
Let us consider an incompressible, homogeneous, and ideal uid, of constant density , in a domain (t). At the time t 2 [0; T), the domain occupied by the uid is, therefore, (t) = f(x; z) 2 R2 R;d(t; x) z (t; x)g; (2.1) We note that x = (x; y) is the horizontal cartesian coordinate, z is the upward vertical coordinate, t is the time, r = (@x; @y)T is the horizontal gradient and rx;z = (@x; @y; @z)T is the classical gradient. There are also two functions, : [0; T) R2 ! R and d : [0; T) R2 ! R, such that z = (t; x) and z = d(t; x) represent the free surface equation and the bottom equation respectively. The local height of the water is h(t; x) = (t; x) + d(t; x): We denote by U(t; x; z) 2 R3, the velocity of the uid particle located at (x; z) at time t, u(t; x; z) 2 R2 and w(t; x; z) are its horizontal and vertical components respectively.
Incompressibilty of the uid then takes the following form div [U] = 0: (2.2) Irrotationality of the uid is useful but not necessary [59], given as follows, curl [U] = 0: (2.3) In addition to the equations (2.2) and (2.3), we need boundary conditions. One of them is the so-called kinematic boundary condition at the free surface and expressed the fact that the uid particle do not across the free surface, it read where m(x; z = (t; x)) is a point at the free surface and nfs is the unit outward normal vector to the free surface given by: nfs := 1 p 1 + (r)2 (r; 1)T :
This leads to the following explicit form: @ @t + u r = w: (2.5) The next boundary condition is needed at the bottom, assumed to be impermeable, U nb = @m @t nb: (2.6) Here m = m(x; z = d(t; x)) represents a point at the bottom, and nb is the unit outward normal at the bottom nb := 1 p 1 + (rd)2 (rd; 1)T ; this leads to the following explicit form.
Variational principle and Euler-Lagrange equations
Let the uid be ideal, incompressible, homogeneous and have a free unknown surface, while the other part of the boundary is given. Then, the following variational principle holds [11]: Variational principle: The motion of the uid must provide a stationary value to the following functional action: S(U) = Z t1 t0 Ldt; on the set of all velocity elds that satisfy the constraint (2.2) and the condition (2.4). Here, is the density of the uid (taken to be constant) and L is the classical Lagrangian density dened as: L := K + P where K and P are respectively the kinetic and the potential energies of a shallow uid moving under the force of gravity g:
The nondimensionalized equations
In order to study the asymptotic behavior of the solutions to the Euler-Lagrange equations (2.19)-(2.24), it is convenient to introduce nondimensionalized quantities based on the typical scales of the problem. Five main lengths are involved, namely, a typical water depth H, a typical wavelength Lx, a typical horizontal scale Ly, the order of free surface amplitude A, and the order of the bottom variations B. We dene, therefore, the three small parameters The signications of these three parameters are the following.
is the so-called shallowness parameter (also called dispersive parameter).
» is the amplitude parameter (also called nonlinearity parameter).
represents the topography parameter.
The shallow-water regime corresponds to the conguration where the wave length Lx of the ow is large compared to the typical depth H: For the sake of simplicity, we suppose in the rest of this section that Lx = Ly = L, then, dimensionless parameters are dened as follows: It is obvious that we nondimensionalize x and z by a unit lengths in a horizontal and vertical direction respectively, and the surface elevation and the bottom by two amplitude variations A and B. But the nondimensionalization of the time T and the potential dier, they are obtained by the linear wave theory (see [59] for more details). They read
Asymptotic expansion
Asymptotic expansion is an essential passage in the derivation of asymptotic models from Euler equations, water wave equations, or, as in our case, the Euler-Lagrange equations. To understand ow’s behavior, it is necessary to introduce the parameters given in the previous section, namely the parameters of shallowness , nonlinearity « , and bottom variation . Many asymptotic regimes can be distinguished based on these three parameters.
Shallow water regime which corresponds to the case when 1. Adding assumptions with respect to the nonlinearity parameter, there are other asymptotic regimes.
Small/large amplitude regime corresponds to the case when » = O() and » = O(1) respectively.
Small/large bottom variations correspond to = O() and = O(1) respectively.
Intermediate depth corresponds to the situation 1.
Deep water regime when > 1.
We are only interested in this work in the case of shallow water limit (see [59, 60] for more details in the other asymptotic regimes).
Taylor-series-type expansion
Keeping in mind that we are in shallow water regime which means that the shallowness parameter is too small ( 1). From then on, we can consider the following formal asymptotic expansion with respect to X(t; x; z) : X(t; x; z) = X0(t; x; z) + X1(t; x; z) + 2X2(t; x; z) + : (2.34) Applying this asymptotic expansion to , u and v and substituting it in equations (2.28)(2.32). Formally, letting goes to zero,
Table of contents :
Notations
I Introduction
1 Introduction
1.1 Contexte et motivation
1.2 Descriptif des travaux de thèse
1.3 Perspectives de recherche
II Depth averaged models
2 Derivation of shallow water equations
2.1 Introduction
2.2 Physical and geometrical set-up
2.3 Variational principle and Euler-Lagrange equations
2.4 The nondimensionalized equations
2.5 Asymptotic expansion
2.5.1 Taylor-series-type expansion
2.5.2 Mass conservation equation
2.5.3 Momentum conservation equation
2.5.4 Asymptotic model
2.6 Depth-averaged equations
2.6.1 Serre-Green-Naghdi equations for a moving bottom
2.6.2 Serre-Green-Naghdi equations for xed bottom
2.6.3 Boussinesq equations
2.6.4 Nonlinear shallow water equations
2.7 Derivation via Euler equations
2.7.1 Asymptotic expansions
2.7.1.1 Asymptotic expansion of the uid velocity
2.7.1.2 Pressure decomposition
2.7.2 Depth-averaged models
2.7.2.1 Mass conservation equation
2.7.2.2 Momentum conservation equation
2.7.2.3 Depth averaged models
2.8 Conclusion
3 Linear wave theory: dispersion and shoaling
3.1 Introduction
3.2 Linear rst order stokes theory
3.2.1 Linear dispersion properties
3.2.2 Linear shoaling properties
3.3 Linear theory for dispersive models
3.3.1 Improved models
3.3.1.1 BBM trick
3.3.1.2 Beji-Nadaoka model
3.3.1.3 Nwogu Model
3.3.2 Dispersion properties
3.3.2.1 Dispersion properties for Boussinesq models
3.3.2.2 Comparison of linear dispersion between Boussinesq equations
3.3.3 Shoaling properties
3.3.3.1 Shoaling properties for Boussinesq models
3.3.3.2 Comparisons between Boussinesq models
3.4 Conclusion
III Section averaged models
4 Generalized SGN equations for open channels and river ows
4.1 Introduction
4.2 The three-dimensional Incompressible Euler equations
4.2.1 Geometric set-up and the Euler equations
4.2.2 Boundary conditions
4.2.2.1 Free surface boundary conditions
4.2.2.2 Wet boundary conditions
4.3 Width-averaged and depth-averaged asymptotic expansions
4.3.1 Dimensionless Euler equations
4.3.2 Validity of the asymptotic and the section-averaging process
4.3.3 3D-2D model reduction and asymptotic expansions
4.3.3.1 Asymptotic expansions of the uid velocity
4.3.3.2 Width-averaged Euler equations
4.3.4 2D-1D like model reduction and asymptotic expansions
4.3.4.1 Asymptotic expansion of the uid velocity
4.3.4.2 Pressure decomposition
4.4 A new non-linear dispersive model
4.4.1 Eq. of the conservation of the mass
4.4.2 Eq. of the conservation of the momentum
4.4.3 The dispersive model for arbitrary non rectangular channel/ river
4.4.4 The dispersive model for a rectangular section
4.5 Energy
4.6 Improved cSGN equations
4.6.1 Reformulation of the cSGN equations
4.6.2 Improved dispersion frequency
4.6.3 Stokes rst-order theory and the choice of the parameter
4.7 A well-balanced nite volume approximation in the case of a nonuniform rectangular section
4.7.1 Numerical method
4.7.2 Propagation of a solitary wave
4.8 Conclusion
5 Simulation of complex free surface ows using SGN type models
5.1 Introduction
5.2 Numerical algorithms
5.2.1 The splitting scheme
5.2.2 Spatial discretization
5.2.3 Iterative methods: the Uzawa algorithm
5.2.4 Iterative methods: the Gauss-Seidel approach
5.2.5 Boundary conditions for the iterative algorithms
5.2.5.1 Solid wall boundaries
5.2.5.2 Fluvial inow-outow
5.3 Numerical validation
5.3.1 Solitary wave solution
5.3.2 Stationary solution
5.3.3 Dam-break
5.3.4 Favre waves
5.4 The two dimensional SerreGreen-Naghdi system
5.4.1 Reformulations of the system
5.4.2 Energy
5.4.3 Variational formulation
5.5 Conclusion
Bibliographie
