# Mathematical analysis of the Blade element momentum theory

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## Space-time parallel methods

For any of the aforementioned DA methods, numerical computation of the state estimate is as relevant as its accuracy. However, the former also requires to be carry out in a reasonable amount of time, which is possible with the help of space or time parallel methods, a natural approach to speed-up the numerical resolution of PDEs using parallel computing. Following Gander [33, 34], we briefly describe some of them.
During the nineteenth century, Fourier analysis was the main tool for studying PDEs, though it is restricted to simple geometries as circles or rectangles. With the purpose of extending Dirichlet’s principle to arbitrary domains, Schwarz [82] proposed to solve the Laplace equation by decomposing the domain into two overlapping subdomains where Fourier analysis apply, and then solving alternately a reduced problem on each, in a procedure known nowadays as the Alternating Schwarz method. This decomposition is the underlying principle of Domain decomposition methods.

### Space-time parallel methods and DA

Trémolet and Le Dimet [88] were among the first to address the parallelization of Variational data assimilation problems in meteorology. In a continuous setting, they proposed a Domain decomposition approach combined with the Adjoint method, by assigning to each subdomain a local version of (1.3), with an extra term on the local cost functional to enforce the continuity of the state between adjacent domains.
Twenty years later, new strategies in this topic also include the time direction. For instance, Rao and Sandu [79] apply a quasi-Newton solver to the 4D-Var problem, but they time-parallelize first the computation of the gradient. A more sophisticated approach is proposed by D’Amore and Cacciapuoti [26], who combines the Parareal algorithm with the Multiplicative Parallel Schwarz method (MPS) to solve 4D-Var.
The methodology associated with the Parareal algorithm has also been applied to optimization problems, but not necessarily related to DA. For instance, Maday, Salomon and Turinici [66] proposed a specific algorithm to solve optimality systems in the case of quantum control. By defining intermediate states and then solving a family of local optimization problems in parallel, these values are updated after solving sequentially the forward and adjoint equations, which is computationally cheap compared with the optimization procedure. Since the original cost functional is not parallelizable, the method uses a different one which depends on the adjoint variable and can be also decomposed as the sum of cost functionals for each subinterval. Ultimately, its optimal solution coincides with that of the original problem.

Blade element momentum (BEM) theory

Blade element momentum (BEM) theory is a mechanical model widely used to evaluate turbine performance, according to the mechanical/geometric characteristics of their blades and the current to which they are exposed. Developed by Glauert [44], it follows from the combination of two different models: Blade element theory (BET) and Momentum theory (MT).
By cutting the blade into sections that are treated according to a planar model, Blade element theory studies the turbine behavior from a local point of view [32].
The fundamental quantities of this model are the coefficients CL and CD called drag and lift, which are introduced to account for the drag and lift forces expressed in the cut plane. The results are then integrated along the blade to obtain the quantities of global interest. In constrast, Momentum theory [78] (also known as disk actuator theory or Axial momentum theory) is a global theory that macroscopically studies the behavior of a fluid column passing through a turbine.
BEM theory then relies on a decomposition of the fluid/turbine system into a macroscopic part via the MT and a local planar part via the BET. The latter considers a radial decomposition for the blades and fluid column (Figure 1.3a), by splitting the rotor area into concentric rings of infinitesimal thickness that do not interact with each other.

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#### Helmholtz formulation

Equation (3.16) defines a time-harmonic field, whose solution has the form (x, t) = Re{ tot(x)e−i!t}, where the amplitude tot satisfies !2 tot + div (gzbr tot) = 0. (3.17).
We wish to rewrite the equation above as a scattering problem. Since a variable bottom zb(x) := z0 + zb(x) (with z0 a constant describing a flat bathymetry and zb a perturbation term) can be considered as an obstacle, we thus assume that zb has a compact support in and tot satisfies the so-called Sommerfeld radiation condition.
In a bounded domain as , we impose the latter thanks to an impedance boundary condition (also known as first-order absorbing boundary condition), which ensures the existence and uniqueness of the solution [75, p.108]. We then reformulate (3.17) as ( div ((1 + q)r tot) + k2 0 tot = 0 in , r( tot − 0) · ˆn − ik0( tot − 0) = 0 on @.

List of Figures
List of Tables
Résumé de la Thèse
Introduction Générale
Contributions de cette thèse
État de l’art
General Introduction
Contributions of this thesis
1 State of the art
1.1 Data assimilation (DA)
1.1.1 Sequential methods
1.1.2 Variational methods
1.2 Space-time parallel methods
1.2.1 The Parareal algorithm
1.2.2 Space-time parallel methods and DA
1.3 Bathymetry estimation
1.3.1 Wave modeling
1.4 Blade element momentum (BEM) theory
2 Time-parallelization of sequential data assimilation problems
2.1 The Luenberger observer
2.2 Time-parallelization setting
2.2.1 Framework
2.2.2 The Diamond strategy
2.3 Parallelization
2.3.1 The Parareal algorithm
2.3.2 Combination with Luenberger observer
2.3.3 Complexity analysis
2.4 Numerical experiments
2.4.1 Diagonalized system
2.4.2 Evolution of k`
2.4.3 Observed efficiency
2.5 Perspectives
3 Bathymetry optimization
3.1 Derivation of the wave model
3.1.1 From Navier-Stokes system to Saint-Venant equations
3.1.2 Helmholtz formulation
3.2 Description of the optimization problem
3.2.1 Weak formulation
3.2.2 Continuous optimization problem
3.2.3 Continuity of the control-to-state mapping
3.2.4 Existence of optimal solution
3.3 Boundedness/Continuity of solution to Helmholtz problem
3.3.1 C0-bound for the general Helmholtz problem
3.3.2 C0-bounds for the total and scattered waves
3.4 Discrete optimization problem
3.4.1 Convergence of the Finite element approximation
3.4.2 Convergence of the discrete optimal solution
3.5 Numerical experiments
3.5.1 Numerical methods
3.5.2 Example 1: a wave damping problem
3.5.3 Example 2: an inverse problem
3.6 Perspectives
4 Mathematical analysis of the Blade element momentum theory
4.1 The Blade element momentum theory
4.1.1 Variables
4.1.2 Glauert’s modeling
4.1.3 Simplified model
4.1.4 Corrected model
4.2 Analysis of Glauert’s model and existence of solution
4.2.1 Simplified model
4.2.2 Corrected model
4.2.3 Multiple solutions
4.3 Solution algorithms
4.3.1 Usual algorithm
4.3.2 Alternative algorithms
4.4 Optimization
4.4.1 Simplified model and usual design procedure
4.4.2 Asymptotical analysis of the corrected model
4.5 Numerical experiments
4.5.1 A practical example
4.5.2 Solution algorithms
4.5.3 Optimization
4.6 Perspectives
Appendix 4.A Convergence in the simplified case
References

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