Power Optimization of Wind Turbines Subject to Navier-Stokes Equations

In this thesis, we first develop a second-order corrected-explicit-implicit domain decomposition scheme (SCEIDD) for the parallel approximation of convection-diffusion equations over multi-block sub-domains. The stability and convergence properties of the SCEIDD scheme is analyzed, and it is proved that this scheme is unconditionally stable. Moreover, it is proved that the SCEIDD scheme is second-order accurate in time and space. Furthermore, three different numerical experiments are performed to verify the theoretical results. In all the experiments the SCEIDD scheme is compared with the EIPCMU2D scheme which is first-order in time. Then, we focus on the application of numerical PDEs in wind farm power optimization. We develop a model for wind farm power optimization while considering the wake interaction among wind turbines. The proposed model is a PDE-constrained optimization model with the objective of maximizing the total power of the wind turbines where the three-dimensional Navier-Stokes equations are among the constraints. Moreover, we develop an efficient numerical algorithm to solve the model. This numerical algorithm is based on the pattern search method, the actuator line method and a numerical scheme which is used to solve the Navier-Stokes equations. Furthermore, the proposed numerical algorithm is used to investigate the wake structures. Numerical results are consistent with the field-tested results. Moreover, we find that by optimizing the turbines operation while considering the wake effect, we can gain an additional 8% in the total power. Finally, we relax the deterministic assumption for the incoming wind speed. The developed model is ultimately a PDE-constrained stochastic optimization model. Moreover, we develop an efficient numerical algorithm to solve this model. This numerical algorithm is based on the Monte Carlo simulation method, the pattern search method, the actuator line method and the corrected-explicit-implicit domain decomposition scheme which we develop for the parallel approximation of three-dimensional Navier-Stokes equations. The developed numerical algorithm, the parallel scheme, and the model are validated by a benchmark used in the literature and the experimental data. We find that by optimizing the turbines operation and considering the randomness of incoming wind speed, we can gain an additional 9% in total power

A pencil distributed finite difference code for strongly turbulent wall-bounded flows

We present a numerical scheme geared for high performance computation of wall-bounded turbulent flows. The number of all-to-all communications is decreased to only six instances by using a two-dimensional (pencil) domain decomposition and utilizing the favourable scaling of the CFL time-step constraint as compared to the diffusive time-step constraint. As the CFL condition is more restrictive at high driving, implicit time integration of the viscous terms in the wall-parallel directions is no longer required. This avoids the communication of non-local information to a process for the computation of implicit derivatives in these directions. We explain in detail the numerical scheme used for the integration of the equations, and the underlying parallelization. The code is shown to have very good strong and weak scaling to at least 64K cores

Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

Numerical Methods for Solving Convection-Diffusion Problems

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For different problems, a convection-diffusion equation may be be written in various forms. The most popular formulation of convective transport employs the divergent (conservative) form. In some cases, the nondivergent (characteristic) form seems to be preferable. The so-called skew-symmetric form of convective transport operators that is the half-sum of the operators in the divergent and nondivergent forms is of great interest in some applications. Here we discuss the basic classes of discretization in space: finite difference schemes on rectangular grids, approximations on general polyhedra (the finite volume method), and finite element procedures. The key properties of discrete operators are studied for convective and diffusive transport. We emphasize the problems of constructing approximations for convection and diffusion operators that satisfy the maximum principle at the discrete level --- they are called monotone approximations. Two- and three-level schemes are investigated for transient problems. Unconditionally stable explicit-implicit schemes are developed for convection-diffusion problems. Stability conditions are obtained both in finite-dimensional Hilbert spaces and in Banach spaces depending on the form in which the convection-diffusion equation is written