Emmanuel Branlard

Development of new tip-loss corrections based on vortex theory and vortex methods

A new analytical formulation of the tip-loss factor is established based on helical vortex filament solutions. The derived tip-loss factor can be applied to wind-turbines, propellers or other rotary wings. Similar numerical formulations are used to assess the influence of wake expansion on tip-losses. Theodorsens theory is successfully applied for the first time to assess the wake expansion behind a wind turbine. The tip-loss corrections obtained are compared with the ones from Prandtl and Glauert and implemented within a new Blade Element Momentum(BEM) code. Wake expansion is seen to reduce tip-losses and have a greater influence than wake distortion.

Aeroelastic large eddy simulations using vortex methods: unfrozen turbulent and sheared inflow

Vortex particles methods are applied to the aeroelastic simulation of a wind turbine in sheared and turbulent inflow. The possibility to perform large-eddy simulations of turbulence with the effect of the shear vorticity is demonstrated for the first time in vortex methods simulations. Most vortex methods formulation of shear, including segment formulations, assume a frozen shear. It is here shown that these formulations omit two source terms in the vorticity equation. The current paper also present unfrozen simulation of shear. The infinite support of the shear vorticity is accounted for using a novel approach relying on a Neumann to Dirichlet map. The interaction of the sheared vorticity with the wind turbine is shown to have an important impact on the wake shape. The obtained wake shape are closer to the one obtained using traditional computational fluid dynamics: Results with unfrozen shear do not have the severe upward motion of the wake observed in vortex methods simulation with frozen shear. The interaction of the shear and turbulence vorticity is shown to reduce the turbulence decay otherwise observed. The vortex code implemented is coupled to an aeroelastic code and examples of aeroelastic simulations under sheared and turbulent inflow are presented.

Investigation of a new model accounting for rotors of finite tip-speed ratio in yaw or tilt

The main results from a recently developed vortex model are implemented into a Blade Element Momentum(BEM) code. This implementation accounts for the effect of finite tip-speed ratio, an effect which was not considered in standard BEM yaw-models. The model and its implementation are presented. Data from the MEXICO experiment are used as a basis for validation. Three tools using the same 2D airfoil coefficient data are compared: a BEM code, an Actuator-Line and a vortex code. The vortex code is further used to validate the results from the newly implemented BEM yaw-model. Significant improvements are obtained for the prediction of loads and induced velocities. Further relaxation of the main assumptions of the model are briefly presented and discussed.

Particularities of Vortex Particle Methods

Vortex particle methods present two aspects that are absent in the segment formulation: the handling of the stretching term, and the problem of the divergence of the particle vorticity field. The particularities of the method are discussed in this chapter. The chapter begins by presenting the different aspects of the particle approximation: the notion of vortex blobs, the mathematical and physical interpretation of the particle approximation, the advection and stretching of the particles. The second part of the chapter treats of the divergence of the vorticity field: the possibility to minimize the error growth, the different corrections existing and the criteria to apply such correction.

The Different Aspects of Vortex Methods

This chapters discusses the different aspects of vortex methods enumerated in the following. The 1st part presents the fundamental equations and concepts of vortex methods. The 2nd part defines the discretization and initialization problem: the information carried by the particle and the way the vortex elements are discretized and initialized from a given vorticity field. The 3rd part presents the Viscous-Splitting algorithm/assumption which is used by some vortex methods to solve the convection and diffusion steps separately. The 4th part discusses the convection and stretching of vortex elements. The Lagrangian markers are convected by resolution of the trajectory equation using different numerical schemes. The stretching is applied by computation of the gradient of the velocity field. Different schemes exist. The 5th part presents grid-free and grid-based methods and provides references to coupled-Lagrangian–Eulerian solvers). The 6th part presents viscous diffusion and provides the general solution of the diffusion equation and different numerical implementations of viscous diffusion in vortex methods. The 7th part mentions bodies, boundaries and boundary conditions. Viscous and inviscid solid boundaries are discussed. Bodies are then presented as a particular case. The 8th part presents the regularization (also called kernel smoothing or mollification) required for the convergence of the (grid-free) method. The 9th part presents the topic of spatial adaptation (also called, or linked to, redistribution, rezoning or reinitialization) which is required due to the Lagrangian stretching of the computational domain which may results in “holes”. The 10th part briefly mentions the possibility to model turbulence via subgrid-scale models in vortex methods. The 11th part discusses the accuracy of vortex methods and provides guidelines and diagnostics to maintain or monitor accuracy within simulations.

Numerical Implementation of Vortex Methods

This chapter provides details of implementation of vortex methods. The interpolation/projection methods required for grid-based methods are described in a first part and Matlab codes for this step are provided. Tree-codes and fast-multipole methods are presented in a third part where the coefficients up to the second order are given. The third part provides references to Poisson solver methods and numerical implementations. Different numerical integration schemes are given in a fourth part. Vorticity splitting and merging schemes are discussed in a fifth part. Two subtleties of vortex methods are discussed in the end of the chapter: the possibility to represent vortex segments by vortex particles, and the choice of distribution of control points along the span or the chord. In particular, the 3/4 chord collocation point, or Pistolesi’s theorem is discussed.

Lifting Bodies and Circulation

This chapter presents the notions related to lifting bodies. In particular, the definitions of the lift force, the center of pressure, and the angle of attack are discussed. The notions of bound, shed and trailed vorticity are introduced. Different engineering models devised to modify the polar data of an airfoil are presented, such as: the extension to full polar, the determination of fully-separated polars, and dynamic stall models. Vorticity-based theories of lifting bodies are briefly mentioned. Lifting-surface and lifting-line theories of a wing are presented since they are relevant for the numerical implementation and validation of vortex-methods for wind turbines. The lifting-line approximation is also extensively used in this book for the representation of wind turbine blades.

Theoretical Foundations for Flows Involving Vorticity

This chapter introduces the fluid mechanics foundations that are relevant for this book. The fluid mechanics equations are given for inertial and non-inertial frames. The chapter presents vorticity kinematics and dynamics and the main theorems involving vorticity. The equations presented are necessary to the development of vorticity-based methods, both analytical and numerical. Some classical results of vortices in viscous and inviscid fluid are provided due to their relevance for the validation of numerical vortex menates and 3D axisymmetric flows are developed in details. They are conveniently used for the study of rotor. Two-dimensional potential flows and conformal mapping solutions are introduced. They are relevant for the implementation and validation of vortex methods and the derivation of Prandtl’s tip-loss factor. A Matlab code to compute the Karman-Trefftz map is provided.

Blade Element Theory (BET)

This chapter introduces the blade element theory and presents the formulae for different applications including: flows with rotational symmetry, rotors with infinite number of blades and cases without the influence of drag. The equations are also written under the lifting-line assumption using the expressions of the induction factors. The formulae are used to derive the blade-element momentum theory formulae in Chap. 10. They are also used throughout the book in comparison to the Kutta–Joukowski results of Chap. 8.

Flows with a Spread Distribution of Vorticity

This chapter derives the velocity fields and properties for some analytical flows involving a spread distribution of vorticity. Different examples of axisymmetric vorticity patches are presented, e.g. the rigid-body vortex patch and the inviscid vortex patch. They are convenient analytical solutions to study and validate numerical vortex methods. The inviscid vortex patch is studied in details in this chapter since it is highly used in Part VII to investigate the different aspects of vortex methods. The velocity field of a rectangular vorticity patch (or 2D vortex brick) is also provided.