PDF《Part 1: Flow model》

Modelling the effect of wind farming on mesoscale flow - Part 1: Flow model A J Brand ECN Wind Energy, P.

O. Box 1, NL

1755 RG Petten, Netherlands [email protected] Abstract. This paper presents a new method for determining the interaction between a wind farm and the prevailing wind for wind energy siting studies. It is shown that neutral planetary boundary layer flow with wind farming essentially is steady and two- dimensional;

and that the convective forces, the Coriolis forces and the vertical and spanwise gradients of the turbulent momentum fluxes all have the same order of magnitude. In addition it is shown that a numerical representation in the form of backward differences allows for an implicit solution of the two horizontal velocity components in vertical direction, iterating on the turbulent viscosity, in combination with a marching solution in the horizontal directions. 1. Introduction Offshore wind farms tend to be placed closer together over the years, as already illustrated by OWEZ and Q7-WF (separated

15 km) in the Netherlands or Horns Rev I and II (separated

23 km) in Denmark. Since these separation distances are between

5 and

10 times the wind farm's horizontal scale, the velocity deficit due to an upstream wind farm may be considerable [1]. If so, energy production loss and mechanical load increase are expected to be significant. For this reason wind farm wake studies have gained attention recently. In this paper we present a new method for determining the interaction between a wind farm and the prevailing wind. First, section

2 gives a brief description of prior work on modelling wind farm wakes. Next, section

3 presents the new flow model, and finally, in section

4 we summarize the new model and introduce the future developments. 2. Prior work A wind farm wake study requires simulation of mesoscale atmospheric flow together with energy extraction/redistribution due to wind turbines. The studies that have been published so far can be subdivided into two categories: self-similar approaches and mesoscale approaches. In a self-similar approach [2][3] the convective force and the spanwise turbulent flux gradients are assumed to dominate the flow, allowing for standard wake-like solutions. In a mesoscale approach, on the other hand, the flow is assumed to be dominated by the Coriolis force and the vertical turbulent flux gradients, opening the door to either extra surface drag approaches [4] or more generic mesoscale approaches [5][6][7]. As will be shown in section 3.2.1 of this paper, neither the self-similar wake approach nor the extra surface drag approach is valid because over the separation distance between wind farms the convective and the Coriolis forces are of equal order of magnitude so that neither can be neglected. Although this was already implicitly recognized in the more generic approaches, these studies lack realistic formulations for the turbulence and the wind turbines. 3. Flow model 3.1. Overview In section 3.2 first we derive the governing equations from the planetary boundary layer equations and the continuity equation by introducing length and velocity scales, modelling Reynolds stresses with turbulent viscosity and wind turbines with body forces. Since these equations are essentially steady and two-dimensional (the vertical wind speed scale is several orders of magnitude smaller than a horizontal wind speed scale), the pressure gradients can be treated as geostrophic wind speed components. We subsequently cast the governing equations in non-dimensional form, employing linear transformations in the horizontal directions and an exponential transformation in the vertical direction. Next, in section 3.3 we derive a numerical representation of the non-dimensional governing equations by using finite differences. The resulting scheme is implicit in vertical direction, iterating between the two horizontal velocity components and the turbulent viscosity, and allows for a marching solution in the horizontal directions. We then address the turbulence parameterization (Baldwin- Lomax) in section 3.4, and the wind turbine parameterization (via the rotor thrust) in section 3.5. Finally, we present the boundary conditions (no-slip at the bottom and geostrophic at the top of the numerical domain;