PDF《Part 1: Flow model》

vanishing but non-zero turbulent viscosity both at the bottom and the top) and initial conditions (logarithmic x-wise and linear y-wise velocity profiles;

corresponding turbulent viscosity profile) in section 3.6. 3.2. Governing equations 3.2.1. Dimensional form The mean (in the sense of Reynolds averaged) flow in the neutral planetary boundary layer is described by the momentum equations [8, section 5.2.1]: x f z ' w ' u y ' v ' u x ' u ' u v f x p

1 z u w y u v x u u t u + ? ? ? ? ? ? ? ? ? + ? ? ρ ? = ? ? + ? ? + ? ? + ? ? φ (1) y f z ' w ' v y ' v ' v x ' u ' v u f y p

1 z v w y v v x v u t v + ? ? ? ? ? ? ? ? ? ? ? ? ρ ? = ? ? + ? ? + ? ? + ? ? φ (2) z f z ' w ' w y ' v ' w x ' u ' w z p

1 z w w y w v x w u t w + ? ? ? ? ? ? ? ? ? ? ? ρ ? = ? ? + ? ? + ? ? + ? ? (3) in combination with the continuity equation:

0 z w y v x u = ? ? + ? ? + ? ? ;

(4) where ? u , v and w are the components of the mean velocity;

? ' u , ' v and ' w are the velocity fluctuations;

? ρ is the air density;

? p is the mean pressure;

? φ f is the Coriolis parameter;

? x f , y f and z f are the components of a mean body force representing wind turbines;

and ? the covariances represent the turbulent momentum fluxes. The equations (1), (2), (3) and (4) constitute a system of

4 equations with

4 unkowns, which can be solved once boundary conditions are set. We come back to the boundary conditions in section 3.3. In order to estimate the magnitude of the individual terms in the equations (1), (2), (3) and (4) we introduce length and velocity scales [8, section 2.4] that correspond to wind farming in the planetary boundary layer (table 1). A length scale a distance related to motions or objects in the planetary boundary layer and a velocity scale is the velocity variation over a given length scale. Table 1. Scales in the planetary boundary layer with wind farming Scale Value Magnitude z0 Surface roughness length

1 mm -

1 cm D Rotor diameter

100 m St Turbine separation

10 D

1 km Sf Wind farm size

10 St

10 km Lx Wind farm separation

10 Sf

100 km Ly Wind farm wake width

10 Sf

100 km Lz Planetary boundary layer height

1 km Ux x-Velocity variation

10 m/s Uy y-Velocity variation

10 m/s Uz Vertical velocity variation 0.1 Ux Lz / Lx

1 cm/s Δp Pressure variation

1 hPa ΔT Temporal variation

1 day u Velocity variation

1 m/s ρ Air density

1 kg/m3 f? Coriolis parameter 10-4 1/s First we address the length scales (figure 1). The vertical length scales include those of the surface layer (proportional to the surface roughness length), the turbine layer (proportional to the turbine hub height and therefore to the rotor diameter) and the top layer (proportional to the planetary boundary layer height). Since we consider wind farming in the planetary boundary layer, the vertical length scale Lz of our flow problem is proportional to the height of the planetary boundary layer. The horizontal length scales include the length of the turbine near wake (proportional to the rotor diameter), the turbine far wake and the related turbine separation (up to 10D), the horizontal scale of a wind farm (typically consisting of

10 rows/columns), and the wind farm wake (up to

10 horizontal wind farm scales). Since we consider motions in the planetary boundary layer due to wind farm wakes, our flow problem has two horizontal length scales: the x-wise length scale Lx which we define to be proportional to the streamwise separation between wind farms, and the y-wise length scale Ly proportional to the width of the wake of a wind farm. Now the vertical length scale Lz is smaller than the horizontal length scales Lx and Ly, and the y-wise length scale Ly is of the same order of magnitude as (but smaller than) the x-wise length scale Lx. Table