Wind-Tunnel Simulation of the Wake of a Large Wind Turbine in a Stable Boundary Layer. Part 1: The Boundary-Layer Simulation

Abstract

Measurements have been made in both a neutral and a stable boundary layer as part of an investigation of the wakes of wind turbines in an offshore environment, in the EnFlo stratified flow wind tunnel. The working section is long enough for the flow to have become very nearly invariant with streamwise distance. In order to be systematic, the flow profile generators of Irwin-type spires and surface roughness were the same for both neutral and stable conditions. Achieving the required profiles by adjusting the flow generators, even for neutral flow, is a highly iterative art, and the present results indicate that it will be no less iterative for a stable flow (as well as there being more conditions to meet), so this was not attempted in the present investigation. The stable-case flow conformed in most respects to Monin–Obukhov similarity in the surface layer. A linear temperature profile was applied at the working section inlet, resulting in a near-linear profile in the developed flow above the boundary layer and ‘strong’ imposed stability, while the condition at the surface was ‘weak’. Aerodynamic roughness length (mean velocity) was not affected by stability even though the roughness Reynolds number ${<}1$, while the thermal roughness length was much smaller, as is to be expected. The neutral case was Reynolds-number independent, and by inference, the stable case was also Reynolds-number independent.

Appendix: Effect of Temperature on LDA Measurements

The refractive index, $n$, of air is dependent on temperature, and so the path of a constituent ray, and therefore that of a beam, may change. In a turbulent flow the path followed by a beam will vary along its length according to the variation of temperature with time along its length. As this is not known a simplified analysis is all that can be made. We follow the line of analysis set out by Resagk et al. (2003). If $s$ is the distance along the ray from its origin at the probe lens, and $\mathbf{\underline{r}}$ the position vector of a point along the ray then

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}s}\left( {n\frac{\hbox {d}\mathbf{\underline{r}}}{\hbox {d}s}} \right) = \mathrm{grad}\;n, \end{aligned}$$

(19)

(Born and Wolf 1999). The case of practical concern here is when the gradient is perpendicular to the ray direction, $\xi$. Then, if $\eta$ is the lateral displacement, this equation reduces to $\hbox {d}^{2}\eta /\hbox {d}\xi ^{2}=(1/n) \hbox {d}n/\hbox {d}\eta$, and since the departure of $n$ from unity is in any case small (see below), we obtain

$$\begin{aligned} \frac{\hbox {d}^{2}\eta }{\hbox {d}\xi ^{2}}\approx \frac{\hbox {d}n}{\hbox {d}\eta } = \beta , \end{aligned}$$

(20)

say. Making the further assumptions that a focused beam can be represented by two rays converging at an angle of $\pm \phi$ and that the gradient $\beta$ is constant across the beam leads to the lateral displacement,$\Delta \eta$, of the focus according to

$$\begin{aligned} \Delta \eta =\frac{\beta }{2}F^{2}\left( {\frac{2\delta }{F\phi }-1} \right) \end{aligned}$$

(21)

where $2 \delta$ is the beam separation at the lens and $F$ is the distance to the focus. It can also be shown that the angle of the beam, $\alpha$, at the focus is given by $\beta F$.

Now, supposing the gradients are moderate enough for both beams (at small internal angle $2 \Phi$) to be affected equally, then Eq. 21 gives the lateral deflection of the measuring volume from the probe axis, and there is no movement along the probe axis. There is also no change in the angle between the beams and therefore no change in fringe spacing. Conversely, assuming the gradient across one beam is equal and opposite in magnitude to that across the other, leads to no lateral deflection of the measuring volume but does lead to a displacement from the probe according to

$$\begin{aligned} \Delta \xi =\frac{\beta F^{2}}{2(\Phi +\alpha )}\left( {\frac{2\delta }{F\phi }-1} \right) , \end{aligned}$$

(22)

where $\xi$ now denotes the distance along the probe axis, rather than along a beam. The change in beam internal angle is $2\alpha$, and the fractional change in Doppler frequency $\delta f/f=- \alpha /(\Phi +\alpha )$.

Now, taking $\hbox {d}n/\hbox {d}T \approx 9.8 \times 10^{-7} \hbox { K}^{-1}$, and noting $dT/d\eta$ has units Km$^{-1}$, so that $\beta = 9.8 \times 10^{-7}$ d $T$ /d $\eta$ (m$^{-1}$) and the probe parameters lead to $\alpha =4.9\times 10^{-8}\;\hbox {d}T\hbox {/d}\eta ,\,\Delta \eta =1.2 \times 10^{-9}\;\hbox {d}T\hbox {/d}\eta \,(\hbox {m})$ and $\Delta \xi =1.5\times 10^{-8}\;\hbox {d}T\hbox {/d}\eta \,(\hbox {m})$. From Fig. 7 $\partial {\phi }'/\partial z$ is not larger than about $3\hbox { K m}^{-1}$, so even if instantaneously $\partial \phi (t)/\partial z$ was, say, three order of magnitude larger then, $\alpha \approx 1.5 \times 10^{-4}$, and $\Delta \eta \approx 3.7\, \upmu \hbox {m}$ and $\Delta \xi \approx 46 \,\upmu \hbox {m}$, or about 3 % of the respective measuring volume dimensions. (If only one beam is supposed affected, then this fraction is about 1.5 %.) The fractional change in Doppler frequency is about 0.18 %. In conclusion, the effects of temperature fluctuations are expected to be negligible.