2.1. Uniform flow through turbine

For uniform flow through a turbine Bernoulli's equation is first applied to the streamtubes which bypass the turbine. This gives

(2.1)\begin{equation} p_{1} + \tfrac{1}{2}\rho u^{2}\phi_{i}^{2} = p_{4} + \tfrac{1}{2}\rho u^{2}\beta_{i+1}^{2}, \end{equation}

for $i\geq 0$, where $\rho$ is the fluid density, $p_{1}$ is the static pressure upstream of the turbine, $p_{2}$ and $p_{3}$ are the static pressures immediately upstream and downstream of the turbine, respectively, and $p_{4}$ is the static pressure downstream of the turbine. It is assumed for $p_{1}$ and $p_{4}$ that the pressure has equalised in the spanwise direction. This can be rewritten as

(2.2)\begin{equation} p_{1} - p_{4} = \tfrac{1}{2}\rho u^{2} (\beta_{i+1}^{2} - \phi_{i}^{2}). \end{equation}

As the pressure drop $p_{1} - p_{4}$ is constant for all bypass streamtubes we can equate the right-hand side of (2.2) for two different bypass streamtubes (indexed $i$ and $j$) to obtain a relationship between the velocities

(2.3)\begin{equation} \beta_{j} = \sqrt{\beta_{i}^{2} - \phi_{i-1}^{2} + \phi_{j-1}^{2}}, \end{equation}

for $i,j \geq$ 1. We can fix $i=1$ so only the downstream bypass flow speed for the streamtube that immediately bypasses the turbine ($\beta _{1}$) is required to know all the other downstream bypass flow speeds ($\beta _{2}$ to $\beta _{N+1}$). Furthermore, if the bypass flow speeds are known then all downstream areas of bypass streamtubes are known via the conservation of mass

(2.4)\begin{equation} R_{i}\phi_{i} = R_{\beta_{i+1}}\beta_{i+1}, \end{equation}

for $i \geq 1$. Now considering the momentum balance for the entire domain between the upstream and downstream locations the thrust of the turbine is obtained as

(2.5)\begin{equation} \frac{T}{A} = \sum^{n}_{i=0} R_{i}\phi_{i}^{2}u^{2}\rho - \sum_{i=0}^{n+1} R_{\beta_{i}}\beta_{i}^{2}u^{2}\rho + \frac{1}{B}(p_{1} - p_{4}). \end{equation}

Using (2.2) with $i=0$ to remove the pressure terms in the equation

(2.6)\begin{equation} \frac{T}{A} = \left( \sum^{n}_{i=0} R_{i}\phi_{i}^{2} - \sum_{i=0}^{n+1} R_{\beta_{i}}\beta_{i}^{2} + \frac{1}{2B} (\beta_{1}^{2} - \phi_{0}^{2})\right) u^{2}\rho. \end{equation}

Applying the Bernoulli equation upstream and downstream of the disc to obtain a relation between the pressure drop across the turbine and the velocity of the flow gives

(2.7)\begin{equation} (p_{1} - p_{2}) + (p_{3} - p_{4}) = \tfrac{1}{2} \rho u^{2} ((\alpha_{av}^{2} - \phi_{0}^{2}) + (\beta_{0}^{2} - \alpha_{av}^{2}) ). \end{equation}

Using the static equilibrium across the turbine the thrust is just the pressure drop across the turbine so from (2.7) we simplify to

(2.8)\begin{equation} \frac{T}{A} = p_{2} - p_{3} = \frac{1}{2}\rho u^{2} (\beta_{1}^{2} - \beta_{0}^{2} ). \end{equation}

We will set the flow speed $u$, turbine area $A$ and density $\rho$ equal to 1 for simplicity in the rest of the calculation. Now equating (2.6) and (2.8) gives

(2.9)\begin{equation} \frac{1}{2}(\beta_{1}^{2} - \beta_{0}^{2}) = \sum^{n}_{i=0} R_{i}\phi_{i}^{2} - \sum_{i=0}^{n+1} R_{\beta_{i}}\beta_{i}^{2} + \frac{1}{2B} (\beta_{1}^{2} - \phi_{0}^{2}). \end{equation}

From the conservation of mass applied to the streamtube which encompasses the turbine, the following expression is obtained:

(2.10)\begin{align} R_{0}\phi_{0} &= R_{\beta_{1}}\beta_{1} + R_{\beta_{0}}\beta_{0} \nonumber\\ &= \beta_{1}\left(\frac{1}{B} - \sum^{n+1}_{i=2} R_{\beta_{i}} - R_{\beta_{0}}\right) + \alpha_{av} \nonumber\\ &= \beta_{1}\left(\frac{1}{B} - \sum^{n+1}_{i=2} R_{\beta_{i}} - \frac{\alpha_{av}}{\beta_{0}}\right) + \alpha_{av}. \end{align}

This can be rewritten in terms of $\alpha _{av}$ as

(2.11)\begin{equation} \alpha_{av} = \frac{\beta_{0} \left( R_{0}\phi_{0} - \beta_{1} \displaystyle\left(\frac{1}{B} -\sum^{n+1}_{i=2} R_{\beta_{i}}\right) \right)}{(\beta_{0}-\beta_{1})}. \end{equation}

Re-arranging (2.9) can give

(2.12)\begin{align} \beta_{0}^{2} + 2\left( R_{0}\phi_{0}^{2} - R_{\beta_{0}}\beta_{0}^{2} - R_{\beta_{1}}\beta_{1}^{2} + \sum^{n}_{i=1} R_{i}\phi_{i} (\phi_{i} - \beta_{i+1})\right) + \frac{1}{B}(\beta_{1}^{2}-\phi_{0}^{2}) - \beta_{1}^{2}= 0. \end{align}

By the conservation of mass of the core flow $R_{\beta _{0}}\beta _{0} = \alpha _{av}$, and along with $R_{\beta _{1}}\beta _{1} = R_{0}\phi _{0} - \alpha _{av}$ equation (2.12) can be rewritten as:

(2.13)\begin{align} & \beta_{0}^{2} - 2\alpha_{av}\beta_{0} +2\beta_{1}\alpha_{av} + 2\left( R_{0}\phi_{0}(\phi_{0} -\beta_{1}) + \sum^{n}_{i=1} R_{i}\phi_{i} (\phi_{i} - \beta_{i+1})\right) \nonumber\\ &\quad + \frac{1}{B}(\beta_{1}^{2}-\phi_{0}^{2}) - \beta_{1}^{2}= 0. \end{align}

Let

(2.14)\begin{equation} C_{1} = 2\left( R_{0}\phi_{0}(\phi_{0} -\beta_{1}) + \sum^{n}_{i=1} R_{i}\phi_{i} (\phi_{i} - \beta_{i+1})\right) + \frac{1}{B}(\beta_{1}^{2}-\phi_{0}^{2}) - \beta_{1}^{2}, \end{equation}

and

(2.15)\begin{equation} C_{2} = \left( R_{0}\phi_{0} - \beta_{1}\left(\frac{1}{B} -\sum^{n+1}_{i=2} R_{\beta_{i}}\right) \right). \end{equation}

Now substitute (2.11) in for $\alpha _{av}$ to obtain

(2.16)\begin{equation} \beta_{0}^{3} - (2C_{2} +\beta_{1})\beta_{0}^{2} + (2C_{2}\beta_{1} + C_{1})\beta_{0} - C_{1}\beta_{1}=0, \end{equation}

which is a cubic equation of $\beta _{0}$ that can be solved analytically if we fix $\beta _{1}$, as $C_{1}$ and $C_{2}$ are functions of known terms and downstream bypass flow terms which can be obtained from $\beta _{1}$ using (2.3) and (2.4). The power generated by the turbine assuming no internal losses is

(2.17)\begin{equation} P=T\alpha_{av}u = \alpha_{av}(\beta_{1}^{2} - \beta_{0}^{2}) \left(\tfrac{1}{2}\rho A u^{3}\right)= C_{P} \left(\tfrac{1}{2}\rho A u^{3}\right), \end{equation}

reintroducing the area, density and velocity terms for clarity. Note that the power coefficient of the turbine, $C_{P}$, is a function of $\alpha _{av}$ (2.11), $\beta _{1}$ and $\beta _{0}$, where only one of these needs to be given to set the operating condition of the turbine and thus solve the problem. In LMADT it is common to introduce a disc resistance (or local thrust) coefficient $k$ to describe the operating condition of the turbine such that

(2.18)\begin{equation} T = \tfrac{1}{2}k\rho A (\alpha_{av}u)^{2}, \end{equation}

therefore $k= (\beta _{1}^{2} - \beta _{0}^{2})/\alpha _{av}^{2}$, by equating (2.17) and (2.18). Hence, the value of $C_{P}$ can be calculated analytically for a given set of $k$, $B$ and a given inflow profile ($R_{i}$ and $\phi _{i}$).

The generalised approach means we are not constrained by the number of bypass streamtubes calculated. This removes, for example, the need to reduce the number of streamtubes through mixing assumptions used in previous multi-row studies which could result in substantial differences in the predicted power (Juniper & Nishino Reference Juniper and Nishino2020).

2.2. Non-uniform flow through turbine

When considering non-uniform flow through a turbine the calculations are much the same as the uniform case presented in § 2.1. A choice is required, however, on how to model the actuator disc. For uniform flow through a turbine there is little decision to be made as all methods considered for the non-uniform case produce equal results. The first method, is to keep the resistance, $k$, uniform across the turbine. The second is to define the velocity profile across the turbine. The final approach considered is to maintain thrust, and therefore ${\rm \Delta} p$, to be uniform across the turbine.

The first method is often used as it gives a more accurate representation of an ideal porous disc which is often used as a representation of a turbine in experimental work (e.g. Myers & Bahaj Reference Myers and Bahaj2012) and in computational studies (e.g. Nishino & Draper Reference Nishino and Draper2015).

Draper et al. (Reference Draper, Nishino, Adcock and Taylor2016) obtain a uniform local resistance coefficient $k$ for an unblocked turbine; however, this is a consequence of defining the flow velocity across the turbine. For the blocked case they restrict their attention to the case where $\alpha _{i}/\phi _{i}$ is constant across the disc, i.e. the velocity profile through the turbine is self-similar upstream. Here, $\alpha _{i}$ is a scalar such that $\alpha _{i}u$ is the velocity of the $i$th streamtube that passes through the turbine. Draper et al. (Reference Draper, Nishino, Adcock and Taylor2016) note that in flow that is not highly sheared or highly blocked the above method results in a near uniform resistance.

Chamorro & Arndt (Reference Chamorro and Arndt2013) adopted the third method, uniform thrust, constructing a correction factor to the Lanchester–Betz–Joukowsky limit for a turbine in a neutrally stratified atmospheric boundary layer. This method will be adopted in our calculation for non-uniform flow through a turbine. In support of this approach Revaz & Porté-Agel (Reference Revaz and Porté-Agel2021) found through large-eddy simulation that a model which computes non-uniform force distributions over the turbine and a model which assumes uniform forces over the turbine can both make accurate predictions for the flow in the far wake and power of the turbine. An exploration of the impact of non-uniform force for LMADT is presented in Appendix A.2.

For non-uniform flow through the turbine we start off as before by applying Bernoulli's equation to streamtubes that bypass the turbine

(2.19)\begin{equation} p_{1}-p_{4} = \tfrac{1}{2}\rho u^{2} (\beta_{i+1}^{2} - \phi_{i}^{2}), \end{equation}

which can be equated to obtain a relationship between all the bypass flow velocities

(2.20)\begin{equation} \beta_{i+1} = \sqrt{\beta_{j+1}^{2} + \phi_{i}^{2} - \phi_{j}^{2}}, \end{equation}

for $i,j \geq M$, where $M$ is the index of the streamtube that is split by the turbine, as depicted in figure 3. Applying Bernoulli's equation to streamtubes that pass through the turbine gives us

(2.21)\begin{equation} p_{2}-p_{3} = {\rm \Delta} p = \tfrac{1}{2}\rho u^{2} (\phi_{a}^{2} - \beta_{a}^{2} - \phi_{M}^{2} + \beta_{M+1}^{2}), \end{equation}

and, as ${\rm \Delta} p$ is held constant,

(2.22)\begin{equation} \beta_{a} = \sqrt{\beta_{b}^{2} + \phi_{a}^{2} - \phi_{b}^{2}}, \end{equation}

for $a,b \leq M$. So all streamtube velocities are related once one downstream velocity in the bypass and one downstream velocity in the core flow are known. Likewise the conservation of momentum is defined as

(2.23)\begin{align} 0 &= (\beta_{M}^{2} - \beta_{M+1}^{2}) + \frac{1}{B}(\beta_{M+1}^{2} - \phi_{M}^{2}) + 2\left(\sum_{i=0}^{M-1} R_{i}\phi_{i}(\phi_{i} - \beta_{i}) \vphantom{\sum_{i=M+1}^{n}}\right. \nonumber\\ &\quad \left. + \sum_{i=M+1}^{n} R_{i}\phi_{i}(\phi_{i} - \beta_{i+1}) + R'\phi_{M}(\phi_{M}-\beta_{M}) + R^{*}\phi_{M}(\phi_{M}-\beta_{M+1}) \right), \end{align}

where $R_{M} = R' + R^{*}$ with $R'$ as the upstream area of the streamtube which passes through the turbine and $R^{*}$ being the upstream area of the streamtube which directly bypasses the turbine.

The power $P$ is calculated as

(2.24)\begin{equation} P = \sum_{i=0}^{M}R_{\alpha_{i}}A\alpha_{i}u {\rm \Delta} p, \end{equation}

where $R_{\alpha _{i}}A$ is the size of the $i$th streamtube at the turbine plane; however, $R_{\alpha _{i}}$ is not calculated in this analysis. Therefore, this is rewritten using the conservation of mass to obtain an equivalent expression based on known upstream values

(2.25)\begin{equation} P = Au{\rm \Delta} p \left(R'\phi_{M} + \sum_{i=0}^{M-1} R_{i}\phi_{i}\right). \end{equation}

We define the power coefficient as

(2.26)\begin{equation} C_{P} = \frac{P}{\dfrac{1}{2}\rho A \overline{u^{3}}} = \frac{P}{\dfrac{1}{2}\rho A} \frac{R'+\displaystyle\sum^{M-1}_{i=0}R_{i} }{u^{3} \left(R'\phi_{M}^{3} +\displaystyle\sum_{i=0}^{M-1} R_{i}\phi_{i}^{3}\right)}, \end{equation}

which is the power normalised using the cube of the upstream velocity of the flow that passes through the turbine, $\overline {u^{3}}$. To solve this system, we need to know what portion of upstream flow passes through the turbine, allowing $M$ to be determined. For this reason $\alpha _{av}$, the average velocity through the turbine, is used as the input parameter, thus allowing $M$ to be determined by conservation of mass. After $\alpha _{av}$ is selected a range of $\beta _{M+1}$ and $\beta _{M}$ values are iterated over to find the relevant root. The combinations of all $\beta _{M+1}$ and $\beta _{M}$ values do not need to be checked. It is possible to solve the system by first solving a corresponding uniform case and then solving for the non-uniform core flow; this is done by averaging the flow through the turbine such that it is uniform and has the same mass as the non-uniform and correcting (2.18) to account for the additional momentum in the non-uniform case. This fixes $\beta _{M+1}$, which will be equal between the uniform and non-uniform cases. Then $\beta _{M}$ can be found using (2.21) and the conservation of mass in the core flow.

3.1. Effective blockage with sheared bypass flow

A study into the impact of non-uniform inflow on the power of a turbine has been conducted by Draper et al. (Reference Draper, Nishino and Adcock2014), in which they found that a faster (or slower) bypass flow, compared with the core flow, could increase (or decrease) the power extracted by a turbine. This was explained by a change in the ‘effective blockage’ of the turbine, i.e. the faster (or slower) bypass flow inhibits (or enhances) the expansion of the core flow for a given geometrical blockage.

The effective blockage of a turbine is the blockage ratio required to attain the same $C_{P\,{{max}}}$ with a uniform inflow. Since $C_{P\,{{max}}}$ for the case with uniform inflow is known to be $16/(27(1-B)^{2})$ (Garrett & Cummins Reference Garrett and Cummins2007; Dehtyriov et al. Reference Dehtyriov, Schnabl, Vogel, Draper, Adcock and Willden2021), the effective blockage, $B_{eff}$, is calculated as

(3.1)\begin{equation} B_{eff} = 1 - \left(\frac{16}{27 C_{P\,{{max}}}} \right)^{{1}/{2}}. \end{equation}

The results of this work by Draper et al. (Reference Draper, Nishino and Adcock2014) was limited by the top-hat inflow profile shape, as shown in figure 4(a), where $r$ is the width of the core flow and $\phi u$ is the velocity of the bypass. It was unclear whether this top-hat inflow profile would accurately predict the flow induced blockage of sheared bypass flows. Draper et al. (Reference Draper, Nishino, Adcock and Taylor2016) investigated the effective blockage of a turbine experiencing a variety of sheared inflow profiles. However, their study considers shear flow across the whole domain so cannot fully clarify the impact of a sheared bypass compared with a uniform bypass on the power extraction of turbine.

Figure 4. Contours of effective blockage ratio, $B_{eff}$, for (a,c) uniform bypass and (b,d) sheared bypass cases, for fixed geometric blockage ratios (a,b) $B=0.2$ and (c,d) $B=0.1$.

In this study a comparison of the effective blockage was conducted between the cases with a top-hat inflow profile and a flow profile with a uniform core and linearly sheared bypass flow, as shown in 4(b). For the sheared case the upstream number of streamtubes is set at 201, which was found to be sufficient to accurately obtain the values of $C_{P\,{{max}}}$ and thus $B_{eff}$ (see figure 5). To make a fair comparison, the mass flux in the bypass is held constant between the uniform bypass and linearly sheared cases.

Figure 5. Value of $C_{P\,{{max}}}$ plotted against number of bypass streamtubes $N$ for $B=0.2$, $r=1$ and six different $\phi$ values.

The effective blockage for various bypass speeds and bypass sizes is plotted in figure 4 for two different blockage ratios $B=0.2$ and $0.1$. When $\phi = 1$ we obtain the Garrett & Cummins (Reference Garrett and Cummins2007) result for a blocked turbine experiencing uniform inflow. Across all bypass sizes and bypass speeds the uniform bypass scenarios predict a higher effective blockage than the sheared bypass flow cases. This is more prominent for the lower blockage case of $B=0.1$. The over-prediction by the uniform bypass case occurs as for a given background pressure gradient the uniform bypass case restricts the expansion of the core flow more than the sheared bypass case; thus less flow is diverted around the turbine. The additional constraint from the uniform bypass flow can be illustrated from the increased effective blockage ratio $B_{eff}$ in the uniform case compared with the sheared bypass in figure 4.

The reduction in $B_{eff}$ across all $r$ and $\phi$ values for the linear shear, in comparison with the uniform bypass, implies that the performance of a turbine placed in another turbine's wake may be consistently over-predicted when the upstream bypass flow is modelled as uniform instead of sheared. This result has implication for the use and benchmarking of wake models which maintain a top-hat flow profile in downstream like that of Jensen (Reference Jensen1983) and Frandsen et al. (Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and Thøgersen2006) when applied to the analysis of multi-row turbine arrays. These top hat wake models are popular analytical models for turbine wakes which are still used today for turbine array optimisation (e.g. Duc et al. Reference Duc, Coupiac, Girard, Giebel and Göçmen2019).

3.2. Effective blockage with sheared core flow

In the previous section the impact of resolution/flow averaging on the power coefficient of a laterally bounded turbine with a sheared bypass flow was identified. In this section the impact of flow averaging for a shear flow passing through a turbine will be explored by comparing two flow profiles, sheared core and uniform core.

The base inflow profile which spans the domain will be a linear shear profile of the form

(3.2)\begin{equation} \phi = (1-\delta) + (2\delta)\frac{y}{A}, \end{equation}

where $y/A$ is the non-dimensional spanwise distance from the centre of the turbine and $\delta$ is a parameter to define the significance of the shear. The portion of the flow which passes through the turbine will either be kept as a shear profile or be averaged to become a uniform flow profile, as illustrated in figure 6(b). All results presented below are for geometric blockage ratios of $B=0.1$ and $B=0.2$. The spanwise resolution of the flow was defined by fixing the number of upstream streamtubes which pass through the turbine at 20, as it was found to be adequate to ensure convergence with a difference of less than 0.2 % compared with calculating 500 streamtubes through the turbine.

Figure 6. Difference in the effective blockage between the uniform flow through a turbine and the shear flow through a turbine, for fixed geometric blockage ratios (a) $B = 0.1$ and (b) $B=0.2$.

The effective blockage is plotted against $\delta$ in figure 6 for the case of sheared flow through the turbine and uniform flow through the turbine. Across all $\delta$ values the uniform core flow case exaggerates the impact of the faster (if $\delta$ is positive) or slower (if $\delta$ is negative) bypass flow on the effective blockage. For negative $\delta$ values where the slower bypass leads to effective blockages lower than the geometric blockage ratio, a uniform core flow will under-predict the power extracted when compared with the sheared core flow case. Conversely, for positive $\delta$ values where the faster bypass increases the effective blockage ratio, a uniform core flow will over-predict the power extracted when compared with the sheared core flow case.

The theoretical approach detailed in § 2.2 only needs to consider which streamtube passes through the turbine and which enters the bypass. Hence, the spanwise position of each streamtube within the core or bypass flow does not impact the results, as long as the streamtube with velocity $\phi _{M}u$ is still split by the turbine. Therefore, although simple linear shears were considered for this analysis, the trends observed here can be expected in cases with more complicated flow profiles as long as the above condition is satisfied.

4.1. Local and global power coefficients

A distinction is made here between what we define as the ‘internal’ and ‘external’ performance of a multi-row array. To demonstrate this distinction in a simple manner, we first consider a special case of multi-row arrays where the streamwise distance between each row is large enough for the wake of each turbine to be fully recovered. This special case has been studied in a tidal turbine array context by Vennell (Reference Vennell2010) who combined the LMADT of Garrett & Cummins (Reference Garrett and Cummins2007) with an analytical tidal channel model (Garrett & Cummins Reference Garrett and Cummins2005). We present an equivalent special case in the wind turbine array context by combining results from the LMADT (with uniform inflow for each turbine) and an analytical ‘external’ model derived from the two-scale momentum theory for large wind farms (Nishino & Dunstan Reference Nishino and Dunstan2020).

We define the array-average ‘local’ or ‘internal’ power coefficient as

(4.1)\begin{equation} C_{P}^{*} = \frac{\displaystyle\sum^{n}_{i=1} P_{i}}{\dfrac{1}{2}\rho u_{av}^{3} n A}, \end{equation}

where $P_{i}$ is the power at the $i$th turbine and $n$ the number of turbines in the array. This represents the total power of turbines for a given (fixed) mass flux through the local flow domain. Here, we assume that the average velocity, $u_{av}$, corresponds to the ‘farm-layer average’ velocity, $U_{F}$, in the two-scale momentum theory (as the inflow profile is uniform for all turbines and thus we cannot define the farm layer in this simple example). By keeping the mass flux constant through the local flow domain we restrict our attention to the ‘internal’ performance of the array, which can increase due to local blockage effects. This is in contrast to ‘global’ performance which can be impacted by wind-farm blockage effects (Bleeg et al. Reference Bleeg, Purcell, Ruisi and Traiger2018; Kirby, Nishino & Dunstan Reference Kirby, Nishino and Dunstan2022) reducing the power of a wind farm as the mass flux through the whole array decreases. Alongside the local power coefficient $C_{P}^{*}$, we also define the array-averaged ‘global’ power coefficient as

(4.2)\begin{equation} C_{P}^{G} = \frac{\displaystyle\sum^{n}_{i=1} P_{i}}{\dfrac{1}{2}\rho U_{0}^{3} n A}, \end{equation}

where $U_{0}$ is undisturbed or ‘natural’ wind speed observed when the whole wind farm does not exist. Therefore, $C_{P}^{G}$ is a measure of the ‘overall’ efficiency of the farm, represented by the ratio of power extracted by the turbines and the power of the undisturbed flow through the total turbine swept area. Here, $C_{P}^{G} = \beta ^{3} C_{P}^{*}$, where $\beta = u_{av}/U_{0}$. To illustrate the impact of local blockage on $C_{P}^{G}$ and $C_{P}^{*}$, we compare how theses power coefficients change as we vary the lateral turbine spacing in a multi-row turbine array.

We calculate $C_{P}^{*}$ directly from LMADT, whereas $C_{P}^{G}$ is estimated by combining LMADT results with the two-scale momentum theory of Nishino & Dunstan (Reference Nishino and Dunstan2020). To estimate the unknown velocity ratio, $u_{av}/U_{0}$, from the theory, we consider that there are multiple rows of turbines placed equidistantly with a large streamwise spacing of $20D$, where $D$ is the turbine diameter, and assume that this distance is large enough for each turbine wake to be fully recovered (see e.g. Sedaghatizadeh et al. Reference Sedaghatizadeh, Arjomandi, Kelso, Cazzolato and Ghayesh2019). This allows us to calculate the array density $\lambda$ (which is the ratio of the total turbine swept area to the farm area) as a function of the lateral turbine spacing $L$, i.e. $\lambda = {\rm \pi}D/80L$. We further assume a simple relationship between $B$ and $L$ as $B = {\rm \pi}D/20L$ for convenience, meaning that the height of the local flow domain (which may be defined when there is a strong capping layer acting like a rigid lid) is $5D$. This means that we assume $\lambda = 0.25B$ in the following analysis.

To solve the two-scale coupled problem, we also need to calculate from LMADT the array-averaged ‘local’ or ‘internal’ thrust coefficient, defined as

(4.3)\begin{equation} C_{T}^{*} = \frac{\displaystyle\sum^{n}_{i=1} T_{i}}{\dfrac{1}{2}\rho u_{av}^{2} n A}, \end{equation}

where $T_{i}$ is the thrust at the $i$th turbine. The two-scale coupled momentum balance equation from Nishino & Dunstan (Reference Nishino and Dunstan2020) is

(4.4)\begin{equation} C_{T}^{*} \frac{\lambda}{C_{f0}}\beta^{2} + \beta^{\gamma} = 1 + \zeta(1-\beta), \end{equation}

where the momentum availability factor (right-hand side) has been approximated using a linear wind extractability model (Patel, Dunstan & Nishino Reference Patel, Dunstan and Nishino2021). Here, we also set the natural bottom friction coefficient as $C_{f0} = 0.002$ to be typical of an offshore wind-farm scenario. We set the bottom friction exponent as $\gamma = 2$, which is a reasonable value and allows for the momentum equation (4.4) to be solved analytically. Therefore, $\beta$ is reduced to a function of an external (mesoscale) flow parameter $\zeta$. Here, $\zeta$ is the momentum response parameter for the atmosphere, which represents the ability of the atmosphere to sustain wind speed against the resistance caused by the wind farm, i.e. the extractability of wind. Typical values of which are expected to be between 5 and 25 for a realistic large offshore wind farm (Patel et al. Reference Patel, Dunstan and Nishino2021).

Figure 7 shows the values of $C_{P}^{*}$ and $C_{P}^{G}$ (for three different atmospheric response strengths, $\zeta$ = 5, 15 and 25) across a range of turbine resistance coefficients and blockage ratios. In each plot the red line shows the optimal resistance coefficient, $k_{opt}$, which is defined as the value of $k$ at which the maximum power coefficient is achieved at each blockage ratio $B$. If $k$ is fixed the local power coefficient $C_{P}^{*}$ always increases with $B$, whereas the global power coefficient $C_{P}^{G}$ tends to decrease at different rates depending on $\zeta$. Importantly, the value of $k_{opt}$ for $C_{P}^{*}$ increases rapidly with $B$ (from 2 at $B$ = 0 to 5.4 at $B$ = 0.2) but $k_{opt}$ for $C_{P}^{G}$ tends to decrease, especially when $\zeta$ is small.

Figure 7. Contour plots of (a) $C_{P}^{*}$ and (b–d) $C_{P}^{G}$ of a multi-row turbine array without wake interactions between rows, for (b) $\zeta = 5$, (c) $\zeta = 15$ and (d) $\zeta = 25$ for various blockage ratios ($B$) and resistance coefficients ($k$). Optimal resistance $k_{opt}$ is shown by a red line.

Figure 8 shows the maximum $C_{P}^{*}$ and corresponding $C_{P}^{G}$ values obtained with $k=k_{opt}^{*}$ at each blockage ratio $B$, where $k_{opt}^{*}$ is the $k_{opt}$ value for $C_{P}^{*}$. Increasing $B$ (or decreasing the lateral turbine spacing $L$) causes $C_{P\,{{max}}}^{*}$ to increase, as observed in figure 7(a), however, $C_{P}^{G}$ at $k=k_{opt}^{*}$ decreases monotonically for all values of $\zeta$ between 5 and 25 (as the increase in $C_{P}^{*}$ is not large enough to compensate for the decrease in $\beta ^3$). Note that $B = {\rm \pi}D/20L$ has been assumed in this example and hence $B = 0.157$ corresponds to $L/D =1$, meaning that blockage ratios of higher than 0.157 are unphysical in this example (unless we allow for variable turbine hub heights in each row).

Figure 8. Maximum local power coefficient and corresponding global power coefficients of a multi-row turbine array without wake interactions between rows, plotted against the blockage ratio ($B$) for a range of momentum response parameters ($\zeta$) (with the assumption that $\lambda = 0.25B$, $C_{f0} = 0.002$ and $\gamma = 2$).

Note that, while a streamwise spacing of $20D$ was selected for the above analysis, larger spacing may be required for each turbine wake to be fully recovered in a real wind farm depending on the turbulence intensity. However, the trends in the relationship between the local and global power coefficients observed in this exploration remain consistent across choices of spacing.

In the following sections we consider the more general case where the streamwise distance between rows is not large enough for the wake to be fully recovered. To model this we use an inviscid–viscous approach, outlined in the next sub-section, but for simplicity no longer consider the external momentum balance, therefore we only focus on the ‘internal’ performance of the array represented by $C_{P}^{*}$.

4.2. Inviscid–viscous approach

For the following multi-row analysis we utilise the inviscid–viscous approach as described in Ouro & Nishino (Reference Ouro and Nishino2021). As illustrated earlier in figure 1, this approach treats the region around the turbine as an inviscid flow region which is modelled with LMADT, followed by a viscous mixing zone downstream which is modelled separately. This approach is in line with the results of West & Lele (Reference West and Lele2020) who, in a large-eddy simulation study of flow around an actuator disc, showed that inviscid flow processes dominate around the turbine and viscous processes dominate further downstream. We adopt the same simplification in this analysis as in Ouro & Nishino (Reference Ouro and Nishino2021) that mixing takes place uniformly across the viscous mixing zone and will be defined by a single parameter $0\leq m\leq 1$ such that $u_{out} = m u_{av} + (1-m) u_{in}$, where $u_{in}$ is a streamtubes velocity entering the viscous mixing zone, $u_{av}$ is the average velocity across the whole domain and $u_{out}$ is the velocity as it exits the viscous mixing zone. For example when $m=1$ the whole flow becomes uniform and when $m=0$ no mixing occurs.

4.3. Instant mixing assumption and its validation

Previous utilisations of the inviscid–viscous approach for an infinitely large array of turbines (Nishino & Draper Reference Nishino and Draper2019; Ouro & Nishino Reference Ouro and Nishino2021) only calculate a fixed number of bypass streamtubes. As the number of streamtubes in the system increases at each turbine row (meaning that an infinite number of streamtubes would be required to model an infinitely large array), an assumption was made to keep the number of bypass streamtubes fixed. The assumption is that the bypass flow in the streamtube adjacent to the turbine (the streamtube with velocity $\beta _{1}$ in figure 2) is always fully mixed in the viscous mixing zone (regardless of the value of the mixing parameter $m$) with the bypass flow in the streamtube immediately outside (the streamtube with the velocity $\beta _{2}$). In the case with uniform flow through the turbine the streamtubes with velocity $\beta _{1}u$ and $\beta _{2}u$ are mixed to a single streamtube with the new velocity coefficient

(4.5)\begin{equation} \beta' = \frac{R_{\beta_{1}}\beta_{1}+R_{\beta_{2}}\beta_{2}}{R_{\beta_{1}}+R_{\beta_{2}}}, \end{equation}

therefore the number of bypass streamtubes remains constant in the upstream and downstream.

The extended LMADT for a generalised number of bypass streamtubes outlined in § 2 allows for the impact of this instant mixing assumption to be accurately quantified. For this investigation both an aligned and a staggered turbine arrangement are considered with the first turbine row experiencing a fully uniform inflow. For a fixed blockage ratio and mixing rate, $B=0.2$ and $m=0.6$, the array-averaged power coefficient $C_{P}^{*}$ of four rows of turbines is plotted in figure 9 against resistance coefficient $k$ for three different levels of instant mixing. Firstly, the ‘one bypass’ case is for instant mixing such that there are only up to two upstream streamtubes (i.e. the number of streamtubes increases from two to three in the inviscid zone but decreases from three to two due to the instant mixing between the two bypass flows in the viscous zone), the ‘two bypass’ case allows for three upstream streamtubes, and the ‘three bypass’ case allows for four upstream streamtubes. The three bypass case is therefore the case where no instant mixing occurs for any of the four rows. The plots show that the only significant difference in predictions occurs in the staggered case when the upstream number of streamtubes is fixed at two (which is understandable as we need at least three upstream streamtubes to represent the wake of a staggered turbine upstream). Therefore as previous studies utilising this method (Nishino & Draper Reference Nishino and Draper2019; Ouro & Nishino Reference Ouro and Nishino2021) have a minimum of two upstream streamtubes for the aligned case and at least three for the staggered, their predictions and conclusions are not significantly impacted by the instant mixing assumption.

Figure 9. The $C_{P}^{*}$ predictions for four rows of (a) aligned and (b) staggered turbines for various numbers of upstream bypass streamtubes. Blockage and mixing rate are fixed at $B=0.2$ and $m=0.6$.

4.4. Efficiency of aligned and staggered arrays

Figures 10 and 11 show the maximum $C_{P}^{*}$ and the corresponding optimal local thrust coefficient, $C_{T\ opt}^{*}$, for an aligned and staggered array of 12 turbine rows, as an example of a multi-row analysis. The analysis was conducted without employing the instant mixing assumption. The optimal local thrust coefficient, $C_{T\ opt}^{*}$, is the $C_{T}^{*}$ value at which the array obtains the maximum $C_{P}^{*}$. These example cases are with a uniform inflow profile at the first turbine row, a constant $k$ for all 12 rows and it is assumed that mixing rate $m$ also remains constant. For the aligned case in figure 10(a) we observe that the array achieves a maximum $C_{P}^{*}$ at higher blockage and mixing rates, with $m=1$ being optimal for all blockage ratios. For the staggered case the efficiency of the array can slightly improve when the mixing rate is less than one. This is because a downstream turbine can take advantage of the increased speed of flow bypassing upstream turbines. These trends agree well with the theoretical and computational results of Ouro & Nishino (Reference Ouro and Nishino2021) for infinitely large arrays of turbines.

Figure 10. Contour plots of the maximum $C_{P}^{*}$ for a given mixing rate $m$ and blockage $B$ across 12 turbine rows for (a) aligned and (b) staggered turbine arrangements.

Figure 11. Contour plots of the thrust coefficient, $C_{T}^{*}$, required for obtaining maximum $C_{P}^{*}$ for a given mixing rate $m$ and blockage $B$ across 12 turbine rows for (a) aligned and (b) staggered turbine arrangements.

In both turbine arrangements the regions of higher $C_{P\,{{max}}}^{*}$ are accompanied by higher $C_{T\ opt}^{*}$. However, the gradient of increasing $C_{T\ opt}^{*}$ does not exactly match that for increasing $C_{P\,{{max}}}^{*}$ so there are opportunities for improving $C_{P}^{*}$ of an array without increasing $C_{T}^{*}$ (meaning that the global power coefficient $C_{P}^{G}$ could be improved by finding an optimal balance between increasing $C_{P}^{*}$ and decreasing $C_{T}^{*}$ for a given ‘external’ condition, as discussed earlier in § 4.1). For example, in a staggered arrangement by using an array layout with blockage $B=0.2$ and $m=0.86$ we would obtain a near identical power coefficient as the $B=0.19$ and $m=0.6$ case with a ${\sim }5.5\,\%$ reduction in $C_{T}^{*}$. As the blockage ratio $B$ is inversely proportional to the spanwise distance between turbines whereas the mixing rate $m$ is expected to increase with the streamwise distance between turbines, the above results imply that the optimal staggered array layout (to maximise $C_{P}^{G}$) would depend on a complex interaction between the local blockage, wake mixing and ‘external’ flow conditions (such as the momentum response parameter $\zeta$ for the case of wind farms).

4.5. Varying turbine resistance across multiple rows

The analytic model presented in § 2 allows for more complex explorations of multi-row turbine array optimisation such as the investigation of varying turbine resistance, $k$, across multiple rows, an investigation that would become too computationally expensive with other modelling techniques. Varying $k$ across multiple rows may provide improvements in the maximum farm-averaged $C_{P}^{*}$. Let $C_{P\,{{max}}}^{*V}$ be the maximum $C_{P}^{*}$ for varying $k$ across multiple rows and $C_{P\,{{max}}}^{*F}$ be the optimal $C_{P}^{*}$ when $k$ is uniform for all rows. Figure 12 depicts $C_{P\,{{max}}}^{*V}$/$C_{P\,{{max}}}^{*F}$ for a range of blockage ratios and mixing rates, for the case of three rows of aligned or staggered turbines. As can be seen from figure 12 the magnitude of the improvements and where they are achieved differ between the aligned and staggered arrangements. The improvements over the uniform $k$ scenario for the aligned case can be up to $2\,\%$ in this example, and these improvements can be best obtained at low mixing and blockage ratios. For the staggered arrangement the improvements are minor, of less than $1\,\%$ in this example, and are best seen for high blockage ratios and low mixing rates, scenarios in which the impact of an upstream turbine on the core flow of a downstream turbine is at its greatest.

Figure 12. Contour plots of $C_{P\,{{max}}}^{*V}$/$C_{P\,{{max}}}^{*F}$, the ratio of the farm-averaged $C_{P\,{{max}}}^{*}$ achieved by varying resistance coefficient, $k$, across three rows of turbines, to that obtained from the uniform $k$ scenario. (a) Aligned; (b) staggered.

For the aligned case we see a peak in potential improvement over the uniform $k$ case when $m$ is low and $B\approx 0.05$. This peak occurs due to the interaction between the flow speed through the turbine, flow speed in the wake and blockage effects. At low mixing rates the interaction between upstream and downstream turbines is at its greatest, allowing for more opportunities to optimise their combined performance. The basic mechanism is that we can increase the average performance by reducing the resistance of upstream turbines to obtain more favourable inflow for downstream turbines. At very low $B$ this mechanism does not allow for much optimisation as changing $k$ (from its optimal value for the uniform $k$ scenario) has a large negative impact on $C_{P}^{*}$ of that turbine and the downstream turbines cannot compensate for this. As $B$ increases to $B\approx 0.05$ a small reduction in $k$ (when $k$ is less than $k_{opt}$ for the uniform $k$ case) has a much smaller negative impact on $\alpha _{av}$ than a positive impact it has on $\beta _{0}$, allowing us to better optimise the inflow for a downstream turbine without sacrificing much upstream turbine performance. As $B$ increases further, varying $k$ has less impact on the flow, meaning that a large change in $k$ is required to alter the inflow for a downstream turbine in a significant manner, and this would not offset the great performance reduction the upstream turbine experiences due to large $k$ variations.

All these results are obtained under the assumption that the mixing rate, $m$, is held constant across the entire array. In reality, the mixing rate would change as we alter the resistance coefficient of the turbine, for example. Implementing a more physically realistic mixing model would provide greater insight on how we should design and operate multi-row turbine arrays. Nevertheless, the analysis presented above highlights the opportunity to explore large parameter space using this computationally fast analytic model, with a single simulation taking less than $1\times 10^{-6}$ seconds.