2.1. Momentum Conservation

We build our momentum conserving wake models starting from the conservation form of the Reynolds-averaged Navier–Stokes (RANS) equations for high Reynolds numbers in the streamwise direction. After neglecting viscous terms and considering the wakes to be in zero-pressure gradient flow, e.g. not in a high blockage environment as a shallow tidal channel, the equation reads

(1)\begin{equation} \frac{\partial U_w (U_0 - U_w)}{\partial x} + \frac{\partial V_w (U_0 - U_w)}{\partial y} + \frac{\partial W_w (U_0 - U_w)}{\partial z} = \frac{\partial u'u'}{\partial x} + \frac{\partial u'v'}{\partial y} + \frac{\partial u'w'}{\partial z}, \end{equation}

with ($U_w,V_w,W_w$) being the vector of mean wake velocities in the streamwise, transverse and vertical directions respectively, $U_0$ is the free-stream velocity and $u'u'$, $u'v'$ and $u'w'$ denote time-averaged turbulent fluctuation correlations. We integrate (1) at any streamwise location of a control volume that embeds the turbine and expands over the $y$ and $z$ directions from $-\infty$ to $\infty$, which together with the assumption that the shear stresses vanish when the distance from the wake centre increases, provides the resulting RANS equation:

(2)\begin{equation} \frac{\textrm{d}}{\textrm{d}\,x} \int_{-\infty}^{\infty} (U_{w}(U_{0} - U_{w}) -u'u')\,\mathrm{d} A \approx 0 . \end{equation}

The streamwise variation of $u'u'$ is much reduced when compared to the convective term (Reference Bastankhah and Porté-AgelBastankhah & Porté-Agel, 2014), hence (2) can be simplified to obtain the momentum integral (Reference Tennekes and LumleyTennekes & Lumley, 1972) as

(3)\begin{equation} \rho\int_{-\infty}^{\infty} (U_{w}(U_{0} - U_{w}))\,\mathrm{d} A = T. \end{equation}

This equilibrium condition states that the momentum deficit flux in the wake is proportional to the thrust force $T$ exerted by the turbine. The thrust force can be related to the thrust coefficient ($C_T$) from the actuator disc theory as

(4)\begin{equation} T = \tfrac{1}{2}C_{T}\rho A_{0} U_{0}^2. \end{equation}

where $\rho$ indicates the fluid density.

2.2. Top-hat Wake Distribution

Analogously to the Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and ThøgersenFrandsen et al. (Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and Thøgersen2006) top-hat HAT wake model, we assume the distance downstream of the rotor the flow in the wake required to reach the free-flow pressure is negligible. The cross-sectional area of the wake just after the initial wake expansion is considered rectangular as the turbine rotor's cross-sectional area at $x=x_a$, i.e. $A_{w}(x_a)= A_{a}$. The value of the wake onset area $A_a$ can be determined using the actuator disc theory as $A_a = \beta A_0$, with $\beta$ representing the relative initial wake expansion at $x_a=0$ in terms of the turbine rotor's cross-section $A_0$, independently of whether such cross-section is circular (HATs) or rectangular (VATs). For a $C_T$ lower than the unity, the actuator disc theory states that the velocity over the plane at a downstream distance $x_a$ is $U_{0}(1-2a)$ whilst at the rotor centre plane the velocity is $U_{0}(1-a)$, with $a=(1-\sqrt {1-C_{T}})/2$ being the so-called induction factor. Hence, the value of $\beta$ is determined as

(5)\begin{equation} \beta = \frac{A_{a}}{A_{0}} = \frac{1-a}{1-2a} = \frac{1}{2}\frac{1 + \sqrt{1-C_{T}}}{\sqrt{1-C_{T}}}. \end{equation}

The wake area at any streamwise location is determined similarly to the approach presented by Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and ThøgersenFrandsen et al. (Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and Thøgersen2006) for HATs with $D_w \propto (x-x_a)^{1/2}$ and $H_w \propto (x-x_a)^{1/2}$, considering the wake width expands unevenly over the horizontal and vertical directions according to

(6)\begin{equation} D_{w} = D_0 \left(\beta + k_{w_{y}}\frac{x}{D_0}\right)^{1/2}, \quad H_{w} = H_0 \left(\beta + k_{w_{z}}\frac{x}{H_0}\right)^{1/2}. \end{equation}

We now apply momentum balance to a control volume that embeds the operating turbine expanding some distance upstream and downstream to obtain the velocity deficit ($\Delta U = U_0-U_w$) for the top-hat model with an uneven expansion over the horizontal and vertical directions

(7)\begin{equation} \frac{\Delta{U}}{U_{0}} = \frac{1}{2}\left(1-\sqrt{1-\frac{2 C_{T}}{ \left(\beta + k_{w_{y}}\dfrac{x}{D_0}\right)^{1/2} \left(\beta + k_{w_{z}}\dfrac{x}{H_0}\right)^{1/2} }}\right). \end{equation}

Values of $k_{w_y}$ and $k_{w_z}$ are the wake expansion rates in the $y$- and $z$-directions respectively, which can be calculated as 2.0$I_u$ with $I_u= (u'u')^{0.5}/U_0$ being the streamwise turbulence intensity (Reference Frandsen, Barthelmie, Pryor, Rathmann, Larsen, Højstrup and ThøgersenFrandsen et al., 2006). In our study, we derived this top-hat wake model to estimate the initial wake width in the super-Gaussian and Gaussian models.

2.3. Super-Gaussian and Gaussian Wake Models

In the derivation of a super-Gaussian model, the wake is assumed to feature a velocity deficit ($\Delta U$) distribution that evolves in the streamwise direction according to the local scale of velocity $C(x)$, determined as the maximum normalised velocity deficit ($\Delta U_{max}/U_0$); and spatial length scales $f_y(y)$ and $f_z(z)$ that define an three-dimensional super-Gaussian distribution according to exponents $n_y$ and $n_z$, respectively. In the case of HATs (Reference Blondel and CathelainBlondel & Cathelain, 2020) or VATs, the wake starts with a rectangular (or nearly top-hat) shape immediately and evolves with an elliptical shape until attaining a Gaussian shape in the far wake ($n_y = n_z = 2$) assuming it is self-similar (Reference Bastankhah and Porté-AgelBastankhah & Porté-Agel, 2014). Thus, $n_y$ and $n_z$ are deemed to vary in the streamwise direction following an exponential decay as

(8)\begin{equation} n_y = a_y \exp \left( - b_y \frac{x}{D_0} \right) + c_y, \quad n_z = a_z \exp \left( - b_z \frac{x}{H_0} \right) + c_z . \end{equation}

Here, we denote the normalised local coordinates $\tilde {y} = (y - y_c)/D_0$ and $\tilde {z} = (z-z_c)/H_0$, with $y_c=0$ and $z_c=z_h$ being the centre of the turbine's rotor (Figure 1). Our super-Gaussian model is derived considering an uneven evolution of the wake shape over the $y$- and $z$-directions, i.e. $a_y \neq a_z$ and $b_y \neq b_z$ as shown later in § 4. Note that, in (8), we propose that the evolution of the shape exponent over the $y$-direction scales with $D_0$ whilst in the vertical direction this is with $H_0$. The evolution of $\Delta U$ according to the superposition of super-Gaussian shapes $f_y$ and $f_z$ can be estimated as

(9)\begin{equation} \frac{\Delta U}{U_0} = \frac{U_0 - U_w}{U_{0}} = C(x) \exp \left( -\frac{\tilde{y}^{n_y}}{2\tilde{\sigma_y}^2} \right) \exp \left( -\frac{\tilde{z}^{n_z}}{2\tilde{\sigma_z}^2} \right). \end{equation}

The characteristic wake widths $\sigma _y$ and $\sigma _z$ again scale depending on $D_0$ and $H_0$, respectively, i.e. $\tilde {\sigma _y} = \sigma _y/D_0$ and $\tilde {\sigma _z} = \sigma _z/H_0$. These wakes grow linearly in the downstream direction according to the wake expansion rates $k_y^*$ and $k_z^*$ respectively, which read

(10)\begin{equation} \tilde{\sigma_y} = k_y^{ * } \frac{x}{D_0} + \varepsilon_{y} , \quad \tilde{\sigma_z} = k_z^{ * } \frac{x}{H_0} + \varepsilon_{z}, \end{equation}

with $\varepsilon _y$ and $\varepsilon _z$ representing the initial wake expansion at $x_a$. In our model, the wake expansion rates are determined as a function of the streamwise turbulence intensity as $k_y^*=k_z^*=0.45I_u$, which are slightly greater than the values normally adopted for HAT. The wake velocity deficit (9) can be re-written as

(11)\begin{equation} U_{w} = U_{0}\left[ 1-C(x) \exp {\left(-\frac{\tilde{y}^{n_y}}{2 \tilde{\sigma_y}^2} \right)} \exp {\left(-\frac{\tilde{z}^{n_z}}{2 \tilde{\sigma_z}^2} \right)} \right] .\end{equation}

We need to determine the velocity scale $C(x)$ as the unknown variable from this algebraic equation. Considering the wake area to be $A_w=D_w H_w$ ($\textrm {d}A=\textrm {d}y\,\textrm {d}z$) depicted in Figure 1, we equate the momentum integral (3) to the turbine thrust force (4) to obtain the definition of the wake velocity (11) as

(12)\begin{equation} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} C(x) \exp {\left(-\frac{\tilde{y}^{n_y}}{2 \tilde{\sigma_y}^2}-\frac{\tilde{z}^{n_z}}{2 \tilde{\sigma_z}^2}\right)}\left[1-C(x) \exp {\left(-\frac{\tilde{y}^{n_y}}{2 \tilde{\sigma_y}^2}-\frac{\tilde{z}^{n_z}}{2 \tilde{\sigma_z}^2}\right)}\right] \mathrm{d} y\,\mathrm{d} z = \frac{1}{2}C_{T} A_{0}. \end{equation}

In the integration of (12) we consider that $\int _{-\infty }^{\infty } \exp (-\tilde {y}^{n_y}/(2 \tilde {\sigma _y}^2) \,\textrm {d}y$ = $2/n_yD_0 \varGamma (1/n_y)(2\tilde {\sigma _y}^2)^{1/n_y}$, $\int _{-\infty }^{\infty } \exp (-\tilde {y}^{n_y}/\tilde {\sigma _y}^2) \,\textrm {d}y = 2/n_yD_0 \varGamma (1/n_y) \tilde {\sigma _y}^{2/n_y}$, and analogously for the integrals in $z$-direction, with $\varGamma (1/2) = \sqrt {{\rm \pi} }$ and $\varGamma (1) = 1$. For convenience, from hereafter we denote $\eta$ = 1/$n_y$+1/$n_z$. Equation (12) can be integrated to determine the normalised maximum velocity deficit, $C(x)$, as

(13)\begin{equation} C(x)^2 - C(x) 2^\eta + \frac{C_T n_y n_z}{8 \tilde{\sigma_y}^{2/n_y}\tilde{\sigma_z}^{2/n_z} \varGamma(1/n_y) \varGamma(1/n_z)} = 0. \end{equation}

From the two possible roots of this quadratic equation, the solution that provides a physically acceptable value for the characteristic velocity scale is

(14)\begin{equation} C(x) = 2^{\eta-1} - \sqrt{2^{2\eta-2} - \frac{C_T n_y n_z}{8 \tilde{\sigma_y}^{2/n_y}\tilde{\sigma_z}^{2/n_z} \varGamma(1/n_y) \varGamma(1/n_z)} }. \end{equation}

Hence, our super-Gaussian model for VAT wakes is determined from (11) and (14) as

(15)\begin{equation} \frac{\Delta{U}}{U_{0}} = \left( 2^{\eta-1} - \sqrt{2^{2\eta-2} - \frac{C_T n_y n_z}{8 \tilde{\sigma_y}^{2/n_y}\tilde{\sigma_z}^{2/n_z} \varGamma(1/n_y) \varGamma(1/n_z)}} \right) \left[ \exp {\left(-\frac{\tilde{y}^{n_y}}{2 \tilde{\sigma_y}^2} \right)} \exp {\left(-\frac{\tilde{z}^{n_z}}{2 \tilde{\sigma_z}^2} \right)} \right]. \end{equation}

Note that if a Gaussian velocity profile is considered for both $f_y$ and $f_z$, i.e. $n_y=n_z=2$, then $\tilde {y}^2/ \tilde {\sigma _y}^2 = y^2/\sigma _y^2$ and $\tilde {z}^2/ \tilde {\sigma _z}^2 = z^2/\sigma _z^2$, yielding a velocity deficit expression

(16)\begin{equation} \frac{\Delta{U}}{U_{0}} = \left( 1- \sqrt{1-\frac{C_{T}A_{0}}{2{\rm \pi}\sigma_{y}\sigma_{z}}} \right) \left[ \exp {\left(-\frac{{y}^2}{2 {\sigma_y}^2} \right)} \exp {\left(-\frac{z^2}{2 {\sigma_z}^2} \right)} \right]. \end{equation}

2.3.1. Initial Wake Width

We need to determine the initial wake width to estimate the values of $\varepsilon _y$ and $\varepsilon _z$ adopted in the calculation of the wake widths $\tilde {\sigma }_y$ and $\tilde {\sigma }_z$. These are determined from the mass flow deficit rate ($m = \rho \iint \Delta U \,\textrm {d}A$, with $\rho$ being the density of the fluid) immediately behind the turbine's rotor $x=0$, equating the prediction from the momentum-conserving top-hat model (7) to the value from the super-Gaussian model (15), which ensures mass and momentum are conserved over the wake width. The expression of mass flow deficit rate for the super-Gaussian wake model reads

(17)\begin{equation} m_{{SG}} = \left( 2^{\eta-1} - \sqrt{2^{2\eta-2} - \frac{C_T n_y n_z}{8 \varepsilon_y^{{2}/{n_y}} \varepsilon_z^{{2}/{n_z}} \varGamma({1}/{n_y}) \varGamma({1}/{n_z}) }} \right) \frac{D_0H_0 2^{\eta + 2} \varepsilon_y^{{2}/{n_y}} \varepsilon_z^{{2}/{n_z}} \varGamma({1}/{n_y}) \varGamma({1}/{n_z}) }{n_yn_z}. \end{equation}

Alternatively, the mass flow deficit rate from the top-hat model (7) is

(18)\begin{equation} m_{{TH}_{x = 0}} = \left(1-\sqrt{1-\frac{2C_{T}}\beta} \right) \frac{D_a H_a}{2}. \end{equation}

Considering the wake onset widths $\varepsilon _y$ and $\varepsilon _z$ are equal to $\varepsilon$, we obtain

(19)\begin{equation} \varepsilon (n_{y,z}\geq 2) = \left( \frac{\beta n_yn_z}{2^{2\eta + 2} \varGamma\left({1}/{n_y}\right) \varGamma\left({1}/{n_z}\right)} \right)^{1/(2\eta)}. \end{equation}

It should be noted that these values of $n_y$ and $n_z$ are those at $x=0$, as $\varepsilon _y$ and $\varepsilon _z$ are solely evaluated as the initial wake width. The initial wake width adopting a Gaussian distribution is

(20)\begin{equation} \varepsilon (n_{y,z} = 2) = \sqrt{\frac{\beta}{4{\rm \pi}}}. \end{equation}

The initial wake expansion rates are just a function of $\beta$, i.e. only depend on the thrust coefficient $C_T$. It is noteworthy that our estimation of the initial wake width using a super-Gaussian model (19) differs from the definition $\varepsilon = \sqrt {\beta }/4$ proposed by Reference Abkar and DabiriAbkar & Dabiri (Reference Abkar and Dabiri2017) which is approximately 12% smaller and corresponds to the value used in HAT wake models. Our formulation overcomes limitations derived from an inaccurate definition of $\varepsilon$ which can yield non-physical estimations of the velocity deficit at short distances behind the turbines. Furthermore, the velocity scale obtained in our Gaussian model (16) differs from that proposed in Reference Abkar and DabiriAbkar & Dabiri (Reference Abkar and Dabiri2017), as the latter assumed the velocity deficit at the wake onset to be that of HATs (Reference Bastankhah and Porté-AgelBastankhah & Porté-Agel, 2014), while our theoretical framework is derived considering the rectangular section of VAT rotors. This consideration, together with the initial wake width (20), allows us to compute the velocity deficit field at any downstream location for any $C_T$ and rotor aspect ratio. In Table 1 we present the values of the super-Gaussian wake model (19) parameters, which provided the lowest error in the velocity predictions, and wake recovery rates used in the Gaussian model. The computed root-mean-square error in the velocity predictions shown later in Section 4 varied less than 2% when decreasing or increasing the super-Gaussian model input parameters $a_y$, $a_z$, $b_y$ and $b_z$ by $\pm$20%, similarly to the sensitivity analysis presented in Reference AbkarAbkar (Reference Abkar2019) for a Gaussian model.

Table 1. Details of the parameters adopted in the super-Gaussian and Gaussian wake models.

4.1. Case 1: VAT of Varying Aspect Ratio from Shamsoddin & Porté-Agel (Reference Shamsoddin and Porté-Agel2020)

Our super-Gaussian and Gaussian theoretical wake models are firstly validated for the three full-scale VATs of different aspect ratio originally presented in Reference Shamsoddin and Porté-AgelShamsoddin & Porté-Agel (Reference Shamsoddin and Porté-Agel2020) which used LES–ALM. The approach flow follows a logarithmic distribution with a roughness height of 0.1 m and friction velocity equal to 0.52 m s$^{-1}$ (Reference Shamsoddin and Porté-AgelShamsoddin & Porté-Agel, 2016). In Figure 3 we present the horizontal ($y$) and vertical ($z$) profiles of the normalised velocity deficit, $\Delta U$, at streamwise locations $x/D = 1$, 3, 5, 7 and 9 for case 1a ($\xi =1.0$). There is a notable three-dimensionality in the near wake at $x/D = 1$, with the vertical profile featuring a top-hat-like shape whilst the wake shape in the $y$-direction is closer to a Gaussian distribution. Here, the super-Gaussian wake model captures well both the wake shape and maximum velocity deficit. In contrast, the Gaussian model estimates well the velocity deficit in the centre of the wake but is unable to reproduce the wake shape. The profiles at $x/D=3$ show that both models underestimate the velocity deficit at this location. At $x/D \geq 5$, the super-Gaussian model agrees well with the LES results, while the Gaussian model underestimates the velocity deficit magnitude, except for $x/D=9$ in which there is a good agreement.

Figure 3. Normalised velocity deficit profiles for case 1a with $C_T = \textit{0.80}$ and $\xi = \textit{1.0}$. Comparison of our proposed super-Gaussian (red line) and Gaussian (black line) analytical wake models, with LES results from Reference Shamsoddin and Porté-AgelShamsoddin & Porté-Agel (Reference Shamsoddin and Porté-Agel2020) (circles).

The results for case 1b with $\xi = 2.0$ are presented in Figure 4 at the same downstream sections as in Figure 3, with the profiles nearer to the VAT exhibiting a more pronounced top-hat distribution. Compared to LES, a larger wake width is predicted by the super-Gaussian model at $x/D$ = 1, likely due to an over-estimation of the initial wake width value $\varepsilon$ (similarly to the results obtained in the Gaussian model derivation for HATs by Reference Bastankhah and Porté-AgelBastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014) in which a one-fifth reduction of $\varepsilon$ yielded better results), or from the influence of the proximity of the turbine bottom tip to the ground surface that generates flow acceleration ($\Delta U \leq 0$) which is not captured in the wake models. Nevertheless, the super-Gaussian model yields a good prediction of the wake width in the vertical profiles further downstream as well as in the rest of the cases analysed, i.e. captures well the wake evolution over the horizontal and vertical directions. In cases 1a and 1b, an uneven wake shape is observed in the far wake at $x/D$ = 9, agreeing with the evolution of the ratio $n_z/n_y$ presented in Figure 2, with a symmetric (or circular) wake shape attained at $x/D \approx 16$ and 32 for $\xi = 1.0$ and 2.0, respectively.

Figure 4. Normalised velocity deficit profiles for case 1b with $C_T = \textit{0.80}$ and $\xi = \textit{2.0}$. Same legend as Figure 3.

Case 1c considers a turbine with a small aspect ratio of 0.25, i.e. the VAT rotor's height is four times smaller than its diameter. Wake predictions from our theoretical models presented in Figure 5 evidence that the super-Gaussian model is able to accurately predict the near and far wakes. However, the Gaussian model underestimates the maximum velocity deficit in the near wake for $x/D \leq 3$ but attains a good accuracy further downstream as the wake becomes nearly symmetric after $x/D = 7$, in agreement with Figure 2. The presented results show that VAT wakes evolve from an elliptical distribution to a circular one. This is observed in Figure 6, with contours of velocity deficit over $yz$-planes at several streamwise locations for cases 1a ($\xi =1.0$) and 1b ($\xi =2.0$), only showing values of $\Delta U/U_0 \geq 0.05$. With $\xi = 1.0$, the wake immediately downstream of the turbine remains wider in the vertical direction. At $x/D = 2$, its shape is almost symmetric whilst at $x/D = 5$ the wake has a larger extent in the transverse direction, reaching an almost circular shape at 10$D$. As per the case with $\xi = 2.0$, the wake is highly three-dimensional at all locations, progressively evolving towards a circular shape.

Figure 5. Normalised velocity deficit profiles for case 1c with $C_T = \textit{0.80}$ and $\xi = \textit{0.25}$. Same legend as Figure 3.

Figure 6. Velocity deficit contours over $yz$-planes at various streamwise locations for cases 1a (top) and 1b (bottom) which depict the evolution of the wake from an elliptical to a circular shape, as indicated in Figure 2. Shaded area represents the VAT rotor's cross-section.

4.2. Case 2: VATs with Varying $C_T$ from Abkar & Dabiri (Reference Abkar and Dabiri2017)

We now consider another LES set-up comprising a single VAT with $D = 26$ m and $H = 24$ m, i.e. $\xi = 0.92$, operating at TSRs, $\lambda$, of 3.8 (case 2a) and 2.5 (case 2b). In these operating conditions, the VAT yields $C_T$ equal to 0.64 and 0.34, and we are interested in comparing these as VAT wakes develop a different dynamics depending on $\lambda$. Furthermore, in case 2c the turbine has a $C_T$ equal to 0.64 and operates at $\lambda = 3.8$, as in case 1a, but with $\xi = 1.85$. Turbines are equipped with three NACA 0018 blades with a chord length $c = 0.75$ m.

The validation of normalised velocity deficit predictions obtained with our analytical wake models compared with LES results from Reference Abkar and DabiriAbkar & Dabiri (Reference Abkar and Dabiri2017) are presented in Figure 7 with horizontal and vertical profiles across the turbine wake centre at $x/D = 1$, 3, 6, 9 and 12. Immediately downstream the VAT, at $x/D =1.0$, the super-Gaussian model is able to capture the three-dimensional wake with the Gaussian model predicting a narrower wake core. At $x/D = 3$, the super-Gaussian model achieves excellent agreement with the LES data whilst the Gaussian model slightly under-predicts $\Delta U/U_0$ in both vertical and horizontal profiles. At locations further downstream, both models predict well the velocity deficit.

Figure 7. Normalised velocity deficit profiles for case 2a with $C_T = \textit{0.64}$, $\lambda = \textit{3.8}$ and $\xi = \textit{0.92}$. Comparison of our proposed super-Gaussian (red line) and Gaussian (black line) analytical wake models, with the LES results from Reference Abkar and DabiriAbkar & Dabiri (Reference Abkar and Dabiri2017) (circles).

Figure 8 presents the results obtained for case 2b in which the VAT has a lower $C_T$ than cases 2a and 2c. The super-Gaussian model again attains a good velocity field prediction at all locations, notably improving the velocity estimates from the Gaussian model, especially at $x/D = 3$ and 6, but both provide a reasonable agreement of the wake width. Due to the low $C_T$ in case 2b, the wake shape features an almost Gaussian distribution over the horizontal and vertical directions already at $x/D \geq 6$, which supports the self-similarity assumption in the Gaussian shape distribution and thus both theoretical models achieve a great accuracy.

Figure 8. Normalised velocity deficit profiles for case 2b with $C_T = \textit{0.34}$, $\lambda = \textit{2.5}$ and $\xi = \textit{0.92}$. Same legend as Figure 7.

The turbine in case 2c has almost the same aspect ratio as in case 1b but with a lower thrust coefficient, which would lead to a faster wake recovery. Profiles of velocity deficit are shown in Figure 9 evidencing the accuracy of the super-Gaussian model in capturing the velocity deficit magnitude and its shape throughout the wake, notably improving the Gaussian model's prediction in the near wake at $x/D = 3$. The wake is observed to expand unevenly over the vertical and horizontal directions as in Figure 4, but the wake recovery is faster for $C_T = 0.64$ than with $C_T = 0.80$, as would be expected.

Figure 9. Normalised velocity deficit profiles for case 2c with $C_T = \textit{0.64}$, $\lambda = \textit{3.8}$ and $\xi = \textit{1.85}$. Same legend as in Figure 7.

4.3. Prediction of the Velocity Scale

To summarise the accuracy of our theoretical models in terms of the predictions of VAT wakes for the six cases analysed in comparison to LES results, the values of maximum velocity deficits in the streamwise direction are presented in Figure 10, which represents the velocity scale in the theoretical models (15). Overall, the super-Gaussian model attains an excellent agreement with LES in cases 1c, 2a, 2b and 2c over the whole wake length, whilst for cases 1a and 1b there is an underestimation of the velocity deficit until $x/D \approx 5$, as a consequence of the models over-predicting the wake width (see Figure 7). The Gaussian model consistently provides good estimates of $\Delta U$ in the far wake but fails to achieve a good agreement in the near wake. Whilst further improvement in cases 1a and 1b would be obtained tuning the super-Gaussian parameters, we aimed at building a consistent model with a given set of parameters that yields the best predictions for the six benchmarks together. A further advantage of our Gaussian model is that the proposed initial wake width (20) enables the calculation of the velocity field in all the wake region, overcoming the limitations from previous models (Reference AbkarAbkar, 2019) that could not provide physical estimates in the near wake for ranges of $C_T$ values.

Figure 10. Evolution of the maximum normalised maximum velocity deficit in the streamwise direction for all cases analysed. Comparison of our proposed super-Gaussian (red line) and Gaussian (black line) wake models, with the LES results (circles) from Reference Shamsoddin and Porté-AgelShamsoddin & Porté-Agel (Reference Shamsoddin and Porté-Agel2020) in cases 1a to 1c and Reference Abkar and DabiriAbkar & Dabiri (Reference Abkar and Dabiri2017) in cases 2a to 2c.

Figure 11. Evolution of the super-Gaussian coefficients $n_z$ and $n_y$ in the downstream direction for various aspect ratios $\xi = H/D$.