Potential of Wake Scaling Techniques for Vertical-Axis Wind Turbine Wake Prediction

Abstract

Analytical wake models are widely used to predict wind turbine wakes. While these models are well-established for horizontal-axis wind turbines (HAWTs), the analytical wake models for vertical-axis wind turbines (VAWTs) remain under-explored in the wind energy community. In this study, the accuracy of two wake scaling techniques is evaluated to predict the change in the normalized maximum wake velocity deficit behind VAWTs by re-scaling the maximum wake velocity deficit behind an actuator disk with the same thrust coefficient. The wake scaling is defined in terms of equivalent diameter, considering the geometrical properties of the wake-generating object. Two different equivalent diameters are compared, namely the momentum diameter and hydraulic diameter. Different approaches are used to calculate the change in the normalized wake velocity deficit behind a disk with the same thrust coefficient as the VAWT. The streamwise distance is scaled with the equivalent diameter to predict the normalized maximum wake velocity deficit behind the desired VAWT. The performance of the proposed framework is assessed using large-eddy simulation data of VAWTs operating in a turbulent boundary layer with varying operating conditions and aspect ratios. For all of the cases, the momentum diameter scaling provides reasonable predictions of the VAWT normalized maximum wake velocity deficit.

1. Introduction

The growing energy demand coupled with net-zero emission goals leads to an increase in the use of renewable energy sources worldwide. In this context, wind energy plays a crucial role in the production of sustainable electricity through horizontal-axis wind turbines (HAWTs) and vertical-axis wind turbines (VAWTs). Onshore and offshore wind farms mainly consist of HAWTs as a well-established and widely used concept. On the other hand, the deployment of VAWTs is progressing at a slower pace, mainly due to their financial viability in large-scale projects [1]. To address this, the research on VAWTs through experimental campaigns [2], numerical simulations [3], and field measurements [4] aims to provide a better understanding of VAWT wake flows. This body of research has demonstrated some promising features for VAWTs, such as the capability of increasing the energy density without changing their footprint [5], and enhancing the performance of current wind farms through co-locating VAWTs in their layout [6]. Moreover, eddy-resolving simulations coupled with a turbine model (actuator line or actuator swept-surface model) are proven to be effective in analyzing the interaction of the wake of a VAWT with the atmospheric boundary layer (ABL) [3,7]. Despite providing insight into the details of the wake flow, these simulation techniques are computationally expensive, making them impractical for applications like determining the optimal configuration of VAWTs in a wind farm.

In the wind energy community, analytical wake models are widely used in applications such as wind farm layout optimization and control strategies [8]. Despite being less accurate than turbulence-resolving numerical simulations, these models are derived from the equations governing the conservation of flow properties, which provide fundamental insight into wake flow physics. For the HAWT, several analytical wake models are proposed based on conservational laws (either mass or mass and momentum) and different shape functions (either top-hat or Gaussian) to predict the wake velocity deficit distribution [9]. Following the theoretical framework for the HAWT analytical wake models, there have been recent developments for the analytical wake models to predict the VAWT’s wake distribution. From these developments, one can mention the VAWT top-hat wake model [10], the VAWT Gaussian wake model for the self-similar region of the wake [7], and the VAWT super-Gaussian wake model that captures the wake development in both near and far wake [1]. To predict the wake development, all of the above-mentioned analytical wake models for VAWTs rely on an estimation of the wake expansion rate (under different operating conditions), which is currently estimated with empirical relationships [1].

Seeking universal and self-similar behavior is a core concept in the literature of classic turbulent flows (e.g., jets and wake) to describe the turbulent flow mean behavior by using the appropriate length scales and shape functions [11,12]. Among this vast body of literature, the focus of this article is on how the shape of wake-generating objects influences the wake flow behind them. Based on the experimental data of the wake behind a sphere and a porous disk (with the same drag) in unstratified flow, Bevilaqua and Lykoudis [13] observed that the wake flow structure depends on the shape of the wake-generating object. Later, Meunier and Spedding [14] performed a series of experiments for the wake behind different bluff bodies (different shapes and drag coefficients) in stratified flow. They showed that normalizing the lengths with an effective diameter (a function of the drag coefficient and the diameter of the bluff body) leads to a common behavior in the mean wake velocity field. This leads to an important conclusion that the wake of a bluff body only remembers the momentum flux imposed by the object on the flow and not the aspects of its geometry. Inspired by this key observation and in a study focused on the effect of aspect ratio on the VAWT wakes, Shamsoddin and Porté-Agel [15] showed that by normalizing both the streamwise and lateral lengths by the momentum diameter, a collapse of wake velocity deficit profiles of different aspect ratios is obtained. In the same context, Huang et al. [16] performed an experimental investigation on the streamwise momentum recovery for actuator surfaces of different shapes. They observed that by normalizing the streamwise distance with the hydraulic diameter, the normalized mean streamwise velocity behind three actuator surfaces (circle, square, and rectangle) shows a common behavior. It should be noted that the concept of momentum and hydraulic diameter will be discussed in more detail in Section 2.

This study aims to profit from the recent models for the wake expansion of HAWTs and the concept of wake scaling (through introducing an equivalent diameter) for actuator surfaces of arbitrary shapes to predict the maximum wake velocity deficit behind a VAWT. By proposing a new approach for estimating the wake expansion behind a VAWT, the results of this study have direct implications for analytical wake modeling of VAWTs and contribute to the wake modeling literature. Moreover, the outcomes of this study can be of interest in a broader range of fluid mechanics applications, such as the wake behind bluff bodies.

This paper is structured as follows. Section 2 provides a brief description of the relevant analytical wake models, followed by two different methods to estimate the equivalent diameter for wake scaling. In Section 3, the performance of the presented wake scaling techniques for predicting the wake maximum velocity deficit behind a VAWT is assessed using LES data. The concluding remarks are presented in Section 4. Finally, within Appendix A, a brief introduction of the LES framework used in this study to estimate the inputs of the wake expansion model is included.

2. Methodology

2.1. Analytical Wake Models

Analytical wake models, as simple and computationally efficient tools, are widely used to predict wind turbine wake flows under various operating conditions [9,17]. Below is a brief review of the key analytical wake models, for HAWT and VAWT, used in the current study.

2.1.1. HAWT

Several numerical and experimental observations showed that the wake velocity deficit behind a wind turbine follows a self-similar Gaussian distribution in the far wake [18,19]. Following this observation and based on mass and momentum conservation, Bastankhah and Porté-Agel [20] proposed the Gaussian wake model for the wake velocity deficit as follows:

$${\displaystyle \frac{\mathrm{\Delta }U\left(x\right)}{U_{\infty }}}=\left(1-\sqrt{1-{\displaystyle \frac{C_T}{8{(\sigma \left(x\right)/d)}^2}}}\right)\times \exp \left(-{\displaystyle \frac{r^2}{2\sigma {\left(x\right)}^2}}\right),$$

(1)

where $U_{\infty }$ is the free-stream velocity, r is the radial direction from the center of the wake, $C_T$ is the turbine thrust coefficient, d is the rotor diameter, $\sigma \left(x\right)$ is the total wake width, and $\mathrm{\Delta }U\left(x\right)=U_{\infty }-U_w\left(x\right)$, in which $U_w\left(x\right)$ is the wake velocity in the streamwise direction. A linear expansion for the wake width was proposed ($\sigma \left(x\right)=k^{*}(x-x_{NW})+d/\sqrt{8}$) with an expansion rate of $k^{*}$ and a wake width of $d/\sqrt{8}$ at the end of the near wake ($x_{NW}$). To estimate the wake expansion rate, empirical relations were proposed based on numerical [21] and experimental data [22] with limited range of applicability regarding the thrust coefficient and incoming turbulence intensity. Recently, Vahidi and Porté-Agel [23] proposed a physics-based model for the wake width growth downstream of a turbine, including both the effects of incoming flow turbulence and turbine-induced turbulence. The physics-based model is derived based on the Taylor diffusion theory, Gaussian wake model, turbulent mixing layers, and the analogy between wind turbine wake and scalar diffusion from a disk source. For each downwind location, the model solves the spreading equation for the mixing layer lateral characteristic length (${\sigma }_{ey}$) from the end of the expansion region ($x_0$):

$${\sigma }_{ey}\left(x\right)=\underset{\mathrm{Ambient}\mathrm{flow}}{\underbrace{\sqrt{S{c}_t}{\sigma }_v{T}_{Lv}\sqrt{2\left({\displaystyle \frac{T}{T_{Lv}}}-(1-exp\left({\displaystyle \frac{-T}{T_{Lv}}}\right))\right)}}}+\underset{\mathrm{Turbine}-\mathrm{induced}}{\underbrace{2S^{\prime }(U_{\infty }T-(x-x_0))}},$$

(2)

where $S{c}_t$ is the turbulent Schmidt number for mixing layers, ${\sigma }_v$ is the standard deviation of the lateral velocity component, $T_{Lv}$ is the Lagrangian integral time scale for the lateral velocity time series, and $S^{\prime }$ is the mixing layer spreading rate. To derive Equation (2), an exponential form for the velocity auto-correlation function was assumed [24]. The travel time (T) is defined based on the wake advection velocity ($U_{adv}\left(x\right)=0.5(U_{center}\left(x\right)+U_{\infty })$ with $U_{center}\left(x\right)$ the wake centerline velocity) from the end of the expansion region ($T={\int }_{x_0}^x\left({\displaystyle \frac{dx}{U_{adv}\left(x\right)}}\right)$). The relation derived from the analogy between wake expansion and scalar diffusion from a disk source (${\sigma }_{wake}/{\sigma }_e=1.95exp(-6.19{\sigma }_e/d)+10.96exp(-20.05{\sigma }_e/d)+1.03$) is used to calculate the wake width at each downwind distance from the corresponding mixing layer characteristic length. For an axisymmetric wake, Equation (2) can be used for the vertical direction with the respective properties. The total wake width can be defined as the geometrical mean of wake widths in the spanwise and vertical directions (${\sigma }_{wake,tot}=\sqrt{{\sigma }_{wake,y}{\sigma }_{wake,z}}$). In the proposed model, the wake centerline velocity in the near wake is assumed to be constant and equal to the value derived from the one-dimensional momentum theory ($U_{\infty }\sqrt{1-C_T}$), and in the far wake, the wake centerline velocity is calculated using the Gaussian wake model. To calculate the end of the near wake, Vahidi and Porté-Agel [23] proposed the following relation for the near wake length under neutral atmospheric conditions:

$${\displaystyle \frac{x_{NW}}{d}}={\displaystyle \frac{{\sigma }_{e,NW}}{d}}{\displaystyle \frac{1+\sqrt{1-C_T}}{2\left(\sqrt{S{c}_t}\left(0.63I_u\right)+S^{\prime }(1-\sqrt{1-C_T})\right)}}+{\displaystyle \frac{x_0}{d}},$$

(3)

where ${\sigma }_{e,NW}/d$ defines the threshold for the onset of the wake self-similar region, and $I_u$ is the streamwise turbulence intensity. The physics-based model is used through this paper with parameters introduced in the original derivation without any parameter tuning ($S{c}_t=0.5$, $x_0/d=1$, ${\sigma }_{e,NW}/d=0.18$, $S^{\prime }=0.043$).

Recently, in a study focused on a new streamwise scaling for the wake of HAWT under different incoming turbulence intensities, Vahidi and Porté-Agel [25] proposed a relation for the collapsed data of the normalized maximum wake velocity deficit as a function of the streamwise distance normalized by the near wake length. This relation, hereafter referred to as the stand-alone model, only requires the near wake length as input to predict the maximum wake velocity deficit behind a turbine and is as follows:

$${\displaystyle \frac{\mathrm{\Delta }U\left(x^{\prime }\right)}{U_{\infty }(1-\sqrt{1-C_T})}}=1.75{(x^{\prime }+0.5)}^{-1.37},$$

(4)

with $x^{\prime }=x/x_{NW}$. In this study, Equation (3) is used to estimate the near wake length for the stand-alone model based on the thrust coefficient and the incoming turbulence level.

2.1.2. VAWT

Based on the self-similar behavior of the VAWT’s wake, operating in a turbulent boundary layer with different tip speed ratios, Abkar and Dabiri [7] proposed an analytical wake model for the wake velocity deficit of VAWTs based on the mass and momentum conservation (similar to the Gaussian wake model for HAWT). For a VAWT with a height of H and diameter of D (also referred to as width in the literature), the following equation shows the variation in the maximum wake velocity deficit ($\mathrm{\Delta }{U}_{max}\left(x\right)$) in the self-similar region:

$${\displaystyle \frac{\mathrm{\Delta }{U}_{max}\left(x\right)}{U_{\infty }}}=1-\sqrt{1-{\displaystyle \frac{C_T}{2\pi ({\sigma }_y{\sigma }_z/A_f)}}},$$

(5)

where $A_f=DH$ is the turbine frontal area, and ${\sigma }_y$ and ${\sigma }_z$ are the standard deviation of the wake velocity deficit profiles in the lateral and vertical directions, respectively. The wake widths in the lateral and vertical directions scale based on the D and H, respectively. In this model, these wake widths are assumed to grow in a quasi-linear manner:

$${\sigma }_y=k_y^{*}x+ϵD,{\sigma }_z=k_z^{*}x+ϵH,$$

(6)

where the parameter $ϵ$ defines the standard deviation of the wake at the rotor location, and $k_y^{*}$ and $k_z^{*}$ are the wake expansion rate in the lateral and vertical directions, respectively. In a recent study focused on developing a more comprehensive representation of the wake of VAWT with analytical models, Ouro and Lazennec [1] proposed using $ϵ=\sqrt{\beta /4\pi }$ as the initial wake width for a Gaussian distribution, where $\beta =0.5(1-\sqrt{1-C_T})/\sqrt{1-C_T}$, and an empirical linear relation for the wake expansion, applicable to the Gaussian model, as a function of streamwise turbulence intensity ($I_u$):

$$k_y^{*}=k_z^{*}=0.35I_u.$$

(7)

2.2. Equivalent Diameter for Wake Scaling

This section introduces the concept of equivalent diameter for actuator surfaces with arbitrary shapes. The key idea of the equivalent diameter is to introduce a relevant length scale for actuator surfaces with different geometrical features that lead to a universal behavior in the wake mean quantities. Below are the derivation and the underlying assumptions for the momentum and hydraulic diameter as two relevant definitions of equivalent diameter.

2.2.1. Momentum Diameter

The concept of momentum diameter for wake-generating objects was proposed originally by Meunier and Spedding [14] to observe a universal behavior in the wake of six different bluff bodies (with different drag coefficients). Later, this concept was used by Shamsoddin and Porté-Agel [15] in the context of VAWTs with varying aspect ratios. To derive the momentum diameter (similar to the above-mentioned studies), by neglecting the pressure and viscous terms in the momentum equation, one can write the momentum integral for a three-dimensional turbulent wake as follows [11]:

$$\rho {\int }_{yz}{U}_w(U-U_w)dA=T,$$

(8)

where the integral is computed on the plane perpendicular to the incoming flow ($yz$), $\rho$ is the air density, and T is the total thrust (drag) force exerted on the flow. Inspired by the concept of momentum thickness for planar wakes [11], one can write the total thrust for the wake-generating object as follows:

$$T=\rho U_{\infty }^2{\displaystyle \frac{\pi }{4}}{\delta }^2,$$

(9)

with $\delta$ as the momentum diameter. Considering the definition of the thrust coefficient, we can obtain the following relation:

$$T=\rho U_{\infty }^2{\displaystyle \frac{\pi }{4}}{\delta }^2={\displaystyle \frac{1}{2}}\rho A_f{U}_{\infty }^2{C}_T,$$

(10)

where $A_f$ is the frontal area of the wake-generating object. By rearranging Equation (10):

$$\delta =\sqrt{{\displaystyle \frac{1}{2}}{C}_T}\sqrt{{\displaystyle \frac{4}{\pi }}{A}_f}.$$

(11)

For wake-generating objects with the same thrust coefficient, one can define the equivalent momentum diameter as $D_{eq,Mom}=\sqrt{(4/\pi )A_f}$.

2.2.2. Hydraulic Diameter

The second equivalent diameter is introduced by Huang et al. [16] in a study focused on the streamwise momentum recovery behind actuator surfaces of arbitrary shapes. The derivation is based on assuming a control volume behind an actuator surface, sufficiently far from the surface to neglect the effect of pressure in the momentum equation ([Fig.3] in [16]). By assuming a top-hat velocity distribution along the cross-section and a uniform distribution for the shear stress along the perimeter of the cross-section, the mass and momentum conservation can be written as follows:

$$\left\{\begin{array}{c}\mathrm{M}\mathrm{a}\mathrm{s}\mathrm{s}:(U+dU)A_f-U{A}_f-VPdx=0\hfill \\ \\ \mathrm{M}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{u}\mathrm{m}:\rho {(U+dU)}^2{A}_f-\rho U^2{A}_f-\rho U_{p}VPdx=\tau Pdx,\hfill \end{array}\right.$$

(12)

where V is the cross-flow velocity, P is the actuator surface perimeter, $U_p$ is the streamwise velocity along the lateral interface of free stream and the wake ($U<U_p<U_{\infty }$), and $\tau$ is the shear stress. By deriving the expression for cross-flow velocity form the mass conservation and replacing it in the momentum balance equation and neglecting second-order terms, the simplified expression is derived as:

$$2\rho U{A}_{f}dU-\rho U_p{A}_{f}dU=\tau Pdx.$$

(13)

From Equation (13), one can derive the rate of momentum recovery along the streamwise direction:

$${\displaystyle \frac{dU}{dx}}={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{P}{A_f}}{\displaystyle \frac{\tau }{2U-U_p}}.$$

(14)

Equation (14) indicates that the streamwise momentum recovery depends upon the ratio of the cross-sectional area ($A_f$) and wake perimeter (P). In analogy with the concept of hydraulic diameter [26], the equivalent diameter can be defined as $D_{eq,Hyd}=4A_f/P$ for wake-generating objects.

For the sake of clarity, Figure 1 shows a schematic representation of the introduced approaches for calculating the equivalent diameter, namely the momentum and the hydraulic diameter.

3. Results

This section presents the predictions for the maximum wake velocity deficit behind a VAWT estimated by mapping the maximum wake velocity deficit behind an actuator disk of the same thrust coefficient. This mapping is carried out by re-scaling the streamwise distance with different equivalent diameters, namely the momentum and hydraulic equivalent diameter. The proposed framework is validated against high-fidelity simulations of VAWTs operating in a turbulent boundary layer. The first set is the LES from Shamsoddin and Porté-Agel [15], which consists of three cases with different aspect ratios ($\xi =H/D$) with the same inflow characteristics and thrust coefficient. In this set, the VAWT was modeled with the actuator line model, and the sub-grid scale turbulent fluxes were calculated using the Lagrangian scale-dependent dynamic model. The second set, simulated by Abkar and Dabiri [7], includes two cases with the same aspect ratio and different thrust coefficients. In their study, Abkar and Dabiri [7] conducted the simulations using the anisotropic minimum dissipation as the sub-grid scale model and the actuator line model to include the effect of the VAWT on the flow. The main characteristics of these cases are reported in

Table 1. Properties of LES cases used for validation. For each case, the aspect ratio ($\xi =H/D$), VAWT diameter (D), thrust coefficient ($C_T$), the incoming streamwise velocity ($U_{\infty }$), and the streamwise turbulence intensity ($I_u$) at the equator level are given. The last two columns present the scaling parameter given different definitions of equivalent diameter.

Case	Aspect Ratio ($\mathit{\xi }$)	D (m)	${\mathit{C}}_{\mathit{T}}$	${\mathit{U}}_{\mathit{\infty }}(\mathit{m}/\mathit{s})$	${\mathit{I}}_{\mathit{u}}$	${\mathit{D}}_{\mathit{eq},\mathit{Mom}}/\mathit{D}$	${\mathit{D}}_{\mathit{eq},\mathit{Hyd}.}/\mathit{D}$
Shamsoddin and Porté-Agel (2020) [15]	2	50	0.8	9.6	0.083	1.596	1.333
	1	50	0.8	9.6	0.083	1.128	1.0
	0.25	50	0.8	9.6	0.083	0.564	0.4
Abkar and Dabiri (2017) [7]	0.92	26	0.64	7.0	0.1	1.084	0.96
	0.92	26	0.34	7.0	0.1	1.084	0.96

. These two sets of simulations provide a wide range of features that influence the VAWT wake (formation and development with the streamwise distance) and have been used in other studies to assess the performance of analytical wake models designed for VAWTs [1].

In order to predict the change in the normalized maximum wake velocity deficit for a VAWT as a function of the streamwise distance, we follow the following procedure. First, the normalized maximum wake velocity deficit behind an actuator disk with the same thrust coefficient and the same diameter as the VAWT is calculated. For this purpose, two models are used in this study: the physics-based wake expansion model (Equation (2)) and the stand-alone model (Equation (4)). The physics-based wake expansion model is robust to variations in the incoming turbulence levels and turbine operating conditions. The inputs for this model (the unfiltered velocity standard deviation and Lagrangian integral time scale) are derived from a coarse simulation with the same domain size as the two reported cases, respecting the grid aspect ratio. Appendix A provides details about the LES framework and how to estimate the model inputs from the LES data. The stand-alone model only requires an estimation of the near wake length to predict the maximum wake velocity deficit, for which Equation (3) is used to calculate the near wake length. Second, the equivalent diameter, as the scaling parameter, is calculated based on the approaches introduced in Section 2.2. This parameter depends only on the shape of the VAWT and defines an equivalent disk for a given frontal shape (the equivalent diameter of a circle is equal to its actual diameter). Third, the streamwise distance of the actuator disk is scaled with the ratio of ($D_{eq}/D$) to map the result to the VAWT space. This step is based on the fact that the maximum wake velocity deficit of actuator surfaces with arbitrary shapes behaves similarly when the streamwise distance is scaled with the equivalent diameter. These steps can be summarized as follows:

• Step 1—Calculate the normalized maximum wake velocity deficit for an actuator disk with the same thrust coefficient and diameter as the VAWT.
• Step 2—Calculate the VAWT equivalent diameter based on its shape features.
• Step 3—Re-scale the downwind distance from Step 1 with the $D_{eq}/D$ to map the result from an actuator disk to the desired actuator surface domain.

Figure 2 and Figure 3 show the change in the maximum normalized wake velocity deficit, at the VAWT equator level, as a function of the downwind distance, computed from the physics-based wake expansion model and stand-alone model and re-scaled with the equivalent momentum and equivalent hydraulic diameter, the Gaussian wake model for VAWT with the empirical wake growth rate (Equation (7)), and that obtained from LES data. These figures provide the chance to assess the sensitivity of the proposed framework to variations in the aspect ratios and turbine thrust coefficients. While Figure 2 examines the performance of the wake scaling techniques for a constant thrust coefficient and three different aspect ratios, Figure 3 focuses on the cases with the same geometry and different thrust coefficient. As shown in these figures, the physics-based model coupled with the equivalent momentum diameter provides a reasonable prediction for the maximum wake velocity deficit for VAWTs with varying aspect ratios and different thrust coefficients. On the other hand, the hydraulic momentum diameter scaling underestimates the variability of the maximum wake velocity deficit compared to the LES data for all of the cases. The stand-alone model coupled with the equivalent momentum diameter shows reasonably accurate predictions for the maximum wake velocity deficit in comparison with LES data. It should be mentioned that for the case with $C_T=0.64$ in Figure 3, the stand-alone model coupled with the equivalent momentum diameter shows an overestimation of the wake velocity deficit in the far wake after downwind distance greater than 8D, which can be attributed to the sensitivity of the stand-alone model to the near wake length estimation [25]. The comparison between the empirical wake growth rate for VAWT (as introduced in Section 2.1.2) and the LES data shows that the prediction of this model is sensitive to the wake expansion rate definition. While the empirical relation provides reasonable predictions in the far wake of downwind distances greater than 10D, it underestimates the wake velocity deficit in the distances closer to the turbine. Based on the presented results, the physics-based wake expansion model and the stand-alone model coupled with the equivalent momentum diameter provide reasonably accurate predictions of the VAWT maximum wake velocity deficit under different operating conditions and aspect ratios from the downwind distances of about 5D.

After evaluating the performance of different wake scaling techniques, it is important to discuss the reasons for the differences in the predictions when the streamwise distance is scaled with the introduced equivalent diameters. The equivalent momentum diameter is derived based on the momentum integral, without any specific assumption for the wake velocity and shear stress distribution behind the actuator surface, and includes the exerted thrust force by the wake-generating object on the flow. On the other hand, the equivalent hydraulic diameter is derived based on the assumption of top-hat velocity distribution behind the actuator surface and a uniform shear stress distribution on the wake edges. While these assumptions can be valid in low turbulence levels and relatively close to the actuator surface (the same as the experiments reported by Huang et al. [16], turbulence level around 1%), they break down for an actuator surface located in a turbulent boundary layer. Moreover, the assumption of the top-hat velocity profile behind an actuator surface breaks down as the turbulence level increases and the wake moves downstream. Under these conditions, the wake starts to have a Gaussian self-similar behavior and transforms from an elliptical shape to a circular one [1,7].

The outcome of this study is important since it can be used to estimate the wake expansion behind VAWTs in a turbulent boundary layer, which is useful for applications such as VAWT layout optimization and co-locating the HAWT and VAWT in wind farms. The proposed framework predicts the maximum wake velocity deficit of a VAWT from an actuator disk maximum wake velocity deficit with the same thrust coefficient and inflow characteristics by re-scaling the streamwise distance. Therefore, it can replace the empirical relations for predicting the wake expansion behind VAWTs and be used as an input for more sophisticated analytical wake models for VAWT to provide the three-dimensional representation of the wake distribution. From a broader perspective, the presented analysis and results may also have implications for the fluid mechanics aspect of bluff body wakes.

4. Summary and Concluding Remarks

In this study, a new framework for estimating the change in maximum wake velocity deficit behind a VAWT has been proposed. This framework is built on two main elements: (a) a model to estimate the wake velocity deficit behind an actuator disk and (b) the concept of equivalent diameter for streamwise scaling of the wake behind actuator surfaces of arbitrary shapes. In essence, the framework estimates the wake behind an actuator disk with the same inflow characteristics and thrust coefficient as the VAWT and then re-scales the streamwise distance with the equivalent diameter as the relevant length scale that includes the VAWT geometrical features. Two equivalent diameters have been tested, namely the equivalent momentum and hydraulic diameters.

The predictions of the proposed framework have been validated against five LES cases of a VAWT operating in a turbulent boundary layer, covering a range of aspect ratios and turbine thrust coefficients. According to the results, the wake expansion model coupled with the equivalent momentum diameter yields reasonable predictions of the maximum wake velocity deficit for all of the LES cases and outperforms the wake expansion model re-scaled with the equivalent hydraulic diameter and the VAWT Gaussian wake model with an empirical wake growth rate relation. It should be mentioned that, with knowledge of the inflow characteristics, the proposed framework expands the applicability of the wake expansion models, for the wake velocity deficit estimation, from actuator disks to actuator surfaces of arbitrary shapes. The proposed approach is fast and reasonably accurate for estimating the maximum wake velocity deficit for engineering applications, and it is not meant to replace more sophisticated analytical wake models.