Wind turbine wakes can cause energy losses in wind farms and increase blade fatigue loads. Hence, understanding wind turbine wakes is important for designing energy-efficient wind farm layouts. Wake recovery is the process describing the flow's return to an undisturbed state via turbulent mixing. The wind energy science community (including the main author of the present work) often refers to the shear at the wake edges as the main driver behind the wake recovery, as the production of the turbulent kinetic energy depends on the square of the mean shear . Other authors have analyzed the mean kinetic energy budget of a wind farm using wind tunnel measurements and large-eddy simulations LES, e.g.,; they concluded that the vertical shear stress component of the Reynolds stress is the main driver behind energy transport of the freestream into the wake. computed transport tubes of the streamwise momentum and energy in wind farms using LES and showed that the energy is transported sideways and top-down, where the dominant direction depends on the turbine lateral spacing. While the shear and the vertical Reynolds stresses are indeed important, they are not the precise reason why wake recovery occurs, since the Reynolds-averaged Navier–Stokes (RANS) equation for streamwise momentum includes gradients of Reynolds stresses (stress divergence) that cause turbulent mixing. If a Reynolds stress is represented by a velocity gradient following the hypothesis of , then it becomes clear that the gradient of the shear is responsible for wake recovery and not the shear or Reynolds stresses themselves. This brief communication is meant to clarify the main mechanisms behind wake recovery, through use of a simple illustrative model of the far wake (Sect. ) and by analyzing LES results (Sect. ) to confirm the trends of the simple model.

The wind turbine wake can be split into near- and far-wake regions . The near wake is a result of the wind turbine blade forces, and it is characterized by complex vortex structures that break down into smaller turbulent eddies further downstream. The near-wake velocity deficit is mainly a footprint of the wind turbine thrust force distribution, and it diffuses downstream in a smoother velocity deficit profile. We define the far wake as the region where the mean velocity deficit has become self-similar. In other words, the far wake has forgotten how it was generated, and only information of the total wind turbine extracted momentum is known. We derive a simple model for the far wake with the aim of creating an illustrative example of the main wake recovery mechanism. The model is not meant to be used for the prediction of a wind turbine wake flow. We start with the Reynolds-averaged Navier–Stokes (RANS) momentum equation for incompressible and high-Reynolds-number flow, for the streamwise direction: 1DUDt=-1ρ∂P∂x-∂u′u′‾∂x-∂u′v′‾∂y-∂u′w′‾∂z+f, where U is the mean streamwise velocity, ρ is the air density, P is the mean pressure, f is the wind turbine thrust force that we choose to represent by an actuator disk (AD) model , t is the time, and xj=(x,y,z) are the streamwise, lateral, and vertical coordinates. The normal Reynolds stress u′u′‾ and the shear Reynolds stresses u′v′‾ and u′w′‾ need to be modeled; we apply the well-known hypothesis of : 2u′u′‾=23k-2νT∂U∂x,u′v′‾=-νT∂U∂y+∂V∂x,u′w′‾=-νT∂U∂z+∂W∂x, with k as the turbulent kinetic energy that can be absorbed in the pressure and νT as the eddy viscosity. In the latter two expressions, we will neglect the ∂/∂x contributions to simplify the illustrative model.

Around the AD, a strong adverse pressure gradient is present that reduces the streamwise velocity upstream and downstream of the rotor. In the absence of the Reynolds stresses, one can derive the well-known 1D (axial) momentum solution for the streamwise velocity at the AD and at the far wake . The latter can be written as a velocity deficit, ΔU, and can be related to the thrust coefficient, CT: ΔU/UH=1-1-CT, with UH as the freestream velocity. In a turbulent flow, the divergence of the Reynolds stress tensor recovers the streamwise velocity back to the freestream velocity. The 1D momentum solution for the velocity deficit can be seen as the maximum deficit that one could obtain in turbulent flow of an AD.

It can be shown that for zero pressure gradient and a constant eddy viscosity, the far-wake velocity deficit is self-similar and can be modeled by a Gaussian function, as shown by . and used wind tunnel measurements and large-eddy simulations of a wind turbine wake to show that the far-wake velocity deficit can indeed be approximated by a Gaussian function: 3ΔUwake(y,z)=ΔUmaxexp⁡-y2+z-zH22σ2, where ΔUmax is the maximum deficit that is normally a function of the downstream distance but can be considered a constant for fixed downstream position x, zH is the wind turbine hub height, and σ is the spatial scale (standard deviation) of the Gaussian wake profile. We model the far-wake velocity as a combination of a Gaussian velocity deficit and a logarithmic inflow similar to : 4U(y,z)=Uin(z)-ΔUwake(y,z)=u*κln⁡zz0-ΔUwake(y,z), where Uin(z) represents a neutral atmospheric surface layer with u* as the friction velocity and z0 as the roughness length. The shear stresses and their contribution to the momentum equation (known as stress divergence) become 5u′v′‾=-νT∂U∂y=-νTσ2yΔUwake(y,z),6u′w′‾=-νT∂U∂z=-u*2-νTσ2z-zHΔUwake(y,z),7-∂u′v′‾∂y=νT∂2U∂y2=νTσ21-y2σ2ΔUwake(y,z),8-∂u′w′‾∂z=νT∂2U∂z2+∂U∂z∂νT∂z=νTσ22-zHz-z-zH2σ2ΔUwake(y,z). Here, we have assumed that the eddy viscosity is unaffected by the wake and equal to the logarithmic inflow: νT=u*κz by assuming a neutral atmospheric surface layer to be valid. This is a strong assumption and does not hold for non-neutral atmospheric conditions and for tall wind turbines that may operate beyond the surface layer. The assumption of a linear inflow eddy viscosity is the same as assuming a constant eddy viscosity in the far wake in order to derive a 1D Gaussian profile as a function of y, for each height z. The eddy viscosity of a real wind turbine far wake is expected to be non-uniform; RANS simulations of a single wind turbine wake typically show a Gaussian-like lateral eddy viscosity profile with its maximum in the wake center. , , and have proposed modified Gaussian velocity deficit profiles to better match measured far-wake results of a circular cylinder, an axisymmetric wake of a disk, and an axisymmetric wake of a plate, respectively, in order to account for the effect of non-uniform eddy viscosity on the self-similar velocity deficit. It should be noted that these measurements were performed for low-Reynolds numbers that are 3 orders of magnitude lower than that of utility-scale wind turbines (using D=100 m), which makes their conclusions not directly applicable to our flow of interest. Therefore, RANS simulations of a single utility-scale wind turbine are performed in Appendix . The RANS simulations indicate that the wake-generated eddy viscosity has a minor impact on the far-wake velocity deficit shape. The assumed Gaussian velocity profile of the simple model also results in self-similar shear stresses and stress divergence terms. argued that higher-order velocity moments of an axisymmetric wake can be shown to develop downstream over large distances and continue to contain information of the near wake. It remains unclear if this conclusion can be applied to a utility-scale wind turbine wake due to the mismatch in Reynolds number.

Using the normalized coordinates ỹ=y/D and z̃=(z-zH)/D, along with the four normalized parameters ΔŨmax=ΔUmax/UH, σ̃=σ/D, z̃H=zH/D, and z̃0=zH/z0, the streamwise velocity, shear stresses, and stress divergence terms can be written in a dimensionless form: 9gỹ,z̃≡ΔUwakeUH=ΔŨmax⋅e-ỹ2+z̃2/2σ̃2,10fỹ,z̃≡U(y,z)-UHUH=ln⁡z̃/z̃H+1ln⁡z̃0-gỹ,z̃,11fα′ỹ,z̃≡-e1/2σu*κzHUHu′uα′‾-u′uα′‾in=1+z̃z̃Hỹσ̃δ2α+z̃σ̃δ3αe1/2gỹ,z̃,12fα′′ỹ,z̃≡-σ2u*κzHUH∂∂xαu′uα′‾=1+z̃z̃H1-ỹ2σ̃2δ2α+z̃z̃+z̃H-z̃2σ̃2δ3αgỹ,z̃, with u′uα′‾in=-δ3αu*2 as the inflow shear stress and δiα as the Kronecker delta. In addition, the Greek index α is used to show that summation is not performed over the indices and will be used throughout this article.

The results of the far-wake model are depicted in Fig. , in terms of normalized streamwise velocity, f; normalized Reynolds shear stresses, fα′; and normalized Reynolds stress divergence, fα′′, as a function of the lateral (α=2) and vertical distance (α=3). Note that the prime indicates the derivative of f times a normalization factor, i.e., fα′≠∂f/∂xα. The results in Fig.  are made with ΔŨmax=0.4, which could reflect a certain downstream distance, although the overall behavior is not influenced by ΔŨmax. Figure a shows the first and second derivatives of the wake deficit that represent the negative shear stress -u′v′‾ and stress divergence -∂u′v′‾/∂y. The stress divergence is negative at the wake edges and positive at the wake center, which shows how momentum outside the wake is transported to the wake center; this is the main mechanism for (far) wake recovery. A similar observation can be made in the vertical wake recovery depicted in Fig. b; however, more momentum from above is transported to the center with respect to the bottom due to the eddy viscosity of the logarithmic inflow that increases linearly with height. It can easily be shown that the integral of the negative stress divergence (depicted by the red areas in Fig. a) is equal to the integral of the positive stress divergence (depicted by the blue area in Fig. a). This must hold, because stress divergence is momentum transport and should not result in a loss or gain of total momentum. The same balance of stress divergence is present in the vertical wake recovery depicted in Fig. b. The amount of lateral U-momentum transfer, Mlateral, and vertical U-momentum transfer at the bottom, Mvertical,b, and top of the wake, Mvertical,t, can be quantified by the bottom and top peaks of u′v′‾ and u′w′‾, respectively; since we can write 13Mlateral=∫-∞y-∂u′v′‾∂ydy+∫y+∞∂u′v′‾∂ydy=u′v′‾|y--u′v′‾|y+=2e-1/2νTσΔUmax14Mvertical,b=∫0z-∂u′w′‾∂zdz=u′w′‾|z-+u*2,Mvertical,t=∫z+∞∂u′w′‾∂zdz=-u′w′‾|z+-u*2, where y- and y+ are the lateral locations of the peaks in u′v′‾, and z- and z+ are the vertical locations of the bottom and the top peaks in u′w′‾, respectively. For the analytic model, we have y-=-σ, y+=σ, and z- and z+ are solutions of the cubic equation 2(z/zH)-(z/zH)[(z-zH)/σ]2=1; see Eqs. () and (). The analytical model predicts that the momentum transfer from above is larger than the momentum transfer from below, as shown by the peak values of u′w′‾ in Fig. , i.e., Mvertical,t>Mvertical,b. performed a related analysis on a large wind farm LES data set by integrating the horizontally averaged vertical kinetic energy flux over the rotor area; the obtained result was shown to be in order of the power extracted by wind turbines.

The fact that wake recovery requires a change of shear also becomes clear when considering a homogeneous shear flow , a temperature diffusion equation, or the Ekman spiral; these examples are further discussed in Appendix .

The wake recovery in terms of the stress divergence of u′uα′‾ is post-processed from an LES of a single wind turbine wake in a neutral pressure-driven atmospheric boundary layer. The LES is the same as used by ; numerical details can be found in . The wind turbine represents a Vestas V80 wind turbine that has a rotor diameter and hub height of 80 and 70 m, respectively. The wind turbine forces are modeled as an AD and has an effective thrust coefficient of 0.77. The inflow wind speed and total turbulence intensity at hub height are 8.0 m s-1 and 5.7 %, respectively.

Figure  shows the normal, lateral, and vertical stress divergence that contribute to the streamwise momentum equation at hub height and at a vertical plane through the rotor center. The normal stress divergence has the largest (negative) values in the near wake (Fig. a and d) but is about 5 times smaller than the shear stress divergence based on volumetric integrals of the three streamwise stress divergence terms: 15Mα=∫V∂u′uα′‾∂xαdV, where V is a box around the wind turbine located at (x,y,z)=(0,0,zH) with dimensions -2≤x/D≤20, -2≤y/D≤2, and 0≤(z-zH)/D≤3.125. We obtain M2/M1=5.5, M3/M1=5.1. Hence, the LES data show that it is mainly the shear stress divergence that leads to wake recovery by bringing momentum from the freestream into the wake center (best visible in Fig. b and f). The shear stress divergence represents wake meandering and turbulent cross diffusion, which is slightly larger in the lateral direction compared to the vertical direction due to the ground (M3/M2=0.93), although the atmospheric conditions and presence of neighboring wind turbines may influence the dominant direction of wake recovery. The normal stress divergence represents the streamwise back-and-forth movement of the wake and streamwise turbulent diffusion, which is much less compared to the lateral and vertical wake recovery.

The LES-based lateral and vertical wake recovery is depicted in Fig. , at three different downstream locations: x/D=2.5, 5, and 7.5. Results of the streamwise velocity, negative shear stress, and shear stress divergence are shown; they are normalized in the same way as performed for the analytical far-wake model as defined by Eqs. ()–() and depicted in Fig. . The normalized standard deviation, σ̃, used for normalization of the shear stress and its divergence is obtained by a Gaussian fit with the velocity deficit at each downstream location (σ̃=0.37, 0.39, 0.45 and σ̃=0.36, 0.37, 0.44 for the lateral and vertical wake recovery at x/D=2.5, 5 and 7.5, respectively). Furthermore, we have used u*=0.333 m s-1 and UH=8.0 m s-1. A similar behavior of the LES-derived velocity deficit, shear stresses, and the stress divergence is obtained at x/D=5 and 7.5 compared to the analytic far-wake model, as depicted in Fig. . As discussed previously, the lateral wake recovery is slightly stronger than the vertical wake recovery; the latter is stronger at the top of the wake with respect to the bottom of the wake, as expected and predicted by the simple far-wake model. The results in the near wake (x/D=2.5) are different in LES, as also expected, where the wake center momentum gain has a double bell shape due to the more complex shear stress profile.

The fact that the normal stress divergence is an order of magnitude smaller than the combination of lateral and vertical shear stress divergence indicates that RANS turbulence model closures do not need to model the anisotropy of the normal Reynolds stresses if the velocity deficit is the only quantity of interest. This implies that one could rely on the isotropic hypothesis of , as long as the turbulence model is able to predict correct shear stresses and give realizable Reynolds stresses – for example by using a flow-dependent eddy viscosity coefficient that limits the turbulence length scale . Whether this also applies in stratified atmospheric conditions is a subject for further studies.

The main mechanisms of wake recovery are explained by the stress divergence, considering both a Gaussian-based analytical far-wake model and LES of a single wind turbine in neutral atmospheric conditions. The LES data show that the divergence of the lateral and vertical shear stresses combined are an order of magnitude larger than the divergence of the normal stresses; i.e., ∂u′v′‾/∂y and ∂u′w′‾/∂z – and not simply “shear” – are the main contributors to wake recovery. The analytical model qualitatively captures the behavior of the stress divergence observed in the far wake of the LES results, which shows that the second derivatives ∂2U/∂y2 and ∂2U/∂z2 induce wake recovery. This also indicates that RANS turbulence model closures only need to be able to model the shear stresses accurately if the velocity deficit is the sole quantity of interest.