# Empirical Gaussian model#

FLORIS's "empirical" model has the same Gaussian wake shape as other popular FLORIS models. However, the models that describe the wake width and deflection have been reorganized to provide simpler tuning and data fitting.

## Wake shape#

The velocity deficit at a point $$(x, y, z)$$ in the wake follows a Gaussian curve, i.e.,

$\frac{u}{U_\infty} = 1 - Ce^{-\frac{(y-\delta_y)^2}{2\sigma_y^2} -\frac{(z-z_h-\delta_z)^2}{2\sigma_z^2}}$

where the $$(x, y, z)$$ origin is at the turbine location (at ground level). The terms $$C$$, $$\sigma_y$$, $$\sigma_z$$, $$\delta_y$$, and $$\delta_z$$ all depend on the downstream location $$x$$.

$$C$$ is the scaling factor for the Gaussian curve, defined as

$C = \frac{1}{8\sigma_{0_D}^2}\left(1 - \sqrt{1 - \frac{\sigma_{y0} \sigma_{z0} C_T}{\sigma_y \sigma_z}}\right)$

Here, $$C_T$$ is the turbine thrust coefficient, which includes any reduction in thrust due to yaw or tilt of the turbine rotor. $$\sigma_{y0}$$ and $$\sigma_{z0}$$ define the wake width at the turbine location $$x=0$$, which are based on the user-specified rotor-diameter normalized initial width $$\sigma_{0_D}$$. Note that this contrasts with FLORIS's other Gaussian models, where $$\sigma_{y0}$$ and $$\sigma_{z0}$$ are defined at the end of the near wake/beginning of the far wake, at some $$x_0 > 0$$. The normalization term $$8\sigma_{0_D}^2$$ provides consistency with actuator disc theory.

## Wake expansion#

The wake lateral and vertical widths, $$\sigma_y$$ and $$\sigma_z$$, respectively, are a function of downstream distance $$x$$. The expansion of the wake is described by a user-tunable, piecewise linear function. This is simplest to express as an integral of a piecewise constant wake expansion rate $$k$$, i.e.,

$\sigma_{y}(x) = \int_{0}^x \sum_{i=0}^n k_i \mathbf{1}_{[b_{i}, b_{i+1})} (x') dx' + \sigma_{y0}$

Here, $$\mathbf{1}_{[a, b)}(x)$$ is the indicator function, which takes value 1 when $$a \leq x < b$$, and 0 otherwise. The above function ensures that expansion rate $$k_i$$ applies only between breakpoints $$b_{i-1}$$ and $$b_i$$, allowing $$n+1$$ varying rates of linear expansion at different downstream ranges, determined by the $$b_i$$. Note that $$b_0 = 0$$ and $$b_{n+1} = \infty$$ by design.

A slight modification is made to the above so that the wake width varies smoothly. As stated above, the wake expansion rate contains jump discontinuities that create "sharp" changes in the wake width. To avoid this, the indicator function $$\mathbf{1}_{[a, b)}(x)$$ is replaced with a pair of "smoothstep" functions that vary smoothly with width parameter $$d$$. In the limit as $$d\rightarrow 0$$, the approximation becomes exact.

While the form of this wake expansion model seems complex, it is very simple to tune to fit data: the user provides the $$n+1$$ expansion rates $$k_i, i=0,\dots,n+1$$ (defined as a list in the wake_expansion_rates field of the input yaml) and the $$n$$ 'break points' $$b_i, i=1,\dots,n$$ where those expansion rates should go into effect (specified in terms of rotor diameters downstream as a list in the breakpoints_D field of the input yaml.

As well as these, the initial width $$\sigma_{0_D}$$ should be provided by setting sigma_0_D and the logistic function width $$d$$ as smoothing_length_D (both specified in terms of rotor diameters).

We expect that the default values for $$\sigma_{0_D}$$ and $$d$$ should be satisfactory for most users. Further, we anticipate that most users will not need more than $$n+1=3$$ expansion rates (along with $$n=2$$ break points) to describe the wake expansion.

## Wake deflection#

The deflection of the wake centerline $$\delta_y$$ and $$\delta_z$$ due to yawing and tilting, respectively, follow a simple model

$\delta = k_\text{def} C_T \alpha \operatorname{ln}\left(\frac{x/D - c}{x/D + c} + 2\right)$

Here, $$k_\text{def}$$ is a user-tunable deflection gain and $$\alpha$$ is the misalignment. When computing the lateral wake deflection $$\delta_y$$ due to yaw misalignment, $$\alpha$$ should be the yaw misalignment specified in radians, clockwise positive from the wind direction. When computing the vertical wake deflection $$\delta_z$$ due to rotor tilt, $$\alpha$$ should be the tilt angle specified in radians, clockwise positive when the rotor is tilted back.

Finally, $$c$$ in the above deflection model is a 'deflection rate'. This specifies how quickly the wake will reach it's maximum deflection $$k_\text{def} C_T \alpha \operatorname{ln}(3)$$ for a given yaw/tilt angle.

User-tunable parameters of the model are as follows:

• A separately tunable deflection gain $$k_\text{def}$$ for each of lateral deflections (due to yaw misalignments) and vertical deflections (due to nonzero tilt), specified using horizontal_deflection_gain_D and vertical_deflection_gain_D (specified in terms of rotor diameters)

• The deflection rate $$c$$, specified using deflection_rate.

We anticipate that most users will be able to use the default value for $$c$$, and set vertical_deflection_gain_D to the same value as horizontal_deflection_gain_D (which can also be achieved by providing vertical_deflection_gain_D = -1).

## Wake-induced mixing#

Finally, turbines contribute to mixing in the flow. In other models, this extra mixing is accounted for by adding to the turbulence intensity value. In the empirical model, explicit dependencies on turbulence intensity are removed completely to aid in tuning. Instead, a non-physical "wake-induced mixing factor" is specified for turbine $$j$$ as

$\text{WIM}_j = \sum_{i \in T^{\text{up}}(j)} \frac{A_{ij} a_i} {(x_j - x_i)/D_i}$

where $$T_T^{\text{up}}(j)$$ is the set of turbines upstream from the turbine $$j$$. Here, $$A_{ij}$$ is the area of overlap of the wake of turbine $$i$$ onto turbine $$j$$; $$a_i$$ is the axial induction factor of the turbine $$i$$; and $$(x_j - x_i)/D_i$$ is the downstream distance of turbine $$j$$ from the turbine $$i$$, normalized by turbine $$i$$'s rotor diameter.

Wake-induced mixing can affect both the velocity deficit and wake deflection. To account for wake-induced mixing, the wake width of turbine $$j$$ is adjusted to

$\sigma_{y}(x) = \int_{0}^x \sum_{i=0}^n k_i \ell_{[b_{i}, b_{i+1})}(x') + w_v \text{WIM}_j dx' + \sigma_{y0}$

Here, $$w_v$$ is the velocity deficit wake-induced mixing gain, which the user can vary by setting wim_gain_velocity to represent different levels of mixing caused by the turbines.

The wake deflection model is similarly adjusted to

$\delta = \frac{k_\text{def} C_T \alpha}{1 + w_d \text{WIM}_j}\operatorname{ln}\left(\frac{x/D - c}{x/D + c} + 2\right)$

where $$w_d$$ is the wake-induced mixing gain for deflection, provided by the user by setting wim_gain_deflection.

Yaw misalignment can also add turbulence to the wake. In the empirical Gaussian model, this effect, referred to as "yaw-added wake recovery" in other models, is activated by setting enable_yaw_added_recovery to true. Yaw-added mixing is represented by updating the wake-induced mixing term as follows:

$\text{WIM}_j = \sum_{i \in T^{\text{up}}(j)} \frac{A_{ij} a_i (1 + g_\text{YAM} (1-\cos(\gamma_i)))}{(x_j - x_i)/D_i} + a_j g_\text{YAM} (1-\cos(\gamma_j))$

Note that the second term means that, unlike when enable_yaw_added_recovery is false, a turbine may affect the recovery of its own wake by yawing.

## Mirror wakes#

Mirror wakes are also enabled by default in the empirical model to model the ground effect. Essentially, turbines are placed below the ground so that the vertical expansion of their (mirror) wakes appears in the above-ground flow some distance downstream, to model the reflection of the true turbine wakes as they bounce off of the ground/sea surface.

## Added mixing by active wake control#

As the name suggests, active wake control (AWC) aims to enhance mixing to the wake of the controlled turbine. This effect is activated by setting enable_active_wake_mixing to true, and awc_modes to "helix" (other AWC strategies are yet to be implemented). The wake can then be controlled by setting the amplitude of the AWC excitation using awc_amplitudes (see the AWC operation model). The effect of AWC is represented by updating the wake-induced mixing term as follows:

$\text{WIM}_j = \sum_{i \in T^{\text{up}}(j)} \frac{A_{ij} a_i} {(x_j - x_i)/D_i} + \frac{\beta_{j}^{p}}{d}$

where $$\beta_{j}$$ is the AWC amplitude of turbine $$j$$, and the exponent $$p$$ and denominator $$d$$ are tuning parameters that can be set in the emgauss.yaml file with the fields awc_wake_exp and awc_wake_denominator, respectively. Note that, in contrast to the yaw added mixing case, a turbine currently affects only its own wake by applying AWC.