Gaussian-Curl Hybrid (GCH) Model

• GCH is a reduced-order wake model that combines a Gaussian deficit with curl-based vorticity corrections to capture yaw effects.
• It serves as the low-fidelity component in a hybrid LES–ML framework, enabling fast full-field wake predictions for energy production optimization.
• The model demonstrates computational efficiency with significant cost reduction compared to LES while maintaining improved fidelity through ML corrections.

Searching arXiv for the specified paper and closely related GCH/FLORIS wake-model papers. Gaussian–curl hybrid (GCH) is a reduced-order wake model implemented in FLORIS v3.4 that combines a steady-state Gaussian wake deficit with a curl-based vorticity correction to represent yaw-induced wake steering. In the South Fork wind-farm study, GCH is used as the low-fidelity component of a hybrid LES–ML workflow: large-eddy simulation (LES) supplies the high-fidelity ground truth, while GCH supplies inexpensive full-field wake predictions that an autoencoder-based convolutional neural network maps to LES-equivalent fields for annual energy production (AEP) estimation and layout optimization (Anjiraki et al., 9 Sep 2025).

1. Formulation and physical interpretation

The Gaussian component of GCH assumes a time-averaged wake velocity field with a Gaussian deficit profile and an optional lateral shift $\delta$:

$$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$

Here, $u_g$ is the local mean velocity within the Gaussian wake, $U_\infty$ is the ambient or freestream wind speed, $C$ is the centerline velocity deficit coefficient, $\delta$ is the lateral wake deflection, and $\sigma_y,\sigma_z$ are wake-width scales in the lateral and vertical directions. In FLORIS, $C$ typically depends on thrust coefficient $C_T$, rotor induction, and turbulence; the study used FLORIS defaults. Wake expansion is represented by

$\sigma_i = k_i\,\Delta x + \epsilon,$

with $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$0, where $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$1 is the wake expansion coefficient, $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$2 is the downstream distance from the rotor plane, and $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$3 is the initial wake width near the rotor (Anjiraki et al., 9 Sep 2025).

The curl component augments the Gaussian deficit through a vorticity-based correction associated with the curled wake model. Physically, yaw misalignment generates a pair of counter-rotating streamwise vortices that steer and skew the wake. In FLORIS’s GCH implementation, the induced velocity field is prescribed through a streamwise vorticity distribution whose strength scales with yaw angle $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$4, rotor thrust $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$5, and rotor geometry $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$6. This correction produces the lateral deflection $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$7 that enters the Gaussian deficit and introduces cross-flow components consistent with curled wake observations in LES. The study does not reproduce the explicit curl equations or coefficients, but it states that GCH captures yaw steering qualitatively while tending to overpredict deflection and, because of rapid wake recovery, suppress secondary wake steering downstream (Anjiraki et al., 9 Sep 2025).

2. Inputs, parameters, and modeling assumptions

The study uses GCH as it is distributed in FLORIS v3.4. Required inputs include ambient wind speed $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$8 and direction from the local wind rose, turbine yaw angle $$\frac{u_g}{U_\infty} = 1 - C \,\exp\!\left( -\left[ \frac{(y - \delta)^2}{\sigma_y^2} + \frac{z^2}{\sigma_z^2} \right] \right).$$9, rotor diameter $u_g$0, hub height, power curve, and general turbine characteristics. Thrust coefficient $u_g$1 and power coefficient $u_g$2 are used internally by FLORIS to set deficits, and turbulence intensity together with stability factors enters implicitly through the wake-expansion coefficients $u_g$3 (Anjiraki et al., 9 Sep 2025).

The assumptions are those of an analytical, steady-state, time-averaged wake model. The inflow is treated as homogeneous, the terrain as flat, and multiple wakes are handled through analytical superposition internal to FLORIS. The model does not explicitly include nacelle or tower geometry. In contrast, the LES reference fields include nacelle and tower effects through actuator-surface and line models. The study emphasizes that no additional site-specific calibration was performed: “The GCH model parameters used herein were adopted directly from the sample input configuration distributed with the FLORIS (v3.4) software package… the study kept the default settings to evaluate the ML model performance without further calibration” (Anjiraki et al., 9 Sep 2025).

These modeling choices define the role of GCH in the workflow. It is not used as a high-fidelity predictive endpoint but as a fast, physics-informed baseline whose errors can be learned and corrected. This suggests that the study treats GCH less as a stand-alone surrogate for LES than as a structured prior for data-driven refinement.

3. Role within the LES–ML hybrid framework

Within the hybrid framework, GCH generates low-fidelity full-domain wake fields for given turbine layouts and wind conditions. These fields are supplied to an autoencoder-based CNN with a U-Net-style encoder–decoder, skip connections, dropout, and a parameter-processing branch composed of an MLP and transformer encoder. The network input is a low-fidelity wake slice $u_g$4 from GCH together with a control vector

$u_g$5

where 79 is the vertical grid count and $u_g$6 are the in-plane dimensions. The output is an ML-predicted high-fidelity field $u_g$7 trained to match the LES field $u_g$8 using RMSE loss (Anjiraki et al., 9 Sep 2025).

The reported training regime uses 14 paired GCH–LES fields for training and 1 held-out test case, for approximately 60,000 epochs, with the Adam optimizer at learning rate $u_g$9, ReduceLROnPlateau with factor $U_\infty$0 and patience of 100 epochs, and dropout $U_\infty$1 after convolution blocks. For inference and AEP computation, the flow domain is rotated so that the streamwise axis aligns with the current wind direction. The paper does not explicitly report normalization or scaling procedures beyond network-architecture specifics (Anjiraki et al., 9 Sep 2025).

In this arrangement, GCH provides the coarse physical structure of the wake field, while LES supplies the target morphology of wake asymmetry, width, recovery, and yaw-induced distortion. A plausible implication is that the ML component is exploiting the inductive bias of the analytical wake model rather than learning wind-farm flow structure from scratch.

4. Fidelity relative to LES and computational cost

The study contrasts GCH and LES both qualitatively and quantitatively. Relative to LES, GCH produces “broad lateral spread and highly symmetric turbine wakes,” overpredicts near-wake deficits below rated speeds, underpredicts deficits above rated speeds, and recovers too quickly, with almost complete recovery by approximately $U_\infty$2, whereas LES shows narrower and more asymmetric wakes with recovery extending to approximately $U_\infty$3. For yawed conditions, GCH captures skew but tends to overpredict deflection and to suppress secondary steering effects on downstream turbines because of its rapid recovery (Anjiraki et al., 9 Sep 2025).

Across 14 training cases on the full 3D domain, GCH versus LES yields $U_\infty$4, $U_\infty$5, $U_\infty$6, and $U_\infty$7. On the unseen test case (layout 7, case 15), the ML-corrected prediction achieves $U_\infty$8, $U_\infty$9, $C$0, and $C$1, whereas GCH alone gives $C$2, $C$3, $C$4, and $C$5. For the optimized layout under representative wind conditions, the same pattern persists: at $C$6 and $C$7, ML gives $C$8 and GCH $C$9; at $\delta$0 and $\delta$1, ML gives $\delta$2 and GCH $\delta$3; at $\delta$4 and $\delta$5, ML gives $\delta$6 and GCH $\delta$7 (Anjiraki et al., 9 Sep 2025).

The cost differential is central to the study’s motivation.

Per-scenario method	CPU hours
LES	$\delta$8
GCH	$\delta$9
ML inference	$\sigma_y,\sigma_z$0

The paper states that ML+GCH reduces computational cost by approximately $\sigma_y,\sigma_z$1-fold relative to LES while achieving LES-like accuracy in the corrected fields. The correction step adds negligible cost over GCH alone (Anjiraki et al., 9 Sep 2025).

5. AEP computation and wake-informed layout optimization

AEP is computed as

$\sigma_y,\sigma_z$2

where $\sigma_y,\sigma_z$3, $\sigma_y,\sigma_z$4 and $\sigma_y,\sigma_z$5 are the numbers of wind-direction and wind-speed bins, $\sigma_y,\sigma_z$6 is the number of turbines, $\sigma_y,\sigma_z$7 is the power of turbine $\sigma_y,\sigma_z$8 for wind-direction bin $\sigma_y,\sigma_z$9 and speed bin $C$0, and $C$1 are the probabilities of the corresponding direction and speed bins from the Wind Integration National Dataset Toolkit. The power values are taken from the SWT-3.6-120 manufacturer power curve rather than from the classical relation $C$2. The effective wind speed $C$3 used to query the power curve is defined as an area-averaged rotor value, with rotor area

$C$4

Wake superposition and deficit combination are handled internally by FLORIS; the specific superposition method is not detailed (Anjiraki et al., 9 Sep 2025).

The optimization procedure is greedy. It starts from a randomly initialized layout on a $C$5 grid of feasible positions within a fixed project footprint. For each candidate move, one turbine at a time, FLORIS–GCH generates the low-fidelity flow field, the trained ML model produces the LES-equivalent field, and AEP is computed from the wind rose and power curve. The move is accepted if it increases AEP, and the procedure iterates across turbines and positions (Anjiraki et al., 9 Sep 2025).

For the South Fork wind farm, the baseline AEP is reported as $C$6 and the optimized AEP as $C$7, corresponding to a $C$8 improvement without expanding land use. The optimized layout tends to increase streamwise spacing and spread turbines laterally to reduce wake–wake interactions. The paper notes that this is consistent with prior findings that streamwise spacing strongly affects farm performance (Anjiraki et al., 9 Sep 2025).

6. Relation to other wake models, limitations, and interpretation

The study places GCH within a broader reduced-order wake-model lineage. It notes that the Jensen/Park model is widely used and available in FLORIS, that Gaussian wake models provide physically motivated deficit shapes, and that curled wake and GCH models incorporate yaw-induced vorticity effects missing in simpler Gaussian or Jensen formulations. It also notes that cumulative curl extensions aim to address momentum-conservation and wake-recovery issues identified for GCH. The original GCH development is described as building on Gaussian models and the curled wake mechanism (Anjiraki et al., 9 Sep 2025).

The limitations highlighted in the study are specific and consequential. GCH can overpredict or underpredict near-wake deficits depending on wind-speed regime, recovers too rapidly compared with LES, can bias AEP upward by delivering higher velocities to downstream turbines, overpredicts yaw deflection, and can suppress secondary wake steering downstream. It also omits nacelle and tower geometry and has limited representation of complex terrain and atmospheric stability effects. Sensitivity to yaw angle $C$9, spacing, and turbulence intensity through $C_T$0 is acknowledged, and the default FLORIS parameters used in the study may not be optimal for every site (Anjiraki et al., 9 Sep 2025).

A common misconception would be to interpret GCH, in the form used here, as a substitute for LES. The evidence reported in the study does not support that interpretation. Instead, GCH alone is presented as a fast but materially biased reduced-order model, whereas the hybrid ML-corrected workflow delivers near-LES fidelity in mean wake fields while preserving the computational efficiency required for iterative design. In the South Fork case, this hybridization is the basis for a practical design loop in which analytical wake modeling, learned correction, AEP assessment, and greedy optimization are tightly coupled (Anjiraki et al., 9 Sep 2025).