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arxiv: 2605.04782 · v1 · submitted 2026-05-06 · ⚛️ physics.flu-dyn

Recognition: unknown

Modelling Farm-to-Farm Interaction Using a Fast Linearised Numerical Approach

Alexia Everley , Hossein A. Kafiabad , Majid Bastankhah
Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:09 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords wind farmswake interactionslinearised modelvertical displacementhub heightturbulent entrainmentfarm-to-farm modellingwake recovery
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The pith

A fast linearised 2D model shows ground proximity drives upward displacement of wind farm wakes, making downstream farms with higher hub heights more vulnerable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a computationally efficient method that solves linearised two-dimensional incompressible flow equations to simulate how wakes from one wind farm affect another. It validates the approach against large-eddy simulations for tandem farm layouts and then varies inter-farm distance and hub-height ratios to quantify performance losses. The central result is that wakes shift upward because the ground blocks downward expansion, creating asymmetric turbulent mixing that entrains more air from above. This shift means downstream farms using taller turbines suffer stronger wake interference than those with shorter turbines at the same locations.

Core claim

The linearised two-dimensional incompressible equations, solved with Fourier transforms in the horizontal direction and finite-difference discretisation in the vertical, predict that wakes from an upstream wind farm displace upward in a tandem configuration. This displacement arises because proximity to the ground restricts downward wake expansion and produces asymmetric turbulent entrainment. Parametric runs across inter-farm distances and hub-height ratios indicate that the upward shift causes downstream farms with higher hub heights to experience greater power losses from upstream wakes than farms with lower hub heights.

What carries the argument

Linearised two-dimensional incompressible equations solved by Fourier transforms horizontally and finite differences vertically, which rapidly computes farm-to-farm wake recovery and vertical displacement without full turbulence simulation.

If this is right

  • Downstream wind farms should use lower hub heights relative to upstream farms to reduce wake interference.
  • Optimal inter-farm spacing depends on the vertical wake shift, which grows with distance from the ground.
  • The fast model enables rapid exploration of many farm layouts before committing to expensive full simulations.
  • Wake recovery rates in tandem configurations are controlled by the same asymmetric entrainment that produces the upward displacement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Farm planners could test staggered hub-height arrangements within a single large array to exploit the vertical displacement effect.
  • The two-dimensional assumption might be relaxed by adding a spanwise Fourier mode to check whether lateral spreading alters the upward shift.
  • If atmospheric stability changes the effective ground blocking, the model would need an adjusted entrainment coefficient to remain predictive.
  • Real-time control systems for wind farms could use the linearised solver to forecast wake positions from measured upstream conditions.

Load-bearing premise

The linearised two-dimensional incompressible equations remain accurate enough to capture wake recovery and the observed vertical displacement even in real turbulent flows at the tested distances and height ratios.

What would settle it

Measure the vertical position of the wake centerline at the downstream farm location in a large-eddy simulation or field experiment for a hub-height ratio of 1.2 at an inter-farm spacing of 10 diameters; if the model over- or under-predicts the upward shift by more than the reported validation error, the displacement mechanism is not supported.

Figures

Figures reproduced from arXiv: 2605.04782 by Alexia Everley, Hossein A. Kafiabad, Majid Bastankhah.

Figure 1
Figure 1. Figure 1: Normalised streamwise velocity deficit at hub height for the (a) aligned baseline (B) and view at source ↗
Figure 2
Figure 2. Figure 2: Root mean square error (RMSE) for a) Nx = 16384, Nz varied and b)Nz = 4000, Nx varied. True value is that taken for Nx = 16384 with Nz = 4000 and results are presented on a log-log scale. computed with the finest grid, Nx = 16384 and Nz = 4000. Figure 2a shows the convergence with respect to the vertical resolution Nz. The RMSE decreases approximately linearly on the log-log scale as Nz increases, dropping… view at source ↗
Figure 3
Figure 3. Figure 3: Normalised streamwise velocity at hub height for inter-farm distances 5km, 10km and 15km view at source ↗
Figure 4
Figure 4. Figure 4: Contours of normalised streamwise velocity perturbation in the view at source ↗
Figure 5
Figure 5. Figure 5: (a) Budget analysis conducted at z = zh for (1a), where the term R is the residual. Each term is non-dimensionalised by U 2 h /Lf . (b) Schematic of u profile at the end of the first farm. The left line represents the scenario for zh = Dz/2 and the right line zh = Dz/10 view at source ↗
Figure 6
Figure 6. Figure 6: Contours of normalised vertical velocity perturbation in the view at source ↗
Figure 7
Figure 7. Figure 7: Ratio of power availability, P, of a downstream wind farm operating in the wake of an upstream wind farm as a function of the normalized inter-farm spacing Lx/Lf , for different hub-height ratios zh2/zh1. P is normalized by the power of the downstream wind farm operating in isolation. P as a function of the inter-farm separation. The hub heights of the upstream and downstream wind farms are denoted by zh1 … view at source ↗
read the original abstract

This paper presents a computationally efficient, linearised numerical method for modelling aerodynamic interactions between wind farms. The linearised two-dimensional incompressible equations are solved using Fourier transforms in the horizontal direction and finite-difference discretisation in the vertical. Model predictions are validated against large-eddy simulation (LES) data, focusing on a tandem wind farm configuration where a downstream wind farm operates within the wake of an upstream array. A parametric study is then conducted to examine the impact of this wake on the performance of the downstream farm across a range of inter-farm distances and hub-height ratios. We demonstrate that the upward vertical displacement of these wakes is driven by asymmetric turbulent entrainment caused by the farm's proximity to the ground, which restricts downward wake expansion. Consequently, the results suggest that, due to this upward wake displacement, downstream wind farms with higher hub heights may be more strongly affected by upstream farms than those with lower hub heights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a computationally efficient linearised two-dimensional numerical method for modeling aerodynamic interactions between wind farms. The approach solves the linearised incompressible Navier-Stokes equations using Fourier transforms in the horizontal direction and finite-difference discretisation in the vertical. Predictions are validated against large-eddy simulation (LES) data for tandem wind farm configurations, followed by a parametric study examining the effects of inter-farm distances and hub-height ratios on downstream farm performance. The key result is that wakes exhibit upward vertical displacement due to asymmetric turbulent entrainment restricted by the ground, implying that downstream wind farms with higher hub heights are more strongly affected by upstream wakes.

Significance. If the results hold, this work provides a valuable fast modeling tool for wind farm wake interactions that could facilitate extensive parametric studies and optimization in wind energy applications. The finding regarding hub-height sensitivity to wake displacement offers a new perspective on farm layout design that may improve energy yield predictions in multi-farm setups.

major comments (2)
  1. [§4] §4 (validation against LES): The abstract states validation against LES for the tandem configuration, but no quantitative agreement metrics (e.g., L2 norms on velocity deficit profiles, wake centroid vertical shift, or vertical momentum flux) are referenced. Without these, the support for the asymmetric entrainment mechanism and the subsequent hub-height claim cannot be fully assessed.
  2. [§5] §5 (parametric study and mechanism attribution): The demonstration that upward wake displacement is driven by ground-restricted asymmetric turbulent entrainment is diagnosed from the linearised 2D solutions. However, linearisation suppresses nonlinear advection and 3D turbulent structures that dominate real entrainment; the paper should include a direct comparison of model-predicted vs. LES wake centroid trajectories across the tested hub-height ratios to confirm the causal explanation is not an artifact of the approximation.
minor comments (2)
  1. The abstract refers to 'a range of inter-farm distances and hub-height ratios' but does not list the specific values used; these should be stated explicitly for reproducibility.
  2. Figures showing vertical profiles would benefit from explicit annotation of the ground plane and hub-height locations to clarify the displacement effect.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive comments, which help clarify the validation and strengthen the interpretation of our results. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [§4] §4 (validation against LES): The abstract states validation against LES for the tandem configuration, but no quantitative agreement metrics (e.g., L2 norms on velocity deficit profiles, wake centroid vertical shift, or vertical momentum flux) are referenced. Without these, the support for the asymmetric entrainment mechanism and the subsequent hub-height claim cannot be fully assessed.

    Authors: We agree that explicit quantitative metrics will improve the assessment of the validation results. We will add L2-norm errors for the velocity deficit profiles, wake centroid vertical shifts, and vertical momentum flux comparisons in Section 4. We will also revise the abstract to reference these metrics, providing clearer quantitative support for the model accuracy and the proposed entrainment mechanism. revision: yes

  2. Referee: [§5] §5 (parametric study and mechanism attribution): The demonstration that upward wake displacement is driven by ground-restricted asymmetric turbulent entrainment is diagnosed from the linearised 2D solutions. However, linearisation suppresses nonlinear advection and 3D turbulent structures that dominate real entrainment; the paper should include a direct comparison of model-predicted vs. LES wake centroid trajectories across the tested hub-height ratios to confirm the causal explanation is not an artifact of the approximation.

    Authors: We acknowledge that the linearised model omits nonlinear advection and three-dimensional turbulent structures, which are important for real entrainment processes. The upward wake displacement in the model arises specifically from the ground boundary condition that restricts downward entrainment, an effect retained in the linearised equations with the imposed eddy viscosity. We will add plots of wake centroid trajectories from the linear model across the hub-height ratios in the revised parametric study section. However, we do not have LES data available for the varying hub-height configurations, so a direct model-versus-LES comparison of trajectories cannot be provided. revision: partial

standing simulated objections not resolved
  • Direct comparison of linear model wake centroid trajectories versus LES for the tested hub-height ratios, due to the absence of LES data for those specific configurations.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from standard equations

full rationale

The paper derives its fast linearised numerical method directly from the standard linearised 2D incompressible Navier-Stokes equations, discretised via established Fourier transforms in the horizontal and finite differences in the vertical. Results for wake recovery, vertical displacement, and farm-to-farm interactions are obtained by solving these equations for given inputs (inter-farm distances, hub-height ratios) and are validated against independent LES data rather than being presupposed. The attribution of upward wake displacement to asymmetric entrainment is an inference drawn from the computed solutions, not a definitional or fitted input. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted to a subset and then relabelled as predictions, and no ansatz is smuggled via prior work. The central claims therefore remain independent of their own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the linearised incompressible equations and the interpretation of LES comparison data; no explicit free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Linearised two-dimensional incompressible Navier-Stokes equations are adequate for farm-to-farm wake interactions
    Basis of the entire numerical method as stated in the abstract.
  • standard math Fourier transforms horizontally and finite differences vertically yield an accurate and efficient discretisation
    Discretisation choice enabling the fast solver.

pith-pipeline@v0.9.0 · 5461 in / 1440 out tokens · 38911 ms · 2026-05-08T17:09:11.315881+00:00 · methodology

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Reference graph

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