Physics-based modelling of wind-turbine wakes turbulence in neutral atmospheric boundary layers

This paper presents a physics-based model for predicting wake-added turbulence intensity in wind farms by analyzing turbulent kinetic energy and Reynolds stress budgets, offering a more accurate and practical alternative to existing empirical or axially symmetric approaches.

The Gist

Imagine you are standing in a vast, open field on a breezy day. You see a giant wind turbine spinning. As the wind hits the blades, it does two things: it slows down (creating a "shadow" of slower air behind the turbine) and it gets choppy and chaotic (creating turbulence).

This paper is about understanding that choppy, chaotic air (turbulence) behind the turbine and creating a better way to predict it.

Here is the story of the research, broken down into simple concepts:

1. The Problem: The "Blind Spot" in Wind Farm Design

Wind farm engineers use computer models to figure out where to place turbines. They know how to predict the "slow air" (velocity deficit) behind a turbine quite well. Think of it like knowing how far a shadow stretches behind a person walking in the sun.

However, they are less good at predicting the turbulence (the shaking, swirling air) behind the turbine.

• Why does this matter? Turbulence is like a double-edged sword. It helps the wind mix back in (helping the next turbine get power), but too much of it can shake the next turbine apart or make it inefficient.
• The old way: Previous models were like "guessing games." They used simple formulas or assumed the turbulence was perfectly round (like a ball) spreading out behind the turbine. But in reality, the air near the ground behaves differently than the air high up, so the turbulence is actually lopsided and messy, not a perfect ball.

2. The Solution: A Physics-Based "Recipe"

The authors (a team from France and the UK) decided to stop guessing and start looking at the actual physics of how energy moves in the air.

Instead of just saying "turbulence spreads out," they looked at the "budget" of energy. Imagine the wind turbine is a factory that takes smooth wind and turns it into chaotic, swirling energy.

• The Factory Process: They analyzed exactly how much energy is created by the spinning blades, how much is carried away by the wind, how much is spread out to the sides, and how much is lost as heat (dissipation).
• The New Model: They built a mathematical "recipe" based on these real physical processes. It accounts for the fact that the air near the ground is rougher (like driving on a gravel road) compared to the smooth air higher up (like driving on a highway).

3. The Analogy: The River and the Rock

Think of the wind as a river flowing smoothly.

• The Turbine is a large rock dropped in the river.
• Velocity Deficit: The water directly behind the rock moves slower.
• Wake-Added Turbulence: The water swirling and churning behind the rock.

Old Models assumed the churning water behind the rock was a perfect circle, spreading out evenly in all directions.
This New Model realizes that because the riverbed is rough (the ground), the churning water behaves differently near the bottom than near the surface. It creates a lopsided, complex swirl. The new math captures this "lopsidedness" by looking at the forces pushing the water around.

4. How They Tested It

To make sure their new recipe worked, they didn't just do math on paper. They used two powerful tools:

1. Super-Computer Simulations (LES): They created a virtual wind tunnel on a supercomputer. They simulated wind blowing over a "virtual turbine" (a porous disk) in different conditions (smooth air vs. rough, bumpy air).
2. Wind Tunnel Experiments: They compared their computer predictions against real-world data from physical wind tunnels where small turbines were tested.

The Result: Their new model matched the super-computer simulations and the real-world wind tunnel data much better than the old "guessing" models. It correctly predicted that the turbulence looks different near the ground than it does high up.

5. Why This Matters for the Future

This isn't just about wind turbines; it's about clean energy efficiency.

• Better Layouts: If we can predict the turbulence better, we can place wind turbines closer together without them breaking each other or losing power.
• More Power: By understanding exactly how the wind recovers after hitting a turbine, we can squeeze more energy out of the same amount of wind.
• Simplicity: Even though the math behind it is complex, the final result is a simple formula that engineers can use easily in their daily design software.

Summary

The authors took a complex, messy problem (predicting chaotic wind behind turbines) and solved it by looking at the fundamental laws of physics rather than relying on simple guesses. They created a new "map" for the wind that shows exactly where the turbulence is strong and where it is weak, helping us build better, more efficient wind farms.

Technical Summary

1. Problem Statement

Wind farm flow solvers (engineering or analytical models) typically rely on two sub-models: one for velocity deficit and one for wake-added turbulence intensity (TI). While velocity deficit models are well-developed, wake-added turbulence models remain less prevalent and often lack physical rigor.

• Limitations of Existing Models: Current models are mostly empirical or assume axisymmetry, which fails to capture the inherently three-dimensional and asymmetric nature of turbulent wakes in sheared atmospheric boundary layers (ABL).
• Consequence: Inaccurate turbulence estimation leads to poor predictions of flow interactions, wake recovery rates, and overall energy yield.
• Gap: There is a need for a physics-based, three-dimensional analytical model that accounts for inflow shear and boundary layer effects without relying on heavy numerical integration (which hinders optimization tasks).

2. Methodology

The authors developed a new physics-based analytical model through a multi-stage process involving Large Eddy Simulations (LES), budget analysis, and simplification of transport equations.

A. Data Generation (LES)

• Solver: Used waLBerla-wind, a solver based on the Lattice Boltzmann Method (LBM) with a D3Q27 stencil and cumulant collision operator.
• Setup: Simulated porous actuator disks (representing wind turbines) in neutral ABLs.
• Variables:
  • Thrust Coefficients ($C_T$): 0.4, 0.6, 0.8.
  • Ground Roughness: Varied to achieve hub-height turbulence intensities ($I_x$) between 5% and 13% (representing offshore to moderately rough onshore conditions).
  • Domain: 36D (length) $\times$ 12D (width) $\times$ 5D (height).
• Validation: The LES setup was validated against wind tunnel experiments (Bastankhah & Porté-Agel; Stein & Kaltenbach).

B. Turbulent Kinetic Energy (TKE) Budget Analysis

The authors analyzed the TKE transport equation to identify dominant terms in the wake:

1. Production ($P_k$): Dominant in shear regions (tip vortices).
2. Advection ($A_k$): Dominant in the far wake.
3. Turbulent Transport ($T_k$): Acts as a diffusion mechanism.
4. Dissipation ($\varepsilon_k$): Significant throughout.

• Neglected Terms: Viscous diffusion and pressure transport were found negligible (<10% of dominant terms).
• Key Observation: The wake-added TKE transitions from a bimodal distribution (near wake) to a more uniform distribution (far wake), and self-similarity is only valid in the very far wake ($x > 9D$).

C. Derivation of the Analytical Model

The authors derived a simplified Partial Differential Equation (PDE) and subsequently an explicit algebraic solution:

1. Simplified PDE: Reduced the TKE transport equation by assuming streamwise advection dominance and using gradient diffusion hypotheses for transport.
2. Far-Wake Approximation: To obtain a closed-form analytical solution, they applied far-wake assumptions:
   • Assumed self-similarity for the advection term.
   • Simplified the transport term by modeling it as a linear superposition of contributions from the wake core, intermediate region, and edges.
   • Assumed the eddy viscosity approaches the ambient value ($\nu_t \approx \nu_{t,\infty}$) in the far wake.
3. Final Form: The model expresses wake-added TKE ($k_w$) as a function of velocity gradients, velocity deficit ($\Delta u$), and calibrated parameters ($\zeta_{k,w}$ and $\gamma_k$).
4. Extension to Reynolds Stress ($u'u'$): The same methodology was applied to the streamwise normal Reynolds stress, incorporating a pressure-strain correlation model to account for the difference between TKE and $u'u'$.

3. Key Contributions

• First Physics-Based 3D Analytical Model: This is the first analytical model for wake-added turbulence that explicitly accounts for non-axisymmetry caused by inflow boundary layer shear.
• Closed-Form Solution: Unlike previous physics-based models (e.g., Bastankhah et al.) that require numerical integration, this model provides an explicit algebraic expression, making it computationally efficient for wind farm layout optimization and real-time control.
• Dual Output: The model simultaneously predicts TKE and the streamwise normal Reynolds stress ($u'u'$), the latter being critical for standard engineering wake expansion calculations.
• Robust Calibration: The model parameters are derived from LES data across a wide range of thrust coefficients and turbulence intensities, ensuring broad applicability.

4. Results and Validation

The model was validated against both LES data and two independent wind tunnel datasets.

• Comparison with LES:
  • The analytical model accurately reproduces the 3D structure of wake-added TKE and $u'u'$ in the far wake.
  • It correctly captures the transition from a bimodal (near wake) to a uniform (far wake) distribution.
  • Minor overestimation occurs in the near-wake ($x/D < 4$), which is expected due to the far-wake assumptions used in the derivation.
• Wind Tunnel Validation (Bastankhah & Porté-Agel):
  • The model showed excellent agreement with experimental lateral and vertical profiles of velocity deficit and turbulence intensity.
  • It outperformed the empirical Ishihara & Qian (I&Q) model, particularly in the vertical profiles below the hub height.
• Wind Tunnel Validation (Stein & Kaltenbach):
  • Tested against Smooth (SBL) and Rough (RBL) boundary layers.
  • The model successfully reproduced the Reynolds stress behavior for both conditions.
  • The I&Q model significantly overestimated turbulence in the RBL case, whereas the proposed physics-based model remained robust.
• Reynolds Number Independence: Additional simulations confirmed that the model's predictions are largely independent of the Reynolds number, validating its applicability from wind-tunnel scales to utility-scale turbines.

5. Significance

• Improved Accuracy: By moving away from purely empirical fits and axisymmetric assumptions, the model offers a more physically consistent tool for predicting flow interactions and energy yield in complex wind farms.
• Computational Efficiency: The closed-form nature of the solution allows it to be integrated into fast engineering solvers without the computational cost of RANS or LES, facilitating large-scale wind farm optimization.
• Foundation for Future Work: The framework established here (budget analysis + simplification) provides a pathway to extend physics-based modeling to include atmospheric stability (stratification) and meandering effects, which are currently neglected for simplicity.

In conclusion, this work bridges the gap between high-fidelity physics (LES) and practical engineering tools, providing a robust, predictive, and computationally efficient model for wake-added turbulence in neutral atmospheric conditions.