[Paper Review] An entrainment-based model for annular wakes, with applications to airborne wind energy

Sam Kaufman‐Martin, Nicholas D. Naclerio+2 more|arXiv (Cornell University)|Mar 23, 2020

Wind Energy Research and Development36 references15 citations

TL;DR

This paper develops an entrainment-based reduced-order model for annular wakes generated by airborne wind energy (AWE) kites, using three coupled ordinary differential equations solved numerically and a simplified two-equation analytical model under the no-radial-drift assumption. The models accurately predict wake evolution and core disappearance, enabling efficient wind farm layout optimization for AWE arrays.

ABSTRACT

Several novel wind energy systems produce wakes with annular cross-sections, which are qualitatively different from the wakes with circular cross-sections commonly generated by conventional horizontal-axis wind turbines and by compact obstacles. Since wind farms use arrays of hundreds of turbines, good analytical wake models are essential for efficient wind farm planning. Several models already exist for circular wakes; however, none have yet been proposed for annular wakes, making it impossible to estimate their array performance. We use the entrainment hypothesis to develop a reduced-order model for the shape and flow velocity of an annular wake from a generic annular obstacle. Our model consists of a set of three ordinary differential equations, which we solve numerically. In addition, by assuming that the annular wake does not drift radially, we further reduce the problem to a model comprising only two differential equations, which we solve analytically. Both of our models are in good agreement with previously published large eddy simulation results.

Motivation & Objective

• To address the lack of theoretical models for turbulent annular wakes, which are critical for planning large-scale airborne wind energy (AWE) farms.
• To develop a reduced-order model based on the entrainment hypothesis to predict the spatial evolution of annular wake shape and velocity.
• To derive both a numerical model (three ODEs) and an analytical model (two ODEs under no-radial-drift assumption) for wake development behind annular obstacles.
• To validate the models against large eddy simulation results from Haas and Meyers (2019), showing strong agreement for large downstream distances.
• To enable scalable, low-cost wake modeling for AWE array optimization by identifying universal scaling behavior independent of the entrainment coefficient E.

Proposed method

• Uses the entrainment hypothesis, assuming radial inflow proportional to velocity differences: we = E(V∞ − Vw) and wi = E(Vi − Vw), with E as the entrainment coefficient.
• Derives a system of three ordinary differential equations (ODEs) for wake velocity Vw, wake span Sw, and total diameter Dw as functions of downstream distance x, based on mass and momentum conservation.
• Imposes the no-radial-drift assumption to reduce the system to two ODEs, enabling analytical solution via scaling transformation X = Ex.
• Solves the full model numerically and the analytical model in closed form, with initial conditions derived from one-dimensional momentum theory.
• Validates both models against large eddy simulation data from Haas and Meyers (2019), focusing on wake growth and core disappearance.
• Demonstrates that the wake velocity at core disappearance (Vw(xnc)) is independent of E, and that solutions can be universally scaled via X = Ex.

Experimental results

Research questions

• RQ1 How does the wake velocity and geometry of an annular wake evolve downstream behind an annular obstacle, and can this be modeled analytically or numerically using entrainment principles?
• RQ2 What is the downstream location xnc at which the central core region of the annular wake disappears, and how does it depend on the entrainment coefficient E and geometry S/D?
• RQ3 Can the wake behavior be universally scaled such that solutions for different E values are related by a simple transformation, and what are the implications for wind farm modeling?
• RQ4 How accurate are the analytical and numerical models compared to high-fidelity large eddy simulations, especially for large x/D?
• RQ5 Can the models be used to transition seamlessly to a circular wake model after core disappearance, and what computational benefits does this enable?

Key findings

• The full numerical model (three ODEs) shows good agreement with large eddy simulation results, particularly for x/D > 5, validating the entrainment-based approach for annular wakes.
• The analytical model (two ODEs under no-radial-drift assumption) provides a closed-form solution for xnc, the location where the core region disappears, and this xnc is inversely proportional to the entrainment coefficient E.
• The wake velocity Vw at the core disappearance location xnc is independent of E, indicating a universal behavior across different turbulence levels.
• A reference solution Vw(X) exists where X = Ex, making all annular wakes with the same S/D geometrically similar regardless of E, enabling scalable modeling.
• The outer wake boundary (Dw/2) remains reasonably accurate in the analytical model for large x/D, though the core region is less accurately captured.
• After core disappearance, the wake can be seamlessly transitioned to the standard circular wake model of Luzzatto-Fegiz (2013), preserving accuracy while reducing computational cost.