Classical Momentum Theory Research Articles

A method for fast and accurate prediction of wind turbine thrust coefficients using classical momentum theory and power curve

Abstract The planning and development of windfarms require accurate prediction of the thrust coefficient (cT ) of wind turbines, which significantly affects the downstream wake. Traditional methods, such as blade element momentum theory (BEMT), often necessitate detailed geometric information of wind turbines for cT computation, information that is not frequently available, especially in the early stages of windfarm planning. This paper aims to address this challenge by presenting a novel and efficient approach to predict cT for horizontal-axis wind turbines (HAWTs). The proposed method integrates classical momentum theory with power curve data to estimate the average axial induction factor (a), thereby enabling the calculation of cT without requiring detailed geometric information of HAWTs. The method was validated against thirty-five existing pitch-controlled HAWTs, with R2 values ranging from 0.9604 to 0.9989. This validation confirms the accuracy of the method, making it a viable alternative to traditional techniques that demand comprehensive wind turbine geometric details. The method has demonstrated both rapidity and precision in cT computation for turbine wake analysis, ensuring high levels of prediction accuracy and potentially lowering the barrier to entry for windfarm development. Unlike existing models predominantly focused on wind turbine power curves, cT modelling has largely been overlooked. This study makes a unique contribution to the field by proposing a novel method for cT prediction, thereby filling a critical gap in windfarm planning and development. However, while the study shows promising results, further research is warranted to explore its applicability in diverse windfarm scenarios and turbine configurations.

An analytical linear two-dimensional actuator disc model and comparisons with computational fluid dynamics (CFD) simulations

Abstract. The continuous up-scaling of wind turbines enabled by more lightweight and flexible blades in combination with coning has challenged the assumptions of a plane disc in the commonly used blade element momentum (BEM)-type aerodynamic codes for the design and analysis of wind turbines. The objective with the present work is thus to take a step back relative to the integral 1-dimensional (1-D) momentum theory solution in the BEM model in order to study the actuator disc (AD) flow in more detail. We present an analytical, linear solution for a two-dimensional (2-D) AD flow with one equation for the axial velocity and one for the lateral velocity, respectively. Although it is a 2-D model, we show in the paper that there is a good correlation with axis-symmetric and three-dimensional (3-D) computational fluid dynamics (CFD) simulations on a circular disc. The 2-D model has thus the potential to form the basis for a simple and consistent rotor induction model. For a constant loading, the axial velocity distribution at the disc is uniform as in the case of the classical momentum theory for an AD. However, an important observation of the simulated flow field is that immediately downstream of the disc the axial velocity profiles change rapidly to a shape with increased induction towards the edges of the disc and less induction on the central part. This is typically what is seen at the disc in full non-linear CFD AD simulations, which is what we compare with in the paper. By a simple coordinate rotation the analytical solution is extended to a yawed disc with constant loading. Again, a comparison with CFD, now with a 3-D simulation on a circular disc in yaw, confirms a good performance of the analytical 2-D model for this more complicated flow. Finally, a further extension of the model to simulate a coned disc is obtained using a simple superposition of the solution of two yawed discs with opposite yaw angles and positioned so the two discs just touch each other. Now the validation of the model is performed with results from axis-symmetric CFD simulations of an AD with a coning of both 20 and −20∘. In particular, for the disc coned in the downwind direction there is a very good correlation between the simulated normal velocity to the disc, whereas some deviations are seen for the upwind coning. The promising correlation of the results for the 2-D model in comparison with 3-D simulations of a circular disc with CFD for complicated inflow like what occurs at yaw and coning indicates that the 2-D model could form the basis for a new, consistent rotor induction model. The model should be applied along diagonal lines on a rotor and coupled to an angular momentum model. This application is sketched in the outlook and is a subject for future research.

A Simplified Numerical Model for the Prediction of Wake Interaction in Multiple Wind Turbines

In the design of wind energy farms, the loss of power should be seriously considered for the second wind turbine located inside the wake region of the first one. The rotation of the first wind-front rotor generates a high-vorticity wake with turbulence, and a suitable model is required in computational fluid dynamics (CFD) to predict the deficit of energy of the second turbine for the given configuration. A simplified numerical model based on the classical momentum theory is proposed in this study for multiple wind turbines, which is proposed with a couple of tuning parameters applied to Reynolds-averaged Navier-Stokes (RANS) analysis, resulting in a remarkable reduction of computational load compared with advanced methods, such as large eddy simulation (LES) where two parameters reflect on axial and rotational wake motion, simply tuned with the wind-tunnel test and its corresponding LES result. As a lumped parameter for the figure of merit, we regard the normalized efficiency on the kinetic power output of computational domain, which should be directed to maximize for the optimization of wind farms. The parameter surface is plotted in a dimensionless form versus intervals between turbines, and a simple correlation is obtained for a given hub height of 70% diameter and a fixed rotational speed tuned from the experimental data in a wide range.

A strongly-coupled model for flexible rotors

A fluid–structure model describing the equilibrium state of a flexible blade rotor with its own wake is derived for various external axial flow conditions. The model is based on three building blocks. The two-dimensional lifting-line theory is first used to compute the local aerodynamic loads and the blade circulation profile. The blade deformation is then obtained by solving the nonlinear equations for bending and twisting angles deduced from a one-dimensional beam model. Finally, the wake is obtained using a Joukowski model. In this wake model, the wake of each blade is modeled by two small-core-size counter-rotating vortices emitted from the rotor axis and blade tip. The velocity field induced by these vortices is computed using the Biot–Savart law. We show that, in the rotor frame, we can obtain a stationary vortex structure for almost any vertical flight regimes. This wake solution can then be used to compute the induced velocity in the rotor plane and apply the two-dimensional lifting-line theory again. By iterating a few times this loop, we converge toward a nonlinear solution of the problem for which the aerodynamics loads, blade deformation and wake structure are compatible.As illustration, this newly-developed model is applied to two rotors. We analyze the effects of the external wind conditions, geometry and material properties of the blades on the blade deformation and wake characteristics. We show that we can describe slow descending regimes for which the classical momentum theory does not apply.

Modelling yawed wind turbine wakes: a lifting line approach

Yawing wind turbines has emerged as an appealing method for wake deflection. However, the associated flow properties, including the magnitude of the transverse velocity associated with yawed turbines, are not fully understood. In this paper, we view a yawed turbine as a lifting surface with an elliptic distribution of transverse lift. Prandtl’s lifting line theory provides predictions for the transverse velocity and magnitude of the shed counter-rotating vortex pair known to form downstream of the yawed turbine. The streamwise velocity deficit behind the turbine can then be obtained using classical momentum theory. This new model for the near-disk inviscid region of the flow is compared to numerical simulations and found to yield more accurate predictions of the initial transverse velocity and wake skewness angle than existing models. We use these predictions as initial conditions in a wake model of the downstream evolution of the turbulent wake flow and compare predicted wake deflection with measurements from wind tunnel experiments.

An inconsistency in the actuator disc momentum theory

AbstractThe classical momentum theory for actuator discs with a constant normal load Δp addresses only the axial momentum balance. Now the radial balance is also derived, which, in the absence of a radial disc load, leads to the requirement that the integral of the static pressure along the disc centre line must be zero. The axial and radial balances appear to be mutually inconsistent: the axial balance provides the well‐known results, but the radial balance cannot be satisfied. The inconsistency can be removed by assuming that the actuator disc force field is composed of more than the constant normal load Δp. It is shown that the modelling by Lee and Greenberg (Journal of Fluid Mechanics 1984; 145: 287) of the load and flow of an actuator strip in hover implies the existence of a singular suction force at the edge. Their results deviate, without explanation so far, from the predictions of classical momentum theory. Addition of this edge force to the radial and axial momentum balances provides consistent results. A more general explanation or solution for the inconsistency is not yet are cable. Comparison the present inviscid analysis with a Navier–Stokes calculation by Mikkelsen et al. (Wind Energy 2001; 4: 121) shows that the integrated pressure along the disc centre line is indeed non‐zero, but that the radial balance is still satisfied. It is not clear why this deviates from the analysis. Copyright © 2004 John Wiley & Sons, Ltd.

Reply by the Authors to J.G. Leishman

Reply by the Authors to J.G. Leishman