11 Sep 2023
 | 11 Sep 2023
Status: this preprint is currently under review for the journal WES.

An analytical linear two-dimensional Actuator Disc Model and comparisons with CFD simulations

Helge Aagaard Madsen

Abstract. We present an analytical, linear solution for a two dimensional (2-D) actuator disc (AD) flow with one equation for the axial velocity and for the lateral velocity, respectively. Although it is a 2-D model we show in the paper that there is a good correlation to axis-symmetric and 3-D CFD simulations on a circular disc. The 2-D model has thus the potential to be the basis for a simple and consistent rotor induction model.

For a constant loading the axial velocity distribution at the disc is uniform as is 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 AD simulations. A comparison of the axis-symmetric AD CFD simulation of the axial velocity at the circular AD shows very good correlation with the axial velocity in the 2-D model when computed at a downstream distance of 5–10 % of the half disc width and the velocity scaled with a factor of 1.05 to account for the decay in velocity downstream the disc.

By a simple coordinate rotation the analytical solution is extended to a yawed disc with constant loading. Again a comparison with CFD, now a three-dimensional (3-D) simulation on a circular disc in yaw, confirms a good performance of the analytical 2-D model for this more complicated flow case for yaw angles up to 60°, however with increasing deviations above a yaw angle of 30°. The comparison comprises the normal velocity to the disc in the plane of the 3-D CFD simulations with the skewed inflow.

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 20° and -20°., respectively. 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 when coupled to an angular momentum model and applied along diagonal lines on a rotor. This application is sketched in the outlook and is subject for future research.

Helge Aagaard Madsen

Status: open (until 09 Oct 2023)

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  • RC1: 'Comment on wes-2023-99', David Wood, 19 Sep 2023 reply

Helge Aagaard Madsen

Helge Aagaard Madsen


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Short summary
We present a linear analytical solution for a two-dimensional (2-D) actuator disc (AD) for a plane disc, a yawed disc and a coned disc. Comparisons of the 2-D model with three-dimensional CFD AD simulations for a circular yawed disc and with an axis-symmetric CFD simulation of a coned disc show good correlation for the normal velocity component to the disc. This indicates that the 2-D AD model could form the basis for a consistent, simple new rotor induction model