We introduce an improved formulation of the double-multiple streamtube (DMST) model for the prediction of the flow quantities of vertical axis wind turbines (VAWT). The improvement of the new formulation lies in that it renders the DMST valid for any induction factor, i.e., for any combination of rotor solidity and tip speed ratio. This is done by replacing the Rankine–Froude momentum theory of the DMST, which is invalid for moderate and high induction factors, with a new momentum theory recently proposed, which provides sensible results for any induction factor. The predictions of the two DMST formulations are compared with VAWT power measurements obtained at Princeton's High Reynolds number Test Facility, over a range of tip speed ratios, rotor solidities, and Reynolds numbers, including those experienced by full-scale turbines. The results show that the new DMST formulation demonstrates a better overall performance, compared to the conventional one, when the rotor loading is moderate or high.

The study of vertical axis wind turbines (VAWTs) has received renewed attention in the last decade. There were noticeable research efforts devoted to VAWTs from the mid-1970s to the mid-1980s, primarily led by Sandia National Laboratories and NASA

An important prerequisite for the successful realization of wind farms is the development of engineering flow models that combine low computational cost and sufficient accuracy, so that they can be used as design and optimization tools. In the case of HAWTs, blade element momentum (BEM) algorithms have been shown to fulfill these conditions and have subsequently become standard aerodynamic tools of the HAWT industry. A significant amount of research has been devoted to the development of analogous models for the case of VAWTs.

This is not a trivial matter, however, as the aerodynamics that govern VAWTs are inherently more complex than HAWTs. The effective angle of attack experienced by a VAWT blade section is not constant, as in the case of HAWTs, but depends on the blade's instantaneous orbital position as well as on the tip speed ratio (ratio of turbine tip to free-stream velocities). In addition, at relatively low tip speed ratios a blade section may experience large and rapid variations in effective angle of attack over the course of one rotation cycle. This leads to the highly unsteady and nonlinear flow phenomenon known as dynamic stall

Despite these inherent complexities, a number of simplified analytical predictive methodologies have been proposed over the years (e.g., vortex, cascade, fixed-wake, streamtube approaches;

Nevertheless, such treatment of the rotor fails to model other important aspects of the flow physics: DMST assumes zero expansion of the streamtubes, and it neglects the wake–blade interaction and the effect of the downstream half of the rotor on the upstream half. For these reasons, DMST algorithms are known to fail to accurately capture the local aerodynamic loads on the rotor

Despite its usefulness, however, DMST is inapplicable to highly loaded VAWTs, i.e., characterized by high values of rotor solidity and tip speed ratio. That is because rotor loading correlates with the induction factors of the streamtubes. At an induction factor of 50 % the core of the DMST model, the classical momentum theory of Rankine–Froude breaks down, predicting zero wake velocity and infinite wake width. For even larger induction factors the wake velocities and wake widths assume nonphysical negative values, while drag is greatly underpredicted

In HAWT BEM models, this inconsistency of the momentum theory is rectified by using empirical values for the drag, the so-called Glauert correction

In this article, we propose a resolution to this issue by substituting the Rankine–Froude momentum theory of the DMST with the momentum theory proposed by

To quantify the accuracy of the proposed methodology, we compare predictions of a conventional DMST model equipped with both the momentum theories of Rankine–Froude and

The structure of the article is as follows: the most relevant steps of the DMST model are outlined in Sect.

In a DMST model, the rotor is divided into a front (upstream) and rear (downstream) half-cycle. The flow through a rotor of radius

Schematic diagram of DMST geometrical configuration with

Using the above simplified flow description, the DMST model is able to provide predictions based on two methodologies: the momentum theory and the aerodynamic load analysis.

The momentum theory aspect of conventional BEM models (including the DMST) builds upon the classical Rankine–Froude actuator disc theory

The actuator disk theory assumes potential flow everywhere apart from the immediate vicinity of the disc, a nonrotating actuator disc, and no base suction in the wake. The latter assumption implies that the wake can be treated using potential flow theory up to a far point where the pressure becomes equal to the free-stream pressure, i.e., the boundary condition of the wake becomes

By applying mass and momentum balance to a control volume enclosing the actuator disk and normalizing the resulting drag with the term

Note that for

Porous plate drag coefficient versus plate open area ratio,

Equations (

The drag coefficient is predicted to be

which, as shown Fig.

Figure

If we express

Equations (

Velocity and force diagram on a top-down view of a VAWT rotor.

The other aspect of the BEM method deals with the local aerodynamics of a blade segment (airfoil). Figure

The relative velocity experienced by the blade,

It is noted that

The drag and lift coefficients of the airfoils can be combined to yield the local tangential and normal force coefficients

Finally, the torque

The DMST model calculates the induction factor

The cycle-averaged thrust coefficient corresponding to

By equating Eq. (

After the induction factor

In order to compare the effect of the two momentum theories in the DMST, the power of a VAWT model was tested at Princeton's High Reynolds number Test Facility (HRTF). The HRTF is a variable-pressure, low-velocity wind tunnel that can be operated at static pressures of up to

The VAWT models (see Fig.

In a DMST algorithm, the number of streamtubes,

Normalized error as a function of the number of streamtubes shown for both conventional and current DMST models, for a three-bladed turbine at

From Fig.

Figure

Comparison of current

As the tip speed ratio increases, the current DMST model provides power predictions which are in better agreement with the measurements, compared to the conventional one. The reason for this improvement can be seen if we compare the contributions of the front and rear disks for each model. As expected, the front power contributions are very similar, since the input velocity

To assess this difference in wake velocity in the above case, in Fig.

Front half-cycle wake velocity profile,

Figure

Measured (diamonds) and predicted (current model: solid lines; conventional model: dashed lines) power coefficients,

In Fig.

Measured (diamonds) and predicted (current model: solid lines; conventional model: dashed lines) power coefficients,

A double-multiple streamtube (DMST) model for vertical axis wind turbines (VAWT) is presented, where the classical Rankine–Froude momentum theory is replaced with the momentum theory of

The predictions of the two DMST formulations were compared with VAWT measurements acquired at Princeton's HRTF facility, covering a range of rotor solidities, tip speed ratios, and Reynolds numbers. The data represent both lightly and heavily loaded rotors, in dynamically similar conditions to field-scale VAWTs. The results showed that the new momentum theory improves the predictions of the DMST, especially as the tip speed ratio increases. It was found that this improvement is explained by a more realistic representation of the wake velocities, or equivalently input velocities to the second rear part of the rotor, from the new momentum theory.

Despite its simplicity and lack of certain flow physics, the DMST model proved reliable in its predictions of the mean power coefficient of the VAWT for the tested range of parameters. This could be in part due to the fact that our tested tip speed ratios are rather low, while DMST inaccuracies tend to emerge at high tip speed ratios where friction and wake effects are more significant

Data can be provided upon request. Please contact Anis A. Ayati (ayati.anis@gmail.com).

AAA and KS wrote the article, implemented the modeling, and performed the simulations. MAM and SD performed the experiments and data analysis. MH supervised the research.

The authors declare that they have no conflict of interest.

Anis A. Ayati gratefully acknowledges the support of the Akademia program at the Faculty of Mathematics and Natural Sciences, University of Oslo.

This research has been supported by the National Science Foundation under grant CBET-1652583 (Program Manager Ron Joslin).

This paper was edited by Raúl Bayoán Cal and reviewed by Hubert Branger and two anonymous referees.