Double-Multiple Streamtube Model Overview

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Several aerodynamic prediction models are currently used for studying vertical axis wind turbines. The aerodynamic methods appropriated for the conception of wind turbines that are presently available to designers can be classified into three categories:

• CFD methods, based on the solution of the Navier-Stokes equations (viscous approach)
• Methods based on vortex models
• Methods based on momentum models

The main objective of each type of method is the calculation of the flow through the rotor and the determination of the aerodynamic loads and performance (torque and power) generated by the turbine.

Computational Fluid Dynamics (CFD) methods, based on the solution of the Navier-Stokes equations, are the most accurate (qualitatively and quantitatively) in terms of flow analysis, but their major drawback is the very high computational cost (require large computational resources and very long time to provide converged solutions).

The vortex-based models use vortex sheets or discrete vortices along with the Kutta Joukowski law to compute the flow field around the turbine. The time evolution of the vortex filaments that are shed from the blades and convect downstream is modeled and used to predict the induced velocity at various points in the flow field. The major advantage of the vortex models is their better accuracy in regard to the prediction of the blade forces than the one based on the streamtube models. However, these models require also considerable computer time, although a few orders of magnitude lower than that specific to the Navier-Stokes models.

Fast and accurate predictions of VAWT performances can be obtained using momentum models. Aerodynamic streamtube models are based on the conservation of momentum principle in a quasi-steady flow by equating the forces on the rotor blades to the change in streamwise momentum through the turbine. They include three types of analytical models.

• The single streamtube model, which assumes that the entire rotor, represented by an actuator disk, is enclosed in one streamtube.
• The multiple-streamtube model, in which the volume swept by the rotor is divided into a series of adjacent streamtubes.
• The upwind and downwind rotor halves model, which replaces the rotor by two actuator disks in series.

The first analytical model to consider a multiple­streamtube system divided into two parts was called the Double-Multiple Streamtube (DMS) model. This model uses two constant interference factors in the induced velocities and accounts for vertical variations in the freestream velocity.

This was further improved by considering the variation in the upwind and downwind induced velocities as a function of the azimuthal angle for each streamtube. This new model is referred to as the DMSV (Double-Multiple Streamtube Variable) model and is the one implemented in the CARDAAV computer code.

Figure: Principle of the Double-Multiple Streamtube model.

In the Double-Multiple Streamtube model, it is assumed that the vertical axis wind turbine can be represented by a pair of actuator disks in tandem at each level of the rotor. Different induced velocities are considered at the upstream and downstream halves of the volume swept by the rotor. The flow through the wind turbine is regarded as being subdivided into a large number of aerodynamically independent streamtubes. The effects of turbulence or gustiness are neglected and only the mean freestream velocity is considered. However, in order to evaluate the sensitivity of the Darrieus turbine to atmospheric wind shear, the common power law freestream velocity profile is used.

The flow in each streamtube is considered to be acted upon by two actuator disks: the first one representing the upwind half of the surface swept by the rotor blades, and the second one representing the downwind half of the rotor. As a result of the forces exerted by the actuator disks on the fluid, the fluid velocity changes along the streamtube. The velocity decreases in the flow direction so that the downwind component is less than the equilibrium velocity and the latter smaller than the upwind velocity. Along any given streamtube the fluid velocity variation is reproduced through five different velocities, related by two interference factors u and u' (for the upwind and downwind halves of the rotor, respectively).

By applying in each streamtube, the momentum equation to the control volumes containing the actuator disks, the forces on the disks and the induced velocities can be determined. The forces on the disks are drag-type forces and can be calculated by using the Blade Element Theory (BET), which involves the static aerodynamic drag and lift coefficients of the blade airfoil. During the rotation of the blade the local speed and angle of attack vary and usually surpass on certain intervals the static stall angle of incidence. Therefore, the airfoil static aerodynamic coefficients are used only below this angle of incidence and dynamic stall models are employed to calculate these coefficients beyond stall (in dynamic stall regime).