(12)

Here   is the velocity at the edge of boundary layer, and ‘r’ is a roughness factor. For highly polished surfaces, r may be taken to be zero.

This model also predicts that transition has occurred if the laminar boundary layer separates, and forms a separation bubble near the leading edge of the rotor. 

Chen-Thyson Transition Model and Michel’s Criterion:

In this model, transition is said to occur at the chordwise location where the local Reynolds number based on the momentum thickness   is related to the Reynolds number based on length  by,

  (13)

In order to avoid an abrupt transition, Chen and Thyson recommend that the eddy viscosity be multiplied by the factor:

  (14)

Upstream of onset point of transition region,   is set to zero. The quantity G is computed from:

  (15)

The transition Reynolds number is defined as:

  (16)

And,

  (17)

It should be noted that the quantity Rx is based on the local freestream velocity (the magnitude of the vector sum of wind speed, induced velocity, and the blade velocity due to rotation  ). Thus, for wind turbines,

  (18)

The non-dimensional velocity   is computed as,

  (19)

Where r is the local radial distance from the hub, R is the tip radius, and   is non-dimensional x coordinate. Induced velocity vi is estimated to a first order from the momentum theory. The Reynolds number based on the momentum thickness is also computed using the free-stream velocity, not the boundary layer edge velocity.

Methodology for Yaw Simulation

A numerical procedure for modeling off axis wind (yaw) conditions has been developed. As in axial flow calculations, the yaw calculations only need to model the aerodynamics of a single blade. Other blades will experience the same load and flow pattern 1/N revolutions later, where N is the number of blades. For a three-bladed rotor the computational domain covers a 120 portion of the rotor disk at a specific running time. The present procedure thus retains the efficiency of the hybrid method even for yaw conditions. In contrast to the hybrid method, a full Navier-Stokes solver would require the modeling of all blades, greatly increasing the computational effort. 

When developing the first-principles based analysis for modeling rotors in cross flow, there are three kinds of non-axial flow (yaw) effects that should be addressed. First is the difference in the flow between the advancing and retreating sides due to the “edge-wise” velocity component in the plane of rotor disk. As shown in figure 1, the turbine blade experiences a higher relative velocity on the advancing side than on the retreating side. This fluctuation in velocity produces fluctuations in the blade loads, and the power generated.

The second effect that must be modeled is the skewness of tip vortex wake as shown in figure 2. This results in an azimuthally non-uniform induced flow field at the rotor plane. Furthermore, the vorticity strength in the wake will vary with time, as the loads on the blade vary with time. This is in contrast to axial flow, where the blade loading is independent of the azimuthal location of the blade.

Finally, the analysis must include aeroelastic deformation of the rotor blades, and blade teetoring and flapping motion, if any. Since the rotor tested by NREL uses stiff blades, the blade was assumed to be rigid without any cyclic pitching or flapping of the blades. 

The present Hybrid methodology has been extended to simulation of yaw conditions. This method uses a skewed wake geometry to model yawed flow conditions. The skew angle is determined by the ratio between the inflow velocity (which is a combination of the  induced velocity from momentum theory and the normal component of wind velocity), and the edgewise component of wind velocity.