1. P M V Subbarao Professor Mechanical Engineering Department
Vortex Lattice Method
P M V Subbarao
Professor
Mechanical Engineering Department
The Field Generated by A VAWT is A System of Vortex Filaments

2. The Shed Vortex
The relationship between the strength of the bound vortex of blade element i at time step j and the strength of the shed vortex from this element is given by the Kelvin's law.
This Law requires that the circulation around any closed curve remains constant over time.
It satisfies the following equation
Where, si,j is the strength of the shed vortex at element i and time step j.

3. The journey of Shed Vortices
In reality, shed vortices are released continuously at the trailing edge.
The newest shed vortices are placed at x % from the current location of the trailing edge to the location of the trailing edge at the previous time step.
The other shed vortices are transported by the calculated velocity.

4. Occurrence of Trailing Vortices
In order to satisfy Helmholtz's theorems, a trailing vortex must emit from the blade at each blade element.
This emission occurs at a location, where the strength of each discrete bound vortex changes to generate the varying circulation a (lift) along the span.
The strength of a trailing vortex is the difference between the strengths of the bound vortices from where it emits.
It can be seen in

5. The velocity field wrt to Rotating VAWT
The relative wind velocity at the turbine blades is a vector sum of the flow velocity at the blade due to its own motion, and the flow velocity.

6. Generation of Velocity Field : A Primary Reaction to Vortex Shedding
In the Lagrangian formulation, the vortices are allowed to drift with the flow velocity.
Neglecting the viscosity outside the boundary layers of the blades, the vortices are propagated according to:
The velocities at each vortex position has to be ....
Evaluate efficiently by using a suitable method.

7. Biot–Savart Law The contribution from the infinitesimal length vortex
The contribution from the finite length vortex
The Gaussian kernel
Here, r is the position
 is the circulation of vortex
r denotes the complex conjugate of r

8. Velocity Field generated by finite length Vortices
The velocity at an arbitrary point P induced by a single vortex element (AB) is obtained by the integration of Biot-Savart law. ds
The simplified form of this integral is:
Where, h is the perpendicular distance from the arbitrary point to the vortex lament.
⃗r0, ⃗r1 and ⃗r2 follow the direction defined in the figure.
In above equation , when h approaches zero, the induced velocity becomes infinite which is inconsistent with the physical reality.

9. Viscous Treatment to Velocity field
In reality, viscous effects are encountered that reduce the velocity to zero at the center of the vortices.
To model this behaviour, many empricial models are suggested in literature.
To model this behaviour, the Vatistas viscous core model is implemented to account for viscous effects of the vortex core.

10. Real Velocity Field Generated by Finite Length Vortices
The equation for velocity field to include the viscous effect.
Here, rc is the viscous core radius, equal to approximately 5-10% of the chord length of the blade.

11. Computation of Total Velocity Field
The velocity due to VAWT blades at each vortex marker at each time step is calculated by summing the induced velocity calculated from all shed vortex laments and trailing vortex filaments.
The net velocity field is obtained by adding the free stream velocity.
During every time step the positions of all vortex laments are updated using a suitable integration method.
Adams-Bashforth integration method.
Location of ith Vortex Marker

12. Rotor and blade element coordinate system
Rotor coordinate system
Top view
Blade element coordinate system

13. Blade loads
Cn and Ct can be expressed in terms of lift and drag coefficients (Cl,Cd) and the relative velocity (Vrel).
Normal Force Coefficient
Tangential Force Coefficient
The torque coefficient of one blade element is calculated as
Where, A is the total frontal area of the rotor. le is the length of the blade element.
c is the chord length.

14. Performance
Then, the average power coefficient (CP) for the entire rotor during one revolution is given as
Where,
NI is the number of time steps per revolution
NE is number of blade elements and
R is the tip-speed ratio.

15. Convergency investigation : H-type vertical axis: NACA0012 : Re=40000

16. Verification of VLM

17. Performance of Single Blade VAWT

18. Performance of two Blade VAWT

19. Performance of Three Blade VAWT

20. Dynamic vortex shedding for a straight-bladed vertical axis turbine