1. P M V Subbarao Professor Mechanical Engineering Department
Vortex Model for the Aerodynamic Analysis of Vertical Axis Wind Turbines -II
P M V Subbarao
Professor
Mechanical Engineering Department
Generation of Vortices is the Action

2. The Ultimate Third Law
Change of circulation will be cancelled with a wake circulation.
Figure displays the counter acting circulation to create lift.
This circulation will be transported downwind, where its effect vanish with the distance.

3. Prandtl’s Lifting Line Theory
A finite blade may be represented by a single equivalent line of vorticity known as the "bound vortex," since it is in a sense bound to the blade.
The lift of a blade can be represented as a single circulation.
Following from that an entire blade can be projected as a single line of circulations.

4. Finer Implementation of Prandtl’s Lifting Line Theory
Many variations and applications of Prandlt theory are developed.
A number of individual lifting lines may be used, rather than a single line.
This approach adapts itself well to numerical solutions of nonuniform flows.
In current practice the vortex method can be implemented into two ways:
A single line or
As a mesh consisting of multiple panels.

5. Imagine a 3D world & Reality of 3D Imagination
Imagine that instead of just vortex lines, the 3D world can have vortex filaments.
Twisty vortex strings that produce infinite V -type velocities as distance from the filament goes to zero.
These twisty vortices are similar to tornados.
How to quantify?

6. Helmholtz Theories for a 2D world & Violation by 3D VAWT
Helmholtz's theorem #1: A vortex line cannot end in the fluid. It must form a closed path, end at a boundary, solid or free surface, or go to infinity.
Every blade is of finite length and its domain of influence in a 2D world is finite.
Helmholtz Theorem #2 : The circulation around a given vortex line (i.e., the strength of the vortex tube) is constant along its length.
Any blade used in a VAWT will have variable lift along the blade length and hence variable circulation.
Helmholtz Theorem #3: An inviscid fluid that is initially irrotational will remain irrotational for all time.
The wind flow is a viscous and turbulent flow.

7. The vortex lattice system for a blade element
Each of the blade element is a bound vortex lament called a Lifting line.
According to the airfoil theory , the lift can also be formulated in terms of the two-dimensional sectional
lift coefficient (Cl), the chord length (c) and the relative velocity (Vrel) as

8. The vortex lattice system for a blade element
The strength of the bound vortex can be expressed as

9. 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.

10. 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.

11. 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