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

The Vision of A Genious

The Sign Notation

Vortex Method of the Turbine The vortex method is commonly used for solving the flow of vertical axis turbines. The main idea behind the model is to use the vorticity as the discretization variable, instead of the velocity. The vorticity is obtained by taking the curl of the flow velocity: The vorticity equation is obtained from the curl of the Navier–Stokes equations: In the two-dimensional case becomes:

Free Vortex Model The vortices are generated by blades of a VWAT are defined as free vortices. The individual vortices are used as discretization variables. The flow velocity is obtained from the model as a superposition of the potential flow solution and the contribution from the vortices: As the turbine is not confined with walls and the blade is approximated as a single point, the potential flow solution  is equal to the asymptotic flow velocity V0.

Vortex Models Vortex models are a different kind of aerodynamic models with respect to the ones named earlier. Instead of being based on momentum and mass conservation, the vortex models use the principle of the Kelvin theorem. Circulation and vorticity are the two primary measures of action generated by a rotating VWAT in a wind flow. Circulation, which is a scalar integral quantity, is a macroscopic measure of rotation for a finite area of the fluid. Vorticity, however, is a vector field that gives a microscopic measure of the rotation at any point in the fluid.

Circulation The circulation, , about a closed contour in a fluid is defined as the line integral evaluated along the contour of the component of the velocity vector that is locally tangent to the contour.

Kelvin theorem & Kutta-Joukowski theorems The change of circulation in time has to be equal to zero. Kutta-Joukowski theorem states that the strength of each bound vortex is equal to the product of velocity Vrel, density ρ and circulation . Normally a change of the angle of attack will lead to a different lift force, which will mean that the circulation has to change as well. In order to compensate this change, another circulation has to be released in the form of a wake.

The Thought Process What would happen if a motionless VAWT is impulsively start in a 2D world of wind? There will be a generation of vorticity/circulation associated with the lift by the moving airfoil. But, in the initial motionless state  = 0 everywhere and D/Dt = 0, so we should have  = 0 everywhere for all time. What gives ?

Your Thinking is Right!!! For some region close to the wing to have positive circulation, some negative circulation path must exist around a vortex not bound to the airfoil. This other negative-G vortex is called a starting vortex. It has equal but opposite strength to the net vortex strength that’s bound to the moving airfoil. Does this really happen?