1. An Analytical Model for A Wind Turbine Wake
P M V Subbarao
Professor
Mechanical Engineering Department
Quantification of Wind Turbine Wakes

2. Interference of Wakes in A Wind FarmV

3. Need for Wake Models
It is very important to determine the optimum distance between neighbouring turbines, when developing wind farms.
A too large distance will result in lower capacity of A wind farm and higher installation cost.
A too short distance will result in lower power per turbine and increased fatigue problems for turbines located in the wake of other turbines.
Fatigue is in particular a problem for the second turbine in a row.
The downstream turbine potentially facing risks, being hit by the tip vortices from the turbines located at the front of the wind farm.
It is important to determine the lifetime of the tip vortices and the parameters that govern their breakdown into small-scale turbulence.
This location is called as Fully Turbulent wake zone and no dange of high fatigue.

4. Signatures of Wind Turbine on Wind & Recovery : Betz Theory

5. Flow Diagnosis of WT Wake

6. Models for Wind Turbine Wake

7. Signatures of Wind Turbine on Wind & Recovery : Fluid Dynamic Theory
The flow field in immediate near wake is approximated by solving the filtered three-dimensional incompressible Navier–Stokes equations.
Define filtered velocity as:
Continuity equation
Navier–Stokes equations
Two body forces (fWT and fturb) are explicitly introduced in the simulations to model the effect of the wind turbine and atmospheric turbulence.

8. Real Signatures of Wind Turbine on Wind & Recovery HAWT : =6 & V=10m/s
Iso-surfaces of vorticity magnitude (ω =6)
Contours of vorticity magnitude in a cut-plane

9. Evolution of Stream-wise velocity & Turbulence in Wakes: HAWT : =6 & V=10m/s
Comparison of the (a) power and (b) thrust coefficients of computations (red line with triangles) with measurements (dots). (c) Time-averaged streamwise velocity profiles and (d) time-averaged streamwise turbulent stresses computed along a horizontal line through the rotor centre. The profiles are extracted when the modelled wind turbine operates at U0=10 m s−1 at its design performance of λ=6. Symbol – denotes the time-averaged quantity. The rotor is located at x/R=0.

10. Evolution of tip vortices & Instability
A WT introduces helical tip voritices into the wake.
Stability Analysis shows that helical tip vortices are inherently unstable and that they will break down at some distance behind the wind turbine.
The breakdown position of tip vortices is assumed to take place where linear amplification reaches its maximum. 
σ is the dimensional growth
rate of an instability

11. Breakdown Position
An Analytical expression for determining the axial position of a given amplitude amplification is derived as :
The breakdown position of tip vortices is assumed to take place where linear amplification reaches its maximum.
It is seen to be reached when the amplitude amplification equals the ratio between the original perturbation and the undisturbed wind speed, i.e. when

12. Evolution of Turbulence Level