# The Effects of Fluctuating Air Streams on the Output of a Wind Turbine

The Effects of Fluctuating Air Streams on the Output of a Wind Turbine
Mechanical Engineering Department, The University of Jordan, Amman 11942, Jordan

Corresponding Author Email:
Bnanalgallab@gmail.com
Page:
27-34
|
DOI:
https://doi.org/10.18280/ejee.210105
5 December 2018
|
Accepted:
1 February 2019
|
Published:
15 April 2019
| Citation

OPEN ACCESS

Abstract:

The objective of this work is to model the fluctuating air stream potential of producing wind energy and study different operating parameters of wind turbines, by using conservation principles continuity and momentum, with wind turbine equations. A combination of two numerical methods which include Runge-Kuttha technique with shooting technique were utilized using MATLAB to obtain numerical solutions. It is found that the increasing of corrugated surface amplitudes decreases velocities inside boundary layers and reduce power output compared to those obtained by Blasius or traditional Pet theory.

Keywords:

boundary layer, electricity production, surface topography, velocity fluctuations, wind energy

1. Introduction
2. Math

Considering steady state wind loads for short periods can incorporate steady state wind model. In addition, the involving of incompressible wind model is usual; this is due to small forces of wind compared to a ramjet or turbojet.

2.1 Governing equations

The corrugated surface can be described by the following equation:

$y=σ(x)=asin(\frac{2πx}{λ})$ (1)

where a*is the amplitude of the wavy surface. It may take values from zero to one and even greater than 1 because it reflects -in physics- the height of hills or buildings relative to distance between them. If a is greater than one; forests of trees near each other are considered, if it is smaller than one; distributed or extended hills are considered, λ is the characteristic length associated with the wave and is the surface profile of the tidal waves.

The laminar boundary-layer governing equations are:

The continuity equation:

$\frac{∂u}{∂x}+\frac{∂v}{∂y}=0$ (2)

The x-momentum equation:

$u\frac{∂u}{∂x}+v\frac{∂u}{∂y}=-\frac{1}{ρ}\frac{∂p}{∂x}+v(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2})$ (3)

The y-momentum equation:

$u\frac{∂v}{∂x}+v\frac{∂v}{∂y}=-\frac{1}{ρ}\frac{∂p}{∂y}+v(\frac{∂^2v}{∂x^2}+\frac{∂^2v}{∂y^2})$ (4)

This is a system of elliptic, nonlinear partial differential equations of the second order. They are accompanied by initial and boundary conditions. Here u, v are the axial and normal velocity components in the x and y directions respectively, p is the fluid pressure and v is the Kinematic viscosity which equals ($v=\frac{u}{ρ}$).

The corresponding boundary conditions for the present problem are:

$u=0$, $v=0$ at $y=0$

$u=u∞$ at $y→∞$ (5)

Here, the following non-dimensional variables are:

$y*=y-σ(x)$   $x*=\frac{x}{λ}$

$a*=\frac{a}{λ}$    $v*=v-λxu$     $u*=u$ (6)

The flow is governed by the following dimensionless form of the governing equations after using the proper non-dimensional transformation variables (6) which can be written as:

The x-momentum equation:

$u*\frac{∂u*}{∂x*}+v*\frac{∂u*}{∂y*}=\frac{1}{ρ}\frac{∂p}{∂y*}σx+(1+σx^2)\frac{∂^2u*}{∂y*^2}$ (7)

The y-momentum equation:

$σx(u*\frac{∂u*}{∂x*}+v*\frac{∂u*}{∂y*})=-\frac{1}{ρ}\frac{∂p}{∂y*}+vσx(1+σx^2)\frac{∂^2u*}{∂y*^2}-σxxu*^2$ (8)

By Combining equations (7) and (8) and eliminating $\frac{∂p}{∂y}$ from them, the resultant equation is expressed as follows:

$u*\frac{∂u*}{∂x*}+v*\frac{∂u*}{∂y*}=v(1+σx^2)\frac{∂^2u*}{∂y*^2}-\frac{σxσxxu*^2}{1+σx^2}$ (9)

where, $σx=\frac{dσ(x)}{dx}$, $σxx=\frac{d^2σ(x)}{dx^2}$

Then, the following transformations are introduced and the use of well-known relations between velocity components u, v and stream function ψ is integrated, in order to reduce the boundary layer equation to an equation with a single dependent variable by changing the partial differential equation (PDE) to ordinary differential equation (ODE), let:

$η=\sqrt{\frac{u∞}{vx}}y$, $ψ(x,y)=\sqrt{u∞vx}f(η)$ (10)

where, η is a similarity variable and ψ is the stream function. The transformations given in Equation (10) are introduced into Equation (9), the momentum equation is transformed to the following form:

$(1+σx^2)f'''+\frac{1}{2}ff''-\frac{σxσxx}{1+σx^2}xf'^2=0$ (11)

The corrugated surface function transformation to non-dimensional derivatives is represented as follows:

$y=σ(x)=asin\frac{2πx}{λ}$

$σ*(x)=σ(x)$ (12)

$σx*=σx$ (13)

$\frac{σxx*}{λ}=σxx$ (14)

That results in the following resultant equation:

$f'''-\frac{σx*σxx*}{(1+σx*2)^2}x*f'^2+\frac{1}{2(1+σx*^2)}ff''=0$ (15)

The boundary conditions associated by equation (2.15) take the following form:

$f(η)=0$, $f'(η)=0$ at $η=0$

$f'(η)=1$ at $η=∞$ (16)

The following equations are used to calculate $η_1$, $η_2$ at specific Rex, y and x.

$η_1=\frac{y_1}{x}\sqrt{Rex}$ (17)

$η_2=\frac{y_2}{x}\sqrt{Rex}$ (18)

where, x* is the distance along corrugated surface as dimensionless distance, it describes the shape of corrugated surface as it ends on x*, it also ranges from (0-1), x is the dimensional distance on earth, $y_1$ equals the hub height minus the turbine radius, and $y_2$ is the hub height plus the radius.

2.2 Power calculations

The actual wind power extracted by the rotor blades can be calculated under the effect of a corrugated surface using the following resultant formula.

The general power equation is represented below .

$P(u(0,R))=\frac{1}{2}ρAu^3(0,R)Cp$

$P=\int_A\frac{1}{2}ρu^3CpdA=\int^{y_2}_{y_1}\frac{1}{2}ρu^3CPdy$ (19)

The potential power is calculated by considering the integration on turbine blades center to outside radius where area is (2πrdr) and the radius changes from 0 to R Which can be expressed as the area enclosed by $f'$(η) curve and limited by the turbine swept area.

$u=u∞f'(η)η=\frac{y}{x}\sqrt{Rex}\to dη=\frac{dy}{x}\sqrt{Rex}$

$P=\int^{η_2}_{η_1}\frac{1}{2}ρu∞^3\frac{x}{\sqrt{Rex}}f'^3(η)dη$  (20)

$P\frac{\sqrt{Rex}}{\frac{ρu∞^3x}{2}}=\int^{η_2}_{η_1}f'^3(η)dη$

The non-dimensional wind power is expressed as follows:

$P*=\int^{η_2}_{η_1}f'^3(η)dη$ (21)

Another power terms must be introduced here ̏ pet power ̋ as the power calculated by substituting freestream velocity outside BL, it is the power of the freestream without involving BL and the associated friction effects.

$Ppet=0.5*ρ*U∞^3*Cp$ (22)

That results in the upcoming non-dimensional term of power: $P*=\frac{P}{Ppet}$

And ̏ Blasius power ̋ when talking about two-dimensional steady flow of an incompressible constant property fluid over a semi-infinite flat plate. So calculating power ratio according to Blasius will have the following formula:

$P*=\frac{P}{PBlasius}=\frac{\int^{η_2}_{η_1}f'^3(η)dη}{\int^{η_2}_{η_1}f'^3(η)dη}$ (23)

Tip-Speed-Ratio (TSR) can be calculated based on radius (R), rotational speed or rotor tip speed (ω) of an elected wind turbine and wind velocity. The following formulas will simplify the equations for calculating TSR.

Wind velocity $U=\int^{η_2}_{η_1}f'(η)dη$

$TSR=\frac{w*R}{U}$ (24)

3. Figures

## figure_1.png Figure 1. The dimensionless velocity f ʹ(η), sheer stress f ʹʹ (η) and stream function f(η) profiles when x*=0.5 and a*=0

## figure_2.png Figure 2. The dimensionless velocity f ʹ(η), sheer stress f ʹʹ (η) and stream function f(η) profiles for different values of a*=(0,0.5,1.5) and x*=0.5

## figure_3.png Figure 3. The dimensionless velocity f ʹ(η), sheer stress f ʹʹ (η) and stream function f(η) profiles for different values of a*= (0.2, 0.75) and x*=0.5

## figure_4.png Figure 4. The dimensionless velocity f ʹ(η), sheer stress f ʹʹ (η) and stream function f(η) profiles with respect to (η) when x*= (0.25, 0.75) and a*=0.1

## figure_5.png Figure 5. The dimensionless velocity f ʹ(η), sheer stress f ʹʹ (η) and stream function f(η) profiles with respect to (η) when x*= (0, 0.5, 1) and a*=0.1

## figure_6.png Figure 6. Shear stress f ʹʹ (η) for different values of x* and a*=0.1

## figure_7.png Figure 7. Velocity profile f ʹ(η) for different values of a* and x*=0, 0.5, 1

## figure_8.png Figure 8. Shear stress f ʹʹ (η) for different values of aand x*=0, 0.5, 1

## figure_9.png Figure 9. Wall shear stress for different values of aand x*

## figure_10.png Figure 10. Wind velocity versus x* for different values of a*where Hhub=100, R=10, $u_∞=5$ and Rex= 4*105

## figure_11.png Figure 11. Wind velocity versus afor different values of x* where Hhub=100, R=10, $u_∞=5$

## figure_12.png Figure 12. Wind velocity for different values of Rex and $u_∞$, where Hhub=100, R=10, x*=0.5 and a*=0.5

## figure_13.png Figure 13. A representation of BL thickness based on the value of (y) at which f ʹ(η) =1

## figure_14.png Figure 14. Wind power ratio versus wave amplitude at different Reynolds numbers

## figure_15.png Figure 15. Wind power ratio against different turbine heights and Rex at a*= 0.1

## figure_16.png Figure 16. Power ratio versus height for the different amplitudes (a*=1.7, 0.9, and 0.01) with Rex =4*105

## figure_17.png Figure 17. Power ratio versus radius of wind turbine blades for the following amplitueds (a*= 0.5, 1.5)

## figure_18.png Figure 18. Power ratio versus Tip-Speed-ratio at different amplitudes

## figure_19.png Figure 19. Power ratio (P/Ppet) versus Reynolds number for different wave amplitudes where x*= (0, 0.5, 1), R=10, $u_∞=1$ and Hhub=100

## figure_20.png Figure 20. Power ratio (P/Ppet) versus Reynolds number for different x* and a* where R=10, $u_∞=1$ and Hhub=100

## figure_21.png Figure 21. Wind power ratio (P/P pet) against wave amplitude for different Rex

## figure_22.png Figure 22. Power output (P/P pet) versus amplitude for different power coefficients

4. Conclusions

Flow over a corrugated surface can affect the hydraulic characteristic of the flow through BL, it is found that the velocity decreases as amplitude values increase, since the velocity is affected slightly when a* is quite small by the roughness of the surface, and almost nothing is hindering wind speed. So, wind speed is smallest when amplitude or roughness of the surface is the most, this is due to higher shear stresses between fluid layers that comes from higher stresses at corrugated surface,

The larger the amplitude is the lesser the velocity will be; the grater the values of shear stress, stream function and wall shear stress (τ) will become. As τ reflects the tangential force extended by the flowing fluid resulted by the friction effects on the surface and between fluid layers.

As moving away from the ground, surface effect diminishes rapidly and vanishes completely outside the BL, where the flow velocity returns to the freestream velocity and becomes f ʹ(ƞ)=1. It is clear that the larger the amplitude is; the grater the BL thickness will be.

The wind power expressed based on Blasius power and Pet power, decreases when considering a corrugated surface compared to that flat one or that stream flow which propagate far away of the surface outside BL. As amplitude values getting larger, the amount of harvested power is lesser, wind power increases as the height of wind turbine increases, and decreases as the radius of wind turbine decreases.

It is found that as amplitude increases the wind velocity decreases, as a result, the wind power decreases as well as Tip-Speed-Ratio. Wind power output is grater when TSR is not so large nor so small.

It is clear that higher Rex value means high inertia forces compared to viscous forces, that in turn means that the wind velocity increases when Rex value is larger as well as wind power. It also refers to a small boundary layer thickness, which limits the viscous effects on fluid layer in a small region near the corrugated surface. It is also obvious that the more the value of u∞, the larger the values of wind velocity, where u∞ is calculated based on the value of Rex, x and v.

Since the power coefficient of a turbine is a measure of however efficiently the turbine converts the energy within the wind into electricity. The less efficient a turbine is the less of the kinetic energy is converted into electricity.

The following are suggestions for further investigations based on the previous work:

1) Finding the effect upon wind turbines, wind farms or wind energy generation considering turbulent conditions when Rex>5*105.

2) Studying the effect of fluctuating wind steams on the structure of wind turbine and its base.

3) Integration of corrugated surface function through calculations can be introduced by taking different functions instead of sine functions to describe shape of obstacles such as cosine functions or any combination of both.

Nomenclature
 A Swept rotor area in square meter m2 a* Amplitude-to-length ratio of the wavy surface (wave amplitude) Dimensionless Cp Power Coefficient of wind turbine Dimensionless Hhub Hub height of the wind turbine M P Wind power in watt W P Pressure anywhere in the field N/m2 R Radius of wind turbine blade M U, V Dimensionless velocity components along the (x, y) axes Dimensionless X, Y Cartesian coordinate component in the direction along and normal to the tangent of the surface M Ω Rotor speed Rad/s β (°) Pitch angle Degrees (°) Λ Characteristic length associated with the wavy surface M v Kinematic viscosity m2/s

'              Differentiation with respect to η

f '           First order component of f

F            Zeroth order component of f

*            Dimensionless quantities

References

 Hébrard E, Listowski C, Coll P, Marticorena B, Bergametti G, Määttänen A, Montmessin F, Forget F. (2012). An aerodynamic roughness length map derived from extended Martian rock abundance data. Journal of Geophysical Research Atmospheres 117: E04008. https://doi.org/10.1029/2011JE003942

 Desmond CJ. (2014). The consideration of forestry effects in wind energy resource assessment. https://doi.org/10.13140/RG.2.2.32022.04166.

 Cabezon D, Hansen KS, Barthelmie RJ. (2009). Analysis and validation of CFD wind farm models in complex terrain Wakes induced by topography and wind turbines. Proc. EWEC.

 István L, Csákberényi-Nagy G, Túri Z, Kapocska L, Tóth T, József Barnabás T. (2014). Analysis of factors affecting wind-energy potential in low built-up urban environments. Conference: Air and Water Components of the EnvironmentAt: Cluj-Napoca, Romania. https://doi.org/10.13140/2.1.4231.5200

 Sengupta TK. (1984). Turbulent boundary layer over rigid and moving wavy surfaces. Project: Drag Reduction. https://doi.org/10.13140/RG.2.2.35849.75366

 Cattin R, Schaffner B, Kunz S. (2006). Validation of CFD wind resource modeling in highly complex terrain. Meteotest, Fabrikstrasse 14, 3012 Bern, Switzerland, 1-10.

 Mohr M, Jayawardena W, Arnqvist J, Bergström H. (2014). Wind energy estimation over forest canopies using WRF model. Project: Forest wind - Wind Power in Forests II.

 Drew DR, Barlow JF, Lane SE. (2013). Observations of wind speed profiles over Greater London, UK, using a Doppler lidar. Journal of Wind Engineering and Industrial Aerodynamics. https://doi.org/10.1016/j.jweia.2013.07.019

 Segalini A, Fransson JHM, Alfredsson PH. (2013). Scaling laws in canopy flows: A wind-tunnel analysis. Boundary-Layer Meteorology 148(2). https://doi.org/10.1007/s10546-013-9813-2

 Okulov VL, Naumov I, Mikkelsen R, Sørensen JN. (2015). Wake effect on a uniform flow behind wind-turbine model. Journal of Physics, Conference Series 625(1). https://doi.org/10.1088/1742-6596/625/1/012011

 Odemark Y, Segalini A. (2014). The effects of a model forest canopy on the outputs of a wind turbine model. Journal of Physics Conference Series 555(1): 012079. https://doi.org/10.1088/1742-6596/555/1/012079

 Al-Abadi A, Youjin K., Ertunç. Ö, Delgado A. (2016). Turbulence impact on wind turbines: experimental investigations on a wind turbine model. Journal of Physics Conference Series 753(3). https://doi.org/10.1088/1742-6596/753/3/032046

 Juan A, Tameemi S. (2013). Wind energy analysis wind energy modelling. Conference: Energy Systems Modelling. https://doi.org/10.13140/2.1.3075.1047

 Agravat S, Manyam NVS, Mankar S, Tirumalachetty H. (2015). Theoretical study of wind turbine model with a new concept on swept area. Energy and Power Engineering 7(04): 127-134. https://doi.org/10.4236/epe.2015.74012

 Mitiku T, Manshahia MS. (2018). Project: Energy Harvesting: Modelling and Optimization of Renewable Energy Systems Using Computational Intelligence.