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

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.[17] 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 [18].

$P(u(0,R))=\frac{1}{2}ρAu^3(0,R)Cp$

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

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.