Wind Turbine Performance under Fluctuating Pressure Gradient of Laminar and Turbulent Air Flows

Air or any other fluid moves according to three fundamental laws and principles of physics: conservation of mass, conservation of energy, and conservation of momentum [7].

Conservation of mass: states that an air mass is neither created nor destroyed. From this principle, it follows that the amount of air mass coming into a wind turbine should equal to that leaving it. In this study, the air was assumed to be incompressible, which mean that the change of air density that occurs as a result of pressure loss and flow is negligible.

Conservation of energy: states that the total energy of an isolated system remains constant, which means that the energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. This is considered the basis of one of the main expressions of aerodynamics, the Bernoulli equation. Bernoulli's equation in its simple form shows that, for an elemental flow stream, the difference in total pressures between any two points is equal to the pressure loss between these points.

Conservation of momentum is based on Newton's second law; that a body maintains in its state of rest or uniform motion unless compelled by another force that changes its state.

These fundamental principles may be portrayed in terms of mathematical equations; that are expressed, in one form, as unsteady Navier-Stokes equations. Equation (1) indicates that the amount of air mass coming into a wind turbine is equal to that leaving, Equation (2) indicates that the inertia forces is equals to the difference between viscous forces and the integration of the fluctuating pressure gradients of Wind.

incompressible parallel flow, unsteady subjected fluctuation pressure gradients assumption has been made, in order to represent wind velocity and power changes, two flow conditions are considered: one represents laminar flows near cut in velocity and second represents turbulent flows near rated speed velocity.

The dimensionless eddy viscosity (b = 1 + ɛm+) represents laminar flows for b = 1 and turbulent flows for b greater than 1, the value of dimensionless eddy viscosity (ɛm+ = ɛm /ν) are in the range of 0.2 * 10⁵ [8].

$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=0 \quad \frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{v}}{\partial \mathrm{y}}=0$ (1)

$\frac{\partial \mathrm{u}}{\partial \mathrm{t}}=-\int \frac{1}{\rho} \frac{\partial \mathrm{P}}{\partial \mathrm{x}}+v \frac{\partial}{\partial \mathrm{y}}\left(\mathrm{b} \frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)$ (2)

The fluctuating pressure gradient is assumed to be:

$-\frac{1}{\rho} \frac{\partial p}{\partial x}=p_{0} \cos (n t)$        (3)

where, Po is the air pressure gradient fluctuation amplitude and n is the number of air pressure gradient fluctuation cycles.

The above equations are a system of second order nonlinear differential equations, which are accompanied by initial and boundary conditions. Here u is the axial velocity, p is the Air pressure, v is the kinematic viscosity which equal to, $\rho$  is the density of air and b indicate the air turbulence level.

The corresponding initial and boundary conditions are:

$\mathrm{u}=u_{\infty}$ at $\mathrm{t}=\mathrm{o}$

$\mathrm{u}=0,$ at $\mathrm{y}=0$ for $\mathrm{t}>0$  (4)

$\mathrm{u}=u_{\infty},$ when $\mathrm{y} \rightarrow \infty$ for $\mathrm{t}>0$

$\hat{\mathrm{u}}$ Variable has been defined as the summation of the velocity and the integration of the fluctuating pressure gradients of the Wind:

$\hat{\mathrm{u}}=\mathrm{u}+\int \frac{1}{\rho} \frac{\partial \mathrm{P}}{\partial \mathrm{x}} d t$ (5)

Substituting Equation (5) into Equation (2), one gets.

$\frac{\partial \hat{\mathrm{u}}}{\partial \mathrm{t}}=v \frac{\partial}{\partial \mathrm{y}}\left(\mathrm{b} \frac{\partial \hat{\mathrm{u}}}{\partial \mathrm{y}}\right)$     (6)

The following non-dimensional variables are adopted to rewrite the governing equation in a proper non-dimensional form:

$u^{*}=\frac{\hat{\mathrm{u}}}{u_{\infty}} \quad \mathrm{y}^{*}=\frac{y u_{\infty}}{v} \quad \mathrm{t}^{*}=\frac{t}{t_{o}}$ (7)

Hence,

$t_{o}=\frac{v}{u_{\infty}^{2}}$    (8)

Substitute above Equations (7-8) in Equations (6) one gets:

$\frac{\partial\left(\mathbf{u}^{*}\right)}{\partial\left(\mathbf{t}^{*}\right)}=\mathbf{b} \frac{\partial^{2}\left(\mathbf{u}^{*}\right)}{\partial\left(\mathbf{y}^{*}\right)^{2}}$      (9)

The following transformation is introduced to reduce the boundary layer equation to an equation with a single independent variable which is, in other words, changing the partial differential equation (PDE) to an ordinary differential equation (ODE):

Let,

$\eta=\frac{A \mathrm{y}^{*}}{t^{*} n} \quad f(\eta)=\mathrm{u}^{*}$    (10)

Hence,

$\frac{\partial(\eta)}{\partial\left({y}^{*}\right)}=\frac{A}{t^{* n}} \quad \frac{\partial(\mathfrak{n})}{\partial\left(\mathfrak{t}^{*}\right)}=-n \mathrm{A} \mathrm{y}^{*} t^{*^{-}(1+n)}$   (11)

Assuming that

$f(\eta)=\mathrm{u}^{*}$     (12)

And hence,

$\frac{\partial\left(\mathbf{u}^{*}\right)}{\partial\left(\mathbf{y}^{*}\right)}=\frac{\partial(f)}{\partial(\mathbf{\eta})} \frac{\partial(\mathbf{\eta})}{\partial\left(\mathbf{y}^{*}\right)}=f^{\prime} \frac{A}{t^{* n}}$

$\frac{\partial^{2}\left(u^{*}\right)}{\partial\left(y^{*}\right)^{2}}=\frac{\partial\left(f^{\prime}\right)}{\partial(\eta)} \frac{\partial(\eta)}{\partial\left(y^{*}\right)}=f^{\prime \prime} \frac{A^{2}}{t^{* 2 n}}$ (13)

$\frac{\partial\left(\mathfrak{u}^{*}\right)}{\partial\left(t^{*}\right)}=\frac{\partial(f)}{\partial(\eta)} \frac{\partial(\eta)}{\partial\left(t^{*}\right)}=f^{\prime} *\left(-n A \mathrm{y}^{*} t^{*^{-}(1+n)}\right)$

Substitute above Equations (13) in Equations (9), the governing equation would be as follows:

$f^{\prime}\left(-n A y^{*} t^{*^{-}(1+n)}\right)=b f^{\prime\prime} \frac{A^{2}}{t^{* 2 n}}$

$-n \frac{\mathrm{n}}{t^{*}} f^{\prime}=\mathrm{b} \frac{\mathrm{A}^{2}}{t^{*^{2 n}}} f^{\prime\prime} \rightarrow 2 n=1 \rightarrow n=\frac{1}{2}$

Then

$f^{\prime \prime}+\frac{n}{2 b A^{2}} f^{\prime}=0$     (14)

To simplify the equation, A is assumed to be $\frac{1}{\sqrt{2}}$, hence the governing equation becomes:

$f^{\prime \prime}+\frac{n}{b} f^{\prime}=0$   (15)

where,

$\eta=\frac{y}{\sqrt{2 v t}}$    (16)

Equation (15) is a linear homogenous second order ordinary differential equation and can be adopted as a replacement to the partial differential Equation (2).

Now the transformed initial and boundary Conditions associated with Equations (4) are:

at $t=0 \rightarrow \eta=\infty$ and $u^{*}=1 \rightarrow f(\infty)=1$

at $\mathrm{y}=0 \rightarrow \eta=0 \quad$ and $\quad u^{*}=0 \rightarrow f(0)=0$ (17)

at $y=y_{\infty} \rightarrow \eta=\infty$ and $u^{*}=1 \rightarrow f(\infty)=1$

Solving equation (15) by adopting the above initial and boundary conditions is outlined below:

Equation (15) can be rewritten as

$\frac{d f^{\prime}}{f^{\prime}}=-\eta \mathrm{d} \eta$

Integrating both sides,

$\ln f^{\prime}=-\frac{\eta^{2}}{2 b}+c_{0}$ or $f^{\prime}=c_{1} e^{-\frac{\eta^{2}}{2 b}}$

Integrating again,

$f=c_{1} \int_{\infty}^{\eta} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma+c_{2}$

The lower limit is arbitrary, and $\infty$ is a good choice.

The boundary condition $f(\infty) \rightarrow 1$ requires that: $c_{2}=1$. The boundary condition $f(0)=0$ requires:

$0=c_{1} \int_{\infty}^{0} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma+1$

$c_{1}=\frac{-1}{\int_{\infty}^{0} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}=\frac{1}{\int_{0}^{\infty} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}$

Then $\eta$

$f=\frac{\int_{\infty}^{\mathrm{n}} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}{\int_{0}^{\infty} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}+1=1-\frac{\int_{\sigma}^{\infty} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}{\int_{0}^{\infty} e^{-\frac{\sigma^{2}}{2 b}} \mathrm{d} \sigma}$

Assuming

 $z=\frac{\sigma}{\sqrt{2 b}}$

Then

$\mathrm{d} \mathrm{z}=\frac{\mathrm{d} \sigma}{\sqrt{2 b}}$

The new initial and boundary conditions associated with these equations are:

$\begin{aligned} \sigma &=\eta \rightarrow \mathrm{z}=\frac{\mathrm{n}}{\sqrt{2 b}} \\ \sigma &=0 \rightarrow \mathrm{z}=0 \\ \sigma &=\infty \rightarrow \mathrm{z}=\infty \end{aligned}$

Then

$f=1-\frac{\int_{\frac{\pi}{\sqrt{2 b}}}^{\infty} e^{-\frac{(z \sqrt{2 b})^{2}}{2 b} \sqrt{2} \mathrm{d} z}}{\int_{0}^{\infty} e^{-\frac{(z \sqrt{2 b})^{2}}{2 b}} \sqrt{2} \mathrm{d} z}=1-\frac{\int_{\frac{\pi}{2 b}}^{\infty} e^{-z^{2}} \mathrm{d} z}{\int_{0}^{\infty} e^{-z^{2}} \mathrm{d} z}$

But

$\int_{0}^{\infty} e^{-z^{2}} \mathrm{d} \mathrm{z}=\frac{\sqrt{\pi}}{2}$

Then 

$f=1-\frac{2}{\sqrt{\pi}} \int_{\frac{n}{\sqrt{2 b}}}^{\infty} e^{-z^{2}} d z$

Hence,

$\operatorname{erfc}\left(\frac{\mathrm{n}}{\sqrt{2 b}}\right)=1-\operatorname{erf}\left(\frac{\eta}{\sqrt{2 b}}\right)=\frac{2}{\sqrt{\pi}} \int_{\frac{\mathrm{n}}{\sqrt{2 b}}}^{\infty} e^{-\mathrm{z}^{2}} \mathrm{d} \mathrm{z}$

Then

$f(\eta)=\operatorname{erf}\left(\frac{\eta}{\sqrt{2 b}}\right)$

And since

$f(\eta)=\mathfrak{u}^{*}$ and $u^{*}=\frac{u+\int \frac{1}{\rho} \frac{\partial P}{\partial x} d t}{u_{\infty}}$  

Then

$\mathrm{u}=\operatorname{erf}\left(\frac{\mathrm{n}}{\sqrt{2 b}}\right) *\left(u_{\infty}\right)+\frac{p_{0} \sin (n t)}{n}$   (18)

where,

$\eta=\frac{\mathrm{y}}{\sqrt{2 v t}}$

This equation shows the laminar and turbulent wind speed inside boundary layers, also it shows the effect of oscillating pressure gradient to different wind boundary layers.

The wind power extracted by the rotor blades can be calculated under the effect of the fluctuating pressure gradient by the following equations.

$P(u(0, R))=\frac{1}{2} \rho A u^{3}(0, R) \lambda=\frac{1}{2} \rho \lambda \int_{A 1}^{A 2} u^{3} d A$    (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.

$P(u(0, R))=\frac{1}{2} \rho \lambda \int_{0}^{R}\left(\operatorname{erf}\left(\frac{\eta}{\sqrt{2 b}}\right) *\left(u_{\infty}\right)+\frac{p_{0} \sin (n t)}{n}\right)^{3} 2 \pi r d r$   (20)

Rewriting Equation (20) with the dimensionless variable $\eta$, where wind turbine radius (r) plus wind turbine hub height (Hb) is replacing (y) in Equation (18), then:

$\eta=\frac{\left(r+H_{b}\right)}{\sqrt{2 v t}}$ and hence $d \eta=\frac{1}{\sqrt{2 v t}} d r$

Then

$P(u(0, R))=\frac{1}{2} \rho \lambda \int_{\eta_{1}}^{\eta_{2}}\left(e r f\left(\frac{\eta}{\sqrt{2 b}}\right) *\left(u_{\infty}\right)+\frac{p_{0} \sin (n t)}{n}\right)^{3} 2 \pi \sqrt{2 v t}\left(\eta \sqrt{2 v t}-H_{b}\right) d \eta$   (21)   

where,

$\eta_{1}=\frac{H_{b}}{\sqrt{2 v t}} \quad \eta_{2} =\frac{r+H_{b}}{\sqrt{2 v t}}$

This power is compared with a power version calculated without the pressure gradient fluctuations effect, on other words, calculating power using Equation (21) with Po=0.

The equation becomes:

$P(u(0, R))=\frac{1}{2} \rho \lambda \int_{\eta_{1}}^{\eta_{2}}\left(e r f\left(\frac{\eta}{\sqrt{2 b}}\right) *\left(u_{\infty}\right)\right)^{3} 2 \pi \sqrt{2 v t}\left(\eta \sqrt{2 v t}-H_{b}\right) d \eta$      (22)

The ratio between the above two versions is an efficient way to study the effect of pressure gradient fluctuations on the output power of wind turbine and is adopted throughout this work.

Throughout this study, a model that simulates wind turbine output power under the fluctuating air pressure gradient where the flow is either laminar or turbulent has been built, the model has been solved analytically. After examining the results, one can conclude that:

1- When the flow is either laminar or turbulent the effects of fluctuating pressure gradient is enhancing the wind turbine output power.

2- For turbulent and laminar flow: as the number of air pressure gradient fluctuation cycle is (n) increased, the enhancement resulting from the fluctuation pressure gradient would be minimized till reaching a state that the increase in (n) would insignificantly affects the wind turbine output power. The best performance of wind turbine occurred at low value of (n).

3- For turbulent and laminar flow: as air pressure gradient fluctuation amplitude (Po) is increased, the performance of wind turbine is enhanced.

4- In turbulent Flow: as turbulence intensity is increased the advantage from the fluctuation in air pressure gradient become more apparent.

The following are suggestions for further investigations:

1- Study the effects of laminar and turbulent fluctuating pressure gradient on wind turbine from Blade Theory point of view.

2- Studying the effects of the fluctuating on the structure of wind turbine and its base.

3- Study the effects of laminar and turbulent fluctuating pressure gradient on wind turbine using the experimental methods such like wind tunnel experiment at different conditions and turbulence intensities.