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Extended Wierenga Equation

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Extended Wieringa Equation

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The Extended Wieringa equation gives the gust factor as a function of turbulence intensity, length of time step, and gust averaging time:

GF_ExtendedWieringa

where:

 

 

 

T

is the length of time step [s]

 

t

is the gust averaging time [s]

 

I

is the turbulence intensity [unitless]

 

GF

is the gust factor [unitless]

Derivation

Wieringa (1973) provides an equation to estimate the gust factor for any gust averaging time:

GF_Wieringa

where:

 

 

 

GF(t, T)

is the gust factor for averaging time t and time step T [unitless]

 

I

is the turbulence intensity [unitless]

 

T

is the length of time step [s]

 

t

is the gust averaging time [s]

The above equation above does not appear explicitly in Wieringa's paper; it combines two equations that appear in the paper, specifically the one the paper refers to as equation 8:

GF_WieringaEqn8

and the one the paper refers to as equation 10:

GF_WieringaEqn10

The Wieringa paper stipulates that the equation is valid only for gust averaging times less than one-seventh the time step. The graph below shows its behavior in the valid and invalid range of gust averaging times.

Graph-GustFactorWieringa

Openwind needs to be able to estimate gust factors for a wider range of gust averaging time, so we seek to extend this equation by switching to a logarithmic curve for gust averaging times greater than one-seventh the time step, meaning for (T / t) < 7. We therefore aim to develop a composite equation of the form:

GF_ExtendedWieringa_proposal

 

We will choose the two logarithmic curve fit parameters such that the two portions of the composite equation produce the same value at (T / t) = 7, and so that the gust factor drops to unity when t = T. We will therefore require the logarithmic curve to pass through two points, with the first at (T / t) = 7:

GF_ExtendedWieringa_point1

and the second point being at (T / t) = 1:

GF_ExtendedWieringa_point2

Now we can find our first curve fit parameter using:

GF_ExtendedWieringa_alpha

And using the second point on the logarithmic curve we can calculate the second curve fit parameter:

GF_ExtendedWieringa_beta

Now substituting these equations into our original proposed composite equation we can write:

GF_ExtendedWieringa_subsitution

And we can simplify that to the following:

GF_ExtendedWieringa

The graph below shows the results of the extended Wieringa equation for the case of ten-minute time steps (T = 600s) and four values of turbulence intensity. The green line segments come from the original Wieringa equation, whereas the blue line segments come from the logathmic extensions.

Graph-GustFactorExtendedWieringa

The graph below compares the extended Wieringa equation to the other two gust factor equations Openwind implements, namely the Gauss-Weibull equation and the AWST gust factor equation, for the case of ten-minute time steps and 10% turbulence intensity:

Graph-GustFactorComparison

See also

Gauss-Weibull equation

UL gust factor equation

Gust factor

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