Continuous gusts

Continuous gusts or stochastic gusts are winds that vary randomly in space and time. Models of continuous gusts are used to represent atmospheric turbulence, especially clear air turbulence and turbulent winds in storms. The Federal Aviation Administration (FAA) and the United States Department of Defense provide requirements for the models of continuous gusts used in design and simulation of aircraft.^[1][2]

Models of continuous gusts

A variety of models exist for gusts^[3] but only two, the Dryden and von Kármán models, are generally used for continuous gusts in flight dynamics applications.^[2][4] Both of these models define gusts in terms of power spectral densities for the linear and angular velocity components parameterized by turbulence length scales and intensities. The velocity components of these continuous gust models can be incorporated into airplane equations of motion as a wind disturbance.^[5] While these models of continuous gusts are not white noise, filters can be designed that take a white noise input and output a random process with the Dryden or von Kármán models.^[6][7]

Assumptions of continuous gust models

The models accepted by the FAA and Department of Defense represent continuous gusts as a wind linear and angular velocity field that is a random process and make a number of simplifying assumptions in order to describe them mathematically. In particular, continuous gusts are assumed to be:^[8]

• A Gaussian process
• A stationary process, so the statistics are constant in time
• Homogeneous, so the statistics do not depend on the vehicle path
• Ergodic
• Isotropic at high altitude, so the statistics do not depend on the vehicle attitude
• Varying in space but frozen in time

These assumptions, while unrealistic, yield acceptable models for flight dynamics applications.^[9] The last assumption of a velocity field that does not vary with time is especially unrealistic, since measurements of atmospheric turbulence at one point in space always vary with time. These models rely on the airplane's motion through the gusts to generate temporal variations in wind velocity, making them inappropriate for use as inputs to models of hovering, wind turbines, or other applications that are fixed in space.

The models also make assumptions about how continuous gusts vary with altitude. The Dryden and von Kármán models specified by the Department of Defense define three different altitude ranges: low, 10 ft to 1000 ft AGL; medium/high, 2000 ft AGL and above; and in between. The turbulence intensities, turbulence scale lengths, and turbulence axes depend on the altitude.^[10] The Department of Defense also provides models for the gust angular velocity but gives criteria based on airplane stability derivatives for when they can be omitted.^[11]

Dryden model

The Dryden model is one of the most commonly used models of continuous gusts. It was first published in 1952.^[12] The power spectral density of the longitudinal linear velocity component is

${\displaystyle \Phi _{u_{g}}(\Omega )=\sigma _{u}^{2}{\frac {2L_{u}}{\pi }}{\frac {1}{1+(L_{u}\Omega )^{2}}}}$

where u_g is the gusts' longitudinal linear velocity component, σ_u is the turbulence intensity, L_u is the turbulence scale length, and Ω is a spatial frequency.^[13]

The Dryden model has rational power spectral densities for each velocity component. This means that an exact filter can be formed that takes white noise as an input and outputs a random process with the Dryden model's power spectral densities.^[6]

von Kármán model

The von Kármán model is the preferred model of continuous gusts for the Department of Defense and the FAA.^[1][2] The model first appeared in a 1957 NACA report^[14] based on earlier work by Theodore von Kármán.^[15][16][17] In this model, the power spectral density of the longitudinal linear velocity component is

${\displaystyle \Phi _{u_{g}}(\Omega )=\sigma _{u}^{2}{\frac {2L_{u}}{\pi }}{\frac {1}{\left(1+(1.339L_{u}\Omega )^{2}\right)^{\frac {5}{6}}}}}$

where u_g is the longitudinal linear velocity component, σ_u is the turbulence intensity, L_u is the turbulence scale length, and Ω is a spatial frequency.^[2]

The von Kármán model has irrational power spectral densities. So, a filter with a white noise input that outputs a random process with the von Kármán model's power spectral densities can only be approximated.^[7]

Altitude dependence

Both the Dryden and von Kármán models are parameterized by a length scale and turbulence intensity. The combination of these two parameters determine the shape of the power spectral densities and therefore the quality of the models' fit to spectra of observed turbulence. Many combinations of length scale and turbulence intensity give realistic power spectral densities in the desired frequency ranges.^[4] The Department of Defense specifications include choices for both parameters, including their dependence on altitude, which are summarized below.^[10]

Low altitude

Low altitude is defined as altitudes between 10 ft AGL and 1000 ft AGL.

Length scales

At low altitude, the scale lengths are functions of altitude,

${\displaystyle 2L_{w}=h}$

${\displaystyle L_{u}=2L_{v}={\frac {h}{(0.177+0.000823h)^{1.2}}}}$

where h is the altitude AGL. At 1000 ft AGL, L_u = 2L_v = 2L_w = 1000 ft.

Turbulence intensities

At low altitude, the turbulence intensities are parameterized by W₂₀, the wind speed at 20 ft.

Turbulence severity	${\displaystyle W_{20}}$
Light	15 kts
Moderate	30 kts
Severe	45 kts

${\displaystyle \sigma _{w}=0.1W_{20}}$

${\displaystyle {\frac {\sigma _{u}}{\sigma _{w}}}={\frac {\sigma _{v}}{\sigma _{w}}}={\frac {1}{(0.177+0.000823h)^{0.4}}}}$

At 1000 ft AGL,

${\displaystyle \sigma _{u}=\sigma _{v}=\sigma _{w}=0.1W_{20}}$

Medium/high altitude

Medium/high altitude is defined as 2000 ft AGL and above.

Turbulence intensity vs. altitude for medium/high altitudes.

Length scales

For the Dryden model,

${\displaystyle L_{u}=2L_{v}=2L_{w}=1750{\text{ft}}}$

For the von Kármán model,

${\displaystyle L_{u}=2L_{v}=2L_{w}=2500{\text{ft}}}$

Turbulence intensities

At high altitude,

${\displaystyle \sigma _{u}=\sigma _{v}=\sigma _{w}}$

They are parameterized by the probability of exceedance or the turbulence severity. A plot of turbulence intensity versus altitude showing lines of constant probability of exceedance and ranges corresponding to different turbulence severities is provided in the military specifications.^[18]

Between low and medium/high altitude

From 1000 ft AGL to 2000 ft AGL, both the length scale and turbulence intensity are determined by linear interpolation between the low altitude value at 1000 ft and the medium/high altitude value at 2000 ft.^[6][7]

Turbulence axes

Above 1750 ft, the axes of the turbulence coincide with the wind frame axes. Below 1750 ft, the vertical turbulence axis is aligned with the Earth frame z-axis, the longitudinal turbulence axis is aligned with the projection of the mean wind vector onto the Earth frame's horizontal plane, and the lateral turbulence axis is determined by the right hand rule.^[19]

See also

• Clear air turbulence
• Dryden Wind Turbulence Model
• Von Kármán wind turbulence model

Notes

1. 14 CFR Part 25: Appendix G (2011). "Airworthiness Standards: Transport Category Airplanes". U.S. Code of Federal Regulations. Government Printing Office.
2. MIL-STD-1797A 1990, p. 678.
3. MIL-STD-1797A 1990, pp. 695–697.
4. Hoblit 1988, Chap. 4.
5. Etkin 2005, pp. 543–562.
6. "Dryden Wind Turbulence Model (Continuous)". MATLAB Reference Pages. The MathWorks, Inc. 2010. Retrieved May 24, 2013.
7. "Von Karman Wind Turbulence Model (Continuous)". MATLAB Reference Pages. The MathWorks, Inc. 2010. Retrieved May 24, 2013.
8. Etkin 2005, pp. 531–543.
9. Hoblit 1988, Chap. 12.
10. MIL-STD-1797A 1990, pp. 673, 678–685, 702.
11. MIL-STD-1797A 1990, p. 680.
12. Liepmann, H. W. (1952). "On the Application of Statistical Concepts to the Buffeting Problem". Journal of the Aeronautical Sciences. 19 (12): 793–800. doi:10.2514/8.2491.
13. MIL-HDBK-1797 1997, p. 678.
14. Diedrich, Franklin W.; Joseph A. Drischler (1957). "Effect of Spanwise Variations in Gust Intensity on the Lift Due to Atmospheric Turbulence": NACA TN 3920.
15. de Kármán, Theodore; Leslie Howarth (1938). "On the Statistical Theory of Isotropic Turbulence" (PDF). Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 164 (917): 192–215. Bibcode:1938RSPSA.164..192D. doi:10.1098/rspa.1938.0013.
16. von Kármán, Theodore (1948). "Progress in the Statistical Theory of Turbulence" (PDF). Proceedings of the National Academy of Sciences. 34 (11): 530–539. Bibcode:1948PNAS...34..530V. doi:10.1073/pnas.34.11.530. PMC 1079162. PMID 16588830.
17. von Kármán, T.; Lin, C. C. (1951). "On the Statistical Theory of Isotropic Turbulence". In von Mises, Richard; von Kármán, Theodore. Advances in Applied Mechanics. Academic Press, Inc. pp. 1–19.
18. MIL-STD-1797A 1990, p. 673.
19. MIL-STD-1797A 1990, p. 702.

References

• Etkin, Bernard (2005). Dynamics of Atmospheric Flight. Mineola, NY: Dover Publications. ISBN 0486445224.
• Hoblit, Frederic M. (1988). Gust Loads on Aircraft: Concepts and Applications. Washington, DC: American institute of Aeronautics and Astronautics, Inc. ISBN 0930403452.
• MIL-STD-1797A (1990). Flying Qualities of Piloted Aircraft (PDF). Dept. of Defense.