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Generate continuous wind turbulence with Von Kármán velocity spectra

Environment/Wind

The Von Kármán Wind Turbulence Model (Continuous) block uses the Von Kármán spectral representation to add turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters. This block implements the mathematical representation in the Military Specification MIL-F-8785C and Military Handbook MIL-HDBK-1797.

According to the military references, turbulence is a stochastic
process defined by velocity spectra. For an aircraft flying at a speed *V* through
a frozen turbulence field with a spatial frequency of Ω radians
per meter, the circular frequency *ω* is calculated
by multiplying *V* by Ω.
The following table displays the component spectra functions:

MIL-F-8785C | MIL-HDBK-1797 | |
---|---|---|

Longitudinal | ||

$${\Phi}_{u}\left(\omega \right)$$ |
$$\frac{2{\sigma}_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339{L}_{u}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |
$$\frac{2{\sigma}_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339{L}_{u}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |

$${\Phi}_{p}\left(\omega \right)$$ |
$$\frac{{\sigma}_{w}^{2}}{V{L}_{w}}\cdot \frac{0.8{\left(\frac{\pi {L}_{w}}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1+{\left(\frac{4b\omega}{\pi V}\right)}^{2}}$$ |
$$\frac{{\sigma}_{w}^{2}}{2V{L}_{w}}\cdot \frac{0.8{\left(\frac{2\pi {L}_{w}}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1+{\left(\frac{4b\omega}{\pi V}\right)}^{2}}$$ |

Lateral | ||

$${\Phi}_{v}\left(\omega \right)$$ |
$$\frac{{\sigma}_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339{L}_{v}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left(1.339{L}_{v}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |
$$\frac{2{\sigma}_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678{L}_{v}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left(2.678{L}_{v}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |

$${\Phi}_{r}\left(\omega \right)$$ |
$$\frac{\mp {\left(\frac{\omega}{V}\right)}^{2}}{1+{\left(\frac{3b\omega}{\pi V}\right)}^{2}}\cdot {\Phi}_{v}\left(\omega \right)$$ |
$$\frac{\mp {\left(\frac{\omega}{V}\right)}^{2}}{1+{\left(\frac{3b\omega}{\pi V}\right)}^{2}}\cdot {\Phi}_{v}\left(\omega \right)$$ |

Vertical | ||

$${\Phi}_{w}\left(\omega \right)$$ |
$$\frac{{\sigma}_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339{L}_{w}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left(1.339{L}_{w}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |
$$\frac{2{\sigma}_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678{L}_{w}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left(2.678{L}_{w}\frac{\omega}{V}\right)}^{2}\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$ |

$${\Phi}_{q}\left(\omega \right)$$ |
$$\frac{\pm {\left(\frac{\omega}{V}\right)}^{2}}{1+{\left(\frac{4b\omega}{\pi V}\right)}^{2}}\cdot {\Phi}_{w}\left(\omega \right)$$ |
$$\frac{\pm {\left(\frac{\omega}{V}\right)}^{2}}{1+{\left(\frac{4b\omega}{\pi V}\right)}^{2}}\cdot {\Phi}_{w}\left(\omega \right)$$ |

The variable *b* represents the aircraft wingspan.
The variables *L _{u},
L_{v}, L_{w}* represent
the turbulence scale lengths. The variables

The spectral density definitions of turbulence angular rates are defined in the references as three variations, which are displayed in the following table:

$${p}_{g}=\frac{\partial {w}_{g}}{\partial y}$$ |
$${q}_{g}=\frac{\partial {w}_{g}}{\partial x}$$ |
$${r}_{g}=-\frac{\partial {v}_{g}}{\partial x}$$ |

$${p}_{g}=\frac{\partial {w}_{g}}{\partial y}$$ |
$${q}_{g}=\frac{\partial {w}_{g}}{\partial x}$$ |
$${r}_{g}=\frac{\partial {v}_{g}}{\partial x}$$ |

$${p}_{g}=-\frac{\partial {w}_{g}}{\partial y}$$ |
$${q}_{g}=-\frac{\partial {w}_{g}}{\partial x}$$ |
$${r}_{g}=\frac{\partial {v}_{g}}{\partial x}$$ |

The variations affect only the vertical (*q _{g}*)
and lateral (

Keep in mind that the longitudinal turbulence angular rate spectrum,
Ф* _{p}*(

The variations lead to the following combinations of vertical and lateral turbulence angular rate spectra.

Vertical | Lateral |
---|---|

Ф ω)Ф ω)−Ф ω) | −Ф ω)Ф ω)Ф ω) |

To generate a signal with the correct characteristics, a unit variance, band-limited white noise signal is passed through forming filters. The forming filters are approximations of the Von Kármán velocity spectra which are valid in a range of normalized frequencies of less than 50 radians. These filters can be found in both the Military Handbook MIL-HDBK-1797 and the reference by Ly and Chan.

The following two tables display the transfer functions.

MIL-F-8785C | |
---|---|

Longitudinal | |

$${H}_{u}\left(s\right)$$ |
$$\frac{{\sigma}_{u}\sqrt{\frac{2}{\pi}\cdot \frac{{L}_{u}}{V}}\left(1+0.25\frac{{L}_{u}}{V}s\right)}{1+1.357\frac{{L}_{u}}{V}s+0.1987{\left(\frac{{L}_{u}}{V}\right)}^{2}{s}^{2}}$$ |

$${H}_{p}\left(s\right)$$ |
$${\sigma}_{w}\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{{L}_{w}{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$ |

Lateral | |

$${H}_{v}\left(s\right)$$ |
$$\frac{{\sigma}_{v}\sqrt{\frac{1}{\pi}\cdot \frac{{L}_{v}}{V\text{}}}\left(1+2.7478\frac{{L}_{v}}{V}s+0.3398{\left(\frac{{L}_{v}}{V}\right)}^{2}{s}^{2}\right)}{1+2.9958\frac{{L}_{v}}{V}s+1.9754{\left(\frac{{L}_{v}}{V}\right)}^{2}{s}^{2}+0.1539{\left(\frac{{L}_{v}}{V}\right)}^{3}{s}^{3}}$$ |

$${H}_{r}\left(s\right)$$ |
$$\frac{\mp \frac{s}{V}}{\left(1+\left(\frac{3b}{\pi V}\right)s\right)}\cdot {H}_{v}\left(s\right)$$ |

Vertical | |

$${H}_{w}\left(s\right)$$ |
$$\frac{{\sigma}_{w}\sqrt{\frac{1}{\pi}\cdot \frac{{L}_{w}}{V}}\left(1+2.7478\frac{{L}_{w}}{V}s+0.3398{\left(\frac{{L}_{w}}{V}\right)}^{2}{s}^{2}\right)}{1+2.9958\frac{{L}_{w}}{V}s+1.9754{\left(\frac{{L}_{w}}{V}\right)}^{2}{s}^{2}+0.1539{\left(\frac{{L}_{w}}{V}\right)}^{3}{s}^{3}}$$ |

$${H}_{q}\left(s\right)$$ |
$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot {H}_{w}\left(s\right)$$ |

MIL-HDBK-1797 | |
---|---|

Longitudinal | |

$${H}_{u}\left(s\right)$$ |
$$\frac{{\sigma}_{u}\sqrt{\frac{2}{\pi}\cdot \frac{{L}_{u}}{V}}\left(1+0.25\frac{{L}_{u}}{V}s\right)}{1+1.357\frac{{L}_{u}}{V}s+0.1987{\left(\frac{{L}_{u}}{V}\right)}^{2}{s}^{2}}$$ |

$${H}_{p}\left(s\right)$$ |
$${\sigma}_{w}\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{{\left(2{L}_{w}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$ |

Lateral | |

$${H}_{v}\left(s\right)$$ |
$$\frac{{\sigma}_{v}\sqrt{\frac{1}{\pi}\cdot \frac{2{L}_{v}}{V}}\left(1+2.7478\frac{2{L}_{v}}{V}s+0.3398{\left(\frac{2{L}_{v}}{V}\right)}^{2}{s}^{2}\right)}{1+2.9958\frac{2{L}_{v}}{V}s+1.9754{\left(\frac{2{L}_{v}}{V}\right)}^{2}{s}^{2}+0.1539{\left(\frac{2{L}_{v}}{V}\right)}^{3}{s}^{3}}$$ |

$${H}_{r}\left(s\right)$$ |
$$\frac{\mp \frac{s}{V}}{\left(1+\left(\frac{3b}{\pi V}\right)s\right)}\cdot {H}_{v}\left(s\right)$$ |

Vertical | |

$${H}_{w}\left(s\right)$$ |
$$\frac{{\sigma}_{w}\sqrt{\frac{1}{\pi}\cdot \frac{2{L}_{w}}{V}}\left(1+2.7478\frac{2{L}_{w}}{V}s+0.3398{\left(\frac{2{L}_{w}}{V}\right)}^{2}{s}^{2}\right)}{1+2.9958\frac{2{L}_{w}}{V}s+1.9754{\left(\frac{2{L}_{w}}{V}\right)}^{2}{s}^{2}+0.1539{\left(\frac{2{L}_{w}}{V}\right)}^{3}{s}^{3}}$$ |

$${H}_{q}\left(s\right)$$ |
$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot {H}_{w}\left(s\right)$$ |

Divided into two distinct regions, the turbulence scale lengths and intensities are functions of altitude.

The same transfer functions result after evaluating the turbulence scale lengths. The differences in turbulence scale lengths and turbulence transfer functions balance offset.

According to the military references, the turbulence scale lengths
at low altitudes, where *h* is the altitude in feet,
are represented in the following table:

MIL-F-8785C | MIL-HDBK-1797 |
---|---|

$$\begin{array}{l}{L}_{w}=h\\ {L}_{u}={L}_{v}=\frac{h}{{\left(0.177+0.000823h\right)}^{1.2}}\end{array}$$ |
$$\begin{array}{l}2{L}_{w}=h\\ {L}_{u}=2{L}_{v}=\frac{h}{{\left(0.177+0.000823h\right)}^{1.2}}\end{array}$$ |

The turbulence intensities are given below, where *W*_{20} is
the wind speed at 20 feet (6 m). Typically for light
turbulence, the wind speed at 20 feet is 15 knots; for moderate turbulence,
the wind speed is 30 knots; and for severe turbulence, the wind speed
is 45 knots.

$$\begin{array}{l}{\sigma}_{w}=0.1{W}_{20}\\ \frac{{\sigma}_{u}}{{\sigma}_{w}}=\frac{{\sigma}_{v}}{{\sigma}_{w}}=\frac{1}{{\left(0.177+0.000823h\right)}^{0.4}}\end{array}$$

The turbulence axes orientation in this region is defined as follows:

Longitudinal turbulence velocity,

*u*, aligned along the horizontal relative mean wind vector_{g}Vertical turbulence velocity,

*w*, aligned with vertical._{g}

At this altitude range, the output of the block is transformed into body coordinates.

For medium to high altitudes the turbulence scale lengths and intensities are based on the assumption that the turbulence is isotropic. In the military references, the scale lengths are represented by the following equations:

MIL-F-8785C | MIL-HDBK-1797 |
---|---|

L = _{u}L = _{v}L =
2500 ft_{w} | L =
2_{u}L = 2_{v}L =
2500 ft_{w} |

The turbulence intensities are determined from a lookup table
that provides the turbulence intensity as a function of altitude and
the probability of the turbulence intensity being exceeded. The relationship
of the turbulence intensities is represented in the following equation: *σ _{u}*=

The turbulence axes orientation in this region is defined as being aligned with the body coordinates:

At altitudes between 1000 feet and 2000 feet, the turbulence velocities and turbulence angular rates are determined by linearly interpolating between the value from the low altitude model at 1000 feet transformed from mean horizontal wind coordinates to body coordinates and the value from the high altitude model at 2000 feet in body coordinates.

**Units**Define the units of wind speed due to the turbulence.

Units Wind Velocity Altitude Air Speed `Metric (MKS)`

Meters/second Meters Meters/second `English (Velocity in ft/s)`

Feet/second Feet Feet/second `English (Velocity in kts)`

Knots Feet Knots **Specification**Define which military reference to use. This affects the application of turbulence scale lengths in the lateral and vertical directions

**Model type**Select the wind turbulence model to use:

`Continuous Von Karman (+q -r)`

Use continuous representation of Von Kármán velocity spectra with positive vertical and negative lateral angular rates spectra.

`Continuous Von Karman (+q +r)`

Use continuous representation of Von Kármán velocity spectra with positive vertical and lateral angular rates spectra.

`Continuous Von Karman (-q +r)`

Use continuous representation of Von Kármán velocity spectra with negative vertical and positive lateral angular rates spectra.

`Continuous Dryden (+q -r)`

Use continuous representation of Dryden velocity spectra with positive vertical and negative lateral angular rates spectra.

`Continuous Dryden (+q +r)`

Use continuous representation of Dryden velocity spectra with positive vertical and lateral angular rates spectra.

`Continuous Dryden (-q +r)`

Use continuous representation of Dryden velocity spectra with negative vertical and positive lateral angular rates spectra.

`Discrete Dryden (+q -r)`

Use discrete representation of Dryden velocity spectra with positive vertical and negative lateral angular rates spectra.

`Discrete Dryden (+q +r)`

Use discrete representation of Dryden velocity spectra with positive vertical and lateral angular rates spectra.

`Discrete Dryden (-q +r)`

Use discrete representation of Dryden velocity spectra with negative vertical and positive lateral angular rates spectra.

The Continuous Von Kármán selections conform to the transfer function descriptions.

**Wind speed at 6 m defines the low altitude intensity**The measured wind speed at a height of 20 feet (6 meters) provides the intensity for the low-altitude turbulence model.

**Wind direction at 6 m (degrees clockwise from north)**The measured wind direction at a height of 20 feet (6 meters) is an angle to aid in transforming the low-altitude turbulence model into a body coordinates.

**Probability of exceedance of high-altitude intensity**Above 2000 feet, the turbulence intensity is determined from a lookup table that gives the turbulence intensity as a function of altitude and the probability of the turbulence intensity's being exceeded.

**Scale length at medium/high altitudes**The turbulence scale length above 2000 feet is assumed constant, and from the military references, a figure of 1750 feet is recommended for the longitudinal turbulence scale length of the Dryden spectra.

### Note

An alternate scale length value changes the power spectral density asymptote and gust load.

**Wingspan**The wingspan is required in the calculation of the turbulence on the angular rates.

**Band-limited noise sample time (seconds)**The sample time at which the unit variance white noise signal is generated.

**Noise seeds**There are four random numbers required to generate the turbulence signals, one for each of the three velocity components and one for the roll rate. The turbulences on the pitch and yaw angular rates are based on further shaping of the outputs from the shaping filters for the vertical and lateral velocities.

**Turbulence on**Selecting the check box generates the turbulence signals.

Input | Dimension Type | Description |
---|---|---|

First | scalar | Contains the altitude in units selected. |

Second | scalar | Contains the aircraft speed in units selected. |

Third | 3-by-3 matrix | Contains a direction cosine matrix. |

Output | Dimension Type | Description |
---|---|---|

First | Three-element signal | Contains the turbulence velocities, in the selected units. |

Second | Three-element signal | Contains the turbulence angular rates, in radians per second. |

The frozen turbulence field assumption is valid for the cases of mean-wind velocity and the root-mean-square turbulence velocity, or intensity, are small relative to the aircraft's ground speed.

The turbulence model describes an average of all conditions for clear air turbulence because the following factors are not incorporated into the model:

Terrain roughness

Lapse rate

Wind shears

Mean wind magnitude

Other meteorological factions (except altitude)

U.S. Military Handbook MIL-HDBK-1797, 19 December 1997.

U.S. Military Specification MIL-F-8785C, 5 November 1980.

Chalk, C., Neal, P., Harris, T., Pritchard, F., Woodcock, R., “Background Information and User Guide for MIL-F-8785B(ASG), `Military Specification-Flying Qualities of Piloted Airplanes',” AD869856, Cornell Aeronautical Laboratory, August 1969.

Hoblit, F., *Gust Loads on Aircraft: Concepts and
Applications*, AIAA Education Series, 1988.

Ly, U., Chan, Y., “Time-Domain Computation of Aircraft Gust Covariance Matrices,” AIAA Paper 80-1615, Atmospheric Flight Mechanics Conference, Danvers, MA., August 11-13, 1980.

McRuer, D., Ashkenas, I., Graham, D., *Aircraft Dynamics
and Automatic Control*, Princeton University Press, July
1990.

Moorhouse, D., Woodcock, R., “Background Information and User Guide for MIL-F-8785C, `Military Specification-Flying Qualities of Piloted Airplanes',” ADA119421, Flight Dynamic Laboratory, July 1982.

McFarland, R., “A Standard Kinematic Model for Flight Simulation at NASA-Ames,” NASA CR-2497, Computer Sciences Corporation, January 1975.

Tatom, F., Smith, R., Fichtl, G., “Simulation of Atmospheric Turbulent Gusts and Gust Gradients,” AIAA Paper 81-0300, Aerospace Sciences Meeting, St. Louis, MO., January 12-15, 1981.

Yeager, J., “Implementation and Testing of Turbulence Models for the F18-HARV Simulation,” NASA CR-1998-206937, Lockheed Martin Engineering & Sciences, March 1998.