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Generate continuous wind turbulence with Dryden velocity spectra

Environment/Wind

The Dryden Wind Turbulence Model (Continuous) block uses the Dryden 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, Military Handbook MIL-HDBK-1797, Military Handbook MIL-HDBK-1797B.

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.
MIL-F-8785C and MIL-HDBK-1797/1797B provide these definitions of longitudinal,
lateral, and vertical component spectra functions:

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

Longitudinal | ||

$${\Phi}_{u}\left(\omega \right)$$ |
$$\frac{2{\sigma}_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{1+{\left({L}_{u}\frac{\omega}{V}\right)}^{2}}$$ |
$$\frac{2{\sigma}_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{1+{\left({L}_{u}\frac{\omega}{V}\right)}^{2}}$$ |

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

Lateral | ||

$${\Phi}_{v}\left(\omega \right)$$ |
$$\frac{{\sigma}_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+3{\left({L}_{v}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left({L}_{v}\frac{\omega}{V}\right)}^{2}\right]}^{2}}$$ |
$$\frac{2{\sigma}_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+12{\left({L}_{v}\frac{\omega}{V}\right)}^{2}}{{\left[1+4{\left({L}_{v}\frac{\omega}{V}\right)}^{2}\right]}^{2}}$$ |

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

Vertical | ||

$${\Phi}_{w}\left(\omega \right)$$ |
$$\frac{{\sigma}_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+3{\left({L}_{w}\frac{\omega}{V}\right)}^{2}}{{\left[1+{\left({L}_{w}\frac{\omega}{V}\right)}^{2}\right]}^{2}}$$ |
$$\frac{2{\sigma}_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+12{\left({L}_{w}\frac{\omega}{V}\right)}^{2}}{{\left[1+4{\left({L}_{w}\frac{\omega}{V}\right)}^{2}\right]}^{2}}$$ |

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

where:

*b*represents the aircraft wingspan.*L*represent the turbulence scale lengths._{u}, L_{v}, L_{w}*σ*represent the turbulence intensities._{u}, σ_{v}, σ_{w}

The spectral density definitions of turbulence angular rates are defined in the specifications as three variations:

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

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

The longitudinal turbulence angular rate spectrum,

$${\Phi}_{{p}_{g}}(\omega )$$

is a rational function. The rational function is
derived from curve-fitting a complex algebraic function, not the vertical
turbulence velocity spectrum, *Φ _{w}*(

The variations result in these combinations of vertical and lateral turbulence angular rate spectra.

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

Φ ω)Φ ω)−Φ ω) | −Φ ω)Φ ω)Φ ω) |

To generate a signal with correct characteristics, a band-limited white noise signal is passed through forming filters. The forming filters are derived from the spectral square roots of the spectrum equations.

MIL-F-8785C and MIL-HDBK-1797/1797B provide these transfer functions:

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

Longitudinal | ||

$${H}_{u}(s)$$ |
$${\sigma}_{u}\sqrt{\frac{2{L}_{u}}{\pi V}\cdot}\frac{1}{1+\frac{{L}_{u}}{V}s}$$ |
$${\sigma}_{u}\sqrt{\frac{2{L}_{u}}{\pi V}}\cdot \frac{1}{1+\frac{{L}_{u}}{V}s}$$ |

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

Lateral | ||

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

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

Vertical | ||

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

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

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

The military specifications result in the same transfer function 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 and MIL-HDBK-1797B |
---|---|

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

Typically, at 20 feet (6 meters) the wind speed is 15 knots
in light turbulence, 30 knots in moderate turbulence, and 45 knots
for severe turbulence. See these turbulence intensities, where *W*_{20} is
the wind speed at 20 feet (6 meters).

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

The turbulence axes orientation in this region is defined:

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.

Turbulence scale lengths and intensities for medium-to-high altitudes the are based on the assumption that the turbulence is isotropic. MIL-F-8785C and MIL-HDBK-1797/1797B provide these representations of scale lengths:

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

L= _{u }L=_{v } L =
1750 ft_{w} | L = _{u}2L = _{v} 2L=
1750 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 and 2000, 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

Airspeed

`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 Dryden selections conform to the transfer function descriptions.

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

**Wind direction at 6 m (degrees clockwise from north)**Measured wind direction at a height of 6 meters (20 feet) 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 exceeding the turbulence intensity.

**Scale length at medium/high altitudes (m)**Turbulence scale length above 2000 feet, assumed constant. MIL-F-8785C and MIL-HDBK-1797/1797B recommend 1750 feet for the longitudinal turbulence scale length of the Dryden spectra.

### Note

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

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

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

**Noise seeds**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 this parameter 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 the 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, is small relative to the aircraft ground speed.

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

Terrain roughness

Lapse rate

Wind shears

Mean wind magnitude

Other meteorological factors

See Airframe/Environment Models/Wind Models in `aeroblk_HL20`

for
an example of this block.

Chalk, Charles, T.P. Neal, T.M. Harris, Francis E. Pritchard,
and Robert J. Woodcock. *Background Information and User
Guide for MIL-F-8785B(ASG)*, *“Military
Specification-Flying Qualities of Piloted Airplanes.”* AD869856.
Buffalo, NY: Cornell Aeronautical Laboratory, 1969.

*Flying Qualities of Piloted Aircraft*.
Department of Defense Handbook. MIL-HDBK-1797. Washington, DC: U.S.
Department of Defense, 1997.

*Flying Qualities of Piloted Aircraft*.
Department of Defense Handbook. MIL-HDBK-1797B. Washington, DC: U.S.
Department of Defense, 2012.

*Flying Qualities of Piloted Airplanes*.
U.S. Military Specification MIL-F-8785C. Washington, D.C.: U.S. Department
of Defense, 1980.

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

Ly, U. and Y. Chan. “Time-Domain Computation of Aircraft Gust Covariance Matrices,” AIAA Paper 80-1615, presented at the 6th Atmospheric Flight Mechanics Conference, Danvers, Massachusetts, August 1980.

McFarland, Richard E, A Standard Kinematic Model for Flight Simulation at NASA-AMES. NASA CR-2497. Mountain view, CA: Computer Sciences Corporation, 1975.

McRuer, Duane, Dunstan Graham, and Irving Ashkenas. *Aircraft
Dynamics and Automatic Control* Princeton University Press,
1974, R1990.

Moorhouse, David J. and Robert J. Woodcock. *Background
Information and User Guide for MIL-F-8785C, "Military Specification—Flying
Qualities of Piloted Airplanes."* ADA119421. Wright-Patterson
AFB, OH: Air Force Wright Aeronautical Labs, 1982.

Tatom, Frank B., George H. Fichtl, and Stephen R. Smith. “Simulation of Atmospheric Turbulent Gusts and Gust Gradients,” AIAA Paper 81-0300, presented at the 19th Aerospace Sciences Meeting, St. Louis, Missouri, January 1981.

Yeager, Jessie, *Implementation and Testing of Turbulence
Models for the F18-HARV Simulation* NASA CR-1998-206937.
Hampton, VA: Lockheed Martin Engineering & Sciences, 1998.