# Dryden Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Dryden velocity spectra

• Library:
• Aerospace Blockset / Environment / Wind

## Description

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. For more information, see .

## Limitations

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

## Ports

### Input

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Altitude, specified as a scalar, in selected units.

Data Types: `double`

Aircraft speed, specified as a scalar, in selected units.

Data Types: `double`

Direction cosine matrix, specified as a 3-by-3 matrix representing the flat Earth coordinates to body-fixed axis coordinates.

Data Types: `double`

### Output

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Turbulence velocities, returned as a three-element vector in the same body coordinate reference as the DCM input, in specified units.

Data Types: `double`

Turbulence angular rates, specified as a three-element vector, in radians per second.

Data Types: `double`

## Parameters

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Units of wind speed due to turbulence, specified as:

UnitsWind VelocityAltitudeAir Speed
`Metric (MKS)` Meters/secondMetersMeters/second
`English (Velocity in ft/s)` Feet/secondFeetFeet/second
`English (Velocity in kts)` KnotsFeetKnots

#### Programmatic Use

 Block Parameter: `units` Type: character vector Values: `'Metric (MKS)'` | `'English (Velocity in ft/s)'` | `'English (Velocity in kts)'` Default: `'Metric (MKS)'`

Military reference, which affects the application of turbulence scale lengths in the lateral and vertical directions, specified as `MIL-F-8785C`, `MIL-HDBK-1797`, or `MIL-HDBK-1797B`.

#### Programmatic Use

 Block Parameter: `spec` Type: character vector Values: `'MIL-F-8785C'` | `'MIL-HDBK-1797'` | `'MIL-HDBK-1797B'` Default: `'MIL-F-8785C'`

Wind turbulence model, specified as:

 ```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.

#### Programmatic Use

 Block Parameter: `model` Type: character vector Values: ```'Continuous Von Karman (+q +r)'``` | ```'Continuous Von Karman (-q +r)'``` | ```'Continuous Dryden (+q -r)'``` | `'Continuous Dryden (+q +r)'` | `'Continuous Dryden (-q +r)'` | `'Discrete Dryden (+q -r)'` | `'Discrete Dryden (+q +r)'` | `'Discrete Dryden (-q +r)'` Default: ```'Continuous Dryden (+q +r)'```

Measured wind speed at a height of 20 feet (6 meters), specified as a real scalar, which provides the intensity for the low-altitude turbulence model.

#### Programmatic Use

 Block Parameter: `W20` Type: character vector Values: real scalar Default: `'15'`

Measured wind direction at a height of 20 feet (6 meters), specified as a real scalar, which is an angle to aid in transforming the low-altitude turbulence model into a body coordinates.

#### Programmatic Use

 Block Parameter: `Wdeg` Type: character vector Values: real scalar Default: `'0'`

Probability of the turbulence intensity being exceeded, specified as `10^-2 - Light`, `10^-1`, `2x10^-1`, `10^-3 - Moderate`, `10^-4`, `10^-5 - Severe`, or `10^-6`. 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 being exceeded.

#### Programmatic Use

 Block Parameter: `TurbProb` Type: character vector Values: `'2x10^-1'` | `'10^-1'` | `'10^-2 - Light'` | `'10^-3 - Moderate'` | `'10^-4'` | `'10^-5 - Severe'` | `'10^-6'` Default: `'10^-2 - Light'`

Turbulence scale length above 2000 feet, specified as a real scalar, which is 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.

#### Programmatic Use

 Block Parameter: `L_high` Type: character vector Values: real scalar Default: `'533.4'`

Wingspan, specified as a real scalar, which is required in the calculation of the turbulence on the angular rates.

#### Programmatic Use

 Block Parameter: `Wingspan` Type: character vector Values: real scalar Default: `'10'`

Noise sample time, specified as a real scalar, at which the unit variance white noise signal is generated.

#### Programmatic Use

 Block Parameter: `ts` Type: character vector Values: real scalar Default: `'0.1'`

Random noise seeds, specified as a four-element vector, which are used 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.

#### Programmatic Use

 Block Parameter: `Seed` Type: character vector Values: four-element vector Default: `'[23341 23342 23343 23344]'`

To generate the turbulence signals, select this check box.

#### Programmatic Use

 Block Parameter: `T_on` Type: character vector Values: `'on'` | `'off'` Default: `'on'`

## Algorithms

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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-8785CMIL-HDBK-1797 and MIL-HDBK-1797B
Longitudinal

`${\Phi }_{u}\left(\omega \right)$`

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

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

`${\Phi }_{p}{}_{{}_{g}}\left(\omega \right)$`

`$\frac{{\sigma }_{w}^{2}}{V{L}_{w}}\cdot \frac{0.8{\left(\frac{\pi {L}_{w}}{4b}\right)}^{1}{3}}}{1+{\left(\frac{4b\omega }{\pi V}\right)}^{2}}$`

`$\frac{{\sigma }_{w}^{2}}{2V{L}_{w}}\cdot \frac{0.8{\left(\frac{2\pi {L}_{w}}{4b}\right)}^{1}{3}}}{1+{\left(\frac{4b\omega }{\pi V}\right)}^{2}}$`

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

`${\Phi }_{w}\left(\omega \right)$`

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

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

`${\Phi }_{q}\left(\omega \right)$`

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

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

where:

• b represents the aircraft wingspan.

• Lu, Lv, Lw represent the turbulence scale lengths.

• σu, σv, σw represent the turbulence intensities.

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 (qg) and lateral (rg) turbulence angular rates.

The longitudinal turbulence angular rate spectrum,

`${\Phi }_{{p}_{g}}\left(\omega \right)$`

is a rational function. The rational function is derived from curve-fitting a complex algebraic function, not the vertical turbulence velocity spectrum, Φw(ω), multiplied by a scale factor. The variations exist because the turbulence angular rate spectra contribute less to the aircraft gust response than the turbulence velocity.

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

VerticalLateral

Φq(ω)

Φq(ω)

−Φq(ω)

−Φr(ω)

Φr(ω)

Φr(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-8785CMIL-HDBK-1797 and MIL-HDBK-1797B
Longitudinal

`${H}_{u}\left(s\right)$`

`${\sigma }_{u}\sqrt{\frac{2{L}_{u}}{\pi V}\cdot }\frac{1}{1+\frac{{L}_{u}}{V}s}$`

`${\sigma }_{u}\sqrt{\frac{2{L}_{u}}{\pi V}}\cdot \frac{1}{1+\frac{{L}_{u}}{V}s}$`

`${H}_{p}\left(s\right)$`

`${\sigma }_{w}\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{1}{6}}}{{L}_{w}{}^{1}{3}}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$`

`${\sigma }_{w}\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{1}{6}}}{{\left(2{L}_{w}\right)}^{1}{3}}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$`

Lateral

`${H}_{v}\left(s\right)$`

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

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

`${H}_{r}\left(s\right)$`

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

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

Vertical

`${H}_{w}\left(s\right)$`

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

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

`${H}_{q}\left(s\right)$`

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

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

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

Note

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.

## References

[1] 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.

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

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

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

[5] Hoblit, Frederic M., Gust Loads on Aircraft: Concepts and Applications. Reston, VA: AIAA Education Series, 1988.

[6] 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, MA, August 1980.

[7] McFarland, Richard E. "A Standard Kinematic Model for Flight Simulation at NASA-Ames." NASA CR-2497. Mountain View, CA: Computer Sciences Corporation, 1975.

[8] McRuer, Duane, Dunstan Graham, and Irving Ashkenas. Aircraft Dynamics and Automatic Control. Princeton, NJ: Princeton University Press, 1974, R1990.

[9] 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.

[10] 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, MO, January 1981.

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