# Von Karman Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Von Kármán velocity spectra

• Library:
• Aerospace Blockset / Environment / Wind

## Description

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. For more information, see Algorithms.

## Limitations

• The frozen turbulence field assumption is valid for the cases of mean-wind velocity.

• 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 because the following factors are not incorporated into the model:

• Terrain roughness

• Lapse rate

• Wind shears

• Mean wind magnitude

• Other meteorological factions (except altitude)

## 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.

Data Types: double

### Output

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Turbulence velocities, returned as a three-element signal, 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 Von Kármán 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 Von Karman (+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. This length is assumed constant.

From the military specifications, 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.

#### Programmatic Use

 Block Parameter: L_high Type: character vector Values: real scalar Default: '762'

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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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-8785CMIL-HDBK-1797
Longitudinal

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

$\frac{2{\sigma }_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339{L}_{u}\frac{\omega }{V}\right)}^{2}\right]}^{5}{6}}}$

$\frac{2{\sigma }_{u}^{2}{L}_{u}}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339{L}_{u}\frac{\omega }{V}\right)}^{2}\right]}^{5}{6}}}$

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

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

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

Lateral

${\Phi }_{v}\left(\omega \right)$

$\frac{{\sigma }_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339{L}_{v}\frac{\omega }{V}\right)}^{2}}{{\left[1+{\left(1.339{L}_{v}\frac{\omega }{V}\right)}^{2}\right]}^{11}{6}}}$

$\frac{2{\sigma }_{v}^{2}{L}_{v}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678{L}_{v}\frac{\omega }{V}\right)}^{2}}{{\left[1+{\left(2.678{L}_{v}\frac{\omega }{V}\right)}^{2}\right]}^{11}{6}}}$

${\Phi }_{r}\left(\omega \right)$

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

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

Vertical

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

$\frac{{\sigma }_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339{L}_{w}\frac{\omega }{V}\right)}^{2}}{{\left[1+{\left(1.339{L}_{w}\frac{\omega }{V}\right)}^{2}\right]}^{11}{6}}}$

$\frac{2{\sigma }_{w}^{2}{L}_{w}}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678{L}_{w}\frac{\omega }{V}\right)}^{2}}{{\left[1+{\left(2.678{L}_{w}\frac{\omega }{V}\right)}^{2}\right]}^{11}{6}}}$

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

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

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

The variable b represents the aircraft wingspan. The variables Lu, Lv, Lw represent the turbulence scale lengths. The variables σu, σv, σw represent the turbulence intensities:

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

Keep in mind that the longitudinal turbulence angular rate spectrum, Фp(ω), is a rational function. The rational function is derived from curve-fitting a complex algebraic function, not the vertical turbulence velocity spectrum, Фw(ω), multiplied by a scale factor. Because the turbulence angular rate spectra contribute less to the aircraft gust response than the turbulence velocity spectra, it may explain the variations in their definitions.

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

VerticalLateral

Фq(ω)

Фq(ω)

−Фq(ω)

−Фr(ω)

Фr(ω)

Фr(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

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

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

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

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

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

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

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

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

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

MIL-HDBK-1797
Longitudinal

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

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

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

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

Lateral

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

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

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

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

Vertical

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

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

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

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

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

Note

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

## References

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

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

[3] 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, August 1969.

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

[5] Ly, U. and Y. Chan. "Time-Domain Computation of Aircraft Gust Covariance Matrices." AIAA Paper 80-1615. Presented at the Atmospheric Flight Mechanics Conference, Danvers, MA, August 11-13, 1980.

[6] McRuer, Duane, Irving Ashkenas, and Dunstan Graham. Aircraft Dynamics and Automatic Control. Princeton: Princeton University Press, July 1990.

[7] Moorhouse, David J. and Robert J. Woodcock. "Background Information and User Guide for MIL-F-8785C, 'Military Specification-Flying Qualities of Piloted Airplanes'." ADA119421. Flight Dynamic Laboratory, July 1982.

[8] McFarland, R. "A Standard Kinematic Model for Flight Simulation at NASA-Ames." NASA CR-2497. Computer Sciences Corporation, January 1975.

[9] Tatom, Frank B., Stephen R. Smith, and George H. Fichtl. "Simulation of Atmospheric Turbulent Gusts and Gust Gradients," AIAA Paper 81-0300. Aerospace Sciences Meeting, St. Louis, MO., January 12-15, 1981.

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