# Von Karman Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Von Kármán velocity spectra

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.

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.

### Low-Altitude Model (Altitude < 1000 feet)

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-8785CMIL-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 W20 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, ug , aligned along the horizontal relative mean wind vector

• Vertical turbulence velocity, wg , aligned with vertical.

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

### Medium/High Altitudes (Altitude > 2000 feet)

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-8785CMIL-HDBK-1797
Lu = Lv = Lw = 2500 ftLu = 2Lv = 2Lw = 2500 ft

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: σu = σv = σw.

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

### Between Low and Medium/High Altitudes (1000 feet < Altitude < 2000 feet)

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.

## Parameters

Units

Define the units of wind speed due to the turbulence.

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

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.

## Inputs and Outputs

InputDimension TypeDescription

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.

OutputDimension TypeDescription

First

Three-element signalContains the turbulence velocities, in the selected units.

Second

Three-element signalContains the turbulence angular rates, in radians per second.

## Assumptions and Limitations

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)

## References

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.