# Dryden Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Dryden velocity spectra

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.

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.

### Low-Altitude Model (Altitude Under 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 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 W20 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, 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 Above 2000 Feet)

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-8785CMIL-HDBK-1797 and MIL-HDBK-1797B
Lu = Lv = Lw = 1750 ftLu = 2Lv = 2Lw = 1750 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 (Between 1000 and 2000 Feet)

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.

## Parameters

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.

## Ports

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 the 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.

## 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

## Examples

See Airframe/Environment Models/Wind Models in `aeroblk_HL20` for an example of this block.

## References

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.

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