Dryden Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Dryden velocity spectra

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Libraries:
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.

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

h — Altitude

scalar

Altitude, specified as a scalar, in selected units.

Data Types: double

V — Aircraft speed

scalar

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

Data Types: double

DCM — Direction cosine matrix

3-by-3 matrix

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

Data Types: double

Output

V_wind — Turbulence velocities

three-element vector

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

Data Types: double

ω_wind — Turbulence angular rates

three-element vector

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

Data Types: double

Parameters

Units — Wind speed units

Metric (MKS) (default) | English (Velocity in ft/s) | English (Velocity in kts)

Units of wind speed due to turbulence, specified as:

Units	Wind Velocity	Altitude	Air Speed
Metric (MKS)	Meters/second	Meters	Meters/second
English (Velocity in ft/s)	Feet/second	Feet	Feet/second
English (Velocity in kts)	Knots	Feet	Knots

Programmatic Use

Block Parameter: units

Type: character vector

Values: 'Metric (MKS)' | 'English (Velocity in ft/s)' | 'English (Velocity in kts)'

Default: 'Metric (MKS)'

Specification — Military reference

MIL-F-8785C (default) | MIL-HDBK-1797 | MIL-HDBK-1797B

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'

Model type — Turbulence model

Continuous Dryden (+q -r) (default) | Continuous Von Karman (+q +r) | Continuous Von Karman (-q +r) | Continuous Von Karman (+q -r) | Continuous Dryden (+q +r) | Continuous Dryden (-q +r) | Discrete Dryden (+q -r) | Discrete Dryden (+q +r) | Discrete Dryden (-q +r)

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)'

Wind speed at 6 m defines the low altitude intensity — Measured wind speed

15 (default) | real scalar

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'

Wind direction at 6 m (degrees clockwise from north) — Measured wind direction

0 (default) | real scalar

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 exceedance of high-altitude intensity — Turbulence intensity

10^-2 - Light (default) | 10^-1 | 2x10^-1 | 10^-3 - Moderate | 10^-4 | 10^-5 - Severe | 10^-6

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'

Scale length at medium/high altitudes (m) — Turbulence scale length

533.4 (default) | real scalar

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 — Wingspan

10 (default) | real scalar

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'

Band limited noise sample time (seconds) — Noise sample time

0.1 (default) | real scalar

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 — Noise seeds [ug vg wg pg]

[23341 23342 23343 23344] (default) | four-element vector

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]'

Turbulence on — Turbulence signals

on (default) | off

To generate the turbulence signals, select this check box.

Programmatic Use

Block Parameter: T_on

Type: character vector

Values: 'on' | 'off'

Default: 'on'

Algorithms

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_u, L_v, L_w represent the turbulence scale lengths.

• σ_u, σ_v, σ_w 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 (q_g) and lateral (r_g) turbulence angular rates.

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(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.

Vertical	Lateral
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-8785C	MIL-HDBK-1797 and MIL-HDBK-1797B
Longitudinal
$$H_u(s)$$	$${\sigma }_u\sqrt{\frac{2L_u}{\pi V}\cdot }\frac{1}{1+\frac{L_u}{V}s}$$	$${\sigma }_u\sqrt{\frac{2L_u}{\pi V}}\cdot \frac{1}{1+\frac{L_u}{V}s}$$
$$H_p(s)$$	$${\sigma }_w\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{L_w{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$	$${\sigma }_w\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{{\left(2L_w\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$
Lateral
$$H_v(s)$$	$${\sigma }_v\sqrt{\frac{L_v}{\pi V}}\cdot \frac{1+\frac{\sqrt{3}{L}_v}{V}s}{{\left(1+\frac{L_v}{V}s\right)}^2}$$	$${\sigma }_v\sqrt{\frac{2L_v}{\pi V}}\cdot \frac{1+\frac{2\sqrt{3}{L}_v}{V}s}{{\left(1+\frac{2L_v}{V}s\right)}^2}$$
$$H_r(s)$$	$$\frac{\mp \frac{s}{V}}{\left(1+\left(\frac{3b}{\pi V}\right)s\right)}\cdot H_v(s)$$	$$\frac{\mp \frac{s}{V}}{\left(1+\left(\frac{3b}{\pi V}\right)s\right)}\cdot H_v(s)$$
Vertical
$$H_w(s)$$	$${\sigma }_w\sqrt{\frac{L_w}{\pi V}}\cdot \frac{1+\frac{\sqrt{3}{L}_w}{V}s}{{\left(1+\frac{L_w}{V}s\right)}^2}$$	$${\sigma }_w\sqrt{\frac{2L_w}{\pi V}}\cdot \frac{1+\frac{2\sqrt{3}{L}_w}{V}s}{{\left(1+\frac{2L_w}{V}s\right)}^2}$$
$$H_q(s)$$	$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot H_w(s)$$	$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot H_w(s)$$

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

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-8785C	MIL-HDBK-1797 and MIL-HDBK-1797B
$$\begin{array}{l}{L}_w=h\\ L_u=L_v=\frac{h}{{\left(0.177+0.000823h\right)}^{1.2}}\end{array}$$	$$\begin{array}{l}2L_w=h\\ L_u=2L_v=\frac{h}{{\left(0.177+0.000823h\right)}^{1.2}}\end{array}$$

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).

$$\begin{array}{l}{\sigma }_w=0.1W_{20}\\ \frac{{\sigma }_u}{{\sigma }_w}=\frac{{\sigma }_v}{{\sigma }_w}=\frac{1}{{\left(0.177+0.000823h\right)}^{0.4}}\end{array}$$

The turbulence axes orientation in this region is defined as:

• Longitudinal turbulence velocity, u_g, aligned along the horizontal relative mean wind vector.

• Vertical turbulence velocity, w_g, 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-8785C	MIL-HDBK-1797 and MIL-HDBK-1797B
L u = L v = L w = 1750 ft	L u = 2L v = 2L w = 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:

σ_u = σ_v = σ_w.

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.

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.

Extended Capabilities

C/C++ Code Generation
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Version History

Introduced before R2006a