Von Karman Wind Turbulence Model (Continuous)

Generate continuous wind turbulence with Von Kármán velocity spectra

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Libraries:
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, Military Handbook MIL-HDBK-1797, and Military Handbook MIL-HDBK-1797B. 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

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 Von Karman (+q -r) (default) | 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)

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)'

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 — Turbulence scale length

762 (default) | real scalar

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 — 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

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-8785C	MIL-HDBK-1797 and MIL-HDBK-1797B
Longitudinal
$${\Phi }_u\left(\omega \right)$$	$$\frac{2{\sigma }_u^2{L}_u}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339L_u\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$	$$\frac{2{\sigma }_u^2{L}_u}{\pi V}\cdot \frac{1}{{\left[1+{\left(1.339L_u\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$
$${\Phi }_p\left(\omega \right)$$	$$\frac{{\sigma }_w^2}{V{L}_w}\cdot \frac{0.8{\left(\frac{\pi L_w}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1+{\left(\frac{4b\omega }{\pi V}\right)}^2}$$	$$\frac{{\sigma }_w^2}{2V{L}_w}\cdot \frac{0.8{\left(\frac{2\pi L_w}{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{1+{\left(\frac{4b\omega }{\pi V}\right)}^2}$$
Lateral
$${\Phi }_v\left(\omega \right)$$	$$\frac{{\sigma }_v^2{L}_v}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339L_v\frac{\omega }{V}\right)}^2}{{\left[1+{\left(1.339L_v\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$	$$\frac{2{\sigma }_v^2{L}_v}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678L_v\frac{\omega }{V}\right)}^2}{{\left[1+{\left(2.678L_v\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$
$${\Phi }_r\left(\omega \right)$$	$$\frac{\mp {\left(\frac{\omega }{V}\right)}^2}{1+{\left(\frac{3b\omega }{\pi V}\right)}^2}\cdot {\Phi }_v\left(\omega \right)$$	$$\frac{\mp {\left(\frac{\omega }{V}\right)}^2}{1+{\left(\frac{3b\omega }{\pi V}\right)}^2}\cdot {\Phi }_v\left(\omega \right)$$
Vertical
$${\Phi }_w\left(\omega \right)$$	$$\frac{{\sigma }_w^2{L}_w}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(1.339L_w\frac{\omega }{V}\right)}^2}{{\left[1+{\left(1.339L_w\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$	$$\frac{2{\sigma }_w^2{L}_w}{\pi V}\cdot \frac{1+\frac{8}{3}{\left(2.678L_w\frac{\omega }{V}\right)}^2}{{\left[1+{\left(2.678L_w\frac{\omega }{V}\right)}^2\right]}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$$
$${\Phi }_q\left(\omega \right)$$	$$\frac{\pm {\left(\frac{\omega }{V}\right)}^2}{1+{\left(\frac{4b\omega }{\pi V}\right)}^2}\cdot {\Phi }_w\left(\omega \right)$$	$$\frac{\pm {\left(\frac{\omega }{V}\right)}^2}{1+{\left(\frac{4b\omega }{\pi V}\right)}^2}\cdot {\Phi }_w\left(\omega \right)$$

The variable b represents the aircraft wingspan. The variables L_u, L_v, L_w represent the turbulence scale lengths. The variables σ_u, σ_v, σ_w 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 (q_g) and lateral (r_g) turbulence angular rates.

Keep in mind that the longitudinal turbulence angular rate spectrum, Ф_p(ω), is a rational function. The rational function is derived from curve-fitting a complex algebraic function, not the vertical turbulence velocity spectrum, Ф_w(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.

Vertical	Lateral
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
$$H_u\left(s\right)$$	$$\frac{{\sigma }_u\sqrt{\frac{2}{\pi }\cdot \frac{L_u}{V}}\left(1+0.25\frac{L_u}{V}s\right)}{1+1.357\frac{L_u}{V}s+0.1987{\left(\frac{L_u}{V}\right)}^2{s}^2}$$
$$H_p\left(s\right)$$	$${\sigma }_w\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{L_w{}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$
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
$$H_w\left(s\right)$$	$$\frac{{\sigma }_w\sqrt{\frac{1}{\pi }\cdot \frac{L_w}{V}}\left(1+2.7478\frac{L_w}{V}s+0.3398{\left(\frac{L_w}{V}\right)}^2{s}^2\right)}{1+2.9958\frac{L_w}{V}s+1.9754{\left(\frac{L_w}{V}\right)}^2{s}^2+0.1539{\left(\frac{L_w}{V}\right)}^3{s}^3}$$
$$H_q\left(s\right)$$	$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot H_w\left(s\right)$$

	MIL-HDBK-1797 and MIL-HDBK-1797B
Longitudinal
$$H_u\left(s\right)$$	$$\frac{{\sigma }_u\sqrt{\frac{2}{\pi }\cdot \frac{L_u}{V}}\left(1+0.25\frac{L_u}{V}s\right)}{1+1.357\frac{L_u}{V}s+0.1987{\left(\frac{L_u}{V}\right)}^2{s}^2}$$
$$H_p\left(s\right)$$	$${\sigma }_w\sqrt{\frac{0.8}{V}}\cdot \frac{{\left(\frac{\pi }{4b}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{{\left(2L_w\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(1+\left(\frac{4b}{\pi V}\right)s\right)}$$
Lateral
$$H_v\left(s\right)$$	$$\frac{{\sigma }_v\sqrt{\frac{1}{\pi }\cdot \frac{2L_v}{V}}\left(1+2.7478\frac{2L_v}{V}s+0.3398{\left(\frac{2L_v}{V}\right)}^2{s}^2\right)}{1+2.9958\frac{2L_v}{V}s+1.9754{\left(\frac{2L_v}{V}\right)}^2{s}^2+0.1539{\left(\frac{2L_v}{V}\right)}^3{s}^3}$$
$$H_r\left(s\right)$$	$$\frac{\mp \frac{s}{V}}{\left(1+\left(\frac{3b}{\pi V}\right)s\right)}\cdot H_v\left(s\right)$$
Vertical
$$H_w\left(s\right)$$	$$\frac{{\sigma }_w\sqrt{\frac{1}{\pi }\cdot \frac{2L_w}{V}}\left(1+2.7478\frac{2L_w}{V}s+0.3398{\left(\frac{2L_w}{V}\right)}^2{s}^2\right)}{1+2.9958\frac{2L_w}{V}s+1.9754{\left(\frac{2L_w}{V}\right)}^2{s}^2+0.1539{\left(\frac{2L_w}{V}\right)}^3{s}^3}$$
$$H_q\left(s\right)$$	$$\frac{\pm \frac{s}{V}}{\left(1+\left(\frac{4b}{\pi V}\right)s\right)}\cdot H_w\left(s\right)$$

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

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.

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

The turbulence axes orientation in this region is defined as follows:

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

• Vertical turbulence velocity, w_g, aligned with vertical relative mean wind vector.

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-8785C	MIL-HDBK-1797 and MIL-HDBK-1797B
L u = L v = L w = 2500 ft	L u = 2 L v = 2 L w = 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: 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 (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.

References

[1] U.S. Military Handbook MIL-HDBK-1797B, April 9, 2012.

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

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

[4] 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.

[5] Hoblit, Frederic M., Gust Loads on Aircraft: Concepts and Applications. 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 Atmospheric Flight Mechanics Conference, Danvers, MA, August 11-13, 1980.

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

[8] 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.

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

[10] 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.

[11] 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.

Extended Capabilities

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Version History

Introduced in R2006b