The Dryden wind turbulence model, also known as Dryden gusts, is a mathematical model of
continuous gusts
Continuous gusts or stochastic gusts are winds that vary randomly in space and time. Models of continuous gusts are used to represent atmospheric turbulence, especially clear air turbulence and turbulent winds in storms. The Federal Aviation Admini ...
accepted for use by the
United States Department of Defense
The United States Department of Defense (DoD, USDOD or DOD) is an executive branch department of the federal government charged with coordinating and supervising all agencies and functions of the government directly related to national sec ...
in certain aircraft design and simulation applications. The Dryden model treats the linear and angular velocity components of continuous gusts as spatially varying
stochastic processes and specifies each component's
power spectral density
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
. The Dryden wind turbulence model is characterized by
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
power spectral densities, so exact filters can be designed that take
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
inputs and output stochastic processes with the Dryden gusts' power spectral densities.
History
The Dryden model, named after
Hugh Dryden, is one of the most commonly used models of continuous gusts. It was first published in 1952.
Power Spectral Densities
The Dryden model is characterized by power spectral densities for gusts' three linear velocity components (''u
g'',''v
g'',''w
g''),
where ''σ
i'' and ''L
i'' are the turbulence intensity and scale length, respectively, for the ''i''th velocity component, and ''Ω'' is a spatial frequency. These power spectral densities give the stochastic process spatial variations, but any temporal variations rely on vehicle motion through the gust velocity field. The speed with which the vehicle is moving through the gust field ''V'' allows conversion of these power spectral densities to different types of frequencies,
where ω has units of radians per unit time.
The gust angular velocity components (''p
g'',''q
g'',''r
g'') are defined as the variations of the linear velocity components along the different vehicle axes,
though different sign conventions may be used in some sources. The power spectral densities for the angular velocity components are
The military specifications give criteria based on vehicle
stability derivatives to determine whether the gust angular velocity components are significant.
Spectral Factorization
The gusts generated by the Dryden model are not
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
processes and therefore may be referred to as
colored noise. Colored noise may, in some circumstances, be generated as the output of a
minimum phase
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
The most general causal LTI transfer function can be uniquely factored into a series of a ...
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
through a process known as spectral factorization. Consider a
linear time invariant system with a white noise input that has unit
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
,
transfer function
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
''G''(''s''), and output ''y''(''t''). The power spectral density of ''y''(''t'') is
where ''i''
2 = -1. For rational power spectral densities, such as that of the Dryden model, a suitable transfer function can be found whose magnitude squared evaluated along the imaginary axis is the power spectral density. The
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
documentation provides a realization of such a transfer function for Dryden gusts that is consistent with the military specifications,
Driving these filters with independent, unit variance, band-limited white noise yields outputs with power spectral densities that match the spectra of the velocity components of the Dryden model. The outputs can, in turn, be used as wind disturbance inputs for aircraft or other dynamic systems.
Altitude Dependence
The Dryden model is parameterized by a length scale and turbulence intensity. The combination of these two parameters determines the shape of the power spectral densities and therefore the quality of the model's fit to spectra of observed turbulence. Many combinations of length scale and turbulence intensity give realistic power spectral densities in the desired frequency ranges. The Department of Defense specifications include choices for both parameters, including their dependence on altitude.
See also
*
Continuous gusts
Continuous gusts or stochastic gusts are winds that vary randomly in space and time. Models of continuous gusts are used to represent atmospheric turbulence, especially clear air turbulence and turbulent winds in storms. The Federal Aviation Admini ...
*
Von Kármán wind turbulence model
The von Kármán wind turbulence model (also known as von Kármán gusts) is a mathematical model of continuous gusts. It matches observed continuous gusts better than that Dryden Wind Turbulence Model and is the preferred model of the United Sta ...
Notes
References
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Atmospheric dynamics
Aviation meteorology
Wind