Turbulence spectrum: Difference between revisions

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[[User:CynthiaBluteau|CynthiaBluteau]] ([[User talk:CynthiaBluteau|talk]]) 01:06, 14 October 2021 (CEST) Add example spectrum with inertial and viscous subranges sketched
Spectra in the frequency domain are converted into the spatial domain via [[Taylor's Frozen Turbulence]] hypothesis.  
[[file:Fig9 SV94.png]]
 
Taylor's Frozen Turbulence for converting temporal to spatial measurements.  
Convert time derivatives to spatial gradients along the direction of profiling using  
Convert time derivatives to spatial gradients along the direction of profiling using  


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and  
and  
<math> \Psi(k) = U_P \Psi(f) </math> .
<math> \Psi(k) = U_P \Psi(f) </math> .
If a sampled quantity, say <math>\zeta</math>, has a spectrum, <math>\Psi(k)</math>, then this spectrum provides the wavenumber distribution of the variance of <math>\zeta</math>.
For example,
<math>\overline{\zeta^2}=\int_0^{k_N} \Psi(k)\, \mathrm{d}k </math>
provides the total variance of <math>\zeta</math>, where <math>k_N = \frac{1}{2} f_s / U_P</math> is the Nyquist wavenumber and <math>f_s</math> is the sampling rate <math>\zeta</math>.
The variance located in the wavenumber band of <math>k_1</math> to <math>k_2</math> is
<math>\int_{k_1}^{k_2} \Psi(k)\, \mathrm{d}k \ \ .</math>
Thus, a spectrum has units of variance per wavenumber.





Latest revision as of 19:35, 1 December 2021


Short definition of Turbulence spectrum
Turbulence energy cascade and its expected spectral representation

This is the common definition for Turbulence spectrum, but other definitions maybe discussed within the wiki.


Spectra in the frequency domain are converted into the spatial domain via Taylor's Frozen Turbulence hypothesis. Convert time derivatives to spatial gradients along the direction of profiling using

[math]\displaystyle{ \frac{\partial}{\partial x} = \frac{1}{U_P} \frac{\partial}{\partial t} }[/math] .

Convert frequency spectra into wavenumber spectra using

[math]\displaystyle{ k = f/U_P }[/math] and [math]\displaystyle{ \Psi(k) = U_P \Psi(f) }[/math] .

If a sampled quantity, say [math]\displaystyle{ \zeta }[/math], has a spectrum, [math]\displaystyle{ \Psi(k) }[/math], then this spectrum provides the wavenumber distribution of the variance of [math]\displaystyle{ \zeta }[/math]. For example,

[math]\displaystyle{ \overline{\zeta^2}=\int_0^{k_N} \Psi(k)\, \mathrm{d}k }[/math]

provides the total variance of [math]\displaystyle{ \zeta }[/math], where [math]\displaystyle{ k_N = \frac{1}{2} f_s / U_P }[/math] is the Nyquist wavenumber and [math]\displaystyle{ f_s }[/math] is the sampling rate [math]\displaystyle{ \zeta }[/math]. The variance located in the wavenumber band of [math]\displaystyle{ k_1 }[/math] to [math]\displaystyle{ k_2 }[/math] is

[math]\displaystyle{ \int_{k_1}^{k_2} \Psi(k)\, \mathrm{d}k \ \ . }[/math]

Thus, a spectrum has units of variance per wavenumber.


  • Missing the y-axi variables
  • Lowest frequency and wavenumber resolvable