Turbulence spectrum: Difference between revisions
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<math> \Psi(k) = U_P \Psi(f) </math> . | <math> \Psi(k) = U_P \Psi(f) </math> . | ||
If a | 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, | For example, | ||
<math>\overline{\zeta^2}=\int_0^{\ | |||
<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. | |||
* Missing the y-axi variables | * Missing the y-axi variables | ||
* Lowest frequency and wavenumber resolvable | * Lowest frequency and wavenumber resolvable | ||
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
