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 measured 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^{\infty} \Psi(k)\, \mathrm{d}k </math> | |||
* Missing the y-axi variables | * Missing the y-axi variables | ||
* Lowest frequency and wavenumber resolvable | * Lowest frequency and wavenumber resolvable | ||
Revision as of 19:23, 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 measured 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^{\infty} \Psi(k)\, \mathrm{d}k }[/math]
- Missing the y-axi variables
- Lowest frequency and wavenumber resolvable
