KikiSchulz at 22:22, 9 November 2021

2021-11-09T22:22:20Z

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	where <math>G_{22}= \hat{\kappa}_1^2 F_{22}</math> is the universal shear spectrum which must integrate to 2/15.		where <math>G_{22}= \hat{\kappa}_1^2 F_{22}</math> is the universal shear spectrum which must integrate to 2/15.

			You want more? Go to [[Spectra in the inertial subrange]]

KikiSchulz: Created page with "{{DefineConcept |description=Often called dissipation spectra, velocity spectra multiplied by $\kappa^2$ or $\kappa_1^2$ |article..."

2021-11-09T22:11:49Z

Created page with "{{DefineConcept |description=Often called dissipation spectra, velocity spectra multiplied by <math>\kappa^2</math> or <math>\kappa_1^2</math> |article..."

New page

{{DefineConcept
|description=Often called dissipation spectra, [[Spectra of velocity|velocity spectra]] multiplied by <math>\kappa^2</math> or <math>\kappa_1^2</math>
|article_type=Fundamentals
}}
The spectra of the gradients of velocity are closely related to the rate of dissipation, <math>\varepsilon</math>, and are often called dissipation spectra.
These spectra are the [[Spectra of velocity|velocity spectra]] multiplied by <math>\kappa^2</math> or <math>\kappa_1^2</math>, whichever is appropriate.
The rate of dissipation is related to the gradient of the three-dimensional velocity spectrum by

<math>
\begin{equation}
\begin{split}
\varepsilon &= 2\nu \int_0^{\infty} \kappa^2 E(\kappa)\, \mathrm{d} \kappa = 2\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa^2 F(\hat{\kappa})\, \mathrm{d} \kappa \\
&=2\nu \left(\varepsilon\nu^5 \right)^{1/4} L_K^{-3} \int_0^{\infty} \hat{\kappa}^2 F(\hat{\kappa})\, \mathrm{d} \hat{\kappa} \\
&= 2\varepsilon \int_0^{\infty} G(\hat{\kappa})\, \mathrm{d} \hat{\kappa}
\end{split}
\end{equation}
</math>

Thus, the universal (non-dimensional) gradient spectrum is <math>G=\hat{\kappa}^2 F</math> , and its integral over all wavenumbers must equal 1/2.
The along-profile gradient of the along-profile velocity fluctuations often called the rate of strain (or, simply strain), is related to the rate of dissipation by

<math>
\begin{equation}
\begin{split}
\varepsilon &= 15\nu \int_0^{\infty} \kappa_1^2 E_{11}(\kappa_1)\, \mathrm{d}\kappa_1 = 15\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa_1^2 F_{11} (\hat{\kappa}_1)\, \mathrm{d} \kappa_1 \\
&= 15\varepsilon \int_0^{\infty} G_{11} (\hat{\kappa}_1)\, \mathrm{d} \hat{\kappa}_1
\end{split}
\end{equation}
</math>

where <math>G_{11}=\hat{\kappa}_1^2 F_{11}</math> is the universal (and non-dimensional) rate of strain spectrum, which must integrate to 1/15.
Similarly, the shear spectrum is related to the rate of dissipation by

<math>
\begin{equation}
\begin{split}
\varepsilon &= \frac{15}{2} \nu \int_0^{\infty} \kappa_1^2 E_{22}(\kappa_1)\, \mathrm{d}\kappa_1 = \frac{15}{2}\nu \left(\varepsilon\nu^5 \right)^{1/4} \int_0^{\infty} \kappa_1^2 F_{22}\, (\hat{\kappa}_1) \mathrm{d} \kappa_1 \\
&= \frac{15}{2}\varepsilon \int_0^{\infty} G_{22} (\hat{\kappa}_1)\, \mathrm{d} \hat{\kappa}_1
\end{split}
\end{equation}
</math>

where <math>G_{22}= \hat{\kappa}_1^2 F_{22}</math> is the universal shear spectrum which must integrate to 2/15.

Spectra of velocity gradients - Revision history

KikiSchulz at 22:22, 9 November 2021

KikiSchulz: Created page with "{{DefineConcept |description=Often called dissipation spectra, velocity spectra multiplied by \kappa^2 or \kappa_1^2 |article..."

KikiSchulz: Created page with "{{DefineConcept |description=Often called dissipation spectra, velocity spectra multiplied by $\kappa^2$ or $\kappa_1^2$ |article..."