Spectra of velocity gradients

Short definition of Spectra of velocity gradients
Often called dissipation spectra, velocity spectra multiplied by ${\kappa }^2$ or ${\kappa }_1^2$

This is the common definition for Spectra of velocity gradients, but other definitions maybe discussed within the wiki.

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The spectra of the gradients of velocity are closely related to the rate of dissipation, $\epsilon$, and are often called dissipation spectra. These spectra are the velocity spectra multiplied by ${\kappa }^2$ or ${\kappa }_1^2$, whichever is appropriate. The rate of dissipation is related to the gradient of the three-dimensional velocity spectrum by

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \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} }

Thus, the universal (non-dimensional) gradient spectrum is $G={\hat{\kappa }}^{2}F$ , 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

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \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} }

where $G_{11}={\hat{\kappa }}_1^2{F}_{11}$ 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

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \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} }

where $G_{22}={\hat{\kappa }}_1^2{F}_{22}$ is the universal shear spectrum which must integrate to 2/15.

You want more? Go to Spectra in the inertial subrange