Spectra of velocity gradients

Short definition of Spectra of velocity gradients
Often called dissipation spectra, velocity spectra multiplied by <math>\kappa^2</math> or <math>\kappa_1^2</math>

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, <math>\varepsilon</math>, and are often called dissipation spectra. These spectra are the 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.