Spectra in the inertial subrange

Short definition of Spectra in the inertial subrange
In the inertial subrange, the three-dimensional velocity spectrum follows a power-law behaviour and this makes it possible to easily derive the one-dimensional spectra, in this range

This is the common definition for Spectra in the inertial subrange, but other definitions may be discussed within the wiki.

In the inertial subrange, the three-dimensional velocity spectrum follows a power-law behaviour and this makes it possible to easily derive the one-dimensional spectra, in this range. Using ( ?) within the inertial subrange gives

${\tilde{F}}_{11}({\hat{\kappa }}_1)=\underset{{\hat{\kappa }}_1}{\overset{\mathrm{\infty }}{\int }}\frac{F(\hat{\kappa })}{\hat{\kappa }}(1-\frac{{\hat{\kappa }}_1^2}{{\hat{\kappa }}^2})\mathrm{d}\hat{\kappa }=\frac{18}{55}C{\hat{\kappa }}_1^{-5/3}=C_1{\hat{\kappa }}_1^{-5/3}$

where $C_1=18C/55\approx 27/55$ is frequently called the one-dimensional Kolmogorov constant, and the tilde is used to indicate these equations apply only in the inertial subrange. It is not possible to measure the three-dimensional spectrum and, thus, it is not possible to estimate $C$ directly. Consequently, there is research interest in estimating </math>C_1</math> because it is the only practical way to determine the three-dimensional Kolmogorov constant $C$. Sreenivasa (1995) compiled the values of the one-dimensional Kolmogorov constant reported from a wide range of measurements in the atmosphere, ocean, wind tunnels and pipes. The mean value (excluding low Reynolds number measurements) is $0.53$ and the standard deviation is $0.055$ (Figure 1). A crude estimate of the $95\mathrm{\%}$ confidence interval is $C_1=0.53\pm 0.03$.

Figure 1. Figure 3 from Sreenivasa (1995) for the estimates of the one-dimensional Kolmogorov constant, $C_1$, derived from experimental measurements of along-stream velocity measurements and/or the rate of strain.

Using ${\tilde{F}}_{22}=\frac{4}{3}{\tilde{F}}_{11}$, the one-dimensional spectrum for the velocity components that are orthogonal to the direction of profiling is

${\tilde{F}}_{22}({\hat{\kappa }}_1)=\frac{4}{3}{C}_1{\hat{\kappa }}_1^{-5/3}$

The gradient spectra in the inertial subrange are

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \tilde{G}_{11} (\hat{\kappa}_1) = C_1 \hat{\kappa}_1^{1/3} \end{equation} }

and

Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} \tilde{G}_{22} (\hat{\kappa}_1) = \frac{4}{3} C_1 \hat{\kappa}_1^{1/3} \end{equation} }