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 maybe discussed within the wiki.

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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

<math> \tilde{F}_{11} (\hat{\kappa}_1) = \int_{\hat{\kappa}_1}^{\infty} \frac{F(\hat{\kappa})}{\hat{\kappa}} \left(1 - \frac{\hat{\kappa}_1^2}{\hat{\kappa}^2} \right)\, \mathrm{d}\hat{\kappa} = \frac{18}{55} C \hat{\kappa}_1^{-5/3} = C_1 \hat{\kappa}_1^{-5/3} </math>

where <math>C_1=18C/55\approx27/55</math> 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 <math>C</math> 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 <math>C</math>. 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 <math>0.53</math> and the standard deviation is <math>0.055</math> (Figure 1). A crude estimate of the <math>95\%</math> confidence interval is <math>C_1=0.53\pm0.03</math>.

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

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

<math> \tilde{F}_{22} (\hat{\kappa}_1) = \frac{4}{3} C_1 \hat{\kappa}_1^{-5/3} </math>

The gradient spectra in the inertial subrange are

<math> \begin{equation} \tilde{G}_{11} (\hat{\kappa}_1) = C_1 \hat{\kappa}_1^{1/3} \end{equation} </math>

and

<math> \begin{equation} \tilde{G}_{22} (\hat{\kappa}_1) = \frac{4}{3} C_1 \hat{\kappa}_1^{1/3} \end{equation} </math>