Spectra of velocity

Short definition of Spectra of velocity
Theoretically derived spectrum of velocity fluctuations in the inertial subrange.

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

The spectrum of velocity fluctuations has only been derived theoretically for the inertial subrange. This is the range of eddy sizes at which the flow is isotropic – they have lost the orientation of the largest eddies – but, their size is still large enough to not be significantly affected by viscosity. In this range kinetic energy is transferred to smaller scales through inertial interaction of the eddies but no energy is lost through friction. The three-dimensional spectrum of velocity, in the inertial subrange, is

[math]\displaystyle{ E(\kappa)=C\varepsilon^{2/3} \kappa^{-5/3} }[/math]

where [math]\displaystyle{ \kappa }[/math] is the magnitude of the angular wavenumber in units of [math]\displaystyle{ \mathrm{rad\,m^{-1}} }[/math] and [math]\displaystyle{ C\approx1.5 }[/math] is the three-dimensional Kolmogorov constant. There is no theoretical derivation for the velocity spectrum at wavenumbers beyond the inertial subrange. It is common to express the entire spectrum by

[math]\displaystyle{ E(\kappa)=C\varepsilon^{2/3} \kappa^{-5/3} f_{\eta} \left(\kappa_{L_K}\right) }[/math]

where [math]\displaystyle{ f_{\eta} }[/math] characterizes the spectrum in the dissipation range, has a value of unity in the inertial subrange ([math]\displaystyle{ \kappa L_K \ll 1 }[/math]), and [math]\displaystyle{ L_K=\left(\nu^3/\varepsilon\right)^{1/4} }[/math] is the Kolmogorov length. It is thought that the velocity spectrum can be described by a universal non-dimensional spectrum, [math]\displaystyle{ F }[/math], defined by

[math]\displaystyle{ E(\kappa) = \left(\varepsilon \nu^5 \right)^{1/4} F(\hat{\kappa}) }[/math]

where [math]\displaystyle{ \hat{\kappa} =\kappa L_K }[/math] is the non-dimensional wavenumber.

It is currently not possible to measure the three-dimensional spectrum of velocity. It is only possible to measure the one-dimensional spectrum of velocity – the spectrum derived from a profile in a single direction. The one-dimensional spectrum of the component of velocity that is parallel to the direction of profiling is

[math]\displaystyle{ E_{11} (\kappa_1 )= \int_{\kappa_1}^{\infty} \frac{E(\kappa)}{\kappa} \left( 1- \frac{\kappa_1^2}{\kappa^2} \right) \, \mathrm{d} \kappa }[/math]

where [math]\displaystyle{ \kappa_1 }[/math] is the angular wavenumber in the direction of profiling. The universal spectrum associated with [math]\displaystyle{ E_{11} }[/math] is given by

[math]\displaystyle{ E_{11} (\kappa_1) = \left( \varepsilon \nu^5 \right)^{1/4} F_{11} (\hat{\kappa}_1) }[/math]

and, therefore,

[math]\displaystyle{ 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} }[/math]

Similarly, the one-dimensional spectrum of the component of velocity that is orthogonal to the direction of profiling is

[math]\displaystyle{ E_{22} (\kappa_1 ) = \frac{1}{2} \int_{\kappa_1}^{\infty} \frac{E(\kappa)}{\kappa} \left( 1 + \frac{\kappa_1^2}{\kappa^2} \right) \, \mathrm{d} \kappa }[/math]

and its universal spectrum is defined by

[math]\displaystyle{ E_{22} (\kappa_1) = \left(\varepsilon\nu^5 \right)^{1/4} F_{22} (\hat{\kappa}_1) }[/math]

so that

[math]\displaystyle{ F_{22} (\hat{\kappa}_1) = \frac{1}{2} \int_{\hat{\kappa}_1}^{\infty} \frac{E(\hat{\kappa})}{\hat{\kappa}} \left( 1 + \frac{\hat{\kappa}_1^2}{\hat{\kappa}^2} \right) \, \mathrm{d} \hat{\kappa} }[/math] .

These two one-dimensional spectra are related to each other by

[math]\displaystyle{ F_{22} (\hat{\kappa}_1)= \frac{1}{2} \left( F_{11} (\hat{\kappa}_1) - \hat{\kappa}_1 \frac{\mathrm{d}F_{11}(\hat{\kappa}_1)}{\mathrm{d}\hat{\kappa}_1} \right) }[/math]

and, thus, [math]\displaystyle{ F_{22}=\frac{4}{3} F_{11} }[/math] in the inertial subrange. These relationships hold for any direction of profiling, as long as we refer to the velocity component that is parallel to the direction of profiling by the subscripts ([math]\displaystyle{ _{11} }[/math]) and the (mutually orthogonal) pair of velocity components that are orthogonal to the direction of profiling using the subscripts ([math]\displaystyle{ _{22} }[/math]). Thus, the second orthogonal velocity component has the spectrum [math]\displaystyle{ E_{33}\equiv E_{22} }[/math] .

You want more? Check out Spectra of velocity gradients