Velocity inertial subrange model

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The inertial subrange separates the energy-containing production range from the viscous dissipation range.

This is the common definition for Velocity inertial subrange model, but other definitions maybe discussed within the wiki.

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Model for steady-flows

This theoretical model predicts the spectral shape of velocities in wavenumber space.

<math>\Psi_{Vj}(\hat{k})=a_jC_k\varepsilon^{2/3}\hat{k}^{-5/3}</math>

Sketch of velocity power density spectrum in log-log space. The inertial subrange's -5/3 slope is highlighted. The vertical axis represents <math>\Psi_{Vj}(\hat{k})</math>. Large scale turbulence anisotropy in low energy flow may alter the expected spectral shape

Here <math>\hat{k}</math> is expressed in rad/m and <math>Vj</math> represents the velocities <math>V</math> in direction <math>j</math>. <math>C_k</math> is the empirical Kolmogorov universal constant of C = 1.5 <ref name="Sreenivasan"> {{#arraymap:K. R. Sreenivasan|,|x|x|, |and}}. 1995. On the universality of the Kolmogorov constant. Phys. Fluids. doi:10.1063/1.868656 </ref>. Amongst the three direction, the spectra deviates by the constant <math>a_j</math>: <ref name="Pope"> {{#arraymap:S.B Pope|,|x|x|, |and}}. 2000. Turbulent flows. Cambridge Univ. Press. doi:10.1017/CBO9780511840531 </ref>

• In the longitudinal direction, i.e., the direction of mean advection (j=1), <math>a_1=\frac{18}{55}</math>
• In the other directions <math>a_2=a_3=\frac{4}{3}a_1</math>

Models influenced by surface waves

Need to add equations and figures from Lumley & Terray<ref name="Lumley_Terray">

{{#arraymap:J. Lumley and E. Terray|,|x|x|, |and}}. 1983. Kinematics of turbulence convected by a random wave field. J. Phys. Oceanogr. doi:<2000:KOTCBA>2.0.CO;2 10.1175/1520-0485(1983)<2000:KOTCBA>2.0.CO;2

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Inertial subrange collapse and anisotropy

Near boundaries or low energy environments--defined as flows with a small separation between the large turbulent overturns <math>L</math> and the smallest (Kolmogorov)-- tends to adversely impact our ability to estimate <math>\varepsilon</math> from the lower wavenumbers. In certain cases, the velocity spectra may not have a sufficiently developed inertial subrange to estimate <math>\varepsilon</math> <ref name="Gargett.etal1984"> {{#arraymap:A. E. Gargett, T. R. Osborn, and P.W. Nasmyth|,|x|x|, |and}}. 1984. Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid. Mech.. doi:10.1017/S0022112084001592 </ref><ref name="Bluteau.etal2011"> {{#arraymap:C.E. Bluteau, N.L. Jones, and G. Ivey|,|x|x|, |and}}. 2011. Estimating turbulent kinetic energy dissipation using the inertial subrange method in environmental flows. Limnol. Oceanogr.: Methods. doi:10:4319/lom.2011.9.302 </ref>.

Anisotropic velocity spectra are exhibited when the largest turbulence scales are less than XX times the Kolmogorov length scale, may inhibit using the vertical velocity component to derive <math>\varepsilon</math>. In these situations, it may be possible to use the longitudinal velocity component (see Bluteau et al. 2011<ref name="Bluteau.etal2011"/>), which requires the user to rotate the velocity in the direction of the mean flow.

Example of how turbulence anisotropy influences the velocity spectral shapes. This instrument was located very close to the bed (0.15 m) in a shallow waterway less than 2 m deep, which results in the vertical velocity's inertial subrange being reduced by the flattening of the spectra at wavenumbers of 10 cpm (0.1m scales). Strong stratification (or shear) is another mechanism that shortens the inertial subrange at the lower wavenumbers<ref name="Bluteau.etal2011"/>. The wavenumber at which its impact is felt is approximately <math>L_o/3</math> where <math>L_o</math> is the Ozmidov length scale.