Velocity inertial subrange model: Difference between revisions

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


[[File:InertialSubrangeSchematic.png|thumb|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 [[Anisotropic turbulence|turbulence anisotropy]] in low energy flow may alter the expected spectral shape]]
[[File:InertialSubrangeSchematic.png|thumb|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 [[#anisotropy|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
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
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* In the other directions <math>a_2=a_3=\frac{4}{3}a_1</math>
* In the other directions <math>a_2=a_3=\frac{4}{3}a_1</math>


== Inertial subrange for flows influenced by surface waves ==
== Inertial subrange collapse and <span id="anisotropy">anisotropy</span> ==
{{FontColor|fg=white|bg=red|text=Need to add equations and figures from Lumley & Terray}}<ref name="Lumley_Terray">
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">{{Cite journal
{{Cite journal
|authors= A. E. Gargett, T. R. Osborn, and P.W. Nasmyth
|authors=J. Lumley and E. Terray
|journal_or_publisher= J. Fluid. Mech.
|journal_or_publisher=J. Phys. Oceanogr
|paper_or_booktitle= Local isotropy and the decay of turbulence in a stratified fluid
|paper_or_booktitle=Kinematics of turbulence convected by a random wave field
|year= 1984
|year=1983
|doi=10.1017/S0022112084001592
|doi= 10.1175/1520-0485(1983)<2000:KOTCBA>2.0.CO;2
}}</ref><ref name="Bluteau.etal2011">{{Cite journal
}}
|authors= C.E. Bluteau, N.L. Jones, and G. Ivey
</ref>
|journal_or_publisher= Limnol. Oceanogr.: Methods
|paper_or_booktitle= Estimating turbulent kinetic energy dissipation using the inertial subrange method in environmental flows
|year= 2011
|doi=10:4319/lom.2011.9.302
}}</ref>.


== Inertial subrange collapse and anisotropy ==
[[Velocity inertial subrange model#anisotropy|Anisotropic velocity spectra]] are exhibited when the largest turbulence scales are less than {{FontColor|fg=white|bg=red|text=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 [[Rotation of the velocity measurements|rotate the velocity]] in the direction of the mean flow.  
Near boundaries or low energy environments, are defined as flows with a small separation between the large turbulent overturns <math>L</math> and the smallest (Kolmogorov).


{{FontColor|fg=white|bg=red|text=Add example spectra, and link to Kolmogorv, Maybe refer to SV94}}
[[File:Anisotropy.png|center|thumbnail|600px|Example of how  [[Velocity inertial subrange model#anisotropy| 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.]]


== Notes ==
== Notes ==

Latest revision as of 20:24, 8 July 2026


Short definition of Velocity inertial subrange model
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 may be discussed within the wiki.



Model for steady-flows

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

ΨVj(k^)=ajCkε2/3k^−5/3

Sketch of velocity power density spectrum in log-log space. The inertial subrange's -5/3 slope is highlighted. The vertical axis represents ΨVj(k^). Large scale turbulence anisotropy in low energy flow may alter the expected spectral shape

Here k^ is expressed in rad/m and Vj represents the velocities V in direction j. Ck is the empirical Kolmogorov universal constant of C = 1.5 [1]. Amongst the three direction, the spectra deviates by the constant aj: [2]

  • In the longitudinal direction, i.e., the direction of mean advection (j=1), a1=1855
  • In the other directions a2=a3=43a1

Inertial subrange collapse and anisotropy

Near boundaries or low energy environments--defined as flows with a small separation between the large turbulent overturns L and the smallest (Kolmogorov)-- tends to adversely impact our ability to estimate ε from the lower wavenumbers. In certain cases, the velocity spectra may not have a sufficiently developed inertial subrange to estimate ε [3][4].

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 ε. In these situations, it may be possible to use the longitudinal velocity component (see Bluteau et al. 2011[4]), 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[4]. The wavenumber at which its impact is felt is approximately Lo/3 where Lo is the Ozmidov length scale.

Notes

  1. ↑ K. R. Sreenivasan. 1995. On the universality of the Kolmogorov constant. Phys. Fluids. doi:10.1063/1.868656
  2. ↑ S.B Pope. 2000. Turbulent flows. Cambridge Univ. Press. doi:10.1017/CBO9780511840531
  3. ↑ A. E. Gargett, T. R. Osborn, and P.W. Nasmyth. 1984. Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid. Mech.. doi:10.1017/S0022112084001592
  4. ↑ 4.0 4.1 4.2 C.E. Bluteau, N.L. Jones, and G. Ivey. 2011. Estimating turbulent kinetic energy dissipation using the inertial subrange method in environmental flows. Limnol. Oceanogr.: Methods. doi:10:4319/lom.2011.9.302
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