Velocity inertial subrange model: Difference between revisions
| Line 9: | Line 9: | ||
<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:InertialSubrange.png|thumb| | [[File:InertialSubrange.png|thumb|Sketch of velocity power density spectrum in log-log space. The vertical axis represents <math>\Psi_{Vj}(\hat{k})</math>]. The inertial subrange's -5/3 slope is highlighted.] | ||
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 | ||
Revision as of 21:38, 11 November 2021
| 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 maybe discussed within the wiki.
Inertial subrange for steady-flows
This theoretical model predicts the spectral shape of velocities in wavenumber space.
[math]\displaystyle{ \Psi_{Vj}(\hat{k})=a_jC_k\varepsilon^{2/3}\hat{k}^{-5/3} }[/math]
[[File:InertialSubrange.png|thumb|Sketch of velocity power density spectrum in log-log space. The vertical axis represents [math]\displaystyle{ \Psi_{Vj}(\hat{k}) }[/math]]. The inertial subrange's -5/3 slope is highlighted.]
Here [math]\displaystyle{ \hat{k} }[/math] is expressed in rad/m and [math]\displaystyle{ Vj }[/math] represents the velocities [math]\displaystyle{ V }[/math] in direction [math]\displaystyle{ j }[/math]. [math]\displaystyle{ C_k }[/math] is the empirical Kolmogorov universal constant of C = 1.5 [1]. Amongst the three direction, the spectra deviates by the constant [math]\displaystyle{ a_j }[/math]: [2]
- In the longitudinal direction, i.e., the direction of mean advection (j=1), [math]\displaystyle{ a_1=\frac{18}{55} }[/math]
- In the other directions [math]\displaystyle{ a_2=a_3=\frac{4}{3}a_1 }[/math]
Inertial subrange for flows influenced by surface waves
Need to add equations and figures from Lumley & Terray[3]
Notes
- ↑ K. R. Sreenivasan. 1995. On the universality of the Kolmogorov constant. Phys. Fluids. doi:10.1063/1.868656
- ↑ S.B Pope. 2000. Turbulent flows. Cambridge Univ. Press. doi:10.1017/CBO9780511840531
- ↑ J. Lumley and E. Terray. 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
