Time and length scales of turbulence

The length and time scales of turbulence must be considered when analysing turbulence measurements. The scales at which the turbulence subranges, both the viscous and inertial subrange, exist depend on the flow properties. These characteristics of the flow influence various decisions when processing velocity measurements for turbulence computations such as:

• Segmenting datasets
• Detrending time series

Lengthscales

Example velocities from an instrument impacted by tides and surface waves, which are presented as variance preserving spectra. The instrument is tens of meters off the bottom on the continental shelf in stratified waters with a buoyancy period of 9 min. Turbulence analysis is concerned with time scales larger than the buoyancy period

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High wavenumber limit

For instance, the wavenumber (length scale) limits depend on the sought quantity [math]\displaystyle{ \varepsilon }[/math], and also the kinematic viscosity [math]\displaystyle{ \nu }[/math]:

[math]\displaystyle{ L_K=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4} }[/math]

Low wavenumber limit

Of the order of the largest eddy sizes...

• Stratified waters Lo
• Near a boundary [math]\displaystyle{ L_Z=0.39z_w }[/math] via von karman's constant of 0.39 and zw the distance to said boundary.
• Sheared flows can also swap Lo with Ls (Corrsin). Ls = Lz in a pure log-law of the wall flow!

Timescales

Can use the equation [math]\displaystyle{ \tau=(L^2/\varepsilon)^{1/3} }[/math] and dump your favourite turbulent length scale. In stratified waters, this amounts to 1/N.