Nomenclature

Frame of reference

• Define frame of reference, and notation. Use u,v,w and x,y, and z?
• Decomposition of total, mean, turbulent and waves.

Turbulence Spectrum

Taylor's Frozen Turbulence for converting temporal to spatial measurements $({\overline{u}}_1\frac{\partial }{\partial x}=\frac{\partial }{\partial t})$

• Missing the y-axis.

Symbol	Description	Eqn	Units
$f$	Frequency	$\frac{\omega }{2\pi }$	Hz
$f_n$	Nyquist frequency	$f_n=0.5f_s$	Hz
$f_s$	Sampling frequency	$\frac{1}{\mathrm{\Delta }t}$	Hz
$k$	Wavenumbers (angular)	$k=\frac{f}{\overline{u}}=2\pi \hat{k}$	rad/m
$\hat{k}$	Wavenumbers	$\frac{k}{2\pi }$	cpm
${\hat{k}}_{\mathrm{\Delta }}$	Nyquist wavenumber, based on sampling volume's size $\mathrm{\Delta }l$	${\hat{k}}_{\mathrm{\Delta }}=\frac{0.5}{\mathrm{\Delta }l}$	cpm
${\hat{k}}_n$	Nyquist wavenumber, via Taylor's hypothesis (temporal measurements)	${\hat{k}}_n=\frac{f_n}{u}$	cpm
$\mathrm{\Delta }t$	Sampling interval	$\frac{1}{f_s}$	s
$\omega$	Angular frequency	$2\pi f$	rad/s

Theoretical Length and Time Scales

Symbol	Description	Eqn	Units
$ϵ$	Turbulent kinetic energy dissipation		W/kg
$\eta$	Kolmogorov length scale (smallest overturns)	$\eta ={\left(\frac{{\nu }^3}{ϵ}\right)}^{1/4}=\frac{1}{2\pi {\hat{k}}_K}$	m