von-Karman Pao Spectrum

The von-Karman Pao spectrum is given by the formula (see Bailly and Juve, 1999 ):

\begin{equation*} E(\kappa) = \alpha \frac{u’^2}{\kappa_\text{e}}\frac{ (\kappa/\kappa_\text{e})^4 }{\left[ 1 + (\kappa/\kappa_\text{e}) \right]^{17/6}}\exp{\left[ -2 \left(\frac{\kappa}{\kappa_\eta}\right)^2 \right]} \label{eq:vkp} \end{equation*}

where $\alpha$ is a scaling constant, $u’$ is the RMS value of the velocity fluctuations, $\kappa_\text{e}$ is related to the wavenumber where energy is maximum, $\kappa$is the wave number, and $\kappa_\eta = \epsilon^{1/4}\nu^{-3/4}$ is the Kolmogorov wave number (smallest turbulent structures). Note that $\epsilon$ and $\nu$ are the dissipation rate and molecular viscosity, respectively. For isotropic turbulence (Bailly and Juve, 1999), $\alpha \approx 1.453$$, $$\epsilon \approx u’^3 / L$ with $L$ being the integral length scale $L \approx 0.746834/\kappa_\text{e}$. The maximum energy occurs at $\sqrt{12/5}\kappa_\text{e}$.