Streamwise turbulent intensity under unstable atmospheric stratification explained by a spectral budget

etc15 Tracking Number 351

Because of its non-conformity to Monin-Obukhov Similarity Theory (MOST), the effects of thermal stratification on scaling laws describing the stream-wise turbulent intensity $\sigma_u$ normalized by the turbulent friction velocity ($u_*$) continues to draw research attention. The streamwise turbulent intensity happens to be of utmost importance as a direct measure of the intensity of turbulence and an analytical model able to predict its nature would be considered useful for a copious number of practical applications- ranging from industrial pipe flow to air pollution modeling among many. A spectral budget method used previously by \cite{Banerjee2013} was demonstrated as a suitable workhorse to analytically explain the `universal' logarithmic scaling law exhibited by $\sigma_u^2/u_*^2$ for neutral conditions as reported in different high Reynolds number experiments. In the present work \cite{Banerjee2014}, that theoretical framework has been expanded to assess the variability of $\sigma_u/u_*$ under unstable atmospheric stratification. At least three different length scales- the distance from the ground ($z$), the height of the atmospheric boundary layer ($\delta$), and the Obukhov length ($L$) are all found to be controlling parameters in the variation of $\sigma_u/u_*$. Analytical models have been developed and supported by experiments for two limiting conditions: $z/\delta<0.02$, $-z/L<0.5$ and $0.02 <0.5$. Under the first constraint, the turbulent kinetic energy spectrum is predicted to follow three regimes: $k^0$, $k^{-1}$ and $k^{-5/3}$ divided in the last two-regimes by a break-point at $kz=1$, where $k$ denotes wavenumber. The $\sigma_u/u_*$ is shown to follow the much discussed logarithmic scaling reconciled to Townsend's attached eddy hypothesis $\sigma_u^2/u_*^2= B_1-A_1 log(z/\delta)$, where the coefficients $B_1$ and $A_1$ are modified by MOST for mildly unstable stratification. Under the second constraint, the turbulent energy spectrum tends to become quasi inertial, displaying a $k^0$ and a $k^{-5/3}$ with a breakpoint predicted to occur $0.3