class: center, middle # ABL project: progress so far ## Ashwin Vishnu Mohanan <!-- <img src="https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/dp_eturb.svg" style="height: 5em;"/> --> ### Slides: [fluid.quest/talks](https://fluid.quest/talks) <img src="https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/dp_su.gif" style="height: 3em;"/> <img src="https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/dp_kth.svg" style="height: 3em;"/> <div style="font-size: 0.8em;"> <a rel="license" href="http://creativecommons.org/licenses/by/4.0/"><img alt="Creative Commons License" style="border-width:0; height: 1em;" src="https://i.creativecommons.org/l/by/4.0/88x31.png" /></a>  All text in this presentation is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0 International License</a>. Images and screenshots are copyright material. </div> --- # In short .pull-left[ ## Milestones so far - **Channel flow**: Neutral stratification; Open and closed; DNS and LES; with and without rotation (Coriolis force) - **LES subgrid scale models**: constant Smagorinsky, Vreman, shear improved Smagorinsky - Outflow (top) sponge layer - DNS with thermal BC - constant and dynamic fluxes at the bottom (Moeng-like) ] .pull-right[  ] --- # In short ## What's next - Implement GABLS1 case of moderately stable stratified boundary layer - Adding stability functions $\phi_m(z/L)$ and $\phi_h(z/L)$ into the boundary conditions. - Improved outflow (top) boundary condition? - SGS models for thermal diffusion. Simple turbulent Prandtl number based or more versatile ones. **Requires**: variable properties (both momentum and thermal diffusivity simultaneously) bug fix in Nek5000. - Bug fixes on penalty --- .pull-left[ # GABLS1 case description - <https://gabls.metoffice.com/>, Beare et al. (2006) Boundary-Layer Meteorology - Domain: 400m × 400m × 400m - Resolution between 12.5 m and 1m - Geostrophic wind, $U_g = 8$ m /s - Coriolis parameter, $f = 1.39e-04$ - Initial condition: - constant pot. temperature at 265K for $0 < z < 100$ m, then increasing at 0.01 K/m. - random noise perturbations on both velocity and pot. temperature fields. ] .pull-right[   ] --- # The wall model We wish to model: - Turbulent momentum flux / shear stress $\tau_{ij} = -\rho \overline{u'_iu'_j } \approx - \rho (u'w' + v'w')$ - Turbulent heat flux $q_* \approx \overline{w'\theta'}$ - Boundary condition, following Moeng (1984)  See Gadde et al. (2020) Boundary Layer Meteorology --- # Monin-Obhukhov Similarity Theory (MOST) ## Governing parameters and assumptions Turbulence in the surface layer is determined by: 1. length scale \\( \sim z \\) 1. turbulent momentum flux or stress at surface \\( \tau = \rho u_*^2 \\) 1. turbulent temperature flux at surface \\( \sim Q_0 \\) 1. buoyancy parameter \\(\sim g/\theta_0 \\) 1. momentum flux or shear \\( \sim \partial U / \partial z \\) ## Buckingham Pi theorem - \\( m = 5 \\) parameters - \\( n = 3 \\) dimensions: length, time, temperature implies the model can be rewritten using \\( m - n = 2 \\) independent dimensionless quantities. --- ## M-O similarity functions .pull-left[ Similarity variable taken as \\( z/L \\), where *Monin-Obukhov length*: \\[ L = -u_*^3 \theta_0 / \kappa g Q_0 \\]   \\( L \\) being a function of the turbulence statistic \\( u_* \\) restricts a closed form solution of the M-O functions. ] .pull-right[ The 1968 Kansas experiment measured the similarity function for different z and L and confirmed the similarity theory:  ] --- ## M-O functions and log-law In **neutral** conditions, \\( z/L \to 0 \implies \phi_m \approx 1\\) and we recover the classical log-law. \\[ \frac{\partial U}{\partial z} = \frac{u_*}{\kappa z} \\] \\[ U(z) = \frac{u_*}{\kappa} \left[ \ln z - \ln z_0 \right] \\] Compare the term \\( \ln z_0 \\) to roughness parameter \\(B\\). For **stable** and **unstable** regimes, curve fitting gives,  --- # Computing stability functions * Stability parameter $z/L$ depends on the fluxes - Either we iterate between Obhukhov length and the fluxes get an agreement. - Issue when the SGS scale is too large then one can have the same flux ($q = w'\theta'$) for two different $z / L$. * Or use one-to-one relation between flux and gradient Richardson number. ---