Governing Equations for 3D atmospheric flow

Incompressible Reynolds Averaged Navier Stokes (RANS) mass and momentum conservation equations with anelastic approximation are solved numerically for wind farm simulations

\[\dfrac{\partial \rho u_i}{\partial x_i}\]

\[\dfrac{\partial \rho u_i}{\partial t} + \dfrac{\partial \rho u_i u_j}{\partial x_j} = -\dfrac{\partial p}{\partial x_i} + \dfrac{\partial}{\partial x_i}\left[ \mu^t \left(\dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i} - \dfrac{2}{3}\dfrac{\partial u_k}{\partial x_k}\delta_{ij}\right)\right] + \rho g_i\]

where \(\boldsymbol{u}=(u,v,w)^T\) is the velocity, \(\boldsymbol{x}=(x,y,z)^T\) is the position vector, \(\rho\) is the air density, \(p\) is the pressure, \(\boldsymbol{\delta}\) is the Kronecker symbol and \(\boldsymbol{g}\) is the acceleration of gravity. The turbulent viscosity \(\mu^t\) is modeled using a \(k-\varepsilon\) turbulence model with linear production.

Depending on the input parameters and modeling choices, a transport equation for the potential temperature \(\theta\) is solved

\[\dfrac{\partial \rho \theta}{\partial t} + \dfrac{\partial \rho u_i \theta}{\partial x_i} = \dfrac{\partial }{\partial x_i}\left[\dfrac{\mu^t}{\sigma_t}\dfrac{\partial \theta}{\partial x_i}\right]\]

with \(\sigma=0.7\) the thermal Schmidt number.

More details about the theoretical framework and numerical methods can be found in code_saturne’s online documentation.