sandbox/Antoonvh/fan

Mixing an uniformly stratified fluid

Disk-shaped fans of size V = \pi R^2 \times w at a height h above the surface (h>R) are located on a L \times L infinite-grid layout (L>R is the periodicity length scale). Each fan accelerates the hosting (Bounsinesq) atmosphere thet is initially at rest and has a reference density \rho_0 according to a kinetic-energy input rate P (i.e. effective power). The resulting fan’s jets are affected by the presence of a constant vertical density gradient. Due to gravity it is so-called `stable’ and can be associated with a the Brunt-Vaisala frequency (N).

We can define a kinematic-power scale (i.e. \mathcal{P} with units \mathrm{m}^2 / \mathrm{s}^3),

\displaystyle \mathcal{P}=\frac{P}{\rho_0V}.

The six system parameters R, w, h, L, \mathcal{P} and N, with units of length and/or time give rise to four dimensionless groups. E.g.:

Three geometrical groups,

\displaystyle \Pi_1 = w/R \displaystyle \Pi_2 = L/R \displaystyle \Pi_3 = h/R

Being the dimensioness width, periodicity length scale and height. Furthermore, there is a group that concerns the mixing capacity of a fan:

\displaystyle \Pi_4 = \frac{\mathcal{P}}{R^2N^3}

Noting that the `mixing capacity’ should be interpreted with respect to the geometrical parameters.

Additional complexity

The orientation of the fan is described by it’s angles with respect to some reference coordinate system. In general a fan can be rotated over three angles (say \theta, \phi, \psi). If,

\displaystyle \{\theta, \phi, \psi\} = \overrightarrow{f}(t)

Then the formulation of \overrightarrow{f}(t) is an important system specification that affects mixing aswell. However, addional complexity arises as the functional form of f may not live in an ordered parameter space.