POLYFLOW - Too many holes: Calibrating the parameters for the Darcy equation
Let us assume a flow through a large channel, whose exit consists of several small holes of length L, as illustrated in the figure Darcy_equation_holes.gif attached below. Not only the generation of the grid can be difficult and tedious, but this may also require the definition of an unmanageably large number of boundary sides.
Figure: <a target=_blank href="http://www.fluentusers.com/support/solutions/1016/Darcy_equation_holes.gif">http://www.fluentusers.com/support/solutions/1016/Darcy_equation_holes.gif</a>http://www.fluentusers.com/support/solutions/1016/Darcy_equation_holes.gif
How can one model this?
A reasonable alternative consists of replacing all holes with a non-isotropic porous media, where the Darcy equation is solved. For such a modelling, a single subdomain has to be generated at the exit with the flow channel. In a POLYDATA session, the simulation will consist of two sub-tasks: a flow in the channel and a flow through a porous media, with an interface between both.
Let v and p denote the locally averaged velocity vector and pressure in the porous media. The Darcy equation is written as
v = -(K/eta).grad(p)
where K is the tensorial permeability, while eta is the (constant) fluid viscosity.
For an isothermal flow, the ratio K/eta is of interest, and needs to be properly identified. This is the calibration procedure.
2. Calibration procedure.
The first part of the calibration procedure consists of evaluating the pressure drop through a single hole, by using the actual fluid properties. Typically, this is a 2D axisymmetric flow calculation. The fluid flow rate q through that hole should be given as the ratio of the overall flow rate Q to the total number of holes n. This will produce a given pressure drop Dp.
The second step consists of identifying the appropriate value of the coefficient K/eta. This can easily be done in a one-dimensional form. Let S denote the area of the channel right upstream of the exit, i.e. right before entering the porous media. The average velocity at the inlet to the porous media is given by
v = Q/S = (k/eta).(Dp/L)
This scalar form of the equation allows the identification of the quantity (k/eta). Especially for non-isothermal situations, a relevant value should be selected for eta, in order to incorporate the appropriate level of viscous heating.
3. Non-isotropic porous media
In the original flow channel, it is obvious that the fluid flow is oriented along a preferred direction. Hence, a permeability tensor should be considered. The diagonal component of that tensor K along the preferred flow direction should receive the value k calculated above. The other diagonal components of K should be assigned very low (but non vanishing) values, e.g. 10^-12, while the non-diagonal components may receive zero values.