Permeable or porous media are used in CFD to model complex components that are not completely solid. In the real world, these are, for example, wire meshes, perforated facades and claddings, louvers, tube banks (stacks of horizontal cylinders), and so on. Models of these structures can have such a complex geometry that they cannot be meshed efficiently, the generated mesh can be extremely fine or, in some cases, of a poor quality. Such cases lead to a hardware and time-consuming computation or to an inaccurate computation. Therefore, it is preferable to use a model of permeable media for such structures.
Based on physical reasoning from experimental measurements, we assume that in the permeable zone, the energy is removed from the flow as a pressure drop. We assume that with growing velocity through the permeable zone, the pressure drop grows. The pressure drop across the zone can be expressed by the polynomial function of the velocity, where the linear part is the viscosity term, and a quadratic part is the inertial term (the dynamic head):
Next, we implement the permeability effect in the Navier-Stokes equations (N-S equations).
Permeability in N-S Equations
A brief introduction to the numerical modeling of permeability follows. The permeability effect is added as a source term to the right side of the N-S equations. It is important not to include the pressure drop directly in our equations, but the source term should be expressed in terms of the pressure drop. The source term S is applied at the cell centroids of the permeable zone, the S term is zero in the cells where the permeable zone is not defined, see the image below.
The source term is a force expressed by the pressure drop related to the cell volume. After some modifications, the source term for the equation in the flow direction can be written in the following form:
The length (thickness) of the permeable medium L expresses the thickness of the permeable medium in the flow direction.
Now, we have the source term, which describes the pressure loss in the permeable medium. Next, it is necessary to specify the coefficients:
To do that, another relation is needed, which is the Darcy's law. The Darcy's law is valid for a slow laminar flow through permeable media for small Re numbers. It is given by the relation:
| α[m2] | Permeability |
| μ[Pa.s] | Dynamic viscosity of the fluid |
| Ux[m/s] | Velocity of the fluid in the x-direction |
Comparing this to the general pressure drop relation, an equation for C1 is obtained:
Darcy's law gives us a relation for C1 as a function of dynamic viscosity, permeability and length of the permeable media, further we need to specify the coefficient C2. There are several ways of how to do this. Either you can use the empirical data obtained from the measurement of the pressure drop and the velocity or flow rate across the permeable zone. The coefficient C2 can be fitted from the polynomial regression of the measured data. Or you can use some published data, for example, the empirical data for a permeable disk, compare it with the geometry (compare the number of holes and their geometry), and derive the coefficients. Some approaches for the coefficient determination can be found here, this approach is described in our Knowledge Base Article.