Using Porous Media to Model Flow in Tubes

In most porous media the additional source terms in the momentum equation, which are used to create the streamwise pressure gradient, dominate the convection and viscous fluxes. For this reason, the convection and viscous fluxes are not explicitly neglected in Simcenter STAR-CCM+.

To omit explicitly the convection and viscous fluxes in porous media, the Porous Media Flux Option is available under the Physics Conditions node of porous regions. This option is deactivated by default. When it is activated, the fluxes are omitted. The flow solver is less robust with this option activated, since approximations are made in the linearization of the discretized equations. Therefore it is recommended that the default setting is used for truly porous media.

In some situations, you can model the flow in a tube without properly resolving the flow. In this case, if the porous media tensor coefficients are used to specify the pressure drop, they are unlikely to dominate the convection and viscous fluxes. The resulting pressure drop does not precisely balance the specified body force. By explicitly neglecting the convection and viscous fluxes, the momentum equation reduces to:

Figure 1. EQUATION_DISPLAY

\frac{d p}{d x} = - (P_{i} v^{2} + P_{v} v)

(245)

where $P_{i}$ and $P_{v}$ are the equivalent scalar coefficients of the inertial and viscous porous resistance tensors, respectively. See Eqn. (239), Eqn. (240), and Eqn. (241).

This relation is often erroneously termed “Darcy’s Law,” which in fact expresses a linear relationship between the flow rate and the pressure drop through porous media such as groundwater flow. Similar linear relations exist for fully developed, incompressible, laminar pipe flow, where the pressure drop per unit length can be expressed as a function of the mean velocity $u$ , dynamic viscosity $μ$ , and pipe radius $r_{0}$ :

Figure 2. EQUATION_DISPLAY

\frac{Δ P}{Δ L} = - \frac{8 μ u}{r_{0}^{2}}

(246)

( $u = Q / A$ , where $Q$ is the volumetric flow rate and $A$ is the cross-sectional area of the pipe.)

In this type of flow, the viscous fluxes dominate. These fluxes could be neglected by using a slip-wall boundary condition. The convective fluxes cancel each other out, unless there are significant entrance effects, in which case the pipe-flow relation can become non-linear in their presence. The convective fluxes also become more significant in the presence of varying fluid properties, such as in compressible flow.