Every CFD model requires a balance of model accuracy and model runtime. Depending on the accuracy needed, the CFD user must choose which areas of the geometry or physics to simplify in order to speed up the simulation without introducing significant modelling/numerical errors. Certain models can benefit greatly in runtime from geometry simplification in ways that will have a negligible impact on solution accuracy. One method of geometry simplification is the use of Porous Media.
Porous Media is used in CFD to represent volumes where both a structured solid and fluid are interspersed, i.e. a porous material. This media then accounts for the macro scale effects of flow resistance (pressure drop) and heat transfer by allowing fluid to flow through it without having to use lots of mesh to capture the small details. Some examples include water flowing through a filter, air flowing through a plant canopy, or a fluid flowing through a heat exchanger. In order to avoid modelling the detailed objects separately (individual filter fibers or plant leaves could take tens/hundreds of millions of cells and is unfeasible for most analysis), the porous media can instead be modelled in large groups and assigned bulk properties which account for resistance to fluid flow reducing the mesh count by a factor of 1000 or more.
Two variables are used to define the resistance to flow that are needed to characterize the material’s bulk properties. Those variables are permeability (viscous loss coefficient) and an inertial pressure loss coefficient. Combining these variables with information from the flow field (velocity components) and fluid properties (viscosity and density), creates a momentum source term which reduces the momentum of the fluid as it passes through the porous media, thus accounting for flow resistance. The source term is defined as:
where Si is the pressure loss per unit length (e.g. psi/ft), α is permeability, μ is the fluid viscosity, ρ is density, vi is the fluid velocity vector through the media, and C2 is the inertial pressure loss coefficient (note: porous media can be isotropic or non-isotropic). For the most accurate modelling of a porous media, the permeability and the pressure loss coefficient should be obtained experimentally by measuring pressure loss (e.g. psi) across the porous object while varying the velocity. A second-order polynomial could be fit to this data where the linear and quadratic coefficients from the polynomial would provide equations for the values of permeability and pressure loss coefficients. In the absence of experimental data, a numerical experiment can be set up to model a small section of the material at the detailed level. Below is an example of a numerical experiment which models the pressure loss as air flows through a plant canopy which was 0.5 m thick (L = 0.5m).
The velocity was varied from 0.25 to 2.5 m/s and the total pressure was recorded both upstream and downstream of the canopy. In this example, the linear coefficient is -0.0367 and the quadratic coefficient is 12.631:
A = -0.0367
B = 12.631
Note: the graph above is showing positive values for y which indicate that the total pressure is being reduced. Since |A| << |B|, this shows that pressure loss due to viscous effects is very small compared to inertial loss. In reality, the coefficients cannot be less than zero. If a coefficient is less than zero, then this indicates imperfect data (it would produce a pressure gain at some velocities). A correction should be made to set the negative coefficient to 0 or a very small positive number such as 0.001. The coefficients α and C2 can be calculated by:
Using air properties:
μ = 1.83e-5 [kg/m-s]
ρ = 1.204 [kg/m3]
And setting A = 0.001, produces:
α= 1.830e-2 [m2]
C2 = 20.98 [1/m]
Numerical Experiment Geometry and Flow Visualization
System Level Model using Porous Media
In this example, porous media was able to accurately represent the presence of the plant canopy and produce a resistance to the fluid flow without having to explicitly model each plant leaf. In addition to flow resistance, porous media also allows for thermal conduction which depends on the material porosity and other properties.
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