Is there a relation between Free area ratio ([porosity] of Perforated plate) and f (of Darcy Forchheimer)

Hello all of you,

I tried to determine a relation between Free area ratio (porosity) of Perforated plate and the f coefficient of Darcy Forchheimer (in case of a tree): see this project: 10 trees by jjansman | SimScale

I found the following results: Simulating a tree
Is there some official/analytic relation between the two?

Thanks for any feedback.

All the best,



Yes, there is a correlation. The perforated plate formulation is available in this book. If you go through the formulation, you will notice that this is based on a Forchheimer model (i.e. d is assumed to be 0).

The Forchheimer term depends on the free area ratio, orifice/discharge coefficients, and thickness of the porous media.


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I will check the book:

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Thanks for this link. A great reference book, IMHO. Can you provide a chapter, section or page number in the book on this porosity/Forchheimer question? I am overwhelmed by the detail of the book, but Forchheimer is not mentioned and thus I was not really able to find a link between porosity and Forchheimer.

It starts around page 1111. If you look at the basic formulation, you will notice that it’s a Forchheimer model (delta P is related to U² only) but with different variables. You can work out a correlation between f and the perforated plate model from there.

Thanks Ricardo, formula (11) and (13) are indeed what I was looking for: φ is porosity and ζ is f (of Forchheimer). I now get similar results (Simulating a tree ) in SIMSCALE as with this Heat atlas section.


I think I was evaluating it wrong. ζ is not f (of Forchheimer) according to your web page (ζ looks to be closer to Cd): How to Predict Darcy and Forchheimer Coefficients (formula (5))
So I need to determine what L is (I assume the size of my object in: 10 trees by jjansman | SimScale )

L would be the length of the porous media in the flow direction.

But I connect the porous medium to an object (in my case a stacked cylinder object with largest diameter of 15.4m), so I assume that the porous material has the L of the largest cylinder (as I am not able to define an L within the Perforated plate (or Darcy-Forchheimer medium)?
I was always wondering how thick a porous object is;-) But I might not understand it all.

The tool will determine it based on the direction of the flow that you set and the geometry.

I expected something like that. As it is are three stacked cylinders, the direction is not that important (for both the perforated plate as the Darcy-Forchheimer medium). So still wondering what the tool would see as L? Which cylinder as I have three different cylinder diameters stacked on top of each?
Perhaps for each cylinder it makes three different porous media?
Perhaps my questions are rhetorical. But it fascinates me how porosity and f interrelated:
I seem not able to untangle it all.

The formulation, as the name suggests, assumes that you are working with a perforated plate, so the length in the flow direction is (normally) going to be easy to tell.

With the base formulation of perforated plate you should be able to derive Forchheimer with a certain thickness, if you’d like to use it for different applications.

There are some models out there that focus on trees specifically.

Thanks Ricardo for you patience.
The problem with that page is the low Cd of 0.2; while I see in literature higher values (see also on my page of my literature study of Cd). I am still totally not understanding all these things.

That is why I tried the (similated) empirical: compare a stacked cylinder with Perforated plate and Darcy Forchheimer medium and see when their behavior in wind velocity was the same (and that is the graph I derived).
So I am now using this emperical way, but I still don’t understand the formula behind it (and thus not the dynamics when I change eg the height of trees).

I think I will step down on this for a few days; see if time will provide me with some more clarity;-)
At this moment I blame myself for not understanding it yet. Thanks again for your patience.

All the best,


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