Coefficients of the Darcy-Forchheimer equation for trees
 

Hello all of you,


I am looking for the d and f coefficients of the Darcy-Forchheimer equation for some typical trees (like oak).
In some places I hear that d=0 for trees and that f is around 0.13 for an oak tree. But is that true? Using these values I get unexpected results, aka resulting in a seemingly too high porosity.


Do people have typical d and f coefficients for some tree species?


Thanks for hints/ideas/links.


All the best,


Victor

What value is Cd and f for determing Forchheimer formula
 

I have read several documents now on Cd definition/value:

• The windspeed is important for the value of Cd.
• Difference between true area (the area of visible branches and leafs) and enclosed area (the area of the crown as whole).
• What is the relation of literature Cd’s with LAD or LAI?
• The Cd can be when the tree is a static/rigid object or dynamic when the tree form changes due to the wind velocity.
• The size of the tree also determines the Cd: the larger the tree the larger the Cd.
• Sometimes it looks that Cd is perhaps the Cm (intertial resistance coeffcient).

Looking at this webpage:
dp/dx=-rhoLADCd*|u|*u
From this formula, f would be
f = 2LADCd = 2LAI/HCd
<H is height of tree; LAI Leaf Area Index>


SIMSCALE uses a Cdsim = 0.2 (regardless of the u and H). LAD is depending on the height, but its behavoir is the reverse to what the Cd has.
Hu (2018, page 23) provides an average Cdha of 0.655 for u’s between 4 and 10m/sec (unknown H).
Bekkers (2022, Fig. 9) has a Cdbek of around 0.77 (at u = 5m/sec and H=6.5m).
Ren (2023, page 7) has a Cdren of around 0.704 (at u = 10m/sec and H=6.8m).
Koizuma (2010) gives an idea of Cdkoi for poplar (H = 12.5m):

• Without leaves the Cdkoi is around 0.2 (Koizuma, 2010, Fig. 7) for u’s between 4 and 11/sec.
• A poplar has a Cdkoi (Koizuma, 2010, Fig. 6) between 0.55 (u = 4m/sec) and 0.3 (u = 11m/sec).

Changing the tree model to a more solid object with an fsim = 0.7 (at u = 6.44m/sec @ 10m) might hopefully result in a better match with the real tree of Ren (2023, Fig. 13).
Still not sure!!! Can you help?




References:
Bekkers, Casper C.A. et al.: Drag coefficient and frontal area of a solitary mature tree. In: Journal of Wind Engineering and Industrial Aerodynamics 220 (2022), pp. 1-11.
Ha, Taehwan: Development of 3D CFD models and observation system design for wind environment assessment over a clear-cut in mountainous region. PhD 2018.
Koizuma, Akio et al.: Evaluation of drag coefficients of poplar-tree crowns by a field test method. In: Journal of Wood Science 56 (2010), issue 3, pp. 189-193.
Ren, Xinyi et al.: The influence of wind-induced response in urban trees on the surrounding flow field. In: Atmosphere 14 (2023), issue 1010, pp. 1-23.

d 按
 

Hello, I am not a native English speaker, this is my first time to come to the forum, so if my language if offended you, please forgive me, I did not mean any harm.
My senior thesis was on using porousSimpleFoam to calculate urban tree wind fields.
I have read many papers on the d and f coefficients in Darcy's model, and I can give you a little information:
The first thing is that a lot of people just treat d as 0;
The second point is that f is about 0.16 (I calculated it myself).

But one thing is that at present, my supervisor has asked me to use the data in this paper to calculate:
“Improving accuracy in simulation of urban wind flows by dynamic downscaling WRF with OpenFOAM”

To be honest, I think the data in this paper is not quite right, because the data in this paper and the data in the paper it cites seem to be contradictory.

These are my thoughts, thank you.

I forgot to mention that the data given in that paper is D=440.8 and F=12.2
I don't think that's quite right, but is it possible that I'm missing something

No problem at all as your text read very well. So no need to apologies!
That is indeed what I have found
I find this f value quite small; although it is similar to what SIMSCALE uses as default (=0.2). When using f=0.2 in SIMSCALE I got trees that looked too porous:
https://www.simscale.com/projects/vr..._3_iets_hoger/
an f=0.45 matched a tree that had been measured in real live (Koizuma).
But as you can seen my query, I am still not 100% sure!!!
I will also have a look at this paper. Thanks.


All the best,


Victor

Hello FanJin,


Have you progressed on your research? I would be interested. I think I now have a sounder relation between porosity and Forchheimer coefficient: http://www.archaeocosmology.org/eng/...m#contribution


I still have problems of consistency in usage within literature around Cm (f) and Cd:
http://www.archaeocosmology.org/eng/...htm#literature





All the best,


Victor

Hello victor,

I'm sorry I didn't check the notifications in time, but lately I haven't been focusing on CFD simulation, I've been writing code for tree positioning. But I have to say, your article is very good, and I run into a similar problem: there is no single standard form of Darcy Forchheimer's Law. For example, I came across the following two forms in my note

So I think experimentally determining the coefficient is probably the best way (but it takes too much time).
And since the Forchheimer term doesn't have a standard form, I chose the Ergun_equation to calculate the inertia term.
(Because it fits in well with the results of the paper)

In general, I finally chose to use the formula 3 in my notes website to calculate, which means that I did not use Cd (drag coefficient), LAD and other parameters to calculate the Forchheimer term. Consult my note
After I write the code, I will run the CFD simulation next week, and I will give the results here.
I hope this helps.

Fanjin

Hello Fanjin,


I was planning to restructure the formulas I have a bit. I think there is some unclarity (or by me or by others;-). So we have formula that describe:

• Δp
  So here the length of the porosity is on the right side. So Δp is over the full length L.
• we have the differentiate of Δp.
  This is de change (which is assumed to be constant) of Δp over L. I want to describe this not as Δp/Δx, but as dΔp/dx. This is IMHO closer what is done.
• formula looked at the Drag force F (https://blossoms.mit.edu/sites/defau...ce-Handout.pdf )
  In this case we have the area on the left side. And I understand here that (dΔp/dx)*A=F
  I write (dΔp/dx)*A, if there is no L on the right side. If there is an L on the right side, it should be: Δp*A=F

I think using the above points, it might become clearer. So I am change the formula a bit to reflect the above.



It looks you are still involving the Darcy coefficient (d), while I understand from literature that that coefficient is very close to zero. So it is neglected. I assume that that neglection is ok. What do you think?


By the way, I think that using the porosity might even be handier. The porosity can be determined by photographing the tree (by determining how much sky can be seen through the leaves of the crown; which is depending on wind speed...). This is IMHO much easier/better(?) than using the LAI (or LAD). There is a relation between porosity and f: http://www.archaeocosmology.org/eng/...m#contribution


I am wondering what you think?


All the best,


Victor

Hello Victor,

The neglection of d is absolutely ok.It's just that in terms of formula derivation, for the sake of precision, I kept it. But in the actual simulation, I also directly ignored.

I completely agree with what you said about using the porosity parameter to calculate, just as I said before, Ergun-equation also describes Forchheimer terms by porosity and pore size.

But if I may offer one thought,your article,the relationship between f and porosity is mentioned in the article, which is a one-to-one correspondence. I mean, I think not only porosity will affect f, but also pore diameter.

Under the same porosity, there can be different pore diameters, and different pore diameters have different effects on the inertial obstruction of the fluid, that is, the Forchheimer term is different, so f is also different.

I hope you don't mind me saying that I have no ill will towards your research results, just a few trivial thoughts.

From the power point of view, the mathematical formula presented in the article should be appropriate.

As for the formula adjustment you mentioned, forgive me for being blunt, do you have more detailed context? I don't quite understand what you mean, it seems that you meant to say second-order differentiation.

I hope this helps.

Fanjin

Hello Fanjin,


I will think about this idea of the pore diameter. In SIMSCALE (which I use) one can only provide the porosity (%). But I can understand though that this pore diameter is important.


By the way, I am very glad with your feedback. So please do. I am not an expert and I am still learning!


About the formulas: I find the formulas somewhat confusing, so I try to define it a little stricter. So I now define Δp as the pressure difference over a length L (aka the full porous medium). And some formula are using Δp/Δx while I would use dΔp/dx. I find that usage more clear (there is no difference in the effect...).
So no second order derivative;-)


I hope by using this stricter formulation I can untangle the different usages in literature better (as least more understandable for myself). Perhaps this evening I will adjust the typography of the formulas;-)


All the best,


Victor

By the way, I thought that Cd was equivalent to Forchheimer coefficient (f). but that might not be correct. Cd seems to be dimensionless, while f looks to be 1/m.
I have not yet fully resolved this issue!
<I though have updated the formulas using 'my' convention: http://www.archaeocosmology.org/eng/...htm#literature >


All the best,


Victor

f is dimensionless.


Please don't try to outsmart the formulas. A pressure drop over length L is the same as dp/dx


Cd plays a simiar role as f but they have different use cases. Cd is applied to flowrate, f is applied to pressure drop

Here they give a different view on the dimension of f:
https://www.simscale.com/knowledge-b...ical-approach/ (formula 4) and https://www.researchgate.net/publica..._Approximation (text below formula 1). In these f looks to be [1/m].


I get the feel I still don't understand it well enough. Perhaps your "Cd plays a similar role as f but they have different use cases. Cd is applied to flowrate, f is applied to pressure drop" is the clue, but I seem not to grasp this.

I wish I could. Is there any 'simple' link/document/etc explaining this?



I want to understand it.


All the best,


Victor

LuckyTran,


You say "Please don't try to outsmart the formulas. A pressure drop over length L is the same as dp/dx" but a 'pressure drop over L' is the total drop over that length L, while dp/dx is a gradient. Or not? So it are different measures. Or do I misunderstand?

Hello Victor

I am curious about your simulation.
Would you mind telling me about your simulation condition?
How you set the velocity and the pressure initial condition.
I am not practised in CFD, so I want to learn from you.
And this is my simulation result:simulation result


I simply set up a cubic basin space, added a porous medium in the middle, and then the inlet speed was given to 3m/s. I wondered how the pressure and speed in the inner region should be given according to the actual situation. From what I've seen in the tutorial, they don't seem to pay much attention to the initial values of the inner areas.
I think your simulation is better than mine.


Fanjin

Hello FanJin,


I am also not an expect, sorry!
This document is quite handy, IMHO: https://theairshed.com/pdf/COST%2073...May%202007.pdf


I do not use 'initial conditions', I let the CFD compute from no initials, it might need some steps to get equilibrium, but until now the CFD found these for me (I used a residual of 10-2).


So you have your porous object in the middle. Why do you have a 'cubic basin space' around it? Or is this 'cubic basin space' the 'external flow volume' (aka the space where the wind goes through)?

oh,yes. the 'cubic basin space' is a external flow volume, which is simulate the urban wind flow. My final simulation task is considering the porous tree, buiding, and atmospheric environment.

So that's the outflow field from the trees.

I just have a problem right now, I don't think the initial value of the internal field should affect the final result, as you said, 'need some steps to get equilibrium'.

But when I give two different initial internal field pressure conditions (0 and 101325pa (atmospheric pressure)), the final result is very different, although the residuals are both around 10-2.

I'll take a good look at what you gave me. Thank you！

A pressure drop of 10 Pa over 1 meter is the same thing as a pressure gradient of 10 Pa per meter.

You are confusing pressure difference versus pressure difference per length: i.e. dp versus dp/dx. There is no need to differentiate a gradient and get a weird second order derivative and make up new physics. Just stare at the existing formula until you understand them.

I can imagine if you compare 0 and 101325 pa. Is there a difference between 95000 and 103000 Pa (so realistic air pressures)?

In some pictures see in literature, you see that the Δp depicts (say 1mbar) the pressure difference over a certain length (say length = 0.1m). In that case de pressure gradient (δΔp/δx) is 10mbar/m. So the Δ is defining a difference (between two locations) and not a derivative (δy/δx). So there are NO second derivative.