F parameters: what values for a tree?

Using SimScale Fluid Flow / CFD
1

<I had put my message in the wrong category, hope it is better here!>

I am a still looking for a consistent method to derive the f parameter values for a tree (d=0, seems consistent).

Reading the report: https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=928986
how does formula (7) translated into the d and f parameters of Darcy-Forchheimer?
Formula (7) is: dP=rho/2Cdryankappau^2L

<Cdryan is the Cd in above artcile ~ 2.8>

Looking here:
How to Predict Darcy and Forchheimer Coefficients :
dP = rho/2fu^2*L (formula below formula (4))

Thus fryan looks to be Cdryan*kappa.

So with a tree (say Robin Red Holly):
fryan = 2.8*4.3 = 12.0

Looking here:
Surface Roughness & Porosity | Advanced Modelling PWC | SimScale

This document details multiple ways of modeling porous media and surface roughness under advanced modelling in PWC analysis using SimScale.

Est. reading time: 8 minutes

dp/dx=-rhoLAI/hCdsim*|u|*u

From this formula f would be
fsim = 2LAI/HCdsim

<Cdsim is SIMSCALE’s Cd ~ 0.2; H is height of tree; LAI Leaf Area Index>

So for an oak tree of 15.5m height and LAI is 5.2:
fsim=25.2/15.50.2=0.134
<I also see this when using: Readyuse-Perforated Plate.xlsx - Google Sheets )

So this is a factor of 100 between the two.

Possible explanation of difference:

a) The Cdsim is around 0.2 (table 4 in Surface Roughness & Porosity | Advanced Modelling PWC | SimScale)
While Cdryan is around 2.8 (chapter 5).

b) In SIMSCALE’s formula there is H of the object (H)!

c) LAI (around 3 to 5.2) is quite close to kappa (around 2 to 4.3)

d) I must be doing/understand something wrong.

e) When using the low values of fsim (=0.134) in simulation, I get IMHO quite (too?) porous objects. I did not yet try the fryan=12 (and compare that with the results of the tree: trees by rtir | SimScale , as that tree matches quite well with a real tree: Atmosphere | Free Full-Text | The Influence of Wind-Induced Response in Urban Trees on the Surrounding Flow Field )

Any help to get a better understanding on my side, is welcome. Thanks.

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At last I have found some more detailed background on f (https://s-space.snu.ac.kr/bitstream/10371/140809/1/000000150897.pdf (TAEHWAN HA, 2018)
In this document f is the ‘Inertial resistance coefficient Ci’.
The main difference with SIMSCALE’s paper is the value of Cd (drag coefficient). In SIMSCALE Cdsim = 0.2 while Ha (2018, page 23) has on average a Cdha ~ 0.64 (~3.25 larger).

This larger Cdha might produce better simulations, see point e):

I will study literature a little more before attempting a simulation (I don’t have many simulations left in my Community Plan).

Again; if you have more feedback, I am interested!

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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.

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I think I made the error to use a CAD-model of a summer oak tree (H=15.5m) that had gaps (blobs) and then adding the Forchheimer medium. Better to use a CAD-model in the form of stacked-solid-cylinders (no gaps) and then the Forchheimer medium.

Still need to analyses how things precisely behave. I am trying to do this on my webpage:
http://www.archaeocosmology.org/eng/Boomsim.htm
But there are still some uncertainties (the f=0.425 of the stacked-solid-cylinders tree looks to match the blobbed tree ; but it is quite different from SIMSCALE’s outcome f=2LAI/HCd=25.2/15.50.2=0.134).

So still working on this. Ideas are welcome.
All the best,

Victor

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Cd is depending on the speed. I see graphs where indeed a Cd is mentioned of around 0.2, but that is for relatively high speeds (20m/sec). I am looking at an average speed of 7m/sec (Bft 3, 6) and that might be the reason why SIMSCALE’s Cd is larger…
Any ideas here?

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This reference (https://www.researchgate.net/publication/44385014_Evaluation_of_drag_coefficients_of_poplar-tree_crowns_by_a_field_test_method) gives a graph with the dependency of Cd on wind velocity: Fig. 4 (see also my rendering of that graph: Simulating a tree ).

So the (default) Cd of SIMSCALE (figure 9, Table 4) is 0.2. Can someone tell me for what wind velocity this is? Thanks very much.

All the best,

Victor

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