Simulating a tree

This web page will look at how to simulate a leafed and leafless tree (as solid objects or as a porous object with a Darcy-Forchheimer medium or a Perforated plate medium. The outcome can be seen in this summary.

Valideren van CAD-model van een boom

Bebladerde boom (voor mei t/m september)

Kale boom (voor november t/m maart)

Determining the Drag force coefficient through literature

On this web page the Drag force and Pressure-difference convention are built around:

Δp: pressure difference
So Δp is over the full length L.
Derivative of Δp is dΔp/dx: pressure gradient
Here dΔp/dx can change (which most of the time is assumed to be constant) over L. On this website this is not writen as Δp/Δx, but as dΔp/dx.
F: drag force
If there is an L on the right side of Δp formula: F = Δp*A
In the case we have the area A on the right side of F formula and no L on the right side of dΔp/dx formula: F = (dΔp/dx)*A
In most case A is assumed to be constant over L.

In Hagen (1971, formula [2]): D_h = F = ρ/2*u²*H*C_dhag_{<H is the height of the windbreak>}
With dΔp/dx =D_h/A → assume A=H²*π/4 → dΔp/dx = ρ/2*u²*[C_dhag/(H*π/4)]
C_d = C_dhag/(H*π/4)
Hagen (1971) gives an idea of C_dhag for an slatted fence (H = 2.44m) with 40% porosity (and u's between 5.5 and 11m/sec): C_dhag = 1.17
C_d = 0.61

In Bitog (2012, formula (1)): Δp = m*ρ*u²*C_dbit
_{<m is the
thichness of the tree crown, aka L>}
dΔp/dx = ρ/2*u²*[2*C_dbit]
C_d = 2*C_dbit
Bitog (2012) gives an idea of C_dbit for an oak (H = 5.5m): C_dbit= 0.55
C_d= 1.1

In Roubos (2014, formula under Fig. 127) : Q_w;rep = F = ρ*u²*A*C_wrou_{<A is the area of the tree crown>}
With dΔp/dx =Q_w;rep/A → dΔp/dx = ρ/2*u²*[2*C_wrou]
C_d = 2*C_wrou
Roubos (2014) gives an idea of C_wrou for an oak (H = 15m): C_wrou= 0.25
C_d = 0.5

In Koizuma (2010, forumula1) : P_w = F = ρ/2*u²*A*C_d_koi_{<A is the area of the tree crown>}
With dΔp/dx =P_w/A → dΔp/dx = ρ/2*u²*[C_d_koi]
C_d = C_d_koi
Koizuma (2010) gives an idea of C_d_koi for poplar (H = 12.5m):

For leafless trees the C_d_koi is around 0.2 (Koizuma, 2010, Fig. 7) for u's between 4 and 11/sec.
A leafed poplar has a C_d_koi (Koizuma, 2010, Fig. 6) between 0.55 (u = 4m/sec) and 0.3 (u = 11m/sec).

C_d = 0.55

In Ha (2018, page 24): P_w = F = ρ/2*u²*A*[C_dha]
_{<A is the area of the tree crown>}
With dΔp/dx =P_w/A → dΔp/dx = ρ/2*u²*[C_d_ha]
C_d=C_dha
Ha (2018, page 23) provides an average C_dha of 0.169 for u's between 2 and 10m/sec and leafless deciduous trees (average H=17m).
Ha (2018, page 23) provides an average C_dha of 0.655 for u's between 4 and 10m/sec and leafed deciduous trees (average H=17m).
C_d_{=
0.655}
In Wikipedia: C_d_wiki is given for several objects
Wikipedia (Figure 2021, accessed Febr. 2024) has a C_d_wiki of around 0.59 for an eliptical sphere (at Reynold number between 10⁴ and 10⁶).
C_d = 0.59

_{In Bekkers (2022, forumula1) : D = F = ρ/2*u²*A*C_d_bek_{<A is the area of the tree crown>}

With}_{dΔp/dx
=D/A}_→_{dΔp/dx =
ρ/2*u²*[C_d_bek]}_{C_d
= C_d_bek}_{Bekkers

(2022, Fig. 9) has a C_d_bek
of around 0.77 (at u = 5m/sec and H=6.5m).}_{C_d
= 0.77}In SIMSCALE (2023, Table 4)
SIMSCALE uses a C_dsim = 0.2 (regardless of the u and H).
C_d = 0.2

In Ren (2023, formula 4): S_i = dΔp/dx = -ρ/2*|u|*u*[C_d_ren]
C_d= C_d_ren
Ren (2023, page 7) has a C_d_ren of around 0.704 (at u = 10m/sec and H=6.8m).
C_d= 0.704

Overview of found Draft force coefficients

The above values of C_d (green is what the given values for leafed deciduous trees are in the reference) are put in below table:

Some general aspects of C_d

Determining the Forchheimer coefficient through simulated emperical way

First simulation iteration

Second simulation iteration

Here is an analysis done for f = 0.1 (leafless tree), 0.35 (like close to SIMSCALE), 0.45 and 0.55 (so only stacked-cylinder objects):
Several Firchheimer
contributions

Proposed to do a third iteration with; f = 0.09 (leafless tree), 0.375, 0.4 and 0.425

Third simulation iteration

Here is an analysis done for f = 0.9, 0.375, 0.4 and 0.425:
Several Forchheimer
coefficient

So f=0.425 looks to be ok-ish for a leafed tree (@ u(10)=6.44msec and z₀=0.5m).

Fourth simulation iteration

Here is an analysis done for f = 0.09 (leafless tree), 0.4, 0.425 and 0.45:
Several Forchheimer
coefficient

So the leafed tree with f=0.45 looks best @ u(10)=7.17m/sec (or u(H)=8m/sec) and z₀=0.25m.

Compare stacked-cylinder tree with real tree

Using the stacked-cylinder object with f_VR=0.45 (colored) and compare it with the anemometer measurements from a real Ulmus parvifolia (black lines) (Ren, 2023, Fig. 16a2), there is a good resemblance:
Comapring Ren with f=0.145

Conditions were: H=15.5m, z₀=0.25m, u(H)=8m/sec

In SIMSCALE (2022 and 2023): S = -ρ/2*|u|*u*f = -ρ/2*|u|*u*[2*LAD*C_dsim]
f_sim = 2*LAD*C_dsim = 2*LAI/H*C_dsim
<H is height of tree; LAI is Leaf Area Index>

Using f_VR = 2*LAI/H*C_dVR, would make a C_dVR of 0.87 (= f_VR*H/2/LAI = 0.45*15.5/2/4).

Proposed Forchheimer coefficient for leafed and leafless summer oak

The CFD behavior (H=15.5m, z₀=0.25m, u(H)=8m/sec) of the stacked-cylinder object with Darcy-Forchheimer medium or the blobbed object without Darcy-Forchheimer medium, provides a good match with the velocity distribution of the real leafed tree.

In case of the stacked-cylinder (summer oak) object the Forchheimer coeffient should be:

For other H, LAI and u:
f_leafed = f_{refleafed *}_{H_ref
/}_{LAI_ref}_{* LAI}_{_leafe}_{/ H}_{_leafed}₌1.73 * u_comp * LAI_leafed / H_leafed

u_comp=( u_leafed(H_leafed) / u_ref(H_ref) )^0.3 (using emperical optimisation) = 0.52 * u_leafed(H_leafed)^0.31

Porosity and Forchheimer coefficient relation through simulated emperical way

Remark: I don't know what the ruwheidsdichtheid is. Also the relation between C_d and f is not 100% sure yet.

A porosity = 0.87*f²-1.54*f+0.85 formula was derived by matching the results of Darcy Forchheimer coefficent (f) and Perforated plate (Free area ratio=p[orosity]) simulations of a stacked-cylinder object in SIMSCALE (honeypot):

This is a formula derived from above picture:
porosity as function of f

Reijs (2024, f - blue), Hagen (1971, C_d - grey), Stichlmair (2010, C_d - formula (11) and (13); thick plates are assumed to have relatively small holes: yellow), SIMSCALE (f - green) are quite similar.
Hagen and Stichlmair don't worh with trees, but fences or plates.
Remark: some further evaluation is needed to add the f - C_d dependency in above picture.

Simulating a tree

Introduction

Valideren van CAD-model van een boom

Bebladerde boom (voor mei t/m september)

Kale boom (voor november t/m maart)

Determining the Drag force coefficient through literature

Overview of found Draft force coefficients

Some general aspects of C_d

Determining the Forchheimer coefficient through simulated emperical way

First simulation iteration

Second simulation iteration

Third simulation iteration

Fourth simulation iteration

Compare stacked-cylinder tree with real tree

Proposed Forchheimer coefficient for leafed and leafless summer oak

Porosity and Forchheimer coefficient relation through simulated emperical way

Conclusions

Referenties

Acknowledgements

Simulating a tree

Introduction

Valideren van CAD-model van een boom

Bebladerde boom (voor mei t/m september)

Kale boom (voor november t/m maart)

Determining the Drag force coefficient through literature

Overview of found Draft force coefficients

Some general aspects of Cd

Determining the Forchheimer coefficient through simulated emperical way

First simulation iteration

Second simulation iteration

Third simulation iteration

Fourth simulation iteration

Compare stacked-cylinder tree with real tree

Proposed Forchheimer coefficient for leafed and leafless summer oak

Porosity and Forchheimer coefficient relation through simulated emperical way

Conclusions

Referenties

Acknowledgements

Some general aspects of C_d