Giorgio Crasto

University of Cagliari - ITALY

Forest ModellingForest Modelling

A canopy model for WindSim 4.5

Giorgio Crasto

University of Cagliari - ITALY

Why the need for a canopy modelWhy the need for a canopy model

0,0

0,5

1,0

1,5

2,0

0,00 0,50 1,00 1,50u/uhc

h/h

c

Roughness Lenght z0 0,6m

0,0

0,5

1,0

1,5

2,0

0,0 0,5 1,0 1,5u'w'/u*2

h/h

c

Roughness Lenght z0 0,6m

Giorgio Crasto

University of Cagliari - ITALY

In the new canopy model of WindSim the In the new canopy model of WindSim the Canopy Layer is solved by the use of Canopy Layer is solved by the use of porosity and drag forces (sinks of porosity and drag forces (sinks of momentum)momentum)

The canopy is described by:The canopy is described by: Height of the canopy – hHeight of the canopy – hcc

The canopy is discretized with a caertain number The canopy is discretized with a caertain number of cells (uniform grid?) of cells (uniform grid?)

Underwood roughness length – zUnderwood roughness length – z00

Porosity – Porosity – Drag coefficients – C1 and C2Drag coefficients – C1 and C2

Giorgio Crasto

University of Cagliari - ITALY

Concept of PorosityConcept of Porosity

Given a Given a generic volumegeneric volume occupied by fluid and occupied by fluid and solid phases, its porosity solid phases, its porosity is defined as the ratio is defined as the ratio

volumeof fluidphases

Vol

Giorgio Crasto

University of Cagliari - ITALY

Given a Given a surface surface partially permeable, also a partially permeable, also a porosity for a 2D windbreak can be defined asporosity for a 2D windbreak can be defined as

S

areaholes

totalarea

Concept of PorosityConcept of Porosity

Giorgio Crasto

University of Cagliari - ITALY

Concept of PorosityConcept of Porosity

Once done an hypothesis on the structure Once done an hypothesis on the structure of the porous region, there is a of the porous region, there is a relation relation linking linking and and SS

x EAST

z HIGH

2

3S 1 1

Darcy’s velocity

Giorgio Crasto

University of Cagliari - ITALY

Every element of the canopy introduces Every element of the canopy introduces in the flow a sink of momentumin the flow a sink of momentum

So inside the canopy the momentum RANS So inside the canopy the momentum RANS equations present an additional termequations present an additional term

x

y

z

x xi t U

i i i

y yi t U

i i i

z zi t U

i i i

U UU p i S

x x x

U UU p j S

x x x

U UU p k g S

x x x

xU 3

NS

m

Advection Diffusion Additional forces: pressure gradient, gravity,

adittional sources

Giorgio Crasto

University of Cagliari - ITALY

The sinks of momentum depend upon The sinks of momentum depend upon porosity and velocity porosity and velocity

A classical approach (for high porous A classical approach (for high porous materials) is to consider the sinks of materials) is to consider the sinks of momentum proportional to the velocitymomentum proportional to the velocity

In the Darcy’s law the advective and In the Darcy’s law the advective and diffusive terms are neglected, so the gravitydiffusive terms are neglected, so the gravity

VCk

Darcy’s law

DV UCP UCS VU

Giorgio Crasto

University of Cagliari - ITALY

Sinks of momentum implemented in the Sinks of momentum implemented in the new version of WindSimnew version of WindSim

Sinks of momentum [N/mSinks of momentum [N/m33]: two terms]: two terms

UUC2ρUC1ρSU

viscous force

pressure force

Since the high Reynolds numbers

pressure >> viscous forces

Giorgio Crasto

University of Cagliari - ITALY

In the canopy model of WindSim 4.5 is possible to set the In the canopy model of WindSim 4.5 is possible to set the drag coefficients C1 and C2 wheredrag coefficients C1 and C2 where

s

1C1 dimensionsnoC2

typermeabili][mk

kkρ

μC1

2

Present modelPresent model

Giorgio Crasto

University of Cagliari - ITALY

How to estimate the porosity k?How to estimate the porosity k?

The relation k vs. The relation k vs. is proposed:is proposed:2

22

k C [m ]1

0

0,002

0,004

0,006

0,008

0,01

0,012

0 20 40 60 80 100

porosity (%)

2C 0,00450799m

In this way it’s possible to estimate the value of the drag coefficient C1 starting from a

given value of porosity

Giorgio Crasto

University of Cagliari - ITALY

0,0

0,5

1,0

1,5

2,0

0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75u/uhc

h/h

c

porosity 15% , C1 1.450E-01, C2 0 porosity 50%, C1 1.002E-02, C2 0

porosity 84%, C1 1.393E-03 , C2 0 porosity 100%

Giorgio Crasto

University of Cagliari - ITALY

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7

speed [m/s]

heig

ht [m

]

canopy 04b - C1 1,002E-02 canopy 06b - C1=C2=0,5E-02

Giorgio Crasto

University of Cagliari - ITALY

0,0

0,5

1,0

1,5

2,0

0,0 0,5 1,0 1,5u'w'/u*2

h/h

c

porosity 15%, C1 1.450E-01, C2 0 porosity 50%, C1 1.002E-02, C2 0

porosity 84%, C1 1.393E-03, C2 0 porosity 100%

Giorgio Crasto

University of Cagliari - ITALY

ConclusionsConclusions With the proposed canopy model is possible to With the proposed canopy model is possible to

estimate properly the dissipation of momentum estimate properly the dissipation of momentum and the production of turbulence inside a forestand the production of turbulence inside a forest

Further validations are needed in order to state Further validations are needed in order to state more precise guidelines for the choice of the more precise guidelines for the choice of the parameters C1 and C2 used togetherparameters C1 and C2 used together

Masts inside forestsMasts inside forests Wind farms close to forestsWind farms close to forests

A possible treatment of a canopy would be with A possible treatment of a canopy would be with additional sources in the transport equations of k additional sources in the transport equations of k and and