Velasco, D., Bateman, A., DeMedina,V. 

Hydraulic and Hydrology Dept.Universidad Politécnica de Catalunya (UPC). Barcelona.

España.

RCEM 20054th IAHR Symposium on River, Coastal and Estuarine Morphodynamics

University of Illinois, Urbana, Illinois,

October 4 - 7, 2005

A new integrated, hydro-mechanical model applied to flexible vegetation in riverbeds.

Grupo de Investigación en Transporte de Sedimentos

Background

Kutija V., Hong (1996), Erduran, K.S., Kutija V (2003). Flexible cylinders (cantilever deflection equation) Eddy viscosity approach and mixing lenght theory Discretization of vertical axis Z to apply an Unsteady Reynolds-x equation

→ converge to a steady solution

López,F., García,M.H. (2001), Fischer-Antze (2001) 3D turbulent κ-ε model Rigid, vertical cylinders

Cui,J.,Neary V.S.(2002), Choi,S.,Kang,H.(2004) 3D turbulent RANS and LES model Rigid, vertical cylinders

Variation of drag coeficients Cd as a function of Re number is never included.Incomplete Calibration (flexible vegetation flume flume data never used)

OBJECTIVES of the present work .

1) Creation of QUICKVEGMODEL, an integrated finite differencies model to calculate vertical velocity profile for vegetated channels, which includes a subroutine for flexible plant deformation.

Input data is required for :

a) vegetation properties: Plant geometry and Stiffness modulus (E)

b) Hydraulic conditions: Drag Coeficients (Cd)

2) Verification of the “Large Deformation” model for plants

Strain –stress tests in laboratory for different stems

3) Adjustment of QUICKVEGMODEL parameters (calibration) using experimental data

z ,region laminar 4. ZONE )( )(

.).(..0

zp ,region shearless 3. ONE Z )( ).(..0

pzk ,region internal 2. ZONE )( )( ).(..0

kz h ,region external 1. ONE Z )( ).(..0

o

o

)(

zzzU

zhSog

zzhSog

zzzhSog

zzhSog

Cd

Cd

Cdxz

xz

zG

).(. pressure cHydrostati

0V

W U

V U: flow 1D

0yx

:Conditions Uniform

0t

:flowSteady

zhgp

W

Vertical Integration of Reynolds equation between coordinates z=h and z

Eq. [1]

Eq. [2]

Eq. [3]

Eq. [4]

1.) Description of QUICKVEGMODEL

k= deflected plant height p=penetration depth (turbulent shear stresses xz=0)

Viscous forces neglected in zones 1,2 and 3

Cd (Drag coefficient) is a function of Re and shape

viscositykinematic .

Re

BU

Evaluation of Cd law as a discrete points {Re(j), Cd(j)} aproximation in a log-log graph

Classical 2-D body resistance laws are not appropiate to stems and leaves

0.001

0.01

0.1

1

10

100

1 10 100 1000 10000

.

Re

BUd

dC

{Re(i), Cd(i)}

{Re(1), Cd(1)}

{Re(n), Cd(n)}

dC

=water density

U= velocity

a=distance between plants (interdistance)

B=plant width

Cd= Drag Coefficient

dzzUa

zBzCz

k

z

dCd ).(.)(

).(..21

)( 22

Eq. [5]

Drag stresses (absorbed by vegetation)

Turbulence closure model

Mixing lenght theory (Karman-Prandtl)

zU

zU

lxz

... 2

pzfor )(

pzfor )'.()(

o

o

lzl

pzlzl

TN- h'=0.18 m

q=0.136 m3/sp=0.086 ma=0.006 m

lo

κ’

Linear law of mixing lenght above penetration point p

(based on experimental own data)

Eq. [6]

0 0.2 0.4 0.6 0.80

50

100

150

200

250

U (m/s)-1 0 1 2

x 10-3

0

50

100

150

200

250Y23Q4

Z (m

m)

-4 -2 0 2

x 10-4

0

50

100

150

200

250

XY/ (m2/s2)

-4 -2 0 2

x 10-4

0

50

100

150

200

250

YZ/ (m2/s2)

XZ/, XY/ (m2/s2)

Z (m

m)

Uo

kp

2/12

....2

tan

odo BC

aSogtconsU

For zone 3 (z<p)Eq. [7]

l (m)

INPUT DATA

Hydraulic data: h,So, Vegetation data: h’,a, B(z),e,E; Resistance coef: {Re(j),Cd(j)}; Turbulent parameters: lo,κ’,sc

OUTPUT DATA

q, U(z), xz (z)

p, k, stem deformation y(x)

Deflected plant height k i

Hydrodynamic SUBROUTINE : Optimization of penetration depth pi

Penetration depth p i, Velocity U i(z)

Turbulent stresses xzi(z)

Initial Conditions: U(z)=Uo, xz(z)=0

Drag Force F i(z)

Mechanical

SUBROUTINE

Deflected plant height k i+1

\k i+1-k i\ < tolk

NO

i=i+1YES

k

Z

h

p

xz sc.So

Flux Diagram QUICKVEGMODEL

Areaxz

X

Z

h

xz predi,j

xz corri,j pi,j

ki

Hydrodynamic SUBROUTINE

k

i,jxz

i,jxz

i,jxz dzzpredzcorrArea

0

.)()(

This subroutine is based on a predictor (xz pred) and corrector (xz corr) scheme involving turbulent shear xz. The well-balanced solution is calculated as a Minimum for functional Area xz , defined as the integrated difference between prediction and correction:

Mechanical Subroutine

Numeral Code which reproduces load-deformation process in a stem FOR LARGE DEFORMATIONS

IE

Mf

s

y

.2

2

Equation of elasticity in beams (Timoshenko) Explicit finite differences scheme to

solve deformations y(x)

Load values F(x) obtained from hydrodynamical module

Conservation of total stem length h’

Iterative force distribution to converge to the deflected plant height {ki}

Mechanical

SUBROUTINE

Secondary Moments effect

A11 -Run for vertical, rigid cylinders from Tsujimoto (1990) experiments

RESULTS OF QUICKVEGMODEL.

PARTICULAR COMPUTATIONAL PARAMETERS:

Cd(z)=1.5

lo=2.5 mm, κ’= 0.17 and sc=1.0

Is there a set of general computational

parameters ???

Calibration !!!!!

X

y

Strain –stress tests applied to stems

Stem attached horizontally

Incremental load steps in the extreme of the stem

Image processing to obtain the deflection profile y(x)

Estimation of Stiffness Modulus E, (N/m2) to adjust measured to calculated data

2) VERIFICATION OF THE “LARGE DEFORMATION” MODEL FOR PLANTS : Numerical calibration of the Mechanical Subroutine

3.1) Experimental Setup : Vegetative Cover

1) Artificial

PVC plastic plants

2) Natural

Barley grass

Density M:

205, 70, 25 plants/m2

Density M:

22850 leaves/m2

3) ADJUSTMENT OF QUICKVEGMODEL PARAMETERS

3D sensor

Velocity sensor:

3D-Acoustic Dopler NDV(25 Hz)

Control Volume

3.2) Experimental Setup : Measurement Instruments

0 0.2 0.4 0.6 0.80

50

100

150

200

250

U (m/s)-1 0 1 2

x 10-3

0

50

100

150

200

250Y23Q4

Z (

mm

)

-4 -2 0 2

x 10-4

0

50

100

150

200

250

XY/ (m2/s2)

-4 -2 0 2

x 10-4

0

50

100

150

200

250

YZ/ (m2/s2)

XZ/, XY/ (m2/s2)

Z (

mm

)

Uo

k

Steady- Uniform regime conditions:

Unit Discharge q, water depth h, Energy slope So

Vertical profile of velocity U(z)

Deflected plant height k

3.3) Experimental Data

Multi- parametric optimization : minimization in a modified conjugate gradients

technique of the quadratic, residual Function Ф:

Drag Coeficients points:

{Re(j),Cd(j)}

Turbulent parameters:

lo (mixing length),

κ’ (momentum diffusion constant)

sc (secondary currents factor)

3.4) Optimization of Parameters:

applied to 14 runs with vegetation

5

12

exp,

2

exp,,

2

214

12

2

))((

)()(

)()(

)()(

j d

dcalcd

weir

weiri

calci

i adv

advi

calci

jC

jCjC

qerrorqq

qerrorqq

where qcalc = calculated unit discharge ( q=∫U(z).dz )

qadv = measured unit discharge using ADV data

qweir = measured unit discharge using Weir data

Cd,calc=calculated drag coef. Cd,exp=measured drag coef.

=standard deviation

0.001

0.01

0.1

1

10

100

1 10 100 1000 10000

CdMeasured

Adjusted

BU .

Re

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 0.05 0.1 0.15 0.2 0.25 0.3

V (ADV data)

V (

Wei

r d

ata)

Drag Coeficients points:

Re Cd

2 22.510 3.1490 0.37632 0.0091275 0.004

Turbulent parameters:

lo=0.01 m

κ’=0.040

sc=0.54

Disappointment between weir data and ADV data

Adjusted drag coeficients.

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

U (m/s)

z (m

)

0 0.05 0.1 0.15 0.20

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

U (m/s)

z (m

)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

U (m/s)

z (m

)

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

U (m/s)

z (m

)

MeasuredCalculatedTop of deflected canopy

TN-h'=0.09 m- Q-2 TN-h'=0.09 m- Q-3 TN-h'=0.09 m- Q-4 TN-h'=0.125 m- Q-2

TN-h'=0.125 m- Q-3 TN-h'=0.125 m- Q-4 TN-h'=0.18 m- Q-2 TN-h'=0.18 m- Q-3

TN-h'=0.18 m- Q-4

T3 -Q-1

T3 -Q-2 T3 -Q-3

T3 -Q-4 T3 -Q-5 Velocity U

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

XZ

(N/m2)

z (m

)

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

XZ

(N/m2)

z (m

)

-1 0 1 2 30

0.1

0.2

0.3

0.4

XZ

(N/m2)

z (m

)

-0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

XZ

(N/m2)

z (m

)

0 1 2 30

0.05

0.1

0.15

0.2

0.25

XZ (N/m2)

z (m

)

-1 0 1 2 30

0.1

0.2

0.3

0.4

XZ (N/m2)

z (m

)

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

XZ (N/m2)

z (m

)

-0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

XZ (N/m2)

z (m

)

-1 0 1 20

0.1

0.2

0.3

0.4

XZ (N/m2)

z (m

)

-0.6 -0.4 -0.2 0 0.20

0.05

0.1

0.15

0.2

XZ (N/m2)

z (m

)

-0.2 0 0.2 0.4 0.60

0.05

0.1

0.15

0.2

XZ (N/m2)

z (m

)

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

XZ (N/m2)

z (m

)

-0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

XZ (N/m2)

z (m

)

-0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

XZ (N/m2)

z (m

)

MeasuredCalculatedTop of deflected canopy

TN-h'=0.125 m- Q-3

TN-h'=0.09 m- Q-3 TN-h'=0.125 m- Q-2

TN-h'=0.125 m- Q-4

TN-h'=0.09 m- Q-2 TN-h'=0.09 m- Q-4

TN-h'=0.18 m- Q-2 TN-h'=0.18 m- Q-3

TN-h'=0.18 m- Q-4 T3 -Q-1 T3 -Q-2 T3 -Q-3

T3 -Q-4 T3 -Q-5

Reynolds Stresses xz

Calculated unit discharge qcalc vs. measured qmea.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.02 0.04 0.06 0.08

qcalc

qm

ea

+15 %

-15 %

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Measured (m)

Deflectedplant heightk

penetrationdepth p

+15 %

-15 %

Calculated deflected plant height k and penetration depth p vs. measured data

ACCURACY OF RESULTS

CONCLUSIONS AND LIMITATIONS

An integrated numerical model of flow through flexible vegetation, QUICKVEGMODEL, is developed on the basis of momentum equilibrium (Reynolds equation).

A Mechanical subroutine, which calculates the plant deformation, is coupled with an hydrodynamical subroutine (mixing length model of turbulence) to obtain velocity and shear stress profiles.

An experimental study (including natural and artificial vegetation) in a rectangular flume is used to calibrate computational parameters and resistance law Cd(Re)

LIMITATIONS: Medium or High density of vegetation is needed to accomplish basic

hypothesis. The presence of real convective currents in the flow is introduced in the

model (sc), but it is hard to evaluate experimentally. A more intense experimental campaign is also needed to verify the general

drag coefficient law Cd(Re). Computational time to accuracy ratio is satisfactory and

QUICKVEGMODEL is going to be applied into general 1D and 2D hydraulic models.