Post on 11-Jan-2016
description
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.