1. Velasco, D., Bateman, A., DeMedina,V. Hydraulic and Hydrology Dept. Universidad Politécnica de Catalunya (UPC). Barcelona. España. RCEM 2005 4th 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

2. 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)

3. 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 (C d ) 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

4. 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

5. C d (Drag coefficient) is a function of Re and shape Evaluation of C d 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  =water density U= velocity a=distance between plants (interdistance) B=plant width C d = Drag Coefficient Eq. [5] Drag stresses (absorbed by vegetation)

6. Turbulence closure model Mixing lenght theory (Karman-Prandtl) TN- h'=0.18 m q=0.136 m3/s p=0.086 m a=0.006 m lolo κ’κ’ Linear law of mixing lenght above penetration point p (based on experimental own data) Eq. [6] UoUo k p For zone 3 (z<p) Eq. [7] l (m)

7. 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 p i Penetration depth p i, Velocity U i (z) Turbulent stresses  xz i (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 \ < tol k NO i=i+1 YES  sc. So Flux Diagram QUICKVEGMODEL

8.  xz pred i,j  xz corr i,j p i,j kiki Hydrodynamic SUBROUTINE 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:

9. Mechanical Subroutine Numeral Code which reproduces load-deformation process in a stem FOR LARGE DEFORMATIONS 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 {k i } Mechanical SUBROUTINE Secondary Moments effect

10. A11 -Run for vertical, rigid cylinders from Tsujimoto (1990) experiments RESULTS OF QUICKVEGMODEL. PARTICULAR COMPUTATIONAL PARAMETERS: Cd(z)=1.5 l o =2.5 mm, κ’= 0.17 and  sc =1.0 Is there a set of general computational parameters ??? Calibration !!!!!

11. 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/m 2 ) to adjust measured to calculated data 2) VERIFICATION OF THE “LARGE DEFORMATION” MODEL FOR PLANTS : Numerical calibration of the Mechanical Subroutine

12. 3.1) Experimental Setup : Vegetative Cover 1) Artificial PVC plastic plants 2) Natural Barley grass Density M: 205, 70, 25 plants/m 2 Density M: 22850 leaves/m2 3) ADJUSTMENT OF QUICKVEGMODEL PARAMETERS

13. 3D sensor Velocity sensor: 3D-Acoustic Dopler NDV(25 Hz) Control Volume 3.2) Experimental Setup : Measurement Instruments

14. UoUo k Steady- Uniform regime conditions: Unit Discharge q, water depth h, Energy slope S o Vertical profile of velocity U(z) Deflected plant height k 3.3) Experimental Data

15. 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 where q calc = calculated unit discharge ( q=∫U(z).dz ) q adv = measured unit discharge using ADV data q weir = measured unit discharge using Weir data C d,calc= calculated drag coef. C d,exp= measured drag coef.  =standard deviation

16. Drag Coeficients points: Turbulent parameters: lo=0.01 m κ’=0.040  sc =0.54 Disappointment between weir data and ADV data Adjusted drag coeficients.

17. Velocity U

18. Reynolds Stresses  xz

19. Calculated unit discharge q calc vs. measured q mea. +15 % -15 % +15 % -15 % Calculated deflected plant height k and penetration depth p vs. measured data ACCURACY OF RESULTS

20. 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 C d (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 C d (Re).  Computational time to accuracy ratio is satisfactory and QUICKVEGMODEL is going to be applied into general 1D and 2D hydraulic models.