Koftis, Theoharris

Papakyritsis, Christos

Prinos, Panayotis

English

Hydraulic Engineering Repository

Conference Paper, Published VersionKoftis, Theoharris; Prinos, Panayotis; Papakyritsis, ChristosReynolds Stress Modeling of Flow in Compound Channelswith Vegetated FloodplainsZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99434Vorgeschlagene Zitierweise/Suggested citation:Koftis, Theoharris; Prinos, Panayotis; Papakyritsis, Christos (2014): Reynolds StressModeling of Flow in Compound Channels with Vegetated Floodplains. In: Lehfeldt, Rainer;Kopmann, Rebekka (Hg.): ICHE 2014. Proceedings of the 11th International Conference onHydroscience & Engineering. Karlsruhe: Bundesanstalt für Wasserbau. S. 215-224.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding. 1 INTRODUCTION In natural rivers, vegetation grows on floodplains, generating complex velocity field within the compound channel. Due to the velocity difference and the momentum exchange between the vegetated and non-vegetated area, strong shear layer and vortices occur (Liu et al. 2013). Therefore, knowledge of the mech-anism of momentum exchange between the main channel and the vegetated floodplain is significant due to the effect on the discharge capacity of the channel, on erosion processes and on biological and issues.  In the present study three dimensional computations of the VARANS (Volume-Averaged Reynolds-Averaged Navier-Stokes) equations, in conjunction with a Reynolds Stress (RS) model, are performed for a non-symmetrical compound channel of a trapezoidal main channel and a vegetated floodplain, corre-sponding to the experimental setup of Yang et al. (2007). The drag effect of the vegetation on the current is taken into account through additional terms in both the momentum and the RSM equations based on a vegetation dynamics approach. The additional terms are related to the drag coefficient Cd and the plant density α, defined as the frontal area per unit volume (m-1). The results are compared against the experi-ments of Yang et al. (2007). Moreover, the analytical method of Shiono and Knight (1991) is applied for the depth-averaged velocity, together with simple Manning calculations. The cross-sectional flow field is presented regarding the streamwise velocity, the shear stresses, the turbulent anisotropy and the secondary currents, revealing the momentum exchange mechanism at the interface region, between main channel and floodplain.    Reynolds Stress Modeling of Flow in Compound Channels with Vegetated Floodplains T. Koftis, P. Prinos & C. Papakyritsis Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece ABSTRACT: Flow in compound channel with vegetated floodplain is complex and efficient modeling of such flow should include the effects of vegetation on velocity, secondary flow and bed shear stress. In the present study, computations of the Volume-Averaged Reynolds-Averaged Navier-Stokes equations, in conjunction with a Reynolds Stress turbulence model based on a vegetation dynamics approach, are per-formed for a non-symmetrical compound channel of a trapezoidal main channel and a vegetated flood-plain. The numerical results agree well with the available experimental data, while the model is capable to reproduce the evolution of vortices with the stronger one found in the interface region between the main channel and the vegetated floodplain. The cross-sectional flow characteristics reveal the momentum ex-change mechanism between main channel and floodplain due to increased shear stresses and turbulence anisotropy near the vegetation interface. Also, the analytical SKM method of Shiono and Knight (1991) is applied for the depth-averaged velocity, together with simple Manning calculations. Keywords: Vegetation, Compound channel, Secondary flow, Shear stress, Turbulence models ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-82152 GOVERNING EQUATIONS 2.1 Reynolds Stress Turbulent Model In this section the macroscopic VARANS equations are presented briefly and emphasis is given to the additional terms, due to vegetation, used in both VARANS and the Reynolds Stress (RS) turbulence model. The volume averaged continuity and momentum, equations, for fully-developed open-channel flow are written respectively as follows (Finnigan 2000, Souliotis and Prinos 2010):  𝜕〈𝑈𝑖〉𝜕𝑥𝑖 = 0 (1) 〈𝑈𝑗〉 𝜕〈𝑈𝑖〉𝜕𝑥𝑗 = − 1𝜌 𝜕〈𝑃〉𝜕𝑥𝑗 + 𝜕𝜕𝑥𝑗 〈−uıuȷ�����〉 − 𝜕𝜕𝑥𝑗 〈𝑈𝚤�𝑈𝚥� 〉 + 𝑆𝑚𝑖   (2) where Ui = fluid velocity in the xi direction (U, V, W in the direction x, y and z respectively), ρ = fluid density, P = effective pressure, 〈−uıuȷ�����〉= Reynolds stresses and Smi= extra drag term due to the presence of vegetation. The symbol 〈 〉 indicates averaged values over a fluid volume. The third term of the right hand side is an “additional dispersive” term, due to correlation of spatial deviations of the mean velocity components, which can be assumed negligible in flows with high vegetation density. The pressure term in Eq. (2), for the streamwise velocity U, in a channel with slope S0 is calculated as: 1𝜌 𝜕〈𝑃〉𝜕𝑥𝑗 = −g𝑆0 (3) The extra drag term in Eq. (2) , is modelled according to Ayotte et al. (1999), as: 𝑆𝑚𝑖  = 12𝜑 𝐶𝑑𝛼|𝑈|〈𝑈𝑖〉 (4) where Cd= drag coefficient, α = plant density, defined as the frontal area per unit volume (m-1) and φ = vegetation porosity. Similar terms, accounting for vegetation effects, are included in the transport equa-tions of the modified Reynolds stress turbulence model, based on the Ayotte et al. (1999) model (not pre-sented here for the sake of brevity). The extra term, used in the transport equations for the normal stress-es, Sstr. is written as: 𝑆𝑠𝑡𝑟.  = 13𝜑 0.5𝐶𝑑𝛼|〈𝑈〉|3 (5) The additional dissipation term 𝑆𝜀 in the 𝜀 equation (𝜀 = dissipation rate of turbulence kinetic energy), ac-counting for vegetation effects, is calculated as:  𝑆𝜀  = 12𝜑 𝑡𝑒𝑓𝑓−1 𝑑𝑖𝑖 (6) where dii= the foliage contribution associated with work against pressure and viscous drag on the vegeta-tion (Ayotte et al. 1999) and teff= time scale variable, based on geometrical and turbulence characteristics (Uittenbogaard 2003). A more detailed analysis of the modified approach of the RSM Ayotte et al. (1999) model can be found in Souliotis and Prinos (2010). 2.2 Analytical SKM Method In this section the analytical SKM is presented. Based on the momentum Eq. (2) for the streamwise veloc-ity U, the equation for the depth-averaged velocity Ud is derived as follows: 𝜌 �𝜕𝐻(𝑈𝑊)𝑑𝜕𝑧 � = 𝜌𝑔𝐻𝑆0 + 𝜕𝐻𝜏𝑥𝑧�𝜕𝑧 − 𝜏𝑏 − 12𝜑 𝜌𝐶𝑑𝛼𝐻𝑈𝑑2 (7) where the index d refers to depth averaged quantity, H=water depth, ?̃?𝑥𝑧= turbulent shear stress, 𝜏𝑏= bed shear stress. The stresses ?̃?𝑥𝑧 and 𝜏𝑏 are calculated as ?̃?𝑥𝑧 = 𝜌𝜀?̃?𝑧𝜕𝑈𝑑/𝜕𝑧, 𝜏𝑏 = (𝑓/8)𝜌𝑈𝑑2 where 𝜀?̃?𝑧 is the turbulent viscosity (𝜀?̃?𝑧 = 𝜆𝑈∗𝐻, 𝜆=0.07 is turbulence constant and 𝑈∗= shear velocity) and f is the Darcy – Weisbach friction coefficient. The left hand side of Eq. (7) denotes the secondary flow. The first term of the right hand side is the gravity term, the second term is the turbulent shear stresses, the third term is the bed shear stress and the last one is the extra drag term due to vegetation. The analytical solution to Eq. (7) is given as 𝑈𝑑 =(𝐴1𝑒𝛾𝑧 + 𝐴2𝑒−𝛾𝑧 + 𝑘)1/2 where Α1 and Α2 constants (different for the vegetated and the non vegetated 2165 CONCLUSIONS The main conclusions from the numerical study of Reynolds stress modeling of flow in compound chan-nels with vegetated floodplains can be summarized in the following: − The turbulence penetration through the vegetation interface is evident with increased stresses and tur-bulence anisotropy near the vegetation interface, due to the momentum exchange between main chan-nel and floodplain.  − The secondary flow is considerable especially in the main channel near the vertical interface with the vegetated floodplain (~8% of Umean). The numerical model is able to reproduce the evolution of vorti-ces with the stronger one found in the interface region. The vortical pattern is in accordance with the experimental findings of Yang et al. (2007). − The analytical SKM method overestimates the mean velocities of such channels and the depth aver-aged velocity profiles is not efficiently reproduced near the vertical interface due to the weakness of the model to describe accurately the momentum exchange between the main channel and the vegetated floodplain.  − The separate channels method, based on the vertical interface, estimates better the mean velocity, in comparison with the numerical one, for the lower relative depths (Dr=0.15 and 0.30), while for the higher one (Dr=0.56) the simple channel method estimates better the mean velocity.  NOTATION 𝛼  plant density, defined as the frontal area per unit volume (m-1) γ parameter used in the SKM method for the non vegetated region 𝜀  dissipation rate of turbulence kinetic energy 𝜀𝑥𝑧�  turbulent viscosity  𝜆  turbulence constant =0.07 𝜈  fluid viscosity ρ fluid density 𝜏𝑥𝑧�  turbulent shear stress τb bed shear stress 𝜑  vegetation porosity Δx,Δy,Δz  dimensions of the computational cells the direction x, y and z respectively b main channel width dii  the foliage contribution associated with work against pressure and viscous drag on the vegetation d  vegetation cylinder diameter f Darcy – Weisbach friction coefficient g gravity acceleration h  main channel bankfull height k parameter used in the SKM method for the vegetated region smc  main channel side slope teff  time scale variable used for calculation of Sε 〈−uıuȷ�����〉 Reynolds stresses A cross sectional area of flow A1, A2 constants used in the SKM method for the vegetated and the non vegetated region B channel width Cd drag force coefficient Dr relative depth ratio (H-h/H) H total depth flow P effective pressure Ph wetted perimeter Rh hydraulic radius (A/Ph)  Re Reynolds number S0 channel bed slope Smi  extra drag term in momentum equation due to the presence of vegetation Sstr  extra term in transport equations of the Reynolds normal stresses Sε  extra drag term in transport equation of dissipation rate of turbulence kinetic energy U, V, W fluid velocity in the direction x, y and z respectively Ud depth averaged streamwise velocity Umean average velocity of the channel cross-section 𝑈∗ friction velocity  222REFERENCES Ayotte, K.W., Finnigan, J.F., Raupach, M.R.  (1999). A second-order closure for neutrally stratified vegetative canopy flow.  Boundary Layer Meteorology, Vol. 90, pp. 189-216. Finnigan, J. (2000). Turbulence in plant canopies. Annual Review of Fluid Mechanics, Vol. 32, pp. 519-571. Liu, Ch., Luo, X., Liu, X., Yang, K. (2013). Modeling depth-averaged velocity and bed shear stress in compound channels with emergent and submerged vegetation. Advances in Water Resources, Vol. 60, pp. 148-159. Nezu, I, Nakagawa, H. (1984). Cellular Secondary Currents in Straight Conduit. Journal of Hydraulic Engineering, Vol. 110, pp. 173–193. Shiono, K., Knight, D. (1991). Turbulent open-channel flows with variable depth across the channel. Journal Of Fluid Mechan-ics, Vol. 222, pp. 617-646. Souliotis D., Prinos P. (2010). 3-D Numerical computations of turbulence in a partially vegetated shallow channel. Riverflow 2010 - Proceedings of the International Conference on Fluvial Hydraulics, Braunschweig, Germany. Andreas Dittrich Edi-tor, pp. 83-90 Tang, X.N., Knight D.W., Sterling M. (2011). Analytical model of streamwise velocity in vegetated channels. Proceedings of the ICE - Engineering and Computational Mechanics, Vol. 164, pp. 91 –102. Uittenbogaard, R. (2003). Modelling turbulence in vegetated aquatic flows. In Proceedings of: Riparian Forest Vegetated Channels Workshop, Trendo, Italy. Yang K.J., Cao S.Y., Knight D.W. (2007). Flow patterns in compound channels with vegetated floodplains. Journal of Hydrau-lic Engineering, Vol. 133, pp. 148–159. White, F. M. (1991). Viscous Fluid Flow, 2nd edition, McGraw-Hill, New York, USA, 614 pages. White, B., Nepf, H. (2008). A vortex-based model of velocity and shear stress in a partially vegetated shallow channel. Water Resources Research, 44, W01412, pp. 1-15. 223

Reynolds Stress Modeling of Flow in Compound Channels with Vegetated Floodplains

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Reynolds Stress Modeling of Flow in Compound Channels with Vegetated Floodplains

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