A General Analytical Model for Lateral Velocity Distributions in Vegetated Channels

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1 Rier Flow - Dittrich, Koll, Aberle & Geisenhainer (es) - Bunesanstalt für Wasserbau ISBN A General Analtical Moel for Lateral Velocit Distributions in Vegetate Channels X. Tang School of Ciil Engineering, The niersit of Birmingham, Egbaston, Birmingham, B5 TT, K State Ke Laborator of raulics an Mountain Rier Engineering, Sichuan niersit, 665 M. Sterling & D.W. Knight School of Ciil Engineering, The niersit of Birmingham, Egbaston, Birmingham, B5 TT, K ABSTRACT: This paper presents a metho for preicting the istribution of epth-aerage elocit in channels with either emergent or submerge egetation. A general analtical solution to the epthintegrate Renols-Aerage Naier-Stokes (RANS) equation is gien, where the rag force ue to egetation is moelle as an aitional momentum sink term. The metho inclues the effects of be friction, rag force, lateral turbulence an seconar flows, ia four coefficients f, CD, λ & Γ respectiel. The analtical solution gies goo preictions of lateral elocit istribution when compare with two sets of experimental ata of egetate channels with submerge egetation an two sets of ata with emergent egetation. The lateral istribution of be shear stress can also be obtaine through the Darc- Weisbach friction factor. The preicte elocit an be shear stress istributions can proie a basis for moelling floo coneance an seiment transport in channels with egetation. Kewors: Vegetation, Open channel, Be shear, Velocit INTRODCTION Man natural riers hae significant egetation, which introuces aitional hraulic resistance to the flow an reuces the local flow elocit. A strong shear laer often exists between the fast flowing water in the central region of a channel an the slower flowing water in the egetate bounaries. In these circumstances, the exchange of mass an momentum between egetate an non-egetate regions will affect the channel coneance as well as the elocit an bounar shear stress istributions. A metho for preicting these is therefore clearl require when esigning floo alleiation schemes, as well as for stuies on bank protection an seiment transport. There hae been seeral stuies on the flow in composite channels with egetation, base either on a simplifie one-imensional (-D) approach with an empirical Darc-Weisbach friction factor, or on a simple e iscosit moel for the turbulence (e.g. Darb 999; Lope & Garcia, ; elmio, 4). These one-imensional moels o not escribe lateral istribution of elocit. More recentl White & Nepf (8) propose a ortex-base moel for preicting the lateral istribution of elocit an shear stress in a partiall egetate channel. This moel showe goo agreement with the experimental ata, espite the fact that the influence of seconar flow being ignore. The present paper proposes a new general analtical solution for preicting the lateral istributions of elocit an be shear stress for both submerge an non-submerge egetation. The moelling is base on a epth-aerage form of the streamwise Renols Aerage Naier-Stokes (RANS) equation with an aitional momentum term to eal with the rag force arising from the presence of the egetation. The metho relies on four hraulic parameters relate to the be friction factor (f), lateral e iscosit (λ) an epth-aerage seconar flow (Γ), an a special parameter for the egetation, namel a spatiall aerage rag coefficient (C D ). The preicte results, base on this analtical solution, are shown to agree well with the experimental epth-aerage elocit ata from egetate channels. The experimental ata for flow with emergent egetation are taken from White & Nepf (8) an Pasche (984), an that for submerge egetation from Shimiu & Tsujimoto (993) an Yan (8). The particular case of 469

2 flow through emergent egetation is also iscusse elsewhere b Tang et al. (). FLOW IN A VEGETATED CANNEL. Goerning equation of epth-aerage flow For stea flow in a prismatic open channel, the equation for the streamwise momentum ma be combine with the continuit equation to gie: τ x τ x ρ ( V ) + ( W ) = gs o + + ρ () where {VW} = elocit components in the {x} irections, x-streamwise parallel to the channel be, -lateral an -normal to the be, ρ = flui ensit, g = graitational acceleration, S o = channel be slope, an {τ x, τ x }= Renols stresses on planes perpenicular to the an irections respectiel, as illustrate in Figure. For flow in a egetate channel, the rag force ue to egetation (plants) ma be introuce into Eq. () (timeaerage RANS) b an aitional momentum sink term to gie (Tang et al ): τ τ x x ρ ( V ) + ( W ) = ρgs + + F () where F is the rag force per unit flui olume ue to the egetation, an represente b: F o = ρ( CDβA ) (3) where C D is the rag coefficient, β is a shape factor of egetation, an A is the projecte area of the egetation in the streamwise irection per unit olume. It shoul be note that Eq. () is a time aerage momentum equation for the streamwise flow in the x-irection. F correspons to a rag force component in the x-irection, an as such is linke onl to the primar flow (). It is appropriate that an aitional ispersie term ue to correlation of spatial eiations of the mean elocit components (Finnigan, ) is negligible, hence not consiere ue to uniformit in the een istribute egetate region. B integrating Eqs () an (3) oer the water epth,, (proie W = when = an, as assume b Shiono & Knight, 99), the epthaerage momentum equation becomes: [ ( ρv ) ] = ρgso τ x + τ b ρ( CDβA ) h (4) where the oerbar or the subscript refers to a epth-aerage alue, τ b is the be shear stress, an an are the epth-aerage elocit oer the egetate height (h) an the total flow h epth (), efine b = an h = respectiel. For the non-submerge egetation, = ue to h = ; otherwise < for submerge egetation, as illustrate in Figure. B taking into account the porosit, δ (= -φ), for the blockage effects of egetation on the flow, where φ is the olumetric egetation ensit, efine as the ratio of the olume of egetation to the flow, it follows that φ = h * (π/4)d n, where D represents a characteristic iameter of the egetation, n is the number of plants per unit bottom area, an h * represents the ratio between the height of egetation (h) an the flow epth (), efine b min [, h]/. Therefore h * < if the egetation is submerge whereas h * = for emergent egetation. The projecte area of the egetation in the streamwise irection, A = 4φ/(πD). It is worth noting that for submerge egetation, in Eq. (4) is replace b, where their relationship can be etermine b Stone & Shen s () equation as follows: h = k (5) where k is a coefficient efine b [(-Dn ½ )/(- Dh * n ½ )], which is about. for most practical cases. Eq. (4) ma be rewritten in the form of effectie water olume as: ( V ) τ ρδ = ρδgso + δ δ τ b ρ( CDβA )( k h* ) Figure. Be an wall shear τ τ b x x τ τ (6) 47

3 4 8 f γ = + C D β A ' (b) λ f 8 δ For the non-egetate channel, i.e. A = an δ =, Eqs (a&b) become: Figure. Sketch of open channel flow with submerge egetation Therefore Eq. (6) is a general epth-aerage momentum equation in the streamwise irection for shallow-water flow with egetation. It shoul be note that h * = an k = for non-submerge egetation. For the non-egetate channel, i.e. A V = an δ =, Eq. (6) then takes the same form as that gien b Shiono & Knight (99). B making the following assumptions: f τ b = ρ ; τ x = ρε x ; 8 εx = λ * ; Γ = [ ( ρv ) ] (7) B using the notation A ' =A k h 3 *, Eq. (6) becomes f ρ gs ρ ρ( C Dβ A ') 8 δ / f + ρλ = Γ 8 (8) These parameters are iscusse more full b Shiono & Knight (99, 99), Knight & Shiono (996), Tominaga & Knight (4), Abril & Knight (4), an Knight et al. () for nonegetate channels an b Rameshwaran & Shiono (7), an Tang et al. () for egetate channels.. Analtical solution of epth-aerage flow For a egetate channel, where the rag coefficient C D, ensit of egetation (φ), local friction factor (f), e iscosit (λ) an seconar flow term (Γ) are gien, an analtical solution to Eq. (8) for can be obtaine as follows: γ γ [ A e + A e ] = + k (9) gso Γ / ρ k = (a) f / 8 + /(δ ) C β A ' D 4 8( gso Γ / ρ) f k = ; γ = () f λ 8 It can be seen that Eq. () is the same as that gien b Shiono & Knight (99). The unknown constants, A to A in Eq. (9), can be obtaine through appling appropriate bounar conitions to the cross-section shown in Figure 3. For each panel the two unknown constants, A to A can be eliminate using the following bounar conitions (Knight et al. 4, Tang & Knight, 8): The no-slip conition, i.e. = at remote bounaries or = gien alues ( or ) as shown in Figure 4; The continuit of the elocit at each omain junction, i.e. (i) (i+) = ; The continuit of unit force (τ x ) at each omain junction, i.e. [ τ x ] (i) = [τ x ] (i+) It follows for a continuous epth omain that the continuit of unit force implies ( i) ( i+ ) μ = μ with μ = λ f () where the superscript (i) inicates the number of an iniiual panel. Knight et al. (4) point out that the continuit of /, preiousl use b most researchers [Shiono & Knight, 99; Erine et al. ], is onl appropriate for certain cases, notabl where f an λ are the same in the two ajoining omains. b () x b () Figure 3. Cross-section of a egetate channel with notation 47

4 For a rectangular channel, the lateral epth-mean elocit for half the channel becomes: k = cosh( γ ) + k cosh( γb) (3) where b is the half channel with, an k with γ are gien b Eqs (a) an (b) respectiel. Therefore Eq. (9) gies the lateral istribution of epth-mean elocit an bounar shear stress [ia Eq. (7)] in a egetate channel with either emergent or submerge egetation. It shoul be note that the solutions (9) - () for are also suitable for non-egetate channels, in which case A = an φ =. x Figure 4. Sketch of partiall egetate laborator channel b White & Nepf (8) 3 APPLICATION OF ANALYTICAL SOLTION TO FLOW IN A CANNEL WIT SBMERGED VEGETATION 3. Introuction to the experiments Two sets of experiments in rectangular channels with submerge egetation were use in this stu. One was with the channel partiall egetate, an the other was with egetation place across the whole with of the channel. In the first, Shimiu an Tsujimoto (993) conucte an experiment in a.4 m wie,.5 m long tilting flume with a be slope of /. Detaile turbulence measurements were carrie out with a two-colour fibre-optic Laser Doppler Anemometer (LDA) in a flow with a epth of.8 m an half the channel fille with a x () mm arra of.8 mm iameter circular cliners. The height of these wooen ros was.6 m, thus representing a moel for submerge egetation, as illustrate in Figure 4. The corresponing olume ensit of moel plants (φ) was about.3. As the channel was configure b Vinl Chlorie plates, it was assume to be a hraulicall smooth channel. The secon set of experiments were unertaken b Yan (8) in a.4 m wie, m long tilting flume. In the series F, where the be slope of channel was set to be.7%, etaile turbulence measurements were carrie out with a LDA for a flow with a epth of. m an with the whole channel fille with an arra of 6 mm iameter circular ros, in a x 5 mm pattern. The height of the aluminum ros was.6m, thus also representing a moel for submerge egetation, as illustrate in Figure. The olume ensit of moel plants (φ) in this case was about.4. The Manning coefficient (n) of the channel was reporte to be about.. 3. Moelling results of lateral elocit istribution In orer to appl Eq. (9) with () to preict the epth-aerage elocit, the rag coefficient (C D ), local friction factor (f), e iscosit (λ ) an seconar flow term (Γ) are require, as well as the shape factor β. Each of these parameters is now aresse in etail below. The friction factor (f) is obtaine using the stanar Colebrook-White equation, gien b: f 3. log = Re f k s +.3 (4) where Re is the local Renols number, efine b 4 /ν, an the equialent san roughness height, k s, was assume to be.6 mm, an was obtaine through the relationship n = k s /6 /(8.5 g / ), with n corresponing to a hraulicall smooth channel surface (i.e. n =.9) (Ackers, 993). Thus the local friction factors (f) were estimate to be.73 an.66 for the aboe two experimental cases respectiel, base on (4). The shape factor β was set equal to., as clinrical ros were use in the experiment to simulate the egetation. Tanino & Nepf (8) showe that the rag coefficient (C D ) ecreases as the Renols number base on the ro iameter, (Re D = D/ν) increases. oweer, alues are tpicall in the range.-.5 for φ <.9 for Re D up to O( 3 ). Therefore it is assume that C D can be taken as., which it is consiere appropriate for the stuie case (φ =.3), whereas for the 47

5 high ensit case (φ =.4), C D was assume to be.. In orer to calculate the seconar flow term, Γ, recourse was mae to the work of Shiono & Knight (99), who, base on the compoun channel ata of the K-FCF flume, suggeste Γ /(ρgs o ) =.5 for the non-egetate main channel. It is worth noting that the relationship is ali for a compoun channel, but it is appropriate to be use for the channel with partiall egetate channel, which has a similar role as a flooplain, as emonstrate b Shimiu & Tsujimoto (993). oweer, the work of Ghisalberti & Nepf (4) an Yang et al. (7) hae emonstrate that not onl is the flow aroun egetation elements complex but there is eience of weak seconar current cells within a egetate flooplain. The effect of the seconar flow on the egetate flooplain was ignore, i.e. Γ =. This can also supporte b the elocit contour plot shown b Shimiu & Tsujimoto (993). The final parameter alue to be aresse relates to the e iscosit, λ. The alue of λ was taken as.7, which is close to the stanar alue.67 (= κ/6, where κ is the on Karman constant). oweer, preious work has illustrate that the alue of λ can change significantl. The results of the simulation for aboe two cases are shown in Figures 5 an 6, where Figure 6 shows onl the simulation for half the channel ue to the smmetr of the channel. Figure 5 shows that a strong shear laer exists between the slow flowing egetate region an the fast flowing non-egetate region. The simulate results of Figures 5 an 6 agree well with the experimental ata. Figures 5 an 6 illustrate the abilit of the analtical solution to proie reasonabl goo results for submerge cases (m/s) Preiction Data (m). Figure 5. Comparison of preicte istributions with experimental ata (Shimiu & Tsujimoto, 993) (m/s) Preiction Experimental Data (m) Figure 6. Comparison of preicte istributions with experimental ata (Yan, 8) 4 APPLICATION OF ANALYTICAL SOLTION TO FLOW IN A CANNEL WIT PARTIALLY EMERGENT VEGETATION White & Nepf (8) carrie out etaile twoimensional elocit measurements with a Laser Doppler Velocimetr (LDV) in a. m wie, 3 m long flume, partiall fille with a.4 m wie arra of 6.5 mm iameter wooen circular cliners. The wooen ros pierce the water surface, thus representing a moel for emergent egetation, as illustrate in Figure 4. The flow epth arie from 55 mm to 5 mm, ensuring large aspect ratios (> 8) to minimie the impact of the sie walls on the flow. Three olume ensities of moel plants (φ =.,.45 an.) were use in their experiments. In this stu, onl one case X (φ =.) was use here, with the corresponing flow parameters: flow epth, =.78 m, S o =.75%, f =.5 an αc D =.77. Further examples of simulations for other experimental runs ma be foun in Tang et al. (). The results of simulating emergent egetation for the case X are shown in Figure 7, where the moelling parameters assume in the analtical solution are outline aboe in section 4.. The agreement with the experimental ata of White an Nepf (8) is again seen to be quite goo. The secon set of experimental ata use to ealuate the analtical solution pertains to that of Pasche (984) an Pasche & Rouse (985). These ata relate to laborator experiments which were conucte in an asmmetric compoun channel with a egetate flooplain. The emergent egetation was moelle using mm iameter clinrical wooen ros. The channel cross section ha a bank full height (h) of.4 m, with the remaining imensions shown in Figure 8. Manning s coefficient n was reporte to be. for the smooth part of the channel for all 473

6 experiments. In what follows, ata from two experiments (Case I: flow epth () =.5 m, S o =.5% an Case II: =.4 m, S o =.%) will be use. The corresponing olume ensities (φ) were set at.63% an.54% respectiel. In keeping with the aboe analsis, the shape factor β was set equal to., as clinrical ros were again use in the experiment to simulate the egetation. It is worth noting that the local friction factors for the main channel an flooplain were obtaine as alues of.5 an.3 respectiel (Tang et al. ). For flow oer a linearl sloping be without egetation in the main channel, is gien b (Shiono an Knight, 99) α ( α + ) = A ξ + A ξ + ωξ + (5) [ ] 3 4 η where the constants α, ω an η are gien b ( s ) / ( 8 ) / s + / α = + + f (6) λ gs ω = (7) / / ( + s ) f λ f s 8 s 8 Γ η = (8) / f 8 ρ + s where ξ is the local epth gien b - ( - b)/s (for >) an + (+b)/s (for <) as shown in Figure 8. Similar to the flat be case, A 3 an A 4 are unknown constants for each panel, but are obtaine b appling appropriate bounar conitions, as outline in section.. Also see Knight et al. (4 & 7) for further etails. The e iscosit (λ) was base on the assumptions outline below. In the main channel, λ was taken as.7. oweer, preious work has illustrate that on the flooplain the alue of λ can change significantl. ence, recourse was mae to the work of Abril an Knight (4), who state that λ fp = (-. +. Dr -.44 ) λ mc (9) in which the subscripts mc an fp refers to the main channel an the flooplain respectiel. The results of the analsis are shown in Figure 9, which again illustrates the abilit of the analtical solution to proie reasonabl goo results oer a wie range of egetation ensities (the RMSSE arie between.3 an.7). Simulations for other experimental runs can be also foun in Tang et al. () (m/s) Preiction Data (m) Figure 7. Comparison of preicte istributions with experimental ata (White & Nepf, 8) ξ() s h Figure 8. Cross-section of egetate compoun channel b Pasche (984): nit: m [m/s] Preiction [m] [m/s] Figure 9. Comparison of moelle istributions with Pasche s experimental ata Experimental Data Channel be (I) φ =.63% Preiction Experimental Data Channel be (II) φ =.54% [m] 474

7 5 CONCLSIONS The following conclusions ma be rawn from this stu: A general analtical solution (9) for epthaerage elocit in a egetate channel has been obtaine. It is base on the epth-integrate form of the Renols- Aerage Naier-Stokes equation, (8), with the rag force ue to egetation being moelle as an aitional momentum term. The analtical solution (9) with () simulates the lateral epth-aerage elocit istribution in egetate channels (either emergent or submerge egetation). It ma also be use for non-egetate channels, in which case A = an δ =. The preicte elocit istributions agree well with the experimental ata for flow in a egetate channel for both emergent an submerge egetation. The propose analtical solutions (9) can be use to preict bounar shear istributions through the Darc-Weisbach friction factor. REFERENCES Abril, J.B., Knight, D. W. 4. Stage-ischarge preiction for riers in floo appling a epth-aerage moel. Journal of raulic Research, IAR, 4(6), Ackers, P Flow formulae for straight two-stage channels. Journal of raulic Research, IAR, 3(4), Darb, S. E Effect of riparian egetation on flow resistance an floo potential. Journal of raulic Engineering, ASCE, 5(5), Erine, D.A., Babaean-Koopaei, K., Sellin, R..J.. Two-imensional solution for straight an meanering oerbank flows. Journal of raulic Engineering, ASCE, 6(9), Finnigan, J.. Turbulence in plant canopies. Annual Reiew of Flui Mechanics, 3, Ghisalberti, M., Nepf,.M. 4. The limite growth of egetate shear laers. Water Resources Research, 4; W75, oi:.9/3wr776. elmio, T. 4. Flow resistance ue to lateral momentum transfer in partiall egetate riers. Water Resources Research, 4; W56, oi:.9/4wr358. Knight, D.W., Mc Gahe, C., Lamb, R. an Samuels, P.G.. Practical Channel raulics: Roughness, Coneance an Afflux, CRC/Talor & Francis, Knight, D.W., Omran, M., Abril, B. 4. Bounar conitions between panels in epth-aerage flow moels reisite. Rier Flow 4, Proc. n Int. Conf. on Fluial raulics, 3-5 June, Napoli, Ital [Es M. Greco, A. Carraetta & R.D. Morte], Vol., Knight, D.W., Shiono, K. 99. Turbulence measurements in a shear laer region of a compoun channel. Journal of raulic Research, IAR, 8(), 99, 75-96, [Discussion in JR, 99, 9(), 59-76]. Knight, D.W., Shiono, K Rier channel an flooplain hraulics, In Flooplain Processes, [Es M.G. Anerson, D.E. Walling & P.D. Bates], Chapter 5, J. Wile, Lope, F., Garcia, M.. Mean flow an turbulence structure of open-channel flow through non-emergent egetation. Journal of raulic Engineering, ASCE, 7(5), Rameshwaran, P., Shiono, K. 7. Quasi two-imensional moel for straight oerbank flows through emergent egetation on flooplains. Journal of raulic Research, IAR, 45(3), Shimiu, Y., Tsujimoto, T Comparison of floo flow structure between compoun channel an channel with egetation one. Proceeings of 5 th IAR Congress, Toko, Vol. A-3-4, Shiono, K., Knight, D.W. 99. Mathematical moels of flow in two or multi stage straight channels. Proc. Int. Conf. on Rier Floo raulics, (E. W.R. White), Wallingfor, September, J. Wile & Sons, Paper G, Shiono, K., Knight, D.W. 99. Turbulent open channel flows with ariable epth across the channel. Journal of Flui Mechanics,, (an JFM, Vol. 3, 693). Stone, B M., Shen, T.. Mean flow an turbulence structure of open-channel flow through non-emergent egetation. Journal of raulic Engineering, ASCE, 8(5), Tang, X., Knight, D. W. 8. Lateral epth-aerage elocit istribution an be shear in rectangular compoun channels. Journal of raulic Engineering, ASCE, 34(9), Tang, X., Knight, D. W., Sterling, M.. Analtical moel of streamwise elocit in egetate channels. Engineering an Computational Mechanics, ICE (in press). Tanino, Y., Nepf,. M. 8. Laborator inestigation of mean rag in a ranom arra of rigi, emergent cliners. Journal of raulic Engineering, ASCE,34(): Tominaga, A., Knight, D.W. 4. Numerical ealuation of seconar flow effects on lateral momentum transfer in oerbank flows. Rier Flow 4, Proc. n Int. Conf. on Fluial raulics, 3-5 June, Napoli, Ital [Es M. Greco, A. Carraetta & R.D. Morte], Vol., White, B. L., Nepf,. M. 8. A ortex-base moel of elocit an shear stress in a partiall egetate shallow channel. Water Resources Research, 45; W4, oi:.9/6wr565. Yan, J. 8. Experimental stu on flow resistance an turbulence of open channel flows with egetation. PhD Thesis, ehai niersit, China, (in Chinese). Yang, K., Cao, S., Knight, D. W. 7. Flow patterns in compoun channels with egetate flooplains. Journal of raulic Engineering, ASCE, 33(),