3-D Numerical computations of turbulence in a partially vegetated shallow channel

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1 Rver Flow Dttrch, Koll, Aberle & Gesenhaner (eds) Bundesanstalt für Wasserbau ISBN D Numercal computatons of turbulence n a partally vegetated shallow channel D. Soulots & P. Prnos Hydraulcs Laboratory, Department of Cvl Engneerng, Arstotle Unversty of Thessalonk; Thessalonk, 54006, Greece ABSTRACT: In the present study three dmensonal computatons, based on the RANS equatons and turbulence models of k-ε and RSM type, have been performed for calculatng the flow and turbulence characterstcs, such as flow velocty, turbulent knetc energy, shear stresses and eddy vscosty, n a partally vegetated shallow channel for varous vegetaton heghts and denstes, based on laboratory experments. Computed results are compared wth avalable expermental measurements and expressons from a vortex-based model proposed by the same nvestgators. In addton, the momentum exchange between the vegetaton and the free channel n the lateral drecton, and the dstrbuton of flow and turbulence characterstcs, manly at the nterface regon, are studed. Keywords: Coherent vortces, Shear stresses, Turbulence models, 3-D computatons 1 INTRODUCTION Open channel shallow flows, partally flled wth vegetaton, s a problem of ncreased mportance n the last decades whch affects many envronmental problems such as flow adacent to terrestral and aquatc vegetaton, dscharge capacty of the channels and the mass and momentum exchange between the free flow and the vegetaton regon whch n turn nfluences the sedment and eroson phenomena. In the past years several expermental measurements and numercal computatons have been performed by varous researchers for the study of flow n partally vegetated shallow channels. Ikeda et al. (1991) modeled the sedment transport n a vegetaton regon usng a two-dmensonal eddy vscosty model to predct the velocty dstrbuton n the lateral drecton of a channel. More recently Nezu & Ontsuka (000) measured the turbulence characterstcs and observed the development of coherent vortces, due to Kelvn-Helmholtz nstablty, at the nterface regon between the vegetaton and the free channel flow. The presence of the coherent vortces has been also observed from Large Eddy Smulaton (LES) models by Nadaoka & Yag (1998) and also Xaohu & L (00). An expermental study has been performed by Whte & Nepf (007) for shallow flow n a channel partally obstructed by an array of crcular cylnders. They showed that vortces are developed due to the ncreased shear at the nterface and also that the regon s dvded n two regons. The nner regon whch establshes the penetraton of momentum nto the vegetaton and the outer regon whch establshes the scale of the vortces. Also Whte & Nepf (008) proposed a model for the vortex-nduced exchange and predcted the dstrbutons of flow velocty and shear stress n shallow channels wth a boundary of emergent vegetaton. They also found expressons for the penetraton length of momentum nsde the vegetaton and the wdth of the boundary layer n the man channel. Several numercal studes on D turbulent vegetated flow have been presented recently by varous nvestgators. In these studes the Volume Averaged Reynolds Averaged Naver Stokes (VARANS) equatons n conuncton wth a turbulence model, based on vegetaton dynamcs or porous meda approach, are employed n order to smulate the effect of vegetaton on channel flow and turbulence characterstcs. L & Yan (007) used the Spalart-Allmaras model to smulate turbulence whle Ayotte et al. (1999) and Cho & Kang (004) developed Rey- 83

2 nolds-stress turbulence models (RSM). Varous researchers (Pedras & de Lemos, 001; Uttenbogaard, 003; Foudhll et al., 005 among others) also developed turbulence models of the k-ε type for the same purpose. In the present study three dmensonal computatons of the VARANS equatons, n conuncton wth the k-ε model of Uttenbogaard (003) and the RSM of Ayotte et al. (1999), whch both are based on a vegetaton dynamcs approach, are used n order to smulate the effect of vegetaton on the channel flow. Although the k-ε model cannot smulate the ansotropy of shear stresses, s used n order to be tested on vegetated flows. It s noted that the RSM model s modfed (Soulots & Prnos 009), for mprovng the performance of the model. Two cases from Whte & Nepf (008) wth C d α=9. m -1 and 8.5 m -1 are smulated by the models (C d = drag coeffcent, α = plant densty). In addton, numercal computatons for two more cases wth less dense vegetaton (C d α = m -1 and 5 m -1 ) have been performed for assessng the effectveness of the models n a wde range of vegetaton denstes. GOVERNING EQUATIONS In ths secton the VARANS equatons are presented together wth the transport equatons for VA turbulence knetc energy k, ts dsspaton ε and the Reynolds stresses u u. In the followng equatons the symbol < > ndcates spatal averaged values and for a general parameter ψ the spatal average quanttes accordng to Nkora et al. (007) are defned as: 1 < ψ > = ψda A f A f (1) where A f =volume of flud contaned n the total volume. In the case of spatal averaged quanttes, A f s parallel to the channel bed and extensve enough to elmnate plant-to-plant varatons n vegetaton structure but thn enough to preserve the characterstc varaton of propertes n the vertcal..1 VARANS equatons The volume averaged contnuty and momentum, equatons, accordng to Raupach & Shaw (198), Fnngan (000) and Soulots & Prnos (009) are wrtten respectvely as: = 0 x () 1 = + < uu > x ρ x x < UU > 0.5Cdα U x (3) where U = flud velocty n the x drecton, ρ = flud densty, P = effectve pressure, u u = Reynolds stresses,. The thrd term n the rhs of Eq. (3) s an addtonal dspersve term, due to correlaton of spatal devatons of the mean velocty components, whch can be assumed neglgble n flows wth hgh densty. The last term n Eq. (3) s an extra drag term due to vegetaton resstance. The Reynolds stresses, , are calculated through the Boussnesq hypothess for the <k>- <ε> model as follows: δ< uu >=< t > + < k> (4) x x ν3 where v t = turbulent vscosty (= C μ <k> /<ε>, C μ = 0.09) and δ = Kronecker delta.. Transport equatons for <k>, <ε> and < uu > The transport equaton for the turbulent knetc energy, <k> and the dsspaton rate <ε> are derved from the VA (Volume Average) of the orgnal transport equatons for k and ε. For the <k>-<ε> model the equaton for <k> s: < k > < v > < > t k = x x σ κ x + < uu > < ε >+ 0.5Cdα U x (5) The last term s an addtonal source term due to vegetaton whle the addtonal dspersve terms resultng from the volume averagng of the convecton and the dffuson terms are assumed neglgble. For the RSM model the turbulence knetc energy <k> s not computed through a transport equaton snce t s obtaned drectly from the normal Reynolds stresses ( < k >= 0.5 < uu >). For the <k>-<ε> (Uttenbogaard, 003) and the RSM models the respectve equatons for <ε> are wrtten n a smlar way to the <k> equaton and they take the respectve forms: < ε> < v > t < ε> < ε> = c ε x x σε x < k> < ε > + c >< uu > + 0.5Cα c t U ( ) -1 ε1 d ε eff < k> x (6) 84

3 < ε > < k > < ε > < Uk > = cε < ukul > x xk < ε > xl < ε> < ε> + c <- uu > - c +0.5t d -1 ε1 k ε eff < k > xk < k > (7) where σ κ, σ ε = constants (= 1.0, 1.3 respectvely), c ε1, c ε = constants (=1.51, 1.9 for the <k>-<ε> model and =1.44, 1.9 for the RSM model), t eff = tme scale varable, based on geometrcal and turbulence characterstcs (Uttenbogaard, 003) and <d > = folage contrbuton assocated wth work aganst pressure and vscous drag on the vegetaton (Ayotte et al., 1999). The last terms n Eq. (6) and (7) are addtonal producton terms of dsspaton ε due to vegetaton. The <ε>-equaton, for the RSM model of Ayotte et al. (1999) has been modfed and a more detaled analyss of the modfed approach can be found n Soulots & Prnos (009). The RSM model calculates the Reynolds stresses < uu > usng a transport equaton whch takes the followng form: < uu > < < > v > uu < > = t U k x k xk σκ xk + <- uu k > +<- uuk > xk xk 1 3 +< φ >- δ < ε >+ 0.5Cdα 3 3 (8) where φ =pressure stran term (FLUENT INC., 001). The last term n Eq. (8) s an extra term due to the vegetaton and s only used n the transport equatons for the normal stresses. 3 NUMERICAL PROCEDURE - CASES STUDIED The FLUENT CFD code, based on a fnte volume technque, s used for the numercal computatons. The code solves Eqns. (), (3), (5), and (6) when the <k>-<ε> model s appled whle Eq. (), (3), (7) and (8) are solved when the RSM model s used. The extra source terms, due to the presence of vegetaton are ntroduced n the above equatons, usng User Defned Functons (UDF) whch are based on a C code. For the grd constructon the GAMBIT program was used. The grd was three-dmensonal, structured and the shape of the cells was orthogonal. Several runs wth dfferent number of computatonal cells showed that the numercal results are grd ndependent. In the present work the grd sze s equal to 5x40x0 (streamwse, lateral, and vertcal drecton respectvely). The grd densty s 85 ncreased near the walls and also at the nterface regon (vegetaton-free channel) due to the ncreased shear. At the entrance and the ext of the flow perodc condtons were used n order to avod the ncreased length whch s needed for the flow to become fully developed. For the free surface, symmetry boundary condtons were used. In Fgure 1 a three dmensonal vew of the vegetated and the free channel regons s shown. Fgure 1: 3-D vew of the total channel regon Expermental measurements from Whte & Nepf (008) are used for the evaluaton of the turbulence models. In these experments a 1.m wde and 13m long flume partally flled wth a 0.4m wde array of cylnders, wth m dameter, whch smulates the vegetaton, were used. The flow depth was vared between 0.066m and 0.068m whle the channel slope S o was.9x10-4. The porosty of the modelled vegetaton vared between a mnmum value of 0.9 and a maxmum of 0.98 whch corresponds to C d α = 74 m -1 and 9. m -1 respectvely. In the present paper two cases from Whte & Nepf (008) wth C d α = 9. m -1 and 8.5 m -1 are used. In addton, numercal computatons for two more cases wth less dense vegetaton (C d α = m -1 and 5 m -1 ) have been performed. 4 ANALYSIS OF RESULTS Intally the varaton of fully developed local mean veloctes and shear stress < uw >, computed by both models, wthn the crosssecton s presented for C d α=9. m -1. Fgure shows the contours of streamwse velocty, made dmensonless wth the maxmum local velocty at the free channel flow (U max ). Insde the vegetaton low values are observed for both models. At the free channel regon hgher values are observed at the center of the channel and near the free surface. Accordng to the <k>-<ε> model the dstrbuton of flow velocty s more unform

4 n the central part of the free channel, n contrast to the RSM model, whch shows that at the channel corner a bulgng of the contours occurs due to the secondary currents predcted by the model n the corner regon. Fgure : UContours of dmensonless streamwse velocty for C max d α = 9. m -1. In Fgure 3 the contours of local shear stress ( < uw > ) are shown. The local shear stress has been made dmensonless wth the depth-averaged shear stress at the nterface ( < uw > nt ).At the nterface regon and for y/h>0.5 hgher values of < uw > are observed due to ncreased velocty gradents n the lateral drecton. Insde the vegetaton shear stresses have very low values whch mean that the presence of vegetaton damps the turbulence wthn the vegetaton and flow may become lamnar. At the centre of the channel, the effect of the sde wall and the nterface regon s reduced, and the values of < uw > are reduced as t s expected. In the followng paragraphs the dstrbutons of depth-averaged mean velocty and turbulence characterstcs (turbulence knetc energy, Reynolds stresses and turbulent vscosty) are presented and compared wth avalable expermental data and emprcal relatons from Whte and Nepf (008). For the sake of brevty the subscrpt denotng depth-averaged values has been removed from all parameters. In Fgure 4 the dstrbutons of wall (bed, sdewalls) shear stresses as well as the depth-averaged shear stress ρ < uw > are shown. They have been made dmensonless wth γhs o, (γ = ρg). For both cases the stresses at the sde wall of the free channel are hgher n comparson to those at the sde wall adacent to the vegetaton, due to the hgh flow velocty n the free channel. The RSM model provdes hgher values at the free channel sde wall. The bed shear stress n the vegetaton regon has low values due to the reduced flow velocty, whle n the free regon, where the dstrbuton of τ bed also follows the development of flow velocty, ncreased τ bed values occur at the central part of the channel. The RSM model presents ncreased τ bed values near the corner regon, due the capablty of the model to compute the secondary currents generated by the ansotropy of the normal stresses. Fgure 3: Contours of dmensonless shear stress < uw > < uw > nt for C d α = 9. m Fgure 4. Dstrbuton of wall shear stress τ wall and depthaveraged shear stress ρ < uw >. Fgure 5 reveals the contrbuton of free channel bed shear stress (τ bed ), wall shear stress at the sde wall of the free channel (τ wall ) and the shear

5 stress at the nterface (τ nt. ), made dmensonless wth the total free channel stress (τ tot. = τ bed + τ wall + τ nt ). In order to get a more clear vew, results from two more cases, wth C d α = and 5 m -1, are used. As t s shown τ bed has domnant role (86-90% of the total stresses), whle the effect of the nterface and the sde wall s lower, wth a vared contrbuton of 6-10% and 4-4.5% respectvely. The ncreased values at the nterface regon n comparson to those at the sde wall can be attrbuted to the momentum exchange between free channel and vegetaton flows. In Fgure 6 the varaton of depth-averaged streamwse velocty, made dmensonless wth the maxmum depth averaged velocty at the free channel flow (U ), n the lateral drecton s shown, for both turbulence models. Insde the vegetated regon the flow velocty s unform and the computed results concde wth the expermental measurements. The momentum equaton, nsde the vegetaton s smplfed to a balance between the drag and the bed gradent. At the nterface regon an nflecton pont s observed due to the momentum exchange and the assocated ncreased shear due to the dfference of flow velocty between free channel and the vegetated area. Shear layer type profles are developed at the free channel regon and, accordng to the <k>-<ε> Uttenbogaard (003) model, the flow becomes unform n a wder central area n comparson to that of the RSM model. The expermental measurements are n agreement wth the RSM results and as t s expected the regon of unform flow velocty s ncreased for the sparser case (C d α =9. m -1 ). The effect of the sde channel wall s sgnfcant for z/l veg. >1.5 accordng to the RSM model, whle the respectve regon accordng to the <k>- <ε> model s decreased (z/l veg. >1.6). Fgure 5. Wall and nterfacal stresses contrbutons to total free channel stress for varous vegetaton denstes. The <k>-<ε> Uttenbogaard (003) model provdes hgher values for τ bed n comparson to the RSM model whle the respectve values for τ wall and τ nt. are lower. However, ths behavor s not observed for the hgh densty case. Fgure 6. Computed and expermental streamwse, depthaveraged velocty dstrbutons. In Fgure 7 the dstrbuton of the depthaveraged shear stress ( < uw > ) s shown. The shear stress has been made dmensonless wth the square of frcton velocty at the nterface U * 87

6 (=max( < uw > ). The profles demonstrate that maxmum values occur at the nterface. Insde the vegetaton < uw > s reduced rapdly, whle n the free channel the respectve decrease s more gradual. The two turbulence models reveal dentcal behavor n the vegetaton regon, whle n the free flow the values of < uw > accordng to the <k>-<ε> Uttenbogaard (003) decrease more rapdly n comparson to those of the RSM Ayotte et al. (1999) model. In ths regon the expermental measurements are n agreement wth the RSM model for both cases whle at the nterface the maxmum values are hgher n comparson wth the computed results. Ths means that shear at the nterface s hgher and hence the length n whch hgh momentum flud penetrates nsde the vegetaton s ncreased as t s shown for the sparse case (C d α = 9. m -1 ). The hgher value of expermental shear stresses, for the denser case, has neglgble effect on the penetraton of hgh momentum flud, whch means that the ncreased vegetaton densty has domnant role. Fgure 7. Dstrbutons of computed and expermental depth-averaged shear stress < uw >. Smlar results are observed from Fgure 8 where dmensonless (wth U * ) dstrbutons of turbulent knetc energy are shown. Insde the vegetaton the RSM model computes neglgble values n comparson to those of the <k>-<ε> model, whch s caused due to an extra source term n <k> equaton whch smulates the presence of vegetaton. For the <k> there are no avalable expermental measurements for comparson wth the computed results. The above results are n agreement wth Whte & Nepf (007) who have shown that two dstnct regons exst n the shear layer. The regon, called nner layer, where momentum can penetrate nto the vegetaton wth length δ I and the regon (outer layer) outsde the vegetaton, wth length δ O, where boundary layer profles are developed. The δ I s taken as the dstance from the nterface to the pont where Reynolds stresses decays to 10% of ts maxmum value (Nepf et al., 007) whle δ O s taken as the dstance from the nterface to the pont where flow velocty reaches ts unform value n the free channel. The dstance δ I can be computed wth hgh accuracy snce the number of the grd ponts s ncreased n ths regon and hence the error s neglgble. Fgure 8. Computed dstrbuton of turbulent knetc energy. In Fgure 9 the nner layer profles are shown and are compared to an hyperbolc profle presented by Whte & Nepf (008): U 1 z z = 1+ tanh UUs δ I o (9) where U 1 = the depth-averaged unform velocty nsde the vegetaton, z o = the nflecton pont (whch concdes wth the nterface) and U s = the depth averaged slp velocty, equal to U zo - U 1. The computed normalzed velocty profles and the hyperbolc profle are n agreement at the n- 88

7 terface, however n the nner layer the comparson s less satsfactory. Insde the vegetaton regon the profles from the turbulence models and from Eq. (9) concde. where z m = the pont at whch the nner and the outer layer slope match and U m = the velocty at z m. In the present work z m = 0 (concdes wth the nterface) and U m s equal to the velocty at the nterface. The comparson shows that the computed veloctes from the turbulence models are developed more rapdly and reach unform values, at the free channel, earler than those computed by the parabolc profle. In Fgure 11 the profles of normalzed eddy vscosty for the two cases are shown. The eddy vscosty has the hgher values for both models at the free channel regon (z/δ O = 0.) where the coherent vortces, due to the shear at the nterface, are ncreased. Insde the vegetaton v t decreases rapdly and for z/δ O < -0. has neglgble values whch means that the vortces are unable to penetrate deeper, due to the ncreased vegetaton densty. Both models demonstrate smlar behavor for the two cases. Fgure 10. Rescaled outer layer profles. Fgure 9. Normalzed nner layer profle. Outsde the vegetaton the hyperbolc profle fals to follow the computed profles. For ths regon the computed outer layer profles are compared (Fgure 10), to a parabolc velocty profle presented by Whte & Nepf (008): U U 1 m z zo z zm = U Um 4 O δδ O (10) Fgure 11. Dstrbutons of normalzed eddy vscosty. 5 CONCLUSIONS In the present study flow and turbulence characterstcs have been computed for a three- dmensonal partally vegetated shallow flow. Dstrbutons of flow velocty, shear stresses and turbulent 89

8 knetc energy, n the lateral drecton, show that at the nterface regon ncreased shear s observed due to the momentum exchange and the nteracton between the vegetaton and the free channel flow. The produced coherent vortces penetrate nto vegetaton and ths penetraton length decreases wth densty whle n the outer regon the effect of vegetaton densty s neglgble and boundary profles of flow velocty are developed. Insde the vegetaton where the coherent vortces are unable to penetrate the flow s unform and the computed values for shear stresses ( < uw > ) and turbulent knetc energy (<k>) are very low. At the central part of the free channel, where the effect of vegetaton s also decreased, reduced varatons of flow and turbulence characterstcs are observed, wth the <k>-<ε> Uttenbogaard (003) model to provde wder areas of constant values. However the modfed RSM model demonstrates n general better behavor when ts numercal results are compared wth expermental measurements for and < uw >. As t s also shown the channel bed provdes the hgher contrbuton to the total wall shear stress and also the wall stress, at the corner regon, demonstrates ncreased varaton n comparson to that of the <k>-<ε> model due to the capablty of the RSM model to predct secondary currents correctly. Nepf, H., Ghsalbert, M., Whte, B., Murphy, E., 007. Retenton tme and dsperson assocated wth submerged aquatc canopes. Water Resources Research, 43, W044, do:10.109/006wr Nkora, V.I., McEwan, I.K., McLean, S.R., Coleman, S.E., Pokraac, D., Walters, R Double-Averagng Concept for Rough-Bed Open-Channel and Overland Flows: Theoretcal background. Journal of Hydraulc Engneerng, ASCE,, 133, Pedras M.H.J., de Lemos M.J.S Macroscopc turbulence modellng for ncompressble flow through undeformable porous meda. Internatonal Journal of Heat and Mass Transfer, 44, Raupach, M.R., Shaw, R.H Averagng procedures for flow wthn vegetaton canopes. Boundary Layer Meteorology,, Soulots, D., Prnos, P Macroscopc turbulence models and ther applcaton n turbulent vegetated flows, Journal of Hydraulc Engneerng, ASCE, (under revew). Whte, B., Nepf, H Shear nstablty and coherent structures n shallow flow adacent to a porous layer. Journal of Flud Mechancs, 593, 1-3; DOI: /S Whte, B., and Nepf, H A vortex-based model of velocty and shear stress n a partally vegetated shallow channel. Water Resources Research, 44, W0141; DOI: /006WR Uttenbogaard R Modellng turbulence n vegetated aquatc flows. Rparan Forest Vegetated Channels Workshop, Trendo, Italy. Xaohu, S., L, C. 00. Large eddy smulaton of free surface turbulent flow n partly vegetated open channels, Internatonal Journal for Numercal Methods n Fluds, 39, REFERENCES Ayotte, K.W., Fnngan, J.F., Raupach, M.R A second-order closure for neutrally stratfed vegetatve canopy flow. Boundary Layer Meteorology, 90, Cho, S. U., Kang, H. S Reynolds stress modellng of vegetated open-channel flows. Journal of Hydraulc Research, IAHR, 4(1), Fnngan, J., 000. Turbulence n plant canopes. Annual Revew of Flud Mechancs, 3, FLUENT INC Fluent 6.0 Documentaton. Lebanon, USA. Foudhll, H., Brunet, Y., Caltagrone, J-P A fnescale k-ε model for atmospherc flow over heterogeneous landscapes. Envronmental Flud Mechancs, 5, Ikeda, S., Izum, N., Ito, R Effects of ple dkes on flow retardaton and sedment transport. Journal of Hydraulc Engneerng, ASCE, 117(11), L, C. W., Yan, K., 007. Numercal nvestgaton of wavecurrent-vegetaton nteracton. Journal of Hydraulc Engneerng, ASCE, 133(7), Nadaoka, K., Yag, H Shallow-water turbulence modellng and horzontal large-eddy computaton of rver flow. Journal of Hydraulc Engneerng, ASCE, 14(5), Nezu, I., Ontsuka, K Turbulent structures n partally vegetated open-channel flows wth LDA and PIV measurements. Journal of Hydraulc Research, IAHR, 39(6),

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