PRACTICAL NUMERICAL SIMULATION METHOD FOR ESTIMATING LOCAL SCOURING AROUND A PIER ~ BOTTOM VELOCITY COMPUTATION METHOD BASED ON DEPTH INTEGRATED MODEL

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1 PRACTICAL NUMERICAL SIMULATION METHOD FOR ESTIMATING LOCAL SCOURING AROUND A PIER ~ BOTTOM VELOCITY COMPUTATION METHOD BASED ON DEPTH INTEGRATED MODEL T. Uchda 1 and S. Fukuoka 1. Research and Development Intatve, Chuo Unversty, Kasuga, Bunkyo-ku, Tokyo, , Japan, (utda@tamacc.chuo-u.ac.p). Research and Development Intatve, Chuo Unversty, Kasuga, Bunkyo-ku, Tokyo, , Japan, (sfuku@tamacc.chuo-u.ac.p) ABSTRACT In ths paper, for estmatng local scourng around a per, the Bottom Velocty Computaton (BVC) method s developed based on the depth ntegrated model. The governng equatons of the BVC method are composed of depth-ntegrated horzontal vortcty and water surface velocty equatons n addton to shallow water equatons and a depth averaged turbulence energy transport equaton. To compute bed varaton, non-equlbrum bed load equaton s proposed. The adequacy of the model s dscussed through the comparsons wth the laboratory expermental results for flows and bed varatons around a per. The per scour s nduced manly by the horseshoe vortex. The horseshoe vortex wth radal dvergng bed-surface flow from the top of the per s produced by the extenson and rotaton of the horzontal vortex flament n approachng flow. On the other hand, bed-surface flow converges and deposts sedment downstream of the per. Ths paper shows that these mportant flow characterstcs around a per are reproduced by the BVC method. And computed scour hole around a per s smlar to that of the measurement. The above ndcates the BVC method wth the bed varaton analyss by non-equlbrum sedment transport model s useful to smulate the severe local scour n front of structures. 1. INTRODUCTION Rver engneers are faced wth urgent task of checkng the safety degree aganst floods, because the safety level of rvers n Japan s predcted to be reduced by the global warmng. A numercal model for bed varatons n rvers durng floods s requred to assess the rsk of rver structures due to local scourng. Local scour around a per has attracted a lot of nterest from engneers and researchers for a long tme. Recently, the horseshoe vortex and local scour around a per have been calculated successfully by full 3D turbulence models coupled wth sedment transport models (Olsen & Melaaen, 1993; Olsen & Kellesvg, 1998; Nagata et al., 5; Roulund et al., 5). However, full 3D turbulence models are stll lmted for applcaton to floods and bed varatons n natural rvers. So, a practcal and relable model for flows and bed varatons durng floods are hghly requred. Depth ntegrated models have been used for computatons of flood flows and bed varatons n rvers (Fukuoka, 5; Wu, 8). A D model based on the shallow water equatons does not gve suffcent accuracy n calculatng local scourng around outer bank n curved channels, contractng sectons, rver confluences, and rver structures. For such condtons, bottom velocty (velocty near bed surface) whch determnes sedment motons s dfferent from that for unform flow, as shown n Fgure 1. For ths reason, many mproved depth ntegrated models have been developed, especally for a helcal flow n a curved or meanderng channel. Those nclude full-developed secondary flow model wthout consderng effects of secondary flow on vertcal dstrbuton of stream-wse velocty (e.g. Engelund, F., 1974; Nshmoto et al., 199), refned secondary flow models wth non-equlbrum (e.g. Ikeda & Nshmura, 1986; Fnne et al., 1999) and mean flow redstrbuton effects (e.g. Blanckaert & de Vrend, 3). As more versatle method for evaluatng vertcal velocty dstrbutons, a quas-3d model, n whch equatons for vertcal velocty dstrbuton are computed, has been developed (e.g. 1

2 Depth ntegrated model 3D model Ishkawa et al., 1986; Fukuoka et al., 199; Jn & Steffler, 1993; Yeh & Kennedy, 1993). The above mentoned models are categorzed n the effcently-smplfed 3D model wth the assumpton of hydrostatc pressure dstrbuton for each obectve of the computaton, as shown n Fgure. However, t s known that these models are stll-nadequate for local scourng nduced by complex flows around a per, because they cannot consder effects of flows due to non-hydrostatc pressure dstrbuton on vertcal velocty dstrbutons. Depth averaged flow path Secondary flow Bed surface flow path Sedment path Fgure 1 Helcal flow due to the centrfugal force n curved channels Full 3D model Hydrostatc pressure dstrbuton [Abbrevated vertcal momentum equaton] 3D model wth the assumpton of hydrostatc pressure dstrbuton Functons of vertcal velocty dstrbuton Quas-3D model (hydrostatc) Quas-3D model (hydrostatc) Helcal flow model due to centrfugal force n curved flow Secondary flow model Abbrevated vertcal velocty dstrbutons D model Fgure Categorzaton of depth ntegrated models by dervatons Ghamry & Steffler () proposed quas-3d model wth non-hydrostatc pressure dstrbuton based on moment equatons. However, the model capablty for complex flow felds around structures s reman ncompletely understood. Recently, we had developed a new depth ntegrated model wthout the assumpton of the vertcal pressure dstrbuton, n whch shallow water equatons and horzontal vortcty equatons are solved to compute vertcal velocty dstrbutons. It was demonstrated that the model can smulate complex flow felds nduced by a trbutary entry (Uchda & Fukuoka, 9). In ths paper, for estmatng local scourng around a per, the Bottom Velocty Computaton (BVC) method s developed based on the depth ntegrated model (Uchda & Fukuoka, 9). To compute bed varaton, non-equlbrum bed load equaton s proposed and combned wth the BVC method. And the adequacy of the model s dscussed through the comparsons wth the laboratory expermental results for flows and bed varatons around a per.

3 . BOTTOM VELOCITY COMPUTATION METHOD.1 Basc concept Fgure 3 shows Bottom Velocty Computaton (BVC) method. We defne the bottom z b as the surface on the shn vortex layer z b on the bed, as shown n Fgure 3. The velocty on the bottom u bx s computed by the BVC method. Wth Stokes' theorem, velocty dfference u x between water surface u sx and bottom surface u bx s represented by depth-ntegrated vortcty y h and spatal dfference of depthntegrated vertcal velocty Wh. The non-dmensonal form s : z s z u sx z s U x Flow Ps y y k h u bx Pb y z b u h y x z b x Fgure 3 The concept of Bottom Velocty Computaton Method * * * h / L W h / x * * * u x yh (1) where, * y= y h/u, h * =h/h, x * =x/l, u * x=u x /U, W * =W/W =Wh /U L, h : representatve water depth (vertcal scale of flow), L : horzontal scale of flow, U : representatve horzontal velocty, W : representatve vertcal velocty. For shallow water flow, the second term s neglected. That means, for many flow felds n rver, we can evaluate the bottom velocty u bx by water surface velocty u sx and depth-ntegrated horzontal vortcty y h. The governng equatons of the BVC method are composed of depth-ntegrated horzontal vortcty and water surface velocty equatons n addton to shallow water equatons and a depth averaged turbulence energy transport equaton. The mportant advantage of the BVC method s the ablty to compute bottom velocty dstrbuton of the complex flow feld around the rver structure due to nonhydrostatc pressure dstrbuton. The local scourng around the structure s calculated by usng the bottom velocty wth contnuty equaton for sedment and non-equlbrum sedment transport equatons.. Vertcal velocty dstrbuton and governng equatons Vertcal velocty dstrbuton and bed shear stress Velocty dstrbutons above very thn vortex layer on bed are computed n the BVC method. The Polynomal equaton of vertcal velocty dstrbuton s assumed. Then, by usng depth averaged velocty U, velocty dfference u between water surface and bed surface, water surface velocty u s, and no vertcal velocty gradents on water surface, the vertcal velocty dstrbuton of a cubc curve s obtaned: u 4 3 u U u () 3

4 where, u u s U, u u s u b, =z s zh, z s : water level, h: water depth. The unform velocty dstrbuton s derved by equlbrum water surface velocty u se U u u and equlbrum velocty dfference u e. For the equlbrum condton, shear stress z s wrtten as: where, b drecton bed shear stress defned as: du vu z b v (3) dz h b UU b C c u u (4) b b where, U U U, u b u b u b. u for the equlbrum condton s gven by: h h u C UU cb ubu (5) b v v Wth the depth averaged eddy-vscosty vu * h, 6, the relatonshp between c b and C s descrbed as: c b C 1 C / (6) / C s descrbed by mannng roughness coeffcent, C gn /h 1/3. Water surface velocty The equatons for the water surface veloctes are derved by assumng very thn layer ust below water surface z s (see Fgure 3), neglectng water surface curvature: u t s u s u x s zs g Ps x (7) where, P s, shearstress actng on z s, s represented by Equaton (8). where, C ps =3. P s v 6 h t 1C ps u se u s 3 u u (8) Depth averaged horzontal velocty and water depth Equatons for water depth h and depth averaged velocty vectors U (shallow water equatons) wth occupancy rato of water descrbed as: h U h t x (9) ht U U h g hx x Uh zs sw (1) h R sw h hx where, sw =shear stress actng on sde wall, R sw =hydraulc radus of sde wall. s horzontal shear stress: S u ' u ' / 3 k (11) t 4

5 where, k=depth averaged turbulence energy, S =stran rato tensor of depth averaged velocty vectors U. The second term s produced by vertcal velocty dstrbuton, whch s defned as: u ' u k s computed by Eq.(13), whch s known as one-equaton model: 13uu uu u u 3u u ' (1) 35 k k U 1 t x h x vh k D k x R sw sw P k (13) where, vc k, C k 3/, C.9, C =1.7/h (Nadaoka & Yag, 1998). Wth usng vertcal velocty dstrbuton (), the producton term P k s derved as: Pk v t C 5h h S S SS S 8u 7u u u (14) where, S =S S, S =S S, S =S S, u =u u, u =u u, S stran rato tensor of u, S stran rato tensor of u. C h 9C 4 C 3, v te /u hv te : equlbrum depth averaged knematc eddy vscosty. The model s equvalent of zero-equaton model for the equlbrum condton of k. And n ths model, the last term of equaton (11) s elmnated, because the normal stress s assumed as hydrostatc pressure dstrbuton. Depth ntegrated horzontal vortcty equatons The velocty dfference u s computed by usng depth ntegrated vortcty for shallow water condtons, as dscussed above: u h (15) 3 where, 3 permutaton symbol ( , =), depth averaged vortcty vectors. The equaton for s descrbed by: h ER t P hd x (16) where, ER u u, D s s b b U U 6 u u ' u' u, 3u / h 5 5 t ' u' ' u', x v.3 Bottom vortcty and producton of vortcty from vortex layer The producton term P of the vortcty equaton (16) represents net vortcty flux from the thn vortex layer on bed. For flat bed condton, the flux s nduced by the turbulent mxng (P b ). In addton, the flux s suppled by the separaton downstream of the abrupt change of bed slope (P s ), as shown n Fgure 3. The producton term s defned as the sum: P P P (17) b s 5

6 P b s approxmated by the product of the knematc eddy vscosty and the vertcal gradent of vortcty near the bed: P b be b C pvtb (18) h where, C p 3, v tb : equvalent depth averaged value of bottom knematc eddy vscosty, v tb max h b, h be, v t ), b b b, be be, v t : depth averaged knematc eddy vscosty, b : bottom vortcty, be : equlbrum bottom vortcty for bottom velocty u b, be 3 c b u b h, b : bottom vortcty. The bottom vortcty s defned by Equaton (19), mnmzng the producton term between b bc and b bq : bc bc bq bq : : b bc bc be bq bq : : b b bc bq (19) where, bc be bc, bq be bq, bc u u )h, bq. The flux suppled by the separaton P s s represented by the product of the bottom layer ntegrated vortcty and the bottom layer averaged velocty: uby / ( uby) ( z y tan) ubx / ( ubx) ( zx tan) P sxx, Ps yy () ( z tan) ( zx tan) y where, x, y: computatonal grd sze of x, y drecton, z x =z xx x, z y =z yy y, z xx, z yy : second order dfferental of bed level computatonal by x, y..4 Numercal schemes The explct conservatve CIP scheme for shallow water flows (Uchda, 6) s adopted for computatons of water depth and depth averaged velocty vectors. In the model, to drectly capture the effect of dstrbuted parameters n the Cartesan coordnate system, each control volume has three knds of varables,.e., the value at the ntersecton of the grd (Pont Value), the averaged value along the sde of the grd (Lne-averaged Value) and the averaged value over the grd (Area-averaged Value). Those values are computed all together based on CIP-CSL scheme (Nakamura et al., 1). All the varables n the governng equatons are set on the same locaton. The utlzaton of multvaluables on a computatonal cell enables us to capture complex geometry even on the Cartesan coordnate system. The above scheme s not adopted for computatons of the equatons for water surface velocty, the depth averaged turbulence energy and vortcty. Those averaged value of the computatonal cell are computed by tradtonal scheme. 3. COMPUTATIONAL METHOD FOR BED VARIATION 3.1 Contnuty equaton and momentum equaton for sedment The tme varaton of bed topography s computed by the contnuty equaton for sedment: z qb ( 1 B) (1) t x where, B : sedment porosty, q B : bedload sedment transport rate. The equaton for the bedload rate s derved by momentum equaton for sedment: 6

7 q t B ub q x B Pu Du m h BP BD * B e () where, u B : sedment partcle velocty of bedload, Pu BP : gan momentum from bed materal, Du BD : loss momentum by partcle deposton, m k sgcoss+1+c M, C M : coeffcent of added mass, k : dynamc frcton coeffcent of sedment, s: specfc gravty of sedment n water, : maxmum grade of bed, h B q B u B, q B q B q B, u B u B u B,, e : component of unt vector of sedment movement and ts equlbrum value. u B, u Be h B are gven by reference to Ashda & Mchue (1974). The frst term n rght sde for momentum exchange between bed load and bed materals s descrbed as: Pu Du q u / L q u L (3) BP BD Be Be e B B / where, q Be : equlbrum sedment transport rate of bed load, L, L e : average partcle step length and that for equlbrum condton. The step length formulae are assumed by reference to Phllps & Sutherland (1989). The second term n rght sde s derved by forces actng on sedment partcles. 3. Bed tractve force and equlbrum bed load The total resstance of bed s evaluated as bed shear stress of equaton (3) n the present model. The total resstance of bed wth sand waves s dvded nto form drag actng on sand waves and surface resstance actng on sedment partcles. The latter s consdered as bed tractve force whch nduces bed load (Ashda & Mchue, 1974). Generally, the surface resstance s evaluated by: bse ( cbsub) (3) where, 1/c bs r 1lnz b k s r 8.5, k s equvalent roughness. However, the above s assumed as a unform flow condton and nadvsable n flow condtons around structures wth local scourng. In ths study, the surface resstance s evaluated by the Boussnesq approxmaton near bed to evaluate nonequlbrum flow condtons: u v tb u vtbcbsub bs cvvtb cvvte (4) z b vte z b h where, c v v tb : bottom knematc eddy vscosty, v te : equlbrum depth averaged knematc eddy vscostythe bed tractve force and the crtcal shear stress of sedment partcle on bed slope are gven by Fukuoka & Yamasaka (1983). The equlbrum bed load s gven by Ashda & Mchue (1974). And where the local bed slope exceeds the repose angle, the sand slde occurs n the model. 4. APPLICATION TO LOCAL SCOURING AROUND A PIER 4.1 Expermental and computatonal condtons The present model s appled to the experment for local scour around a per by Fukuoka et al. (1997) to valdate the model performance. Experments were conducted n the channel wth 7.5 m long, 1.5 m wdth, 1/6 ntal bed slope and unform sand materal d=.8 mm. A per wth. m dameter was nstalled at the center of the cross secton n the channel. The expermental dscharge was.66 m 3 /s, and sedment supply rate at the upstream end was 8.8 cm 3 /s. The average water depth h and Fr number of the experment were h =11.6 cm and Fr=.4, respectvely. The computatonal doman ncludes the overall channel wdth from 5m upstream to 6m downstream of the per. From about.5m upstream to about 4m downstream of the per, unform grd of dx=dy=.5m s used, as shown n Fgure 4. The expermental dscharge and sedment supply rate are gven at the upstream end of the doman and the fxed water level s gven at the downstream end. 7

8 The downstream end water level and mannng roughness n of channel bed are decded to adust longtudnal water surface profles between computaton and measurement. And the equvalent roughness for bed tractve force s gven by k s d. The ntal flow condton of the bed varaton analyss s gven by computaton results for fxed flat bed wth z= at the per center secton wth ntal bed slope 1/ Flow Water surface velocty dstrbuton 9 Mar Fgure 4 Unform grd of dx=dy=.5 m Grd around a per The computatonal doman and grd per Grd around a per..5 m/s Water surface velocty dstrbuton 9 Mar 1 Water surface velocty dstrbuton 9 Mar 1 -. Fgure mn: Flat bed.5 m/s m/s Bed varaton (cm) mn The computed bottom velocty felds on flat bed and scour bed after 3mn Fgure The measured bottom velocty felds for equlbrum stage (65 hours) (Fukuoka et al., 1997) 8

9 4. Computed results and dscusson Fgure 5 shows computed temporally averaged bottom velocty felds on ntal flat bed and computed local scour after 3 mnutes. Fgure 6 shows the measured bottom velocty feld around the per for the equlbrum condton (after 65 hours). In the expermental results, t s fned that radal bottom flows from the top of the per nduces local scour and the bottom flow runs around to the back of the per wth sedment deposton. These mportant characterstcs of the bottom velocty around the per are shown n the present model, especally for the radal bottom flow nduced by the horseshoe vortex. The radal bottom flow after 3 mnutes by the present model s developed from that on ntal flat bed due to bed scourng. Ths phenomenon s smlar as examned n prevous expermental results by Melvlle & Raudkv (1977) and computed results by the full 3D turbulence model (Nagata et al., 5). As ndcated above, the present model s reasonable to evaluate the bottom velocty felds around the per, because the horseshoe vortex s manly developed by the extenson and rotaton of the transverse vortcty n approachng flow Water surface velocty dstrbuton 8 Mar The measured bed topography The computed bed topography Fgure 7 The comparson of between measured (Fukuoka et al., 1997) and computed bed topographes around the per after 3 mnutes Fgure 7 shows the comparson of bed topographes around the per between measured (Fukuoka et al., 1997) and computed results after 3mnutes. And Fgure 8 shows the comparsons of longtudnal 9

10 and transverse bed profles along the center-lne of the per. At the back of the per, computed scour s not developed but computed sedment deposton s developed compared wth those of the measurement. Roulund et al. (5) also ndcated that the scour depth at the back of the per by the full 3D turbulence model wth the bed varaton model by the bed load sedment transport was smaller than that of the experment. To mprove ths ssue, a suspended load sedment transport model s requred, as ndcated by Roulund et al. (5). However, n the upstream part of the per, computed scour hole s smlar to that of the measurement. And the upstream and lateral bed profles of scour hole of the computaton are n good agreement wth those of the experment. As ndcated above, the present model s useful to smulate local scour n front of structures due to horseshoe vortex. (cm) Exp. -15 Cal (cm) The longtudnal bed profles 3 (m) -3-9 Fgure Exp (m) The transverse bed profles The comparson of bed between measured (Fukuoka et al., 1997) and computed bed profles Cal. 5. CONCLUSIONS Ths paper ndcates that the bottom velocty s evaluated by water surface velocty and depthntegrated horzontal vortcty for shallow water flows. Based on ths consderaton, ths study proposes a bottom velocty computaton (BVC) method, n whch depth-ntegrated horzontal vortcty and water surface velocty equatons are solved n addton to shallow water equatons and a depth averaged 1

11 turbulence energy transport equaton. To develop a bed varaton model for local scour around structures, non-equlbrum bed load equaton s derved based on the momentum equaton of the partcle n the bed load sedment. It s demonstrated that the model can reproduce bottom velocty felds around the per and the scour hole nduced by the horseshoe vortex. 6. REFERENCES Ashda, K. and Mchue, M.(197). Study on hydraulc resstance and bed-load transport rate n alluval stream. Proceedngs of the Japan Socety of Cvl Engneers, Vol.6,pp.599, n Japanese. Blanckaert, K. and de Vrend, H. J.(3). Nonlnear modelng of mean flow redstrbuton n curved open channels. Water Resources Research, Vol.39, No.1, 1375, do:1.19/3wr68. Engelund, F.(1974). Flow and bed topography n channel bends. Journal of hydraulcs dvson, Proc. of ASCE, Vol.1, HY11, pp Fnne, J., Donnell, B., Letter, J., and Bernard, R.S.(1999). Secondary flow correcton for depthaveraged flow calculatons. Journal of Engneerng Mechancs, ASCE, Vol.15, No.7, pp Fukuoka, S.(5). Flood hydraulcs and channel desgn. Morkta, Japan, n Japanese. Fukuoka, S. and Yamasaka, M.(1983). Alternatng bars n a straght channel. Proceedngs of the Japanese Conference on Hydraulcs, JSCE, pp.73-78, n Japanese. Fukuoka, S., Myagawa, T. and Tobosh, M.(1997). Measurements of flow and bed geometry around A cylndrcal per and calculaton of ts flud forces. Annual ournal of hydraulc engneerng, JSCE, pp , n Japanese. Ghamry, H. K. and Steffler, P.M.(). Two dmensonal vertcally averaged and moment equatons for rapdly vared flows. Journal of hydraulc research, Vol.4, No.5, pp Ikeda, S. and Nshmura, T. (1986). Three-dmensonal flow and bed topography n sand-slt meanderng rvers. Proceedngs of the Japan Socety of Cvl Engneers, Vol.369/II-5,pp.99-18, n Japanese. Ishkawa, T., Suzuk, K. and Tanaka, M. (1986). Effcent numercal analyss of an open channel flow wth secondary crculatons. Proceedngs of the Japan Socety of Cvl Engneers, Vol.375/II, pp , n Japanese. Jn Y.-C. and Steffler, P.M.(1993). Predctng flow n curved open channels by depth-averaged method. Journal of Hydraulc Engneerng, ASCE, Vol.119, No.1, pp Melvlle, B. W. and Raudkv, A. J. (1977) Flow characterstcs n local scour at brdge pers, Journal of Hydraulc Research, 15(4), Nagata, N., Hosoda, T., Nakato, T. and Muramoto, Y.(5). Three-dmensonal numercal model for flow and bed deformaton around rver structures. Journal of Hydraulc Engneerng, ASCE, No.131, No.1, pp Nadaoka, K. and Yag, H.(1998). Shallow-water turbulence modellng and horzontal large-eddy computaton of rver flow. Journal of Hydraulc Engneerng, Vol.14, No.5, pp Nakamura, T., Tanaka, R., Yabe, T., and Takzawa, K. (1). Exactly conservatve sem-lagrangan scheme for mult-dmensonal hyperbolc equatons wth drectonal splttng technque. Journal of Computatonal Physcs 174, pp Nshmoto, N., Shmzu, Y. and Aok, K. (199). Numercal smulaton of bed varaton consderng the curvature of stream lne n a meanderng channel. Proceedngs of the Japan Socety of Cvl Engneers, Vol.456/II-1,pp.11-, n Japanese. 11

12 Olsen, N. R. B. and Melaaen, M. C. (1993). Three-dmensonal numercal calculaton of scour around cylnders. Journal of Hydraulc Engneerng, ASCE, No.119, No.9, pp Olsen, N. R. B. and Kellesvg, H. M. (1998). Three dmensonal numercal flow modelng for estmaton of maxmum local scour depth. Journal of Hydraulc Research, IAHR, 36(4), pp Phllps, B.C. and Sutherland, A.J. (1989). Spatal lag effects n bed load sedment transport, Journal of Hydraulc Research, Vol.7, No.1, pp Roulund, A., Sumer, B.M., Fredsøe, J. and Mchelsen, J. (5). Numercal and exparmantal nvestgaton of flow and scour around a crcular ple. Journal of Flud Mechancs, 534, pp Uchda, T. (6). A CIP-based method for shallow water flows n complex geometres usng Cartesan grds. Proceedngs of 7th Internatonal Conference on Hydroscence and Engneerng, ICHE. Uchda, T. and Fukuoka, S. (9). A depth ntegrated model for 3D turbulence flows usng shallow water equatons and horzontal vortcty equatons. 33rd IAHR Congress: Water Engneerng for a Sustanable Envronment, pp Yeh, K.-C. and Kennedy, J..F. (1993). Moment model of nonunform channel-bend flow. I: fxed beds. Journal of Hydraulc Engneerng, ASCE, Vol.119, No.7, pp Wu, W. (8). Computatonal rver dynamcs, Taylor & Francs, London. 1