Computation of Flow, Turbulence and Bed Evolution with Sand Waves

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1 Compaion of Flow, Trblence and Bed Evolion wih Sand Waves Yasi SHIMIZU and Sanja GIRI Dr. Eng., Professor, Dep. of Hdralic Research, Hoaido Universi (Norh 3, Wes 8, Kia-K, Sapporo 6, JAPAN) Posdoc Research Fellow, Dep. of Hdralic Research, Hoaido Universi (Norh 3, Wes 8, Kia-K, Sapporo 6, JAPAN) Absrac A verical wo-dimensional morphodnamic model wih non-hdrosaic, free srface flow is repored herein. This nmerical model can simlae flow and rblence characerisics over sand waves. Liewise, model can reprodce he sand wave formaion and migraion process as well as free srface oscillaion simlaneosl in a moving bondar domain wih he implicaion of a generalized coordinae ssem. A nonlinear - model is emploed as a rblence closre. CIP nmerical echniqe is sed o resolve advecion erm of momenm eqaions. An Elerian sochasic formlaion of sedimen echange process in erms of pic p and deposiion fncion is incorporaed o simlae non-eqilibrim sedimen ranspor ha eplicil considers he flow variabili dring morphodnamic compaion. The nmerical model is validaed wih eperimenal daa on flow and rblence over fied dnes as well as laboraor measremen and visalizaion of dne geomer and migraion. Ke Words: Nmerical simlaion, free srface flow, sedimen ranspor, morphodnamics. Inrodcion The microscale sand waves in allvial sreams are sall formed in lower flow regime nder rogh rblen condiion. Several invesigaions have been carried o so far ha have made invalable conribions o improve ndersanding on morphodnamic feares of dnes as well as flow and rblence characerisics indced b sand waves. The comple hdralic and morphodnamic aspecs associaed wih bed form evolion have an imporan bearing on problems; however sill remain poorl ndersood despie nmeros effors ha have been made since long. Undersanding flow and rblence indced b sand waves is of imporance o eplicae heir formaion mechanism and frher developmen. Nmber of phsical sdies in his regard was made in he pas ). For insance, Nelson e al. ) condced flow and rblence measremen over wo-dimensional fied dnes, in which along wih he copling of mean flow and rblence, effecs on bed form insabili and finie amplide sabili were eamined. Schmeecle e al. 3) condced laboraor visalizaion of rblence and sspended sedimen ranspor over wo-dimensional dne. Frhermore, some noable nmerical sdies were also carried o. Shimiz e al. 4) performed a hree-dimensional direc nmerical simlaion of flow and rblence over wo-dimensional fied dnes. The nmerical model was able o reprodce he coheren srcre indced b dne cres adeqael. Mos observaions ielded srong correlaion beween coheren rblence srcre indced b flow separaion, bed resisance and sedimen ranspor mechanism. The flow field and bed configraion are fond o be inerdependen having direc and converse effec on each oher ha creaes difficlies in appropriae phsical inerpreaion and qaniaive deerminaion of he said problem. Nmeros approaches were developed in order o qanif sand wave indced morpholog of allvial channels. Mos earl invesigaions were condced sing heoreical, semi-empirical and empirical approaches ), 6),. Among hem, heoreical sdies are largel based on linear sabili heor which is, in effec, no appropriae approach for he deerminaion of naral phenomena having finie amplide

2 feares ). Scan aenion has been given in regard o nmerical modeling of sand waves morphodnamics 7), probabl becase of nmerical complei and limiaions on is widespread applicaion o real-world problem. Naaama and Shimiz 8) carried o a nmerical sd on sspended sedimen ranspor over sand waves. The compared he compaion resl of ime averaged concenraion of sspended sedimen wih Rose disribion crve as well as laboraor measremens. One of he significan sdies was made b Onda and Hosoda 9), in which a deph-inegraed flow model was proposed wih an allowance for verical acceleraion. The described boom shear sress based on poenial flow analsis considering acceleraion-deceleraion effec near he bed. In order o compe waer srface profile, a nmerical echniqe was sed o simlae ndlar bore and flow over dnes wih an aemp o eliminae he deficienc of hdrosaic assmpion b inrodcing a redcion facor o he verical acceleraion erms in case of high deph gradien. However, his flow model canno reprodce separaion behind dne cres and, hereb, is effec on flow field and sedimen ranspor. Frhermore, he sed sedimen pic p and deposiion fncion ) o simlae sand dne formaion. Finall, he analzed emporal variaion of sand wave specrm as well as dne geomer sing heir nmerical resls. Presen research has been promped b he need o develop a more complee and reliable compaional model o simlae flow and morphodnamic feares of sand waves. An earlier proposed hree-dimensional model 4) seems o be more complee approach for compaion of flow and rblence over dnes. However, copling his flow model wih a morphodnamic modle ma inhibi i from being applied efficienl becase of raher high compaional cos. Conseqenl, aforesaid hree-dimensional hdrodnamic model is simplified o be a verical wo-dimensional model and enhanced b imposing non-hdrosaic, free srface flow condiion as well as copling wih morphodnamic modle. The model is verified wih flow and rblence eperimen over fied dnes as well as laboraor measremen of sand dne geomer and visalizaion of dne migraion process.. Compaion of Flow and Trblence () Governing eqaions The governing eqaions for verical wo-dimensional flow in caresian coordinae ssem reads as: v p v ρ v v v p v ' v' ρ ( ' ' ) ( ' v' ) v' v' ( ) ( ) g () () (3) where and coordinaes in horizonal and verical direcion respecivel; and v componens of veloci in horizonal and verical direcion respecivel, '', 'v' and v'v' Renolds sress ensors, ρ flid densi, g graviaional acceleraion, p pressre. Eq. ()-(3) are ransformed from,, caresian coordinae ssem o a moving bondar fied ξ, η and τ coordinae ssem 4). Pressre erm in momenm eqaions can be comped considering non-hdrosaic componen as follows: Eq.(4) can be rewrien as: H p p p' g ρ d ' (4) p p ρ g( H ) p' () in which, p non-hdrosaic componen of pressre, H he locaion of free srface. Calclaion is performed sbsiing pressre erm (p) b Eq.() in momenm eqaions, conseqenl, comping he non-hdrosaic componen.

3 () Calclaion of waer srface flcaion Compaion of ime-dependen waer srface change is of imporance for he realisic reprodcion of free srface flow over migraing bed forms. Basicall, majori of morphodnamic nmerical models developed earlier have assmed he rigid lid waer srface condiion o achieve nmericall sable solion. In presen sd, he inemaic condiion is imposed along he free srface (a H) o compe he emporal waer srface elevaion. The inemaic condiion, which consrains flid paricles o remain on he waer srface a an ime following he local flow veloci, is epressed as follows: v H H (6) h H b (7) where b bed elevaion and h local flow deph. (3) Trblence closre In convenional - model, rblence sress ensors are evalaed sing linear relaionship. In order o reprodce rblence characerisics more accrael in shear flow wih separaion zone, a nonlinear erm is added o he sandard - model as follows ) : 3 ' ' 3 3 β βαα β β δ δ ij ij ij ij j i S S C S (8) µ C (9) where edd viscosi coefficien, δ ij Kronecer dela, rblen ineic energ, dissipaion rae of. The deail descripion and formlaion of srain ensors S ij, S βij, S βαα, as well as coefficien C β can be fond elsewhere ). - eqaions can be epressed as follows: σ σ h P v () C P C v h κ σ σ () j i j i h P ' ' () where σ, σ, C and C are sandard model consans. (4) Bondar condiions The bondar condiions are no slip a he bed and free flow a he srface. The following epression for near-bed region is adoped: ln p p κ (3) where p veloci a near-bed grid poin, bed shear veloci, κ Karman consan, p disance from bed o neares grid, s /3, s roghness heigh (.d), d sedimen diameer. The periodic bondar condiion is emploed in his compaion. Trblen energ and is dissipaion in near-bed region are evalaed as follows: C µ (4) 3 () In order o accon for he diminion of rblence lengh scale a he free srface laer, following damping fncion o he edd viscosi is sed:

4 ( h ) s f s ep B (6) 3 / s where B. The dissipaion rae of rblen energ a he srface laer is evalaed sing following eqaion: 3/ 4 3/ Cµ s s (7).4 s where s disance from waer srface o firs grid and coefficien C µ.9 (8) () Nmerical algorihm The ransformed eqaions are nmericall solved b spliing hem ino non-advecion and pre advecion phase. The non-advecion phase is comped sing cenral difference mehod. The pressre field is resolved sing SOR mehod. The advecion phase is calclaed sing a high-order Godnov scheme nown as CIP mehod ). In his scheme, a ver small ime incremen, he change in ime of veloci componens a a poin in space can be spli ino he ime evolion of he inhomogeneos erms and he ime evolion a a poin de o he advecion of he field. A brief descripion of CIP nmerical scheme o solve advecion erms in generalized coordinae ssem can be fond elsewhere 4). 3. Sedimen Transpor Model Some earlier invesigaions 3) have revealed ha spaiall or/and emporall averaged sedimen ranspor models are inadeqae o describe he eremel inconsisen nare of sedimen paricle moion pariclarl in he region wih high near-bed rblence. In effec, a direc compaion of sedimen paricle moion, viz. Discree Elemen Model (DEM), which responds o he local and emporall variable flow field, is phsicall more complee approach. However, copling his approach wih an advanced hdrodnamic model, a presen, seems o be raher sophisicaed in view of compaional complei and epenses, in pariclar, for solving comparaivel big scale problem. Compaionall efficien and relaivel comprehensive approach is believed o be he modeling of pic p and deposiion of sedimen paricles in erms of a sochasic formlaion. Conseqenl, an Elerian sochasic formlaion of sedimen ranspor proposed b Naagawa and Tsjimoo ) is incorporaed in or nmerical code. This mehod ielded one of he bes predicions of pic p fncion for fine sedimen grain. Moreover, his approach was effecivel sed earlier 9) as well for bed form developmen compaion. The dimensionless pic p rae is epressed as follows: p 3 s d ( ρ s ρ ) g.3τ (.3 / τ ) (9) Where p s sedimen pic p rae, ρ and ρ s flid and sedimen densi respecivel and τ dimensionless local bed shear sress. The sedimen deposiion rae reads as: pd ps f s (s) () where p d sedimen deposiion rae and f s (s) disribion fncion of sep lengh. Disribion fncion of sep lengh is fond o be eponenial as follows: s f s ( s) ep () Λ Λ where Λ he mean sep lengh and s he disance of sedimen moion from pic p poin. The mean sep lengh can be calclaed as Λ αd, in which α is an empirical consan and proposed o be. In order o consider he insabili of bed forms in pper flow regime, a modle o compe sspended sedimen ranspor is also incorporaed in proposed nmerical model. Considering sspended sedimen ranspor, he bed deformaion rae can be comped sing following sedimen conini eqaion: b A3 ( pd ps ) d ( qsi w f cb ) () λ A

5 where b bed elevaion,λ porosi of sedimen paricle, A and A 3 shape coefficiens of sand grain wih wo- and hree-dimensional geomerical properies, q si pward sspended sedimen fl per ni area, w f falling veloci and c b reference concenraion of sspended sedimen. In case of eqilibrim sspended sedimen ranspor or is absence, he hird erm of Eq. () vanishes. q si is calclaed sing eqaion proposed b Iara-Kishi: in which, Ω q si τ B ρ s ρ gd K α Ω w (3) ρ s f ξ ep( ξ ) dξ π B η τ ep( ξ ) dξ π a a B τ η (4) a () where B, η, α and K are consans, he vales of which are adoped, namel B.43, η., α.4 and K.8. The falling veloci is calclaed sing well-nown Rbe's formla: 36 6 w f Sgd (6) 3 d d where S relaive densi of sedimen in waer, namel.6, inemaic viscosi of waer. 4. Dne Migraion Eperimen An eperimenal invesigaion on sand dne formaion and migraion process was carried o nder he deparmen of hdralic research of Hoaido Universi. Eperimens were performed in laboraor flme of CERI of Hoaido. The oal lengh of he flme was 3m, widh was cm and heigh was 3cm having / slope, reglaed b aomaic conrol ssem. The middle par of flme (m) was covered wih cm hic sedimen laer wih.8mm median diameer, which was rapped b sing sedimen soppers a he par of psream and downsream having lengh of m each. The sand feeding was done simpl b filling p manall he space beween movable bed and sand sopper in he psream bondar, formed becase of ranspored sand. The migraion process of sand dnes was visalized sing high resolion video camera. Maimm lielihood evalaion of sand dne geomer was made a he end of each eperimen. The eperimenal cases and condiion are lised in Table. I is o be noed ha dne migraion feare (pariclarl, lengh of dne) seems o be irreglar for rn A-4 and A-.. Model Verificaion () Mean flow and rblence characerisics In order o evalae he performance of hdrodnamic model, an eperimen ) (along wih daa obained b personal correspondence wih he ahor of said wor) on flow and rblence over fied dnes is reprodced b sing proposed nmerical model. A pical eample of insananeos flow and vorici field, simlaed b nmerical model, is depiced in Fig.. From his, i can be seen ha he nmerical model is able o reprodce he realisic feare of flow separaion and vorices generaion behind he dne cres. The ime-averaged sreamwise and verical veloci profiles in all regions over dnes are reprodced qie well b nmerical model (Fig.). The rblence characerisics, pariclarl in he separaion region behind he dne cres, are fond o be nder-prediced in some poins (Fig.3). I seems ha proposed model is sill nable o resolve rblence in small scales wih compare o hree- dimensional model 4). However, on he whole, resl can be regarded as saisfacor.

6 Table Eperimenal cases and condiion Rn Time [min] Discharge [cm 3 /s 3 ] Flow deph [cm] A-. 7. A A A A () Waer srface flcaion Proposed nmerical model is fond o be able o reprodce realisicall he waer srface flcaion de o form drag indced b migraing dnes. The increase in deph afer dne formaion is fond o be in same order as observed in or eperimens. However, for rn A- and A-, average waer deph seems o be nder-prediced (Table ). The reason is no so clear. Noneheless, i is hogh o be associaed wih nder-predicion of dne geomer in nmerical compaion as well as some eperimenal discrepancies. (3) Dne migraion simlaion An aemp has been made o simlae sand wave formaion and migraion b adding an iniial perrbaion on sand bed. The compared shape and size of dnes afer analogos ime incremen for eperimen and compaion has been lised in Table. The dne geomer is fond o be poorl prediced b nmerical model, pariclarl for low flow rae, i.e. rn A- and A-. On he oher hand, he dne characerisics are no so reglar, especiall for rn A-4 and A-, which is reprodced b nmerical model in he same manner as in eperimens. Overall, comped resls show accepable predicion capabili even for he migraion speed of dnes. Frhermore, simlaed dne migraion process is compared wih he eperimenal visalizaion for he qaliaive assessmen of model performance. Some qaliaive comparison beween simlaed and visalized dne geomer is depiced in Pic.. The nmerical simlaion of dne migraion reveals idenical qaliaive characerisic as in laboraor observaion. Dos indicae paricle simlaion of flow field Fig. Nmerical simlaion of flow separaion and vorici 47.8cm/s (do-eperimen, line-nmerical) v 4.cm/s (cm) (cm) (cm) (cm) Fig. Mean flow comparison (Nelson s eperimen, 993)

7 '' 3.cm/sec (cm) (cm) v'v' 63.7cm/sec (cm) (cm) (cm) 'v'.6cm/sec (cm) Fig.3 Trblence characerisics comparison (do-measred b Nelson e al., line-comped) Table Comparison of measred qaniies wih comped Rn Flow deph [cm] Dne lengh [cm] Dne heigh [cm] Migraion speed [cm/sec] A- 7./4.9 3/ 3./../. A- 9.9/7. / 3./..8/3. A-3./3.6 4/.4/. 4.7/3. A-4./.8 irreglar./. 6.3/6. A- 6.7/6. irreglar 3.6/3. 9.4/7. Noe: vales in nmeraor and denominaor denoe measred and comped respecivel.

8 Iniial bed and waer level (Rn A-3) Rn A- Rn A-3 Rn A- 6. General Conclsion Pic. Comparison beween nmerical simlaion and eperimenal visalizaion of sand dnes Apparenl, a significan aemp has been made here o embod a reliable and phsicall based modeling approach relaing o he flow, rblence and non-eqilibrim sedimen ranspor. The proposed approach is believed o be more appropriae, for comping flow-field and non-eqilibrim sedimen ranspor mechanism wih bed forms, han generall sed morphologic modeling approach. Addiionall, modle o compe sspended sedimen fl cold also be relevan o consider he bed form insabili dring pper flow regime, hogh sill remains o be verified. In general, compaion resls seem o be realisic and promising, which demonsrae scope and sabili of nmerical simlaion echniqe ha cold be a spplemenar ool o comprehend a sophisicaed river engineering phenomenon. Acnowledgemen: We graefll acnowledge Kazae Asahi who carried o eperimens as a par of his gradae sd. Liewise, we acnowledge Jonahan Nelson for providing eperimenal daa. References ) Benne, J.S. and Bes, J.L.:Mean flow and rblence srcre of fied, wo-dimensional bed forms; implicaion for sedimen ranspor and bed form sabili, Sedimenolog, Vol.4, pp.49-3, 99. ) Nelson, J.M., McLean, S.R. and Wolfe, S.R.:Mean flow and rblence over wo-dimensional bed forms, Waer Resor. Res., Vol.9 (), pp , ) Schmeecle, M.W., Shimiz, Y., Hoshi, K. and Taea, K.:Trblen srcres and sspended sedimen over wo-dimensional dnes, Proc. In. Conf. Riv. Coas. Morph. Dn., Genova, pp.6-7, 999.

9 4) Shimiz, Y., Schmeecle, M.W. and Nelson, J.M.:Direc nmerical simlaion of rblence over wo-dimensional dnes sing CIP mehods, J. Hdrosci. Hdr. Eng., Vol. 9 (), pp.8-9,. ) Engelnd, F.:Insabili of erodible beds, J. Flid Mech.,Vol.4, pp.-44, 97. 6) Engelnd, F. and Fredsoe, J.:Sedimen ripples and dnes, Ann. Rev. Flid Mech., Vol.4, pp.3-37, 98. 7) Fredsoe, J. and Tjerr, S.:Morphological compaion of dnes, Proc. In. Conf. Riv. Coas. Morph. Dn., Obihiro, Japan, pp.-3,. 8) Naaama S., and Shimiz, Y.:Nmerical calclaion of sspended sedimen over sand waves, Proc. In. Conf. Riv. Coas. Morph. Dn., Obihiro, Japan, pp.7-4,. 9) Onda, S. and Hosoda, T.:Nmerical simlaion on developmen process of micro scale sand waves and flow resisance, J. Hdrosc. Hdr. Eng, pp.3-6,. ) Naagawa, H. and Tsjimoo, T.:Sand bed insabili de o bed-load moion, J. Hd.Div., ASCE, 6, pp.9-, 98. ) Kimra, I. and Hosoda, T.:A nonlinear -e model wih realizabili for predicion of flows arond blff bodies, In. J. Nm. Meh. Flids, Vol.4, pp , 3. ) Yabe, T., Ishiawa, T., Kadoa, Y. and Ieda, F.:A mlidimensional cbic-inerpolaed psedoparicle (CIP) mehod wiho ime spliing echniqe for hperbolic eqaions, PSJ, Vol.9, pp.3-34, 99. 3) Schmeecle, M.W. and Nelson, J. M.:Direc nmerical solion of bedload ranspor sing local, dnamic bondar condiion, Sedimenolog, Vol., pp.79-3, 3.