FRICTION SLOPE FORMULAE FOR THE TWO-DIMENSIONAL SHALLOW WATER MODEL

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1 Annual Journal of Hydraulc Engneerng, JSCE, Vol.58, 2014, February FRICTION SLOPE FORMULAE FOR THE TWO-DIMENSIONAL SHALLOW WATER MODEL Hdekazu Yoshoka 1, Koch Unam 2 and Masayuk Fuhara 3 1 Member of JSCE, M. of Agrculture, Research Fellow of JSPS, Doctoral Student, Graduate School of Agrculture, Kyoto Unversty (Ktashrakawa-Owake-cho, Sakyo-ku, Kyoto , Japan) 2 Dr. of Agrculture, Assocate Professor, Graduate School of Agrculture, Kyoto Unversty (Ktashrakawa-Owake-cho, Sakyo-ku, Kyoto , Japan) 3 Member of JSCE, Dr. of Agrculture, Professor, Graduate School of Agrculture, Kyoto Unversty (Ktashrakawa-Owake-cho, Sakyo-ku, Kyoto , Japan) Shallow water models, whch assume the ncompressblty of water and hydrostatc pressure dstrbuton, serve as effectve mathematcal tools to descrbe dynamcs of surface water flows. The 2-D shallow water equatons (2-D SWEs) are among the most effectve one. Solutons to the 2-D SWEs are senstve to the choce of the frcton slopes, the terms for turbulence and frcton forces on sde-walls. Ths paper carres out comparatve numercal analyss on two Mannng s frcton slope formulae, a conventonal water depth-based one and a hydraulc radus-based one, n order to see ther nfluences on the solutons to the 2-D SWEs. Computatonal results of the hydraulc bores reveal relatvely hgh accuracy of the hydraulc radus-based formula compared to the water depth-based counterpart. Key Words : surface water flows, shallow water equatons, frcton slope, fnte-element/volume method, hydraulc bore 1. INTRODUCTION Analyss of surface water flows s of central mportance n understandng and predctng a wde range of water engneerng ssues. Dynamcs of surface water flows s descrbed usng the shallow water equatons (SWEs) wth the assumptons of the ncompressblty of water and the hydrostatc pressure dstrbuton. The SWEs consst of non-lnear hyperbolc equatons that descrbe balance laws of mass and momentum. Although the SWEs are smple compared wth the 3-D models 1)-4), lteratures showed that they adequately descrbe varous practcally mportant flows such as rver flows, dam break flash floods, and tdal flows n estuares 5)-7). Analytcal solutons to the SWEs are avalable only for lmted cases such as the flows n a frcton-less channel 8)-11). The SWEs are therefore numercally solved n scentfc and engneerng applcatons. Solutons to the SWEs are senstve to the choce of the frcton slopes, whch have conventonally been parameterzed usng the Mannng s formula 12). A typcal 1-D model uses the hydraulc radus-based formula 13), 14). On the other hand, a typcal 2-D model uses the water depth-based formula, whch s asymptotcally equvalent to the 1-D counterpart for a very shallow water depth. From a physcal pont of vew, such a 2-D model neglects frctonal forces on sde-walls, whch cannot be ustfed for open channel flows. Some researchers proposed hydraulc radus-based, corrected Mannng s formulae. Molls et al. 15) developed a hydraulc radus-based formula for the flows n straght rectangular open channels. Brufau and Garca-Navarro 16) coupled a hydraulc radus-based formula nto a Remann solver and verfed t wth expermental dam break flows. Fe et al. 17) ndcated that the conventonal, water depth-based formula fals to handle subcrtcal flows wth re-crculaton zones, whch on the other hand are reasonably well smulated usng the hydraulc radus-based counterpart. The corrected Mannng s formulae proposed n Dewals et al. 18) and Roger et al. 19) are essentally same wth that of Brufau and Garca-Navarro 16). As ndcated n the lteratures, the choce of the frcton slopes s sgnfcant for open channel flows. Performance of the corrected formula, however, has not been well-studed for expermental

2 transcrtcal flows wth shocks. Ths paper proposes a new 2-D shallow water numercal model, an extended verson of the fnte element/volume method (FEVM) model 20). The proposed model evaluates the frcton slopes usng the hydraulc radus-based formula 16). Two severe tests cases of transent transcrtcal flows 21) are carred out to see performance of the model. The rest of ths paper s organzed as follows. A concse ntroducton for the 2-D SWEs s gven n Secton 2. The shallow water models focused on n ths study are presented n Secton 3. Numercal analyss on the models s carred out n Secton 4. Secton 5 provdes conclusons of ths study. 2. MATHEMATICAL MODEL (1) 2-D SWEs The 2-D SWEs represent conservaton laws of mass and momentum n an ncompressble, hydrostatc surface water body. The 2-D SWEs on the 3-D Cartesan x y z coordnates consst of the contnuty equaton η p q + + = 0 (1) t x y and the momentum equatons 2 p p pq η gh + Sx = 0 (2) t x h y h x and 2 q pq q η gh + S y = 0 (3) t x h y h y where t s the tme, η s the water surface elevaton, h ( = η z) s the water depth, g s the gravtatonal acceleraton, and p and q are the lne dscharges n the x and y drectons, respectvely. S x and S y are the frcton slopes n the x and y drectons, respectvely, whch are 22), 23) specfed later. The turbulent vscosty terms are not ncluded n the momentum equatons Eqs.(2) -(3) snce they smooth out the abrupt water depth changes, such as hydraulc bores and hydraulc umps, whch are mportant features of transcrtcal shallow water flows. (2) FRICTION SLOPES The frcton slopes S x and S y n Eqs.(2) and (3) are parameterzed usng the Mannng s formula n p q 4/3 ( Sx, Sy ) = + ( p, q) λ h h h (4) where n s the Mannng s coeffcent assumed as constant and the length scale λ represents the water depth h or the hydraulc radus R. (3) SOME REMARKS The hydraulc radus R s a well-defned quantty for the cross-sectonally averaged 1-D SWEs n an open channel; however, t s dffcult to fnd ts mathematcally rgorous ustfcaton for the 2-D SWEs snce Eq.(4) s n general understood n a pont-wse sense. To overcome ths dffculty, the hydraulc radus s numercally modeled as shown n the next Secton. Although the past researches dd not pont out, the hydraulc radus-based formula used can therefore be regarded as an engneerng approach at the present stage. Another mportant remark s that the solutons subect to the two frcton slope formulae, the water depth-based one and the hydraulc radus-based one, are expected to have dfferent structures near sde-walls even f same ntal and boundary condtons are specfed. Ths can be ntutvely understood snce λ may take sgnfcantly dfferent values dependng on ts choces. 3. NUMERICAL METHODS Ths study makes comparsons of the two shallow water models, both of them are based on the 2-D FEVM model 20) but employ dfferent frcton slope formulae wth each other. In the 2-D FEVM model, the contnuty equaton Eq.(1) s solved wth the standard Galerkn fnte element method scheme wth trangular elements and the momentum equatons Eqs.(2)-(3) wth an upwnd, cell-centered fnte volume method scheme, utlzng the water surface elevaton and the lne dscharges as unknowns algned n a staggered manner. The water surface elevaton s defned at each node and the lne dscharges on each element. A fully explct temporal ntegraton s acheved n the model wth the help of an effcent approxmate matrx nverson method 24). The model has already been verfed wth a seres of test problems ncludng several standard benchmark problems, such as the dam break flows 25) and oblque hydraulc ump 26), and expermental transcrtcal flows 20), 27). The frst model, referred to as FEVM-h model, uses the conventonal, wdely used water depth-based Mannng s formula ( λ = h n Eq.(4)). The model s dentcal to the 2-D FEVM model. The second model, referred to as FEVM-R model, uses the hydraulc radus-based formula of Brufau and Garca-Navarro 16) ( λ = R n Eq.(4)), whch s brefly explaned n the followng. The hydraulc radus s calculated n each trangular cell. The

3 hydraulc radus n the th trangular cell denoted by R, s gven by 1 Ω, 3 1 S, R = + µ, (5) h = 1 V wth the volume of water on the cell Ω, gven by V = (6) h A where h s the arthmetc mean of the three nodal water depths of Ω, A s the surface area of Ω, µ s the nteger defned as, ( Ω ) ( ) 1 If the th edge of s on sde-wall µ, = (7) 0 Otherwse and S, s the th surface area of the column of the water on Ω. The hydraulc radus R equals to h f all the edges of Ω the averaged water depth are not on sde-walls. The two models are therefore dentcal for the flows wth no sde-wall. 4. NUMERICAL TESTS (1) Hydraulc bore wth no obstacle The two shallow water models are frstly appled to a hydraulc bore runnng up an acryl-made rectangular flume, a transent transcrtcal flow wth a shock. A laboratory experment n a 20 m long flume havng a unform rectangular cross-secton wth the wdth of 0.60 m and the bed slope of 0.01 was carred out. Intally a unform supercrtcal flow wth the water depth of m and the dscharge of l/s was set up. Froude number of the flow was Downstream-end of the flume was suddenly closed n order to generate a hydraulc bore propagatng upstream. Front poston of the shock was estmated from a vdeo mage as 21) * * * t = x x (8) ( ) * where t s the elapsed tme snce the * downstream-end was closed and x s the front poston of the shock measured from the downstream-end. The Mannng s coeffcent n for the two models are determned so that the formula Eq.(4) s consstent wth the ntal unform flow. The values of n for the FEVM-h model and FEVM-R model are consequently determned as s/m 1/3 and s/m 1/3, respectvely. These values of n are consdered to be reasonable because the flume s acryl-made. The entre flume s dscretzed nto a computatonal mesh wth 9,000 elements and 4,961 nodes. The tme ncrement for temporal ntegraton of the models s fxed to s, whch s suffcently small to guarantee the Courant Fredrchs Lewy condton 28). Increasng the mesh resoluton does not sgnfcantly alter the results presented below. Fgs. 1 and 2 show computed water depths at the nterval of 10 s for the FEVM-h model and FEVM-R model, respectvely. Front postons of the shock based on Eq.(8) are also plotted n the fgures as the broken lnes. The bore propagates upstream gradually ncreasng ts downstream water depth. Both the models reasonably predct the shock celerty wth slght devatons, whch are consdered manly due to the hydrostatc pressure assumpton and the absence of energy dsspaton mechansms for turbulent flows at the front of the bore. These are common drawbacks of shallow water models. The FEVM-R model qute accurately captures the front poston at 30 s, 40 s and 50 s. Fg. 3 shows an enlarged vew of the flow feld around the shock at 40 s for the FEVM-R model. The results are qualtatvely more reasonable than those of the FEVM-h model n terms of that the former captures the boundary layers along the sde-walls of the flume. Such boundary layers cannot be reproduced usng a model that does not account for frcton forces on sde-walls. The FEVM-R model captures a depresson of the front of the shock, whch was observed n the experment. The results here suggest the use of the FEVM-R model for open channel flows. (2) Hydraulc bore wth an obstacle The two models are next appled to a hydraulc bore runnng up the flume wth an obstacle. Ths problem s also a transent transcrtcal flow wth shocks, but has more complex flow structures than the prevous one. A brcks-made obstacle wth the wdth of m and the length of m was placed n the flume. The downstream sde of the obstacle was at m from the downstream-end of the flume. One of the long sdes of the obstacle was set along the centerlne of the flume, creatng a m-wdth reach on the rght sde and a m-wdth reach on the left sde. A steady flow wth the same nflow dscharge of l/s was frstly set up. After that, the downstream end of the flume was suddenly closed so that a hydraulc bore propagatng upstream s created. Based on a vdeo mage, the arrval tme of the shock at the downstream-end of the obstacle was estmated as 35 s. In the experment, water surface upstream of the obstacle developed an undular hydraulc ump due to the reflecton effect by the channel separaton. A very shallow water regon was created ust the downstream sde of the obstacle followed up by oblque shocks. The Mannng s coeffcent n for the doman s agan set as s/m 1/3 for the

4 Fgure 1 Computed water depths of the hydraulc bore wth no obstacle usng the FEVM-h model. Fgure 2 Computed water depths of the hydraulc bore wth no obstacle usng the FEVM-R model. FEVM-h model and s/m1/3 for the FEVM-R model. The entre flume s dscretzed nto a computatonal mesh wth 9,353 elements and 5,111 nodes. Steady flow wth the constant dscharge of l/s s taken as the ntal condton for each model. The tme ncrement s set as s. Fgs. 4 and 5 show computed water depths at the nterval of 10 s for the FEVM-h model and the FEVM-R model, respectvely. Also plotted n the fgures s the observed front poston of the hydraulc ump upstream of the obstacle, whch had an nterval due to ts undular nature. Ths undular nature of the hydraulc ump cannot be fully resolved wth the 2-D SWEs because t s an essentally non-hydrostatc phenomenon. Both the models reasonably well reproduce the expermentally observed flows: a hydraulc ump at the upstream of the obstacle and a very shallow water regon at mmedately downstream of the obstacle, followed up by oblque shocks (Fg. 6 for the FEVM-R model). Fg. 6 clearly shows that the front shape of the bore s gradually nclned n the transverse drecton toward the narrower reach as t approaches the obstacle, beng consstent wth the observaton. Flows for the FEVM-h model presented a smlar behavour. The observed and computed front postons of the hydraulc umps agree well (Fgs. 4 and 5). The arrval tme of the shock at the downstream sde of the obstacle s 36.3 s and 36.4 s for the FEVM-R model and FEVM-h model, respectvely; both of them are slghtly dfferent from the observed value 35 s manly due to Fgure 3 An enlarged vew of the shock front at 40 s for the FEVM-R model. the hydrostatc pressure assumpton n the SWEs. Fg. 7 shows the dfference of the water depths d h between the two models, namely, d h = hfevm-r - hfevm-h. (9) The purpose of Fg.7 s to more clearly show the dfferences between the computatonal results of the two models, whch are ndcated n Fgs. 4 and 5. Fg. 7 shows that the FEVM-R model more accurately captures the front poston of the hydraulc ump than the FEVM-h model as drawn wth the red and blue colors upstream of the obstacles n the fgure. Fg. 7 also shows that the shock celerty of the FEVM-R model s slghtly faster than that of the FEVM-h model. For both of the models, the hydraulc bore propagates upstream through the sde reaches of the obstacle and approaches the hydraulc ump, further runnng up the flume after absorbng the ump. 5. CONCLUSIONS I_58 Valdty of the two shallow water models was

5 Fgure 4 Computed water depths of the hydraulc bore wth an obstacle usng the FEVM-h model. Fgure 5 Computed water depths of the hydraulc bore wth an obstacle usng the FEVM-R model. Fgure 6 Water depth dstrbutons around the obstacle for the FEVM-R model. Fgure 7 Dfferences of the computed water depths of the FEVM-h model and the FEVM-R model. examned through numercal smulatons of the hydraulc bores. The FEVM-R model computed more reasonable numercal solutons than the FEVM-h model n the presented cases, reproducng the boundary layers along the sde-walls and the depresson of the front of the hydraulc bore, the latter beng observed n the experment n partcular. Furthermore, the FEVM-R model more accurately I_59

6 captured the front poston of the shocks. The obtaned computatonal results mpled that an approprate modellng of the frctonal slopes s one of the keys to carry out a successful smulaton of shallow water flows. Ths paper showed satsfactory performance of the FEVM-R model for the severe transcrtcal open channel flows. Nevertheless, there stll reman mportant research topcs to further valdate the model. Frstly, applcaton of the model to other benchmark problems and real problems, such as open channel flows wth rregular bed topography and/or wet and dry nterfaces, wll be necessary to see ts performance more n detal. Mathematcal ustfcaton of the treatment of the frcton slopes n the FEVM-R model s another mportant research topc. Investgatons on these research topcs are now undergong and wll be presented elsewhere. ACKNOWLEDGMENT: Ths study s fnancally supported by the JSPS under grant REFERENCES 1) Park, I. R., Km, K. S., Km, J. and Van, S. 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