- A D-A 86 556Technical Report REMR-HV*-12AD-A286 556 12= .•R.-..,

November 1994m

US Army Corpsof EnginwersWaterways ExperimentStation

Repair, Evaluation, MaiUnteance, and Rehabdtaiaon Research Program

HIVEL2D: A Two-Dimensional FlowModel for High-Velocity Channels

by R. L. Stockslt, R. C. Berger

AjPPuvgd For Publi Rgeaess O.vlbulisn ULmasd

94 1128 020

94-36238

Prepaed for Headquarters. U.S. Army Corp of Engineers

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IApai, Evalumlon, MuI I KN nd Technical Report REMR-HY-12.,,ab.mt 48n 0wch Pro November 1994

HIVEL2D: A Two-Dimensional FlowModel for High-Velocity Channelsby R. L Sidul, R. C.B Bgu

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Contents

Prefalcc . . . . . . - - . . . . .. iv

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Figue 1-26

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SF 298

Preface

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Propam. The work wo pwfomed swe Cm) Works Ramwch Work Um.32657. *ModI fow Evshmm and Maaummn of Hqib-Vdockly Cbmw.Th. RENA TedmWca Monger wV Mr, Dan Wwmd (CBCW-EH).

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1 Introduction

Background

Tb. )iydruaiic purwuincs of a b -iv*uM chy e dqm an mammin# a mipicraca Flow rums owi specdW poIds tm af leo.Pfedcuaw the pou.i~al k~casm of sheck. macb asobloit szinii% waves andhydnWic Opas, and damwuvmq lb. sWmuvvo at lbs wo wma~ anchaemis bandt arv ecameuy to evabius wAu lbsm tuia nr walheights Typicafy alymncal eqpmbon or pbyuaca bydimAh mo"i bmvbms uWa to amisk dbm evJalowes

Phyica modii. w wer md to lbs mu~eem wdv of wmy exodq food-coauol channesa. baa wbommima 1~ their diaop bmaw ha nsWmm indi~bsmhre pwem dm thorn fr b" lbs Im Iwas oftmaly disiad-Ohemaci. mcb as dotri W ot bridg Onw. my caimed flow to pay to aiubcnucal saw. dmi r~uaw i foloodmp . 1bdueas man m sm i sand a readily avil"N m fm rmhmtftM ibm d amms ism i A

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A Gmemalal flow model. HIVEL2D. ho boo do epdo as a wdtoolsevaas. b ~vefociry chmubs whic am uinwea x -PI-m tmd dhbmwith hydm4ratiky aMW slWOpe The soma deip 0( hbj-vdockychammels 4,;pinmb on acci pi A -* -- do flow ~de . HJVEL2D is a

dsps-av rq.rwdim iuu (2-0) flow maMi &Wiu qawdk&cRy for

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-.po and Scope

The purpose of "~ rqo"ist to dm,* di inral flow Ndi".H1VELD. and to WlhMMu qV*Wa h-sedocky flo fisd dew di madi IS

A 1 e of simlaniag. Thus mW vwglflaio coisM ofe coupaa so mob- - KJVELZD wMh bi*WU7 ibm- Model Hmksm am also

d~cue.ft is kvxi d HJVELID rn's bie~ ch flow

struaions are suited to suzmdaaion using HIVEL2D and which are not.

A user's manual will be the second report published for this work unit.The user's manual will discuss such topics as gridl gnemraon (FastTABS),hardware requirennms (personal comuers (PC's) and/or workstations), timeinvolved in grid geneation and sinualuion rum, and guidance relauive to gridlayout and parameter choice

2 Ch" I Ivla•'tW

2 Description of Model

The mmde as doesid to siimulw flo typ" aAi h~ig-veloco clamml.mwe mode aS a fimmt Almmi duaipum o da~ 2-D atasliw-wanapioin conmetvwe k(mu Specifialy, the mo a 4ampe to provile te water5ufface in amul arund~ bmsinwy awamm. beidVpwna. cmaflmmi. ba3&.and Wmh ponewc khur corn am highbwlocit cbw. HIVEU2D msapplcable to saab- and sqitutiamb flow ad dw uiniim but - a damertgunw

The sysem ot nodmlir etpamm an whvnd %mm dw Newwo-Raphomitetatve method. The NewwmRapbsnometod wu s klu Orthis modbecame the ooahmw mmn in higb-velocwy cbind flow Field wte pSagaictam,

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Assumptios

HIVEL2D soltre the 4p&isavenqmd meamdy shello-watw squab~maompicitly using finift Woemsan The gova *qumm are pminemu kodetal in Appendix A.

An~ uuiot~um (wWsodtinladd) umemsltm mo m th &rivwab of tdw gov-etnino equaition as th veca" accx atitm mv amgloqthe *bnt a i~sire tothe hontzumW accelonemaar wtd acsktalito due to petty This moane Owithe presme disnbatimr is by*awwlc The bydnawk prminuie upmynlmIis gemaity applcah to tihqwelocImy chinwwis Howwm.r cam mm betaken in apply-O the in rim! to flow ssuww *bur. th wavelemo us nolowq remime to thw deth AlD bydicawi uuxkb ovu'hmf dle WOW. ofshe"t wavvelenth.

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where

p-flui density

z - venical Caresian coordintme

w - the z-direction componen of velocity

S - accekerdon due to grviyM

Newr the crest of a wave, or amy wbete the the wata surfae is convex, thefthu elemnts ridig am the unsu ce has a negative vertical accelenmin. i.e..dwv/t* < 0. Therefoe, fte vertical acceleration acts opposite to gravity andSpressure as reduaced. Convrseuely. am the tuoug whare the wonr surfacie isconcave, the near-Pstrace flui eleen bas a positive aclermion. aid so acsto erhance the presswre The no effect as that the surface ouvatur redcesthe pressisre gradiess aimd the wave wil travel ,mere slowty . The shorte thewavelength. the slower it will tend to propagae. For long wavelengths this

imatis aegli*e

Another coommpamace o( the hyd&rostatic aimmyaon as do the shallow-wwate 4qu2um can Provide 00 lsWONINdnsos w th length Of a kUP- i e.,duustance twi e leading to the trailing alga ot a hydraulic sm. TheAhallow-wate **wboa comai no verial velocity or accelvaidon therforeany reneily that shoul be curatoed in vetical motion is lost. The shalow-wane emqtmu na sualmal the OWu~ as a discoutimairy and all verticaletergy is uiuudwsely dissipume This is idike the real system int which anuunhala puamp my Aormtm dissipose tho vertical mobitn over a longdwsance,

Another sagaificam amou is~ . di the bad slope is guamwicaliy wildw tdiusin 0 - tan - 0. wbo is thu cb uldope. TbWgeoumrkaltyamild slope uawmpn is dkitaguWWs ftwa a bydraulically mild slope thaconveyr flow n s bcniics (i. ea. dw Ablow wmom e~moa ams~ tha theslope a geonmerrcly VANl. althOg it mny he byd ~aslcly IMw). Thsis kaareagombew Mesuophlo = mos highvekociy cbm applicaioms wbese theslope is less Owa 002. Howeve. whes applcaslin is We toa long chaneth bsvft fvoabrsle skip inam of my0O-5. diemadddope

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Discretization

Discrete values of the unknown variables are solved using a Petrov-Galerkin finite element formulation of the governing equations. Details of thefinite element formulation are presented in Appendix B. The model is quitestable as a result of the particular Petrov-Galerkin test function employed,which is weighted upstream along characteristic lines. The finite elementmodel reproduces triangular and quadrilateral bilinear elements and can beeasily modified to capture complex geometries.

HIVEL2D solves the finite element equations using the weak formulation.The weak formulation facilitates the specification of natural boundary condi-tions and allows accurate shock capturing, which is essential in modelingsupercritical flow.

Temporal derivatives are solved implicitly using a finite difference discreti-zation. The temporal derivatives can be approximated using either a first-order(ct = 1.0) or second-order (at = 1.5) backward difference. A first-order differ-ence is used for the spin-up to a steady flow solution, whereas a second-orderdifference is more appropriate for unsteady flow simulation.

Boundary Conditions

Inflow and outflow boundary conditions can be specified as supercritical orsubcritical. Supercritical inflow boundaries require the specification of thethree degrees of freedom (i.e., the x- and y-components of unit discharge andthe depth or the x- and y-components of velocity and the depth). Subcriticalinflow boundaries, juire that only two specific degrees of freedom be given(i.e., the x- and y-components of unit discharge or the x- and y-components ofvelocity). Bot-idary conditions at supercritical outflow boundaries should notbe specified. Subcritical outflow boundaries require the specification oftailwater elev-4tion (i.e., depth plus bed elevation).

Sidewall boundary conditions are enforced using the weak statement. Theboundary conditions enforced are that the mass and momentum flux throughthe sidewalls are zero. Sidewalls also enforce a partial slip condition.

5O*Otw 2 Doe$ of M P

3 Test Cases

Numerous test cases are presented to illustrate the model's ability to simu-late high-velocity channel flow fields. More importanly. results of test casespoint out flow conditions that are not accurately modeled by HIVEL2D. Themodel user must understand when the model's governing equations are ade-quate for the particular problem to be solved and when they are not. It isdemontratd that the numerical scheme accurately solves the depth-averaged2-D equations. The depth-averaged equations appropriately describe mosthigh-velocity channel flow fields.

Several features commonly found in higb-velocity chuimels are included inthe test cases: simulations of chamnel contractioas. expansiom, confluences,bends, hydraulic jumps, bridge piers, bores, and roll waves.

Input parameters for each simulation are provided in tabular form: thetemporal difference coefficient a; the Prrovm-Galerkin weight coefficient 0(for smooth flow) and Os (for flow near sbocks); Maaming's coeffici st n; theeddy viscosity P (for smxxth flow) and P. (for flow near shocks); and theacceleration due to gravity g. Detailed descriptions of the tempora differenceand Petrov-Gakzn weight coeffients an provided in Appendix B. Earlytests were conducted using a version of (IVEL2D that used a woummK user-specified eddy viscosity for the entire flow field. The current version ofHIVEL2D varies the viscosity based on local flow variables. Details of howviscosities are determined an presented in Appendix A.

Channal Contraction

Initially, HIVELZD was run to sinualate supercritical flow in a channelcontraction. A laboraiory investigation of spercritical flow in chammiecontractions was reported by Ippen and Dawson (195l) in which depth con-tours were presented. One psnhular test documented flow conditions in astraight-wall contraction with an approach Froude number of 4.0. Thestraight-wall contraction was a 2-ft- (0.61-m-) wide cham4el, trntsitioning to

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6 pChsprw 3 Tos Cases

a l-ft- (0.30-m-) wide channel at a convergence angle of 6 deg. The gridused for the simulation is provided in Figure 1. The grid consisted of 1661nodes and 1500 elements. The long apprsach length of 20 fIt was used toensure uniform flow conditions at the contraction. Figure 2 shows contoursof flow depth observed in the laboratory and those computed using HIVEL2Dfor a discharge of 1.44 cfs (0.041 cim) and an upstream Froude number of4.0. Input parameters used in the numerical simulation are provided inTable 1. A water-surface mesh generated using these HIVEL2D results isshown in Figure 3.

The contraction is a geometrically simple case, but the results demonstratethe model's ability to capture the oblique standing waves resulting fromchanges in the wall boundaries. Oblique standing waves are created as thesupercritical velocity encounters the converging sidewalls. These standingwaves are reflected from opposing sidewalls for a distance downstream.HIVEL2D captures the standing waves and the depth increase as the wavesintersect at the channel center line.

Table 1

Input Parameter, for the Chanunel Co ation

0 1.0

.0.o 0.1.0.5t 0.011

0.005.0.006 ft'lsec (4.65 x 10' m4slac}

g 32.206 ?ii..ce (9.817 m/iwcIl

Hydraulic Jump

Tests were conducted to compare the results from HIVEL2D with some ofthose olxained in the physical model study of Rio Puerto Nuevo Flood-ControlChannel (Stockstill and Leech 1990) conducted for the U.S. Army EngineerDistrict. Jacksonville. Three models were constructed and tested for thePuerto Nuevo study. These models were selected for comparisons withHIVEL2D calculations simply because the channels' geometric and hydraulicdetails were at hand and several features common to high-velocity channelswere included. Specifically. the Puerto Nuevo study channels containedsupercritical confluences, horizontal wives, expansions, and a tailwater-imposed hydraulic jump. To avoid similitude questions, HIVEL2D was run atlaboratory scale. 7 his resulted in direct comparisom of calculations toobserved quantities.

The Margarita Channel was one of the three models tested for the project.The Margarita Channel consisted of two reverse curves sepamed by a short

7Chamw 3 Test Caso.

tangent, a channel expansion, and tailwater that produced a hydraulic jumpupstream of the width transition. Details of the modeled reach of theMargarita Channel are presented in Figure 4. Input parameters used inmodeling the Margarita Channel are provided in Table 2, and the finite ele-ment grid is shown in Figure 5. Typically when steady-state solutions aresought, a first-order temporal derivative is appropriate and an a = 1.0 isused. Because portions of the Margarita Channel exhibited unsteady flowfeatures, an a = 1.5 was used resulting in a second-order temporal derivative.The finite element grid consisted of 796 nodes and 651 elements. Figure 6compares water-surface profiles along the channel walls for a discharge of1.2 cfs (0.034 cim) with a tailwater depth at the downstream end of the modelof 0.601 ft (0.183 m).

Tabe 2

Input Parameters for the Margarita Channel

conwhim VaIte

a

'.0 0.2.0.5

n 0.010

VrV, 0.008,0.025 felsIc 17.43 x 10', 2.32 x 10' ml/sec)

g 32.209 Wties' 198.17 misac2 )

HIVEL2D not only captured the hydraulic jump in the channel, but alsoreproduced the asymmetric flow patterns downstream of the width transitionthat were observed in the physical model. The short length of the transition(I transverse on 4 longitudinal) in conjunction with the asymmetric flowdistribution produced by the upstream bend resulted in a large eddy in thedownstream channel that produced flow concentrations along the left wall.Depth-averaged velocities at three stations are presented in Figures 7-9.Higher velocities (1.9 fps (0.58 raps)) existed along the left wall whereas theflow along the right wall was essentially stagnant as shown by the dark area inFigure 10. The numerical model results shown in Figure 11 show thatHIVEL2D accurately predicted the asymmetric flow patterns downstream ofthe expansion.

Supercritical Confluence

The numerical model was further tested by comparing computed results ofthe Puerto Nuevo/Guaracanal Channel confluence with those observed in thelaboratory study. Geometric details of the supercritical confluence are pro-vided in Figure 12. Table 3 lists the input parameters, and Figure 13 showsthe finite element grid, which is composed of 491 nodes and 390 elements.Boundary conditions for these tests were supercritical inflow with 3.6 cfs(0.10 cms) in the main channel and 1. 1 cfs (0.03 cms) in the tributary channel

8 Chapter 3 Test Cases

Table 3

Input Parameters for the Suprectical Confluence

a 1.0

_ _ _ _ _ _0.1,0.5

n 0.009

v, v, 0.005.0.001 ftlsoc 14.65 x 101 ml1.c)

g 32.208 ftNse (9.817 m/e"c)

and supercritical outflow. Figures 14 and 15 show that the HIVEL2Dcaptured the overall features of the diamond-shaped standing wave patternresulting from the confluence geometry. The water surface contours shown inFigure 15 illustrate the model's ability to simulate superelevation in the watersurface within the tributary channel bend. Water-surface profiles along theflume walls obtained using H1VEL2D and-those observed in the laboratory arepresented in Figure 16. In most supercritical channels the water surface oscil-lates in time even under steady boundary conditions. The recorded laboratoryresults are the maximum water-surface elevations observed, whereas thenumerical model represents average water-surface elevations. HIVEL2D ade-quately simulated the initial shock wave crest, but the location of each subse-quent wave crest was increasingly in error. This difference is due to the shal-low-water assumption used in the 2-D numerical model. The shallow-waterassumption results in all waves traveling with the celerity of a long wave,whereas three-dimensional flow is actually composed of many wave speeds,the maximum of which is the long-wave celerity. The larger wave speedmeans that the standing wave angles will be greater than the three-dimensionalwaves. Generally, this shallow-water equation limitation should be of littleconsequence since channel wall heights are set to contain the maximum water-surface elevation plus freeboard.

Bridge Pier

A hypothetical bridge pier was tested to demonstrate HIVEL2D's ability tosimulate a Class B bridge in which the piers choke the flow such that theapproaching flow is subcritical even though the channel slope is hydraulicallysteep. The channel was 100 ft (30.5 m) wide by 1,000 (305 m) ft long. Twopiers were used to choke the flow. Each pier was 20 ft (6.1 m) wide by100 ft (30.5 m) long with a square nose and tail. The finite element grid forthe bridge pier problem was composed of 993 nodes and 900 elements(Figure 17).

Water-surface meshes calculated using HIVEL2D are shown in Figures 18and 19. Table 4 lists input parameters for this simulation. The flowapproaching the piers was subcritical and accelerated around the piers tosupercritical. The imposed tailwater resulted in a hydraulic jump immediatelydownstream of the piers. The hypothetical problem demonstrates HIVEL2D's

Chapter 3 Test Cases 9

Table 4

Input Parameters for the Bridge Pler Problem

Camndki Vdm

a 1.0

[.s, 0.2.0.5

n 0.020

v, v, 10.0.10.0 ftlsoc (0.93 m'/slc)

g 32.208 ftise

ability to model subcritical flow, the transition to supercritical flow, and alsothe transition to subcritical flow via the hydraulic jump.

Dam Break Problem

Unsteady flow simulations were conducted to test HIVEL2D's discretiza-tion to the temporal derivatives of the governing equations. Results werecompared to hydraulic flume results reported in Bell, Elliot, and Cbaudhry(1992). A plan view of the flume facility is shown in Figure 20. The flume,constructed of Plexiglas, simulated a dam break through a horseshoe bend.Because this was a 2-D problem and model results were being compared tohydraulic flume results, the limitations of the shallow-water equations them-selves needed to be considered. Initially, the reservoir had an elevation of0.1898 m relative to the channel bed; the channel itself was at a depth (andelevation) of 0.0762 m. The velocity was zero and then the dam wasremoved. The surge location and height were recorded at several stations andcompared to the model at three of these, at stations 4, 6, and 8. Station 4 was6.00 m from the dam along the channel center line in the center of the bend,Station 6 was 7.62 m from the dam near the conclusion of the bend, andStation 8 was 9.97 m from the dam in a straight reach. The model-specifiedparameters are shown in Table 5.

Table 5

Test Conditions For The Dam Break Case

Condllk Vslue

aa

AD, 0.25, 0.50

n 0.009

v. v, 0.001, 0.01 m'lsec

g 9.802 m/sec'

At 0.05 sec

10 Chapter 3 Test Cases

The numterical grid (Figure 2 1) contains 698 elements and 811 nodes. Thisgrid was reached by increasing the resolution until the results no longerchanged. The most critical reach was in the region of the contraction near thedam breach. The basic element length in the channel was 0. 1 m with fiveelements across the channel width. For the smooth channel case, Bell, Elliot,and Chaudhry (1992) used a one-dimenional calculation to estimate theManning's n to be 0.016, but experience at the U.S. Army Engineer Water-ways Experiment Station suggests that this value should actually be 0.009(Brater and King 1976).

The test results for stations 4, 6, and 8 are shown in Figures 22-24. Herethe time-history of the water elevation is shown for the inside and outside ofthe channel for both the numerical model (at a of 1.0 and 1.5) and the flume.The inside wall is designated by squares and the outside by diamonds. Ofparticular importance is the arrival time of the shock front. At station 4 thenumerical prediction of arrival time using a of 1.0 was about 3.4 sec, whichappears to be about 0.05 sec sooner than for the flume. This is roughly 1-2 percent fast. For a of 1.5 the time of arrival was 3.55 sac, which is about0. 1 sec late (3 percent). At station 6 both flume and numerical model arrivaltimes for a of 1.0 were about 4.3 sec, and for station 8 the numerical modelwas 5.6 sec and the flume was 5.65 to 5.8 sec. With a set at 1.5, the time ofarrival was late by about 0.2 and 0.15 sec at stations 6 and 8, respectively.The flume at stations 6 and 8 had a earlier arrival time for the outer wavecompared to the inner wave. The numerical model did not show this. Incomparing the water elevations between the flume and the numerical model, itis apparent that the flume results show a more rapid rise. The numericalmodel is smeared somewhat in time, likely as a result of the first-ordertemporal derivative calculation of a of 1.0. The numerical model with a setat 1.5 shows an overshoot. This is likely a numerical artifact and not basedupon physics even though this looks much like the flume results. The surgeelevations predicted by the numerical model are fairly close if one notices thatthe initial elevation of the flume data was supposed to be 0.0762 m and itappears to be recorded as much as 0.015 m higher at some gauges. Since thevelocity was initially zero, then all of these readings should have been0.0762 m and all should be adjusted to match this initial elevation.

With this in mind, stations 4 and 8 match fairly closely between flume andnumerical model. Station 4 in the flume would still have a greater differencebetween outer and inner wave than that predicted by the model. The differ-ence might be a manifestation of a three-dimensional effect that the modelcannot mimic. The overall timing and height comparisons are good.

Figure 25 shows the spatial profile of the outer wall water-surfaceelevation of the numerical model versus distance downstream from center lineof the dam. The two conditions are for a of 1.0 and 1.5, i.e., first- andsecond-order temporal derivative.

The nodes are delineated by the symbols along the lines. The overshoot ofthe second-order scheme and the damping of the first-order are obvious.

11Chaeer 3 Test Cases

Again, it is probable that the overshoot is a numerical artifact even though this

is much like what the flume would show.

Roll Waves

Roll waves are a phenomenon of a high Froude number environment. Ifthe Froude number is greater than 2, it can be shown that disturbaes cangrow into shocks, which progress downstream faster than the flow. Thisresults in a characteristic sawtooth-shaped water-surface pattern in which thesteep face is downstream. The roll wave tests used a modified version ofHIVEL2D in which a sinusoidal perturbation was input at the upstreamboundary of a long straight channel. The water depth results, shown inFigure 26, demonstrate the gradual steepening of the downstream wave facedownstream from inflow point. While this is a qualitative comparison, themodel does demonstrate the capacity to reproduce this characteristic event.

12 Chaptw 3 Teat Cues

4 Discussion andConclusions

this series of tests demonstrates the ability of the model HIVEL2D tosupply engineering decision makers with a tool to evaluate hydraulic results ofstructural modifications in supercritical channels.

The code itself is relatively flexible with the one major limitation that theboundary conditions are constant over a simulation. The most significantlimitations are therefore imposed by the equations modeled. Tnese are theshallow-water equations employing the mild geometric slope assumption andhydrostatic pressure distribution. The obvious result is that one would notwant to use the model to evaluate features with steep slopes. However, thehydrostatic assumption is more subtle. T1h result of this assumption is thatshorter wavelengths will tend to propagate too quickly in the model. This wasparticularly noted in the supercritical confluence tests. While the water-surfacepredictions were good over the first reflection or so, they became progressivelyworse downstream. Another consequence of the hydrostatic assumption is thatenergy is dissipated too quickly. The result is that energy that should be cap-tured in vertical motion is neglected and undular jumps are instead modeled asstrong jumps. Shallow-water models in which vertical motions are assumednegligible cannot predict the length of a hydraulic jump.

MpW 4 oemnusiand s WWWOn 13

References

Abbott, M. B. (1979). Conputioual hydraulics, elenent of dwe theory of freesurface flows. Pitman Advanced Publishing Limited, London.

Bell, S. W., Elliot, R. C., and Chaudhry, M. H. (1992). "Experimntal resultsof two-dimensional dam-break flows, Journal of Hydraulic Research30(2), 225-251.

Berger, R. C. (1992). "Free-surface flow over curved surfaces," Ph.D. di..,University of Texas at Austin.

Berger, R. C. (1993). "A finite element scheme for shock capturing,*Technical Report HL-93-12, U.S. Army Engineer Waterways ExperimentStation, Vicksburg, MS.

Berger, R. C., and Winant, E. H. (1991). "One dimensional finite elementmodel for spillway flow.' Hydraulic engineering, Proceedings, 1991National Conference, ASCE, Nashville, Tennessee, July 29-August 2, 1991.Richard M. Shane, ed., New York, 388-393.

Brater, E. F., and King. H. W. (1976). Handbook of hydraulics. McGraw-Hill, New York, 7-22.

Carnahan, B., Luther, H. A., and Wilkes, J. 0. (1969). Applied nunericalmedtods. Wiley, 319.

Chapman, R. S. and Kno, C. Y. (1985). 'Application of the two-equation k-gturbulence model to a two-dimensional, steady, free surface flow problemwith separation," Iernational Journal for Numerical Medsods in Fluids 5,257-268.

Courant, R., Isaacson, E., and Rees, M. (1952). 'On the solution of nonlinearhyperbolic differential equations," Czmmmicatdm an Pure and AppliedMathema$ics 5, 243-255.

14 F awme

Dendy, J. E. (1974). *Two methods of Galerkin-type achieving optimum L2

rates of convergence for first-order hyperbolics," SIAM Journal ofNumerical Analysis 11, 637-653.

Drolet, J., and Gray, W. G. (1988). 'On the well posedness of some waveformulations of the shallow water equations,' Advances in Water Resources11, 84-91.

Gabutti, B. (1983). "On two upwind finite difference schemes for hyperbolicequations in non-conservative form,* Computers and Fluids 11(3), 207-230.

Hicks, F. E., and Steffrer, P. M. (1992). *Characteristic dissipative Galerkinscheme for open-channel flow," Journal of Hydraulic Engineering, ASCE,118(2), 337-352.

Hughes, T. J. RL, and Brooks, A. N. (1982). "A theoretical framework forPetrov-Galerkin methods with discontinuous weighting functions: Applica-tions to the streamline-upwind procedure&" Finite elements in fluids. R. H.Gallagher, et al., ed., Wiley, London, 4, 47-65.

Ippen, A. T., and Dawson, J. H. (1951). *Design of channel contractions."High-velocity flow in open channels. A symposium Transactions of theAmerican Society of Civil Engineers 116, 326-346.

Katopodes, N. D. (1986). *Explicit computation of discontinuous channelflow,' Journal of Hydraulic Engineering, ASCE, 112(6), 456-475.

Praagman, N. (1979). Numerical solution of the shallow water equations by a

finite element method. EGNER, Den Helder, The Netherlands.

Moretti, G. (1979). 'ne ?,-scheme," Computers in Fluids 7(3), 191-205.

Rodi, W. (1980). 'Turbulence models and their application in hydraulics - astate of the art review," State-of-the Art Paper, International Association forHydraulic Research, Delft, The Netherlands.

Steger, J. L, and Warming, R. F. (1981). *Flux vector splitting of theinviscid gas dynamics equations with applications to finite differencemethods,' Journal of Computational Physics 40, 263-293.

Stockstill, R. L, and Leech, J. R. (1990). "Rio Puerto Nuevo Flood-ControlProject, San Juan, Puerto Rico; Hydraulic model investigation,* TechnicalReport HL-90-18, U.S. Army Engineer Waterways Experiment Station,Vicksburg, MS.

Verboom, G. K., Stelling. G. S., and Officier, M. J. (1982). 'Boundaryconditions for the shallow water equations," Engineering applications ofcomputational hydraulics. M. B. Abbott and J. A. Cunge, eds., vol. 1,Pitman Publishing, Marshfield, MA, 230-262.

Ak @@ 15

Figural. Numnerical moeamtiloe mesh for the chwinel contractionproblemn

61 1.0

Z 0.0......

z 042-02

g0.26.210.

WI0.5 0 0.5 1.0 1.5 2.0 2.5 130 .365 4.0 4.5 5.0 5.5 6.0 0.5 7-0

DISTAN4CM "WM COMMACMK4 FT

a. Laboratory flumne (modifie fom Ippen and Dason 1951)

0.4

10

0.: L1

0.4 0

S0.6 O20.S1.0

is0.5 0 05 10 15s 2.0 2.5 3.0 .3.5 4.0 4.5 5.0 5&5 6,10 6.5 7.0

OISTAfta FROM CONTRACMION f

b. Numerical model (HIEL2O)

Figure 2. Depth confours of supercritical flow fIn a contraction

Figime 3. Watem urfa mesh of Su crtIca flOw in acoIncirdOn

onrT~rTT*1 IL

doI

HIVEL20 LEFT SIDE

-HIVEL2D RIGHT SIDE

It 0(3 MODEL LEFT SIDE

SMODEL RIGHT SIDE

90 CuRVE I CuRvE 2 CURVE 3

0 1 20 340 40 510 60

DISTANCE ALONG FLUME, FT

Figure 6. Water-surlace peofiles of Margarita Channel flume

0.9

0. B - - -FLUME TESTCOMPUTED

0,7

39:0.6

~0.5

g 0.4IxI

0.2

LONGJTUDINAL VELOCITY. FPS

F"gr 7. Lgata distribuior ol Wingituidinal velocinyk Mergeria Channel atswimn I

D__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

i i ' IVI N 1t r4b 374 It

U-U-

0 Ia.

0z'AT Vj Ii

LA3 NV1mI8

3tdi

Figure 10. Flow conditions in the Margarita Channel physical model

4 .

Figure 11. HIVEL2D results of Margarita Channel. Velocity contours andvectors (darker contours indicate lower velocities)

37.384 FT

2.229-FT WIDTH

15.332 FT

13.903 FT (MAIN) =

18.990 FT (TRIBUTARY) P.C. 17.561 FT

9.788 FTP.T. 13759 FT0.09-F WAL OFSE 1 ,45 FT

0.029-FT WALL OFFSET 9.074 • T 0.029-FT WALL OFFSET

,/X.-0.686-FT WIDTH

z 2I-

1.714-FT WIDTH-0.000 FýT

MODEL LIMIT

Figure 12. Details of supercritical confluence

Figure 13. Numerical model computational mesh for the supercrlticalconfluence problem

Figure 14. Flow conditions in the supercritical confluence physical model

Figure 15. Depth contours in the supercritical confluence computed byHIVEL2D (lighter contours indicate deeper depths)

10.5

9.. 9 - HIWEL20 RIGHT SIDE &-'MODEL RIGHT SIDE

9 .8......

9.6

o 5 10 15 20 25 .30 35 40

DisTANCE ALONG FLUME. FT

a.Right side

10,5

L 9.9 - HivEL2D LEFT SIDE 0----O MODEL LEFT SIDEL .16 1`1111ý1 f l9.8

LJ9.6 -- %

9.5

0 5 10 15 20 25 30 35 40

DISTANCE ALONG FLUME, FT

b. Left skise

Figure 1S. Wster-surface profiles for the superorltlcel confluence problemn

Figure 17. Numerical mode computational mesh for the bridge pier problem

Figure 18. Oblique view of the water-surface mesh of the bridge pier problem(flow is from upper right to lower left)

Figure 19. Side view of the water-surface mesh of the bridge pier problem(flow is from right to left)

2.294 m

3. 738 m

E RESERVOIR STATION 8 STATION 6P STATION 4

0.914 m4

DAM

Figure 20. Details of the dam break test flume (modified from Bell, Elliot, andChaudhry 1992)

Figure 21. Numerical model comfpstiaonal mesh for the dam break problem

0.25-

0.2-

S0.15-

0.1-

0.05-3.0 3.5 4.0 4.5 5.0 5.5

T=&e sa

a. Flume

0.25-

0.2 - o ..eoe *°°*° °°°e°°°°e°** °ee**eess *e**

is 0.15 =

A'0.1-wwa0.05- 1 •r 1

3.0 3.5 4.0 4.5 5.0 5.5

Tm., sw

b. Numedcal model, ct . 1.0

0.25 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0.2

S0.15UA' U 1am' wsv*

0.1 0 w *am..eoee..s.

0.05'3.0 3.5 4.0 4.5 5.0 5.5

c. Numerical model,, u, 1.5

Figuxe 22. Fklme and nmvical m deplh hiMtes for sation 4

0.25-

0.2- *

.~0.15

0.1 __

0.05 -4.0 4.5 5.0 5.5 6.0 6.5

aLFkxm

0.2 *

0.1 6

0.05-4.0 4.5 5.0 553 6.0 6.5

b. Nwnmiecal model, a,~ 1.0

0.25-

0.2 *

0.15 * iww

0.1 *

0.054.0 4-5 5.0 5.5 &.0 6.5

'0& on

c. Numeilcal model. ac a 1.5

Figur 23. Fime and numerical made depth heitstore for staton S

0.25-

0.2-

8S 0.15-

0.1-

0.05-5.5 6.0 6.5 7.0 7.5 8.0

a. Flume

0.25'

0o.2

0.15* Inuu wave

0.1• - W- - l -

0.05-3.5 6.0 65 ;.0 7.5 8.0

Tamr.

b. Numerical model, a, u 1.0

0.25 - __________________0.2-0.2 -.

i 0.15-

° ow wave

0.1000 w v

5.5 6.0 6.5 7.5 8.0

c. Numerical model, a, 1.5

Figure 24. Flume and numerical model depth histories for station 8

E O

.20

0.15

OAS IJ

0

103

06

10.300

9.90

0 50 100 150 200 250 300 350 400 450 500

Distance Along Flume, m

Figure 26. Depth profiles for the roll wave problem

Appendix AGoverning Equations

Vertical integration of the three-dimensional equations of mass and momen-tum conservation for incompressible flow with the assumption that verticalvelocities and accelerations are negligible compared to horizontal motions andthe acceleration of gravity results in the governing equations commonlyreferred to as the shallow-water equations. The dependent variables of thetwo-dimensional fluid motion are defined by the flow depth h, the x-directioncomponent of unit discharge p, and the y-direction component of unit dis-charge q. These variables are functions of the independent variables x and y,the two space directions, and time t. Neglecting free-surface stresses and theeffects of Coriolis force as these are not considered important in high-velocitychannels, the shailow-water equations in conservative form are given as(Abbott 1979; Praagman 1979):.

Oh +p + !- 0 (Al)

ot ox Oy

for the conservation of mass. Conservation of momentum in the x-directionand y-direction are given respectively as:

at9 rXp 12 hcyx) ayp (A2)yOp Op 2 p2+h2

- -h.z + ga coh 7/3

and

I References cited in this Appendix are listed in the Referencs at the end of tde main text.

Apperdkx A Gwwn, q Eaons Al

Tt 8xI h a I h(A3)

-h• + gnqvp2+q2

where

g = acceleration of gravity

o., os,, o,,� -= Reynolds stresses per unit mass where the first subscriptindicates the direction and the second indicates the face onwhich the stress acts

z = channel invert elevation

n = Manning's roughness coefficient

CO = dimensional constant (C,= 1 for SI units and 2.208 fornon-SI units)

The governing equations are given in vector form as:

__ OF 1 OF, H(4

aQ +L" __ + __ + H 0 (A4)7t ax Oy

where

S= P(A5)

q

p

F - h + g2 ho- (A6)

-Pq her.

h

A2 Appendx A Govening Equaftins

q

F, p. - h ,, (A7)

h +÷igh -ho

0

gh z n 2prp2 + q2

H . x - "2h7/3 (A8)

gh.L - g n 2q/p2 + q2

ay Ch '7/3

where

p = uh, u being the depth-averaged x-direction component of velocity

q = vh, v being the depth-averaged y-direction component of velocity

The Reynolds stresses are determined using the Boussinesq approach ofgradient-diffusion:

au

Oyy M 2v t'•"a

Where v, is the viscosity (sum of turbulent and molecular viscosity, commonlyreferred to as eddy viscosity), which varies spatially and is solved empiricallyas a function of local flow variables (Rodi 1980; Chapman and Kuo 1985):

V a4Cb rg n p2+q2 (A1O)Co h1/6

where Cb is a coefficient that varies between 0.1 and 1.0.

Appsdk A Go• nf Eqain A3

This system of equations constitutes a hyperbolic initial boundary valueproblem. Appropriate boundary conditions are determined using the approachof Daubert and Graffe as discussed in Drolet and Gray (1988) and Verboom,Stelling, and Officier (1982). Daubert and Graffe use the method of charac-teristics to determine the required boundary conditions. The number of bound-ary conditions is equal to the number of characteristic half-planes that originateexterior to the domain and enter it. If the inflow boundary is supercritical,then all information from outside the domain is carried through this boundary.Therefore, p and q (or u and v) and the depth h must be specified. If theinflow boundary is subcritical, then the depth is influenced from the flowinside the domain (downstream control) and therefore only p and q (or u andv) are specified. Outflow boundary conditions required are determined byanalysis of information transported through this boundary. If the outflowboundary is supercritical, then all information is determined within the domainand no boundary conditions are specified. However, if the outflow boundaryis subcritical, then the depth of flow at the boundary (tailwater) must bespecified. The no-flux boundary condition is appropriate at the sidewallboundaries and is discussed in detail in Appendix B.

A4 Ap A Govening Equaf

Appendix BFinite Element Formulation

A variational formulation of the governing equations involves finding asolution of the dependent variables Q using the test function * over thedomain 0. The variational formulation of the shallow-water equations inintegral form is:

where t is time and Q, F, F, and H are defined in Equations A5-A8.

The finite element approach taken is a Petrov-Galerkin formulation thatincorporates a combination of the Galerkin test function and a non-Galerkincomponent to control oscillations due to convection. The finite element formof the governing equations is:

i. [q + L__I + LPY + 19j dOj (B2)

-0, for each i

where

e = subscript indicating a particular element

i = subscript indicating a particular test function

- - discrete value of the quantity

The geometry and flow variables are represented using the Lagrange basis 0:

B1Appndxf B Finie Element Formuation

j(B3)

wherej is the nodal location. Bilinear triangular and quadrilateral elementsare used with nodes at the element corners. Figure BI shows the two bilinearelements used in terms of local coordinates • and il.

'7?

-1 -1

-1 1 -I1

Figire B1. Local bilinear elements

The test function used (to be elaborated in the next section) is:

0, = + 1 q+ , (134)

where

0, = Galerkin part of the test function

I = identity matrix

,p = non-Galerkin part of the test function

To facilitate the specification of boundary conditions, the weak form of theequations is developed using an integration by parts procedure. Integration byparts of the terms

62 Appenmx B Pfn Eekwmnt Fomulation

ax (B5)

yields the weak form of the equations. The - is omitted for clarity and thevariables are understood to be discrete values. The weak form is given as:

OLa- L...2 - L2F +oiAN+oBL

e at x ax a (yTi (B6)

+0H] al), + fi(n,+ Fn,) dJ7]=

where (n, n,) = % the unit vector outward normal to the boundary r,, and

8FA = ----.

ag8Q

Natural boundary conditions are applied to the sidewall boundaries through theweak statement. The sidewall boundaries are 'no flux" boundaries. That is,there is no net flux of mass nor momentum through these boundaries. Thisboundary condition is enforced in an average sense through the weak state-ment. Setting the mass flux through the sidewall boundary to zero:

f (pn. + qn) aT-- 0 (B8)r

where

p = x-direction component of unit discharge

q = y-direction component of unit discharge

There is no net momentum flux through the boundaries. Therefore, thex-direction momentum through the boundary is set to zero:

63Appendix B Finite Element Formulawn

f[Qq)n. + (uqý] dT = 0 (B9)r

and the y-direction momentum through the boundary is set to zero:

f [(vp)n. + (vq)nJ dl= 0 (B01)r

where

u = p/h = the depth.-averaged x-direction component of velocity

v = q/h - the depth-averaged y-direction component of velocity

h = the depth of flow

Sidewall drag is treated as a partial slip condition. That is the boundarystress terms in the governing equations, integrated along the sidewall, arespecified via the Manning relation:

*(hon. + hv.,n,)d' 01 gp n I 'n2 + q2 (B11)J h•

and

- r hn. + ha=n d r *gq n p2 + q2 dl(B12)1 J T2. h,1

where

a,, ow aw = Reynolds stresses per unit mass where the first subscriptindicates the direction and the second indicates the faceon which the stress acts

g = acceleration of gravity

CO = dimensional constant (Co= 1 for SI units and 2.208 fornon-SI units)

84 Appendix B Finite Element Formutatlon

Petrov-Galerkin Test Function

For the shallow-water equations in conservative form (Equation B2), thePetrov-Galerkin test function (, is defined as (Berger 1993)':

Wx ay

where 16 is a dimensionless number between 0 and 0.5, and 0 is the linearbasis function. In the manner of Katopodes (1986) the grid intervals arechosen as:

x2 X 2 (B114)Ax=2[•.J +[ ]

and

Ay=2 2 + a]] 1/2 (B14)

where t and r are the local coordinates defined from -1 to 1 (Figure BI).

To find A consider the following:

A- F(Q) (B 16)

aQ

P' A P = A (B17)

where A = IA is the matrix of eigenvalues of A, and P and P' are made upof the right and left eigenvectors.

tReferences cited in this appendix are listed in the References at the end of the main text.

85Appendix B Finite Element Formulation

A A -(B18)

where

XI 0 0

(x2 + 02) ( X2A 0 ~~~(k2 + V212 0(19

0 0k(X3 2 + 212

and

X, = u + c (B20)

X2 = u - c (B21)

X3 = u (B22)

c = (gh)'M (B23)

A similar operation may be performed to define A

This particular test function is weighted upstream along characteristicssimilar to a concept like that developed in the finite difference method ofCourant, Isaacson, and Rees (1952) for one-sided differences. These ideaswere expanded to more general problems by Moretti (1979) and Gabutti(1983) as split-coefficient matrix methods and by the generalized flux vectorsplitting proposed by Steger and Warming (1981). In the finite elementcommuifity, instead of one-sided differences the test function is weightedupstream. This particular method in one dimension (I-D) is equivalent to theSUPG (qtreamline lipwind Petrov-Qalerkin) scheme of Hughes and Brooks(1982) and similar to the form proposed by Dendy (1974). Examples of thisapproach in the open channel environment using the generalized shallow-waterequations are presented for I-D in Berger and Winant (1991) and for 2-D in

B6 Appendix B Finite Element Formulation

Berger (1992). A I-D St. Venant application is given by Hicks and Steffler(1992).

Shock Capturing

Berger (1993) shows that the Petrov-Galerkin scheme is not only a goodscheme for advection-dominated flow, but is also a good scheme for shockcapturing because the scheme dissipates energy at the short wavelengths.When a shock is encountered, the weak solution of the shallow-waterequations must lose mechanical energy. Some of this energy loss is analogousto a physical hydraulic system losing energy to heat, particle rotation, etc; butmuch of it is, in fact, simply the energy being transferred into vertical motion.And since vertical motion is not included in the shallow-water equations, it islost. This apparent energy loss can be advantageous.

To apply a high value of j6, say 0.5, only in regions in which it is needed,since a lower value is more precise, construct a trigger mechanism that candetect shocks and increase 0 automatically. The method employed detectsenergy variation for each element and flags those elements that have a highvariation as needing a larger value of 0 for shock capturing. Note that thiswould work even in a Galerkin scheme since this trigger is concerned withenergy variation on an element basis and the Galerkin method would enforceenergy conservation over a test function (which includes several elements).

The shock capturing is implemented when Equation B24 is true

Tsi > 3' (B24)

where -f is a specified constant and

Ts ED - E (B25)S

where ED1 , the element energy deviation, is calculated by

(E- E1 )yd (B26)

ai

B7Appendwx B Finifte El•ernt Formulation

where

,= element i

E = mechanical energy

a, = area of element i

and E,, the average energy of element i, is calculated by

E ___ (B27)a,

and _

E = the average element energy over the entire grid

S = the standard deviation of all ED,

Through trial a value of 'y of 1.0 was chosen.

Temporal Derivatives

A finite difference expression is used for the temporal derivatives. Thegeneral expression for the temporal derivative of a variable, Q, is:

"[' I ] {ae~i _ _" - __ + (1-_ a) '- "2s

where

a = temporal difference coefficient

j = nodal location

m = time-step

An a equal to 1 results in a first-order backward difference approximation andan a equal to 1.5 results in a second-order backward difference approximationof the temporal derivative.

B8 Appendix 1 FRnite EWlement r son

Solution of the Nonlinear Equations

The system of nonlinear equations is solved using the Newton-Raphsoniterative method (Carnahan, Luther, and Wilkes 1969). Let R, be a vector ofthe nonlinear equations computed using a particular test function •i and usingan assumed value of Q. R, is the residual error for a particular test functioni. Subsequently, RA is forced toward zero as:

ai Aq, =-R* (B29)

where k is the iteration number, j is the node location, and the derivativescomposing the Jacobian are determined analytically. This system of equationsis solved for ,q and then ai. improved estimate for Q.* is obtained from:

' + &(130)

This procedure is continued until convergence to an acceptable residual erroris obtained.

Equation B29 represents a system of linear algebraic equations that must besolved for each iteration and each time-step. A *Profile" solver is imple-mented to achieve efficient coefficient matrix storage. This method stores theupper triangular portion of the coefficient matrix by columns and the lower byrows. Any zeros outside the profile are not stored or involved in the com-putations. The necessary arrays are then a vector composed of the columns ofthe upper portion and a pointer vector to locate the diagonal entries.Triangular decomposition of the coefficient matrix is used in a direct solution.The program to construct the triangular decomposition of the coefficient ma-trix uses a compact Crout variation of Gauss Elimination.

B9Appendix 8 Finite Element Formulation

REPORT DOCUMENTATION PAGE 0M No. oAoA e

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1994 Final report4. TITLE AND SUBTITLE S. FUNDING NUMBERS

HIVEL2D: A Two-Dimensional Flow Model for High-Velocity Channels WU 32657

IL AUTHOR(S)

R. L. StockstillR. C. Berger

7. PERFORMING ORGANIZATION NAME(S) AND ADORESS(ES) B. PERFORMING ORGANIZATIONREPORT NUMBER

U.S. Army Engineer Waterways Experiment Station Technical Report3909 Halls Ferry Road, Vicksburg, MS 39180-6199 REMR-HY-12

9. SPONSORING /MONITORING AGENCY NAME(S) AND ADORESS(ES) 10. SPONSORING / MONITORINGAGENCY REPORT NUMBER

U.S. Army Corps of Engineers, Washington, DC 20314-1000

11. SUPPLEMENTARY NOTES

Available from National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161.

12a. DISTRIBUTION/AVAILABXITY STATEMENT 12b. DISTRIBUTION CODE

Approved for public release; distribution is unlimited.13. ABSTRACT (Maximum 200 worch)

A numerical flow model, HIVEL2D, has been developed as a tool to evaluate high-velocity channels.HIVEL2D is a depth-averaged, two-dimensional flow model designed specifically for flow fields that containsupercritical and subcritical regimes as well as the transitions between the regimes. The model is a finiteelement description of the two-dimensional shallow-water equations in conservative form. Provided in thisreport are a des-ription of the numerical flow model and illustrative examples of typical high-velocity flowfields that the model is capable of simulating. Model verification is obtained by comparison of simulationresults with data obtained from flume studies. Model assumptions and limitations are also discussed.

14. SUBJECT TERMS 1S. NUMBER OF PAGES

Finite element method Numerical model Petrov-Galerkin 53High-velocity channels Oblique standing waves Subcritical flow 16. PRICE CODE

Hydraulic jump (strong shock) (weak shocks) Supercritical flow17. SECURITY CLASSIFICATION 118. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

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