US Army Corps of Engineers Hydrologic Engineering Center

Development of Generalized Free Surface Flow Models Using Finite Element Techniques July 1978 Approved for Public Release. Distribution Unlimited. TP-53

Standard Form 298 (Rev. 8/98) Prescribed by ANSI Std. Z39-18

REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to the Department of Defense, Executive Services and Communications Directorate (0704-0188). Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ORGANIZATION. 1. REPORT DATE (DD-MM-YYYY) July 1978

2. REPORT TYPE Technical Paper

3. DATES COVERED (From - To)

5a. CONTRACT NUMBER

5b. GRANT NUMBER

4. TITLE AND SUBTITLE Development of Generalized Free Surface Flow Models Using Finite Element Techniques

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER 5e. TASK NUMBER

6. AUTHOR(S) D. Michael Gee, Robert C. MacArthur

5F. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army Corps of Engineers Institute for Water Resources Hydrologic Engineering Center (HEC) 609 Second Street Davis, CA 95616-4687

8. PERFORMING ORGANIZATION REPORT NUMBER TP-53

10. SPONSOR/ MONITOR'S ACRONYM(S) 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 11. SPONSOR/ MONITOR'S REPORT NUMBER(S)

12. DISTRIBUTION / AVAILABILITY STATEMENT Approved for public release; distribution is unlimited. 13. SUPPLEMENTARY NOTES Presented at the Second International Conference on Finite Element Techniques in Water Resources, Imperial College, London, July 1978. 14. ABSTRACT The Corps of Engineers' Hydrologic Engineering Center is involved in the development, evaluation, and application of mathematical models. Two finite element hydrodynamic models, one for two-dimensional free surface flow in the horizontal plane and one for the vertical plane are evaluated. Although the models are formulated to solve dynamic flow problems, all work to date has been with steady state solutions. Recent research has focused on mass continuity performance of the models, proper boundary condition specification, and comparison with finite difference techniques. The objective of this research is to develop generalized mathematical models for routine use by the engineering community. This paper presents recent results of evaluation and application of the models. 15. SUBJECT TERMS water resources, hydrodynamics, mathematical models, finite elements 16. SECURITY CLASSIFICATION OF: 19a. NAME OF RESPONSIBLE PERSON a. REPORT U

b. ABSTRACT U

c. THIS PAGE U

17. LIMITATION OF ABSTRACT UU

18. NUMBER OF PAGES 28 19b. TELEPHONE NUMBER

Development of Generalized Free Surface Flow Models Using Finite Element Techniques

July 1978 US Army Corps of Engineers Institute for Water Resources Hydrologic Engineering Center 609 Second Street Davis, CA 95616 (530) 756-1104 (530) 756-8250 FAX www.hec.usace.army.mil TP-53

Papers in this series have resulted from technical activities of the Hydrologic Engineering Center. Versions of some of these have been published in technical journals or in conference proceedings. The purpose of this series is to make the information available for use in the Center's training program and for distribution with the Corps of Engineers. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products.

DEVELOPMENT OF GENERALIZED F R E E SURFACE FLOW MODELS USING FINITE ELEMENT TECHNIQUES

D. Michael Gee, Robert C. MacArthur

The Hydrologic Engineering Center, U.S. Army Corps o f Engineers, Davis, Cal i fornia

INTRODUCTION

The Corps of Engineers' Hydro1 ogic Engineering Center i s involved i n the development, evaluation, and application o f mathematical models. Two f i n i t e element hydrodynamic models, one fo r two-dimensional f r ee surface flow in the horizontal plane and one f o r the ve r t i ca l plane are being evaluated. Although the models are formulated t o solve dynamic flow problems, a11 work t o date has been with steady s t a t e solutions. Recent research has focused on mass continuity performance of the models, proper boundary condition spec i f ica t ion , and comparison with f i n i t e difference techniques. The objective of t h i s research is t o develop general i zed mathematical niodel s for routine use by the engineering communi ty. This paper presents recent r e su l t s of evaluation and application of the models . THE MODEL FOR TWO-DIMENSIONAL F R E E SURFACE FLOW IN SHE HORIZONTAL PLANE

The model for twc-dimensional f r ee surface flow in the horizontal plane solves the govern,ing equations ,in the f o l l owing form:

Continuity

Momen turn a u a u a u ah a a 0 E --+ u- + v-c 9% +' F-. - 3 a 2 u % "2 - z2Losin + a t ax ay ax p 57- P ,y2

where u , v = x and y veloci ty compor~ents respec t ive ly

t = time h = depth a,= bed e levat ion E = tu rbu len t exchange goe f f i c i en t s g = grav i ta t iona l accelera t ion w = r a t e of e a r t h ' s angular ro ta t ion @ = l a t i t u d e C = Chezy roughness coe f f i c i en t s = empirical wind s t r e s s c o e f f i c i e n t Va= wind speed $ = angle between wind d i rec t ion and x - ax i s p = f l u id densi ty

Before so lu t i on , the equations a re r eca s t w i t h flow (ve loc i t y times depth) and depth as the dependent var iables . A l i n e a r shape function i s used f o r depth and a quadra t ic function fo r flow. The Galerkin method of weighted res idua l s i s used and the r e su l t i ng non- l inear system of equations solved w i t h t h e Newton-Rapheson scheme. Detai 1s of t he so1 ution have been published previously by Norton, e t a1 (1973) and King, e t a1 (1975). General discussions of f i n i t e element techniques have been pub1 ished by Zienkiewi cz (1971 ) , Hubner (1 975), and Strang & Fix (1973).

Evaluation of Continuity E r r o r - --- -- The f i n i t e element method y i e l d s a so lu t ion which approximates the t r ue so lu t ion t o the governing p a r t i a l di f f e r e n t i a1 equations. The approximate nature of t h i s so lu t i on becomes evident when mass con t inu i ty i s checked a t various loca t ions in the so lu t ion domain f o r a steady s t a t e simulation. A1 though overal l con t inu i ty i s maintained (inflow equals outflow over the boundary), ca1 cul a ted flows across i n t e rna l s ec t i ons devia te somewhat from the inflow/outflow values. A s tudy was made t o evaluate e r r o r s i n con t inu i ty as a function of network dens i ty. Poor con t inu i ty approximation is important of i t s e l f i f water q u a l i t y simulation is t he goal. In t he present app l ica t ions , however, water sur face e leva t ions and v e l o c i t i e s a r e the var iables of i n t e r e s t . Therefore, t he impact of

continuity e r ro r s on these parameters was a1 so invest igated. Flows on the Rio Grande de Loiza flood plain were

simulated using several networks. This flood plain was se lected because o f i t s complex flow f i e l d and a p r i o r study by the U.S. Army Corps of Engineers (1976) had made t he data readi ly avai 1 able. Model performance had previously been evaluated f o r simple hypothetical and 1 aboratory flows by Norton e t a1 (1973) and King e t a1 (1975). The i o i za flood plain i s about 10 by 10 km (6 by 6 miles) i n extent and i s characterized by var iab le bottom topography, one i n l e t and two ou t l e t s , and several i s l ands . Three o f the networks used i n the study a re shown i n Figs. 1 t o 3 i l l u s t r a t i n g pro- gressive increase i n network de t a i 1.

The solut ion was considered acceptable i f flow a t a l l continuity check l i ne s devia ted from inflow by l e s s than f 5%. Continuity i s checked by i n t eg ra t i ng the normal component of veloci ty times depth along l i n e s spec i f ied by the modeler. The continuity check l i n e s used in t h i s study a re indicated by dark l i ne s on Figs. 1 t o 3. Note t h a t , because the flow divides around the i s l ands , in some cases the sum of flows across two check l i n e s (such as 5 and 6) should be compared with inflow. Various parameters of the problem a re summarized i n Table 1. No attempt was made t o c a l i b r a t e the coe f f i c i en t s used.

The cont inui ty approximation improved w i t h increasing network d e t a i l , as expected. Flow a t the worst check l i n e in the coarses t network ( 7 + 8) improved from 79.3% t o 98.2% of inflow as network de ta i 1 was increased. Network character- i s t i c s , computer execution t imes, and r e s u l t s of the simu- l a t ions w i t h these th ree networks a r e summarized in Table 2. Average depths and v e l o c i t i e s along the cont inui ty check l i ne s are given in Table 3. The check l i n e numbers i n Tables 2 and 3 r e f e r t o the l i n e s ind ica ted on Figs. 1-3.

Table 1 Data f o r Loiza Flood Pla in Simulation

1. Boundary condit ions : a. Inflow ( l i n e 1 ) = 8200 cms (290,000 c f s ) b. 0 u t l e . t ~ ( l i n e s 11 & l 2 ) , water surface e levat ion =

2.5 m (8 f t ) MSL c. All o the r boundaries; e i t h e r tangent.ia1 flow o r

stagnation points 2. Bed, roughness: Chezy C s p a t i a l l y var,ied from 5.5 t o 22

rn1/'/sec (10 t o 40 f t l / ' / sec ) 3. Turbulent exchange coef'fi c i en t s : varied w i t h element s i z e

from 24 t o 48 m2/sec (360 t o 500 f t 2 / s e c )

O U T F

INFLOW

Figure 1 Continuity Check Network 3.1 (Dark Lines Ind ica te Continuity Check Lines)

Table 2 Cont,inuity Performance o f the Networks

T F L O W

1 Network 3.1 3.3 3.5 No. of Nodes 31 0 375 432

No. of Elements 131 162 1 89

1 CDC 7600 Execution 'Time (sec) 22 3 1 45

I Check Line Percent o f Inflow 1 ( inf low) 100.0 100.0 100.0

2 + 3 89.2 90.8 96.2 4 114.9 106.8 104.9

5 + 6 87.5 92.0 96.4 7 t - 8 79.3 90.1 98.2 9 + 10 99.8 99.4 98.7

11 + 12 100.0 100.0 100.0 (outflow)

O U T F L

OUTFLOW

OUTFLOW

1 N FLOW

Figure 3 Cont inui ty Check Network 3.5

Table 3 Flows (as percent of i n f l o w ) , depth, and veloci t ies f o r the networks

L

For most of the check l ines , the improvement in con- t inui ty obtained with increasing network de ta i l was associated with changes in both velocity and depth. In two cases, 1 ines 6 & 8, the velocity changes were substant ial . This region of the flow f i e l d i s characterized by a rapid change of direct ion. The resul ts reinforce the caveat t h a t increased network de ta i l i s important in such regions. Furthermore, i t appears t h a t depth i s somewhat less sens i t ive t o e r rors Sn continuity than i s velocity. Therefore, i f one i s interested in water surface elevations only, a l e s s s t r ingent continuity performance cr i te r ion could be accepted than i f ve loc i t ies are of in t e re s t .

Application t o McNary Dam Second Powerhouse -- St& An example of a "production" type application of the horizontal flow model i s the second powerhouse s i t e select ion study f o r McNary lock and dam on the Columbia River. Flow f i e lds down- stream o f the dam were simulated f o r several possible locations of the second powerhouse. O f i n t e r e s t were ve loc i t ies , both magnitudes and d i rec t ions , in the v ic in i ty of the approach channel t o the navigation lock. The study area and several of the possible second powerhouse locations are shown on Fig. 4. Fini te element networks f o r the ex is t ing condition and f o r the south shore powerhouse with excavated discharge channel are shown i n Figs. 5 and 6. Data are summarized in Table 4. The roughness coef f ic ien t was cal ibrated to reproduce an observed condi t i on.

This study was great ly f a c i l i t a t e d by an automatic re- ordering al gori t h m (Coll ins (1 973)) which has been incorporated in to the model. This algorithm makes modification of a network (compare Figs. 5 and 6 ) straightforward in tha t the e n t i r e network need not be re-numbered. The exis t ing nuiiibering scheme i s u t i l ized fo r input/output and the system of equations in te rna l ly re-ordered t o reduce storage.

1 STUDY I R C A SHOWING POSSISLE 1

FIG 5

FINITE ELEMENT FOR RUNS I

FIG 6

FiUiTE ELEYEKT N E T W K

m R PUN 5 ._A .-.___---_____-.-

Table 4 Data f o r McNary Second Powerhouse Study

1. Upstream boundary condition: a. Spillway: Q = 7000 crns (250,000 c f s ) f o r cal ibra t , ion

runs Q = 0 f o r production runs

b. Exis t ing powerhouse: Q = 6500 cms (230,000 c f s ) c. Second po~erhouse: Q = 7000 cms (250,000 c f s )

2. Downstream boundary condit ion: Water sur face e levat ion = 82.4 m (270.3 f t ) MSL*

3. All o ther boundaries: E i ther tangent ia l flow o r s tagnat ion points

4. Roughness: Chezy C = 55 m'/2/sec (100 f t 1 I 2 / s e c ) 5. Turbulent exchange coef f ic ien t s : Varied with element s i z e

from 4.8 t o 14.4 m2/sec (50 t o 150 f t 2 / s e c )

*For production runs i n which t o t a l r i v e r discharge was 13600 crns (480,000 c f s ) . T h i s e levat ion was varied according t o a known s tage-di scharge re1 a t ionship f o r o ther discharges.

A vector p lo t t i ng rout ine was used t o display simulated flow f i e l d s . Two such p lo t s a r e shown on Figs. 7 and 8. P lo t s of t h i s type a r e considered e s sen t i a l f o r i n t e rp re t i ng and analyzing complex flow f i e l d s .

Continuity e r r o r s were generally l e s s than ?5% w i t h the exception o f t h e cons t r i c t ion near the downstream boundary where e r r o r s were on the order o f -15%. I f f u tu r e de t a i l ed s t ud i e s a r e made, and ve loc i t i e s in t h a t area become important, more network d e t a i l wi l l be provided.

Figure 7 Veloc i t i es f o r Spillway Q = 7000 crns (250,000 c f s ) , Ex is t ing Powerhouse Q = 6500 cms (230,000 c f s ) , S l i p Boundary Conditions

- - - * 1s n rn 4 . - nm L I R I ~ - - . --. \ - - .\ -

-- & C -.. _-• ' 1 --

/ C--- /- - -

'. -. -- .- --- - gure 8 Veloc i t i es f o r Spillway Q = 0 , Exist ing Powerhouse

Q = 6500 cms (230,000 c f s ) , Second Powerhouse Q = 7000 crns (250,000 c f s ) , S l i p Boundary Conditions

The model a1 1 ows two val id types of boundary condit ions a t boundaries where no flow e n t e r s o r leaves the system. One i s the s tagnat ion point where both components o f ve loc i ty a r e zero; the o the r i s t he s l i p boundary condition where the veloci ty on t he boundary i s tangent ia l t o the boundary. The s s l i p condition requires use o f curved-sided e lemmts on the boundaries. Use o f curved boundaries with tangent ia l flow i s favored. Use of s tagna t ion points a1 ong the boundaries r e s u l t s i n a subs t an t i a l l y d i f f e r e n t so lu t ion as shown i n Fig. 9. Not only i s the ve loc i ty d i s t r i b u t i o n a1 te red , but calcula ted head loss i n t he reach i s about 0.21 m (0.7 f t . ) g rea te r than w i t h the s l i p boundary condit ion. Continuity performance f o r t he two simulations was s i m i l a r , though i n o ther problems analyzed by Resource Managenient Associates (1977), the s l i p condition was super ior . Use of d i f f e r e n t boundary condit ions should be investigated i n an at tempt t o i d e n t i f y under what condit ions the modeler should choose s1 i p o r s tagnat ion point boundaries.

I t i s encouraging t o note t h a t the McNary study required no code changes t o the model.

Figure 9 Veloci t ies f o r Spillway Q = 7000 crns (250,000 c f s ) , Exist ing Powerhouse Q = 6500 crns (230,000 c f s ) , Stagnation Point Boundary Conditions

TWO-DIMEb?SIOP!AL M0DEL.S IN THE VERTICAL PLANE

Two-dimensional ( longi tudinal and ve r t i c a l ) hydrodynamic models have been developed t o a i d the Corps i n t he descr ipt ion and ana lys i s of w s e r v o i r water qua l i ty . The importance o f t he longi tudinal as well as t he ve r t i c a l exchange i n long, r e l a t i v e l y narrow and deep impoundments has been s tud ied by Pr i t chard (1971), Anthony and Drummond (1973) and t he Tennessee Val ley Authority (1969). Invest igat ions such as these have shown t h a t the hydrodynamics of a s t r a t i f i e d r e se rvo i r in- f luences the water qua1 i ty and, the re fore , the b io log ica l product ivi ty of deep impoundments. Addi ti onal ob jec t ives f o r the development of multidimensional models a r e t o be ab le t o ~ r e d i c t the e f f e c t s t h a t o u t l e t type and loca t ion , degree of ; t r a t i f i c a t i o n , and rese rvo i r operation have on t h e water q u a l i t y i n downstream r i v e r s and streams.

As well as the general i n t e r e s t i n s imulating flows i n the ve r t i c a l plane, t h i s research has provided the opportunity t o compare the performance of an imp l i c i t f i n i t e d i f fe rence method (FDM) model w i t h t h a t of a f i n i t e element method (FEM) model. The FDM model was developed by Edinger and Buchak (1977) and is named LARM (La t e r a l l y Avera~ed Reservoir Model). The FEM ve r t i c a l model was developed by Norton e t a1 (1973) and King e t a1 (1975). Although i n i t i a l development of t h e v e r t i c a l FEM model was accomplished a t t he same time as t h a t o f the hor i - zontal mode1 previously discussed, f u r t h e r refinement and use of t he ve r t i c a l model has lagged considerably,

The primary object ives of t he comparison o f these two hydrodynamic models were: (1) t o compare the r e l a t i v e ease w i t h w h i c h the required data and boundary condi t ions could be prepared and coded; ( 2 ) t o compare the overal l performance of

the two d i f fe rent approaches with respect t o s tab i 1 i ty , convergence, accuracy and p rac t i ca l i ty ; and f ina l ly , (3) t o compare re la t ive r u n times and simulation costs tetween the two me.thods f o r s,i m i 1 a r problems. The fol 1 ovii ng paragraphs . present the fundamental equations used by the two models.

Governing Equations - Both models incorporate simi 1 a r forms of the so-called phenom- enoloqical equations f o r mom~ntum, along with the continuity euuation and 'a ,form of the convective-diffusion equation fo r thermal or material t ransport i n the ver t ica l . ~ o t e , however, t ha t the FEM mode1 re ta ins the vert ical momentum equation, which i s replaced by the hydrostat ic pressure d is t r ibut ion in the FDM model. Both models u t i l i z e a Cartesian coordinate system with the longitudinal x dimension posi t ive downstream. The vert ical z dimension i s referenced posi t ive upward from the x-axis in the FEM model , whi l e i t i s posi t ive downward from the x-axis i n LARM. Both models allow f o r a variable width in the l a t e r a l y direct ion.

FEM Hydrodynamic Model - Momentum Equation:

Continuity Equation: a k-(bu) + -(bw) = 0 ax a z

Convecti ve-Di ffus ion Equation for Dens i ty: a a P a + b (,* + 2 , ~ ) - D -- (bG) - D Z az (b$)= 0

a t ax a z x ax (7 )

where u , w = f l u i d veloci ty in the x and z d i rec t ions respectively

b = breadth p = pressure

D x , D, = eddy diffusion coef f ic ien ts in the x and z direct ions respectively

Ab = area over which bottom s t r e s s i s e f f e t t i v e As = the area over iqhich the wind s t r e s s i s e f f ec t ive

Other var iables have previ ously been defined.

FDM Hydrodynamic Model LARM Momentum Equation :

Boundary s t r e s s e s a r e found using t he f o l l owing expressions:

a t the bottom:

Hydrostatic Pressure d i s t r i b u t i o n :

Continuity Equation:

Thermal Convecti ve-Di ,ffusi on Equation :

a . + a(uTb) + a ( w W a a T a aT f'b - - (Dxb ) - -- a t ( D b - ) =--- ax a z ax az z a~ P C ~

Equation of s t a t e :

P = P(T) where

p = f l u i d dens i ty Pa = dens i ty o f a i r S~ = boundary shea r s t r e s s q = l a t e r a l inflow per u n i t volume T = temperature

Dx¶ Df = heat t r a n s p o r t d i spers ion coe f f i c i en t s = heat inflow per u n i t volume

cp = s p e c i f i c heat

Other var iab les have been previously defined.

Description o f t h e Test Problem -- - Data col lected by the Tennessee Val l ey Authority (TVA) (1969) were used t o t e s t and compare the two vert . ica1 models i n a rese rvo i r simulation. 'These data were fo r the Fontana Reservoir i n North Carolina. The models were appl ied t o the f i r s t 23 km ('14.5 miles) of t he r e s e r v o i r upstream from the dam. To

2 simp1 i fy geometric requi rements, a u n i form reservo,i r breadth of 638 m (2095 f t ) was used. T h i s breadth was se lec ted t o conserve

rese rvo i r volume. 'The bottom p r o f i l e and e1evat.ions were determined from sediment inves t iga t ion cross sect ions . Con- d i t i ons t ha t ex i s t ed i n the r e se rvo i r during the l a s t week of March, 1966 were used t o provide t he boundary condit.ions f o r the simulation. Water temperature p r o f i l e s , water surface e levat ions , and flows i n t o and out of the t e s t reach were obtained from the 'TVA (1969) data. The rese rvo i r was s t r a t i - f i e d and was approximately 108 rn (353 ft. ) deep a t the dam. Surface heat exchange, wind ve loc i ty , and t r i b u t a r y ,inflows were a11 assumed t o be ze ro f o r t he purposes of t h i s i nves t i - gation; a steady inflow and outflow of 140 cms (5000 c f s ) was used.

Discussion --- The time and e f f o r t necessary t o describe the r e se rvo i r geometry f o r both the FDM and FEM models were comparable. Po achieve calcula ted r e s u l t s a t comparable locat ions i n space, optional quadr i l a te ra l elements were used s o t h a t t h e f i n i t e element network (Fig. 10) was almost iden t ica l t o FDM g r id (not shown). I t is recognized t h a t t h i s network does not exp lo i t the capab i l i t y o f the FEM model t o allow increased geometric resolut ion where des i red , such as near t he r e se rvo i r outflow point , but t h i s s imp l i f i c a t i on was useful f o r com- par i son of r e s u l t s .

The convergence of the FEM so lu t ion was noted t o be somewhat more s e n s i t i v e t o the magnitude of the t u rbu l en t exchange coe f f i c i en t s than the FDM model. The ranges of values of the coe f f i c i en t s over which convergent so lu t ions can be obtained f o r the two models have no t y e t been firmly es tab- l i shed. Additional s e n s i t i v i t y inves t iga t ions s h a l l be under- taken a t a l a t e r time. Ar ia thura i , e t a1 (1977) examined s imi l a r equations and found t h a t s t a b i l i t y and convergence of the solut ion could be r e l a t ed no t only t o s p a t i a l and temporal s t e p s i z e s b u t a l s o t o t he Pec le t number which i s the r a t i o of convective t r anspo r t t o d i f fu s ive t ranspor t .

The flow f i e l d s ca lcu la ted with the FDM and FEM models a r e shown in Figs. 11 and 12 respect ively . The ve r t i c a l s c a l e of Figs. 10-12 i s exaggerated by a f ac to r of 100. Coef f ic ien t s used ( r e f e r t o equations 4-7) were: rxx=24, ~ ~ ~ = 4 . 8 x 1 0 ' ~ ,

EZf =240, Dx=23, DZ=9.3x10-7 m2/sec (260,0.05, 2600, 250, f t / sec) . Although the models have numerous de t a i l ed d i f i e r - ences, pa r t i cu l a r l y i n the descr ip t ion of boundary condi t ions , the calcula ted flow f i e l d s a r e s i m i l a r and reasonable. For the t e s t appl i ca t ion , t he rese rvo i r was thermally s t r a t i f i e d , w i t h the incoming f l u i d cooler and more dense than the f l u i d i n the surface layers. The s t a b l e densi ty gradient i n the region of the thermocline tends t o i n h i b i t ve r t i c a l momentum and mater ia l t r anspor t , y e t c i r cu l a t i on appears i n the upper l ayers . The c i rcu la t ion i n the sur face l ayers i s driven by in te rna l hor i - zontal shearing between the cool water flowing toward t h e o u t l e t and the warmer water above. A s i m i l a r flow pa t te rn i s a l so observed i n t he bottom region below the main flow i n the

F'i gure 70 Finite Element Network for Reservoir Simulation

Figure 11 Reservoir Velocities Calculated with the Finite D i fference Model

Figure 12 Reservoir Velocities Cal cul ated w'i t h the Finite El ement Model

FDM model (Fig. 71). Generally, the FEM solution predicts 1 arger ver t ical velocity components, perhaps due t o the retention of the ver t ical momentum equation. Comparison of the solutions w i t h avai lable f i e l d data will be undertaken once general performance charac ter i s t ics of the two models are fur ther defined.

For these steady s t a t e simulations, the FDM took about 6 times more CDC 7600 computer tirne than the FEM. The primary reason i s t ha t , t o achieve a steady s t a t e solut ion, the FDM model m u s t be r u n through pseudo-time with constant boundary values unti 1 t rans ien ts from i n i t i a l conditions die out (about 75-100 days i n t h i s case). The FEM model, however, has the capabi l i ty of solving the system once w i t h zero time deriva- t ives t o arr ive a t a steady s t a t e solution. Comparative costs f o r dynamic simulations w i 11 depend primarily upon length of time s tep and number of elements used t o define the study regi on.

SUMMARY

The work t o date with the horizontal flow model indicates the fol lowing:

(1) Internal continuity e r ro r s can be reduced t o acceptable levels by increasing network detai 1, par t icu lar ly in areas of large curvature of the velocity f i e ld .

(2) Errors in continuity tend t o be re f lec ted more strongly i n the velocity than the depth.

(3) General application of the model to steady s t a t e simulations i s feas ib le a t present.

The preliminary work w i t h the ver t ical flow models indicates the following:

(1) The f i n i t e element method model i s l e s s cos t ly than the f i n i t e difference model f o r steady s t a t e solutions.

(2) Simulation of flows in which density gradients are important requires careful select ion of turbulent exchange and eddy diffusion coef f ic ien ts .

(3) The f i n i t e element model predicts la rger ver t ica l ve loc i t ies than the f i n i t e difference model, perhaps due t o the re tention of the ver t ica l momentum equation.

(4) More experience with, and development of , the ver t ical models wi 11 be required before "production" appl i - cations can be e a s i l y made.

Indicated areas o f fu r the r work are: (1) Veri f i cation of models ' performance when an adequate

data s e t becomes available. (2) Development of guidance on select ion of turbulent

exchange coef f ic ien ts , re la t ionship t o flow properties e tc . (3) Invest igate models' behavior f o r dynamic simul ations. (4) Evaluate use of stagnation vs. s l i p boundary con-

d i t ions i n the f i n i t e element models. (5) Extend simulations with the ver t ica l models t o

variable breadth problems.

REFERENCES

Anthony, M. and Drummond, G. (1973) "Reservoir Water Q u a l i t y Control ," i n Man-Made Lakes: Their Problems ------ and Environmental Ef fec t s , Geophysical Monograph 17, American Geophysical Union, pp. 549-551.

Aria thur i , R. , MacArthur, R . C . , and Krone, R . B . (1977) "Mathe- mati cal Model of Estuar ia l Sediment Transport ," Tech Report D-77-12, Off ice , Chief of Engineers, U.S. Army Corps of En g i nee rs.

Col l ins , R.J. (1973) "Bandwidth Reduction by Automatic Re- numbering," In te rna t iona l Journal f o r -- Numerical Methods i n Engineering, Vol. 6 , 345,-356.

Edinger, J.E. and Buchak, E.M. (1977) "A Hydrodynamic Two- Dimensional Reservoir Model: Development and Tes t Application t o Sutton Reservoir, Elk River, West Virginia ," Report t o U.S. Army Corps' of Engineers, Ohio River Division.

Gray, W.G., Pinder, G.F., and Brehbia, C.A. (1977) F in i t e Elements i n Water Resources, Proceedings of t he F i r s t In te r - i i X X E a 1 Conference on F in i t e Elements in Water Resources, Pentech Press.

Huebner, K. (1975) The F in i t e Element Method f o r Engineers, - John Wiley & Sons.

King, I.P., Norton, W . R . , and Iceman, K.R. (1975) "A Fin.ite Element Solution f o r Two-Dimensional S t r a t i f,ied Flow Problems ," i n F,in,ite Elements i n Flu-, Vol. 1 , John Wiley & Sons.

Norton, W.R. , King, I .P . , and Orlob, G.T. (1973) "A F in i t e Element Model fo r Lower Granite Reservoir," Report t o U.S. Army Corps of Engineers, Wall a Wall a D i s t r i c t .

P r i t chard , D.W. (1971) "Hydrodynamic Models: Two-Dimensional Model," i n Estuarine Modeling: An Assessment, Environmental Protect ion Agency, Stock No. 5501-0129, U.S. G . P . O . , Washington D.C .

Resource Management Associates (1977) "Case Studies of Con- t i n u i t y S a t i s f a c t i on w i t h an Improved 2-0 Hydrodynami c Model ," Report t o the U.S. Army Corps of Engineers, Hydrologic Engineering Center.

St rang, G. and Fix, G. (1973) An Analysis of the F . i n i E Element Method, Prentice-Hall .

Tennessee Val l ey Authority ( 1969) "Eval ua t i on of Fontana Reservoir Field Measurements," Laboratory Report No. 17-90.

U. S. Army Corps of Engineers, Jacksonvi l l e D i s t r i c t (1976) "F lood Hazard In fo rma t , i on ( 'Technical Appendix), R i o Grande de Loiza, Puer to Rico."

Z ienkiewicz, O.C. (1971) The F i n i t e Element Method i n Eng ineer ing Science, McGraw H i l l .

ACKNOWLEDGEMENTS

The evaluation of the mass continuity performance was funded by the U.S. Federal Highway Administration. The Walla Walla Di s t r i c t , U.S. Army Corps of Engineers funded the McNary second powerhouse study. The comparison of f i n i t e element and f i n i t e difference techniques i s being funded by the U.S. Army Corps of Engineers Waterways Experiment Stat ion. Con- tinued advice of the f i n i t e element model developers: Ian King and Bill Norton of Resource Management Associates, and the f i n i t e difference model developers: John Edinger and Ed Buchak of J. E. Edinger and Associates, is gra tefu l ly acknowledged.

Technical Paper Series TP-1 Use of Interrelated Records to Simulate Streamflow TP-2 Optimization Techniques for Hydrologic

Engineering TP-3 Methods of Determination of Safe Yield and

Compensation Water from Storage Reservoirs TP-4 Functional Evaluation of a Water Resources System TP-5 Streamflow Synthesis for Ungaged Rivers TP-6 Simulation of Daily Streamflow TP-7 Pilot Study for Storage Requirements for Low Flow

Augmentation TP-8 Worth of Streamflow Data for Project Design - A

Pilot Study TP-9 Economic Evaluation of Reservoir System

Accomplishments TP-10 Hydrologic Simulation in Water-Yield Analysis TP-11 Survey of Programs for Water Surface Profiles TP-12 Hypothetical Flood Computation for a Stream

System TP-13 Maximum Utilization of Scarce Data in Hydrologic

Design TP-14 Techniques for Evaluating Long-Tem Reservoir

Yields TP-15 Hydrostatistics - Principles of Application TP-16 A Hydrologic Water Resource System Modeling

Techniques TP-17 Hydrologic Engineering Techniques for Regional

Water Resources Planning TP-18 Estimating Monthly Streamflows Within a Region TP-19 Suspended Sediment Discharge in Streams TP-20 Computer Determination of Flow Through Bridges TP-21 An Approach to Reservoir Temperature Analysis TP-22 A Finite Difference Methods of Analyzing Liquid

Flow in Variably Saturated Porous Media TP-23 Uses of Simulation in River Basin Planning TP-24 Hydroelectric Power Analysis in Reservoir Systems TP-25 Status of Water Resource System Analysis TP-26 System Relationships for Panama Canal Water

Supply TP-27 System Analysis of the Panama Canal Water

Supply TP-28 Digital Simulation of an Existing Water Resources

System TP-29 Computer Application in Continuing Education TP-30 Drought Severity and Water Supply Dependability TP-31 Development of System Operation Rules for an

Existing System by Simulation TP-32 Alternative Approaches to Water Resources System

Simulation TP-33 System Simulation of Integrated Use of

Hydroelectric and Thermal Power Generation TP-34 Optimizing flood Control Allocation for a

Multipurpose Reservoir TP-35 Computer Models for Rainfall-Runoff and River

Hydraulic Analysis TP-36 Evaluation of Drought Effects at Lake Atitlan TP-37 Downstream Effects of the Levee Overtopping at

Wilkes-Barre, PA, During Tropical Storm Agnes TP-38 Water Quality Evaluation of Aquatic Systems

TP-39 A Method for Analyzing Effects of Dam Failures in Design Studies

TP-40 Storm Drainage and Urban Region Flood Control Planning

TP-41 HEC-5C, A Simulation Model for System Formulation and Evaluation

TP-42 Optimal Sizing of Urban Flood Control Systems TP-43 Hydrologic and Economic Simulation of Flood

Control Aspects of Water Resources Systems TP-44 Sizing Flood Control Reservoir Systems by System

Analysis TP-45 Techniques for Real-Time Operation of Flood

Control Reservoirs in the Merrimack River Basin TP-46 Spatial Data Analysis of Nonstructural Measures TP-47 Comprehensive Flood Plain Studies Using Spatial

Data Management Techniques TP-48 Direct Runoff Hydrograph Parameters Versus

Urbanization TP-49 Experience of HEC in Disseminating Information

on Hydrological Models TP-50 Effects of Dam Removal: An Approach to

Sedimentation TP-51 Design of Flood Control Improvements by Systems

Analysis: A Case Study TP-52 Potential Use of Digital Computer Ground Water

Models TP-53 Development of Generalized Free Surface Flow

Models Using Finite Element Techniques TP-54 Adjustment of Peak Discharge Rates for

Urbanization TP-55 The Development and Servicing of Spatial Data

Management Techniques in the Corps of Engineers TP-56 Experiences of the Hydrologic Engineering Center

in Maintaining Widely Used Hydrologic and Water Resource Computer Models

TP-57 Flood Damage Assessments Using Spatial Data Management Techniques

TP-58 A Model for Evaluating Runoff-Quality in Metropolitan Master Planning

TP-59 Testing of Several Runoff Models on an Urban Watershed

TP-60 Operational Simulation of a Reservoir System with Pumped Storage

TP-61 Technical Factors in Small Hydropower Planning TP-62 Flood Hydrograph and Peak Flow Frequency

Analysis TP-63 HEC Contribution to Reservoir System Operation TP-64 Determining Peak-Discharge Frequencies in an

Urbanizing Watershed: A Case Study TP-65 Feasibility Analysis in Small Hydropower Planning TP-66 Reservoir Storage Determination by Computer

Simulation of Flood Control and Conservation Systems

TP-67 Hydrologic Land Use Classification Using LANDSAT

TP-68 Interactive Nonstructural Flood-Control Planning TP-69 Critical Water Surface by Minimum Specific

Energy Using the Parabolic Method

TP-70 Corps of Engineers Experience with Automatic Calibration of a Precipitation-Runoff Model

TP-71 Determination of Land Use from Satellite Imagery for Input to Hydrologic Models

TP-72 Application of the Finite Element Method to Vertically Stratified Hydrodynamic Flow and Water Quality

TP-73 Flood Mitigation Planning Using HEC-SAM TP-74 Hydrographs by Single Linear Reservoir Model TP-75 HEC Activities in Reservoir Analysis TP-76 Institutional Support of Water Resource Models TP-77 Investigation of Soil Conservation Service Urban

Hydrology Techniques TP-78 Potential for Increasing the Output of Existing

Hydroelectric Plants TP-79 Potential Energy and Capacity Gains from Flood

Control Storage Reallocation at Existing U.S. Hydropower Reservoirs

TP-80 Use of Non-Sequential Techniques in the Analysis of Power Potential at Storage Projects

TP-81 Data Management Systems of Water Resources Planning

TP-82 The New HEC-1 Flood Hydrograph Package TP-83 River and Reservoir Systems Water Quality

Modeling Capability TP-84 Generalized Real-Time Flood Control System

Model TP-85 Operation Policy Analysis: Sam Rayburn

Reservoir TP-86 Training the Practitioner: The Hydrologic

Engineering Center Program TP-87 Documentation Needs for Water Resources Models TP-88 Reservoir System Regulation for Water Quality

Control TP-89 A Software System to Aid in Making Real-Time

Water Control Decisions TP-90 Calibration, Verification and Application of a Two-

Dimensional Flow Model TP-91 HEC Software Development and Support TP-92 Hydrologic Engineering Center Planning Models TP-93 Flood Routing Through a Flat, Complex Flood

Plain Using a One-Dimensional Unsteady Flow Computer Program

TP-94 Dredged-Material Disposal Management Model TP-95 Infiltration and Soil Moisture Redistribution in

HEC-1 TP-96 The Hydrologic Engineering Center Experience in

Nonstructural Planning TP-97 Prediction of the Effects of a Flood Control Project

on a Meandering Stream TP-98 Evolution in Computer Programs Causes Evolution

in Training Needs: The Hydrologic Engineering Center Experience

TP-99 Reservoir System Analysis for Water Quality TP-100 Probable Maximum Flood Estimation - Eastern

United States TP-101 Use of Computer Program HEC-5 for Water Supply

Analysis TP-102 Role of Calibration in the Application of HEC-6 TP-103 Engineering and Economic Considerations in

Formulating TP-104 Modeling Water Resources Systems for Water

Quality

TP-105 Use of a Two-Dimensional Flow Model to Quantify Aquatic Habitat

TP-106 Flood-Runoff Forecasting with HEC-1F TP-107 Dredged-Material Disposal System Capacity

Expansion TP-108 Role of Small Computers in Two-Dimensional

Flow Modeling TP-109 One-Dimensional Model for Mud Flows TP-110 Subdivision Froude Number TP-111 HEC-5Q: System Water Quality Modeling TP-112 New Developments in HEC Programs for Flood

Control TP-113 Modeling and Managing Water Resource Systems

for Water Quality TP-114 Accuracy of Computer Water Surface Profiles -

Executive Summary TP-115 Application of Spatial-Data Management

Techniques in Corps Planning TP-116 The HEC's Activities in Watershed Modeling TP-117 HEC-1 and HEC-2 Applications on the

Microcomputer TP-118 Real-Time Snow Simulation Model for the

Monongahela River Basin TP-119 Multi-Purpose, Multi-Reservoir Simulation on a PC TP-120 Technology Transfer of Corps' Hydrologic Models TP-121 Development, Calibration and Application of

Runoff Forecasting Models for the Allegheny River Basin

TP-122 The Estimation of Rainfall for Flood Forecasting Using Radar and Rain Gage Data

TP-123 Developing and Managing a Comprehensive Reservoir Analysis Model

TP-124 Review of U.S. Army corps of Engineering Involvement With Alluvial Fan Flooding Problems

TP-125 An Integrated Software Package for Flood Damage Analysis

TP-126 The Value and Depreciation of Existing Facilities: The Case of Reservoirs

TP-127 Floodplain-Management Plan Enumeration TP-128 Two-Dimensional Floodplain Modeling TP-129 Status and New Capabilities of Computer Program

HEC-6: "Scour and Deposition in Rivers and Reservoirs"

TP-130 Estimating Sediment Delivery and Yield on Alluvial Fans

TP-131 Hydrologic Aspects of Flood Warning - Preparedness Programs

TP-132 Twenty-five Years of Developing, Distributing, and Supporting Hydrologic Engineering Computer Programs

TP-133 Predicting Deposition Patterns in Small Basins TP-134 Annual Extreme Lake Elevations by Total

Probability Theorem TP-135 A Muskingum-Cunge Channel Flow Routing

Method for Drainage Networks TP-136 Prescriptive Reservoir System Analysis Model -

Missouri River System Application TP-137 A Generalized Simulation Model for Reservoir

System Analysis TP-138 The HEC NexGen Software Development Project TP-139 Issues for Applications Developers TP-140 HEC-2 Water Surface Profiles Program TP-141 HEC Models for Urban Hydrologic Analysis

TP-142 Systems Analysis Applications at the Hydrologic Engineering Center

TP-143 Runoff Prediction Uncertainty for Ungauged Agricultural Watersheds

TP-144 Review of GIS Applications in Hydrologic Modeling

TP-145 Application of Rainfall-Runoff Simulation for Flood Forecasting

TP-146 Application of the HEC Prescriptive Reservoir Model in the Columbia River Systems

TP-147 HEC River Analysis System (HEC-RAS) TP-148 HEC-6: Reservoir Sediment Control Applications TP-149 The Hydrologic Modeling System (HEC-HMS):

Design and Development Issues TP-150 The HEC Hydrologic Modeling System TP-151 Bridge Hydraulic Analysis with HEC-RAS TP-152 Use of Land Surface Erosion Techniques with

Stream Channel Sediment Models

TP-153 Risk-Based Analysis for Corps Flood Project Studies - A Status Report

TP-154 Modeling Water-Resource Systems for Water Quality Management

TP-155 Runoff simulation Using Radar Rainfall Data TP-156 Status of HEC Next Generation Software

Development TP-157 Unsteady Flow Model for Forecasting Missouri and

Mississippi Rivers TP-158 Corps Water Management System (CWMS) TP-159 Some History and Hydrology of the Panama Canal TP-160 Application of Risk-Based Analysis to Planning

Reservoir and Levee Flood Damage Reduction Systems

TP-161 Corps Water Management System - Capabilities and Implementation Status

Front CoverIntroductionThe Model for Two-Dimensional Free Surface Flow in the Horizontal PlaneEvaluation of Continuity ErrorsApplication to McNary Dam Second Powerhouse Study

Two-Dimensional Models in the Vertical PlaneGoverning EquationsFEM Hydrodynamic ModelFDM Hydrodynamic Model LARMDescription of the Test ProblemDiscussion

SummaryReferencesAcknowledgementsTechnical Paper Series