tidal wave analysis for estuaries with intertidal flats

Tidal wave analysis for estuaries with intertidal flats

K.-P. H O L Z and G. N I T S C H E LehrstuhlJfir Stromungsmechanik, Hannover University, 3 Hannover, Federal Republic o[ Germany

INTRODUCTION These equations hold for three-dimensional flow in The finite element method has proven a powerful tool for Langrangian formulation, using Cartesian coordinates. the treatment ofestuarine problems. Especially in the field Here p denotes the density of water,f~ the forces per unit of of vertically integrated models for the calculation of volume, ?:ithe velocity components of each particle, and Lj horizontal flow, great progress has been made during the stresses acting on the surface S of the volume t, with its recent years, and many different models have become outward normal uj. available in the meantime. They are formulated For the calculation of far-field tidal flow phenomena, implicitly~.2, explicitly3-5, or as hybrid models 6. some simplifications can be made. With the assumption of Combinations of all these techniques within one program a constant density, vertically averaged horizontal are possible 7. Nevertheless, a comparison with models velocities vi, and by applying the Boussinesq basing on the finite difference technique shows its approximation for the description of turbulent flow, and superiority with respect to economics. This is true as long by neglecting the vertical velocity component, the well- as coastlines are rather regular, the bottom topography known equations for shallow water waves are obtained. not too much structured, and detailed local information is not needed. The more complex the topological situation ( ' f becomes, and when additional grid refinements must be J~,q~,~ + (t,~q~)~j - ~ j q j + g(a + h)h,~ + provided, the flexibility of finite elements makes this

A

method much more attractive. So the modelling of

estuaries with intertidaltidal channels,flats which are structuredbe by 2 .Iv.,,. -Anqijf~ highly irregular seems to a very a~_hV~J~fl~ d r = 0 (3) promising field of application for finite elements.

Many estuaries along the North Sea coast are extremely shallow, the water depth ranging within a few /" feet only. This shallowness allows for various human J[h,,+q~.~}dv=O (4) interferences, such as land reclamation by means of dikes, A or the construction of dams for connecting islands. Moreover, such shallow estuaries are extremely sensitive The integral form follows from the conservation to pollution. Investigations in these areas are of a great formulation. The equations refer to an Eulerian practical interest, coordinate system as indicated by Fig. 1. In the notation

Simulation of the flow in intertidal flats structured by used, q, stands for the vertically averaged flux, v,~ for the deep tidal channels is rather complicated. It can be corresponding velocities, f~ is the Coriolis parameter with: performed in different ways which depend on the main interest of the investigetion. If a tidal wave propagation O / must be analysed in a deep water region which is only c~= _ 1 0 slightly influenced by tile dynamics on the flats, rather simple strategies may be used, but if the flow on the flats The influence of bottom friction as well as that of themselves has to be modelled, more accurate strategies turbulence are parameterized with the friction parameter must be applied. This will be shown by two examples.

TIDAL WAVE EQUATIONS free water surface

The tidal wave equations can be derived directly from the x3 L ~ _ . _ . , ~ basic conservation principles of physics, such as the " horizontal

datum conservation of momentum: x~

°f f f ~f P?idt'- pjldv- rijujdS=O (1) +h)

t v S h = water level

bottom depth and for the conservation of mass: - x k ~ r a ~ . H water depth

°f Dt- pdv=0 (2) ,. Fiyure 1. Notation

0309-1708/82/030142-07 $2.00 142 Adv. Water Resources, 1982, Volume 5, September © 1982 CML Publications

Tidal wave analysis: K.-P. Holz and G. Nitsche

and the turbulence eddy coefficient A n . Integration must space domain. This leads to the matrix equation of the be performed over the horizontal plain A. foi'rn:

The formulation of boundary conditions follows directly from these equations. Applying Gauss' theorem, NZ,, + M(Z)Z - R = 0 (10) and assuming AH to be constant, three boundary integrals are obtained. The vector Z contains all nodal parameters of the

Conservation of mass on the boundary leads to: assembled system. Different time integration techniques 9 can be applied to

f( this equation. The Crank-Nicholson scheme with the qini-q . )ds=O (5j coefficient 1/2~<0~<1 is most commonly used:

The flow of water 0, normal to the boundary has to be (N+AtOM(7 , ) )Z t+a ,=(N-A t (1 -0 )M(Z) IZ ,+ prescribed. It is equal to zero on the coastline. The same holds for the momentum flux m i, which is described by the AtR, +oa, (11)

second condition: and renders a mixed implicit formulation. If the space f( discretization is performed on an element level only, and

viqj- rhi)n~ds = 0 (6) then the fluxes are eliminated by expressing them in terms of the water-level parameters, a hybrid implicit scheme is obtained 6. All these implicit models are unconditionally

As ff~i---0 on the coast, this equation can equally be stable. This does not hold for explicit formulations which fulfilled by setting qi=0 directly, instead of using the are obtained by choosing the coefficient 0=0, or other integral form. This would then correspond to the no-slip techniques 1°. These models are subject to the Courant condition, conditions, which limits the time-step in relation to the

For the last integral space discretization.

flAHqi'JtlJ CHOICE OF NUMERICAL MODEL ~ 0 (7) 1

the influence of friction and turbulence ~ along the The typical topography of estuaries with intertidal flats, as coastline must be specified. Parametric formulations, shown schematically in Fig. 2, is characterized by nearly depending on the actual velocities, have been proposed 6 horizontal plains, structured by narrow deep channels in case that a slip condition is used, otherwise T~ can be with steep banks. The flow in the plains is normally within taken to be zero. the range of Froude numbers 0.0 to 0.05, whereas in the

Since at the open s a mass an momentum flow are channels it might be even equal to 1.0. The transition zone hardly known from nature, generally the water-level must between these quite different flow conditions is near the be prescribed. In most cases, this leads to reasonable banks of the channel with its steep slope. This area, and results, but it should be mentioned that sometimes the channels themselves, have to be modelled very problems mightar i sef romthemomentumfluxcondi t ion carefully. In order to obtain a high resolution of which is not specified, They can be overcome by omitting topography, rather small elements have to be used (Fig. 2), the convective terms in the set of elements next to the open and local tuning of friction and eddy viscosity parameters boundary L8. might become necessary for the calibration of the flow

field. On the tidal flats, as well as in the deep water region, rather big elements can be chosen (Fig. 2). The critical Courant number defining the time-step for explicit

DISCRETIZATION IN SPACE AND TIME computation is given from the discretization in the tidal The tidal wave equations are only weakly non-linear. A channels. linearization may be made by means of a Taylor series A discretization with strongly varying Courant expansion, numbers would make the use of an implicit formulation

rather attractive, at least with respect to computation {cjqi}ij=?jqi.j+~iqj.j-1;i~j(a+h)ij (8) time. The most economic scheme is obtained by the

hybrid formulation, but this does not give full la + hth,.. = - a,~(a + h) + (a + hXa + h),, (9) compatibility for the flux on the inter-element boundaries.

This disadvantage is avoided by the use of more expensive The quantities marked by the bar are assumed to be mixed models. known from the initial conditions of each computational The simulation of ebbing and flooding in boundary time-step, elements on tidal flats turns out to be rather complicated.

After introducing trial solutions in the sense of a finite The most simple method is that of switching offelements element approxiamtion: from further calculations when at least one nodal point is

dry. Elements are re-included when, by horizontal q~=~o~(x~, x2)~i ~ extrapolation from neighbouring nodes, an element

would be re-wetted. This strategy does conserve neither h=qG(x 1, x2)h , momentum nor mass, gives poor results, and is full of

spurious disturbances. The results improve tremendously where q~,, qG are shape functions and ~i, h the nodal values, if at least mass is conserved. One way of assuring this is to the linearized equations {3, 4) can be integrated in the calculate the water mass in an element which became

Adv. Water Resources, 1982, Volume 5, September 143


numbers of nodal points and bandwidths, explicit formulations are much cheaper.

The mentioned strategy gives no momentum conservation. In order to achieve this, the dynamics in the

\ ~ o J partly wetted elements have to be considered too. This - -,'~, \ \ - - " leads to a finite element grid with moving boundaries t 2.

~\ The realization would cause considerable problems for "---" implicit formulations when intertidal islands or

watersheds on tidal fiats must be taken into account. The grid has to be disconnected during falling water, and fitted

~ . , ~ . ~ \.X~X. together again at rising water. This can easily be simulated by explicit schemes.

"-. . " " / / ] EXPLICIT FINITE ELEMENT SCHEME

" , \ .- ,.I , ' Explicit finite element formulations can be obtained by / / ~ • discretizing the time derivative Z, t of equation (10), and

' " / ~ . ) [ subsequent lumping of the matrix N T M 2 by constructing t t special weighting functions ~ or by directly using specially-

~ I ! shaped elements1 o, making use of the characteristics of the [ \ ~ hyperbolic equations. This approach is chosen here. To t \ attain second-order accuracy in space and time, a

composite macro-element, composed of tetrahedral sub- elements, is used. The element is shown in Fig. 3.

- - ~ " " For space and time, linear shape functions are used for a representation of mass flow qi and water level h

qi=tP~(xl, x2, t)?ti~

" h=tp~(xl, x 2, t)h~

The formulation is coordinate invariant. It allows the use of the slip condition by means of boundary integrals. Moreover, the second-order term of the momentum equation can still be represented by taking the boundary integral (equation 7) over the boundary of the composite element, and neglecting the contributions from the internal sub-element boundaries.

The explicit formulation leads to an equation system for three unknown variables at each nodal point, if the linearization of the differential equation is performed at a sub-element level. The equations uncouple for non- boundary elements, when the flux and water-level at the centre of the composite element are taken for linearization. Practical calculations have shown that this simplification is acceptable in deep water regions, whereas on tidal flats the linearization on the sub-element level is more favourable. Moreover, it was found that the scheme is stable within the limitations of the Courant condition, if

b

Figure 2. Schematic estuary. (a) Topography." (b) ~t t

discretization !

partly dry during the last computational step. : f / Extrapolating the velocities then on the boundary to neighbouring and completely wetted elements with respect to time, and using these values for prescribing the mass flow into these elements represents the first step of an iterative procedure. During this it must be checked \ \ whether the calculated water-level on the boundary of the ~ \ wetted elements agrees with that of the remaining water ~ \ volume on the partly dry element. If not, a new estimate b on the mass flow has to be made~ t. The main drawback of Figure 3. Explicit finite element. (a) Composite finite this method lies in the computation time. For increasing element at boundary; (b) sub-element

144 Adv. Water Resources, 1982, Volume 5, September


S I M U L A T I O N O F T I D A L F L A T S

The purpose of these invesitgations was to find a simple ~ ~ ' ~ " ' - h 3 h - but computa t iona l ly economic strategy for the s imulat ion

of tidal flats which could be used for pilot studies. It was then improved to obta in a formulat ion of high accuracy

" i

l l t I I \1 ~ I l [ 1 I ' i

1 / t | I t 1 / 1 ~ " / / 1 1 l , / ; l / ' i I / j i ~l,

t t 1 1 t l ' i / .~,.. / • I

Figure 4. Interpolation at coastline 1 1 / / / ' i I i [

either the second-order viscosity term is used, or a 1 i ~ /'zJ , ' 1 \

numerical diffusion provided. Fo r this purpose, the ~ / / / l i l ~ \ formulat ion ~ 3 ....,

- i

Y. ZjA/S~ . - • . / 1 \ \

Z i = 0 t Z i + ( 1 - ~ ) ' (12) / I ~ \ ~ |

Z A / S j <" " ' " J

was used. The index i denotes the inner node, a n d j the ~ / " / T = 9 h surrounding nodes at distances Sj, with cor responding Figure 5. Computed flow field element areas Aj. The value of ~ could be reduced up to ct = 0.99. . ~ -

The use of very much simplified strategies for flooding - " and ebbing of elements leads to undesirable disturbances. . - - "" "" They were smoothed by t ime-averaging every 10 to 30 . - " ~" time-steps. ~ - " ~,,~ . . . . . .

N,,

R E P R E S E N T A T I O N O F F R I C T I O N "~ ~

The m o m e n t u m equat ions (3) show that the friction / ; " would tend towards infinity for decreasing water-depth. / -~", - This is physically wrong and can be avoided by " ~ ~ .........- in t roducing a condition: "~.~ g g rill~

I ~ / / / ). t \

(a+h)>H,m z*=(a+h) X/~jvjq, / / / /

(a+h)<n,m *t=--~,mX/VjVjq, /

A proper value of Hli m has to be found by calibration of ~ / / " \ the model. More flexibility for the representat ion of ~ / friction is obtained, when a linear app rox ima t ion is used ~ . . . within the range 0 ~< a + h ~< Ht,m- In these investigations, the tangent to the curve r=A/(a+h) at the water -depth H l i m w a s used. Fioure 6. Pathway of particle

A dr. Water Resources, 1982, Volume 5, September 145


$ ) - p , /

i,, 2-=-2 - - - V ,

" ' " i"i;t ! ' ' 'vt t \ ~ " ~ '

," '~ ." ,#, \~ .;7 ~ \ r~.~ ) ' i : ' \~~ ' f ~ ,, ', ,, '; ' ~ ¢ - ~ ' , ' \ 1 % ~ ,/ , L ,;

, ,1 . . ", , , \ " . . ~ % , . . - / . , . " - _ . . . . . '~ \ '

\ % . . " , : l f

- - - . . . . . . . . _ "13,, ' '\ ~ \ "

V, "/I/1/I/1/W////////////. co== ~ l i ~ ","IIIIIIIIIIIII17~-. . ., ,', \ '. ', ,' ,,--,

. ' ' I I / ' , \ \ ' , /

Figure 7. Topographic situation at Minsener Oog island

which could be used even for the study of local details on tidal flats. During each time-step the direction and thetidalflats, and in the channels. Numerical experiments distance of the moving coastline on the partly dry were performed for the system shown in Fig. 2, which is a elements has to be estimated. This can be done either by schematized detail taken out of Fig. 7. an extrapolation from the mean velocities within the

The above-mentioned switching on and offofelements, element, or again by calculating the mass balance for the which are partly dry, is a very simple though not partly dryelements. When doingthis, a fictitious negative successful strategy. Retaining all computational water depth has to be allowed for. By interpolation advantages which mainly follow from the fact that no between the water-level at dry and wetted nodes the new interpolations or iterations are made, this strategy can be position of the coastline is determined. improved by not switching off a whole element, but Both techniques work pretty well, although they are forcing only the flow components qi = 0 for all dry nodal not exact. They can be improved by an it3ration which, points. It is clear in this formulation that the water-level however, is felt to be unnecessary for an explicit gradient for ebbing water will tend in the limit to the slope computation with small time-steps. of the bottom. This will lead to a completely wrong Calculation with moving boundaries generates dynamic behaviour. An improvement is obtained when triangular as well as trapezoidal elements along the the mass conservation equation (4) is continuously solved coastline. This is illustrated by Fig. 4 in which flooding of for partly dry elements, even when the calculated water- an element is assumed. level is fictitious, and beyond the sea-bottom. The Yet this modification of the original grid does not lead fictitious values are used for determining the gradient of to any computational difficulty since in any event new the free surface for the momentum equation, composite elements can be set up. Thus disconnections at

The violation of the conservation principle cannot be intertidal islands and watersheds can easily be simulated avoided except by considering moving boundaries on the too.

146 Adv. Water Resources, 1982, Volume 5, September


/

/ -.., ,.,,-..,,-..~-...,~-... - . . .~ : , . .~- . . . -... - . . . . - . . .~- , .~- - .~ .

/ "--, "--,, x "-,, "-~ "--~"-.Q--~"-.~"-.C'.~"--~"~"--~"-~",~"~"~ ", /

; -; -75f,

" ' ' ' I i j l ' ' ' I . ' 1 v i i 7 i jlk \, , . . 1 i l I ' " ) i / i i / d l / 1 ~ ' , , ' , , , " l / i I i - ~ ' I / / / I / I I I l" I ~ ' " " J " / ' ~ A I ' ~

l / I l l i i l l / I i ¢ i , . : " - - i I ~ I i l 1 1 / J i l l i I I I ~ / i - - - - e l 7

, v i / l l l i l / l / I / " . " . . . . ~ ' t / 1 , I I . " . ' . ' I I / / i / I / i t . , ' - - - ~ ' ~ - " I ~

I l 7 7 il / I I i l i i l i t i i " " ' ~ " " " t l 1 / / I I I I i i t 1 i i t / t v e t I 1

,' 1 1 1 i 1 1 i t i i " ' ' " V / l 7 i / I I 1 1 I i i i i v , , " / / / t

I l l i , , " " ~ " " - I ~ i

Figure 8. FIow lield

This adjustment of the grid and the integration made for the time-history of the water-level at different performed over the flooded element part alone give high stations in the channels and deep water region, a accuracy at the price of a computation time some 2 0 ~ o difference in amplitude of only about 1% is observed. greater than the proposed simple strategy.

Numerical results are shown in Figs. 5 and 6 for the A P P L I C A T I O N system given by Fig. 2. The flow field (Fig. 5) was

computed with the more accurate strategy. A comparison Computation of currents on extended tidal flats must be with the simplified strategy by means of plotted velocities highly accurate, when further investigations concerning seems to show practically no differences. They become sediment transport are to be performed. This was done for eviden.t, however, by integrating the velocities of one an investigation on the island Minsener Oog in the particle over the time (Gig. 6). The positions finally German Bight. This island is an artificial island, formed obtained differ considerably. The velocities given by the from deposits from extended dredging of navigation simple strategy are much too high. It can be concluded channels. The topography is presented in Fig. 7. The from further computations that they are some 30% too island itself is fortified by an irregular system of groynes. high on the tidal fiats. The error decreases in deep water For the computation, a grid net with 764 nodes ,and regions, where it is only about 5%. When a comparison is 1396 elements was used. Only by this refinement local

Adv. Water Resources, 1982, Volume 5, September 147

T i d a l wave analysis: K . -P . H o l z and G. N i t s c h e

deta i ls could be r ep roduced sat isfactori ly. Th e t ime s tep of 4 Kawahara, M., Takeuchi, N. and Yoshida, T. Two step explicit the c o m p u t a t i o n was 15 sec. F i g u r e 8 shows the flow field finite element method for tsunami wave propagation analysis,

Int. J. Num. Meth. En O. 1978, 12, 331 d u r i n g r is ing water. I t is ev iden t tha t the n u m e r i c a l m o d e l 5 Wang, H. P. Multi-levelled finite element hydrology model of represents m a n y detai ls correct ly, even local ly observed block island sound, Finite Elements in Water Resources, (Eds. eddies. Gray, H., Pinder, G. F. and Brebbia, C. A.) Pentech Press,

London, 1977, pp. 4.69-4.93 C O N C L U S I O N 6 Herrling, B. Computation of shallow water waves with hybrid

finite elements, Adv. Water Resourc. 1978, 1,313 A mode l for the c o m p u t a t i o n of t idal wave p r o p a g a t i o n 7 Meissner, U. An explicit-implicit water-level model for tidal

computation of rivers, J. Comp. Meth. Appl. Mech. Eng. 1978, p. on t idal fiats is presented . A brief review is given, 221 c o m p a r i n g a d v a n t a g e s of explici t a n d impl ic i t 8 Leendertse, J. J. Aspects of a computational model for long- f o r m u l a t i o n s with respect to s i m u l a t i o n s t ra tegies of period water-wave propagation, Memorandum RM 5294 PR, in te r t ida l flats. N u m e r i c a l resul ts are g iven for b o t h a 1967, Rand Corporation, St. Monica simplified and an accura te s t ra tegy wi th m o v i n g 9 Gray, W. G. and Lynch, D. R. Time-stepping schemes for finite

element tidal model computations, Adv. Water Resourc. 1977, I, bounda r i e s . The la t ter is h ighly supe r io r to the fo rmer o n e 83 in the t idal flat region. 10 Withum, D., Holz, K.-P. and Meissner, U. Finite element

formulations for tidal wave analysis, J. Comp. Meth. Appl. Mech. En O. 1979, 17/18, 699

R E F E R E N C E S 11 Herrling, B. A finite element model for estuaries with inter-tidal fiats, ASCE Proc. 15th Conf. Coastal En O. 1976, p. 3396

1 Grotkop, G. Finite element analysis of long period water waves, 12 Lynch, D. R. and Gray, W. G. Finite element sumulation of J. Comp. Meth. Appl. Mech. Eno. 1973, 2, 147 shallow water problems with moving boundaries, Finite

2 Taylor, C. and Davis, J. M. Tidal propagation and dispersion in Elements in Water Resources, (Eds. Brebbia, C. A., Gray, W. and estuaries, Fluid Elements in Fluids, (Eds. Gallagher, R. H., Oden, Pinder, G.) Pentech Press, London, 1978, pp. 2.292.42 J. T., Taylor, C. and Zienkiewicz, O. C.) J. Wiley, New York, 12 Holz, K.-P. Explizite finite element formulierung zur berechnung 1975, pp. 95-118 langperiodischer flachwasserwellen, ZA1elTff 1978, 58, 277

3 Connor, J. J. and Wang, J. D. Finite element modelling of 13 Jamet, P. and Bonnerot, R. Numerical solution of the Eulerian hydrodynamic circulation, Numerical Methods in Fluid equationsofcompressible flow by a finite element method which Dynamics, (Eds. Brebbia, C. A. and Connor, J. J.) Pentech Press, follows the free boundary and the interfaces, J. Comp. Phys. 1975, London, 1974, pp. 355 367 18, 21

148 Adv . Water Resources , 1982, V o l u m e 5, S e p t e m b e r

tidal wave analysis for estuaries with intertidal flats

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