Depiction of friction damping in numerical flow modelling

D A V I D STEPHENSON

Department of Civil Engineering, University of the Witwatersrand, Johannesburg, 2001, South Africa

The use of numerical modelling for prediction of water levels and flows is introduced. Models are classified and simplifications appropriate to various configurations are suggested to minimise program- ming effort and computational capacity and time. The use of animated colour graphics is advocated for depiction of results, and for setting up the numerical scheme for solving the hydraulic dif- ferential equations. The effect of friction in damping waves and improving numerical stability is demonstrated with the assistance of a micro computer.

INTRODUCTION

Numerical modelling of water flow in closed conduits, channels and open waters is becoming a viable and econo- mical method of predicting water levels and flow dynamics. Advances in mathematical methods and development of generalised models have enabled the most sophisticated flow systems to be simulated by computer. Varied flow, unsteady flow and dispersion have been the subject of much research)

The more practical engineer with little advanced mathe- matical training may feel at a loss when such models are assessed. The interpretation of results is not as simple as on a laboratory model. Experience and a feel for the accept- ability of results which a hydraulic model affords is absent when it comes to numerical models. It is difficult to present computer model results and to demonstrate the mechanism of operations to clients, and even to many engineers.

Computer graphics have done a lot in alleviating this situation. Water surface profiles can be depicted at specific points in time in two or three dimensions by plotters. The resulting picture of grid lines on a water surface gives a 'still' at a selected time.

Graphics can be taken a step further with computer monitors. Water levels, pressures, flows or flood lines can be depicted graphically in animated form on a screen as computations proceed. Colour monitors offer an even more vivid picture and such animated flows can be produced on readily available micro computers.

The cost of micro computers is now well within the budget of the most modest engineering organisation. The facilities for flow modelling are therefore probably more available than they were when hydraulic models were the vogue. Colour graphics enables easy interpretation of model results, so the main obstacle remaining is the development of appropriate software. Many numerical models require excessive storage and computer time and are therefore more appropriate on main-frame computers. This article docu-

ments some of the simple types of numerical models suitable for micro computers.

It also attempts to overcome some of the more complex problems of numerical modelling of water systems - s u c h as numerical instability, accuracy, calibration and verifica- tion. It uses the graphic facility of colour animation to demonstrate these problems. By using a basic engineering approach it is shown how a model can be assembled and operated. A feeling for numerical correctness and stability can be obtained by following computations on a graphics screen.

Some of the proposed criteria for numerical stability are investigated and the effect of fluid friction is shown to assist, rather than deter, numerical stability if the correct numerical scheme is selected.

FREE SURFACE ONE-DIMENSIONAL MODELS

Even if flow is confined to one direction, the number of possible ways of describing the flow in equation form is large. The list therefore omits 'hydrological' models which attempt to reproduce complex catchments runoff by 'black box' methods (input--output relationships devoid of hydraulic realities), hydrograph methods and cascades of reservoirs. Some of these methods produce very acceptable results, but generally owing to their empirical nature they require extensive 'calibration'. Surface-subsurface flow and Muskingum routing methods, even though they have con- siderable hydraulic plausibility are also omitted, as are finite element methods. 2

The hydraulic models described for free surface flow are various forms of one-dimensional flow reproduction. They stem from the 'Saint Venant' equations:

Continuity;

aQ aA - - + - - = q ~ ( 1 ) ax at

Dynamics;

av av ay - - + v - - + g - - = g ( s o - s r ) - v q i / y (2) at ax ax

These equations omit vertical accelerations (e.g. ref. 3). The equations can also be written omitting the lateral inflow term qi and can be simplified for wide rectangular channels and some natural channels to:

aq ay - - + - - = 0 (3) ax at

av av ay - - + v - - + g - g ( S o - - S f ) (4) at ax ~xx -

0141-1195/86/040211-06 $2.00 © 1986 Computational Mechanics Publications Adv. Eng. Software, 1986, Vol. 8, No. 4 211

where Sf = v I v [ n2/R 4~3 according to the Manning resist- ance law, Q is flow rate,q is flow per unit width,y is water depth, v is velocity, g is gravity, A is cross-sectional area, x is distance in flow direction, t is time, So is bed slope, ~) is friction gradient, n is Manning's roughness, R is hydraulic radius AlP, P is wetted perimeter and for wide rectangular channels R is approximately equal to y.

The magnitudes of the various terms in the dynamic equation vary from one problem to another and the omis- sion of one or more of the terms is possible in many instances and this can lead to simplified solution of the equations.

APPROXIMATIONS TO SAINT VENANT EQUATIONS

A numer of simplificants of the previous equations were compared by Vieira. 4 He employed the following two para- meters to classify flow:

Fraud no. F - c v r ~ (5)

Kinematic wave no. k -gLX/aO/C4/~i2e/3 (6)

where

C = v /x /S fy = Chezy coefficient (7)

=yl/6/n for Manning flow in SI units (8)

L is the catchment length and 0 ~ tan0 ~-sin0 the bed slope, i e is the net transverse inflow rate per unit area.

(i) Kinematic f low:

For large k, the momentum equation reduces to

So = Sf (9)

(ii) Diffusion equation:

For large k and small F,

av vi e " - - - ( 1 0 ) g-~x - g ( S ° - S f ) - y

The author later adds the v av/ax term to the left-hand side as it improves backwatering and does not significantly affect solution.

(iii) Gravity waves:

For k very small

av av ay - - + v - - + g -~x = Vie/y (11) at ax

(iv) Zero depth gradient:

For Fo and k very small

av ay v - - + g - - = 0 (12)

ax ax

Vieira demonstrates graphically the applicability of the various equations in terms of k and F. The author employs a more pragmatic approach below. The schemes which emphasise friction namely (i) and (ii) are examined in detail as the friction term can be employed to enhance stability of simplified numerical schemes.

Diffusion equations The following sections investigate the friction equation

in more detail with a view to illustrating the effect friction has on numerical schemes freedom, and on damping. The

diffusion equation and the kinematic equation are highly dependent on the friction terms.

The first term in the dynamic equation (4) represents unsteady flow. It can be expected to be significant for oscillating waves and rapidly accelerating flow, e.g. dam break type problems. For flood hydrographs where flow increases over hours, the ' t ' term can generally be omitted. The dynamic equation then represents the space rate of change of terms in the Bernoulli equation, i.e.

a ( v2 + - * ) = - - S f (13)

where z is bed elevation above a datum. The equation therefore accurately reproduces backwater effects in steady sub-critical flow (or forewater effects in supercritical flow). The first equation (3), the continuity equation, can be employed to predict time variation in depths. In fact, numerical simulations, starting from an arbitrary initial estimate of flow depths along a channel can produce steady state water levels if the flow is held steady, Provided the numerical method of solution is carefully selected the method is an alternative to the standard step method (e.g. ref. 5) for backwater analysis.

Kinematic equations For long river reaches at relatively steep grades, back-

water effects may be insignificant, and the assumption of uniform flow is adequate. The resulting simplification has a considerable effect on the ease of solution of the equations.

The dynamic equation reduces to

S f = S o (14)

or if the Manning equation is used,

v = S~/2R2/a/n (15)

For wide rectangular channels and overland flow, where the kinematic equations are particularly relevant, R is approximately equal to flow depth y , and using the equa- tion q = vy one can reduce the continuity equation to one involving one dependent variable q (or y) and two inde- pendent variables, x and t. Numerical solution on an x - t grid is relatively simple, and there are many situations where analytical solutions are possible, for time of concen- tration, for hydrograph shape and for flow depth profiles.

The fact that the friction gradient alone provides a rela- tionship between water depth and flow velocity means that upstream or downstream effects can be neglected when it comes to establishing flow rates along a channel. The change in water depth over a time interval is yielded by the con- tinuity equation.

NUMERICAL SCHEMES

Solution of the Saint Venant equation has been discussed by various workers (e.g. ref. 1). Although the characteristics method of solution of the Saint Venant equation was popu- lar before widespread use of computers, finite difference solutions have gained greater popularity and are reputed to work in cases where the characteristic method does not, e.g. supercritical flow and where rectangular grids are preferred.

Finite difference schemes are generally based on a four- point or 'box' grid (Fig. 1). The derivatives in the equations to be solved are approximated by some weighted combina. tion of values at one or more of the four points around the box.

212 Adv. Eng. Software, 1986, I1ol. 8, No. 4

n+l

rl

Figure 1.

j j+l

x-t grid for finite differences

X ) V h,

Thus

_ _ = ( f n + l n Of ~u#+, --f i+,)-I- (1 -- 4b)(j~ n+'-J~n) at At

(16)

[ ~,n+l ¢n+la ~--f=OtLi+l--J/ "~+(1- -O)( f ln l - - f /n) (17)

ax Ax

If ¢ = 1 we have a backward difference scheme, of interest to advective systems. If ~ = 0 we have a forward difference scheme, especially of relevance in backwater calculation. The case 4~ = } is referred to as Preissmann's four-point scheme 6 for 0 = 0.5. If 0 = 0 we have an explicit scheme. If 0 = 1 we have an implicit scheme.

Although the case of 0 = 0 is easiest to solve, since it uses values of ~f/ax at defined grid points, it has a number of disadvantages. It is unstable in many cases, producing oscillations in the answers which can rapidly escalate to yield unrealistic answers. Limits have to be placed on the time increment At or some form of numerical damping is required.

Solution o f numerical equations In general the implicit forms of the equations are pre-

ferred for accuracy and stabilityfl Even the 0 = ~ case can be unstable for rapidly oscillating flow and Amein and Chu 8 indicate that the finite difference equations are easier to handle for 0 = 1. A problem with implicit schemes is that the number of simultaneous equations and correspond- ing number of unknowns increases in proportion to the number of points along the conduits. Since the general method of solution of simultaneous equations involves a matrix whose size increases as the square of the number of variables, computer capacity could become a problem. The effort in iterative solutions will increase in proportion to the number of points to the power of 3. In general inter- locked grids are preferred where y and v or q are computed at successive points in space. 9 This means both forward and backward differences in one dependent variable (y or v) can influence the rate of change of the other.

Methods of solution of the resulting set of simultaneous non-linear equations include generalised Newton iteration method, 7 iterative matrix methods such as Gauss-Seidel successive relaxation, and double sweep methods] The latter relies on the sparseness of the matrix to reduce computations.

Even with double sweep methods the volume of com- puter storage and complexity of programming can deter

the inexperienced modeller, and may limit the type of problem which can be handled by micro computers.

Explicit solutions to the finite difference equation do not require simultaneous equations since each new point is dependent only on previously computed values at neigh- bouring points. Liggett and Cunge l° investigated implicit methods in some detail and suggest the following methods:

(i) Leap-frog method:

Of /?+1 f#n-1 08)

at 2At

O f = gn+, -- t)n_, (19)

~x 2Ax

(ii) Lax scheme:

- - - ( 2 0 )

at At

as_ -4"_, (21) ax 2Ax

Both methods require Ax/At = (gy)a/2 for maximum accur- acy. Chaudhry Ix suggests the following scheme:

(iii) Diffusive scheme:

~__f=/~n+, ~n (22)

at At

a f = fflq -- fin-* (23) ax 2Ax

Solution of kinematic equations A number of numerical schemes for kinematic equations

was investigated by Brakensiek] 2 He investigated an explicit, an implicit and a four-point scheme. He recommended the four-point scheme for accuracy and stability and as it is particularly easy to use for kinematic flow.

Starting from known conditions at points (], n) and Q' + 1, n) (see Fig. 1) a single pass is sufficient to establish y and v (and q = vy).

The kinematic equations used are those for overland or wide stream flow:

aq ay - - + - - = i e (24) ax at

where

q =ocy m (m = 5/3 for Manning flow and ~=~/S/n) (25)

The previous equations can either be solved in two steps (the f'trst one fo ry and this is substituted in the second one for q) or the second equation can be substituted into the first leaving only one equation to solve:

ql/m-x ~q aq + _ _ ie (26) ax ma 1/m ~t

The four-point implicit scheme uses the following forms of finite difference equations:

aq qin+a--q? (27) ~x Ax

Adv. Eng. Software, 1986, VoL 8, No. 4 213

aq u?# + q?+' " " - - q j+ l - - q /

3t 2At (28)

qllm-I (q~+i)2ts + (q]n)2/s = (29)

2

The latter equation is an explicit approximation which is admissible since 2/3 is less than 1, i.e. the scheme is not very sensitive to qUm-1. This simplification yields a series of linear equations for all ]. Starting at the upstream known boundary condition the scheme proceeds downstream evaluating q~.+~ knowing q~÷l, q~ and q~÷a.

~ n + l x , -_n ~-2/5_{._ r_n~-215 • q / ~ t q l + l ) t q # ) q t + l - - q l + l - - q j

n n n

le +---A--XX )- - (-]~3)i~ 3"~ 2At

q~**~ - 1 (q~+i)-=/s + (q~)-2ls - - 4 Ax (20/3) Ata als

(30)

Constant/n/des 13 indicated that an explicit finite difference scheme based on backward differences produced accurate, stable results provided the Courant criterion was sat/sifted:

At/Ax <. l/(dx/dt) (31)

where

dx /dt = moty m-1 ' (32)

i.e.

of D."+ ' -D. " - ( 3 3 )

3t At

Of ~n--~n_ 1 (34)

~x Ax

This scheme is called the backward centred explicit scheme. It is interesting to note that Constant/n/des found the

diffusive scheme suggested by Chaudhry n for dynamic flow, unstable for kinematic flow. This is because only upstream flow can effect kinematic flow depth. The same is the case for the diffusion equations, since it is only the energy gradient which is modified to account for backwater effects downstream.

S T A B I L I T Y O F N U M E R I C A L S C H E M E S

For a numerical scheme to be convergent, Lax stated that it is necessary and sufficient that it be stable. Stability thus means that successive application of the finite difference equation shall not introduce an accumulative error. The accepted criterion for stability of explicit schemes is that of Courant, Friedrichs and Lewy, 14 called the Courant criterion:

At/Ax <. 1/I c + v l (35)

where

c = (gyo) in (36)

The criterion is really only applicable to linear systems and the non-linear friction term renders it of limited applic- ability. Huang and Song Is modified the criterion and did an extensive series of tests for the dynamic equations with head loss.

ALLOWANCE FOR FRICTION

Because the friction term in the flow equation is non-linear it makes solution of implicit type equations more difficult than without the friction term. A number of methods of accounting for the friction term was described by Cunge et al.: 1 The friction gradient is assumed to be of the form

S f = Q I Q I / K 2 (37)

where

K =AR2/a/n (Manning, S.I• units) (38)

R =AlP (39)

If an explicit scheme is not acceptable, for instance if Sf is large compared with ay/ax, then some form of averag- ing of Sf in time is required. Cunge et al. suggest taking the average Q over the distance interval and squaring that, rather than the average of the squares of the Q's over the interval, i.e.:

2 K 2 _ 0 Q n+l

+'- + (40)

4 L\Ki] ,,/,. #+lJ

An alternative which produces a linear equation and also yields the correct sign of Q was suggested by Stephenson 16 for closed conduits:

o [QT+,I (r"'a +,","+' , ' , " i/(/,;s."+,) QIQI /K2= 4 ~ ix rei+l ~/+l

+ - - 1--0

4 [(Q~/Kfl) 2 + (Q~÷I/Kfl÷a)2] (41)

Stability with friction Strelkoff a7 indicates that the direct explicit scheme is

inherently unstable. He indicates the Lax type scheme should satisfy the Courant criterion• For implicit schemes he suggests that to ensure stability with friction:

Ko At < (42)

Agv -o where

Ko = AoCX/-Ro = Qo/,~o (43)

u

.'. a t < - - (44) gs l

Wylie Is suggested that for a simple linear explicit system for open channels that for stability:

At <<. (Axle) (1 --gSfAt/2 V) in (45)

Even this does not guarantee stability according to Wylie. A comparison of various criteria is made below. An 8 km

length of wide rectangular channel in 1000 m lengths at a slope of 1/1000, normal depth 2 m and Manning n = 0.04 is used as a basis. The diffusion equation is used to predict the backwater profile of a 3 m high weir downstream. Inflow is kept steady with top depths set at 3 m.

214 Adv. Eng. Soflware, 1986, VoL 8, No. 4

Strelkoff's criterion gives:

V At < - -

gsr 22/a/1000 I/2

0.04 × 9.8/1000

= 128s

Wylie's criterion suggests:

At <<. (Ax /c ) (1 --gS[At/2V) 1/2

1000 9.8At x 0.04 × 10001/2 - 1

(9.8 × 2) 1/2 1000 x 2 × 22/3

A t < 14.6s

Courant criterion for Saint Venant Equations:

At <<. Ax/I c + v 1

= 1000/(X/'9-~.8 × 2 + 22/a/10001/2 × 0.04)

= 1000/(4.43 + 1.25)

= 176s

Courant criterion for kinematic flow:

At < Ax / (dx /d t )

= A x / a m y rn-I

1/2

Figure 4. BCE scheme after one iteration, dt = 400 s

Figure 5. BCE scheme after two iterations, dt = 400 s

Figure 2. lnitial conditions down 8kin channel

Figure 3. [Cater levels after ten iterations o f backward centred explicit (BCE) scheme, d t = 300 s

= lO00/(V~o/n) my rn-1

: 1 0 0 0 / ( ~ 1 0 0 0 0.04) (5/3)22/3

= 478s

From experiments, backward centred explicit scheme (BCE scheme):

At ~< 300 s(n = 0.04)

~< 150s(n = 0.02)

Diffusion scheme:

At ~< 300 s (n = 0.04), <~ 150 s (n = 0.02)

It is concluded the Courant criterion is the most relevant for the centred explicit scheme even with the friction pre- dominant type of equations. Experiments are, however, required to improve the selection of the best time step.

G R A P H I C A L C O M P A R I S O N

Depiction of the water surface variations on a screen illus- trates rapidly whether a scheme is stable or not. Various schemes were programmed on a micro computer and portrayal of water levels illustrates 'movement of waves', smoothing and enlargement of instabilities. 'Stills' at various stages can be compared.

Figure 2 shows the initial conditions used for Figs. 3-9. The backward centred explicit scheme produced stable

Adv. Eng. Software, 1986, Vol. 8, No. 4 215

Figure 6. Diffusion scheme after one iteration, dt = 200 s

Figure 7. Diffusion scheme after two iterations, d t = 200 s

levels for a time increment o f 300 s and had practically converged to steady state condit ions after ten passes or i terations at each point (Fig. 3). For a t ime increment of 400 s instabilities manifested and magnified (Figs. 4-5) after first and second passes).

Figures 6-9 show that although the diffusion scheme produces stable results for a t ime increment of 200 s the water levels are not realistically depicted, i.e. after three passes the wave front due to the higher flow from the upstream end is numerically diffused down the channel. After ten iterations the water levels had still not settled down to their steady state values.

CONCLUSION

For friction predominant type equations (kinematic and diffusion equations) finite difference equations as follows give the most stable and sensible solutions.

S] n-1 = [ % _ 1 + y~_-~ + (v~5~)2/2g)

-- (zj+l + y~7,~ + (v~,S~)2/2g)]/(xj +1 -- x j_l) (46)

~ t y? ' ' 2

n

i n-l~5/3[~n-1 Yl-11 ~o1_ 1 )I'2I/(X 1 - -Xj_I) 1471

= ( s 7 - , ) , % , }, )2,3/.

where S is negative energy gradient, z is bed level, r is flow depth, v is flow veloci ty ,g is gravi ty ,x is distance, t is time, n is Manning's number. These demonstrations apply to a wide rectangular channel.

The finite difference equations were selected after com- parison of numerous schemes and following the numerical solutions by graphically depicting water levels down a channel on a micro-computer monitor. Damping, smooth- ing and realistic behaviour were examined. The Courant criterion is found to be the best indicator of stability although refinement must be done with the actual model. This is not to say that the proposed scheme applies to other cases and in particular gravity waves and dynamic systems.

The possibility of analysing steady flow situations using simulation presents itself. If the system will converge from an arbitrarily assumed water level to steady state levels in a few iterations, it can be used in place of the standard step method for backwater calculations.

REFERENCES

1 Cunge, J. A., Holly, F. M. and Verwey, A. Practical Aspects of Computational River Hydraulics, Pitmans, Boston, 420 pp., 1980

2 Connor, J. J. and Brebbia, C. A. Finite Element Techniques/'or Fluid Flow, Newnes-Butterworths, 310 pp., 1976

3 Dressier, R. F. and Yevjevich, V. Hydraulic resistance terms modified for the Dressler curved flow equations. J. Hydr. Res. 1984, 22 (3), 145-156

4 Vieira, J. H. D. Conditions governing the use of approximations for the Saint-Venant equations for shallow surface water flow, J. Hydrol. 1983, 60, 43-58

5 Chow, V. T. Open-Channel Hydraulics, McGraw-Hill, New York, 680 pp., 1959

6 Preissmann, A. Propagation des intumescences dans les canaux et rivieres, First Congress of the French Association for Com- putation, Grenoble, 1961

7 Amein, M. and Fang, C. S. Implicit flood routing in natural channels, Proc. ASCE, J. Hydr. Div. 1970, 96 (HY12), 2482- 2500

8 Amein, M. and Chu, H. Implicit numerical modelling of unsteady flows, Proc. ASCE, J. Hydr. Div. 1975, 10l (HY6), 717-731

9 Abbott, M. B. and Ionescu, F. On the numerical computation of nearly horizontal flows, J, Hydr. Res. 1967, 5 (2), 97-117

10 Liggett, J. A. and Cunge, J. A. Numerical methods of solution of the unsteady flow equations. Chapter 4 in Unsteady Flow in Open Channels, Water Ress. Publiens, Fort Collins, 1975

11 Chaudhry, M. H. Applied Hydraulic Transients, Van Nostrand Reinhold, 503 pp., 1979

12 Brakensiek, D. L. Kinematic flood routing, Trans. Am. Soc. Agric. Engrs 1967, 10 (3), 340-343

13 Constantinides, C. A. Numerical techniques for a 2-dimensional kinematic overland flow model, Water S.A. 19810 7 (4), 234- 248

14 Courant, R., Friedrichs, K. O. and Lewy, H. On the partial difference equations of mathematical physies, Math. Ann. 1928, 100, 32 (In German)

15 Huang, J. and Song, C. S. Stability of dynamic flood routing schemes, Proc. ASCE, J. Hydraulic Engg. 1985, 111 (12), 1497-1505

16 Stephenson. Pipeflow Analysis, Elsevier, Amsterdam, 1984 17 Strelkoff, T. Numerical solution of Saint-Venant equations,

Proc. ASCE, J. Hydr. Div. 1970, 96 (HY1), 223-252 18 Wylie, E. B. Unsteady free-surface flow computations, Proc.

ASCE, J. Hydr. Div. 1970, 96 (HYll), 2241-2251

216 Adv. Eng. Software, 1986, Vol. 8, No. 4