ELSEVIER

Advances in Water Resources, Vol. 21, No 1, pp. l-9, 1998 0 1997 Elsevier science Ltd

All rights reserved. printed in Great Britain PII: SO309-1708(96)00009-7 0309-1708/98/$17.00+0.00

Simulation of dam-break flow with grid adaptation

Mizanur Rahman & M. Hanif Chaudhry Civil & Environmental Engineering Department, Washington State University, Pullman, WA 99164, USA

(Received 1 March 1995)

The unsteady free surface flow caused by sudden collapse of a dam produces discontinuities in the flow variables. As the flow surges downstream, it forms a moving bore front with steep gradients of water height and velocity. In the numerical simulation of this flow, proper grid distribution can play a crucial part in the prediction and resolution of the solutions. The use of presently available numerical schemes to solve this problem on a uniform course grid system fails to resolve the characteristic flow features and hence do a poor job in simulating this flow. In this paper, an adaptive grid which adjusts itself as the solution evolves is used for a better resolution of the flow properties. Rai and Anderson’s” method is used to determine the grid speed; however, a difirent partial differential equation based on the conservative principle of grid arc lengths for clustering grids in one-dimensional flow is used along with the St. Venant equations to numerically simulate the flow. Both the subcritical and the supercritical flows under extreme boundary conditions are solved using this technique. With a specified number of grid points, this provides better quality solutions as compared to those obtained with uniformly distributed grids. 0 1997 Elsevier Science Ltd. All rights reserved

Key words: dam break flow, sub-super critical flow, bore front, adaptive grid.

INTRODUCTION

Nonlinear partial differential equations can be used to describe flows with complex flow features and their interactions with each other. Dam-break flow, hydraulic jump and flash floods produce steep gradients of flow properties and can be described by the St. Venant equations.’ The formation and propagation of any such flow features present di%culties for many numerical methods while a combination of more than one can complicate the problem further. This is because these flow phenomena usually occur in different length scales and a grid system designed to resolve a particular physical process cannot resolve the others with smaller scales. Moreover, the discretization error is directly related to grid size in the physical domain. Under such conditions, an adaptive or moving grid is indispensible to accurately and efficiently describe the flow processes. A dynamic grid system can continuously redistribute to accommodate the length scales and their rapid transition with time. With a specified number of points, they should be optimally distributed in order to obtain the best possible solution.

Early works on the numerical simulation of dam- break flow involve using the method of characteristics,2J sometimes with a shock fitting technique.4 The recent efforts, however, concentrate more on using the finite difference/element methods with shock capturing tech- nique. In contrast to shock fitting,4 where the bore is treated separately by shock relations and then the solutions across it are patched together, in shock capturing,5-7 the bore is formed automatically as a part of the solutions. Since most of the time, one- dimensional St. Venant equations are solved on a uniform grid, the bore spreads over several mesh points in a shock capturing technique. This severely limits the ability of finite difference/element methods to produce a better resolution of the dam-break flow solution, especially in the region of sharp gradients. Unless a very large number of grid points are used, a uniform grid system fails to resolve these sharp variations of flow properties. Due to the unsteady nature, local clustering in a particular region is not useful. In supercritical flow, the problem becomes even more complicated because of the boundary conditions and because most of the presently available finite

2 M. Rahman, M. Hanif Chaudhry

difference methods fail to produce a distinct bore - the most important characteristic feature of the dam-break flow. Considering these, the authors believe a dynamic grid adaptation to be an ideal candidate to remedy this problem and hence is used here to simulate the dam- break flow. To the authors’ knowledge, grid adaptation technique has not been used for this particular problem before.

A number of different approaches can be used to construct the solution adaptive grids. One of the earliest approach was to generate the grid motion by taking the time derivative of the governing differential equations of the coordinate mapping such as elliptic equations as developed by Thompson et al.’ Hindman et al9 used this technique to solve two-dimensional time dependent Euler equation. However, he used the boundary point motion rather than local solution for the interior grid adaptation. Dwyer et al.” and Olsen” used techniques based on an algebraic relation between the grid sixes and derivatives of flow variables to cluster grids. In contrast, Rai and AndersonI proposed to simultaneously solve a coupled system of the physical equation and the grid equation to determine grid point location. This is a powerful approach to adapt grid with evolving solution in an optimal way. It allows automatic control over orthogon- ality, smoothness and stretching. However, when the equations are coupled, a simultaneous solution by an implicit scheme requires more memory and time and hence becomes very expensive. Moreover, the discrete system is more likely to be ill-conditioned, due to mesh .tangling (authors experienced this problem) among other reasons. Often, it becomes so severe that time integration can not be carried out. In the present research, this particular problem was encountered, and hence a different approach was sought. According to Rai and Anderson,12 grid locations are directly calculated from the grid speed equation. In contrast, the conservative form of the partial differential equation governing the arc lengthi between two consecutive nodes was used. The governing equations were decoupled from the grid equation and were solved independently by the MacCormack18 explicit finite difference method.

In this paper, the St. Venant equations describing flows produced by dam break are solved on a fixed nonuniform grid. The flow solutions are then used in turn to solve the grid equation. The flow variables at new grid locations are determined which are then used as the known conditions for the next time step. To mention some, Bieterman and Babuska,14 Sanz-Serna and Christie” and Furzelandi6 are among those who have used this concept.

GOVERNING EQUATION

St. Venant equations are the differential form of the conservation of mass and momentum principles in a

fluid flow under the assumptions of hydrostatic pressure distribution and small bottom slope of the channel. For one dimensional flow, they constitute a set of hyperbolic partial differential equation and are represented by’

Q,+F,+T=O (1) where

Q = [h,uhlT (4

F = [uh, u2h + ;gh21T (3)

T = [0, -gh(S, - A#. (4)

In the above equations u is the flow velocity, h is the flow depth, S, is the bottom slope, Sf is the slope of energy grade line, g is the gravitational constant and t is the time.

The energy grade line slope, S,, is given by,

where M,, is the Manning’s roughness coefficient, R is the hydraulic radius and C, is a dimensional constant (1 for SI unit and 1.49 for customary English unit).

For a wide rectangular channel, the above equations can be written as

U,+E,+S=O

where

(6)

U = [h, ulT (7)

E = [uh,iu2 +ghlT (8)

s = [O, -g(S, - S,)]‘. (9)

For grid adaptation calculations, we use the differ- ential form of the ‘Geometric Conservation Law’ (GCL).i2 In one-dimension this equation is represented by

Jt + tit& = 0 (10) where

J is the Jacobian of transformation and < is the horizontal coordinate in the computational plane.

A detailed derivation of eqn (10) is given in Ref. 12 and hence is not included here to conserve space. Equation (10) is similar to the differential statement of flow conservation laws, eqn (1) or (6) except that it governs the time variation of Jacobian J (arc length in 1-D). If this equation is solved along with the governing PDE for fluid flow using the same numerical scheme, it produces a self-consistent solution for the effective arc length in 1-D corresponding to each grid point.

One of the essential ingredients in grid adaptation is the driving mechanism to control the grid motion. The driving mechanism is in turn activated and controlled by

Simulation of dam-break jlow 3

some error indicators, e.g., local truncation error, gradient of dependent variables, etc. However, during the formulation one must be careful so that it concentrates grids in the region of large gradients of flow properties but at the same time does not deprive regions with no/smaller gradient. The distribution also must retain sufficient smoothness and orthogonality; otherwise, the truncation error will increase owing to increased skewness.” This means that points must not move independently, but rather should be coupled by some means to its neighbors. Also the grid points must not move too far or too fast else instability or oscillations may occur.

In the present work, the grid speed is used as the driving mechanism after Rai and Anderson.‘* Suppose let is the error indicator and lelavg is the average of error over the domain. Then the point where lel is greater than lelavg attracts other points and where le( is less than

MacCormack explicit predictor-corrector finite differ- ence scheme.18 This scheme is second-order accurate in time and space. It is widely used in computational fluid dynamics because it is straightforward, easy to program and proven to handle wide variety of problems. The equations to be solved are first discretized in space and then numerically integrated in time. For example, if the flow field variables are known or specified over the region at any particular instant of time, say t,, the spatial gradient of dE/dx can be calculated by dis- cretizing over the nodes using finite differencing. Once this is found, the time derivative in eqn (6) can be computed and used to advance the flow field variables to the new time level tn+l. In the predictor step, eqn (6) is expressed as

!g= _(El----Y-l> _syj

I4 avg repels the neighboring points. This analogy of where the spatial derivative aE/ax is calculated by usilg ‘attraction-repulsion’ is translated into a mathematical backward differencing. The intermediate solution U”+’ form and is represented lin the form of grid speed, &, as is thus given by

(12) i=2,3 )...... (N-1)

where K is a constant, iV is the number of grid points and r is the distance between two grid points in the computational domain. Grid oscillation and excessive clustering of grids can be prevented by limiting the change of Jacobian at each time step to control the grid speed. If J’ is the initial Jacobian and Jk is the Jacobian at kth time step then

Jf/J,! if Jr/J; > 1 J,!/Jf if Jf/Ji < 1’

(13)

If Ri exceeds a preset :maximum value, R,,,, the grid speeds are damped exponentially. Let,

Ri_ = max(Ri), i =: 1,2,3, . . . N

then

(15)

where p is a constant. This provides a strong control over grid points

motion. Since this formulation is not problem depen- dent, it is frequently preferred over others.

NUMERICAL SOLUTION

The numerical approach used to solve the governing equation and the grid equations is based on the

(17)

These proximate flow variables are then utilized to find En+’ and Sn+l. In the corrector step, eqn (6) is expressed as

Note that the spatial derivatives are now discretized using forward differencing and the predicted values from eqn (17). The final solution at the end of time level t,,+] is given by

(19)

This procedure can be used for both time accurate and steady state solutions. The step size may be varied subject to the stability restriction given by Courant- Friedrichs-Lewy (CFL) criterion.” According to this, the numerical domain of disturbance has to be at least as big as the physical domain of influence. Mathematically this is given as

Ax/At 2 max(lul+ m). (20)

To solve the grid equation, the same numerical procedure discussed above was used.

The initial conditions for the dam-break flow are given by specifying the depth and velocity upstream and downstream of the dam as shown in Fig. 1. The flow can be subcritical or supercritical depending on the ratio &,/hi). During the computation, both the upstream and downstream boundary conditions remain unchanged for the cases studies.

As with most of the higher-order numerical schemes,

4 44. Rahman, M. Hanif Chaudhry

1025 m

Fig. 1. Definition sketch.

MacCormack scheme produces high-frequency oscilla- tions in the numerical solution near steep gradients. To dampen these oscillations, some kind of artificial viscosity needs to be added. Jameson et aL2’ developed a procedure that can effectively suppress the high- frequency oscillations. This is given by

Ci = +i+l/2[“i+l - uil - 4i-1/2[“i - ui-ll

where

4i+1/2 = Bmax [@i+l j tiil

(21)

15

OO 500 1000 1500 2(

DISTANCE (hi)

D

‘,I \ 2 Tii = 41.n

OO 500 1000 1500 2000

DISTANCE (hl)

Here B is used to regulate the amount of dissipation, Ci is calculated both in the predictor and corrector steps and added to the flow variable, Vi.

RESULTS AND DISCUSSION

A wide, horizontal, rectangular and frictionless channel shown in Fig. 1 is used to simulate the dam-break flow using dynamic grid adaptation. At time t = O+, the dam is removed instantly. This generates an unsteady flow with a bore propagating down-stream. The flow pattern depends on whether it is subcritical or supercritical, i.e., on the ratio (ho/hi). For the present study, a channel 2 km long is used and discretized into 80 grid points for numerical calculations. All the flow cases experimented are run for 60 s.

The first case examined is a subscritical flow with ho/h1 = O-5. Figure 2 shows the evolution of fluid flow with time. Along with the governing equations, grid equation is solved every time step. This gives better results as compared to those when solved after every few time steps. The driving force (g) for grid adaptation is

1 --

15

s Time = 35.87

OO 500 1000 1500 2000

DISTANCE (M)

15

z^ Time = 60.00

OO 500 1000 1500 2000 4

DISTANCE (M)

Fig. 2. Evolution of dam-break flow with time.

Simulation of dam-break flow

t Time = 60

0 500 1000 1500 2000

Distance (m)

Fig. 3. Comparison of solutions at subcritical flow (h - O/h, = 0.5).

defined in terms of local and average first difference of h (depth), i.e., g = jhr/&I - &/&lavg. This particular form of the driving force is used because of the fact that the truncation error of a first-order differential equation can be approximated by a first derivative in physical space.2’ However, this works well in compres- sing grids in the region of steep gradient and dispersing them otherwise. The grid follows the solution as it evolves. Other choices of error indicator fail to produce similar or better effects.

Figures 3 and 4 compare the adaptive grid solution with other non-adaptive solutions of the same problem. Adaptation technique p:roduces sharper bore front and

the depression wave by concentrating points around them and thereby prevents spreading them over a large number of mesh points unlike other solution techniques.

Figure 5 shows the variation of water depth (h) along the channel for a supercritical flow (h,/h, = 0.05). The grids are dynamically adapted and shown in the same figure. A comparison of this solution with others generated by different non-adaptive schemes is shown in Fig. 6. All the numerical solutions overestimate the bore height. Fennema and Chaudhry’ reported that in Gabutti scheme the bore speed at this low ratio of (ho/hi) becomes too slow and does not propagate at all until some dissipation is added. The iterative version of

15

!- ---- Adaptive Grid

-_--_ Analytical ----- Euler, Non-Iterative

- Euler, Iterative - -..- - Non-Adaptive Grid

Time = 60

1000

Distance (m)

Fig. 4. Comparison of solutions at subcritical flow (ho/h1 = 0.5).

6 M. Rahrnan, M. Hanif Chaudhry

Time = 60.00

1,. I .

500 1000 1500 2000 Distance (m)

IL 0

Fig. 5. Adaptive flow solutions at supercritical flow (ho/h, = 0.05).

Beam and Warming reduces the bore speed slightly. In contrast, the *present numerical scheme with grid adaptation produces correct bore speed and much sharper bore front.

Figure 7 shows the adaptive grid solution for depth ratio of ho/h, = 0.004. This is a supercritical flow case under extreme conditions and some of the standard numerical schemes fail’ to simulate such flows. The numerical solution obtained by Euler Implicit, Trape- zoidal and Adaptive Grid technique are plotted in Fig. 8 along with the analytical solution for comparison. The bore speeds for all three schemes are slow, yet the present dynamic grid technique achieved the fastest speed among them. Moreover, Euler implicit and trapezoidal methods fail to generate any well defined bore front, while the grid adaptation technique does a much better job in this regard.

In the present numerical experiment, the conservative

form of the St. Venant equations with u and uh as the independent variables was used except for the subcritical flow where u and h were used as the independent variables. This study shows that when equations are solved in terms of u and h, the bore speed becomes very slow and its height is overestimated, especially in supercritical flow. A comparison is shown in Fig. 9 for ho/h1 = 0.05. Similar results are obtained with flow for ho/h1 = 0,004.

The above results show that the adaptive procedure calculates I, (grid velocity) and the metrics & accurately and distributes the grid points when and where necessary during the evolution of flow field in the dam-break flow. In this experiment, we use & over & to establish the mapping < = E(x, t) because this technique is easy to extend to multi-dimensional problems. The grid speed & is directly proportional to excess error, i.e., I4 - l&s and inversely proportional to distance r raised to power n. In this experiment, the gradient of depth (h) is used as an error indicator (e); this gave better results than the gradients of u or uh. The performance of the grid adaptation technique is found to depend on the proper choice of values of the parameters n, K, R,, and (&)max and deserves a critical examination. Many researchers have reported problems with integrating the differential equations for the mesh velocities. It is important to numerically satisfy the geometric conservative law (GCL) equation; otherwise it may result in mesh trajectory leaving the domain, crossing each other or oscillating wildly from time step to time step.22 However, this may also occur when the solution of the governing differential equations and the GCL do not advance in time at the right pace. For example, we need to specify a maximum value of & during the calculation. If this is assigned too large a

Grid Adaptive -_-_- Analytical -.-.- - - Euler, Non-Iterative ---- Gabutti - .-. - .- Euler, Iterative

0

Time = 60

Distance (m)

Fig. 6. Comparison of solutions at supercritical flow (ha/hi = 0.05).

Simulation of dam-break flow 7

‘5l--- I

Time=60.00

ho -cr

f 3 \ $5

Lh OO 500 1000 1500 2000

Fig. 7. Adaptive flow solutions at supercritical flow (h - Cl//Q = 0.004).

value, grids may be pushed too far either to the right or to the left, thereby missing either the depression wave or the bore front, may caus’e mesh tangling and sometimes grid oscillation. For best performance, calculations are started with a small value of (tJrnax and then gradually increased with each iteration. The reason behind this is that as grids are clustered together, the global value of (At),, becomes very s’mall and so does the driving force, g. Under these circumstances, grids would not move at all until we raise the value of K in eqn (12) and one of the ways to do so is to raise the value of (&)max.

The value of exponent n has influence on the extent of grid compression at any location. Equation (12) which contains n can be compared with the gravitational force equation. As the value of n increases, less and less number of grid points ertperience the influence. On the contrary, a small value of n causes more and more points to come under the influence resulting in a much

smoother point motion and hence their clustering. An optimum value of n = 0.75 gives the best result. There is no noticeable clustering when n value goes beyond 1.5. The value of R,,, in eqn (15) plays a very important role in determining the performance of dynamic grid. For example, the inaccuracy in the error estimate sometimes leads to excessive stretching or compression of grids. Equation (14) is used to prevent this situation with a R max value of 1.9. This is found by trial and error procedure.

Grid adaptation technique increases the number of iterations (45% more over that of nonadaptive techni- que) to get the solution. However, for 1-D problem this is very minor considering the computer time it takes. Figure 10 shows the error distribution for different grid sizes. As the number of grids increases, the error decrease, but it decreases at a faster rate for adaptive grids.

CONCLUSIONS

In dam-break flow, a very complex unsteady flow field evolves, and, due to its very nature, it forms an important class of flow in hydraulics. Hence, it is important to find a suitable numerical scheme that can efficiently and accurately simulate the flow with all its characteristics features. The simulation of supercritical flow under extreme boundary conditions is complex. Solution grid adaptive technique is found to be very effective to produce better solution and resolution of the steep gradients both for subcritical and supercritical flows. Since the physical equations and the geometric conservation law (GCL) are decoupled, they can be solved independently of each other. This allows the

15

I

_

Adaptive Grid _ Analytical

- Euler, Implicit Trapezoidal

1000

Distance (m)

Fig. 8. Comparison of solutions at supercritical flow (ho/h1 = 0$)04).

8 M. Rahman, M. Hanif Chaudhry

15

Adaptive Grid (St. Vcnsnt Eq with u. uh as var.)

_____------ Adaptive Grid (n. “cnan, Eq. wlh u. has var.1

Time = 60

0 500 1000 1500 2000

Distance (m)

Fig. 9. Comparison of adaptive solutions as obtained using conservative and non-conservative variables.

Error= [C 1 h_ - hcom 1 211’2 2.5

Adaptive

1

30 40 Number 0Ygricl 70 80

points

Fig. 10. Error distribution for different grid sizes.

application of any existing finite different or finite element code without major modifications or changes. A separate subroutine to solve the GCL can be added. Moreover, this technique can be easily extended to multi-dimensional problems. It is flexible enough to allow clustering in x, y or z directions simultaneously or in any one direction independent of others.

ACKNOWLEDGMENT

Foundation for supporting this research through grant no. INT-9302500.

REFERENCES

1. Chaudhry, M. H., Open-Channel Flow. Prentice Hall, Englewood Cliffs, New Jersey, 1993.

2. Sakkas, J. G. & Strelkoff, T., Dam-break flood in a prismatic dry channel. J. Hydraul. Div., ASCE, 99 (1973) 21952216.

The authors would like to thank the National Science 3. Chen, C., Laboratory verification of a dam-break flood

model. J. Hydraul. Div., ASCE, 106 (1980) 535-556.

Simulation of dam-break flow 9

4. Katopodes, N. D. & Strelkoff, T., Computing two- dimensional dam-break flood waves. J. Hydruul. Div., AXE (1978) 1269-1287.

5. Fennema, R. J. & Chaudhry, M. H., Simulation of one- dimensional dam-break: flow. J. Hydruul. Res., 25 (1987) 41-51.

14. Bierterman, M. & Babuska, I., An adaptive method of lines with error control for parabolic equations of the reaction-diffusion type. J. Comput. Phys., 63 (1986) 33- 66.

6. Fennema, R. J. & Chaudhry, H. M., Explicit methods for 2-D transient free-surface flows. J. Hydruul. Engng, 116 (1990) 1013-1035.

7. Katopodes, N. D., A dissipative Gale&in scheme for open-channel flow. J. Hydraul. Engng, 110 (1984) 450-466.

8. Thompson, J. F., Thames, F. C. & Mastin, C. M., Automatic numerical generation of body-fitted curvilinear coordinate systems for fields containing any number of arbitrary two-dimensional bodies. J. Comput. Phys., 15 (1974) 299-319.

15. Sanz-Serna, J. M. & Christie, I. A., Simple adaptive technique for nonlinear wave problems. J. Comput. Phys., 67 (1986) 348-360.

16. Furzelan, R. M., Report TNER.85.022, Thornton Research Center, Shell Research Limited, 1985.

17. Thomson, J. F., Warsi, Z. U. A. & Mastin, C. W., Numerical Grid Generation: Foundations and Application. North Holland, 1985.

18. MacCormack, R. W., The effect of viscosity in hyperve- locity impact cratering. AZAA Paper (1969) 69-351, Cincinnati, Ohio.

9. Hindman, R. G., Kutler, P. & Anderson, D. A., A two- dimensional unsteady Euler-equation solve for flow regions with arbitrary boundaries. AZAA Paper (1979) 76-1465.

19. Courant, R., Friedrichs, K. & Lewy, H., Uber die Partiellin Differenzengle-ichungen der Mathematischen Physik. Math. Ann., 110 (1928) 32-74.

10. Dwyer, H. A., Kel, R. J. & Sanders, B. R., Adaptive grid methods for problems in fluid mechanics and heat transfer. AZAA J., 18 (1980).

20. Jameson, A., Schmidt, W. & Turkel, Numerical solutions of the Euler equations by finite-volume methods using Runge-Kutta time-stepping schemes. AIAA 14th Fluid and Plasma Dynamics Conference, Palo Alto, CA, AIAA pp. 81-1259, 1981.

11. Olsen, J., Subsonic and transonic flow over sharp and 21. Anderson, D. A. & Rai, M. M., The use of solution round nosed nonlifting airfoils. Ph.D. dissertation, Ohio adaptive grids in solving partial differential equations. State University, Columbus, Ohio, 1976. Proceedings of a Symposium on the Numerical Generation

12. Rai, M. M. & Anderson, D. A., Application of adaptive of Curvilinear Coordinate Systems and their Use in the grids to fluid-flow problems with asymptotic solutions. Numerical Solution of Partial Differential Equations. AZAA J., 20 (1982) 496-502. April 13-16, Nashville, TN, 1982.

13. Thomas, P. D. & Lombard, C. K., The geometric conservation law-A link between finite-difference and finite-volume methods of flow computation on moving grids. AZAA Paper (1978), 78-1208.

22. Coyle, J. M., Flaherty, J. E. & Raymond, L., On the stability of mesh equidistribution strategies for time- dependent partial differential equations. J. Comput. Phys., 62 (1986) 26-39.