Scientia Iranica A (2016) 23(3), 876{881

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Dam break model with Eulerian equations usingmapping technique

A. Lohrasbi� and M.D. Pirooz

School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran.

Received 22 April 2015; received in revised form 25 July 2015; accepted 15 September 2015

KEYWORDSDam break;Eulerian equations;Free-surface;Mapping technique.

Abstract. This paper deals with numerical modeling of water ow, which is generated bythe break of a dam. The problem is solved by applying Eulerian equations with mappingtechnique over a horizontal bottom considering unsteady incompressible ow with freesurface. The proposed model has been used to simulate a two-dimensional problem ofthe collapse of a water column inside a rectangular tank. A new mapping is developed totransform the governing equations from the physical domain to a computational domain. Itis shown that the model predicts the changes of wave pro�le and the free surface elevationof water. In addition, the results are compared with experimental data.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

When a dam is breached, catastrophic ooding occursin the downstream channel. Dam break analyses areused to estimate the potential hazards associated witha structure failure. Zienkiewicz et al. [1] studied owsdue to dam breaking by using the Lagrangian descrip-tion. Pohle [2] and Stoker [3] presented a systematicprocedure for the determination of the successive termsin these expansions. However, only the leading-order terms were constructed and analyzed. In bothLagrangian and Eulerian descriptions, the expansionsof the solution were in time power. However, such asolution was successfully derived in a relevant problemconcerning a uniformly accelerating wave-maker byKing and Needham [4]. There are numerous numericaland experimental studies on dam break ows. Relevantto this study is the paper by Stansby et al. [5] inwhich the initial stage of dam break ow, for dry andwet bed cases, is studied, experimentally, and it isobserved that for the dry bed case, a horizontal jet

*. Corresponding author. Tel.: +98 21 44258217E-mail addresses: ar lohrasbi@ut.ac.ir (A. Lohrasbi);mdolat@ut.ac.ir (M.D. Pirooz)

forms at small times. They also performed a numericalanalysis on the dam break problem. However, inorder to avoid the singularity of the numerical solutionat the intersection point, they had to pre-wet thebed in front of the dam by an arti�cial thin uidlayer. Several numerical studies, performed during thepast few years, were based on the solution of Nonlin-ear Shallow Water Equations (NSWE) using di�erentmethods such as the �nite-volume method, the �nite-di�erence method [6-10]. Hunt [11] used a kinematicwave approximation to obtain a closed-form solutionfor a sloping channel and mentioned that his solutionwas valid for large times. The dam break problem wasinterpreted in the context of a liquid column collapsingunder gravity. Penney and Thornhill [12] studiedthe collapse of a uid column, which was surroundedby a lighter uid. The analysis was performed inEulerian variables for both small and moderate times.They derived the initial asymptotics of the solutionfor uid columns of semi-cylindrical and hemisphericalshapes, and showed that these asymptotics are notcorrectly close to the base, where the uid velocityis much higher than that in the rest of the column.Korobkin and Yilmaz [13] studied the initial stages ofthe dam break ow in the framework of potential free

A. Lohrasbi and M.D. Pirooz/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 876{881 877

surface ows. Dutykh and Mitsotakis [14] solved theclassical dam break problem of validating the nonlinearshallow water equations solvers and tried to examinethe validity of the mathematical model under Navier-Stokes simulations. Lohrasbi et al. [15] presentedan approach to solve Naiver-Stokes equations basedon mapping technique about dam break. They usedArbitrary Lagrangian Eulerian (ALE) for transformingphysical domain to computational domain. However,their solution had some limitations on ratio of upstreamwater height to downstream water height and had someerrors, such as shock waves, on free surface. In thisarticle, new mapping is presented that shows accurateresults.

In this paper, in addition to the extraction ofapproximation functions, correction of their coe�cientis also carried out. This modi�cation improves theaccuracy of the model and gives appropriate results tothe previous works.

2. Formulation of the problem

The unsteady plane problem of free surface ow isconsidered, which is generated when a vertical dam infront of a liquid region is suddenly removed. Then,the liquid ow and the shape of its free surface duringthe early stages of the process will be determined.Initial condition of the liquid is shown in Figure 1.The physical domain �V surrounded by a piecewisesmooth boundary �S is shown in Figure 2. Thistwo-dimensional domain is occupied by a non-viscousincompressible uid with the speci�c mass of �. Theproblem under consideration is the unsteady motion ofa surface wave under gravity. The governing equationsare expressed by the unsteady Eulerian equation andthe equation of continuity. The rectangular coordi-nates are denoted by x and y and the correspondingvelocity components are denoted by u and v. As aresult, the equations of conservation of momentum and

Figure 1. Initial condition of the liquid at time t = 0 sec.

Figure 2. The computational grid is shown mapped backto the physical space.

mass for incompressible Newtonian uids are given asfollows:@�u@�t

�����;�

+(�u� �wu)@�u@�x

+ (�v � �wv)@�u@�y

= �1�@�p@�x

@�v@�t

�����;�

+ (�u� �wu)@�v@�x

+ (�v � �wv)@�v@�y

= �1�@�p@�y� �g

@�u@�x

+@�v@�y

= 0; (1)

where wu and wv are the mesh velocities in x and ydirections. The boundary �S consists of two types ofboundaries: one is �S1, on which velocity is given; theother is the free surface boundary �S2, on which thesurface force is speci�ed. The boundary conditions areexpressed as the followings:

�u = �̂u on �S1 �p:n�x = �̂cx on �S2;

�v = �̂v on �S1 �p:n�y = �̂cy on �S2; (2)

where the superscript caret denotes a function whichis given on the boundary and n�x and n�y symbolizethe direction cosines of the outward normal to theboundary with respect to coordinates x and y. Also,�̂cx and �̂cy are the constants of integration. Theabove equations can be rendered dimensionless byintroducing the following variables:

�x = x �d; �y = y �d; �p = p�g �d;

�u = u(�g �d)1=2; �v = v(�g �d)1=2;

�t = t� �d

�g

�1=2

: (3)

Using these transformations, Eq. (1) can be modi�edas follows:@u@t

+(u� wu)@u@x

+ (v � wv)@u@y= �@p

@x;

@v@t

+(u� wu)@v@x

+ (v � wv)@v@y= �@p

@y� 1;

@u@x

+@v@y

= 0: (4)

878 A. Lohrasbi and M.D. Pirooz/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 876{881

3. Free surface formulation

The uid surface equation is written as:

�F = �h(x; t) + �d� �y = 0 on �S2; (5)

where h is the position of the free surface. The kine-matic condition associated with the uid free surface isde�ned as:

D �FD�t

= 0; (6)

then:

@ �F@�t

+ (�u� �wu)@ �F@�x

+ (�v � �wv)@ �F@�y

= 0: (7)

With substituting into Eq. (5), the following is ob-tained:�

@�h@�t� @�y@�t

�+ (�u� �wu)

@�h@�x

+ (�v � �wv)(�1) = 0: (8)

Utilizing the dimensionless form of the free surfacekinematic equation written as:�

@h@t� @y@t

�+ (u� wu)

@h@x

+ (v � wv)(�1) = 0: (9)

Also, Eq. (9) is simpli�ed as:

@h@t

+ (u� wu)@h@x� v = 0: (10)

4. Transformation of equations

For the �nite element method, such problem requiresa complicated interpolation function on the local gridlines, which results in the local loss of accuracy inthe computational solution. Such di�culties requirea mapping or transformation from physical space to ageneralized space. This transformation simpli�es theproblem. Lohrasbi et al. [15] presented this mappingthat transforms the wave propagation model from thephysical domain:

x =3Xi=1

(� + h�i)Fi(�);

y = �(1 + h): (11)

This transformation has limitation on the ratio ofupstream to downstream water levels. So, in this

research and for showing more and better results, theLagrange interpolation formulations is used such as:

x =nXi=1

(� + h�i)�ni=0

� � �i�k � �i (k = 0; 1; 2; :::; n)

y = �(1 + h): (12)

The �i values are considered the �fth-order polynomialfunction of � as follows:

�i =b

"3l5(1� ")3

�2(� � �0)5(2"� 1) + l(� � �0)4

(4� 5"� 5"2) + 2l2(� � �0)3(�1� "+ 5"2)

+ l3"(� � �0)2(3� 5")�: (13)

De�nitions of b, ", l, and �0 are illustrated in Figure 3.

5. Numerical example

For dam break modelling with Navier-Stokes equationsin this approach, the physical domain has been dis-creted to �x = �y = 0:25 m and 200 elements inhorizontal and 4 elements in vertical directions in spaceand �t = 0:01 sec in time as shown in Figure 4.Free surfaces of dam at t = 0:0 to t = 1:1 sec areshown in Figure 5 and Figure 6 and surface pro�lein several times is shown in Figure 7. For better

Figure 3. De�nition of parameters in function �i.

Figure 4. Physical domain meshing.

A. Lohrasbi and M.D. Pirooz/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 876{881 879

Figure 5. Free surface in dam break at time = 0.0, 0.1,0.2, 0.3, 0.4, 0.5 sec.

Figure 6. Free surface in dam break at time = 0.6, 0.7,0.8, 0.9, 1.0, 1.1 sec.

Figure 7. Free surface pro�le in several times.

demonstration of the results, variation of horizontaland vertical velocities is shown in Figures 8-10. Theresults of present numerical model have been comparedwith the results of Zienkiewicz [1] at t = 0:0 (H1=H2 =2:0) in Figure 11. Comparison between the currentmodel and the numerical results of Zienkiewicz [1]shows acceptable results. This model has the capabilityof dam break model, such as H1=H2 = 4:5 that isillustrated in Figure 12. This ratio is the limitationof solution and has some errors, such as shock waves,on free surface.

6. Conclusion

The propagation and deformation of free surface in dambreak over at bathymetry is investigated. Here, adam separating two stationary water levels is suddenlyremoved and the almost vertical waves progress into thetwo domains. This numerical approach is very suitablefor solving Eulerian equation while maintaining theaccuracy of calculations. The new mapping techniquetransforms the physical domain with the vertical wallto a simple rectangle computational domain. Thistransformation has no limitation on the ratio of up-

880 A. Lohrasbi and M.D. Pirooz/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 876{881

Figure 8. Free surface, horizontal and vertical velocitiesin domain in t = 0:1 sec.

Figure 9. Free surface, horizontal and vertical velocitiesin domain in t = 0:6 sec.

Figure 10. Free surface, horizontal and vertical velocitiesin domain in t = 1:1 sec.

Figure 11. Comparison of the present model withZienkiewicz numerical model with H1=H2 = 2:0 (full linefor the current model and dashed line for Zienkiewiczresults).

A. Lohrasbi and M.D. Pirooz/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 876{881 881

Figure 12. Dam break with the present numerical model(H1=H2 = 4:5).

stream to downstream water levels unlike the previousresearch [15]. The model is validated by comparingnumerical results with results obtained numerically.Overall, the conformity of the available data with thecomputations is well and in most cases, the numer-ical model gives excellent results. The method canbe applied to other unsteady waves and it providesappropriate agreement with independent calculationsof other research.

References

1. Zienkiewicz, R., Taylor, L. and Zhu, J.Z., The FiniteElement Method: Its Basis and Fundamentals, 6thEdition, Elsevier, Oxford (2005).

2. Pohle, F. The Lagrangian equations of hydrodynamics:Solutions which are analytic functions of time , NewYork, University U.S.A. (1950).

3. Stoker, J., Water Waves: The Mathematical Theorywith Applications, Pure and Applied Mathematics, IV,Interscience Publishers, Inc., New York, IntersciencePublishers Ltd., London (1957).

4. King, A. and Needham, D. \The initial developmentof a jet caused by uid, body and free surface interac-tion", J. Fluid Mech, 268, pp. 89-101 (1994).

5. Stansby, P.K., Chegini, A. and Barnes, T. \The initialstages of dam-break ow", J. Fluid Mech, 374, pp.407-424 (1998).

6. Zoppou, C. and Roberts, S. \Explicit schemes for dam-break simulations", J Hydraul Eng., 129, pp. 11-34(2003).

7. Zoppou, C. and Roberts, S. \Numerical solution of thetwo-dimensional unsteady dam break", Appl. Math.Fluids Res., 24, pp. 457-475 (2000).

8. Luigi, F. and Toro, E.F. \Experimental and numericalassessment of the shallow water model for two dimen-sional dambreak problems", J. Hydraul. Res., 33, pp.843-864 (1995).

9. Glaister, P. \Solutions of a two dimensional dam breakproblem", Int. J. EngSci., 29, pp. 1357-1362 (1991).

10. Brufau, P. and Garcia-Navarro, P. \Two dimensionaldam break ow simulation", Int. J. Numer. MethodsFluids, 33, pp. 55-57 (2000).

11. Hunt, B. \Asymptotic solution for dam break prob-lem", J. Hydraul. Div, ASCE, 108, pp. 115-126 (1982).

12. Penney, W. and Thornhill, C. \The dispersion, undergravity, of a column of uid supported on a rigidhorizontal plane", Phil. Trans. Roy. Soc. A, 244, pp.285-311 (1952).

13. Korobkin, A. and Yilmaz, O. \Initial stages of dam-breaking ow", Journal of Engineering Mathematics,63, pp. 293-308 (2009).

14. Dutykh, D. and Mitsotakis, D. \On the relevance of thedam break problem in the context of nonlinear shallowwater equations", Discrete Contin. Dyn. Syst., 13, pp.799-818 (2010).

15. Lohrasbi, A., Dolatshahi, M. and Lavaei, A. \Hy-draulic model of dam break using Navier Stokes equa-tion with arbitrary Lagrangian-Eulerian approach",International Journal of Engineering and Technology,8, pp. 249-253 (2015).

Biographies

Alireza Lohrasbi is PhD graduate in Civil Engi-neering, University of Tehran, Iran. His research isin CFD, numerical and analytical modeling of freesurfaces. He has published several technical lecturesin hydrodynamic and hydraulic modeling.

Moharram Dolatshahi Pirooz is Associate Profes-sor in the Department of Civil Engineering, Universityof Tehran, Iran. He is a very active engineer in the�eld of marine structures. He has published severaltechnical lectures and worked with many consultantsin hydrodynamic and hydraulic modeling. He recentlypublished two papers in the �eld of wave heightprediction.