PSEUDO THREE-DIMENSIONAL FINITE ELEMENT ...

Vol 64, No. 4;Apr 2014

PSEUDO THREE-DIMENSIONAL FINITE ELEMENT MODEL FOR UNSTEADY FREE SURFACE FLOW Ali Asghar Mahessar1, Ahsanullah Baloch2 and Abdul Latif Qureshi3 1

Ph.D Scholar, Institute of Water Resources Engineering and Management, Mehran University Engineering and Technology, Jamshoro, Pakistan Email: [email protected] 2 Professor, Department of Computer Science, ISRA University, Hyderabad, Pakistan Email: [email protected] 3 Professor, Institute of Water Resources Engineering and Management, Mehran University Engineering and Technology, Jamshoro, Pakistan Email: [email protected]

ABSTRACT A computational Pseudo Three-dimensional model has been developed for unsteady flow to investigate free surface water profile. Unsteady flow in channels is complex by variation of hydraulic data due to morphological changes and geometrical properties. In order to simulate the flow phenomena with satisfactory resolution in spatial and temporal domain, the Navier-Stokes equations have been solved using Taylor-Galerkin technique. In this paper, unsteady flow due to partial dam breach and dam break with sudden enlargement in channel has been investigated, using semiimplicit Taylor Galerkin finite element scheme. The accuracy and reliability of the model was validated by comparing of computed results with available solution of the unsteady flow problems in open literature. Numerical results show that the technique is capable and accurate tool to simulate hydrodynamic behaviour through computing water depth hydrograph with space and time variation of flow. Key words: Pseudo-3-D, FEM, Unsteady flow, Water level, Hydrograph 1

INTRODUCTION

The worst weather related floods mostly resulting from abnormal rainstorm, sometimes combined with melting snow, which causes the rivers to overflow their banks. There is a long history of flooding from Indus River and their tributaries due to torrential rainfall frequently occur in the monsoon season (Memon, 2004). The flood stages at control points, upstream of Sukkur and Kotri barrages of the Indus River were computed using developed finite element model by Qureshi et al. (2014). The occurrence of a series of floods resulting from dam failures has focused attention on the need for developing a general applicable model for predicting dam break flood waves (Chen, 1980). Flooding can mostly affect flow dynamics resulting in formation of hydraulic jump, shock and wave propagation. Lateral contraction or expansion of downstream channel can be considered to represent the irregular topography in floodplain. Hence, variation of flow depth with time for different locations, formation of negative wave and velocities of negative and positive wave fronts are the important parameters. Due to difficulties in obtaining field data for dam break flow, laboratory experiments in idealized situation were conducted (Selahattin and Hatice, 2012) for understanding the ongoing phenomena and to validate the numerical models. In fact, there are limited numbers of laboratory data and scaled physical models of natural channels compared to numerical studies. 426

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With the advancement of computer technology and numerical solution methods of the shallow water equations, hydrodynamic models based on one-dimensional and twodimensional approaches are increasingly being used for predicting unsteady flows. Currently, numerical solutions of the shallow water equations type are one of the most active topics in the field of hydraulic research work (Xia et al., 2010). The computation of two-dimensional unsteady flow is more complicated than that of one-dimensional flows due to need for efficient solver routines and the inclusion of proper boundary conditions (Fennama and Chaudary, 1990). For two-dimensional unsteady flows, characteristics, explicit and implicit finite difference methods were used by a number of investigators (Abbot, 1979) to simulate the two-dimensional propagation of dam break flood waves. The FEM is flexible in handling general shapes of domain and boundary conditions, which has been observed for solving sediment transport problems (Qureshi and Baloch, 2013). Recently, Pseudo three-dimensional finite element model was developed (Kwok-Woon, 2011) for unsteady free-surface flow, which gave more realistic description of flow than that of one and two-dimensional approaches and computation is less complicated than that of three-dimensional. Pseudo 2-D model was also developed by Sutkar et al. (2013) for simulating flow pattern; the numerical results in terms of flow pattern and average particle velocities in both regimes were compared with experimental results exhibited fairly well. Numerical methods are needed not only for the evolution of interface, but also for determination of velocity and depth characterizing the flow. The application of standard Galerkin finite element methods calls different discrete finite element approximation for velocities and pressure (Brezzi et al., 1991). To investigate algorithmic aspects of this method when applied to the Navier-Stokes equations is the objective of present work. 2.

GOVERNING EQUATIONS

2.1

Pseudo Three-dimensional Navier-Stokes Equations

Pseudo three-dimensional model has been developed for analysing the problem in real situation. This model has been used for computation of velocity, pressure and water depth values. Governing three-dimensional Navier-Stokes equations in Cartesian coordinates and consideration has been given to an implicit and a semi-implicit temporal and spatial discretization according to Taylor Galerkin /projection scheme, as introduced by the pioneers of this algorithm (Townsend and Webster, 1987). Continuity Equation u  v   0  x  y

(1)

Momentum equations in Pseudo three-dimension component wise u  (u, v, w); x  (x, y,0); 427

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2u 2u    u u u p  μ  2  2   ρ u v  t  x  y  x  y  x   

(3)

 2v 2v   v v  v  p  μ  2  2   ρ u v  t  y  x y   x x

(4)

x: ρ

y: ρ

z: ρ

 2w  2w  w  μ 2  ρ 2  t  x  y  

 w w v u  y  x

(5)

Water depth is computed following the idea of (Fei et al., 2001-2) Dh (6) w Dt Where u (x, t) and v (y, t) are local velocities in longitudinal (x) and vertical (y) directions respectively, p (x, t) is isotropic pressure (per unit density), t is time,  is kinematic viscosity μ/ρ  , μ is dynamic fluid viscosity and ρ is fluid density. 2.2

Non-Dimensionalization of Equations

It is convenient to cast the governing systems of equation in non-dimensional form using non-dimensional variables x*, t*, p* and u*. This may be achieved by selecting a suitable choice of characteristic scaling factors in the following manners: Displacement vector (x =) Lx*, time (t =) {L/U} t*, velocity (u=) Uu*, pressure (p =) ρU 2 p*. The above system of equation (1) and (2 - 5) may be rewritten as follows: Continuity Equation

 .u  0

(7)

Momentum Equation u 1   2 u, w  u.u, w   p, w  t Re

(8)

Where, u  (u , v, w) is the velocity vector field, u is velocity in x direction, v is velocity in y direction and w is velocity in z direction). 3.

NUMERICAL SCHEME

In the proposed finite element scheme, two-step Lax-Wendroff predictor-corrector technique has been adopted. The choice of numerical scheme is rest on the accuracy, efficiency and stability of the scheme. Literature exhibits that in an explicit scheme the basic difficulty arrives when using large time steps (Δt) which leads to use the alternate 428

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approach i.e. semi-implicit scheme and fully implicit schemes. Implicit numerical schemes usually enhance the numerical stability and also computationally more expensive unless for steady calculations they are able to increase the convergence rate with significantly larger time steps. The fully discrete semi-implicit system of equation is then as follows (Baloch, 1994). Stage -1(a): Predict the Velocity U at half time step (n+1/2) level using following equation.  2M S   u 2Re  Δt

  

 n 1   S   U j 2  U j n    u U j  NU  U j  LPk  b . t        Re

n

(9)

Stage -1(b): Exercising the above information, the second order accurate velocity U at full time (n + 1) level applying the following formulation. M S   u  Δt 2 R e

Stage-2:







n

 *  S   U j  U j n   u U j  LPk  b . t   NU  U nj 1 / 2   Re 



K Pkn 1  Pkn 

2 LU *j Δt

Stage-3: 2 M n 1 U j  U *   LT Pkn 1  Pkn  Δt

(10)

(11)

(12)

Here, b. t. = boundary terms, j and j are shape functions; for more details ref. (Baloch, 1994). 4.

NUMERICAL RESULTS AND DISCUSSIONS

Under this section, simulation results of two problems using the developed finite element. Taylor-Galerkin schemes are presented. Their problems’ description, numerical predictions and discussions are described as follows: 4.1

Partial Dam Breach

The domain of computation is comprised of a 200 m wide and 200 m long channel. The non-symmetrical breach 75m wide of dam is in the flow direction at 100 m length. The water surface level is 10 m at upstream and 5 m at downstream end considered as initial condition. The bottom is considered as flat and ground resistance to the motion end is neglected (Fennama and Chaudhry, 1990). The schematic diagram of channel with developed finite element mesh is shown in Fig. 1. The mesh has triangular elements of 5590 with total nodes of 11457 and having 37305 DOF (Degree of Freedom). Figure 2 shows initial condition of the problem. This test was also conducted by Fennama and Chaudhry (1990) using finite difference scheme. The test consisted of simulating the 429

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submersion wave due to partial collapse of a dam. There is no analytical solution available in open literature for this problem.

(a) Plan view of 2D channel

(b) Finite Element mesh

Fig. 1: Schematic diagram of 2-D channel for partial dam breach and the Finite Element mesh

Fig. 2: Initial conditions developed by presented FE model The computed water surface profile by the present model shows an unsteady flow generated by the sudden collapse of an asymmetrical 75 m long portion of the barrier (see Fig. 3). The computed water level was obtained in 7.2 sec after breach. A good comparison with results of Fennama and Chaudhry (1990) as shown in Figure 4. The water level at immediately upstream the gate is lower and water level downstream, the gate became bit a higher. Hence, this model can be used to compute viable results for dam break or sudden opening of sluice gates, dike-breach flows or for the analysis of flows having bores or flood propagation waves.

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Fig. 3: Computed water surface profile by the presented Taylor Galerkin model

Fig. 4: Comparison of present model with other schemes 4.2

A Dam-break Flow with a Sudden Enlargement

The breaking of the dam was taken to further validate the model i.e. simulating by the rapid downstream movement of a thin gate at the middle of the flume (see Fig. 5). The selected test case was conducted in the Civil Engineering laboratory of the University Catholique Louvain, Belgium, over a 6 m long flume with a non-symmetrical sudden enlargement from 0. 25 to 0.5 m width, located at 1.0 m downstream of the gate. The initial conditions consisted of 0.1 m high horizontal layer of fully saturated sand over the whole flume and initial layer of model predictions were compared with experimentally measured water levels at six points and levels at two cross-sections (Xia, et al., 2010). The location of selected six points in x and y coordinates are described in Table 1.

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Fig. 5: Sketch of a dam-break flow experiment over a mobile bed Table 1: Selected Points and their X-and Y-Coordinates Point No.

X coordinate (m)

Y coordinate (m)

1

3.0

0.1125

2

4.23

0.1125

3

4.48

0.1125

4

4.96

0.1125

5

4.23

0.36

6

4.96

0.36

The plan view of a two-dimensional sudden enlargement channel is shown in schematic diagram (see Fig. 6). The computational domain mesh has been constructed, which have 8380 elements and 17321 nodes.

Fig. 6: Schematic diagram of a sudden enlargement channel The present model has been used to simulate transient behavior for a real time of 12 seconds. The water surface profiles have been computed at various points. The hydrographs at selected six points (shown in Fig. 5) have been computed which were drawn in Fig. 7. 432

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Figure 7 shows that at point 1, located at nearby contraction section, water level raises suddenly reaches to height of about 0.203m and then falls steadily. Similarly at point 2 and 3 the water surface levels show a rise in water level, however both have increased to about 0.17m, 0.15m respectively. Whereas, for point 4, the water surface level is intial shows arise to 0.13m, then became constant for about 2 seconds, and again shows another a rise up to level of 0.15m (i.e equal to maximum water depth oberved for point 3). This phenomenon demonstrates that after 4 seconds, the water levels at this enlarged location show same levels along the width.

Fig.7: Computed hydrographs at various selected points These hydrographs have also been compared with laboratary experients and numerical predictions of (Xia et al., 2010) to determine the model validity level. In this connections some hydrographs are given in Figures 8 to 10 plotted using Tec-plot software. Figure 8 shows satisfactory results while comparing with laboratory experimented data and hydrograph (water level) computed using FDV by (Xia et al., 2010) at Point P1 i.e. located at nearby contraction section. The water level reaches to highest depth of 0.202m at about 4 seconds. Both numerical models, present and that of Xia et al. (2010) shows good match.

Fig. 8: Comparison of present FE model (Computed) with Laboratory experiment and FDV model (cal.) of Xia et al. (2010) at Point P1 433

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However, models’ graphs wave satisfactorily matches with the experiments. At the start, these are under prediction; whereas near the peak depth shows good matching and then both were over predicted than the experiment. On the whole, it is satisfactory. The hydrograph (water level) predicted at Point 2 which is located at starting of sudden enlargement section for the proposed Finite Element model (given in Fig. 9) shows good agreement while comparing with laboratory experiments data and computed readings using FDV (Xia et al., 2010).

Fig. 9: Comparison of present FE model (Computed) with Laboratory experiment and FDV model (cal…) of Xia et al (2010) at Point P2 The water level predicted using the proposed Finite Element model given at point P5 in Fig. 10 also shows a good agreement while comparing with laboratory experimented data and hydrograph computed by Xia et al. (2010).

Fig. 10: Comparison of Numerical results of present model (Computed) and the FDV model of Xia et al. (Cal.) with Lab. Experiments at Point P5 5.

CONCLUSIONS

The developed Pseudo three-dimensional finite element model using multi-step predictor-corrector Taylor-Galerkin technique has been applied for prediction of water levels for both partial dam breach and a dam break with enlargement in the channel section. Initially the accuracy of this model was validated through comparing the numerical predictions for a partial dam breach against the numerical results of 434

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(Fennama and Chaudary, 1990). The predicted water levels at two selected nodes (20, 15) and (24,15) up to real time of 7.2 seconds show a good matching with Mac Cormack scheme of Fennama and Chaudhry, which concludes that the present model is well validated. For further verification, the proposed model show quantitatively well match with not only laboratory experiments but also with numerical calculations of Xia et al. (2010) at various points in dam break and an enlarged channel section. This suggests that the Pseudo 3D model may be applied to any channel and/or river section through providing suitable boundary conditions. ACKNOWLEDGEMENTS The authors are thankful to the Institute of Water Resources Engineering and Management, Mehran University of Engineering & Technology, Jamshoro, Pakistan for providing facilities to conduct this research work. The authors are thankful to Dr. Robet J. Fennama and Dr. M. Hanif Chaudhry and Dr. Junqiang Xia, Dr. Binliang Lin, Prof. Dr. Roger A. Falconer and Dr. Guangqian Wang. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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