Spillway Flow Modeling by Finite Volume Solution of Slopping Depth Averaged Equations on Triangular Mesh; Application to KAROUN-4

Saeed-Reza Sabbagh-Yazdi, SYazdi@kntu.ac.irDepartment of Civil Engineering,

KNT University of Technology, Tehran, Iran

AbstractThe results of a numerical analyzer for solving super critical free surface flow in

chute spillways with steep slope are presented in this paper. The developed model consists of depth averaged equations for flow on surface with steep slope toward one of the main horizontal axes of Cartesian coordinates. The depth-averaged form of the flow equations converted to discrete form for unstructured meshes using the cell vertex finite volume method. Therefore, the horizontal mesh is firstly transformed into inclined coordinate system. Then the discrete formulations are solved on the transformed mesh. Having computed the velocity vector parallel to the bed surface and flow depth perpendicular to the bed surface, the model computes three components of the velocity in Cartesian coordinate system as well as the water depths parallel to the z axe in vertical direction. The develop model is utilized for simulation of various flow regimes in prismatic canals, which have two different constant bed slopes in certain direction. The numerical results present the ability of the model to simulate very high speed super-critical flows in very steep bed slopes. Finally, application of the developed model on simulation of sub and super critical flow from reservoir to steep chutes and ski jump flip bucket spillways of KAROUN-4 in KHOZESTAN province of IRAN produced encouraging results.

Keywords: Numerical Simulation, Sub and Super Critical Flow, Spillway with Steep Slopes,

1- IntroductionsThe standard form of shallow water

equations is used in several numerical research works used as a mathematical model for simulation of spillway flows (Unami et al 1999). But most of the chute spillways have steep slopes in a certain direction. Therefore, because of the assumption of negligible vertical velocity component (hydrostatic pressure) and mild bed slope, the set of common

shallow water equations is not suitable for simulating most of the real world spillway flow cases. Hence, casting these equations for steep slopes may help overcoming the problem. Such a mathematical model can be developed by rewriting the depth averaged equation in a frame of reference using flow depth normal to the bed surface and two velocity components parallel to the bed surface.

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 26

In this work, a finite volume model suitable for the triangular unstructured mesh is used for converting the mathematical equations to discrete formulations. Proper artificial viscosity terms are added to the formulations to stabilize the explicit solution procedure in the convection-dominated flow regions. Formulations of these additional terms are designed for solving on unstructured meshes in such a way that not only damp out unwanted numerical oscillations but also preserve the accuracy of the solution. In addition, proper numerical techniques are adopted for increasing the efficiency of the computation on unstructured meshes.

2- Mathematical EquationsHere, the set of shallow water

equations is modified for two dimensional depth averaged flow on an inclined surface with assumptions hydrostatic distribution of pressure. This assumption implies negligible velocity component in normal to the flow plane. In the model, common assumptions for practical open channel water flows, like incompressibility, negligible wind stresses and earth rotation effects are considered as well.

The mathematical model is cast for solving supercritical flow in a chute canal with a slope α in a certain direction (in which the bottom elevation variation plays important role in forming flow patterns). In this model the set of depth averaged equations is modified to describe super-critical flow in a coordinate system with an axe normal and two axes of x* ( αCosxx /* = ) and y parallel to the bed surface.

The set of governing equations contains an equation of continuity and two equations of motion in y and x*

directions as follow. Using

αη CoshZb /*+= , the above mentioned depth averaged equations of motion is reduced to following from.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )ρ

τη

ρτη

*****

**

*

*******

***

*

*

***

0

y

x

yghvvh

yvuh

xvh

t

xghvuh

yuuh

xuh

t

yvh

xuh

th

−=∂∂+

∂∂+

∂∂+

∂∂

−=∂∂+

∂∂+

∂∂+

∂∂

=∂

∂+

∂∂

+∂

∂

Where, t is time, u* is velocity component along x* (in x-z plane) which has α angle x ( αCosuu /* = ), while v is the velocity component along y and h* is the flow depth normal to the bed surface (

αhCosh =* ). Here, Zb is bed elevation, g is the gravity acceleration. The global dissipative forces can be defined as,

22**

**

,, vuUUvCUuC fy

fx +==−=−

ρτ

ρτ

Where, Cf represents the effective global dissipative coefficient. The effects of gravitational force to the water body in x/ direction is introduced using the local bed slope (bed elevation gradient) in terms of : ** xZghSinW b ∂∂=α in the above equations. This term plays an important role in formation of flow patterns.

Having computed the velocity vector parallel to the bed surface and flow depth perpendicular to the bed surface numerically, three components of the velocity in Cartesian coordinate system as well as the water depths parallel to the z axe in vertical direction can be computed as ,

αCoshh /* = , αCosuu *= ,

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 27

vv = , αSinuw *=

(3)

Note that, by omitting the superscripts * from the above equations of continuity and motion, will make them similar to the standard form of shallow water equations (SWE), and hence similar numerical procedure can be applied for the solution (Vreugdenhil 1994)‎‎.

Therefore, similar equations can be allied for the sub-critical parts of the solution domain (in which the water surface elevation variation plays important role in forming flow patterns), by considering; 1=αCos and 0=αSin .

3- Numerical FormulationIn this work, the domain is divided

into triangular sub-domains (control volumes), which is formed by triangles meeting every computational node, and then the governing equations are integrated over each sub-domain. The equations of continuity and the motions are integrated over each control volume. Application of the Green's theorem to the integrated equation in general form result is:

( ) ∫∫∫ΩΓΩ

Ω=−+Ω∂

∂ dSGdxFdydt

W

Where iΩ and Γ are the area and boundaries of the control volume, respectively. W represents time dependent terms of above equations while, F and G represent x and y fluxes, respectively. S is the sink/source term of the equation. Its value equals to zero for the continuity equation and equal to global forces for two equations of motion. If nodal values of dependent variables at each triangle vertex are taken as the unknowns at the

central node of the control volume, the discrete explicit form of the equation is evaluated by conversion of the boundary integral into the summation over m edges of the control volume, as

n

nm

ji

nnn SxGyFtWW +

∆−∆

Ω∆−= ∑

=

+

1

1 )(

Where Wn+1 is the value of Wn to be computed after t∆ . The parameters F and G are the average values of the fluxes in x and y spatial derivatives in each edge at the boundary edges of the control volume formed by gathering triangular cells connected to a computational node (Weatherill et al 1993).

Here, x∆ and y∆ should be computed using transformed coordinate in x* direction in sloping supercritical parts of the solution domain.

Explicit solution of the convective equations where the global dissipative terms are negligible, some numerical oscillations grow particularly near the high gradient regions. These numerical noises disturb the solution procedure in the cases with small physical dissipation mechanisms. For the flow problems with gradual changes in dependent variables (flows with no shock waves), the fourth order term (Biharmonic operator) produces enough dissipations to damp out the numerical oscillations and stabilize the explicit solution procedure (Sabbagh-Yazdi & MohamadZadeh 2004).

Time marching of the explicit computations ( t∆ ) should be proportional to the speed of wave propagation of the applied convective equations. This speed can be computed using, λ the maximum Eigen values of Jacobin matrix of homogonous form of the set of governing equations. Since we are dealing with unstructured meshes, the size of control

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 28

volumes varies over the computational domain. Therefore, every control volume has its own time step, it∆ . Hence, the speed of explicit computations is limited to the minimum min)( t∆ in the unsteady flow field. Although the values of the time step, it∆ for every control volume vary during the stages of the numerical solution, it may approach to certain values when the computations converge to the steady state conditions.

4- Project DetailsIn this part, details of a real

spillway flow case, which is chosen for assessment of the ability of the numerical model, is described. KAROUN-4 arch dam is under constructions in KHOZESTAN province in Iran. The height of dam is 75m from the river bed (Figure 1).

a: service spillway location in dam layout, b: service spillway approach channel

c: service spillway plan of three bays, b: service spillway along one bay

Figure 1: General layout and geometrical specification KAROUN-4 dam and spillway

The right bank gated chute spillway (RBS) of Karun-4 project is the main (service) dam spillway for releasing large floods. RBS include: approach channel, gated ogee spillway (control structure), chute (conveyance structure) and flip bucket (terminal structure). The entering flow through the RBS is divided to 3 parts by two middle piers on the crest which will continue on the chute toward downstream until bucket lips as middle

separating walls. Therefore the chute is divided by 3 equal bays. As the variation of thickness of piers in plan view the width of chute bays is also variable. The net width of each chute bays is equal to 10.5m on the spillway crest. At a distance of 21.74m downstream of spillway crest the thickness of piers diminish from 4m to 1.5m. At distance 33.63m downstream from spillway crest, the width of left and

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 29

right bays will change to 11.75m and the width of middle bay will change to 13m.

The approach channel in plan view is a curved shaped channel fully excavated at elevation 996. Its right bank copies the geometric form of the access road. The left bank of the approach channel is much shorter and lower than the right one, practically does not affect the flow. The flow on the spillway is controlled by 3 equal radial gates each one has 10.5m width and 16m height. The Ogee crest level is 1010 and the tangential point of the spillway to the straight line chute is located at elevation 1009.03 which is 5.93m downstream of ogee crest. The ogee profile of RBS in Karun-4 project is designed according to standard profiles of U.S. Army Corps of Engineers. The constant slope of the chute invert profile is 0.3033 and the tangential point of chute invert profile with flip buckets is different for 3 bays. The chutes are terminating to cylindrical flip buckets by which the flow will throw to the air as free trajectory jets. Figure 1 show plan and profile of the final design of RBS.

5- Computational ConditionsThe geometrical dimensions of the

spillway is digitally modeled according to the laboratory model (scale 1:50). The bottom surface of the spillway is modeled by an unstructured triangular mesh which contains 3070 nodes and 5442 triangular cells generated using Delauney triangulation method (Thompson et al 1999) is converted to a three dimensional surface by assigning vertical elevation to each grid point (Figure 2).

Inflow and outflow boundaries can be manually defined or automatically using normal vector and velocity vector at the boundary nodes (Figure 2). In present model different flow conditions at inflow

and outflow boundaries are imposed. ,For sub-critical flows, at the inflow boundary, velocity components are imposed and water depth is extrapolated from the interior points and at the outflow boundary, water depth is imposed and velocity components are imposed form the inside domain. For super-critical flows, at the inflow boundary, water depth and velocity components are imposed and at the outflow boundary all of the variables are extrapolated from the interior points. When mixed flow forms in the channel, the boundary conditions of the flow boundaries have to be defined concerning flow regime at each boundary (Sabbagh-Yazdi & Amin-Nejad 2005). Therefore, the water surface level in the reservoir is set at elevation of MWL = 1032.9m, which is equal to 2065.8cm in laboratory set up using scale of 1:50. This is the condition for flood discharge of 6250 m3/sec in 100% gate opening (Water Research Center, 2003).

At the solid boundary nodes slipping condition are applied by imposing zero to normal velocity components to the wall. Having computed tangential velocity at wall nodes, wall resistance may be computed and applied to boundary nodes. The effective surface for wall global friction stresses may be computed multiplying the flow depth by contribution length of two wall boundary edges connected to each boundary node Various types of boundary conditions are specified in the numerical model for flow and wall boundary conditions (Figure 2).

Proper initial condition may help accelerating the solution procedure. As can be seen in figure 2, in this work constant water elevation is considered for sub-critical part of the flow domain (where the depth averaged flow equations are to be solved in horizontal coordinate) and constant water depth is considered for

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 30

super-critical part of the flow domain (where the depth averaged flow equations are to be solved in sloping coordinate).

X

1.5

2

2.5

3

3.5

4

Y

1

2

3

4

5

H

-2

-1

0

X

Y

Z

H: -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

KAROUN IV Spilway Flow

Dr. S. Yazdi

KAROUN IV Spilway Flow

Dr. S. Yazdi

X

1.5

2

2.5

3

3.5

4

Y

1

2

3

4

5

H

-2

-1

0

X

Y

Z

H: -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

KAROUN IV Spilway Flow

Dr. S. Yazdi

KAROUN IV Spilway Flow

Dr. S. Yazdi

Figure 2: Left; boundary conditions (Wall: black, Flow: grey), Right; initial conditions (Depth)

6- Simulation ResultsThe computed water surface

elevations and flow depths at the entire computational domain are shown in figures 3 and figures 5. The interesting point is the computation of two inclined shock waves starting from the middle part of the chute (from the point that the thickness of the dividing wall decreases, and hence, the width of the two chute bay decrease) and continue to the buckets. This fact proves that developed model can compute expected shock waves,

which are expected to appear due to change in chute width.

The computed stream lines and velocity vectors, which shows the flow direction are plotted in figures 4 and 5. As can be seen, these vector present horizontal flow in sub-critical part of the domain (reservoir) and parallel to the bed surface flow in supercritical part of the flow domain (chute and bucket). This fact proves that developed model can compute the velocity patterns of chute spillways in a three dimensional manner.

X

1.5

2

2.5

3

3.5

4Y

1

2

3

4

5

H

-2

-1

0

X

Y

Z

D: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Dr. S. YazdiDr. S. Yazdi

X

Y

Z

D

10.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.1

Dr. S. YazdiFigure 3, computed water depth , general view (left) close view (right)

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 31

X

1.5

2

2.5

3

3.5

4Y

1

2

3

4

5

H

-2

-1

0

X

Y

Z

V: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

Dr. S. YazdiDr. S. Yazdi

X

Y

Z

V: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6Dr. S. Yazdi

KAROUN IV Spilway Flow

Dr. S. YazdiFigure 4, Computed stream lines, general view (left) close view (right)

X

1.5

2

2.5

3

3.5

4Y

1

2

3

4

5

H-2

-1

0

X

Y

Z

V: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

Dr. S. YazdiDr. S. Yazdi

X

Y

Z

V: 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6Dr. S. Yazdi

KAROUN IV Spilway Flow

Dr. S. YazdiFigure 5, Computed velocity vectors, general view (left) close view (right)

7- Conclusion

A finite volume flow solver is developed using an inclined coordinate system, the horizontal velocity components are transformed to the velocity components parallel to the bed surface. The depth average equations derived for flow in coordinate system

mapped on sloping bed surface are very similar to the standard shallow water equations for flow in horizontal plane. Therefore, unstructured finite volume flow solver for solving sub-critical flow problems is easily adapted for the solution of above mentioned developed mathematical model for super-critical flow on steep chute canals. Using this

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 32

technique, not only horizontal component of the flow velocity are computed correctly, but also vertical velocity component appeared in the simulation results. Consequently the computed velocity vectors are parallel to the bed surface at the entire supercritical part of the flow domain. Combining the developed numerical solver for super critical flow on steep slopes with the standard shallow water equation solver suitable for sub-critical flow provided the ability of simulating mixed sub and super critical flows. Therefore, three dimensional numerical simulations of velocity patterns and shock waves for both low speed flow in reservoir and high speed super critical slopping flow in steep chute and ski jump flip bucket spillway is performed successfully.

8- Acknowledgement

This research founded by Water Resource Management Company of Ministry of Energy of Iran under contract number “DAM2 81171” and their financial support is greatly acknowledged.

9- References

1. Sabbagh-Yazdi SR, Amin-Nejad B, 2005, Chute Spillway Flow Modeling using

Various Cartesian Flux Convections Techniques on Overlapping Unstructured Finite Volumes, Proceeding of XXXI IAHR Congress, Seoul, Korea

2. Sabbagh-Yazdi SR & Mohamad-Zadeh Qomi M, 2004, Finite Volume Solution of Two-Dimensional Convection Dominated Sub-Critical Free Surface Flow Using Unstructured Triangular Meshes, International Journal of Civil Engineering, Vol 2, No 3

3. Thompson JF, Soni BK, Weatherill NP, 1999, Hand book of grid generation, CRC Press

4. Unami K, Kawachi T, Babar M, Itagaki H, 1999, Two-Dimensional Numerical Model of Spillway Flow, Journal of Hydraulic Engineering, ASCE, Vol 125, No 4, 369-375

5. Vreugdenhil CB, 1994, Numerical Methods for Shallow Water Flow, Kluwer Academic Publisher.

6. Water Research Center, 2003, Final Report of Karun-4 HEPP (Comprehensive Hydraulic Model), Tehran, Iran

7. Weatherill, NP, Hassan O, Marcum DL, 1993, Calculation of Steady Compressible Flow-Fields with the Finite Elements Method, 31 Aerospace Sciences Meeting and Exhibit, AIAA-93-0341

Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 33