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Zhou, Bhajantri: Numerical Study of the Effects of Spillway Crest Shape on the Distribution of Pressure and Discharge

Numerical Study of the Effects of Spillway Crest Shape on the Distribution of Pressure and Discharge

Model Validation: Currents and Waves

Fayi Zhou, M. Bhajantri

Parallel Session (parallel16), 31.08.1998, 16:00 - 18:15Numerical Simulation of River Flow

ABSTRACT: The effect of spillway crest shape on the discharge coefficient and pressure distribution was studied using a two-dimensional large eddy simulation(LES) model. The model was developed based on the compressible hydrodynamic equations. The body-fitted explicit finite volume scheme was adopted. Four spillway crest profiles were studied and the numerical results on the discharge coefficients and pressure distribution were compared with the physical experiment.

INTRODUCTION

The optimum spillway crest shape should provide a high discharge coefficient and a fairly uniform and predictable pressure distribution over the crest boundary (ASCE, 1995). Therefore, various spillway crest shapes based on the flow discharge and pressure distribution must be investigated and compared in the design stage. Usually, such investigations are done by physical modelling. However, the measurement of the fluctuating pressures and velocities in physical models is difficult. Besides, the fabrication of spillway models is time consuming and expensive if the model has to be modified repeatedly. With respect to the numerical simulation of spillway flow, researchers such as Ippen (1951), Ellis (1982), Montes (1994), and Khan and Steffler(1996) studied the spillway flow by either a one-dimensional or a two-dimensional potential flow model. Clearly, these approaches can not simulate the real flow phenomena like turbulence, free-surface waves, and the effect of non-uniform entrance velocity distribution. Very few advances seem to have been made in numerical simulation of spillway flows recently and the design of spillways is still largely dependent on physical modelling.

In this paper, a two-dimensional large eddy simulation (LES) model was developed to simulate the free surface flow over a spillway. The LES model has the advantage of obtaining instantaneous values which are the most important ones in pressure studies.

This study focuses on the effect of spillway crest shape on the pressure distribution and the discharge coefficient both of which are the most practical interest to engineers. The incoming flow was steady. The body-fitted control volume was adopted which provided for the accurate treatment of the solid boundary. The second order MacCromack prediction and correction scheme was used to solve the equations. The air entrainment was neglected because our concern is limited to the domain near the crest where the air entrainment was insignificant. Four crest shapes were simulated and the results were compared with those of physical models. Good agreement was obtained between the numerical simulation and physical modeling in terms of the discharge coefficients and averaged pressure distribution.

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GOVERNING EQUATIONS

For general small Mach number flows, including transient flows, the equation of continuity is (Song and Yuan, 1988)

(1)

and the equation of motion can be written as

(2)

The flow which satisfy the Equations 1 and 2 is called weakly compressible flow.

For turbulent flow modeling, the large eddy simulation (LES) model was adopted. In this case the kinematics viscosity in

Equation (1) was replaced by the sum of the kinetic viscosity and the eddy viscosity . Here the Smagorinsky (1963) sub-grid scale turbulence model is used to simulate the eddy viscosity:

(3)

(4)

In Equation (4) the coefficient is actually a variable that has to be determined. It was found that the coefficient must

vary from 0 on the wall to 0.12 outside of the viscous boundary layer. is an index representing the grid size.

The determination of the grid size is important in LES. The minimum grid size near the larger eddy scale can be roughly estimated from the assumption that the energy production is equal to the energy dissipation. The large eddy scale l can be estimated by

(5)



where us is the turbulent velocity scale which can be estimated by

(6)

where U is the mean flow velocity, L is the distance along the spillway from the starting point of the crest curve, and is the thickness of boundary layer which can be calculated by (ASCE, 1995)

(7)

where ks is the roughness of the spillway surface.

BOUNDARY AND INITIAL CONDITIONS

The upstream boundary can be set up on a reservoir section at which the water stage and the incoming discharge can be known. This section should be far away from the spillway to avoid the reflection effect. The downstream boundary should be located based on the range of the interested domain. For the study of the crest shape effect, the downstream condition will have no effect on the upstream flow since the flow over the downstream slope of the spillway is super-critical. Here the downstream section was chosen on a slopping section where the turbulence was fully developed so that zero-gradients of velocity and pressure can be assumed. At the spillway surface, the partial-slip velocity condition was used which meant a wall function was used instead of the non-slip condition. Doing so prevented the generation of the fine mesh near the wall and the saved a huge amount of computer storage space. The wall function can be approximated

by where u* is the friction velocity.

The most difficult boundary to simulate is the free-surface boundary, which requires a kinetic condition and a dynamic condition. The kinetic condition is based on the idea that the free-surface is a material surface and has the form of

(8)

where Zf is the free surface displacement along the normal direction. This equation will be solved using the MacCromack

scheme to modify the new free surface position during the iteration.

The dynamic boundary condition, ignoring the surface tension effect, is the zero stress condition. For this model the dynamic boundary condition, which is zero stress on the free-surface, is simplified as follows:

(9)

where represents the unit vector normal to the free surface.

The initial condition can be provided by solving the Bernoulli equation. An accurate assumed velocity and free surface profile will accelerate the convergence of the 2D turbulent flow computation.



MESH GENERATION

Because the supercritical flow is very sensitive to the geometric boundary condition, the smooth fitting of the boundary profile is essential to prevent a false wave generated by inaccurate representation of the boundary profile. Therefore, a boundary fitted mesh system based on Thompson (1985) was used. The whole computational domain was first decomposed into two zones, as shown in Fig. 1, in such a way that the shape of each zone was relatively simple and the variation of the flow within a zone was relatively small. Zone 1 covered most of the reservoir; while Zone 2 covered a smaller portion of the reservoir and the whole flow region over the spillway. The mesh in zone two was perpendicular to the spillway surface in order to increase the accuracy of the solution near the bottom.

Numerical Solution by Finite Volume Method(FVM)

An explicit finite volume scheme was used for the unsteady flow calculation. To apply a finite volume scheme, it is convenient to first rewrite the governing equations, Equations 1 and 2, in a conservative form as follows:

(10)

where (11)

(12)

(13)

(14)

In the above equations are the sum of viscous and turbulent stresses.

Equation 10 is solved with MacCormack (1969) predictor-corrector scheme. For more detailed description of the scheme as applied to the weakly compressible flow equations, reference is made to Yuan et al (1991).



Solution Procedure

1) Specify the initial condition by solving the Bernoulli equation;

2) Generate the mesh based on the initial free surface profile;

3) Calculate velocity and pressure based on Equation 10 by iteration until the continuity equation of the flow is satisfied;

4) Re-calculate the free surface displacement by Equation 8;

5) The mesh is regenerated based on the new free surface grid location;

6) Go to step 3 until the free surface variation between the previous and current iteration is small enough.

7) Calculate the discharge coefficient Cd by and where he is the depth on the crest and H the water head.

NUMERICAL RESULTS

The computation was carried out on the IRIS workstation. Four crest shapes under three different heads H were calculated:

Profile 1(P1): X3.289 = 5436.489Y

Profile 2(P2): X2.748 = 817.481Y

Profile 3(P3): X2.413 = 245.613Y

Profile 4(p4): X1.834 = 21.845Y

Three heads, 6.87m, 11.27m, and 14.67m respectively, are imposed at the upstream section. The minimum grid size was 0.10m. The minimum large eddy scale near the spillway surface under these flow condition was about 0.11m.

The calculated discharge coefficient was compared with the experimental result and is shown in Table 1. Most of the results agreed well with those from the experiment. But for a lower head and profile 1, the difference was large. This is resulted from the simplified treatment of boundary layer which may develop rather slowly at a flatter crest such as in Profile 1.

The static pressure distributions (expressed in meters of water column height) over the crest were also compared with those from the experiment, as shown in Fig.2a through Fig.2d. Again good agreement was found between the calculated values and experimental data.



CONCLUSIONS

Using the LES simulation, the turbulent flow over the spillway was studied and the crest shape effect on the discharge coefficient and pressure distribution was evaluated. The flatter crest shape had the least discharge capacity and the lowest pressure and therefore, is not recommended for spillway design. Profile 4, with the largest Cd value and no negative pressure, is the optimum profile.

ACKNOWLEDGEMENTS

This work has been financially supported by the United Nations Agriculture Organization for training of the second author. The supercomputing facility was provided by the Minnesota Supercomputer Institute of the University of Minnesota.

APPENDIX I. REFERENCES

Ellis, J. and Pender G.(1982), Chute Spillway Design Calculations, Proc., Inst. Civil Engrs., 73, Pt.2 (June):299-312.

Ippen, A.T. et al. (1951), Proceedings of a symposium on high-velocity flow in open channels. Trans., Amer. Soc. Civil Engrs., 116:265-400.

Khan, A.A., Steffler, P.M.(1996), Vertically averaged and moment equations model for flow over curved beds, Journal of Hydraulic Engineering, ASCE, Vol.12, No.1, pp3-9.

MacCormack, R.W.(1969), Effect of Viscosity in Hypervelocity Impact Cratering , AIAA Paper 69-354.

Montes, J.S.(1994), Potential-Flow Solution to 2D Transition From Mild to Steep Slope, Journal of Hydraulic Engineering, ASCE, Vol.120, No.5, May, 601-621.

Smagorinsky, J., General Circulation Experiment with the Primitive Equations, Monthly Weather Review, Vol.91, No.3(1963), pp99-164.

Song, C.C.S., Chen, X , (1996), Compressibility Boundary Layer and Computation of Small Mach Number Flows, 2nd International Conference on Hydrodynamcis, Hong Kong, Dec.1996.

Song, C.C.S., Yuan, M(1988). A Weakly Compressible Flow Model and Rapid Convergence Method, Journal of Fluid Engineering, Vol.110,441-445.

Thompson, J.F., Warsi, Z.U.A. and Martin, C.W., (1985), Numerical Grid Generation, Foundations and Applications, North-Holland, Amsterdam.

Yuan, M., Song, C.C.S. and He, J., (1991), NumericalAnalysis of Turbulent Flow in a Two-Dimensional Nonsymmetric Plane-Wall Diffuser, ASME, Journal of Fluid Engineering, Vol.113, 210-215.



Appendix II.--Notation

The following symbols are used in this paper:

A=cross-section area;

a = speed of sound;

Cs= coefficient of SGS model;

D= diameter of circular cylinder;

E= flux at i surface in Eq.(12);

F= flux at j surface in Eq.(12);’

G= flux at k surface in Eq.(12);

g= gravitational acceleration;

H= sum of flux E, F,G in Eq.(11);

h = depth of water

J = Jacob Matrix

Ko= elastic modulus of the fluid

M= Mach number

P= coefficient for adjust mesh distribution

p=pressure

Q = coefficient for adjust mesh distribution;

S = area of the control volume surface;

t = time

u--velocity component

V--cross-sectional averaged velocity;

v--velocity component

w--velocity component;

Z-- elevation



--weight of smoothing in Eq.(24)

s--the width of filtering in LES model

--kinematic viscosity coeffcient

t--turbulent viscosity coefficient

--density of fluid

Table 1. Comparison of discharge coefficients between computation and experiment

Head(m)

6.87

11.27

14.67

Profile numerical experiment numerical experiment numerical experiment

P1 0.52 0.63 0.55 0.64 0.58 0.65

P2 0.56 0.66 0.61 0.67 0.63 0.68

P3 0.62 0.68 0.63 0.68 0.64 0.69

P4 0.73 0.70 0.75 0.74 0.76 0.76

Fig.1 Computational domain and mesh system



Fig.2 Time averaged pressure distribution over the spillway surface

Fayi Zhou: University of Alberta, T. Blench Hydraulic Lab.Edmonton, [email protected]

M. Bhajantri: University of Alberta, T. Blench Hydraulic Lab.Edmonton, AlbertaCanada


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