LARGE-EDDY SIMULATION OF FLOW OVER A LOW-HEAD DAM

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21st Century Dam Design —

Advances and Adaptations

31st Annual USSD Conference

San Diego, California, April 11-15, 2011

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Large-Eddy Simulation 355

LARGE-EDDY SIMULATION OF FLOW OVER A LOW-HEAD DAM

Piroz Zamankhan1

ABSTRACT

Flow over a low-head dam is quite complex. A dangerous roller may develop on the down-stream of the dam which makes it difficult to dislodge floating objects. This paper presents the results of a simulation using the large-eddy simulation technique (LES) and the level-set approach to predict the flow over a low-head dam. Qualitative and quantitative comparisons with experimental results provide insights in the capabilities of LES and the level-set approach to predict complex two-phase flows including flow over a low-head dam or a sharp-crested weir. In addition, suggestions are made on how to diminish the hazards associated with rollers.

INTRODUCTION

Rivers can be dangerous because of structures they flow over, around and through (Wyrick et al., 2009). Low-head dams, also known as drowning machines (Leutheusse and Birk 1991), are particularly dangerous river features that boaters and swimmers need to avoid. They exist in many rivers, often built to divert water for open-channel irrigation and power plant cooling water purposes with water depth not exceeding 7.5 m (Vischer and Hager 1998). A number of mill dams were constructed in the 19th century to harness the small amount of hydraulic head needed to run water wheels. Figure 1 (a) shows a typical low head dam.

As depicted schematically in Fig. 1 (b), water flowing over a low-head dam forms a backwash. An object that floats down the river and over the dam will be washed down to a point called the boil. The stream of water splits at the boil, either continuing its motion downstream or going back toward the dam. The circulation pattern illustrated in Fig. 1 (b) can trap objects and humans and keep them submerged. Some low-head dams are built to aerate the water to enhance its quality. In this case, water pouring over the dam will be highly aerated, reducing buoyancy. Figure 1 (c) depicts a boat that got too close to a dam from downstream so that the reverse current caught it and carried it to the trough, where the boat with its passengers was submerged and remained submerged within a few meters of the dam. In a situation like this, the passengers can become entangled in underwater debris and not be able to free themselves. The seriousness of the hazard depends on the height of the dam, the horizontal distance the water falls, the shape of the river bed, the velocity of the river and the volume of water passing over the dam.

1 Faculty of Industrial, Mechanical Engineering and Computer Sciences , University of Iceland Hjardarhagi 2-6, IS- 107 Reykjavik, Iceland, [email protected]

356 21st Century Dam Design — Advances and Adaptations

Figure 1. (a) An image of a typical low-head dam. (b) Circulation pattern for a low-head dam. (c) A time sequence of photographs of a trapped boat with its passengers. The arrows point to the trapped boat. Untrained

rescuers should never approach the roller in a boat. A model experiment involving flow over a low-head dam is presented as a movie in Munson, Young, and Okiishi (2002). The movie clearly shows that on the downstream side of the dam, a backwash was created which trapped items that moved over the dam. Dams do not need to have a deep drop to create a dangerous roller. The roller current is governed mainly by the volume of water and flow current. In the present effort the main objective was to use advanced numerical techniques to make appropriate changes to the downstream geometry of low-head dams to eliminate the named hazard. To achieve the objective a large-eddy simulation technique (LES) (Pope 2002) and a level-set approach (Osher and Fedkiw 2003) was combined to predict incompressible, two-phase flows. This approach makes it possible to accurately compute flows with large density and viscosity variations. To resolve an interfacial motion in a multi-phase flow simulation, an auxiliary function is required. The level-set method can


effectively represent an arbitrarily complex interface. Hence, it is well suited for the computation of flows with surface tension and rapidly changing topology. The model was tested by resolving the flow over a low-head dam. The results were compared with available experimental results in order to verify the validity of the model. The validated model and the model detailed in Zamankhan (2009) were then combined to simulate situations in which dangerous circulatory flow patterns had been eliminated. The outline of this paper is as follows. In section II, the experimental apparatus is briefly described and some experimental results are presented. The mathematical model is presented in section III. Then it is applied to the specific example of low-head dams. Section IV presents qualitative as well as quantitative comparisons between simulation results and experimental results to verify the validity of the model. Section V describes the design by which the circulatory pattern of a low-head dam is eliminated. Concluding remarks are presented in Section VI. The present attempt may point to more effective and less expensive ways to modify low-head dams.

EXPERIMENTAL APPARATUS, PROCEDURE AND RESULTS

Computational fluid dynamics (CFD) modeling is being used widely in the hydro industry. Large eddy simulation (LES) is a mathematical model for turbulence used in CFD. Simulating hydraulic behavior using a numerical model is a tried-and-true method for determining how a river will interact with its environment, namely, a low-head dam. Numerical modeling can provide adequate information when other methods are not applicable. In many cases, CFD modeling is faster and cheaper than physical modeling. Then the question arises: “Why is a physical model ever necessary, and what can be gained from studying it?” The answer is: “Some physical phenomena, particularly when complex geometries are involved, still cannot be accurately predicted using CFD. In these cases, physical models are more than competitive in schedule and cost.” For an open channel model, the relevant dimensionless parameters are the Reynolds and the Froude number. Hydraulic similitude requires matching the aforementioned dimensionless numbers in both scaled model and prototype that play significant roles in the physical processes being examined. In the present case, matching both the Reynolds and the Froude number in model and prototype would require scaling of not only physical dimensions, but scaling of fluid properties, namely, viscosity, and fluid density. Unfortunately, this can almost never be achieved in the real world due to the fact that fluids with suitably scaled properties almost never exist. In this work water is both the model and prototype fluid for economic reasons. Hence the low head dam is designed to adhere to Froude number scaling. Consequently, a 1:15 Froude scale physical model of a low head-dam was built. The model illustrated in Fig. 2 has all the characteristics of the prototype reproduced to scale and satisfies design restrictions. Scale model studies of low head dams are a cost-effective way of investigating performance of proposed system modifications.


One primary function of modeling is to relax the aforementioned restriction. In this regard, CFD models (through which both the Reynolds and the Froude number in model and prototype can easily be matched) do enjoy an advantage over physical models. However, CFD models need to be calibrated to either prototype or physical model measurements. Figure 2 (a) is a photo showing a profile view of a typical flow over a low-head dam with a plunging nappe. Water surface elevations were measured using an instrumentation platform which could be moved to the desired stream-wise locations. Still and video photography were performed for profile views. The length of the model is approximately 180 cm and its height and width are 65 and 50 cm, respectively. The full description of the physical model is beyond the scope of this work and will be taken up elsewhere.


Figure 2. (a) 1:15 Froude scale model of a low-head dam. (b) Dye injection is used to track the flow profile inside the physical model.

Dye injection is the most common flow visualization technique because of ease of handling. A mixture of vegetable food coloring and water was injected from ports on the model inlet. The dye acts as a tracer as shown in Fig. 2 (b), enabling visualization of the flow in the danger zone downstream of the dam. The real low-head dam on the river has a 1.5 m drop and a typical flow rate of 1.50 . The model flow rate was made correspond to this, with the result 0.39 . The following section describes the governing equations including the level-set method.

GOVERNING EQUATIONS

In the open-channel shown in Fig. 2 water flows with a free surface. The air above the flow is at rest and at standard atmospheric pressure. Open-channel flow is possibly one of the most complicated fields in fluid mechanics (Osman Akan 2006). The fluid flow is primarily controlled by the gravity force and boundary friction. Air–water interactions occur at the free surface through which there are continuous exchanges of both mass and momentum between water and atmosphere. Some low-head dams are built for water quality purposes. Their flows are characterized by significant free-surface aeration which may be caused by a combination of wave instabilities and turbulence fluctuations acting on the air–water free surface. In the process called ‘white water’ (Chanson, 2004), air bubbles may be entrained when the turbulent kinetic energy is large enough to overcome both surface tension and gravity effects. The interface between air and water in free-surface aeration may be resolved using the level set approach developed by Osher and Fedkiw (2003). This technique provides an excellent resolution when the ratio of liquid to gas densities is large. The flow in the channel in Fig. 2 is turbulent. The shear Reynolds number is defined as

. Here, is the shear velocity, is the average surface roughness and is the kinematic viscosity of water. Typical roughness height of the material used in the physical model is 0.01–0.02 mm. The shear Reynolds number is in the range

. Thus the channel operates in the transition region between smooth turbulence and fully rough turbulence (Chanson, 2004). Large eddy simulation of the channel flow in the transition region is considered to be a promising approach (Bates, Lane and Ferguson, 2005). Hence, in the current attempt the aim is to combine the best features of the level set methods with those of the LES in order to achieve more accurate simulations of the flow over a low-head dam.


The filtered continuity and momentum equations for an isothermal, 3D incompressible flow of air-water mixture separated by an interface may be given as

(1)

(2) The viscous stress tensor in (2) is defined as

(3) In (1) and (2), is the i-component of the filtered velocity field, denotes the filtered pressure field, is the subgrid scale (SGS) stress tensor, g is the gravitational acceleration, and is a volumetric source term, such as the interfacial force resulting from the surface tension. Here, the unresolved part of is neglected. The SGS stresses are modeled using a model composed of a scale-similar part and an eddy viscosity part introduced by Smagorinsky (1963). That is

, (4) where is the resolved strain rate tensor. The coefficient C is evaluated dynamically following Germano et al. (1991). The fluid properties such as density and viscosity are discontinuous across the interface. However, the velocity across the interface is continuous. The interface is defined as the zero level set of a smooth function which is a signed distance function. The level set function is defined as

(5) The position of the interface can be tracked by solving a transport equation given as


(6) The physical properties of the fluid (such as density and viscosity) in each phase are given by the following expressions

(7)

(8) where the subscripts a and w represent air and water, respectively. Here, is the smooth Heaviside function defined as (Sussman et al. 1999):

(9) where is the interface thickness. The interface will have a uniform thickness as long as

is maintained as a distance function. To keep as a signed distance function in the evolutionary process the level set function must be re-initialized on the regular time. The re-initialization equation for the level set function may be given as (Sussman et al. 1994)

(10) where is the pseudo time, the smooth sign function is defined as

. The interfacial force resulting from the surface tension, , can be given as (Brackbill et al. 1992)

, (11)

where is the curvature defined as , and is the coefficient of surface tension.


The numerical solutions for the turbulent flow of an incompressible air-water mixture flowing over a low-head dam will be presented in the following section.

RESULTS AND DISCUSSION

The present approach involves using the CFD model detailed in the preceding section

• to validate the model with results from a physical model, • to develop and compare alternatives, and • to verify of the proposed changes in the geometry using the physical

model.

To do these, an incompressible flow of an air-water mixture over a low head dam was simulated using a model whose sketch is shown in Fig. 3 (a). The model is identical to the physical model described in Section II. The problem can be simplified by implementing periodic boundary conditions in the z-direction. Figure 3 (b) illustrates a front view of the system with selected nomenclature. The system was initially filled up with water to the level free surface at . The physical properties of air and water are given in Table 1.

Table 1. The material properties Values Properties Symbols

Air water fictitious fluid

Density

Dynamic viscosity

Surface tension

Figures 4 (a) and (b) illustrate inverted black-white photographs showing a profile view of a flow over a low-head dam with a plunging nappe in the physical model. These photographs were taken at intervals. The visualization of fluid flow is based on the advection and decay of dyes which are popular in flow visualization because they are easy to handle. The dye must be thick enough to minimize diffusion in the working substance. In this study, Sudan III was used which possesses the same density as the water. This dye is preferable in rotational flows as it does not diffuse rapidly. The experimental results illustrated in Figs. 4 (a) and (b) are compared with the results of the simulation in order to verify the validity of the model. To do this, numerical simulations were performed in the computational setup shown in Fig. 3. The free surface was flat before the simulations were started.


The dye is injected into steady motion in the x direction with a uniform inlet profile at . The water from to , is dead storage that does not affect the dynamics of the dam significantly. The inlet boundary conditions at is set to . Furthermore, it is assumed that the top

surface of the model is open to the atmosphere, namely . The outlet

boundary condition is set to . The no-slip boundary condition is applied on the walls of the model.

Figure 3. (a) Perspective view of the sketch of computational setup. (b) Longitudinal

schematic of computational setup. Here,

and . Dimensions are in centimeters.



Figure 4. (a) and (b) Two still frames separated by an interval of

approximately from the movie of the physical model. The frames show profile views of the model. Here, 24 frames were taken for one second of movie. Sudan III dye was used for flow visualization. The original color of the dye was dark-red color which appears light in the inverted black and white images. (c) Computed instantaneous dye

concentration contours. Water was used as the working fluid. (d) Computed instantaneous dye concentration contours. A fictitious fluid whose properties are given in

Table 1 was used as the working fluid. (e) Magnified lower part of (d). (f) Computed instantaneous velocity field upstream of the dam face.

The finite volume method (Patankar, 1980) was used to discretize the equations (1-4) on a non-uniform staggered Cartesian grid. The flow in the system is unsteady. Hence, a semi-implicit time-advancement scheme is used to integrate the momentum equations with the second order Crank-Nicolson scheme for the diagonal viscous terms and the second order Adams-Bashforth scheme for the convective terms. The level set and its reinitialization equation are solved using the third-order Runge-Kutta scheme for time advancement and the fifth-order Hamilton-Jacobi scheme for spatial discretization. The simulations are executed on a graphics processing unit (GPU) (Zamankhan, 2011). Figure 4 (c) depicts two computed contours of the dye concentrations each separated by using an air-water mixture whose properties are given in Table 1. Qualitative as well as quantitative comparisons with the measurements in the physical model at the same locations indicate that the CFD model accurately simulates velocity


distribution. Agreement between the results was considered sufficient to justify use of the CFD model for alternative development. Froude scale models do not accurately simulate the tractive forces because they do not simulate viscous forces. The Reynolds number offset ratio of a Froude scale model is

, where is the geometric scale ratio, which is . This means that the CFD model in Fig. 3 would have the same Reynolds number as the prototype if the viscosity of the working fluid would be . It is instructive to use the aforementioned fictitious fluid (whose properties are given in Table 1) to match both the Reynolds and the Froude number in CFD model and prototype. Figure 4 (d) shows computed instantaneous dye concentrations at the same locations as those depicted in Figs. 4 (a) and (b). The discrepancy between the dye concentration distribution in the physical and the computed results depicted in Fig. 4 (d) highlights the disadvantages of using dyes in visualizing vortical structures in transitional flows in physical models. In these cases, particle image velocimetry (PIV) which is an optical method of flow visualization would be advantageous (Zamankhan 2010). Figure 4 (e) shows the magnified lower part of Fig. 4 (d). Visual inspections reveal the occurrence of interfacial instability of Kelvin-Helmholtz type when the dye injects through the inlet into the water at rest in the channel. As can be seen from Fig. 4 (e), the interface separating the dye and dead storage becomes wavy and because of this deformation of the interface the dye moves slightly faster than before in some regions and it moves slightly slower in other regions. Fig. 4 (f) illustrates computed instantaneous fluid velocity vectors illustrated in the channel. Note that the pressure rises when the dye velocity deceases and falls where the dye velocity increases. The pressure gradients are in the directions producing amplification of the disturbances as clearly can be seen from Fig. 4 (e). LES calculation displays quite nicely the vortical structures in the system. The flow visualization technique provided by the CFD modeling discussed above can be used to aid in an improved understanding of the flow dynamics in the danger zone shown in Fig. 2 (b). The physical model was used to simulate flow angle, and the recirculation pattern in the danger zone. Figure 5 (a) illustrates a faster, shallow upstream flow changes to a slower and deeper downstream flow. The nappe is submerged by tailwater and a standing wave can be observed located at from the face of the dam. The solid line aa in Fig. 5 (a) represents the angle of the diverted flow approaching the step like obstacle at the bottom of channel. The flow angle and the location of the standing wave computed using the CFD model were compared with those measured in the physical model. Comparisons were made using aforementioned flow visualization technique.



Figure 5. (a) A still frame from the movie of the physical model which shows dye

concentration in the spillway. The solid line represents the direction of diverted flow. The dashed line represents the level of water upstream of the dam. Water was used as working

fluid. (b) Computed instantaneous dye concentration in the spillway. The dashed line “aa” is the same as the solid line “aa” in (a). (c) Dye concentration in the danger zone of the physical model. Note that the circulation pattern cannot be clearly visualized due to rapid dispersion of the dye. Here, represents the horizontal distance of the standing wave from the dam face and is the vertical distance between the boil and the water

level upstream of the dam. (d) Computed instantaneous dye concentration in the danger zone. The agreement between (c) and (d) is quite impressive. (e) Computed instantaneous velocity field in the danger zone. (f) The circulation patterns of flow in the danger zone.

Note that there exists a pair of counter rotating vortices of different strengths in the danger zone. Here, water was used as the working fluid.

Figure 5 (b) illustrates computed instantaneous dye concentrations at the same locations as those depicted in Fig. 5 (a). Figure 5 (c) shows the stream of water in the physical model splits at the boil. The circulation pattern cannot be clearly visualized due to rapid dispersion of the dye in the recirculating flow in the danger zone. Figure 5 (d) illustrates the CFD model predictions. Agreement between the results was considered quite


impressive. The CFD model also provided the ability to visualize the velocity vectors in the danger zone as shown in Fig. 5 (e). Figure 5 (f) visualize the circulation patterns of flow for a typical low head dam. A careful inspection of Fig. 5 (f) reveals that there exists a pair of counter rotating vortices of different strengths in the danger zone. These vortices could be responsible for the high dispersion of dye as illustrated in Fig. 5 (c). In this section the validated CFD model for the low head dam and the model detailed in Zamankhan (2009) for water-sand mixtures were combined to develop a generalized model for determining how to change the downstream geometry of low-head dams to eliminate the vortices in the danger zone. In brief, Zamankhan (2009) used large-eddy simulation technique for dense particle-fluid flows in order to predict the current-induced scour for both the mono- and bidispersed systems below a horizontal submarine pipeline exposed to unidirectional flow. The simulations were four-way coupled, which implies that both solid-liquid and solid-solid interactions are taken into account. Particles are assumed to behave as viscoelastic solids during interactions with their neighboring particles, and their motion are predicted by a Lagrangian method. The interparticle normal and tangential contact forces between particles are calculated using a generalized Hertzian model. The other forces on a particle that are taken into account include gravitational, pressure gradient force accounting for the acceleration of the displaced liquid, the drag force resulting from velocity difference with the surrounding liquid, and the Magnus and Saffman lift forces. To simulate situations in which dangerous circulatory flow patterns can be eliminated, a set-up including a vertical hopper was used as illustrated in Fig. 6 (a). This figure and Fig. 6 (b) show schematics of the CFD model and selected nomenclature. The xy plane cross section of the hopper is a trapezoid. The lengths of two bases are and , respectively, and the altitude of trapezoid is . The width of the hopper is . The hopper was initially filled with a ternary mixture of spherical particles of same

density . The diameter and volume fraction of the smallest particle size in the mixture was , and . The diameter of particles 2 and 3 (largest particles) was , and , respectively. The total solid volume fraction in the hopper was and the solid volume fraction of the largest particles was . Flow of sand particles was created due to gravitational force their free fall was controlled by a gate whose opening is , as illustrated in Fig. 6 (c). Figure 6 (d) depicts a snapshot of granular mixture out of the hopper. It is known that the tractive stress on a solid particle fluctuates because of turbulence. In addition, the drag force and turbulence are a function of the Reynolds number. CFD models involving transport of noncohesive granular materials must simulate tractive stress. Indeed, the tractive stress causes the drag force required to overcome the gravity forces holding the sand particles in a system as shown in Fig. 6 (e). As stated earlier, Froude scale models do not simulate the tractive forces and sediment transport accurately. To deal with this problem, the fictitious fluid whose physical properties are given in Table 1 was used as the working fluid in the simulations.



Figure 6. (a) and (b) Schematics of the CFD model and selected nomenclature. (c)

Schematic of the hopper. (d) A snapshot of solid particles flowing out of the hopper. (e) Perspective view of flowing solid particles in the spillway. (f) Front view of (e). (g) An instantaneous configuration of particles. (f) and (g) are each separated by . (h) Computed velocity vectors around the hopper. Inset: Magnified version of (h) in order to better visualize the flow around the gate. (i) The average total solid volume fraction at the sampling port in (h). (j) Computed instantaneous dye concentration contours after

modifications. Note that the vortices shown in Fig. 5 (f) were eliminated.


In the present simulations, the steady motion in the x direction with a uniform inlet profile at was used. Figure 6 (f) shows a front view of moveable granular materials. Figure 6 (g) illustrate computed instantaneous configuration of particles. This configuration and that shown in Fig. 6 (f) are each separated by . Large-scale simulations of the sediment transport at interactive rates, which provides insights into these systems with extensive associated complexities, is feasible using multiple GPUs. The CFD model also provided the ability to visualize the velocity vectors around the hopper as shown in Fig. 6 (h). This information may be useful in controlling transport capacity. Figure 6 (i) depicts the average total solid volume fraction at the sampling port shown in Fig. 6 (h).

Figure 6 (j) illustrates computed instantaneous configuration of particles after . The hopper has been removed and the inlet velocity has been increased to . As can be seen from Fig. 6 (j) the hazards associated with rollers in the danger zone have been partially diminished. The present approach may offer suggestions for new ways for eliminating dangerous circulatory flow patterns in a low head dams. The verification of the proposed changes in the geometry using the physical model will be taken up elsewhere. Note that the particle sizes have to be adjusted in the Froude scale model illustrated in Fig. 2 for a Reynolds number that is much lower.

CONCLUSIONS

Relying on data from the CFD and physical models, some changes are proposed in the geometry of the danger zone of a low head dam to eliminate its dangerous circulatory flow patterns. It is shown that large-scale simulations of the aforementioned system at interactive rates using multiple GPUs can provide insights into the system not to be obtained in any other way. The verification of the proposed changes in the geometry using the physical model will be carried out in the future.

ACKNOWLEDGEMENTS

The author is grateful to Hrafn Arnórsson and Gísli Steinn Pétursson for their careful review of this manuscript and suggestions for many revisions.

REFERENCES

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Germano, M., Piomelli, U., Moin, P. and Cabot,W. H., A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760 (1991). Leutheusse, H. J., and Birk, W. M., Drownproofing of low overflow structures, J. Hydr. Engrg. 117, 205 (1991) Munson, B. R., Young, D. F., and Okiishi, T. H., Fundamentals of Fluid Mechanics, John Wiely & Sons, New York, 2002. Osher, S., and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, Springer, New York, 2003. Osman Akan, A., Open Channel Hydraulics, Elsevier, Amsterdam, 2006. Patankar, S. V., Numerical heat transfer and fluid flow, McGraw Hill, New York, 1980. Pope, S. B., Turbulent Flows, Cambridge University Press, Cambridge, 2000. Smagorinsky, J. General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weath. Rev. 91, 99 (1963). Sussman, M., Smereka, P., and Odher, S. J., A level set approach for computing solutions to incompressible two-phase flows, J. Computational Physics, 114, 146 (1994). Sussman, M., Bell, J. B., Almgren, A.S., Howell, L. H., Colella, P., and Welcome, W. L., An adaptive level set approach for incompressible two-phase flows, J. Computational Physics, 148, 81 (1999). Vischer, D. L., and Hager, W. H., Dam Hydraulics, John Wiely & Sons, New York, 1998. Wyrick, J. R., Rischman, B. A., Burke, C. A., McGee, C., Williams, C., Using hydraulic modeling to address social impacts of small dam removals in southern New Jersey, J. Environmental Management, 90, 270 (2009). Zamankhan, P., Analysis of submarine pipeline scour using large-eddy simulation of dense particle-liquid flows, J. Offshore Mechanics and Arctic Engineering, 131, 021604 (2009). P. Zamankhan, Large Eddy Simulation and Piv Experiments of Air-Water Mixing Tanks, Commun Nonlinear Sci Numer Simulat 15, 1511 (2010). Zamankhan, P., Bubbles and solid structures in a vibrated bed of granular materials, Physica A, doi:10.1016/j.physa.2010.11.039 (2010).


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LARGE-EDDY SIMULATION OF FLOW OVER A LOW-HEAD DAM