CHANNEL DROPS: A COMPARISON BETWEEN CFD SIMULATIONS AND EXPERIMENTAL OBSERVATIONS

Transcription

1 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops CHANNEL DROPS: A COMPARISON BETWEEN CFD SIMULATIONS AND EXPERIMENTAL OBSERVATIONS Guido CALENDA, Michele DI LAZZARO, Aldo FIORI, Pietro PRESTININZI, Elena VOLPI * Keywords: energy dissipation; drop structures; sewer; computational hydraulic ABSTRACT Laboratory measurements of the free surface profile were carried out over a physical model of a classical channel drop inserted in a sewer, equipped with a depressed stilling basin. This devise is still in use in several combined sewer systems and may have unfavourable consequences on the water quality of the receiving bodies. The present paper is part of a study to economically modify the existing structure in order to prevent dry weather sedimentation and heavily polluted first flush flow. The experimental free surface profiles observed in different conditions were compared with numerical simulations obtained using the computational flow dynamic software FLOW 3-D with the imposed conditions at the downstream boundary. The standard k-ε turbulence model is used in conjunction with the volume of fluid (VOF) free surface model. The computer simulations faithfully reproduce the water level in the upstream pipe branch and in the dissipation manhole. However, some discrepancies were observed between the measured and simulated jet, and in the position of the hydraulic jump. * Dipartimento di Scienze dell Ingegneria Civile, Università degli Studi Roma Tre 1

2 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS 1. INTRODUCTION 1.1 CHANNEL DROPS WITH STILLING BASIN Channel drops in ancient sewer systems were equipped frequently with a stilling basin, in order to force the hydraulic jump to develop in the manhole, preventing it from drifting in the downstream sewer. An example of the device is shown in figure 1. Figure 1 - Channel drop used in some ancient sewer system. When public attention began to focus on water pollution problems, it soon appeared that the use of such structures, especially in combined sewer systems, involves serious environmental drawbacks. During dry weather flow the stilling basin becomes a regular settling tank, where sediments accumulate day after day, until storm runoff washes the settled solids downstream discharging them in the receiving bodies, producing severe shock water pollution. This fact may explain the occurrence of first flush pollution events, especially after long summer dry periods, even in high sloped sewer systems, where sedimentation is not expected. Final objective of this study is to find an economical device to effectively dissipate the energy after the elimination of the stilling basin by filling the depression of the floor. To this end a physical model of the channel drop has been built in order to investigate on its hydraulic behaviour, and verify the dissipative efficiency of alternative configurations. The use of physically based numerical modelling may save the money and the time expenditure of the hydraulic model studies, allowing a far greater versatility in testing different designs, by easily changing the boundary conditions (i.e., the shape of the model structure) and the flow conditions of the system. However, the reliability of the computational flow dynamic (CFD) 2

3 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops models in predicting the flow patterns under fully turbulent conditions is still an open question. The aim of this study is to check the numerical solution against the experimental results and gain the necessary insight in the behaviour of the numerical code. Thus, in the present work a first comparison between a field experiment on dissipation manholes and the results of computational flow dynamics model Flow 3-D is performed. 1.2 PREVIOUS STUDIES The first experimental study of free overfalls over flat bottoms dates back to the work of Moore (1943) whose results are still widely used. The main purpose of the author was to study the energy dissipation at the base of the fall in a rectangular channel, since the characteristics of the downstream flow and the design of the structure are strongly influenced by this loss. Measurements of surface elevation and velocity were taken, respectively using a point gage and hypodermic needles. The author first demonstrated that, contrary to previous beliefs, a substantial amount of energy was lost at the impact of the jet on the floor of the lower channel, well upstream of the hydraulic jump. If H is the drop height, h c is the critical depth, h p is the water level in the pool behind the falling jet and h 1 is the water level at the base of the fall (see figure 2), according to the hypothesis that no energy losses occur upstream of the hydraulic jump, the ratio between the head E 1 at the base of the fall and the critical depth h c should be given by E h c H Q H 3 = + 1+ = (1) 2 3 h 2gb h h 2 c c c Figre 2 - Diagram showing a drop structure and the water depths considered by Moore (1943). 3

4 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS Interpreting the experimental data, Moore proposes an empirical equation describing the energy loss as a function of the ratio between the drop height H and the critical depth, as shown in figure 3, where the horizontal distance between the experimental and the theoretical curves represents the energy loss divided by the critical depth. Within the range of the physical experiments conducted by Moore, it appears that the energy loss at the base of a free overfall increases with the drop height, and reaches the value of 1/2 for H h c = 10 and almost 2/3 for H h c = 15. Figure 3 - Experimental results on energy losses at drops as a function of drop height: observed measurements (circles), fitting of the data (black curve) and curve of no energy loss. Moore also noticed that the depth of water in the pool behind the jet was considerably greater than that in the downstream channel. The equation of the horizontal momentum coupled with the continuity equation yields h h p c 2 h1 hc = + 3 h (2) c h1 where h p is the water depth in the pool. Moore found that the measures of the water depth h p and of the tailwater h 1 well agreed with equation (2). He also investigated the velocity profile at the base of the fall when an hydraulic jump is forced into the dissipation structure. He found that in this case a rapid decrease of the energy line occurs just upstream the starting point of the hydraulic jump. 4

5 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops Finally, Moore found a good agreement between the measured experimental depths upstream and downstream of the jump and those computed using the so-called Belanger equation, derived from the momentum principle. Other researchers pointed out that in his work Moore didn t use the actual measures of h 1, since he found them difficult to obtain, but inferred these values indirectly from the measures of the kinetic energy of the jet. This could of course affect the estimate of the water depths h 1 and h 2, and partially impair the results. Discussing the paper of Moore, White (1943) predicted the energy losses proposing a theoretical solution based on several assumptions that can be summed up as follows: 1 the circulating flow in the pool at the bottom of the drop is the same as the backward flow in an jet falling with the same angle θ on a flat bottom, when the flow in the upstream and downstream directions is free and unobstructed; 2 the velocity of the supercritical stream at the end of the drop, v, is the same as the velocity in the thicker forward current downstream of the jet impact; and as the velocity of the backward current. This implies that the thickness of the approaching jet is simply the sum of the depths d a and d b (figure 4) 3 the return flow has negligible momentum in the direction of the jet. This considerably simplifies the problem, but leads to a substantial underestimation of the effects of the pool. Figure 4 - Sketch showing the schematization of the impinging jet according to White (1943) Using the horizontal momentum equation the ratio between the water depth in the channel and in the pool is: 5

6 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS d d b a 1 cosθ = (3) 1 + cosθ This ratio is equal to the ratio between the recirculation flow in the pool and the total flow. Though some assumptions in the theory of White seem indeed drastic, the comparison between the computed energy losses obtained by the author and those observed by Moore are in good agreement. However the values of the water level in the pool, h p, successively computed by Gill (1979) using equation (1) with the h 1 values given by White, are consistently lower than those observed by Moore. Rand (1955), using the experimental data of Moore, proposed empirical equations to compute the various hydraulic elements of a vertical drop structure. Some of the drastic assumptions made by White were critically analyzed and replaced by Gill (1979). He modified the velocity expressions in order to explicitly account for the influence of the pool. The impact angle θ of the jet (see figure 4) is smaller than the one assumed by White, and the discharge circulating in the pool is accordingly smaller than that flowing freely in the backward direction in White s scheme (figure 4). Gill made some experiments to obtain the pool depth and the inclination of the falling jet, plotting both the lower and the upper surfaces. He found that his modified theory shows a better agreement with the observations, if compared with White s theory and with the empirical equations proposed by Rand. Rajaratnam and Chamani (1995) analysed the energy losses of the drops, using experimental results to evaluate critically the assumptions of White and Gill. Their work indicates that the loss is mainly due to the mixing of the water in jet with that in the pool. Recently Chamani and Beirami (2002) observed velocities and surface elevations with subcritical and supercritical approach flows. The differences between the predicted parameters of the flow and the experimental results were explained in terms of the assumptions made (neglecting the air entrainment and the bed shear stresses). 2. EXPERIMENTS 2.1 EXPERIMENTAL INSTALLATION The experiments were made in the Hydraulics Laboratory of "Roma Tre" University. The setup of the experiment is schematically represented in figure 5 and a general view is shown in figure 6. A pipe ends with an abrupt drop in the energy dissipation manhole, followed by an outlet conduit. Both the inflow and outflow conduits are plexiglass circular pipes, with diameter D = 0,29 m. The length of the upstream pipe is 6 meters the maximum that could be obtained in the 6

7 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops experimental setting in order to reduce the effects of the geometry transition from the rectangular supplying water tank, located upstream, to the circular section of the pipe. The transition from the tank to the pipe is suitably tapered. The 2 meters long downstream pipe ends with an adjustable weir. Figure 5 - Schematic representation of the experimental installation Figure 6 - General view of the experimental set-up. The model of the manhole, also made of plexiglass, reproduces the typical configuration of the drop structures shown in figure 1. The manhole is 0.31 m wide and 2.2 m long. This length is 7

8 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS equal to the sum of the distance reached by the jet and of the full development of the hydraulic jump with the maximum experimental flow, according to classic experimental equations (Hager, 1999). Different drop heights can be easily obtained by changing the elevation of the bottom. The entire model is rigidly supported by a steel structure, whose slope can be varied by small rotations. The fulcrum is located near the upstream tank, and an oleodynamic equipment, located 9 meter downstream, regulates the vertical position of the beam extremity. By changing the slope, both subcritical and supercritical flows conditions can be obtained in the upstream conduit. 2.2 FLOW CONDITIONS The water is pumped from the laboratory reservoir to the stilling tank above the flume, which it feeds the upstream conduit. The required flow is obtained by regulating a manually operated valve. The adjustable weir at the downstream end of the flume controls the tail water depth. The flow measurements are carried out using a sharp-crested triangular weir (a classical V- notch). Particular attention is paid to dampen the turbulence of the flow approaching the weir by means of baffles. The efflux law of the weir was calibrated running volumetric tests. All the experiments were carried out fixing a constant m/m slope. Two different flow conditions were tested: 25 l/s and 20 l/s. A first experimental series was performed setting a 0.35 m drop, in order to annul the end sill between the manhole floor and the bottom of the downstream pipe. 2.3 ANALYSIS OF THE EXPERIMENTS The measure of water depths were taken at fixed locations along the pipes and the manhole, using a point gauge mounted on a carriage that could move on rails along the whole experimental set-up. The measures in the upstream pipe showed that the flow was subcritical both for the 20 l/s and 25 l/s experimental discharges. The flow conditions in the downstream pipe depend on the downstream boundary conditions. Four different downstream conditions were selected, and the following flow conditions were observed: 1. downstream gate is removed (free outflow at the downstream end of the pipe): supercritical flow develops in the manhole and in the downstream pipe (figure 7); however, the brusque geometry transition from the manhole to the pipe causes a significant local rise of the water surface (figure 8); 2. downstream weir fixed at h g = m: the flow is still supercritical in most of the pipe, but a hydraulic jump develops close to the gate (figure 9); 3. downstream weir fixed at h g = m: the hydraulic jump moves upstream in the pipe; 8

9 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops 4. downstream weir fixed at h g = m: the hydraulic jump in the manhole is completely drowned and the flow is subcritical both in the downstream pipe and in the manhole (figures 10 and 11). The free surface elevations measured in the four cases are represented in figure 12. Similar flow conditions were obtained with a 20 l/s discharge. Figure 7 - Supercritical flow downstream the jump (case 1 and 2) Figure 8 - Supercritical flow in the pipe (case 1 and 2). Note the increase in water depth at the transition from the manhole and the pipe. Figure 9 - Supercritical flow in the downstream pipe and hydraulic jump observed just upstream the weir (case 2). 9

10 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS Figure 10 - Direct hydraulic jump in the manhole (case 4). Figure 11 - View of the flow patterns when a direct hydraulic jump occurs (case 4). 3. NUMERICAL MODEL 3.1 NUMERICAL CODE FLOW-3D is a finite-difference, transient-solution code that solves over a non-uniform Cartesian grid the Navier-Stokes equations: 10

11 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops y (m) Water surface elevation h g = 0m y (m) h g = 0.033m bottom/top observed critical depth x (m) x (m) 0.6 y (m) 0.6 y (m) 0.5 h g = 0.053m 0.5 h g = 0.063m x (m) x (m) Figure 12 - Observed water depths at m 3 /s flow discharge and for different levels of the downstream weir. ui t ui xi u + x = 0 i j 1 p = ρ x i + G i τ 1 τ bi ij + ρ ρ x i (4) where u i is the velocity component in the i-th direction (i = 1 3), t is the time, x i is the spatial coordinate in the i-th direction, p is the pressure, ρ is the local density, G i are the body accelerations, τ ij are the viscous stresses and τ bi are the wall shear stresses (activated only close to solid boundaries). The origins of the FLOW-3D code can be traced back to the Los Alamos National Laboratory and to the research of C. W. Hirt, which led to the creation of the SOLA-VOF, NASA-VOF, and RIPPLE codes (Nichols, et al. 1980; Hirt and Nichols, 1981; Torrey, et al. 1987; Kothe, et al. 1991). The developed free-surface algorithm, called the Volume-of-Fluid (VOF) method, tracks 11

12 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS the movement of the free surface by calculating the fraction of fluid F in each computational cell: its value at any grid-point is 1 if the cell is entirely occupied by the fluid and 0 if it is completely void (figures 13 and 14). The evolution equation of this scalar is solved simultaneously using equation (4): ( Fu ) F + t x j j = 0 (5) Following the VOF method, the fluid is allowed to collide with the solid bodies, to form or destroy bubbles, and to develop transient shocks and jets. The VOF method simulates the sharp interfaces accurately, but does not compute the dynamics in the void or air regions. FLOW-3D allows the user to choose among a variety of turbulent closure schemes, including: Prandtl mixing length, one-equation (turbulence energy) transport, two-equation k-ε transport, renormalized group theory (RNG), and large eddy simulation (FSI 2000) 1. Numerical efficiency in the model is achieved by using explicit solution schemes whenever possible; however the user can select implicit solutions if desired Figure 13 - Visual representation of the fraction of fluid for the discrete elements scheme. Another distinctive feature of the model is the Fractional Area Volume Obstacle Representation (FAVOR) technique (Hirt and Sicilian, 1985), which allows for the definition of solid boundaries within the Eulerian grid. When a solid surface intersects a cell, FAVOR determines the fractions of areas and volume open to the flow in it. In this way, the process of defining boundaries and obstacles is performed independently of the mesh pattern, avoiding sawtooth representation or the use of body fitted grids. In fact, the geometry can be defined using a built-in solid modeller, which includes quadratic functions that represent objects, or it can be externally provided through CAD or ANSYS formats. Once the geometry is defined, the 1 FSI (2000), Flow Science Inc. web page URL: 12

13 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops computational mesh is constructed independently, and, with some limitations, can be thickened in zones of the domain of particular interest. The average values for the flow parameters (pressure and velocity) in each cell are computed at discrete times using a staggered grid technique (Versteeg and Malalasekera 1995). Figure 14 - Fraction of fluid simulation of a cross section in the downstream pipe. The basic algorithm for a time unit advance of the solution consists of the following three steps (Flow Science, 2000): 1. compute the velocities in each cell using the initial conditions or the values at the previous time-step for all the accelerations components (advective, pressure, and others), using an explicit approximation of the momentum (Navier-Stokes) equations; 2. adjust the pressure in each cell to satisfy the continuity equation at the given the time step; 3. update the fluid free surface or interface to obtain the new fluid configuration, using the volume of fluid value in each cell. 13

14 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS To perform step 2 several implicit solver can be selected, while an explicit solver is available only for compressible fluids. Instead, both implicit and explicit solvers can be used for time increment solutions in step NUMERICAL MODEL OF THE EXPERIMENTS The geometry of the physical model, described in section 2.1, was reproduced as a single vectorial solid in a CAD program and imported into Flow3D as a STL (stereolithography) obstacle. The bottoms of both pipes and manhole were assumed parallel to XY plane, and the slope was simulated imposing non-zero gravity component along horizontal directions. The mesh size was uniformly set at 2 cm along the three directions. Several obstacle faces were inserted as mesh constraints along the three directions, in order to implement the FAVOR procedure. These constraints slightly modified the cell size over the whole domain, but the change of the aspect ratio did not exceed 0.2. Boundary conditions were imposed as follows: i. upstream: the flow depth and an uniform cross section distribution of the velocity, in order to reproduce a constant inflow discharge: the depth value was that observed in the physical model, and the velocity was obtained by the discharge-wetted area ratio; ii. downstream two different conditions are considered: - an abrupt drop, simulated by extending the mesh beyond the end of the pipe, and allowing the jet to live the domain with an OUTFLOW type boundary condition; - a weir, simulated as a vertical obstacle at the end of the pipe; iii. the flow never reaches the lateral, bottom, and top boundaries of the numerical domain being always contained by solid obstacles (pipe and manhole walls). Since Flow3D does not include a steady-state solver, a pseudo-steady condition was obtained as an asymptotic solution of the unsteady simulation, having imposed initial dry bed conditions. The program was set to solve the Reynolds Averaged Navier-Stokes Equations (RANSE): among RANSE approaches, the k-ε closure model was chosen, its use being extensively documented in literature. Among the implicit solvers available for the pressure solution, the standard SOR technique was used; an explicit scheme was used instead for global advancing in time. The wall friction is modelled assigning a metric surface roughness: since the experimental apparatus is made of Perspex, a roughness order of only 10-4 m was assumed. The roughness is not expected to a significant role in the problem at and, especially since the wetted walls are very smooth; however the influence of the roughness was checked by running the model using three likely values of the equivalent roughness height (5 10-5, and m) and comparing the results. 14

15 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops 4. RESULTS AND DISCUSSION The models was run with only a 25 l/s flow, and the upstream depth was set at the observed value of m, resulting in an inflow velocity of 0.73 m/s. Two scenarios were simulated: 1. gate height h g = 0 m (gate removed), 2. gate height h g = m. The performance of the code was assessed comparing the depth measured in middle longitudinal section in 19 gauge locations. Since fluctuating levels were observed everywhere, even in stationary conditions, the flow depth in each location was averaged over a period of 7 seconds. The computed water surface elevation (WSE) profiles are compared to the observed values in figure 15 (scenario 1) and in figure 16 (scenario 2) 2. The results show that almost everywhere the computed profiles are not influenced by the assumed roughness, as expected for such smooth walls. What is surprising is that in case 1 the three profiles diverge at the passage from the manhole to the downstream pipe, where the profiles are significantly different. This is a feature that, obviously, cannot be explained in term of roughness of the wall. Probably it depends on some form of physical instability, since in this case the downstream depth is very close to the critical depth. The problem will be further investigated. In the upstream conduit, where the flow is subcritical and it is mainly influenced by and the upstream boundary and by the drop, the computed profiles approximate well the observed water elevations. As shown in figure 17, the direction of the computed jet has a horizontal component greater than that of the experimental jet. This feature was already highlighted by Cook and Richmond (2001). No further analysis has been as yet performed to evaluate the nature of this difference, that may depend on the relatively coarse numerical grid. Downstream of the drop the profile of case (i) and (ii) are different, since they are influenced by the boundary condition at the end of the downstream conduit. In case (i), as already stated, the supercritical current in the downstream conduit is very near to critical conditions, so that even small roughness changes may produce computed patterns that are significantly different, and propagate all along the pipe, influencing the water levels at the downstream end of the manhole. In the manhole, after the jet, the current is always supercritical and the computed profiles, that are not influenced by the roughness, fit well to the observed depths. 2 When in a position (x, y) there is more than one free surface, as it happens in correspondence of the free fall, where three water surfaces are superimposed, the program plots only the height of the lower free surface: therefore only the surface of the pool below the jet appears in figure 15 and

16 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS y (m) Water surface elevation bottom/top observed critical simulated r=5e-5m simulated r=1e-4m simulated r=2e-4m simulated r=3e-4m x (m) Figure 15 - Simulated and observed free surface profiles for h g = y (m) Water surface elevation bottom/top observed critical simulated r=5e-5m simulated r=1e-4m simulated r=2e-4m x (m) Figure 16 - Simulated and observed free surface profiles for h g = 0.63 m. In case (ii) the current in the downstream conduit is subcritical. The computed depth over the weir is somewhat lower than the observed one, and the difference propagates upstream in the conduit and then in the manhole, so that the computed position of the jump is 0.75 m downstream of the observed one. It is likely that also this fact is due to the coarse representation of the weir by a 2 cm mesh, even if the height of the weir was fixed at a value that matches the limit of the meshes. The matter requires a careful assessment and is presently under investigation. 5. CONCLUSIONS The first results of the comparison of the numerical simulation of a channel drop with the experimental measurements show some promising features that encourages further research. The main problems that emerged seem related to the influence of the size of the grid meshes. 16

17 G. CALENDA, M. DI LAZZARO, A. FIORI, P. PRESTININZI, E. VOLPI Channel drops Figure 17 - Angle and range of the jet, compared to the experimental photographic documentation. Different mesh sizes will be investigated, but a serious limit is posed by the heavy computation burden of the program. If satisfactory results will be obtained with the present set-up, changes of the physical model configuration and their numerical simulations will be used to validate the FLOW-3D code for the problem at hand. 6. ACKNOWLEDGEMENTS The present study was supported by the Italian Ministry of University and Research - PRIN 2005/07 (project n _001). 7. REFERENCES Bombardelli, F.A., C.W Hirt,. and M.H. García, Discussion on Computations of Curve Free Surface Water Flow on Spiral Concentrators by B. W. Matthews, C. A. J. Fletcher, A. C. Partridge and S. Vasquez, Journal of Hydraulic Engineering, ASCE 127(7), ,

18 STANDARD DESIGN OF HYDRAULIC STRUCTURES IN URBAN DRAINAGE SYSTEMS Chamani, M.R., and M.K. Beirami, Flow characteristics at drops, Journal of Hydraulic Engineering, ASCE, 128(8), , Cook C.B., and M.C. Richmond, Simulation of tailrace hydrodynamics using Computational fluid dynamics models, Pacific Northwest National Laboratory Richland, Report Number: PNNL-13467, Gill, M.A., Hydraulics of rectangular vertical drop structures, Journal of Hydraulic Research, 17(4), , Hager, W.H., Wastewater hydraulics, theory and practice. Springer, 1999 Hirt, C.W. and B.D. Nichols, Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, Journal of Computational Physics, 39(1), Hirt, C.W. and J.M. Sicilian, A Porosity Technique for the Definition of Obstacles in Rectangular Cell Meshes, Proc. Fourth Int. Conf. Ship Hydro., National Academy of Science, Washington, DC., Sept Kothe, D.B, R.C. Mjosness, and M.D. Torrey, RIPPLE: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory Report LA MS, Moore, W.L., Energy Loss at the base of a free overfall, Transactions ASCE, 108, , Nichols, B.D., C.W. Hirt and R.S. Hotchkiss, SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries, Los Alamos Scientific Laboratory Report LA- 8355, Rajaratnam, N., Chamani, M. R., Energy-loss at drops, Journal of Hydraulic Research, 33(3), , Rand, W., Flow geometry at straight drop spillways, Journal of Hydraulic Engineering, ASCE, 81(1), 1-13, Torrey, M.D., R.C. Mjolsness, L.R. Stein, NASA-VOF3D: A Three-Dimensional Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory Report LA11009-MS, Versteeg, H.K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics The Finite Volume Method, Prentice Hall, p.257, White, M. P., Discussion of Moore (1943), Transactions ASCE, 108, ,