6.2 Dam Break Against a Vertical Wall . . . 116 6.3 Dam Break Against a Vertical Cylinder . . . 132 6.4 Closure . . . 145 The aim of this chapter is to validate the developed solver and evaluate its accuracy in the simulation of dam-break flows by comparing with the experimental data presented in the previous chapter as well as other experimental works in literature. Additionally, the numerical modeling of the physical aspects pertaining to such flows, such as turbu-lence, gate effect, and three-dimensionality, are extensively investigated and the results are reported.
6.1 Solver Configuration
The numerical simulations of dam break flows were conducted using the configurations presented in Table6.1. The configuration is used for all numerical simulation presented
113
Table 6.1: Solver configuration parameters
Parameter Value
Pressure-Velocity coupling PISO method
Number of PISO steps 4
Number of non-orthognality
pressure correction steps 1
Velocity gradient method Face-Averaged Green-Gauss method Pressure gradient method Cell-based least square method
Convection scheme TVD-Van Leer scheme
Turbulence Model Standard k-ε model
Timestepping scheme Implicit Euler method Maximum Courant Number (CFL) 0.40
Water density (kg/m3) 999.7
Air density (kg/m3) 1.246
Water dynamic viscosity (kg/(m s)) 0.00001778 Air dynamic viscosity (kg/(m s)) 0.001307 Gravity acceleration (m/s2) 9.8 Surface Tension Coefficientσ (N/m) 0.0742 Wall adhesion contact angle(◦) 47
UMTHINC:β 4.0
UMTHINC: Volume fraction gradient Node-Averaged Green-Gauss method UMTHINC:decalculation method Developed revised method
with 32 quadrature points UMTHINC:αf calculation method Integrated with 25 points
distributed on face surface Gate motion profile Developed two equation profile Gate motion profile: v0(m/s) 5.055
Gate motion profile:t0(s) 0.036
Gate motion profile: b 55.2
in this chapter unless explicitly stated otherwise.
In order to model the moving gate, in dam break flow, a new method have been developed. In this model, the gate is treated as a zero thickness flat plate snapped to
115 6.1. SOLVER CONFIGURATION Algorithm 1:Algorithm of the developed dam-break gate model
Data: Att=0: Create a list of the mesh internal faces (not boundary faces) coincinding on the gate surface plane designated as gate face list.
whilet < tgatedo
ComputeZgate(t)from the gate motion profile ( equation or dataset) ; Examine the gate face list and omitt face where z-component of the face
center is less thanZgate;
Update the wall velocity of the faces remaining in the list based on the gate motion profile ( equation or dataset) ;
end
liquid air Moving Gate
liquid air Moving Gate
liquid air Moving Gate
Figure 6-1: Illustration of the developed gate model for dam-break flow simulations the cell faces coinciding on the gate surface plane. Such cell faces are treated as double sided wall boundary ( shell boundary with a finite wall velocity ). Thus, the new method requires that the surface mesh on the gate is initially constructed and then volume mesh generated to rest of the domain. The gate motion is introduced by omitting the cell faces which no longer coincide on the gate surface plane according to the prescribed motion profile then updating the wall velocity accordingly. The complete algorithm of the new method is presented in Algorithm 1 coupled with the illustration given in Figure 6-1 wheretgateis the duration of gate motion.
The new method avoids the complexity of dynamic meshing methods which is the straight forward approach in such situations while maintaining accurate modeling of the gate motion. The new method also eliminates the risk of mass loss and waste of computational resources presented by the immersed boundary method which can be rather complicated when applied to unstructured meshes. Additionally, the developed
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
method allows for straight-forward implementation of turbulence wall functions and wall adhesion models without any special treatment/formulation.
117 6.2. DAM BREAK AGAINST A VERTICAL WALL main boundaries are set to wall condition except for square hatch, 0.2m×0.2mdirectly above the water column, which is set to open boundary conditions. Three 2D grids were adopted to investigate mesh dependence of the numerical solution. The minimum and maximum mesh spacing for the three grids are provided in Table 6.2. Figure 6-3 shows the wave front, free surface profile and forces over the impact wall solved on three different grids. The results of the fine and medium grids are fairly close while the coarse mesh shows considerable difference especially for the viscous shear force.
Table 6.2: Mesh spacings of the three grid adopted in the mesh independence test min. spacing (mm) max. spacing (mm) Total number of cells
Coarse 1 10 13205
Medium 0.5 5 47952
Fine 0.25 2.5 213063
(a) Wave front evolution (b) Free surface profile att∗=5.6
(c) Pressure force on the impact wall (d) Shear force on the impact wall Figure 6-3: Mesh independence test
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
Figure 6-4: Computational mesh for the dam-break flow against vertical wall
Therefore in the subsequent numerical computations both two dimensional and three-dimensional, grids with minimum spacing of 0.5mm and maximum spacing of 5mm will be used. The 3D mesh that is used in the subsequent simulations is presented in Fig-ure6-4. The unstructured mesh was generated using the GMSH open-source software and it consists of 1,958,114 cells.
119 6.2. DAM BREAK AGAINST A VERTICAL WALL Figure6-5shows the time history of the wave front position as compared to several recent experimental results as well as a numerical 3D case without gate modeling. The gate effect results in a slower wave and the gate motion profile directly affects its in-stantaneous motion. However, some discrepancy is found between numerical (with gate model) and experimental data between t∗=1.0 and 1.5 . These discrepancies can be attributed to the small differences between the implemented gate profile and the actual gate profile of each experiment.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
t∗=tq g H x H
Present 3D Simulation ... H = L = 0.20m Present 3D Simulation (No Gate) ... H = L = 0.20m Experiment ... H = L = 0.20m
Sueyoshi exp.(2015) ... H = L = 0.20m Lobovsky exp.(2014) ... H = L = 0.60m Hu and Sueyoshi exp(2010) ... H = 2L = 0.40m
Figure 6-5: Wave front propagation at H = 0.2m, compare to literature
Table 6.3: The non-dimensional average velocityv/√
ghof the wave front toe fort∗>1
Numerical/Experimental run Avg. velocity
Present Method (3D), with Gate , H = L = 0.20m 1.42 Present Method (3D), without Gate , H = L = 0.2m 1.65
Experiment, H = L = 0.20m 1.38
Sueyoshi exp. (2015) , H = L = 0.20m 1.33 Lobovsky exp. (2014) , H = L = 0.60m 1.34 Lobovsky exp. (2014) , H = 0.5L = 0.30m 1.56 Hu and Sueyoshi exp. (2010), H = 2L = 0.40m 1.17
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
A quantitative analysis of the average velocity of the wave front aftert∗>1 is given in Table6.3. The numerical case with gate model closely matches with the experiment with a difference no more than 3% which can be explained with the previously outlined reasons. Table6.3 also indicate a close relation between the average non-dimensional wave front velocityv∗and the ratio (HL). The relation implies that the wave front velocity is strongly dependent on the ratio (HL) and inversely proportional to its value.
Table 6.4: Coefficients of the fitted equations for numerical and experimental data of the time history of the wave front position as in Figure6-5
Numerical/Experimental run a2 a1 a0 b0
Present Method (3D) with Gate 0.3203 0.4642 0.6585 0.6593 Present Method (3D) without Gate -0.03902 2.314 0.09814 1.735 Experiment , H = L = 0.20m 0.1841 0.7728 -0.01179 -0.01861
The wave front position, for the numerical results as well as the present experiment, can be fitted into an algebraic equation using MATLAB in the following form
(x
H) =t∗(a2t∗2+a1t∗+a0)
(t∗+b0) (6.1)
wheret∗is the non-dimensional time (t∗=t qg
H) and the coefficientsa0,a1, andb1are given in Table6.4
An expression for the wave front velocity can be readily obtained by differentiation of Equation (6.1). The fitted equation, which only has 4 coefficients , allows for easy comparisons with any subsequent numerical and experimental results.
Figure6-6shows the evolution of the free surface at selected time instants. Pictures of the free surface of the experiment are also presented at the same time instants for comparison. Owing to the effect of the gate, a splash of water is observed in the initial stages as shown by the experimental pictures. Our numerical gate model was able to
121 6.2. DAM BREAK AGAINST A VERTICAL WALL predict a similar pattern as given in Figure 6-6c. Generally, the agreement between the predicted free surface profile and the experimental images is very good. Even the shape of the wave front as it climbs the vertical wall and the maximum height of the wave over the wall show very good matching with the experimental images as shown in Figures6-6cto6-6e.
(a)t∗=1.12(t=0.16s)
(b)t∗=1.82(t=0.26s)
(c)t∗=2.867(t=0.41s)
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
(d)t∗=4.19(t=0.598s)
(e)t∗=6.3112(t=0.90s)
(f)t∗=7.15(t=0.16s)
Figure 6-6: Comparison of the free surface profile(3D Iso-metric view) at selected time instances
Figure6-7gives a comparison of the computed and measured pressure at two mea-surement points located at (0.8,0.1,0.004) and(0.8,0.1,0.025) . It should noted that the numerical solution was slightly time-shifted to have the same impact time as the
ex-123 6.2. DAM BREAK AGAINST A VERTICAL WALL perimental data since the impact time of the numerical solution is based on a gate motion profile computed from the mean values of the motion profile parameters which would not necessarily give an impact time equal to mean the impact time of the experimental data. The agreement between the numerical solution and the measured experimental data is very good at the first location. Some discrepancy is observed at the second lo-cation betweent=0.45s, and 0.8swhich can be attributed to turbulence modeling and requires further investigation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0
1,000 2,000 3,000 4,000 5,000 6,000
time
Pressure
Present Numerical Method Average of Measured Pressure Error-bars of Measured Pressure
(a) Pressure at(0.8,0.1,0.004)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0
1,000 2,000 3,000 4,000 5,000 6,000
time
Pressure
Present Numerical Method Average of Measured Pressure Error-bars of Measured Pressure
(b) Pressure at(0.8,0.1,0.025) Figure 6-7: Pressure measurement on the vertical wall
An investigation into the existence and extent of three-dimensionality in the dam break flow against a vertical wall is conducted here. The investigation include the free surface profile, wave front location and speed as well as pressure dynamics.
Table 6.5: The non-dimensional average velocityv/√
ghof the wave front toe fort∗>1
Numerical/Experimental run Avg. velocity
Present Method (3D), Center-Line , H = L = 0.20m 1.623 Present Method (2D) , H = L = 0.20m 1.623 Present Method (3D), Near Side-Wall , H = L = 0.2m 1.615 CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
In this study, the gate model is not used since it is an obvious source of three-dimesionality so the gate is assumed to be instantaneously removed and other factors are examined. The non-dimensional wave front velocity is given Table6.5for the two-dimesional case and three-two-dimesional case at the center line and near the side walls.
(a)t∗=0.70
(b)t∗=1.05
(c)t∗=1.75
(d)t∗=2.94
125 6.2. DAM BREAK AGAINST A VERTICAL WALL
(e)t∗=3.50
(f)t∗=4.20
(g)t∗=6.30
(h)t∗=7.14
Figure 6-8: Comparison of the free surface profile(front view) at selected time instances.
The left: 2D section at the domain center line(red:3D, blue:2D), middle: 3D perspective front view, right: experiment perspective front view
The wave front velocity adjacent to the tank side-walls is less than its value at the CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
central line by no more than 1%. This indicates the weak extent of three dimensionality before the wave impacts the right vertical wall.
The evolution of the free surface at selected time instants is given Figure6-6. The con-tour line (α =0.5) of the three-dimesional case at the central plane (y = 0.1) and the two-dimensional case are given for comparison. Pictures of the free surface of the ex-periment are also presented at the same time instants for comparison. It is observed that until the wave front impacts with the vertical wall, the free surface is two-dimensional in the x-z plane and the contour lines nearly coincide.
(a)t∗=2.31 (b)t∗=2.35
(c)t∗=2.38 (d)t∗=2.42
Figure 6-9: Velocity vectors in the bottom right corner area of the impacted wall (taken at the x-y plane) at the moment of impact. The figure is a magnified bottom view of the corner where the blue region represents water and the white represents air
127 6.2. DAM BREAK AGAINST A VERTICAL WALL The wave front is nearly straight with some curvature near the side walls due to the no-slip wall boundary. When the wave impacts the vertical wall and the free surface collapses on it self , the flow becomes fully three-dimensional.
Prior to impact, the velocity near the side-wall is slower than the center-line area owing to the no-slip boundary. However after impact, the sides of the wave front become faster than the central area. Additionally, the side walls show some wetting near the impact zone which keeps increasing until the wave reaches its maximum height on the vertical wall. This is observed in both experiment and simulations as shown by Figures 6-6 and 6-8. In order to explain this, we examine the flow behavior in the corners at the moment of impact as shown in Figure6-9. Due to the boundary layer effect at the side walls, a vortex-like flow occurs when the flow impacts the wall[51]. Consequently, the water moves from the center to sides which would result in the wetting of the side walls and the loss of momentum from the inner region to sides. This would also account for the irregular shape of the wave front in Figures6-6cto6-6eas the water climbs the wall.
≈
≈ y z
P1 P2 P3 P4 P5 P6
3.2 6.4
9.6 0.4
1.7 4.6 0.4
Figure 6-10: Schematic drawing of the problem of dam-break against a vertical wall(left), and sensor placement on the right vertical wall(right) (all dimensions are in cm)
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
(a) Sensor P1 (b) Sensor P2
(c) Sensor P3 (d) Sensor P4
(e) Sensor P5 (f) Sensor P6
Figure 6-11: Time history of the pressure at the vertical wall
129 6.2. DAM BREAK AGAINST A VERTICAL WALL
Figure 6-12: Pressure distribution in the transverse direction at the vertical wall
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
Mohd [59] conducted an experiment to investigate the three-dimensionality in pres-sure by placing six prespres-sure sensors at locations along the bottom and right side of the impact vertical wall as shown in Figure6-10. The chosen location were expected to exhibit three-dimesionality if any existed. The comparison between the computed pressure history and experimental data, at the six prescribed location depicted in Fig-ure6-10, is given in Figure6-11. The two-dimensional result is compared with the three dimensional results and experimental data for sensorsP1toP4.
Generally the numerical simulation agree well with experimental data but the nu-merical simulation seems to under-estimate the value of the first and second pressure peak. This could the result of excessive diffusion generated by the two-equationk−ε model which could be mitigated by using a higher order turbulence closure model[68].
It can also be observed that a sudden pressure drop occurs for sensors P4 to P6 in the interval 3<t∗<4 which may the result of small air-pocket or bubbles. Figure6-11a shows very good agreement with experiment and the 2D result seems to generally over-estimate the pressure when compared to 3D numerical results and experimental data.
As we move away from the center-line the difference between the 2D and 3D results in-creases in both magnitude and phase as shown by Figures6-11ato6-11d. Figures6-11d to6-11fshow the pressure variation with height adjacent the side walls. The pressure experimental data shows an increase in the oscillatory behavior as the height increases.
This may be attributed to either the pressure sensors ability to accurately detect such low pressures or simply the excessive bubble formation adjacent to the side wall after impact. As a result, the comparison with the experiment data deteriorates with height.
The pressure distribution in the transverse direction is given in Figure 6-12 at se-lected time instants. At the early stages just after impact, the pressure variation seems localized to side wall areas. As time advances, the pressure variation extends to the in-ner part of the domain. It is also observed that the magnitude of the transverse pressure variation decreases as we move away from the bottom wall. This would confirm that
131 6.2. DAM BREAK AGAINST A VERTICAL WALL source of three-dimensionality is the 3D vortices localized at the lower corners of the impact wall.
Figure 6-13: Effect of tank width on the pressure distribution in the transverse direction at the vertical wall with left ( W = 0.1m) and right (W = 0.3m)
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
The effect of the tank width is explored by running numerical simulations for two additional tank widths 0.1m and 0.3m. In Figure 6-13, the pressure variation in the transverse direction is compared with the existing case in Figure 6-12. For all tank widths, the three dimensionality originate from the side walls and advances across the transverse direct as time progresses. Larger tank width affects the pressure distribution in the transverse direction and results in localized higher pressure regions towards the center. However, smaller tank width gives closer comparison to the 2D solution with localized discrepancy near the side walls. As time advances, the semi-symmetric pres-sure distribution breaks much faster than the other two case due to the interaction of the side-wall vortices.
133 6.3. DAM BREAK AGAINST A VERTICAL CYLINDER The simulation parameters are the same as the previous case outlined in Table6.1, however, the realizablek−ε turbulence model is employed since it has been reported to provide good predictions for separated flows. Same as the experiment two cases are considered: a circular cross-section and a square cross-section. The numerical results of the forces on the obstacle, free surface profile, separation pattern and pressure dynamics are reported for both cross-sections.
Figure 6-15: Computational mesh for the dam-break flow against a circular vertical cylinder
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
Figure 6-16: Computational mesh for the dam-break flow against a square vertical cylin-der
A schematic of the problem setup is depicted in Figure 6-2. The dimensions of the domain and the water column height are the same as the experiment presented in the previous chapter. Similar to the previous case, all domain boundaries are set to wall con-dition except for a square hatch, 0.2m×0.2mdirectly above the water column, which is set to open boundary conditions. The computational grids were constructed with a min-imum spacing of 0.5mm and maxmin-imum spacing of 5mm . Special attention was directed
135 6.3. DAM BREAK AGAINST A VERTICAL CYLINDER to the zone downstream of the cylinder. The mesh was refined enough to capture the re-circulation and separation phenomena expected in that zone. The unstructured meshes, shown in Figures6-15and6-16, were generated using GMSH open-source software and they consist of 2,311,574 cells for circular cylinder case and 2,393,826 cells for square cylinder case.
Figures 6-17 and 6-18 show the evolution of the free surface at selected time in-stants for the circular and square cylinder cases respectively. Pictures of the free surface of the experiment are also presented at the same time instants for comparison. The de-veloped gate model was able to produce the same gate effect as shown in Figures6-17a and6-18a Generally, the agreement between the predicted free surface profile and the experimental images is very good thoughout the simulation time. The wave separation pattern was predicted accurately for both cross-sections as shown in Figures6-17cto 6-17eand Figures6-18cto6-18e. In the circular cross-section case, the impact with the cylinder is quite mild while the impact with the vertical wall is nearly as violent the no-obstacle (vertical wall) case . In the square cross-section case, the opposite of this case is observed, the impact with the cylinder is violent while the impact with the vertical wall is quite mild.
Figure6-19shows a comparison of the separation pattern between the circular and square cross-sections. In the circular cross-section case, the wave impacts with the cylinder and separates from its surface after 50% of its length while for the square cross-section the separation occurs at the leading corners of the square cylinder. Re-attachment can be observed as we move away from the bottom of the tank in the circular case only. The figure shows that the wave impacts the vertical wall, in circular cylinder case, in a manner similar to the no-obstacle case which is an indication that it hasn’t lost much of its momentum in the stream-wise x-direction. However, for the case of square cylinder, the wave impacts the vertical wall in a rotational manner moving from the sides towards the center resulting in a much softer impact.
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
(a)t∗=1.12(t=0.16s)
(b)t∗=1.68(t=0.24s)
(c)t∗=2.03(t=0.29s)
(d)t∗=2.45(t=0.35s)
137 6.3. DAM BREAK AGAINST A VERTICAL CYLINDER
(e)t∗=2.87(t=0.41s)
(f)t∗=4.27(t=0.61s)
(g)t∗=5.95(t=0.85s)
(h)t∗=8.05(t=1.15s)
Figure 6-17: Comparison of the free surface profile at selected time instances between the numerical solution (left), and experiment (right) for the circular obstacle case
CHAPTER 6. VALIDATION CASES: DAM BREAK FLOW
