This chapter presents a review of the work done in the field of solving dam breaks analytically, experimentally and computationally. This chapter helps to provide a base knowledge for the project.
The dam break problem has been an area of interest in research for over 100 years, this is due to the potential severe damage that could occur if a dam were to fail. The solutions to the problem have developed substantially over the years, starting with early analytical solutions with some broad assumptions to advanced 3D CFD solutions in the modern day.
Early Analytical Methods
The dam break problem is generally defined as a rapidly moving column of water generated by the instantaneous release of a given volume of water confined in a rectangular channel (Crespo et al, 2008). To the authors knowledge, the first solution to the dam break problem is Ritter (1892) who solves it by applying expressions of Saint Venant, often called the non-linear shallow water equations. Ritter’s solution is based on an inviscid fluid on a dry bed and friction is also neglected. The solutions provide a parabolic water surface profile that is concave upward. The front travels downstream with a wave speed, c=2√(gd_(0)), where g=acceleration due to gravity, and d0=initial quiescent water depth behind the dam (Crespo et al, 2008). This solution has been found to agree well with experimental values, except from the leading tip of the wave which is significantly affected by friction (reference comparing ritter to experimental). Dressler (1952) introduces the dam break problem as a highly unsteady flow, with a forward positive wave advancing over the channel and a back disturbing wave propagating into the still water above the dam. The case investigated is a 2D, horizontal dry bed, with the water in the dam at rest. Dressler (1952) argues that hydraulic resistance caused by stream bed friction and the resultant turbulence dominate the propagation of the dam break wave and therefore concludes that the solution provided by Ritter (1892) is unrealistic as it neglects this term. Dressler (1952) adds the Chézy resistance term to the shallow water equations and calculates the discharge rate, flow velocities and the locus of critical flow. Dressler (1952) confirms that the front tip region of the wave where the slope is vertical needs to be analysed separately. Whittam (1955) investigates this tip region and finds that the water near the tip builds up and is pushed along by the water behind at a speed significantly less than predicted in previous solutions such as Ritter (1892). Whittam (1955) defines the front tip region as a definite boundary layer and uses an adaptation of the Polhausen method to calculate the velocity at the tip of the wave as a function of time. The Polhausen method is based on the Von Kármán momentum theorem and is well known for its application to boundary layer problems. The velocity at the tip of the wave is found as a function of time and is found to be accurate in comparison with experimental results, however as t (time) gets large the solution starts to disagree but it is noted that a dam does not have an infinite volume so solutions for large t can be ignored.
Experimental Methods
As real-life dam break data is obviously very hard to make, experimental methods are very important in order to validate both analytical and computational methods.
Martin and Moyce (1952) carried out a series of tests including the 2D collapse of rectangular and semi-circular sections, and the 3D axial collapse of right circular cylinders over an initially dry horizontal bed. The velocity of spread of fluid and the rate of fall of the top of the water column are explored in the experiment. For each of the experimental cases, the apparatus was similar. The fluid column was constrained by a very thin waxed paper diaphragm held in position by a thin film of beeswax on a metal strip forming part of the fluid reservoir. A bank of car batteries was used to short the circuit with the metal strip, causing the waxed paper to be free and the therefore allow the dam break to begin (Martin and Moyce, 1952). The theoretical solutions assume that the dam break is instantaneous, so the removal of the dam is a vital part in ensuring the experimental data is accurate. It has been found through experiments that if the removal of the dam is <0.1s then it can be assumed to be an instantaneous release (Bellos et al, 1992). This rule was established after these experiments were done, so it is not possible to see if this is an efficient dam removal method. Martin and Moyce (1952) found the wave front velocity to be proportional to the original water column height which is in agreement with Ritter (1892). Dressler (1954) carried out a series of experiments to confirm his 1952 solution. Dressler (1954) carries out the experiments in a horizontal glass flume, 65m in length and 22.5cm in width. Three different initial water column heights were investigated: 22cm, 11cm and 5.5cm. Three different bed surfaces with varying roughness and obeying Manning’s law were also investigated. A system of springs was used to lift the aluminium gate to allow the motion to start; from photographs, it can be seen that there is no motion in the water until the gate has been completely removed. It was found that there was a bigger disagreement between theoretical and experimental results for smoother channel beds. Dressler (1954) concludes that it can be assumed with high probability that the Chézy resistance function is not wholly adequate to describe highly dependent flows, such as the dam break wave. Dressler (1954) states that the reason it cannot be completely concluded is because his solution (1952) is not an exact solution but only an approximation. Bellos et al (1992) investigate 2D dam break induced flow in a converging-diverging flume. To avoid 3D flow effects flat bottom, vertical side walls and mild side wall contractions and expansions were used. Bellos et al (1992) assume the channel geometry and the bed roughness to be constant throughout and investigate the effect of changing the channel bed slope in the flow direction, the upstream water depth and the initial water column height, on the flow. As expected, the flood routing is faster for larger bed slopes, the wave depression travels faster for larger initial water column heights and the wet bed conditions result in a reflected wave travelling upstream (Bellos et al, 1992). Stansby et al (1998) carried out a rigorous series of dam break experiments over dry and wet beds. The experiments were carried out in a 15.24m long, 0.4m wide and 0.4m high flume and the dam gate is released using a pulley system. For the dry-bed case a horizontal jet forms for small t (time) and for the other cases a mushroom-like jet occurs, neither of which had been observed previously (Stansby et al, 1998). The results are compared with Stoker’s (1992) solution and for small times, the differences between them are quite significant. However, after a bore has formed downstream due to highly complex flow interactions, the results agree extremely well (Stansby et el, 1998).
There is a lack of data in the aforementioned dam break flow experiments
regarding the flow dynamics. Lobovský et al (2014) provide a detailed insight into the dynamics of the dam break wave over a dry horizontal bed under controlled laboratory conditions. Lobovský et al (2014) used a purposely built polymethyl methacrylate prismatic tank with internal dimensions of 1610mm long, 600mm high and 150mm wide. A similar pulley system to Stansby et al (1998) is used to remove the gate, however further detail is added by Lobovský et al (2014) regarding experimental accuracy such as replacing the steel wire because it had plastically deformed due to the high shock from the pull. As this is an important part of the experiment, Lobovský et al (2014) carry out an analysis of the gate removal time and find that 95% of cases fall within the range <0.06s;0.10s>. Lobovský et al (2014) add extra information that has not been included in previous dam break experiments such as preheating the water to 25˚C before each run, which means the water can be considered Newtonian with a density of 997kgm-3, kinematic viscosity of 8.9 x 10-7 and surface tension of 0.072Nm-1. Five pressure sensors were used and arranged with 4 on the centre line of the back wall at varying heights and 1 offset from the centre line. The experiment has been carried out for initial water column heights of 300mm and 600mm, the test has also been repeated 100 times in order to provide statistically sound data. This repetition had not been done before in previous experiments and adds a level of accuracy to the data. On the downstream wall, the peak pressure values are found to be spread out. For the larger initial water column height, 600mm, the scattering in the peak pressure values is found to be significantly larger (Lobovský et al, 2014). A linear relationship is found between the initial water column height and the impact pressure load for the sensor nearest to the channel bed which was subjected to the largest load. The relationship between the other sensors is not linear but it is found that a higher initial water column height corresponds to a higher impact pressure force (Lobovský et al, 2014). The pressure forces are compared with the results of Lee et al (2002) and the results from Lee et al (2002) are found to be outside the 95% confidence interval produced by Lobovský et al (2014) during the impact event. Lobovský et al (2014) suggest that the reason for this is that Lee et al (2002) used pressure sensors with a larger diameter and did not account for the unpredictable nature of only carrying out the experiment once. The results from Lobovský et al (2014) are extremely robust and surpass the accuracy of previous experiments by a significant margin. Lobovský et al (2014) also address the pressure forces due to a dam break in which the literature is lacking. For these reasons as well as the data being easily available online, the results from Lobovský et al (2014) will be used as validation for the model produced in ANSYS Fluent later in the project.