Tsunami waves are often caused by tectonic events, usually originating from seabed displacements at larger water depths. In combination with their wave length, they are predominantly shallow-water waves by definition. While propagating in the deep ocean, tectonic tsunamis have small wave heights and usually cause no damage. They can often be detected by early warning systems facilitating pressure sensors on the seafloor (DART system, Bernard and Titov, 2015). More recent attempts have used satellite altimetry to detect tsunamis in deep water (Ablain et al., 2006). Once reaching shallower waters, shoaling increases the wave height while the wave gets shorter and more devastating. Wave breaking occurs mostly prior to landfall and undular bores might develop (Matsuyama et al., 2007; Grue et al., 2008).

Storm waves, which are much shorter than tsunamis, can be simulated almost in their original scale in the largest wave flumes in the world equipped with flap or piston-type wave makers (Oumeraci, 2010). These facilities are also capable of generating solitary or N-waves, which have been often used in simulating tsunami waves. However, even though leading solitary-like waves might reach the shoreline first, Madsen et al. (2008) revealed in a comprehensive study that solitary waves as a model are inappropriate to represent the bulk tsunami on its geophysical scale by affirming that they are “magnitudes” too short. Later on, in 2010, Madsen and Schäffer (2010) conclude that from its generation in the ocean to its impact at the shore, the relevant length- and time-scales of the bulk tsunami are never defined by the solitary wave tie, that is, given by the ratio of the wave height to the water depth. In 2012, Chan and Liu (2012) relax the statement that a solitary wave is “magnitudes” too short (Madsen et al., 2008) by analyzing the time history of an offshore buoy affected by the Tohoku tsunami. They found that in this case, a solitary wave would be approximately six times too short but not magnitudes. However, generating sufficiently long waves is not (or only on a small scale) possible in common wave flumes due to the limited stroke of these facilities. Hence, pump-driven wave generators are often used to cope with this limitation (Goseberg et al., 2013; Park et al., 2013; Schimmels et al., 2016; Sriram et al., 2016; Tomiczek et al., 2016). In line with Madsen et al. (2008) and Madsen and Schäffer (2010), Chanson (2006) suggested using dam-break waves in analogy to tsunami waves propagating on-shore and proposed an analytical two-dimensional (2D) model to calculate dam-break waves of real fluids, including friction. This author found good agreement between calculations and observations made by video cameras during the 2004 Banda Ache tsunami (Chanson, 2006).

Similar to tsunami inundation, dam-break waves are characterized by a quasi-infinite wavelength, increasing water level over time and simultaneously decreasing depth-averaged flow velocities. Ritter (1897) first derived equations to describe the surface, depth-averaged velocity, and wave tip celerity of an ideal fluid dam-break wave in a horizontal, initially dry flume. However, his equations do not consider friction. Addressing this issue, Dressler (1952) used the Chezy resistance coefficient to propose an improved approximation. Whitham and Lighthill (1955) adapted the Ritter (1897) solution to consider friction in the wavefront tip region, where friction and turbulence cause the wavefront to slow down and thicken. A more recent analytical solution was presented by Chanson (2009) who additionally included a variable Darcy friction factor for the wave tip region and successfully validated the model against large-scale experimental data. The latter solution is applicable to horizontal and sloping channels.

In laboratory testing, dam-break waves are commonly generated by lift or swing gates, either driven by rapidly opening actuators (von Häfen et al., 2018; von Häfen et al., 2019) or using the vertical release method where an elevated reservoir is quickly emptied into a lower basin that is again connected to a propagation flume (Wüthrich et al., 2018).

Some authors investigated dam-break waves propagating on sloped bathymetry. Mostly, past studies focused on the wave propagating down an inclined bottom in analogy to a dam-break wave triggered by a bursting river dam in the mountains rushing down into the valley (positively inclined slope), while few others focused their work on dam-break waves propagating up an inclined bottom, this then in analogy to a tsunami wave inundating a site with rising topography (negatively inclined slope). Studies dealing with positively inclined slopes are not in the scope of this thesis. The interested reader is referred to Dressler and Stoneley (1958), Hunt (1983), Hunt (1984), Nsom et al. (2000), Fernandez-Feria (2006), and Chanson (2009), who proposed the previously mentioned analytical solution to describe the dam-break wave celerity and surface in a positively or negatively inclined channel. Among others, studies dealing with dam-break wave propagation on negatively inclined slopes are those of Yeh et al. (1989) and Yeh (1991) who investigated broken bores running up a slope in laboratory experiments. These authors used water on both sides of a lift gate in initial settings and a constant slope angle of 7.5°. Once the gate was opened, the dam-break wave slumped into the resting water downstream of the gate. Yeh et al. (1989) and Yeh (1991) observed a so-called “momentum exchange” between the dam-break wave and the resting water downstream of the gate and found that the resting water is pushed up the slope first, followed by the bore. Lu et al. (2018) conducted experimental tests similar to those of Yeh et al. (1989) and Yeh (1991) but using a larger distance between the gate and the toe of the slope ( 0.4   m vs. 1.8   m ). They found a linear relationship with a uniform gradient between impoundment depth and maximum run-up height. Also, given the same impoundment depth, the run-up height increases with increasing initial water depth downstream of the gate. Hence, run-up height is the smallest for an initially dry flume. Furthermore, Lu et al. (2018) analyzed the spatiotemporal development of the run-up process in detail. Barranco and Liu (2021) generated dam-break waves in a flume with water on both sides of the dam-break gate and used a variable reservoir length to investigate the features of the waves’ run-up. They found a dependency between the run-up height and the initial impoundment depth and flood duration. By facilitating a numerical model, they proposed equations to predict the inundation depth, run-up height, and duration of the flood in relation to the bore characteristics (Barranco and Liu, 2021).

Following Chanson (2006) and Madsen et al. (2008), dam-break waves are frequently used by a multitude of authors in the context of tsunami engineering: Derschum et al. (2018), Khan et al. (2000), Stolle et al. (2018), Stolle et al. (2019), Stolle et al. (2020b), Stolle et al. (2020a), and von Häfen et al. (2021) used dam-break waves to study debris transport and debris-induced loadings, and Nistor et al. (2017a), Nistor et al. (2017b), Stolle et al. (2016), and Wüthrich et al. (2020) used the vertical release method to investigate debris motion as well. Al-Faesly et al. (2012), Arnason et al. (2009), Aureli et al. (2015), Cross (1967), Farahmandpour et al. (2020), Moon et al. (2019), Ramsden (1996), Shafiei et al. (2016), Soares-Frazão and Zech (2007), Soares-Frazão and Zech (2008), Winter Andrew et al. (2021), and Xu et al. (2020) used dam-break waves to investigate loads on structures like residential houses, breakwaters or idealized cities, and the associated flow regime. Kuswandi and Triatmadja (2019), Maqtan et al. (2018), and Triatmadja et al. (2011) used dam-break waves to investigate scouring around structures during tsunami inundations.

Since tsunami forecasting is challenging and only allows for a short-term warning, evacuation plans need to be applied rapidly after a tsunami hazard is detected (Taubenböck et al., 2013), and no measuring equipment can be installed before the wave reaches the site. Therefore, very limited measurements of flow depths and velocities of real-world tsunami events exist. Fritz et al. (2006b) analyzed survivor videos taken during the 2004 Banda Aceh tsunami. Both survivors observed sites about 3   km inland; one in downtown Banda Aceh, and one in a residential area. Extensive image processing and referencing allowed for extracting surface velocimetry and flow depth from the camera feeds. Fritz et al. (2006b) calculated the Froude numbers for both sites and found that they are close to F r   =   1.0 . Matsutomi et al. (2010) analyzed field data of several tsunami events and calculated Froude numbers in the range of F r   =   0.42–2.0 (also depending on velocity estimation). Chanson (2006) also used camera footage from the Banda Ache tsunami in 2004 to validate his analytical dam-break wave approach and highlights the importance of real-fluid bottom-friction interaction. He reports a wavefront velocity of about 1.5 to 1.6   m / s at a site located about 1.5 to 3.0   km inland. Fritz et al. (2012) conducted field surveys in 2011 in Kesennuma Bay (Japan), which was heavily affected by the Tohoku tsunami. They used video recordings taken during the inundation and calibrated them with real-world coordinates measured in field surveys. Focusing on the hydrodynamics in the Kesennuma Bay narrows, they report a wave trough of − 3  m followed by 12 min of flooding with a wave crest height of up to 9  m before the flow direction reverses into an outflow. Observed standing water surface waves within the navigation channel (typical water depth of 10  m ) support the finding that the Froude number, extracted from the camera recordings, approached approximately unity.

Dam-break waves are widely used to simulate tsunami hydrodynamics, both in numerical and physical tests. This is due to their ability to mimic some relevant features that govern tsunami wave hydrodynamics, such as their very long wavelength. In addition to the applications in the field of tsunami-related research, dam-break waves are frequently used in numerical fluid modeling as it is a common test case to validate the accuracy of numerical formulation. Also, numerous publications exist dealing with the obvious analogy to breaking river dams and the associated flood wave. Due to their wide field of application, a multitude of approximations and analytical solutions exists, predicting the surface elevation and flow features of dam-break waves.

However, the aforementioned literature review on tsunami-related applications revealed a lack of knowledge regarding dam-break waves propagation over a sloping bathymetry adjacent to a horizontal plane (composite bathymetry) in analogy to a tsunami wave reaching a sloping bathymetry, approaching a shore, and subsequently inundating a horizontal site (see Figure 1). To date, it remains unclear how the dam-break-induced flow evolves after it has climbed the slope and then suddenly being exposed to a change in inclination.

This study, hence, intends to shed light on the features of a dam-break waves’ overland flow. Previous studies on tsunami effects, where dam-break waves have been used, did simply set a specific distance between the dam-break gate and their test setup; very rarely was that distance discussed. However, the surface elevation η is a function of space x and time t with the largest gradients for small x (von Häfen et al., 2019). This study, hence, also aims to raise awareness of the fact that the distance parameter x has to be taken into account when planning dam-break-based laboratory tests in analogy to tsunami waves.

This study presents a unique, novel set of physical large-scale dam-break tests propagating over a composite bathymetry; the data set is subsequently used to calibrate a high-resolution numerical model. That model is then used to analyze the parameters governing the hydrodynamics. The following specific objectives are addressed within this study:

1) To raise awareness of and quantify the dam-break waves’ changing hydrodynamics over space and time using simple analytical considerations.

Although there might be additional interest in the sloping region of the composite slope, this study focuses specifically on the flow characteristics along the horizontal plane, as it represents the region where urban siting would typically be present (see Figure 1). The authors deem this region mostly important since improving knowledge in this area is of great interest for reducing the number of severe losses caused by a tsunami event.

Laboratory experiments are conducted to calibrate and validate the numerical model Reef3D (see Section 2.2). The test facility is situated at the Leichtweiß-Institute for Hydraulic Engineering and Water Resources, Technische Universität Braunschweig, Germany. The flume used for the dam-break experiments is 2 m wide and approximately 100 m long. A swing gate separates the flume into a 20-m long reservoir section, where water is impounded, and an 80-m long propagation section. The gate is driven by an electric linear drive, which allows a fully controllable movement of the gate. The gate opens within 0.7 s ensuring an unaffected, quasi-instantaneous dam-break (comparable to the ideal numerical dam-break used within this study, see Section 2-3), leading to a dam-break bore propagating downstream the flume (von Häfen et al., 2019). The flume is equipped with four wave gauges, capacity type (200 Hz, accuracy ∼1%, tailor-made). One of the gauges is located in the reservoir and three downstream the gate in the propagation section of the flume whose positions are indicated in Figure 2. Wave gauges and linear drive are synchronized using a data acquisition system (ADLINK DAQe-2206, 64 channel, 16 bit, 250 kS/s). Capacitance wave gauges have been used successfully in accurately detecting aerated flows as occurring during advancing broken bores (Derschum et al., 2018; Ghodoosipour et al., 2019; Stolle et al., 2019a; Stolle et al., 2019b). Figure 2 shows the flume including a composite bathymetry and measuring devices. Wave reflection at the end of the flume is not recorded by the data acquisition system since the length of the flume is sufficient to stop the measurements before a reflection arrives.

For calibration and validation purposes, four experimental tests are conducted: two with a flat bottom and two with a composite bathymetry installed. Each of these four tests is repeated twice. Table 1 contains the experimental protocol including the impoundment depth d 0 and the positions of the wave gauges. In the case of an installed composite slope (test nos. 3 & 4), the distance between the gate and toe of the slopes x t o e is given, as well as the parameters describing the slope geometry; the height of the horizontal plane w and slope length l s l (see Figure 2). A numerical model (see Section 2.3) is calibrated using the flat bottom test no. 1 (see Table 1) and validated later on against the remaining tests with (no. 2) and without the composite bathymetry (nos. 3 & 4).

Within this study, the software package REEF3D is used. REEF3D is an open-source hydrodynamics framework developed by Bihs et al. (2016), consisting of several numerical modules. The computational fluid dynamics (CFD) solver within REEF3D is used to solve the incompressible unsteady Reynolds-averaged Navier–Stokes (URANS) equations. This approach is selected since the flow is expected to be highly transient, which cannot be represented by the classical RANS formulation. The URANS approach can consider vortices in the front of the dam-break wave and is less costly in terms of computation compared to LES (large eddy simulation, see Iaccarino et al. (2003) for a comparison of the approaches). In addition to these CFD approaches, REEF3D is also used to solve depth-averaged non-hydrostatic shallow water equations (SWE) (Wang et al., 2020), which are much more computationally efficient than CFD calculations and are used by a multitude of authors to approximate dam-breaks; see Brufau and Garcia-Navarro (2000), Liang (2010), Ostapenko (2007), and Ozmen-Cagatay and Kocaman (2010). In order to identify the approach that represents the best compromise between efficiency and accuracy, URANS, SWE, and LES are compared against laboratory data of a flat bottom setup (test no. 1, see Table 1). This approach was chosen as the domain was still fairly large to be efficiently computed by the URANS approach. Figure 3 displays the time histories of the four wave gauges in comparison to the numerical approaches. Deviations in the measured time histories of the wave gauges are small in the reservoir (WG 1), where bottom friction almost does not influence the flow and no turbulences are present. The positive wavefront, however, propagating over the flumes’ bottom, results in larger deviations in the measurements. This is due to small imperfections in the flumes’ concrete bottom and a highly turbulent flow associated with large fluctuations. Calculations are performed on a regular grid with 0.01 m cell size in a two-dimensional domain and for a duration of 20 s (see Appendix 6.1 for a convergence study on cell size). The cell size represents the finest resolution, which leads to just acceptable long computing times (approx. 15 h per test in the numerical flume, see Section 2.3 and Appendix 6.1) on the available computing servers (two dual-socket CPU servers with AMD EPYC 7452; in total 128 cores at 2.35 GHz and 128 GB RAM) with the setting of the numerical model summarized in Appendix 6.2.

All approaches are in good agreement with the experimentally measured data. However, large eddy simulations are computationally costly, hence, the validation concentrates on the URANS and SWE approaches with the following settings (also summarized in Appendix 6.2, showing the control file of REEF3D): for the URANS approach, the WENO (weighted essentially non-oscillatory) scheme (Jiang and Shu, 1996) is used for the convection discretization and the level set method is used to track the free surface (Osher and Sethian, 1988). The scheme can handle large gradients accurately by taking smoothness into account, and it is deemed to be appropriate to handle wet front progress; it was confirmed to be accurate in simulating dam-break scenarios as well (Sun et al., 2012; Cannata et al., 2018; Li et al., 2020). Turbulence is approximated by using the k − ω turbulence model (Wilcox, 2006), where k is the turbulent kinetic energy and ω is the specific turbulent dissipation. Time stepping for the momentum equation is performed by the TVD Runge–Kutta scheme, which is also applied to the level set and reinitialization method. The pressure is calculated using the projection method (Chorin, 1968). In the case of the SWE approach, the REEF3D:SFLOW implementation is used (Wang et al., 2020), which solves for non-hydrostatic pressure using a quadratic approximation (Jeschke et al., 2017) and uses the WENO scheme for convection. Figure 4 shows validation runs of the URANS and the SWE approach against wave gauge time histories of the experimental tests 2–4 (see Table 1). In analogy to Figure 3, the SWE approach shows good results compared to a larger dam-break wave on the horizontal bottom (test 2, see Table 1), cp. Figures 4A–D. However, once a composite bathymetry is modeled (Figure 4E-L), the SWE solution significantly underestimates the water depth along the slope (Figures 4F, G, J, K); it also overestimates the water depth on the adjacent horizontal plane (Figures 4H, L). The SWE cannot resolve vertical velocities, yet these would become relevant, especially where slope change occurs. Equally, these are also important to resolve reflection processes at and along the slope (see Section 3.2.1). However, the URANS approach is in good agreement with the experimental data and will hence be used within this study.

The two-dimensional (2D) numerical flume (see Figure 5) consists of a solid bottom and left boundary, while the right boundary is an outflow to prevent reflections. The sides of the numerical flume are symmetry planes. The length of the reservoir l r e s equals the length of the propagation section ( l r e s = x T o e + l s l + l h o r ). Since—theoretically and neglecting friction—the negative wavefront, propagating upstream in the reservoir section, is half the celerity of the positive wavefront, a reservoir length equal to the length of the propagation section minimizes the influences of an emptying reservoir on the hydrodynamics on the horizontal plane. The simulation time t s i m is hence chosen to be twice the time the negative wavefront needs to reach the end of the flume and is, therefore, a function of the impoundment depth and the reservoir length and reads as follows: t sim = 2 ( l res / g d 0 )   [ s ] . (1)

The simulation time t s i m can be converted into a dimensionless simulation time T s i m in analogy to the expression from Wüthrich et al. (2018): T = t g / d 0 , so that the dimensionless simulation time T s i m reads as follows: T sim = 2 l res / d 0 [ − ] . (2)

An ideal dam-break wave is ensured by an instantaneous release of the water column (starting with the first computational iteration, without simulating any gate). Wave gauges and velocimetry profilers, distributed over the entire domain with increments of 0.1 m in the propagation and 10 m in the reservoir section, yield spatio-temporal information about the flow features and the surface elevation. Simulations are performed on a regular grid with 0.01 m spacing using the URANS approach (see Section 2.2 for detailed settings and reasoning, as well as Appendix 6.2).

The geometry and the position of the composite bathymetry affect the hydrodynamics of the dam-break wave during propagation. To connect the parameters describing the composite bathymetry’s geometry with the hydrodynamics and the underlying physical processes, the parameters are systematically varied. Those parameters investigated in this work are (see Figure 5) the initial impoundment depth d 0 , the distance between the initial water column and the toe of the slope x t o e , the length of the slope l s l , and the height of the horizontal plane w . Three values are examined for each of these four parameters. This results in 3 4 = 81 possible combinations of parameters or a number of tests, respectively. In addition, three reference tests with changing impoundment depth but a horizontal bottom (without composite bathymetry) are performed. Thus, in total, 84 tests are conducted. Table 2 lists the values of the four parameters being varied systematically ( d 0 , x t o e , l s l , and w ). In addition, the table contains the range of these values in dimensionless writing (indicated by capital letters: X t o e , L s l , and W ) normalized by the initial impoundment depth d 0 as commonly applied in studies on dam-break waves (Lauber and Hager, 1998; Chanson, 2009; Nouri et al., 2010; Oertel and Bung, 2012; Goseberg et al., 2013; Goseberg and Schlurmann, 2014; Hooshyaripor et al., 2017; von Häfen et al., 2018; Wüthrich et al., 2018; von Häfen et al., 2019; von Häfen et al., 2021).

The literature review revealed that in most experimental or numerical studies incorporating dam-break waves, there is little information on how the distance between the test setup and the (idealized) gate is chosen. However, dam-break waves’ hydrodynamics change considerably over space and time, which will be briefly highlighted in this section. Based on the analytical approach of Ritter (1897), Figures 6A–C show the computed surface elevation ( η , panel A), depth-averaged flow velocity ( U , panel B), and the Froude number ( F r , panel C) of a dam-break wave over space (abscissa) and time (ordinate). The dam-break location is at x = 0   m , the reservoir extends into negative x -direction, whereas the propagation section is positive in x -direction. The positive wavefront is indicated as a red solid line, while the negative wave, propagating in opposite direction within the reservoir, is shown as a red dashed line.

Figure 6 shows that surface elevation, velocity, and, subsequently, also the Froude number always depend on space and time. This is evident from the changing color gradients at constant x in Figures 6A–C and is emphasized in Figures 6D–F, which show the time series along the transects A and B, shifted to the time of positive wavefront arrival ( t w f ). The larger the distance between the gate and the location being considered (e.g., the location of the test setup), the

1) lower and flatter the surface elevation time history (see Figure 6D),

Hence, the distance between the gate and test setup should preferably be provided in dam-break wave-structure interaction studies; this is to ensure comparability and repeatability of tests. In addition, the distance can be utilized to adjust the local hydrodynamic setting and to achieve conditions as close to in situ conditions as possible. Within the subsequently evaluated numerical test program, the distance between the gate and the toe of the slope ( x t o e ) is hence varied (see Table 2) to adjust for the gate to compound beach distance effect and to provide just comparisons.

Note that friction is neglected by the analytical approach of Ritter (1897). Even though the approach resembles the overall characteristics of a dam-break wave well, friction significantly influences the waves’ tip region (e.g., addressed by Chanson (2009)). Hence, close to the wave tip, the surface elevation will be larger and the depth-averaged velocities and Froude numbers smaller in nature than predicted by Ritter (1897) and presented in Figure 6.

The following sections are using the computational results of the numerical model to investigate the flow features of the dam-break waves over compound bathymetries. First, some qualitative observations will be presented before flow depth and velocities as well as Froude numbers are investigated next.