Run-Up Simulation of Impulse-Generated Solitary Waves
Publication: Journal of Engineering Mechanics
Volume 144, Issue 2
Abstract
A numerical method is presented to accurately simulate paddle-generated impulse waves. An order-one implicit splitting scheme allows advection and diffusion phenomena to be decoupled. Advection is solved using a semi-Lagrangian scheme implemented on a dynamic adaptive octree. This method enables the use of Courant numbers larger than one while the octree is refined at the liquid–air interface. Diffusion is solved on a fixed, unstructured finite-element tetrahedral mesh. Interpolation between octree and finite elements is discussed. Numerical results pertaining to paddle-generated waves in a tilted cavity are successfully compared with experimental data.
Introduction
Impulse waves in natural lakes or reservoirs are generated by landslides or avalanches and may lead to great damage in Alpine valleys (Miller 1960; Schnitter and Weber 1964; Fuchs and Boes 2010; Brideau et al. 2012; Carey et al. 2012). Plane impulse waves have been investigated at the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zurich, leading to a general description of the wave generation and propagation processes used to predict the main wave features as a function of the governing slide impact parameters. Fuchs and Hager (2012), Fuchs (2013), Fuchs and Hager (2015), and Hafsteinsson (2014), generated the waves using a pneumatic piston-type wave generator. The wave heights and wave run-up heights on a linearly inclined slope were measured optically and pointwise using ultrasonic sensors and capacitance wave gauges.
This paper numerically simulates these paddle-generated impulse waves. Because the breaking of these water waves can be observed, Eulerian methods have been widely used, either volume of fluid (VOF) methods (Hirt and Nichols 1981; Gopala and van Wachem 2008; Pilliod and Puckett 2004; Pringle et al. 2016; Wroniszewski et al. 2014), or level-set methods (Sethian 1999; Osher and Fedkiw 2006). Lagrangian meshless methods have also been proven to be a robust alternative (Khayyer et al. 2008; Xu 2016). For Eulerian methods, the use of a fine grid is required to compute the liquid–air interface accurately, which may lead to prohibitive memory requirements and computing time. This limitation is overcome by using dynamic adaptive meshing. A natural and efficient approach to achieve adaptive meshing in the framework of structured grids is to use an octree structure (Samet 2005). Free-surface flow octree solvers have been combined either with the level-set method (Strain 1999; Losasso et al. 2004; Nikitin et al. 2011) or with a VOF approach (Popinet 2003). In Laurmaa et al. (2016), for a given velocity field, a semi-Lagrangian octree scheme was described for the displacement of free surfaces, avoiding the Courant–Friedrichs–Lewy (CFL) condition. this paper couples their approach to a finite-element Navier–Stokes solver in order to compute the velocity and to perform these impulse wave simulations.
The outline of the paper is as follows. First, the model is presented. The Navier–Stokes equations (NSEs) are coupled with the transport equation for the volume fraction of liquid . A splitting scheme (e.g., Glowinski 2003) enables transport and diffusion to be decoupled. Transport of the volume fraction of liquid and of the velocity is performed on an adaptive octree grid using a semi-Lagrangian scheme, thus enabling CFL numbers larger than 1. Diffusion of the velocity is performed on a fixed, unstructured finite-element tetrahedral mesh. This method has already been successfully applied, without the adaptive octree feature, to Newtonian, viscoelastic, multiphase flows by Maronnier et al. (1999, 2003), Bonito et al. (2006, 2008); Caboussat (2006), Caboussat et al. (2012, 2013), and James et al. (2014). Finally, paddle-generated water waves in a tilted cavity are simulated and compared with experimental data available in Hafsteinsson (2014) and Hafsteinsson et al. (2017). Wave generation, propagation, and breaking are all successfully simulated.
Model
Let be the cavity containing the liquid at all times , where is the final time of simulation. Let be the region occupied by the liquid at time , , where : is the volume fraction of liquid. Let be the liquid space-time domain . Let : be the velocity field and let : be the pressure field. Note that and are defined only in the liquid domain. The time-dependent incompressible Navier–Stokes equations are solvedwhere = density; = dynamic viscosity of the fluid; = gravitational field; and is the strain rate tensor. The NSEs are coupled with the advection equationwhich translates the fact that the fluid particles move at velocity . Given the initial volume fraction of liquid , the initial liquid domain is , the initial velocity then has to be prescribed in . The boundary conditions for Eq. (1) and (2) are then as follows. Neglecting forces due to pressure from the gas outside the liquid region and any capillary forces, it is assumed that the interface is stress free. Therefore the following boundary condition holds at the boundary of the liquid region that is not in contact with the boundary of the cavity where = outward unit normal to the free surface. If the liquid is in contact with , either Dirichlet conditions (no-slip or inflow, for instance) or slip conditions can be imposed.
(1)
(2)
(3)
Splitting Scheme for the Navier–Stokes Equations
Maronnier et al. (2003) proposed an implicit time splitting scheme to decouple advection and diffusion phenomena in Eqs. (1)–(3). Advection was handled by a semi-Lagrangian scheme on a structured grid, whereas diffusion (a time-dependent Stokes problem) was solved on a tetrahedral grid with standard continuous, piecewise linear finite elements. Bonito et al. (2006) and James et al. (2014) successfully extended this splitting scheme to more complex viscoelastic and multiphase flows.
Let be the number of time steps and be the time step, such that , . Assume that the approximation : of the volume fraction of liquid is known at time step , which defines , the approximation of , the liquid domain at time . Assume that the approximation : of the velocity field at time is also available. Then and are computed as shown in the following subsections.
Prediction Step: Advection
In the advection step, is first transported along with the velocity to obtain the new volume fraction of liquid , which defines the new liquid region , and a prediction of the velocity field . The advection step consists of solvingbetween and , where the initial conditions are given by
(4)
(5)
From Eq. (5) it can be deduced that the velocity is constant along the trajectories of the fluid particles and that the trajectories are straight lines. Therefore , the position at time of the particle at location at time , is given by , where , and and are given for all by
(6)
(7)
The new liquid region is and a prediction of the velocity : is available.
Correction Step: Diffusion (Stokes)
After the prediction step (advection), the correction step consists of solving the diffusion equations corresponding to Eqs. (1) and (2). Given and the predicted velocity , the correction step thus consists of computing the corrected velocity : satisfying
(8)
(9)
The next two sections provide details on the space discretization and algorithms used to perform the prediction and correction steps.
Space Discretization: Octree and Tetrahedral Finite Elements
The splitting scheme [Eqs. (6)–(9)] allows for the use of different grids/methods to solve the prediction step [Eqs. (6) and (7)] and the correction step [Eqs. (8) and (9)]. The prediction step (advection) is implemented using the adaptive octree, whereas the correction step (diffusion, Stokes) is solved on a fixed, unstructured tetrahedral mesh (Fig. 1). As stated by Laurmaa et al. (2016), the choice of the adaptive octree is motivated by the capture of fine details at the interface while keeping coarser cells in the bulk of the fluid for efficiency.
Diffusion Step: Finite Elements
The use of a fixed, unstructured tetrahedral mesh allows for cavities with potentially complex shapes to be described with accuracy. Moreover, solving Stokes’ problem [Eqs. (8) and (9)] is rather standard in the framework of tetrahedral finite elements. The mini element has been used with a generalized minimal residual method solver and incomplete lower upper (LU) preconditioner (e.g., Quarteroni and Valli 2008); solving Stokes problem on an adaptive octree would be more involved (e.g., Popinet 2003). Finally, a mesh with small tetrahedra around the liquid–air interface is not needed to accurately compute the velocity, because the velocity is continuous in the liquid (the velocity is computed only in the liquid but not in the surrounding air). Thus, even though small cells are needed to compute the volume fraction of liquid in the neighborhood of the liquid–air interface, a coarser tetrahedral mesh can be used to compute the velocity . The numerical results presented hereafter (especially those of Fig. 7) confirm this claim. Let be a fixed, unstructured tetrahedral mesh [Fig. 1(b)] where is the typical size of the tetrahedra. The octree cells in the bulk of the liquid [Fig. 1(a)] should therefore also be of typical size , whereas the cells close to the liquid–air interface will be of smaller size . A trade-off between accuracy and complexity is to take (Maronnier et al. 2003).
Advection Step: Adaptive Octree
Laurmaa et al. (2016) proposed an efficient adaptive octree scheme to compute efficiently the transport of the liquid/air interface. An octree structure is a hierarchical 3D structure based on an axis-aligned hexahedron which is split into eight hexahedra and in which each hexahedron can be split into eight further hexahedra or remain unsplit, resulting in a tree-like structure. The octree structure allows for efficient refining; coarsening; access to neighbor, parent, and children cells; traversal of all cells; and retrieval of a cell containing a specified point. The scheme proposed by Laurmaa et al. (2016) ensures that the octree is always refined to a desired accuracy around the interface (Fig. 2).
The octree has five integer parameters that determine the shape and size of the octree cells; sets the minimal refinement level of liquid cells and sets the maximal refinement level of any cell (Fig. 2). An initial subdivision of the domain can also be specified in order to select the aspect ratio of the cells. The parameters , , and determine the number of initial subdivisions of the cavity in directions , , and , respectively (Fig. 3).
The implementation of Eqs. (6) and (7) on the octree consists of four steps: interface prediction refinement, transport, decompression, and coarsening (Laurmaa et al. 2016). Interface prediction refinement ensures that cells that will receive the transported fluid are at least as refined as the transported cell. The transport step transports the fluid by shifting the center of each cell to and projecting the cell onto the grid. The decompression step ensures that and thus that remains a valid volume fraction of liquid. The coarsening step then coarsens noninterfacial cells to decrease memory usage.
The next section explains how to interpolate and from the adaptive octree to the tetrahedral mesh , so that Eqs. (8) and (9) can be solved on the tetrahedral mesh to obtain the velocity correction , at which point is interpolated from the tetrahedral mesh back onto the octree for the next iteration.
Interpolation between meshes inevitably introduces an extra numerical error, , at each time step whenever using a mesh of size , thus for the whole simulation, the number of time steps being . Because and are linked through the CFL number, the overall interpolation error is , and thus the method is still of order one in time and space. Interpolation between meshes has not been shown to be an issue without the octree (Maronnier et al. 2003). The numerical results presented hereafter (especially those of Fig. 7) show that the overall precision of the method is not destroyed by the octree.
Interpolations between Octree and Finite Elements
Dynamic Mapping between Meshes
To compute interpolations between octree cells and tetrahedra, an efficient way of mapping each cell to its neighboring tetrahedra and vice versa is needed. A cell is said to be the neighbor of a tetrahedron and vice versa if their bounding boxes in the , , and frame have a nonempty intersection.
At initialization, for each octree cell the list of its tetrahedral neighbors is computed and stored, and for each tetrahedron the list of its neighbor cells is computed and stored. When a cell of the octree is split, each of its eight children stores a copy of the neighbor list of the cell and the tetrahedra is removed from the eight lists that are not neighbors with each respective child. The list of neighbors of is then deleted. The neighbor lists of the tetrahedra involved in the update are also updated accordingly. When eight cells are coarsened, the neighbor list of the parent cell is set to be the union of the eight cells’ neighbor lists and the neighbor lists of the involved tetrahedra are updated accordingly.
Interpolation from Octree to Tetrahedral Mesh
The fields and are interpolated from the adaptive octree to the fixed, unstructured tetrahedral mesh . The values of the volume fraction of liquid are only used to determine which tetrahedra are considered to be liquid. Then the Stokes problem [Eqs. (8) and (9)] is solved on the union of all liquid tetrahedra.
An interpolated value is then computed for each tetrahedron in , and if the tetrahedron is considered to be liquid. Let be the axis-aligned bounding box of in . The value is defined as a weighted average of values , where are neighbors of . Weights are given by the volume of the intersection between cell and the axis-aligned bounding box of
In order to solve the Stokes problem [Eqs. (8) and (9)], needs to be defined at the vertices of the tetrahedral mesh . A variation of the inverse distance weighting interpolation (Franke 1982) is used. For each vertex of the mesh , let be the axis-aligned bounding box of the union of the tetrahedra containing (Fig. 4; the dashed cells are involved in the interpolation). In order to handle different sizes of octree cells, the contribution of each cell should be weighted not only by the inverse distance but also by the volume of the intersection between and . Thus the velocity at vertex is computed as follows. If there exists a cell whose center coincides with , then ; otherwise
Note that the amount of fluid contained in a cell which is not included in a liquid tetrahedron is moved into a buffer; it is redistributed during the decompression phase of the next time step.
Interpolation from Tetrahedral Mesh to Octree
Pseudo-2D Paddle-Generated Wave Simulations in Tilted Cavity
A paddle-generated wave in a tilted 3D cavity was used as a benchmark to determine the accuracy of the numerical scheme for solving free surface flows governed by the NSE. Although the wave propagated in a 3D cavity, the cavity was narrow enough that the wave only exhibited 2D features. The experimental wave profile measurements (Hafsteinsson 2014; Hafsteinsson et al. 2017) were provided by the Laboratory of Hydraulics, Hydrology and Glaciology (VAW), ETH Zürich.
The experimental cavity was 14 m long, 0.5 m wide, and 0.7 m high. One of the side walls was lined with observational glass windows and the other wall was lined with smooth polyvinyl chloride along with the bottom. A pneumatic wave-generating piston was mounted at one end of the cavity and was controlled with electrical impulses sent by a computer. The setup was successfully used in several experiments by VAW (Fuchs and Hager 2012; Fuchs 2013; Fuchs and Hager 2015).
Fig. 6 illustrates the experimental setup. The cavity is tilted at angle . A plate pushes the water parallel to the bottom of the cavity with a velocity profile designed to generate solitary waves with a given height relative to the still water depth. The relative wave height is then the ratio of the initial wave height to the still water depth . The run-up height is . In order to define more accurately what is meant by still water depth, define the -axis as parallel to the bottom of the cavity, directed from the paddle to the other end of the cavity. The paddle makes a sweeping motion from to . The still water depth is the maximum water depth when the plate is at position in the middle of the plate sweep. The start and end coordinates of the plate depend on the desired relative wave height . In the following analysis, all combinations of parameters , and , and 0.7 were simulated.
Two types of sensors were used to measure the wave profiles, ultrasonic distance sensors (USDS) and capacitance wave gauges (CWGs). The USDS were placed at the top of the cavity and sent an acoustic signal downward toward the water surface. The signal reflects off the water free surface and the free surface height was deduced from the time taken for the signal to return to the sensor. Hafsteinsson (2014) gave an estimate of 3 mm for expected measurement errors. However, this sensor can give spurious values if the signal is reflected away from the sensor. This can occur in particular if the free surface under the sensor is at a steep angle. Because this happens when , the less accurate CWG sensors were also used.
Capacitance wave gauges are vertical enamel-coated wires attached to the top of the cavity whose capacitance varies linearly with respect to water depth. After calibration, they can be used to determine water depth. The obtained wave profiles are noisy and show jumps of up to 8 mm between sampling points, and Hafsteinsson (2014) estimates the measurement error to be also up to 8 mm. For this reason, they were smoothed with a Savitzky–Golay filter (Savitzky and Golay 1964). The Savitzky–Golay filter smooths the signal by fitting low-level polynomials with the linear least-squares method with successive subsets of adjacent data points. The polynomials were also used to derive a smooth velocity which was used as a Dirichlet boundary condition at the paddle location for the numerical simulation. As parameters for the filter, a data point window size of 71 data points and polynomials of degree 3 were used. Numerical wave profiles were also smoothed for visualization clarity with the Savitzky-Golay filter but to a much lesser extent, with a data point window size of 7 data points and polynomials of degree 3.
A high-speed camera captured photographs which were used to compare static-in-time wave profiles with the simulated wave profiles. The camera was located between Sensors 3 and 4; the CWG wires can be seen hanging from the top of the cavity. The photographs were fitted together by matching initial still water heights and CWG wires with their known positions in the numerical cavity. Wave slices of the simulated wave matched the camera-side window of the experiment cavity. Although the water level at rest is not shown, the initial water level of the simulated wave and the water-air interface of the real wave at the camera-side window did indeed match. Distortion and refraction effects due to the viewing angle of the camera were less than 1 mm between the middle of the photo and its peripheral areas.
The physical properties of the water for the Navier–Stokes simulations were and . Because the wave only exhibited two-dimensional features, the cavity was truncated to a width of 0.125 m (instead of 0.5 m). The base oct of the cavity corresponded to parameters , , and , where , , and are as defined in Fig. 3. Three meshes/octrees—coarse, medium, and fine—were used to check convergence. The coarse setting corresponded to a finite-element tetrahedral mesh of size ; to an octree with parameters and , which resulted in an octree of minimum size (the octree size is in the bulk of the liquid); and to a time step . This setting corresponded to a CFL number close to 4 on the octree, which was a good trade-off between accuracy and computing time. To check convergence, the mesh size, octree size, and time step were divided by two (medium setting) and four (fine setting). Thus the medium setting corresponded to , , , , and , and the fine setting corresponded to , , , , and . Because , the cells in the bulk of the liquid were approximately of the same size as tetrahedra, whereas the cells on the liquid–air interface were four times smaller. Boundary conditions for the velocity were set to the imposed Dirichlet conditions at the paddle location and to slip conditions at the walls of the cavity. The paddle movement was simulated in practice by computing at each step which nodes were intersecting the region that was swept by the paddle and imposing the paddle velocity at those points. The octree was aligned with the water level and therefore not with the cavity. It was refined to level at the bottom of the cavity to capture the cavity slope at the wall in contact with the paddle and at the water free surface, but not on the lateral sides which were aligned with the octree, and hence captured exactly. Fig. 7 justifies the choice of the octree parameters and . The computed wave corresponding to and was reported at time for several values these two parameters. Results indicate that the choice advocated in Fig. 1 yields accurate results.
Figs. 8–19 show results for the three mesh settings: coarse (fineness 0), medium (fineness 1), and fine (fineness 2). For each combination of experimental parameters and , and 0.7 the evolution of the wave height over time was first plotted at four fixed points in the cavity and snapshots of the numerical wave were overlaid on photographs. Four pairs (USDS and CWG) of sensors were placed at coordinates , , , and , the last two locations being visible on the photographs. The sensors provided water height data from times at a sampling rate of 100 Hz. The pneumatic-driven paddle moved with a slight delay compared with the electrical input it received. Wave profiles were therefore shifted to correct for this fact. Despite this, measurements were available at four different points in the cavity, which allows comparing the wave profiles from numerical simulations with those from the experimental profile.
All computations were performed sequentially on a 3.3 Ghz Intel (Santa Clara, California) Xeon E5-2643 with 32 Gb RAM. The computing time when and was approximately 1 h for the coarse setting, 3 h for the medium setting, and 47 h for the fine setting. The ratio is close to , which is the theoretical ratio corresponding to refining the mesh size by a factor of 2 (thus multiplying the number of vertices by a factor ) and the time step by a factor of 2.
Figs. 8–13 show results for . For these experimental parameters, the wave does not break in the data shown. However, for , the wave does break soon after Sensor 4. Very good agreement was achieved between numerical and experimental profiles. For , where the wave presented a sharper and higher crest, the mesh needs to be refined more than for lower, smoother waves in order for the wave profile to be captured. Ultrasonic distance sensors are more accurate than CWGs and should be considered the baseline except when the wave is too steep, making the ultrasonic signal bounce away from the sensor, resulting in spurious artifacts as in the two middle frames in Fig. 12. The simulated wave crest was slightly lower than the real one (Fig. 13), but seemed to converge to the measured wave profile (Fig. 12). Fig. 20 presents the vector plot of the velocity corresponding to Fig. 11(c). Fig. 21 presents the plot of the pressure field corresponding to Fig. 17(b); the pressure differed very little from the hydrostatic pressure.
For , the waves broke and caused a splash. These types of simulations are typically more difficult and cannot be simulated by shallow water models. Turbulence, fine-scale drops of water, and air bubbles occur during wave breaking, and these aspects are not captured by the proposed model. Figs. 14–19 show results for these parameters. Because the USDS 4 sensor was located above (right) the still water surface and was dry at the start of the test, its curve should be disregarded. Because the CWG 4 sensor was not calibrated for measurements in this range and is also limited in measuring air–water mixtures after wave breaking (its curves significantly undercut the USDS curves), its curve should also be disregarded.
In some of the results reported when [e.g., Fig. 16(b)], the free surface appears to contain unphysical perturbation (presence of a ragged interface). Fig. 22 shows a close-up of the free surface, cells, and velocity at the vertices of the finite-element mesh. These oscillations remained localized within one tetrahedron and did not spread during the simulation.
Numerical wave profiles showed excellent agreement with the experimental wave profiles despite the fact that the model does not capture turbulence and the air cushion formed beneath the breaking wave (Fig. 19). For , once again the higher, sharper wave crest was more difficult to capture and the slight error in the solitary wave heights caused a more significant tongue shape during breaking. Despite this, water wave profiles quickly matched again after wave breaking.
Table 1 reports the run-up height—denoted in Fig. 6—for and , and 0.7. The computed values correspond to the finest setting and are 10% smaller than the experimental values. The coarse and middle meshes yielded 40 and 20% errors, respectively. Thus a finer mesh would be needed to obtain more accurate results.
Experimental | Computed | |
---|---|---|
0.3 | 0.176 | 0.165 |
0.5 | 0.244 | 0.220 |
0.7 | 0.315 | 0.280 |
Conclusions and Perspectives
This paper presented a numerical method to accurately simulate paddle-generated impulse waves. Two space discretizations were used. A dynamic octree was used to solve advection, and a fixed, unstructured finite-element tetrahedral mesh was used to solve diffusion. The method allows the use of CFL numbers larger than 1, although the octree was refined in the neighborhood of the liquid–air interface. Interpolation issues between octree and finite elements were discussed. Experiments using paddle-generated waves in a tilted cavity conducted at VAW were reproduced with accuracy.
Acknowledgments
The authors thank the Swiss National Science Foundation and the Commission for Technology and Innovation for their financial support. Viljami Laurmaa was supported by SNF grant 143470. Gilles Steiner was supported by CTI grant 14359.1 PFES-ES. All the computations were performed using the industrial cfsFlow software developed by EPFL and Ycoor Systems. Helgi J. Hafsteinsson is acknowledged for conducting the experiments at VAW. The (unknown) reviewers are thanked for constructive remarks.
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Received: Dec 20, 2016
Accepted: Jul 7, 2017
Published online: Dec 4, 2017
Published in print: Feb 1, 2018
Discussion open until: May 4, 2018
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