Steep Wave Front Research Articles

Field observations of wave-averaged suspended sediment concentrations in the inner surf zone with varying storm conditions

During extreme conditions, the transport of the wave-averaged suspended sediment concentrations in the inner surf zone affects dune erosion. Although large-scale laboratory experiments have provided insight in what drives these sediment concentrations, corresponding field data are lacking. To fill this gap, novel field observations of suspended sediment concentrations are compared to drivers that govern sediment suspension during storm conditions known from literature. A total of 128 time intervals of 20 min are analysed, spread over 10 different high water events with different hydrodynamic conditions. For each time interval, the wave-averaged (i.e. 20 min mean) suspended sediment concentration is computed and compared to three suspension drivers. The studied drivers are (1) bed shear due to near bed velocities that originate from mean currents in combination with wave-induced orbital flow, (2) the horizontal pressure gradients under steep wave fronts that increase the forces on the bed material, and (3) bore-induced turbulence that is generated at the free surface and reaches the bed. The derived bore-induced turbulence generates the greatest correlation with the mean suspended sediment concentrations (r = 0.74, p = 4.47E-23). Samples that deviate from this correlation correspond to time intervals with lower values of derived bore turbulence, less wave energy saturation in the inner surf zone, and stronger mean currents. The correlation with the mean suspended sediment concentrations increases when the shear stress originating from mean currents is used for these time intervals (r = 0.83, p = 1.63E-33). For time intervals during which more energetic conditions persist and the wave energy is saturated in the nearshore, bore turbulence was the dominant mechanism in stirring up sediment. The outcome of this study suggests that, based on the events analysed, dune erosion models may achieve more accurate results if computations of suspended sediment concentrations include a bore-induced turbulence term, or if already included, properly address the relative importance of bore-induced turbulence when compared to bed shearing.

A numerical study of aeration characteristics of a plunging solitary wave on a slope

In this paper, the characteristics of aerated flow under a plunging solitary wave on a 1:20 sloping beach are investigated numerically. The numerical model solves the Reynolds-averaged Navier–Stokes equations for mean flow. The turbulence is described by the k−ε model, in which the turbulence production and dissipation modified by entrained air bubbles are considered by an additional term. A transient equation is solved for air bubble transportation. The numerical model is validated by comparing the air bubble concentration, mean flow velocities, and turbulent kinetic energy against experimental data, demonstrating its capability for simulating transient aerated flows under breaking waves. The validated model is further applied to reveal the detailed interaction of the entrained air bubbles and the turbulent free surface flows during the wave breaking process. Plunging breaking wave consists of four stages, namely, the wave front steepening, the initiation of overturning, the transitional stage, and the quasi-steady bore propagation stage. The results reveal that the overturning and breaking wave front is the main source for turbulence generation and air entrainment in the initiation and transitional stage of breaking wave, respectively. The entrained air bubbles are mainly transported backward and downward by turbulence structures and forming distinct bubble vortex rollers near the bottom. The distribution of air bubble concentration shows a linear correlation to the distribution of turbulence quantities in the initial and transitional stage of breaking wave, demonstrating the important role of local turbulent structures on air entrainment and transportation.

Influence of air chambers on wavefront steepening in railway tunnels

The behaviour of wavefronts propagating in a tunnel with multiple, compact air chambers is investigated with particular reference to the suppression of unacceptable pressure disturbances radiating from exit portals. Attention is focussed primarily on chambers that respond in an over-damped manner to pressure variations in the tunnel. In the early part of the paper, comparisons are made between the qualitative behaviour of wavefront propagation in tunnels with and without air chambers. Thereafter, attention turns to a quantitative assessment of the influence of key design parameters, especially the chamber volume and properties of connectors between chambers and the tunnel. It is shown that the optimum type of connector depends upon the length of the tunnel along which the chambers exist. For short tunnels, relatively low-resistance connectors with linear pressure:flow rate characteristics are preferred, whereas for long tunnels, relatively high-resistance connectors with quadratic pressure:flow rate characteristics are likely to be more suitable. Brief attention is paid to the practical feasibility of creating suitable connectors.

Impact dynamics of debris flow against rigid obstacle in laboratory experiments

Debris-flow impact against rigid obstacle involves instant redirection of flow velocity and elevation of flow depth, and could serve as a stringent approach for verification of the two-phase flow models. In this study, kinetic and dynamic behavior of debris-flow impact with varying flow regime has been analyzed on the basis of laboratory tests with distributed measurement. Debris flows interact with obstacle in the modes of vertical jet (deflected vertically) and momentum jump (reflected upstream). The experimental results are further adopted to verify the predictability of vertical jet model and momentum jump model. Our findings highlight that, due to the elevated confining stress in the impact area, the liquefaction ratio increases, resulting in a more “fluid-like” state of debris flow. The two analytical models could distinguish the runup height around the threshold Froude number 3–4. However, the impact loads under both modes could be very close, because the effective impact in the vertical jet mode only concentrates in the lower portion of runup height. The potential threat of flip-through impact (shock impact by a steep or concave wave front) should also be considered, as its impulse load could be much higher than the predicted load by vertical jet and momentum jump models.

Frequency-dependent amplification of the Sanriku tsunamis in Ryori Bay

In the present study, the local tsunami amplification observed in Ryori Bay, located on the Sanriku coast of Japan, was investigated using numerical simulations. Large-scale tsunami propagation simulations and tsunami inundation simulations for the bay were systematically conducted to estimate and model the 2011, 1933, and 1896 tsunamis that occurred off the Sanriku coast and which resulted in large run-ups. The simulation results, which are moderately consistent with observations, presented larger run-up heights and inundations for the 1933 and 1896 tsunamis (which followed relatively small earthquakes) compared to those of the 2011 tsunami (which followed a larger earthquake). Furthermore, the frequency analysis indicated that the former two tsunamis comprised higher predominant components. A tsunami inundation simulation using parametrized synthetic waveforms was conducted to identify the contributing factors associated with the large amplification and run-ups. The results indicated that the predominant components are significantly amplified in the bay and the initial decrease in the water surface elevation prior to the primary waves of the two tsunamis leads to an increase in their run-up heights. Furthermore, the simulated waveforms of the tsunamis revealed that the 1933 and 1896 tsunamis had their wavefronts changed into a steep wavefront, i.e., a bore-like wave, during their wave developments in the bay, attributed to shoaling, narrowing bay width, and the nonlinear effect of the wave. These results, therefore, indicate that bores which are known to generate large run-up heights were generated in the bay during the two tsunamis.

Nonlinear deformation and run-up of single tsunami waves of positive polarity: numerical simulations and analytical predictions

Abstract. The estimate of an individual wave run-up is especially important for tsunami warning and risk assessment, as it allows for evaluating the inundation area. Here, as a model of tsunamis, we use the long single wave of positive polarity. The period of such a wave is rather long, which makes it different from the famous Korteweg–de Vries soliton. This wave nonlinearly deforms during its propagation in the ocean, which results in a steep wave front formation. Situations in which waves approach the coast with a steep front are often observed during large tsunamis, e.g. the 2004 Indian Ocean and 2011 Tohoku tsunamis. Here we study the nonlinear deformation and run-up of long single waves of positive polarity in the conjoined water basin, which consists of the constant depth section and a plane beach. The work is performed numerically and analytically in the framework of the nonlinear shallow-water theory. Analytically, wave propagation along the constant depth section and its run up on a beach are considered independently without taking into account wave interaction with the toe of the bottom slope. The propagation along the bottom of constant depth is described by the Riemann wave, while the wave run-up on a plane beach is calculated using rigorous analytical solutions of the nonlinear shallow-water theory following the Carrier–Greenspan approach. Numerically, we use the finite-volume method with the second-order UNO2 reconstruction in space and the third-order Runge–Kutta scheme with locally adaptive time steps. During wave propagation along the constant depth section, the wave becomes asymmetric with a steep wave front. It is shown that the maximum run-up height depends on the front steepness of the incoming wave approaching the toe of the bottom slope. The corresponding formula for maximum run-up height, which takes into account the wave front steepness, is proposed.

An exact source-term balancing scheme on the finite element solution of shallow water equations

In this paper, a numerical model is presented based on a semi-discrete Streamline Upwind Petrov–Galerkin scheme for solving the two-dimensional shallow water equations with arbitrary bed topography and wetting–drying fronts. The present finite element method is implemented for moving-boundary problems on fixed numerical meshes. In order to satisfy the C-property for both still water and dry regions, a new exact source-term balancing method is presented within the finite element framework. The formulation is stabilized using two different stabilization terms, and a discontinuity-capturing scheme is used to simulate steep wave fronts. The second-order scheme is applied for both temporal and spatial discretization. The proposed mathematical balancing method is verified through well-known test cases including the traditional dam break problem, evolution of a dam break wave over an obstacle, and oscillation of a bead of water in a parabolically-shaped basin.

Influence of ballast quantity on compression wavefront steepening in railway tunnels

The impulsive pressure wave (micro-pressure wave) emitted from a tunnel portal is one of the environmental problems associated with high-speed railways. The principle underlying the countermeasures for this micro-pressure wave is to reduce the pressure gradient of the compression wavefront at the tunnel portal. This study investigated the influence of the ballast layer on the rate at which compression waves steepen in railway tunnels. Field measurements in an actual Shinkansen tunnel with slab tracks were recorded before and after installing ballast along most of the length of the tunnel. Previously published results on the spread of the compression wavefront in a ballasted track tunnel are also presented for comparison. A rapid method using an empirical equation is shown to accurately estimate the variation in the maximum pressure gradient of the compression wavefront in certain conditions. A more widely applicable estimation method using a numerical analysis is also demonstrated, which describes the distortion of the compression waveform. The results show that both of these methods can estimate the influence of the ballast quantity on mitigating the compression wavefront steepening in Shinkansen tunnels.

Individual violent wave-overtopping events: behaviour and estimation

ABSTRACTTo better understand individual violent wave overtopping, of significance for coastal defence design, three breaking wave types (steep-fronted, plunging and broken) based on focused wave groups, were generated in laboratory and numerical models. High-speed video captured overtopping events and produced velocity vector maps by means of bubble image velocimetry (BIV). Results were compared with a numerical model based on a linear wave detection procedure and a two-phase incompressible Navier–Stokes-based solver. This novel approach revealed that the overtopping waves comprised an initial jet of 0.2 s duration, but dominated by quasi-steady flow. Whilst laboratory surface-elevation time-histories were highly repeatable, overtopping volume repeats were sensitive to the breaker type. Measured volumes were compared with: the numerical model (which over-predicted, but was reasonably accurate for steep-fronted waves); estimations based on BIV results (which provided very close agreement for the steep-fronted waves); and a weir-based analogy (which provided reasonable agreement, but always under-predicted).

An adaptive hybridizable discontinuous Galerkin approach for cardiac electrophysiology.

Cardiac electrophysiology simulations are numerically challenging because of the propagation of a steep electrochemical wave front and thus require discretizations with small mesh sizes to obtain accurate results. In this work, we present an approach based on the hybridizable discontinuous Galerkin method (HDG), which allows an efficient implementation of high-order discretizations into a computational framework. In particular, using the advantage of the discontinuous function space, we present an efficient p-adaptive strategy for accurately tracking the wave front. The HDG allows to reduce the overall degrees of freedom in the final linear system to those only on the element interfaces. Additionally, we propose a rule for a suitable integration accuracy for the ionic current term depending on the polynomial order and the cell model to handle high-order polynomials. Our results show that for the same number of degrees of freedom, coarse high-order elements provide more accurate results than fine low-order elements. Introducing p-adaptivity further reduces computational costs while maintaining accuracy by restricting the use of high-order elements to resolve the wave front. For a patient-specific simulation of a cardiac cycle, p-adaptivity reduces the average number of degrees of freedom by 95% compared to the nonadaptive model. In addition to reducing computational costs, using coarse meshes with our p-adaptive high-order HDG method also simplifies practical aspects of mesh generation and postprocessing.

Formation of Regions with High Energy and Pressure Gradients at the Free Surface of Liquid Dielectric in a Tangential Electric Field

Nonlinear dynamics of the free surface of an ideal incompressible non-conducting fluid with high dielectric constant subjected by strong horizontal electric field is simulated on the base of the method of conformal transformations. It is demonstrated that interaction of counter-propagating waves leads to formation of regions with steep wave front at the fluid surface; angles of the boundary inclination tend to {\pi}/2, and the curvature of surface extremely increases. A significant concentration of the energy of the system occurs at these points. From the physical point of view, the appearance of these singularities corresponds to formation of regions at the fluid surface where pressure exerted by electric field undergoes a discontinuity and dynamical pressure increases almost an order of magnitude.

A hybrid finite-volume and finite difference scheme for depth-integrated non-hydrostatic model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.

Effects of horizontal pressure gradients on bed destabilization under waves

We report on new experiments designed to investigate bed destabilization processes in a two-dimensional wave flume physical model of a beach. The mobile bed consists of non-cohesive granular material of low density. The wave conditions are provided by repeating a cycle of waves made of two bichromatic groups of different period. The horizontal and vertical velocities are acoustically profiled vertically from free-stream elevation down to the still bed level in the surf zone. Additional measurements of the fluid pressure at positions closely aligned horizontally and vertically in and slightly above the sediment bed are undertaken. Mobile bed interfaces, still bed and top interface, are detected via acoustic and optical methods. Both methods are cross-compared and give similar results. Flow turbulence over the bed is analysed, the Reynolds turbulent shear stress is found negligible compared to the orbital flow induced momentum diffusion. The shear stress and the horizontal pressure gradient are computed at near-bed elevation and used in the bed incipient plug flow model of Sleath (Cont. Shelf Res., vol. 19 (13), 1999, pp. 1643–1664). Both the model and the measurements confirm that destabilization occurs when the non-dimensional pressure gradient (or Sleath number) exceeds the threshold value of 0.3 which is simultaneous with strong flow acceleration. The near-bottom fluid shear stress detected during these flow accelerations at steep wave fronts is found experimentally to be negative, which is retrieved with an unsteady plug flow model. This is suggesting that the fluid above the bed resists the sediment layer motion at these particular phases.

Free boundary problems in shock reflection/diffraction and related transonic flow problems.

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection-diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

Modeling dam-break flows in channels with 90 degree bend using an alternating-direction implicit based curvilinear hydrodynamic solver

This paper presents a combined numerical and experimental investigation of dam break induced free-surface flows in channels with 90 degree bend. These types of flows are best described by the two-dimensional shallow water equations (SWE) representing the conservations of mass and horizontal momentums. In this study, the governing equations are solved numerically by means of an alternating-direction implicit (ADI) finite-difference scheme in a curvilinear coordinate and contravariant velocity system. This model is tested by simulating for various flow conditions including dam-break flows onto dry beds in a converging–diverging channel and a channel with 45 degree bend. Good fits of the present model predictions with published laboratory measurements are achieved. To further the validation of the model, a series of physical model tests for dam-break flows in a channel with 90 degree bend were conducted. The predicted time-varying water depths downstream of the dam face are shown to have a fairly good agreement with recorded data from model tests. The present ADI solver is found to be capable of capturing the formation and movement of steep wave fronts.

Formation mechanism of steep wave front in magnetized plasmas

Bifurcation from a streamer to a solitary drift wave is obtained in three dimensional simulation of resistive drift waves in cylindrical plasmas. The solitary drift wave is observed in the regime where the collisional transport is important as well as fluctuation induced transport. The solitary drift wave forms a steep wave front in the azimuthal direction. The phase of higher harmonic modes are locked to that of the fundamental mode, so that the steep wave front is sustained for a long time compared to the typical time scale of the drift wave oscillation. The phase entrainment between the fundamental and second harmonic modes is studied, and the azimuthal structure of the stationary solution is found to be characterized by a parameter which is determined by the deviation of the fluctuations from the Boltzmann relation. There are two solutions of the azimuthal structures, which have steep wave front facing forward and backward in the wave propagation direction, respectively. The selection criterion of these solutions is derived theoretically from the stability of the phase entrainment. The simulation result and experimental observations are found to be consistent with the theoretical prediction.

Two-phase damage models of magma-fracturing

Damage and fracturing in two-phase and porous flows are relevant for geological process such as magma-fracturing during melt migration, which is associated with the propagation of a pore-generating damage front ahead of high-pressure fluid injection. We therefore examine the propagation of porous flow in a damageable matrix by applying the two-phase theory for compaction and damage proposed by Bercovici et al. (2001a) and Bercovici and Ricard (2003). The movement of the fluid and the solid is governed by the two-phase flow laws, while damage (void generation and microcracking) is treated by considering the generation of interfacial surface energy by deformational work. Calculations of one-dimensional (1-D) flow of fluid migrating buoyantly through compacting and damageable matrix show that damage is mitigated in steady-state largely because of the loss of the velocity gradient at the fluid front. However, in time-dependent flows, linear stability analysis shows that the propagation velocity of porosity waves is strongly dependent on damage. In the damage-free case porosity waves are dispersive in that wave-speed decreases with wavenumber (inverse wavelength); however with damage the dispersion flattens and beyond a critical damage reverses (the wave speed increases with wavenumber). Since normal dispersive behavior balances breaking in the nonlinear wave case, such reversed dispersion implies that damage has a profound effect in the nonlinear limit by facilitating wave front steepening and higher wave velocities. Nonlinear solitary wave solutions are obtained numerically and show that the transmission of porosity waves induces high stress and damage that can push the damage front forward. With damage the porosity waves sharpen and calculations suggest that they can transform from shape-conserving solitary waves into faster high amplitude waves, which is also predicted by the linear theory. Such pulse-like sharper waves may prove effective at promoting fluid migration through magma-fracturing.

Longitudinal nonlinear wave propagation through soft tissue

In this paper, wave propagation through soft tissue is investigated. A primary aim of this investigation is to gain a fundamental understanding of the influence of soft tissue nonlinear material properties on the propagation characteristics of stress waves generated by transient loadings. Here, for computational modeling purposes, the soft tissue is modeled as a nonlinear visco-hyperelastic material, the geometry is assumed to be one-dimensional rod geometry, and uniaxial propagation of longitudinal waves is considered. By using the linearized model, a basic understanding of the characteristics of wave propagation is developed through the dispersion relation and in terms of the propagation speed and attenuation. In addition, it is illustrated as to how the linear system can be used to predict brain tissue material parameters through the use of available experimental ultrasonic attenuation curves. Furthermore, frequency thresholds for wave propagation along internal structures, such as axons in the white matter of the brain, are obtained through the linear analysis. With the nonlinear material model, the authors analyze cases in which one of the ends of the rods is fixed and the other end is subjected to a loading. Two variants of the nonlinear model are analyzed and the associated predictions are compared with the predictions of the corresponding linear model. The numerical results illustrate that one of the imprints of the nonlinearity on the wave propagation phenomenon is the steepening of the wave front, leading to jump-like variations in the stress wave profiles. This phenomenon is a consequence of the dependence of the local wave speed on the local deformation of the material. As per the predictions of the nonlinear material model, compressive waves in the structure travel faster than tensile waves. Furthermore, it is found that wave pulses with large amplitudes and small elapsed times are attenuated over shorter spans. This feature is due to the elevated strain-rates introduced at the end of the structure where the load is applied. In addition, it is shown that when steep wave fronts are generated in the nonlinear viscoelastic material, energy dissipation is focused in those wave fronts implying deposition of energy in a highly localized region of the material. Novel mechanisms for brain tissue damage are proposed based on the results obtained. The first mechanism is related to the dissipation of energy at steep wave fronts, while the second one is related to the interaction of steep wave fronts with axons encountered on its way through the structure.

Numerical Modeling of Tsunami Wave Run-Up and Effects on Sediment Scour around a Cylindrical Pier

AbstractThe linked two-dimensional hydrodynamic and sediment-scour models have been enhanced to simulate tsunami wave run-up on a sloping beach to determine its effects on sediment scour around a cylinder pier located in the wave breaking and run-up zone. To resolve the steep wavefront of the tsunami bore, the kinetic flux vector splitting scheme was adopted to solve the hydrodynamic model equations in the unstructured triangular mesh. The models have been validated by comparing model simulations with experimental data. The results indicate that the model predictions of water surface elevations and velocity match well with the measured data. The error for peak flood elevation ranges from 0.01 to 0.11 m, and the maximum error for the peak velocity is 6%. The model simulations adequately characterize the tsunami wave propagations and transformations as the wave approaches the beach from offshore, especially for the sharp tsunami front before it breaks and the tsunami bore runs up in the beach slope. The mod...

Velocity Coupling from Pulse-Test Firings

There has been much dispute about triggering of non-linear combustion instability(CI). The norm has been to use ad hoc models to include second order combustion terms most notably velocity coupling. This term arose due to classical linear analyses predicting stable motors necessitating another driving function. However there is no practical universally accepted experimental method that can obtain the velocity coupling response function. The solid rocket motor (SRM) designer is left to his own devices as to what the contribution of velocity coupling may be. This has led to much confusion and rocket motors that have been predicted stable but in practice were not. Flandro and co workers provide an alternative interpretation into the cause of non-linear combustion instability by including rotational terms into the linear analysis. This has been shown to improve the linear stability prediction by predicting unstable systems i.e. CI is always present but at low amplitudes growing too slowly to form a steep fronted wave. A pulse in this context needs to be of high enough amplitude to induce rapid growth. No criterion exists to predict such an amplitude. This study uses a newly developed analysis methodology using tubular grain SRMs to investigate both interpretations i.e. a linearly stable system with velocity coupling or linearly unstable system. This method when combined with response data can be used to obtain a velocity coupling response function from pulse tests. From this analysis it was possible to obtain reasonable velocity coupling response function. When using these constants to predict it was possible to predict a trigger point. Interestingly this point changes significantly with the initial higher order mode amplitudes. It then becomes vital to understand the likely wave structure of a pulse and what influence it has on the triggering amplitude. The system dynamics did not follow that of the experimental data but this may be due to truncation errors and assumptions made in the mathematical model. However the limiting amplitude is predicted correctly. The Flandro interpretation can be evaluated using this new method and applying a systematic study to evaluate criteria for the triggering amplitude. The linearly unstable values seem to capture the system dynamics better. However, in theory any pulse should cause CI. Regardless whether it is true or perceived triggering there seems to be an amplitude value at which non-linear CI is induced. The boundary layer pumping term here seems to be the key to interpreting triggering. The boundary layer acts as a piston that drives the acoustic oscillations when in phase. Thus for non-linear CI to occur the pulse must be of such an amplitude to cause the boundary layer to oscillate and maintain a high enough amplitude until the boundary layer starts to oscillate in phase. Regardless of interpretation this method shows great promise in allowing SRM designers to obtain quantitative response data and investigate triggering quickly and cheaply.