Regions Of Constant Depth Research Articles

Unsteady draining of reservoirs over weirs and through constrictions

The gravitationally driven flow of fluid from a reservoir following the partial collapse of its confining dam, or the partial opening of its confining lock, is modelled using the nonlinear shallow water equations, coupled to outflow conditions, in which the drainage is modelled as flow over a constricted, broad-crested weir. The resulting unsteady motion reveals systematic draining, on which strong and relatively rapid oscillations are imposed. The oscillations propagate between the outflow and the impermeable back wall of the reservoir. This dynamics is investigated utilising three methods: hodograph techniques to yield quasi-analytical solutions, asymptotic analysis at relatively late times after initiation and numerical integration of the governing equations. The hodograph transformation is used to find exact solutions at early times, revealing that from initially quiescent conditions the fluid drains and yet repeatedly generates intervals during which there are regions of constant depth and velocity adjacent to the boundaries. A novel modified multiscale asymptotic analysis designed for late times is employed to determine the limiting rate of draining and wave structure. It is shown that the excess height drains as and, when the obstacle has finite height, the velocity field decays as , and exhibits a wave structure that tends towards a constant and relatively rapid phase speed. In the case of a pure constriction, for which all the fluid ultimately drains out of the reservoir, the motion adjusts to a self-similar state in which the velocity field decays as . Oscillations around this state have an exponentially increasing period. Numerical simulations with a novel implementation of boundary conditions are performed; they confirm the hodograph solution and provide data for the asymptotic results.

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.

Estimating tsunami runup with fault plane parameters

Forecasting the maximum runup height and inundation area caused by a tsunami event has often been done by solving a numerical model describing the physical process. This approach requires data of fault plane parameters and bathymetry. The time required to prepare and execute the numerical model could be substantial. Therefore, it might not be suitable for the purpose of warning tsunami coastal flooding. In this paper we offer an alternative method that provides analytical relationship between the runup height and the fault plane parameters and the characteristic slope of coastal bathymetry. The method uses Okada's linear elastic dislocation model (Okada, 1985) to estimate the coseismic seafloor deformation and the corresponding (initial) sea surface displacement for a given set of fault plane parameters. Once the tsunami waves are generated, Carrier & Greenspan's solution (1958) is adopted to yield analytical expressions for the time histories of shoreline elevation and velocity. Two types of problems are investigated. In the first scenario, the bathymetry is modeled as a constant slope that is connected to a constant depth region, where a seismic event occurs. In the second scenario, the bathymetry is further simplified as a constant slope, on which a seismic event occurs. As far as the Carrier–Greenspan's solution for shoreline movement is concerned, the former is treated as a boundary-value-problem (BVP) and the latter as an initial-value-problem (IVP). For both problems the relationships between dimensionless runup height and dimensionless fault plane parameters are developed and discussed. In addition, since Carrier–Greenspan's approach is valid only for non-breaking waves, expressions limiting the applicability of the present solutions are obtained. Application of the present solutions is demonstrated with two examples: the 2004 Sumatra and 2010 Chile earthquakes. Acceptable agreements are obtained with the field measurements of these events. The present solutions offer a fast way to estimate the runup heights with the simplified beach geometry.

An experimental study of the interaction of two successive solitary waves in the swash: A strongly interacting case and a weakly interacting case

The interaction of successive solitary waves in the swash zone have been studied using large-scale experiments with a simple bathymetry of a constant depth region, where the water depth was 1.72m, and a plane beach, whose slope was 1:12. Two wave cases were considered where two successive solitary waves of the same height were generated one after the other so that the wave crests were separated by the effective wavelength associated with a single solitary wave. In the weakly interacting case, the swash period associated with a single solitary wave is smaller than the time period separating the successive wave crests, which leads to similar run-ups for the first and second wave, whereas in the strongly interacting case, the swash period associated with a single solitary wave is larger than the time period separating the successive wave crests, which leads to a significant reduction in the run-up of the second wave. The degree to which there is an interaction between the swash uprush of the second wave and the downrush of the first wave is found to be related to the solitary wave slope parameter, which predicts breaker type of the first wave. Previous data from literature are found to support this claim. Measurements of bed shear stress, bed pressure, and the free-surface displacement at a location near the stillwater shoreline are used to describe how the dynamics of the boundary layer differ when the downrush of the first wave meets the incoming second wave in both cases.

Run-up amplification of transient long waves

The extreme characteristics of long wave run-up are studied in this paper. First we give a brief overview of the existing theory which is mainly based on the hodograph transformation (Carrier & Greenspan, 1958). Then, using numerical simulations, we build on the work of Stefanakis et al. (2011) for an infinite sloping beach and we find that resonant run-up amplification of monochromatic waves is robust to spectral perturbations of the incoming wave and resonant regimes do exist for certain values of the frequency. In the setting of a finite beach attached to a constant depth region, resonance can only be observed when the incoming wavelength is larger than the distance from the undisturbed shoreline to the seaward boundary. Wavefront steepness is also found to play a role in wave run-up, with steeper waves reaching higher run-up values.

Run-up of long solitary waves of different polarities on a plane beach

We study the run-up of long solitary waves of different polarities on a beach in the case of composite bottom topography: a plane sloping beach transforms into a region of constant depth. We confirm that nonlinear wave deformation of positive polarity (wave crest) resulting in an increase in the wave steepness leads to a significant increase in the run-up height. It is shown that nonlinear effects are most strongly pronounced for the run-up of a wave with negative polarity (wave trough). In the latter case, the run-up height of such waves increases with their steepness and can exceed the amplitude of the incident wave.

Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water

Using a Boussinesq model with improved linear dispersion, we show numerical evidence that bottom non-uniformity can provoke significantly increased probability of freak waves as a wave field propagates into shallower water, in agreement with recent experimental results [K. Trulsen, H. Zeng, and O. Gramstad, “Laboratory evidence of freak waves provoked by non-uniform bathymetry,” Phys. Fluids 24, 097101 (2012)]. Increased values of skewness, kurtosis, and probability of freak waves can be found on the shallower side of a bottom slope, with a maximum close to the end of the slope. The increased probability of freak waves is typically seen to endure some distance into the shallower domain, before it decreases and reaches a stable value depending on the depth. The maxima of the statistical parameters are observed both in the case where there is a region of constant depth after the slope, and in the case where the uphill slope is immediately followed by a downhill slope. In the case that waves propagate over a slope from shallower to deeper water, however, we do not find any increase in freak wave occurrence.

Context Coding of Depth Map Images Under the Piecewise-Constant Image Model Representation

This paper introduces an efficient method for lossless compression of depth map images, using the representation of a depth image in terms of three entities: 1) the crack-edges; 2) the constant depth regions enclosed by them; and 3) the depth value over each region. The starting representation is identical with that used in a very efficient coder for palette images, the piecewise-constant image model coding, but the techniques used for coding the elements of the representation are more advanced and especially suitable for the type of redundancy present in depth images. Initially, the vertical and horizontal crack-edges separating the constant depth regions are transmitted by 2D context coding using optimally pruned context trees. Both the encoder and decoder can reconstruct the regions of constant depth from the transmitted crack-edge image. The depth value in a given region is encoded using the depth values of the neighboring regions already encoded, exploiting the natural smoothness of the depth variation, and the mutual exclusiveness of the values in neighboring regions. The encoding method is suitable for lossless compression of depth images, obtaining compression of about 10-65 times, and additionally can be used as the entropy coding stage for lossy depth compression.

Transformation of a shoaling undular bore

AbstractWe consider the propagation of a shallow-water undular bore over a gentle monotonic bottom slope connecting two regions of constant depth, in the framework of the variable-coefficient Korteweg–de Vries equation. We show that, when the undular bore advances in the direction of decreasing depth, its interaction with the slowly varying topography results, apart from an adiabatic deformation of the bore itself, in the generation of a sequence of isolated solitons – an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore. Using nonlinear modulation theory we construct an asymptotic solution describing the formation and evolution of this solitary wavetrain. Our analytical solution is supported by direct numerical simulations. The presented analysis can be extended to other systems describing the propagation of undular bores (dispersive shock waves) in weakly non-uniform environments.

On the runup of long waves on a plane beach

Using the records of free surface fluctuations at several locations during the 2011 Japan Tohoku tsunami, we first show that the leading tsunami waves in both near‐field and far‐field regions are small amplitude long waves. These leading waves are very different from solitary waves. We then focus on investigating the evolution and runup of non‐breaking long waves on a plane beach, which is connected to a constant depth region. For this purpose, we develop a Lagrangian numerical model to solve the nonlinear shallow water equations. The Lagrangian approach tracks the moving shoreline directly without invoking any additional approximation. We also adopt and extend the analytical solutions by Synolakis (1987) and Madsen and Schäffer (2010) for runup and rundown of cnoidal waves and a train of multiple solitary waves. The analytical solutions for cnoidal waves compare well with the existing experimental data and the direct numerical results when wave amplitudes are small. However, large discrepancies appear when the incident amplitudes are finite. We also examine the relationship between the maximum runup height and the leading wave form. It is concluded that for a single wave the accelerating phase of the incident wave controls the maximum runup height. Finally, using the analytical solutions for the approximated wave forms of the leading tsunamis recorded at Iwate South station from the 2011 Tohoku Japan tsunami, we estimate the runup height.

A numerical study of swash flows generated by bores

The dynamic processes of bore propagation over a uniform slope are studied numerically using a 2-D Reynolds Averaged Navier–Stokes (RANS) solver, coupled to a non-linear k − ε turbulence closure and a volume of fluid (VOF) method. The dam-break mechanism is used to generate bores in a constant depth region. Present numerical results for the ensemble-averaged flow field are compared with existing experimental data as well as theoretical and numerical results based on non-linear shallow water (NSW) equations. Reasonable agreement between the present numerical solutions and experimental data is observed. Using the numerical results, small-scale bore behaviors and flow features, such as the bore collapse process near the still-water shoreline, the ‘mini-collapse’ during the runup phase and the ‘back-wash bore’ in the down-rush phase, are described. In the case of a strong bore, the evolution of the averaged turbulence kinetic energy (TKE) over the swash zone consists of two phases: in the region near the still-water shoreline, the production and the dissipation of TKE are roughly in balance; in the region farther landwards of the still-water shoreline, the TKE decay rate is very close to that of homogeneous grid turbulence. On the other hand, in the case of a weak bore, the bore collapse generated turbulence is confined near the bottom boundary layer and the TKE decays at a much slower rate.

An analytic solution to the mild slope equation for waves propagating over an axi-symmetric pit

An analytic solution to the mild slope equation is derived for waves propagating over an axi-symmetric pit located in an otherwise constant depth region. The water depth inside the pit decreases in proportion to an integer power of radial distance from the pit center. The mild slope equation in cylindrical coordinates is transformed into ordinary differential equations by using the method of separation of variables, and the coefficients of the equation in radial direction are transformed into explicit forms by using the direct solution for the wave dispersion equation by Hunt (Hunt, J.N., 1979. Direct solution of wave dispersion equation. J. Waterw., Port, Coast., Ocean Div., Proc. ASCE, 105, 457–459). Finally, the Frobenius series is used to obtain the analytic solution. Due to the feature of the Hunt's solution, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth waters. The validity of the analytic solution is demonstrated by comparison with numerical solutions of the hyperbolic mild slope equations. The analytic solution is also used to examine the effects of the pit geometry and relative depth on wave transformation. Finally, wave attenuation in the region over the pit is discussed.

이동로봇의 자율주행을 위한 다중센서융합기반의 지도작성 및 위치추정

Presently, the exploration of an unknown environment is an important task for thee new generation of mobile service robots and mobile robots are navigated by means of a number of methods, using navigating systems such as the sonar-sensing system or the visual-sensing system. To fully utilize the strengths of both the sonar and visual sensing systems. This paper presents a technique for localization of a mobile robot using fusion data of multi-ultrasonic sensors and vision system. The mobile robot is designed for operating in a well-structured environment that can be represented by planes, edges, comers and cylinders in the view of structural features. In the case of ultrasonic sensors, these features have the range information in the form of the arc of a circle that is generally named as RCD(Region of Constant Depth). Localization is the continual provision of a knowledge of position which is deduced from it`s a priori position estimation. The environment of a robot is modeled into a two dimensional grid map. we defines a vision-based environment recognition, phisically-based sonar sensor model and employs an extended Kalman filter to estimate position of the robot. The performance and simplicity of the approach is demonstrated with the results produced by sets of experiments using a mobile robot.

Stereo matching via selective multiple windows

Window-based correlation algorithms are widely used for stereo matching due to their computational efficiency as compared to global algorithms. In this paper, a multiple window correlation algorithm for stereo matching is presented which addresses the problems associated with a fixed window size. The developed algorithm differs from the previous multiple window algorithms by introducing a reliability test to select the most reliable window among multiple windows of increasing sizes. This ensures that at least one window is large enough to cover a region of adequate intensity variations while at the same time small enough to cover a constant depth region. A recursive computation procedure is also used to allow a computationally efficient implementation of the algorithm. The outcome obtained from a standard set of images with known disparity maps shows that the generated disparity maps are more accurate as compared to two popular stereo matching local algorithms.

Influence of variable Froude number on waves generated by ships in shallow water

Passage through the transcritical speed region of a moving disturbance in a shallow channel, is examined using numerical simulations based on a set of forced Boussinesq equations. The transition is accomplished either by accelerating the wave generating disturbance in a region of constant depth or by moving the disturbance with constant speed over a sloping bottom topography. A series of test cases are examined where the transcritical region is traversed both from subcritical to supercritical speed and vice versa. Results show that the generation of upstream solitary waves depends on the time required for the transition, with large waves being generated for long transition times. It is also apparent that the shape of the wave pattern, and the maximum amplitude of the waves, differ significantly depending on whether the Froude number increase or decrease during the transition of the transcritical region. However, the wave pattern is not determined simply in terms of the Froude number. The strength of the forcing term as well as the underlying process which cause the Froude number to vary, i.e., acceleration and depth variation, influence the wave pattern in different ways. The Froude number is none the less a useful indicator for the problem, as all cases with similar Froude number variations share some common characteristic features.

Wave transformation by axisymmetric three-dimensional bathymetric anomalies with gradual transitions in depth

The development of an analytic model (Axisymmetric 3-D Step Model) for the propagation of linear water waves over an axisymmetric bathymetric anomaly in arbitrary water depth is presented. The Axisymmetric 3-D Step Model is valid in a region of uniform depth containing an axisymmetric bathymetric anomaly with gradual transitions in depth allowed as a series of steps approximating arbitrary slopes. The velocity potential is calculated by applying matching conditions at the interface between regions of constant depth. The velocity potential obtained determines the wave field in the domain for monochromatic incident waves of linear form. A second analytic model (3-D Shallow Water Exact Model) is developed for comparison within the shallow water limit. The Axisymmetric 3-D Step Model determines the wave transformation caused by the processes of wave refraction, diffraction and reflection. Wave transformation is demonstrated in plots of the relative amplitude for bathymetric anomalies in the form of pit or a shoal, highlighting areas of wave sheltering and wave focusing. Anomalies of constant volume, but variable cross-section are employed to isolate the effect of the transition slope on the wave transformation. Comparisons to a shallow water model, numerical models, and experimental data verify the results of the Axisymmetric 3-D Step Model for several bathymetries including both pits and shoals. Also included are estimates of the energy reflection induced by an axisymmetric depth anomaly. The 3-D Axisymmetric Step Model has been applied previously to account for nearshore transformation (sloping bathymetry) and associated shoreline changes [C.J. Bender, R.G. Dean, Coastal Engineering 51 (2004) 1143].

Long waves propagating over a circular bowl pit

An analytic solution to the mild slope wave equation is derived for long waves propagating over a circular, bowl-shaped pit located in an otherwise constant depth region. The analytic solution is shown to reduce to a previously derived analytic solution for the case of a bowl-shaped enclosed basin and to agree well with a numerical solution of the hyperbolic mild-slope equations. The effects of the pit dimensions on wave scattering are discussed based on the analytic solution. This analytic solution can also be applied to pits of different general shapes. Finally, wave attenuation in the region over the pit is discussed.

Water wave scattering by a step of arbitrary profile

The two-dimensional scattering of water waves over a finite region of arbitrarily varying topography linking two semi-infinite regions of constant depth is considered. Unlike many approaches to this problem, the formulation employed is exact in the context of linear theory, utilizing simple combinations of Green's functions appropriate to water of constant depth and the Cauchy–Riemann equations to derive a system of coupled integral equations for components of the fluid velocity at certain locations. Two cases arise, depending on whether the deepest point of the topography does or does not lie below the lower of the semi-infinite horizontal bed sections. In each, the reflected and transmitted wave amplitudes are related to the incoming wave amplitudes by a scattering matrix which is defined in terms of inner products involving the solution of the corresponding integral equation system.This solution is approximated by using the variational method in conjunction with a judicious choice of trial function which correctly models the fluid behaviour at the free surface and near the joins of the varying topography with the constant-depth sections, which may not be smooth. The numerical results are remarkably accurate, with just a two-term trial function giving three decimal places of accuracy in the reflection and transmission coefficents in most cases, whilst increasing the number of terms in the trial function results in rapid convergence. The method is applied to a range of examples.

Coastal amplification of a tsunami wave train

Steady-state wave solutions are obtained in closed form for two simple coastal geometries. The first model consists of two regions of constant depth separated by a step, and the second model consists of a constantdepth region joined by a step to a region in which the mean depth decreases linearly to a value of zero at the shore line. The step, which is used to model a continental shelf, can create resonance and large wave amplifications in both cases, but amplifications are found to be much greater for the second model. Furthermore, peak amplifications at resonance are found to not change with incident wave frequency for the first model but to increase with wave frequency for the second model. Finally, since wave amplifications in the second model depend very strongly on incident wave frequencies, periods are calculated for waves near the leading edge of a tsunami wave train moving through water of constant depth.

Propagation of bottom-trapped waves over variable topography

Abstract The behavior of bottom-trapped waves on a topographic slope, as they propagate towards a deeper region of constant depth, is examined using a quasi-geostrophic model. It is seen that there can be no free transmitted wave to the region of constant depth. The bottom-trapped energy is thus segregated in the region of the slope and its vicinity. The details of how this occurs are worked out for a slope shoaling towards the west. For a given frequency and meridional wavenumber, two free bottom-trapped waves can exist on the slope, with the shorter (longer) of the two having an eastward (westward) group velocity. When the short wave propagates to the region of constant depth, a reflected wave is generated. There is no transmitted wave, but a ‘fringe’ which decays away from the interface between the slope and the region of flat topography is produced. Over the flat topography the fringe consists of baroclinic and barotropic motions which lead to bottom-intensification in the immediate vicinity of the slope and to increasingly barotropic currents farther away.