Mild Slope Research Articles

Effect of frontal slope on waveform evolution of a depression interfacial solitary wave across a trapezoidal obstacle

Waveform evolution associated with a depression interfacial solitary wave (ISW) over uniform slope or slope-shelf with different front slopes has been observed in many laboratory experiments and field studies since the 1980s, and by numerical modeling in more recent time. However, attention was mainly focused on waveform validation with KdV theory, wave run-up, reflection, energy dissipation and evolution on emergent mild uniform slope, rather on wave evolution and inversion across the plateau of a slope-shelf with steep front slopes, which are common in field conditions. Since it is often difficult to measure the entire process of waveform evolution in the field, laboratory experiments are adopted as an alternative to compliment field observations. This paper presents the results of laboratory experiments on ISW evolution across a submerged trapezoidal obstacle with four different front slopes, ranging from mild to very steep and vertical. Our results indicate that not only the likelihood of internal hydraulic jump, waveform inversion and the strength of internal wave breaking reduced, but more wave energy dissipated as well, as the front slope steepened. Upon using Hilbert–Huang transform to analyse the wave profiles recorded at four probe locations along the wave flume, the variability in the instantaneous characteristic wave period can be detected while an ISW propagated across a trapezoidal obstacle.

Modeling of Erosion at Göksu Coasts

Numanoğlu Genç, A., İnan, A., Yılmaz, N. and Balas, L., 2013. Modeling of Erosion at Göksu Coasts.Göksu coastal area, located in the south of Silifke County of Mersin on the coastal plain formed by Göksu River, is one of the Specially Protected Areas with an area of almost 15000 hectares and at an altitude of 0–5 m. The Eastern and Western parts of the Göksu river mouth along the Mediterranean coast and the coast of Paradeniz Lagoon are suffering significantly from erosion. As a result of the coastal erosion and the consequent coastal recession problem along the coasts of Göksu, agricultural fields located near the coast are being lost, and especially the narrow barrier beach which separates Paradeniz Lagoon from the sea is getting narrower, creating a risk of uniting with the sea, thus disappearing of the Lagoon. In order to propose solutions for the problem, and to protect the coastal zone and the habitats of Göksu against coastal erosion, coastal dynamic processes and sediment transport characteristics of the region causing the erosion should be investigated and identified. Therefore, longshore sediment transport and suspended sediment transport in Göksu coastal area are examined. First of all, the wind and wave climate of the study area are determined. Then, current pattern, wave propagation, and longshore sediment movements are modeled numerically utilizing three dimensional hydrodynamic transport model HYDROTAM-3D. HYDROTAM-3D is a three (3) dimensional numerical Hydrodynamic and Transportation Model which simulates current patterns due to wind and wave actions, pollutant transport, wave propagation over mild slopes, and longshore sediment transport rates. Model includes hydrodynamics, transport, turbulence, wind and wave climate and wave propagation sub models. HYDROTAM 3D, is developed using “cloud computing” architecture and tightly integrated with Geographic Information System. The study outputs will be used for the determination of the coastal erosion.

津波による旧川位置での河道形成と戻り流れの排水機能

This study investigated the effect of the past drainage system and river mouth to the return flow of the 2011 Great East Japan Tsunami at the Arahama Coast and the Akaiko Coast, Sendai, Japan. At both locations, there were old river mouths, which have been closed for many years. They used to be the down stream end of the past drainage system that was confirmed from historical map. Breaching occurred at the location of these old river mouths due to the strong return flow induced by the 2011 tsunami. A detail GIS analysis was carried out based on topographic data (DEM) to obtain the flow direction and accumulation. The results revealed that the topography appears as a river basin with mild slope towards the old river mouth locations. Analysis of the flow accumulation confirms that the topographical features caused the flow to accumulate through the past drainage system towards the location of the old river mouths, which caused the breaching due to the tsunami return flow. This allowed the return flow to be effectively drained.

橋桁に作用する津波波力と津波流速の関係

Authors carried out hydraulic experiments to clarify the relationship between tsunami fluid force and tsunami velocity acting on a bridge deck subjected to surging breaker bores, plunging breaker bores, waves whose surface inclination is mild slope, and breaker bores with a bridge deck sets up under the still water level. Experimental results show the mechanism that the height of a bridge deck to a tsunami height with meanings of variation of surface inclination of waves causes a significant difference of wave velocity. In addition, it causes variation of Froude number, and leads differences of horizontal and vertical fluid forces acting on a bridge deck.

Prediction of Erosion Rate at Several Land Units in Upper Watershed of Batang Mangau, West Sumatera

Assessment of erosion in a watershed is very important for the purpose of determining whether a watershed has been degraded and damaged or not, so it is useful for the planning of the future for the watershed areas so as to achieve sustainable land use and environmentally friendly.Study of erosion prediction and mapping erosion hazard rate (TBE), and determine the rate of erosion that can be tolerated in a variety of land units, and determining alternative land use followed by the appropriate conservation measures in order to suppress erosion as small as possible or equal to Etol (tolerable erosion ) has conducted research on upstream Mangau Padang Pariaman and Agam districts. The study lasted in September 2011 until January 2012. The experiment was conducted using the survey method. Secondary data obtained from statistical data, maps and the results of previous researches. The results showed that the greatest erosion occurs in people's gardens land units with steep slopes (KrF), with a percentage of 62% slope and slope length of 45 m. While the smallest erosion occurs in wetland units with a gentle slope (SWB), the percentage of 7% slope and slope length of 28 m. Land units experiencing mild erosion rate is 37.80%, of the total study area. Then the land units suffered heavy erosion rate is as much as 20.81%, and further land experiencing very severe erosion hazard level is sebayak 37.86% of the total study area. Actual erosion rate greater than can be tolerated erosion found 4 unit of land is the garden of the people with a rather steep slope (KrD), mixed garden with a rather steep slope (KcD), and their fields with steep slopes (KrF). Furthermore, there are 8 land unit value erosion rate is less than the erosion rates that can be tolerated so that the use of the land does not need changing. Alternative use of land for every land units that have greater erosion speed of the tolerable erosion is the land units of protected forest with steep slopes (HlF), the secondary forest with steep slopes (HsF). Whereas, the land unit of rice paddy soil with relatively steep slopes (SwD), rice paddy soil with gentle slopes (SwB), scrub with relatively steep slopes (SmD), scrub with steep slopes (SmF) and farm people with mild slopes (KrB) fixed defended. Furthermore, the alternative use of the land for the unit of farm people with steep slopes (KrF) and farm people with relatively steep slopes (KrD) and a mixture of garden soil units with relatively steep slopes (KcD) is recommended for improved by adding high density mixed garden and make bench terrrace with good construction and land cover tightly coupled.

Investigation for Bed Stabilization Methods in the Upstream Channel of Haman Weir Using CCHE2D Model

During the four river restoration project, several weirs were constructed in the four rivers to prevent drought and flood, to improve water quality, and to manage water resources. However, due to the weir construction, bed changes are produced in the upstream channel of installed weirs because the incoming flow velocity is reduced and sediment transport capacity is also lowered. Especially, since the Haman Weir is located in the lowest downstream section among newly installed weirs in Nakdong River, bed change and sedimentation problems are expected due to the mild slope and reduced velocity. Therefore, numerical simulation was performed to analyze flow and bed changes in the upstream channel of Haman Weir and to evaluate quantitatively sediment control methods for bed stabilization using CCHE2D model. As a result of flow and bed change simulation after installation of Haman Weir, the flow velocity at the initial condition was faster than the final bed condition with the specific simulation time and it was represented that the locations where bed changes were great were identical for all modeling conditions of flow discharge. In case of 4.5 m of water level lowered from 5.0 m of the management water level at Haman Weir for bed stabilization, the flow velocity was generally faster than the case of the management water level and the continuous erosion was developed at the most narrow channel section as the applied discharge and simulation period were increased. The channel width extension at the most narrow channel section was proposed in this study to prevent and stabilize continuos bed erosion. As a result of numerical analysis, there was no bed erosion after channel width extension and it was presented that the channel geometry extension was effective for bed stabilization at Haman Weir.

SHOALING OF NONLINEAR INTERNAL WAVES ON A UNIFORMLY SLOPING BEACH

The internal waves in the two-layer systems have been numerically simulated by solving the set of nonlinear equations in consideration of both strong nonlinearity and strong dispersion of waves. After the comparison between the numerical results and the BO solitons, as well as the experimental data, the internal waves propagating over the uniformly sloping beach are simulated including the cases of the mild and long slopes. The internal waves show remarkable shoaling after the interface touches the critical level. In the lower layer, the horizontal velocity becomes larger than the local linear celerity of internal waves in shallow water just before the crest peak and the position is defined as the wave-breaking point when the ratio of nonlinear parameter to beach slope is large. The ratio of initial wave height to wave-breaking depth becomes larger as the slope is milder and the wave nonlinearity is stronger. The wave height does not increase so much before wave-breaking on the mildest slope.

Modeling wave-mud interaction on the central chenier-plain coast, western Louisiana Shelf, USA

The strong coupling between hydrodynamics and seafloors on shallow muddy shelves, and resulting bed reworking, have been extensively documented. On these shelves, spectral wave transformation is driven by a complex combination of forcing mechanisms that include nonlinear wave interactions and wave energy dissipation induced by fluid-mud at a range of frequencies. Wave-mud interaction is investigated herein by using a previously validated nonlinear spectral wave model and observations of waves and near-bed conditions on a mildly-sloping seafloor off the muddy central chenier-plain coast, western Louisiana Shelf, United States. Measurements were made along a cross-shelf transect spanning 1km between 4 and 3m water depths. The high-resolution observations of waves and near-bed conditions suggest presence of a fluid mud layer with thickness sometimes exceeding 10cm under strong long wave action (1 meter wave height with 7 s peak period at 4 meter depth). Spectral wave transformation is modeled using the stochastic formulation of the nonlinear Mild Slope Equation, modified to account for wave-breaking and mud-induced dissipation. The model is used in an inverse manner in order to estimate the viscosity of the fluid mud layer, which is a key parameter controlling mud-induced wave dissipation but complicated to measure in the field during major wave events. Estimated kinematic viscosities vary between 10−4-10−3 m2/s. Combining these results of the wave model simulations with in-depth analysis of near-bed conditions and boundary layer modeling allows for a detailed investigation of the interaction of nonlinear wave propagation and mud characteristics. The results indicate that mud-induced dissipation is most efficient when the wave-induced resuspensions of concentrations >10g/L settle due to relatively small bottom stresses to form a fluid mud layer that is not as thin and viscous as a consolidated seafloor in absence of wave action but also not as thick and soft as a near-bed high concentration layer that forms during strong wave action.

An experimental study of large waves in intermediate and shallow water depths. Part I: Wave height and crest height statistics

This paper describes a series of laboratory observations undertaken in a purpose-built wave flume. The objective of the study was to simulate a range of realistic ocean spectra evolving over a number of mild bed slopes (m); the chosen sea states allowing a systematic investigation of the role of local water depth, spectral peak period, significant wave height, spectral bandwidth and bed slope. The focus of the present paper lies in the statistical distribution of both the wave height (H) and the wave crest elevation (ηc); the laboratory data being used to provide both an understanding of the importance of the various parameters and an assessment of the commonly adopted design distributions. The results of the study have shown that both the wave height and the crest height distributions are primarily dependent upon the effective water depth (kpd), where kp is the wave number corresponding to the spectral peak and d the water depth, and the sea state steepness (1/2Hskp), where Hs is the significant wave height. Conversely, the distributions are independent of the mild bed slope (for m1≤/100) and the spectral bandwidth; the latter result being very different to that which arises in deep water. Following detailed comparisons, across a wide parameter range, the wave height distributions in the deeper water depths (kpd>1.0) are shown to be in reasonable agreement with the (Forristall, 1978) and (Glukhovskii, 1966) distributions; the latter becoming more important as the proportion of breaking waves increases. As the water depth reduces, the (Mendez et al., 2004) distribution as well as the (Battjes & Groenendijk, 2000) composite Weibull distribution (CWD) become increasingly applicable. However, comparisons between sea states of varying steepness, but with all other parameters held constant, suggest that the empirical parameters on which the latter solution is based may not be universally applicable. Finally, in the shallowest water depths (kpd≈0.5), none of the existing distributions provide a good description of the measured data; all leading to an over-estimate of the largest wave heights.

A method for estimating spatially variable seepage and hydraulic conductivity in channels with very mild slopes

AbstractInfiltration along ephemeral channels plays an important role in groundwater recharge in arid regions. A model is presented for estimating spatial variability of seepage due to streambed heterogeneity along channels based on measurements of streamflow‐front velocities in initially dry channels. The diffusion‐wave approximation to the Saint‐Venant equations, coupled with Philip's equation for infiltration, is connected to the groundwater model MODFLOW and is calibrated by adjusting the saturated hydraulic conductivity of the channel bed. The model is applied to portions of two large water delivery canals, which serve as proxies for natural ephemeral streams. Estimated seepage rates compare well with previously published values. Possible sources of error stem from uncertainty in Manning's roughness coefficients, soil hydraulic properties and channel geometry. Model performance would be most improved through more frequent longitudinal estimates of channel geometry and thalweg elevation, and with measurements of stream stage over time to constrain wave timing and shape. This model is a potentially valuable tool for estimating spatial variability in longitudinal seepage along intermittent and ephemeral channels over a wide range of bed slopes and the influence of seepage rates on groundwater levels. Copyright © 2012 John Wiley & Sons, Ltd.

Hydraulic resistance to overland flow on surfaces with partially submerged vegetation

This study investigates numerically the characteristics of upscaling the Manning resistance coefficient (nt) for areas covered by partially submerged vegetation elements, such as shrub or tree stems. A number of high‐resolution hydrodynamic simulations were carried out corresponding to scenarios with different domain slopes (S), inflow rates (Q), bed roughness (nb), and vegetation cover fractions (Vf). Using simulations performed at fine space‐time scales, two methods were developed for computing the upscaled Manning coefficient, termed “Equivalent Roughness Surface (ERS)” and “Equivalent Friction Slope (EFS).” Results obtained with these two methods indicate that both yield highly correlated estimates of nt. The effects of four independent variables (Vf, S, Q, and nb) on nt were further investigated. First, as Vf increases, nt also grows. Second, two distinct modes of the relationship between S and nt for a fixed Vf and Q emerge: a positive dependence at low‐flow rates and a negative dependence at high‐flow rates. For a fixed Vf and S, two distinct modes of the relationship between Q and nt are also identified: a positive dependence at mild domain slopes and a negative dependence at steep slopes. A regression analysis shows that the two conflicting trends can occur depending on whether the variability of flow depth with respect to S (or Q) is greater than the ratio of h and S (or Q). Third, a rougher soil bed (i.e., larger values of nb) implies a higher resistance due to vegetation. Last, the study argues that nt increases as h increases and decreases as V increases. A generic regression relation that includes all four of the above variables and the difference nt − nb (i.e., the additional resistance due to partially submerged vegetation representing the sum of the form and wave resistances) was developed. The range of applicability of this relation is given by the following conditions: Vf ≤ 0.5, 0.1 ≤ S ≤ 1.1, and 0.0001 ≤ Q ≤ 0.01. The difference nt − nb computed from the developed regression relation was compared with estimates reported by five different studies. Furthermore, the simulated wave resistance coefficients were compared with those predicted from an equation in a previous study; the estimates were consistent in the range of experimental conditions for which the latter equation was developed. The relationship is sufficiently general and applicable to other flow conditions with partially submerged roughness elements.

Study of Wave Models of Parabolic Mild Slope Equation and Boussinesq Equation

The Parabolic mild slope equation and Boussinesq equation wave models are studied in this paper. First, the wave models Funwave and REF/DIF, which based on Boussinesq equations and the parabolic mild slope equation, respectively, are introduced. And then, two experiments are used to study these two wave models, one is the non-breaking shoal experiment of University of Delaware and the other is the breaking undertow test experiment, which was finished in Ocean University of China by author. Last, the simulation data of two wave models are compared with the measured data. The results show that both Boussinesq equation and the parabolic mild slope equation wave models can simulated nearshore wave condition precisely, but Boussinesq equation wave models has a disadvantage in catching the variation of wave height caused by wave breaking.

SUBHARMONIC GENERATION OF TRANSVERSE OSCILLATIONS INDUCED BY INCIDENT REGULAR WAVES

It is generally accepted that there are transverse oscillation, which are concentrated and confined to the backwall and decay asymptotically offshore, existed in the harbor of constant slope, however, whether these oscillations can be induced by the normally incident waves is not clear. This numerical investigation aims at providing the subharmonic generations of transverse oscillations within the harbor of a plane slope by waves normally impacting on. For the harbor of perfectly plane slopes, the subharmonic transverse oscillations are small on the mild and moderate slopes but evident on the steep slope. This instability can take place only if the incident wave amplitude exceeds a threshold value, and transverse oscillations can even grow up to a larger value than that of longitudinal oscillations. The magnitudes of transverse oscillations are approximately the same, only their growth rates are affected by the incident wave amplitude.

Brink as a device for measurement discharge for partially filled circular channel

Many devices are fabricated to measure discharge in circular open channel. The flow over a free overall (brink phenomenon) can be used as a device to measure discharge in open channels with known the end depth. This paper presents mathematical and experimental studies to evaluate efficiency of the brink as device of measuring low flow rate in horizontal, mild and adverse circular open channels. The general mathematical model is derived using momentum, discharge and Froude number equations for horizontal, mild and adverse slopes for circular open channels. A proposed mathematical model is calibrated by previous and present experimental data. Equation gives accurate discharge from known end water depth is presented. The results of the laboratory experiments showed that the circular flumes can be successfully used to measure the low flow rate in open channels. A relationship of discharge to the end depth for horizontal smooth circular open channels is proposed.

Numerical Simulation of Solitary Waves Using Smoothed Particle Hydrodynamics Method

Understanding shallow water wave propagation is of major concern in any coastal mitigation effort. Many times, a solitary wave replicates a shallow water wave in its extreme sense which includes a tsunami wave. It is mainly due to known physical characteristics of such waves. Therefore, the study of propagation of solitary waves in the near shore waters is of equal importance in the context of non linear water waves. Owing to the significant growth in computational technologies in the last few decades, a significant number of numerical methods have emerged and applied to simulate nonlinear solitary wave propagation. In this study, one such method, the Smoothed Particle Hydrodynamics (SPH) method has been described to simulate the solitary waves. The split-up of a single solitary wave while it crosses a continental kind of shelf has been simulated by the present model. Then SPH model is coupled with the Boussinesq model to predict the time interval between two successive solitary waves on landfall. It has also been shown to be equally efficient in simulating the wave breaking while a solitary wave propagates over a mild slope.

GIS 기반 하천경사 산정 및 하천망에 따른 표출 방식 개발

Recently, a variety of GIS-based tools enabling to generate topographic parameters for hydrologic and hydraulic researches have been developed. However, most of GIS-based tools are usually insufficient to estimate and visualize river channel slopes especially along the river network, which can be possibly utilized for many hydraulic equations such as Manning's formula. Many existing GIS-based tools have simply averaged cell-based slopes for the other advanced level of hydrologic units as likely as the mean watershed slope, thus that the river channel slope from the simple approach resulted in the inaccurate channel slope particularly for the mountain region where the slope varies significantly along the downstream direction. The paper aims to provide several more advanced GIS-based methodologies to assess the river channel slopes along the given river network. The developed algorithms were integrated with a newly developed tool named RiverSlope, which adapted theoretical formulas of river hydraulics to calculate channel slopes. For the study area, Han stream in the Jeju island was selected, where the channel slopes have a tendency to rapidly change the upstream near the Halla mountain and sustain the mild slope adjacent to watershed outlet heading for the ocean. The paper compared the simple slope method from the Arc Hydro, with other more complicated methods. The results are discussed to decide better approaches based on the given conditions.

An exact analytic solution to the modified mild-slope equation for waves propagating over a trench with various shapes

An exact analytic solution to the modified mild-slope equation (MMSE) in terms of Taylor series for waves propagating over an asymmetrical trench with various shapes is given. Because of the use of the MMSE, on one hand, the present analytic solution can be valid in the whole wave range from long waves to short waves, which is clearly superior to all previous long-wave analytic solutions; on the other hand, the present analytic solution can get rid of the limitation of the ‘mild slope’ assumption and be valid for bottom slope as high as 1:1. It is clarified that the improvement in solution accuracy by using the mass-conserving matching condition against the conventional matching condition mainly depends upon the jump quantities at all common boundaries. In addition, in comparison with previous approximate analytic model based on the approximate mild-slope equation, the present model is more accurate and can converge in the whole trench region without any restriction to trench depth. Based on the present MMSE solution, influence of trench dimensions to reflection effect is analyzed, which shows that total reflection effect increases when trench wall becomes steep and the phenomenon of zero reflection mainly occurs for symmetrical trenches.

Numerical study of wave and longshore current interaction

Wave and longshore current interaction was examined based on the numerical models. In these models, water waves in the presence of longshore currents were modeled by parabolic mild slope equation, and wave breaking induced longshore currents were modeled by shallow water equation. Water wave provided the radiation stress gradients to drive current. Wave and longshore current interactions were considered by cycling the wave and longshore current models to a steady state. The experiments for regular and irregular breaking wave induced longshore currents by Hamilton and Ebersole (2001) and Reniers and Battjes (1997) were simulated. The numerical results indicate that the present models are effective for simulating the interaction of wave and breaking wave induced longshore currents, and the numerically simulated longshore current at wave breaking point considering wave and longshore current interaction show some disagreement with those neglecting the wave-current interaction, and the breaking wave induced longshore current effect on wave transformation is not obvious.

An analytic solution for combined wave diffraction and refraction around a vertical cylinder with idealized scour pit

An analytic solution of the mild slope wave equation for wave propagating over a vertical cylinder with a scour pit has been derived. The scour pit is idealized to be geometrically axisymmetrical, and the water depth inside the pit is assumed to decrease in proportion to an arbitrary power of the radial distance from the center of the cylinder. By separation of variables, solution of the mild slope wave equation is reduced to find the general solution of an ordinary differential equation, of which the coefficients are functions of the wavenumber or the wave celerity, and can eventually be transformed into explicit functions of the independent variable by means of Hunt's approximate formula for the dispersion relation of small amplitude waves. By a tactical mapping of the coefficients into polynomials, the ordinary differential equation is successfully solved in the form of Frobenius series. The solution obtained is a significant extension to the long wave solution previously presented by the same authors. The long wave assumption is now removed and the shape of the scour pit is also generalized to a large extent. Based on the solution obtained, the wave run up around the cylinder is carefully studied, and the effects of the incident wave conditions, the extent, the depth, and the shape of the scour pit on wave transformation are also discussed.

Numerical modelling of the mild slope equation using localised differential quadrature method

Although various numerical techniques have been applied over the last few decades to solve the mild slope equation (MSE), each technique has its own limitations, particularly in terms of computational cost, accuracy, and stability. Localised differential quadrature method (LDQM) is here investigated as an alternative new solution to the MSE. Localised DQM, rather than classical DQM, was used to solve the MSE because of its improved performance, lower computational cost and wider range of applicability. To evaluate the proposed method, four examples were studied, covering a range of complexity which included propagation and transformation of waves due to an elliptic shoal, breakwater gap, and non-rectangular harbour resonance. The results were compared with experimental data, analytical solutions, and other numerical methods. The agreement between numerical and benchmark results was good, and in some cases the performance of LDQM exceeded that of other numerical methods. LDQM can lead to accurate results using fewer grid points and lower computational cost if the number of local nodes is optimised. For a large number of local grid points in LDQM, and also for the case of classical DQM, iterative methods such as conjugate gradient should be employed to solve the system of equations.