Wave pressures on a vertical wall due to short-crested waves: fifth-order approximation

Non-linear wave pressure induced by short-crested waves on a vertical wall is an important factor to be considered in the design of coastal structures. The existing models to estimate the wave pressure in engineering design are limited to the third-order solution ([Hsu et al., 1979]). In this paper, an analytical solution up to the fifth-order is derived through perturbation approximation. This analytical closed-form solution is used to investigate the contributions of the higher-order components in short-crested waves. It is found that fifth-order components significantly affect the change of pressure, especially in shallow water and larger waves.     

An analytical approximation for the horizontal drift force acting on a deep draught spar in regular waves

An analytical solution is presented for the horizontal drift force acting on a uniform circular cylinder that is undergoing surge and pitch motion in regular waves. The solution is exact to second order, for a perturbation expansion in terms of wave steepness, but is limited in a strict sense to the hypothetical case when the draught is equal to the water depth. The formulation is used to provide an analytical approximation for investigating the influence of key parameters on the horizontal drift force acting on a deep draught spar.     

2D and 3D coexisting modes of thermal convection in fractured hydrothermal systems - Implications for transboundary flow in the Lower Yarmouk Gorge

Numerical investigations of 2D and 3D modes of large-scale convection in faulted aquifers are presented with the aim to infer possible transport mechanisms supporting the formation of thermal springs through fractured aquicludes. The transient finite elements models are based on idealized structural features that can characterize many hydrothermal systems. The sensitivity analysis of the fault permeability showed that faults cross-cutting the main regional flow direction allow groundwater to be driven laterally by convective forces within the fault planes. Therein thermal waters can either discharge along the fault traces or exit the fault through adjacent permeable aquifers. In the latter case, the resulting flow is helicoidally and transient. The location and the spacing between discharge areas can migrate with time, is not strictly constrained to the damage zones and reflects the wavelength of the multicellular regime in the fault zone.An illustrative example based on simplified structural data of the Lower Yarmouk Gorge (LYG) is presented. The numerical calculations indicate that crossing flow paths result from the coexistence of fault convection, developing for example along NE-SW oriented faults within the Gorge, and additional flow fields. The latter are induced either by topography NS gradients, e.g. perpendicular to the major axe of the Gorge, or by local thermal convection in permeable aquifers below the Eocene aquiclude. Sensitivity analysis of fault hydraulic conductivity (K) and the analytical solutions based on viscous-dependent Rayleigh theory show that K values between 2.3e⁻⁷ m/s and 9.3e^{– 7} m/s (i.e. 7 m/yr and 30 m/yr, respectively) favor coexisting transport processes. The uprising thermal plumes spread over several hundred meters forming clusters of springs, in agreement with observation, and which temperature fall within the measured ranges, i.e. 20 °C−60 °C. To some extent the models also reproduced the transient behavior of the spring temperature. Owing to the idealized nature of the presented models, the numerical results and the associated analytical solution can be applied to study the onset of thermal convection and resulting flow patterns of any fractured hydrothermal basin.     

Third order contribution to the wave-making resistance of a ship at finite depth of water

This paper presents a potential based boundary element method for solving a nonlinear free surface flow problem for a ship moving with a uniform speed in finite depth of water. The free surface boundary condition is linearized by the systematic method of perturbation in terms of a small parameter up to third order. The surfaces are discretized into flat quadrilateral elements and the influence coefficients are calculated by Morino's analytical formula. Dawson's upstream finite difference operator is used in order to satisfy the radiation condition. The second order solution gives better result than the first or third order solution. So the present method with the second order solution can be adopted as a powerful tool for the hydrodynamic analysis of the thin ship in finite depth of water.     

A refined analytical analysis of submerged pipelines in seabed laying

The paper analyzes the elastic deflection of submerged pipelines laid with a stinger by taking into account the overall effects of the ovalization of the cross section. The analysis is performed by means of a singular perturbation technique and the analytical solution obtained is shown to offer different advantages over the finite element method. Two examples demonstrate the effectiveness of the procedure by means of a comparison with results from other less refined analytical solutions and from the commercial finite element code .     

On the Fifth-Order Stokes Solution for Steady Water Waves

This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear (1961), confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.     

Nonlinear progressive waves in water of finite depth — an analytic approximation

An analytical solution using homotopy analysis method is developed to describe the nonlinear progressive waves in water of finite depth. The velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike the perturbation method, the present approach is not dependent on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant improvement of the convergence rate and region is achieved by applying Homotopy-Padé Approximants. The calculated wave characteristics of the present solution agree well with previous numerical and experimental results.     

The second order spectrum of two-dimensional random waves

Ｔｈｅｓｅｃｏｎｄｏｒｄｅｒｓｐｅｃｔｒｕｍｏｆｔｗｏ－ｄｉｍｅｎｓｉｏｎａｌｒａｎｄｏｍｗａｖｅｓ￥ＤｉｎｇＰｉｎｇｘｉｎｇ；ＳｕｎＦｕａｎｄＹｕＺｈｏｕｗｅｎ（ＲｅｃｅｉｖｅｄＪａｎｕａｒｙ１５，１９９３；ａｃｃｅｐｔｅｄＭａｙ３１，１９９３）Ａ...     

Approximate Analytical Turning Conditions for Underwater Gliders: Implications for Motion Control and Path Planning

This paper describes analysis of steady motions for underwater gliders, a type of highly efficient underwater vehicle which uses gravity for propulsion. Underwater gliders are winged underwater vehicles which locomote by modulating their buoyancy and their attitude. Several underwater gliders have been developed and have proven their worth as efficient long-distance, long-duration ocean sampling platforms. Underwater gliders are so efficient because they spend much of their flight time in stable, steady motion. Wings-level gliding flight for underwater gliders has been well studied, but analysis of steady turning flight is more subtle. This paper presents an approximate analytical expression for steady turning motion for a realistic underwater glider model. The problem is formulated in terms of regular perturbation theory, with the vehicle turn rate as the perturbation parameter. The resulting solution exhibits a special structure that suggests an efficient approach to motion control as well as a planning strategy for energy efficient paths.      

A 3-D model for ocean waves over a Columb-damping poroelastic seabed

The evaluation of the wave-induced seabed instability in the vicinity of a breakwater is particularly important for coastal and geotechnical engineers involved in the design of coastal structures. In this paper, an analytical solution for three-dimensional short-crested wave-induced seabed instability in a Coulomb-damping porous seabed is derived. The partial wave reflection and self-weight of breakwater are also considered in the new solution. Based on the analytical solution, we examine (1) the wave-induced soil response at different location; (2) the maximum liquefaction and shear failure depth in coarse and fine sand; (3) the effects of reflection coefficients; and (4) the added stresses due to the self-weight of the breakwater.     

A numerical model for wave propagation in curvilinear coordinates

Using the perturbation method, a time dependent parabolic equation is developed based on the elliptic mild slope equation with dissipation term. With the time dependent parabolic equation employed as the governing equation, a numerical model for wave propagation including dissipation term in water of slowly varying topography is presented in curvilinear coordinates. In the model, the self-adaptive grid generation method is employed to generate a boundary-fitted and varying spacing mesh. The numerical tests show that the effects of dissipation term should be taken into account if the distance of wave propagation is large, and that the outgoing boundary conditions can be treated more effectively by introduction of the dissipation term into the numerical model. The numerical model is able to give good results of simulating wave propagation for waters of complicatedly boundaries and effectively predict physical processes of wave propagation. Moreover, the errors of the analytical solution deduced by Kirby et al. (1994) [Kirby, J.T., Dalrymple, R.A., Kabu, H., 1994. Parabolic approximation for water waves in conformal coordinate systems. Coastal Engineering 23, 185–213.] from the small-angle parabolic approximation of the mild-slope equation for the case of waves between diverging breakwaters in a polar coordinate system are corrected.     

Influence of m-cresol purple indicator additions on the pH of seawater samples: correction factors evaluated from a chemical speciation model

The perturbation of the indicator m-cresol purple on the pH in seawater is illustrated in diagrams, representing measurements in 1-cm and 5-cm cells. The diagrams apply to a measured pH interval of 7.4–8.4 using a 2-mM stock solution of m-cresol purple sodium salt dissolved in seawater. The magnitude of the perturbation is described as correction values, i.e., the change in seawater pH caused by the indicator. The diagrams are based on calculations made by using the equilibrium speciation programme, MARINHALT. From these calculations, and least squares fitting methods, pH correction values are described in terms of the pH difference between each seawater sample and the pH of an indicator stock solution. Calculations are performed for a typical high latitude water and a north Pacific deep water. Diagrams are presented for a salinity of 35 and a temperature of 15°C. Responses to salinities between 32 and 36 and temperatures 15–25°C are illustrated as well. A ±0.05 pH difference between a seawater sample and an indicator stock solution gives a correction of less than 0.001 pH unit for a 1-cm cell. For a 5-cm cell, pH differences between the indicator stock solution and a seawater sample as large as ±0.3 cause corrections smaller than ±0.001 pH unit. Calculations demonstrate that the five-fold lower indicator concentration used with 5-cm cells decreases the perturbation effect by approximately a factor of five relative to 1-cm cells.     

有界赤道大洋波包解及其年际年代际变率

Linearized shallow water perturbation equations with approximation in an equatorial β plane are used to obtain the analytical solution of wave packet anomalies in the upper bounded equatorial ocean. The main results are as follows. The wave packet is a superposition of eastward travelling Kelvin waves and westward travelling Rossby waves with the slowest speed, and satisfies the boundary conditions of eastern and western coasts, respectively.The decay coefficient of this solution to the north and south sides of the equator is inversely proportional only to the phase velocity of Kelvin waves in the upper water. The oscillation frequency of the wave packet, which is also the natural frequency of the ocean, is proportional to its mode number and the phase velocity of Kelvin waves and is inversely proportional to the length of the equatorial ocean in the east-west direction. The flow anomalies of the wave packet of Mode 1 most of the time appear as zonal flows with the same direction. They reach the maximum at the center of the equatorial ocean and decay rapidly away from the equator, manifested as equatorially trapped waves. The flow anomalies of the wave packet of Mode 2 appear as the zonal flows with the same direction most of the time in half of the ocean, and are always 0 at the center of the entire ocean which indicates stagnation, while decaying away from the equator with the same speed as that of Mode 1. The spatial structure and oscillation period of the wave packet solution of Mode 1 and Mode 2 are consistent with the changing periods of the surface spatial field and time coefficient of the first and second modes of complex empirical orthogonal function(EOF)analysis of flow anomalies in the actual equatorial ocean. This indicates that the solution does exist in the real ocean, and that El Ni?o-Southern Oscillation(ENSO) and Indian Ocean dipole(IOD) are both related to Mode 2.After considering the Indonesian throughflow, we can obtain the length of bounded equatorial ocean by taking the sum of that of the tropical Indian Ocean and the tropical Pacific Ocean, thus this wave packet can also explain the decadal variability(about 20 a) of the equatorial Pacific and Indian Oceans.     

The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia

In this paper, a beam without contact with water is called the “dry” beam and the one in contact with water is called the “wet” beam. For a partially (or completely) immersed uniform beam carrying an eccentric tip mass possessing rotary inertia, the conventional analytical (closed-form) solution is achieved by considering the inertial forces and moments of the tip mass and rotary inertia as the boundary conditions at the tip end of the beam. However, it has been found that the approximate solution for the last problem may be achieved by two techniques: Method 1 and Method 2. In Method 1, the basic concept is the same as the conventional analytical method; but in Method 2, the tip end of the beam is considered as a free end, while the inertial forces and moments induced by the tip mass and rotary inertia are considered as the external loads applied at the tip end of the beam. The main differences between the formulation of Method 1 and that of Method 2 are: In Method 1, the “normal” shapes of the “dry” beam are functions of the frequency-dependent boundary conditions but the external loads at the tip end are equal to zero; On the contrary, in Method 2, the “normal” mode shapes of the “dry” beam are determined based on the zero boundary conditions at the tip end of the beam but the external loads at the tip end due to the inertial effects of the tip mass and rotary inertia must be taken into consideration for the free vibration analysis of the “wet” beam. Numerical results reveal that the approximate solution obtained from Method 2 are very close to that from Method 1 if the tip mass moment of inertia is negligible. Besides, the two approximate solutions are also very close to the associated analytical (closed-form) solution or the finite element solution. In general, it is hoped that there exist several methods for tackling the same problem so that one may have more choices to incorporate with the specified cases. It is believed that the two approximate methods presented in this paper will be significant from this point of view.     

A spectral barotropic model of the wind-driven world ocean

A global spectral barotropic ocean model is introduced to describe the depth-averaged flow. The equations are based on vorticity and divergence (instead of horizontal momentum); continents exert a nearly infinite drag on the fluid. The coding follows that of spectral atmospheric general circulation models using triangular truncation and implicit time integration to provide a first step for seamless coupling to spectral atmospheric global circulation models and an efficient method for filtering of ocean wave dynamics. Five experiments demonstrate the model performance: (i) Bounded by an idealized basin geometry and driven by a zonally uniform wind stress, the ocean circulation shows close similarity with Munk’s analytical solution. (ii) With a real land–sea mask the model is capable of reproducing the spin-up, location and magnitudes of depth-averaged barotropic ocean currents. (iii) The ocean wave-dynamics of equatorial waves, excited by a height perturbation at the equator, shows wave dispersion and reflection at eastern and western coastal boundaries. (iv) The model reproduces propagation times of observed surface gravity waves in the Pacific with real bathymetry. (v) Advection of tracers can be simulated reasonably by the spectral method or a semi-Langrangian transport scheme. This spectral barotropic model may serve as a first step towards an intermediate complexity spectral atmosphere–ocean model for studying atmosphere–ocean interactions in idealized setups and long term climate variability beyond millennia.     

潮波方程数值格式与摩阻系数选取

基于一维阻尼潮波传播方程解析解,从求解数值格式及Heuristic稳定性分析方面,讨论了数值解的精度、计算耗时和摩阻系数选取等问题。研究结果表明:1)Courant数小于1时,潮波方程显格式解的精度略高于隐格式解,计算耗时少于隐格式解;2)为减少计算耗时,潮波方程的隐格式解允许较大的时间步长,但解的精度有所降低,须通过减小底床摩阻系数以保证计算精度;3)隐格式解摩阻系数的选取与Courant数有关,Courant数越大,摩阻系数的选取值比实际值越小,通过理论分析结合数值试验得到了相应的关系式。这些研究结论对实际海域的潮波传播的数值模拟具有重要的应用价值。     

多消浪室局部开孔沉箱防波堤反射特性的迭代解析研究

基于势流理论,对多消浪室局部开孔沉箱防波堤的反射特性进行解析研究。研究中采用开孔墙处的二次压力损失边界条件,可以直接考虑波高对于开孔墙处能量损失的影响。利用匹配特征函数展开法和迭代方法得到当前问题的解析解。收敛性验证表明,迭代计算和级数解均具有良好的收敛性。该解析解的计算结果与分区边界元的数值计算结果一致,并且与已有的试验结果符合良好。通过算例分析,研究开孔沉箱防波堤反射系数的主要影响因素。结果表明：与单消浪室开孔沉箱防波堤相比,多消浪室开孔沉箱防波堤可以在更宽的波浪频率范围内保持低反射;增大开孔墙的开孔率,有利于降低多消浪室开孔沉箱防波堤的反射系数;当开孔墙的开孔率沿着入射波方向依次递减时,多消浪室开孔沉箱防波堤的反射系数较小。本文所建立的解析模型简单可靠,可用于工程初步设计中分析开孔沉箱防波堤的水动力性能。     

A spectral barotropic model of the wind-driven world ocean

A global spectral barotropic ocean model is introduced to describe the depth-averaged flow. The equations are based on vorticity and divergence (instead of horizontal momentum); continents exert a nearly infinite drag on the fluid. The coding follows that of spectral atmospheric general circulation models using triangular truncation and implicit time integration to provide a first step for seamless coupling to spectral atmospheric global circulation models and an efficient method for filtering of ocean wave dynamics. Five experiments demonstrate the model performance: (i) Bounded by an idealized basin geometry and driven by a zonally uniform wind stress, the ocean circulation shows close similarity with Munk’s analytical solution. (ii) With a real land–sea mask the model is capable of reproducing the spin-up, location and magnitudes of depth-averaged barotropic ocean currents. (iii) The ocean wave-dynamics of equatorial waves, excited by a height perturbation at the equator, shows wave dispersion and reflection at eastern and western coastal boundaries. (iv) The model reproduces propagation times of observed surface gravity waves in the Pacific with real bathymetry. (v) Advection of tracers can be simulated reasonably by the spectral method or a semi-Langrangian transport scheme. This spectral barotropic model may serve as a first step towards an intermediate complexity spectral atmosphere–ocean model for studying atmosphere–ocean interactions in idealized setups and long term climate variability beyond millennia.     

Discussion of “Analytic solution of long wave propagation over a submerged hump” by Niu and Yu (2011)

Recently, Niu and Yu (2011) presented an analytical solution of the long wave refraction by a submerged circular hump. The geometry of the hump was assumed to be axi-symmetric and the water depth over the hump region was described by a positive constant plus a power function of the radial distance with an arbitrary value of the power exponent, i.e., h = h₁ + βr^s, where h₁ is the water depth at the crest of the hump. Their general hump is an extension of the paraboloidal hump (i.e., s = 2) studied by Zhang and Zhu (1994) and Zhu and Harun (2009). Because of this extension in the topography of the hump, the problem to seek a general analytical solution to the long-wave equation becomes much more complicated and the solution technique need to be more skillful, especially for the case with the exponent s being a rational, see Eq. (17) in Niu and Yu (2011).     

Stability based approach to design cold-water pipe (CWP) for ocean thermal energy conversion (OTEC)

Cold-water pipe (CWP) is a novel, most-challenging component of Ocean Thermal Energy Conversion (OTEC) floating structure which is installed to transport the deep seawater to the board. For commercial scale, the transported seawater flow rate will be in the order of 10² m³/s. This large amount of internal flow may trigger instability which leads to the failure of CWP. Considering this issue, the present paper aims to design commercial-scale OTEC CWP focusing on the effects of internal flow to the stability of the pipe. The design analysis is deliberated to select the pipe material, top joint configuration (fixed, flexible, pinned) and bottom supporting system (with and without clump weight). Initially, the analytical solution is built by taking into account the components of the pipe dynamics. Separately, a fully coupled fluid-structure interaction analysis between the pipe and the ambient fluid is carried out using ANSYS interface. Using scale models, the results obtained from the analytical solution are compared with the ones from numerical analysis to examine the feasibility of the analytical solution. After being verified, the analytical solution is used to observe the dynamic behavior of the CWP for 100 MW-net OTEC power plant in the full-scale model. The results yield conclusions that pinned connection at the top joint is preferable to decrease the applied stress, clump weight installation is necessary to reduce the motion displacement and Fiber Reinforced Plastic (FRP) is the most suitable material among the examined materials.