Quadratic Pressure Research Articles

Gravity wave scattering by retrofitted circular breakwaters using dual boundary integral formulation

In the present study, the gravity wave scattering by circular edged breakwaters in the presence of retrofits is analysed within the framework of linearized water wave theory. Various types of retrofits (say, porous shield) such as quarter-circular retrofit, semi-rectangular retrofit, semi-circular retrofit, and rectangular retrofit are investigated to protect the newly developed or existing circular breakwaters from the incident wave stroke. A dual boundary integral formulation is used to solve the boundary value problem considering the quadratic pressure drop boundary condition for the waves past the thin porous barriers. A generalized computer code is developed to accommodate the domain changes and is validated with the known results available in the literature. Various results such as the scattering coefficients (wave reflection, transmission and wave energy loss) and force coefficients (vertical and horizontal force on the breakwater and retrofit) are computed as a function of non-dimensional wave number and discussed. The sharp-edged retrofits (rectangular and semi-rectangular) display a considerable reduction of wave transmission as compared with smooth-edged retrofits, and the semi-circular breakwater with rectangular retrofit is suggested for further development of the porous structures. The effect of porous shield alone, breakwater alone, both breakwater and shield on incident wave energy distribution is also reported. The minimal wave transmission and higher wave damping coefficients attended occurred by the proposed breakwaters are identified when the shield porosity varies within 0.1≤μ≤0.2.

Water wave interactions with perforated elastic disks: Quadratic pressure discharge condition

A fully coupled numerical model is developed to study wave interactions with perforated elastic disks. The flow past the perforated surface is represented by a quadratic pressure discharge condition with practical validity. The nonlinear nature of the pressure drop condition results in a dependency of hydrodynamic responses on wave steepness.

Three-dimensional (3D) semi-analytical solution of wave-induced fluid resonance in narrow gaps of caisson-type breakwaters

This study presents a 3D semi-analytical solution for wave-induced fluid resonance in narrow gaps of caisson-type breakwaters. The energy dissipation near the breakwater is modelled by introducing a quadratic pressure loss condition on the gap entrance and exit. By introducing this treatment of nonlinear boundary, the pressure loss due to flow separation induced by the sudden contraction and expansion of a fluid domain can be accounted. The present semi-analytical solution is validated by previous analytical results. The dimensionless energy loss coefficient is calibrated with available experimental data and is found to be insensitive to the incident wave angle θ0 (θ0 ≤ π/12) and wave frequency. The analytical predictions are in good agreement with experimental data. With calibrated coefficient, the effects of wave and geometrical parameters on hydrodynamic quantities are investigated. The results show that the gap resonance leads to large wave energy dissipation and wave force, and the wave resonant frequencies are insensitive with dimensionless gap width l/a ranging from 26.6 to 56.6. The abrupt changes of wave reflection and transmission coefficients at high wave frequency are mainly due to the occurrence of multiple reflection and transmission waves and the wave frequency at the occurrence of multiple waves is always larger than resonant wave frequency.

The water vapor self-continuum absorption at room temperature in the 1.25 µm window

The water vapor self-continuum is newly measured at room temperature in the high energy edge of the 1.25 µm window by using highly stable and sensitive cavity ring down spectroscopy (CRDS). Self-continuum cross-sections, CS, are derived between 8290 and 8620 cm−1 at 29 selected spectral points by using pressure ramps (up to 15 Torr) of pure water vapor. Purely quadratic pressure dependence is obtained for the absorption coefficient at each measurement point. Although the spectral measurement points were chosen to minimize the contribution of resonance line absorption, the latter represents between 30 and 70% of the measured absorption in the studied region. The measurements are found consistent with a previous study of the low frequency edge of the 1.25 µm windows and (Campargue et al. J Geophys Res Atmos 2016;121:13,180 – 13,203. doi:10.1002/2016JD025531). The frequency dependence of the retrieved CS values shows an overall good agreement with the MT_CKD values, although an additional broad absorption feature is observed with a center near 8455 cm−1. It is tentatively interpreted as a possible impact of the uncertainties on the resonance line contribution on the derived CS values or as a possible evidence of a band of the bound dimers, (H2O)2.

Pressure and Temperature Transformations of the Molecular Conformation and Crystal Structure of Ferrocene Fe2+(η5-C5H5)2–

X-ray diffraction, optical spectroscopy, and numerical calculations were carried out on pristine ferrocene Fe2+(η5-C5H5)2– single crystals under various pressure/temperature conditions. Structural data were obtained at room temperature and pressures up to 6.5 GPa as well as at ambient pressure and low temperatures up to 100 K. The iron–carbon bond length (l = 2.043 Å under ambient conditions) shows linear temperature dependence and quadratic pressure dependence with the coefficients ∂l/∂T = −2.84 × 10–5 Å K–1, ∂l/∂P = −1.36 × 10–3 Å GPa–1, and ∂2l/∂P2 = −3.04 × 10–4 Å GPa–2, respectively. Jumping of the Fc molecular configuration between D5h and D5d conformations decreases at high pressure/low temperature, leading to complete ordering in low-temperature (LT, <161.3 K) and high-pressure (HP, >3 GPa) phases. The initially large bandwidth of lattice phonon modes increases with pressure to a maximum at 2 GPa and decreases to the common value after molecular ordering at pressures P > 3 GPa. The electronic structure calculations in the HP phase at high pressure show a linear increase in the fundamental gap Eg with the pressure coefficient of 28 meV/GPa. Optical transmission measurements at high pressure demonstrate an almost 2 times smaller pressure coefficient of about 12 meV/GPa and an abrupt decrease in the gap by ∼14.5 meV at P = 2 GPa.

Analysis of oblique wave interaction with perforated caisson breakwaters with partial wave absorption parts

This article presents a three-dimensional (3D) iterative analytical solution for water wave interactions with perforated caisson breakwaters with partial wave absorption parts based on linear potential flow theory. A periodic boundary condition is adopted for developing the series solutions of the velocity potentials by the separation of variables method. A quadratic pressure drop condition is imposed on the perforated walls to introduce wave energy dissipation. The unknown expansion coefficients in the velocity potentials are determined by matching the boundary conditions between adjacent fluid domains combined with iterative calculations. The hydrodynamic quantities, including the reflection coefficient, wave forces, and free surface elevation, are estimated. The convergences of the series solutions and the iterative calculation process are examined. The analytical solution calculations agree well with the known analytical and experimental results for special cases in the literature. Variations in the hydrodynamic quantities versus the main influencing factors are clarified. Partial wave absorption concept design enables the hydrodynamic performance of the present caisson breakwater with a smaller perforated wall area to be similar to that of a fully perforated caisson breakwater. To achieve a lower reflection coefficient and wave forces, the ratio of the wave chamber length to the caisson length is suggested to be 0.3–0.5, and the front wave chamber width should be less than half of the total wave chamber width.

Numerical simulation with a macroscopic CFD method and experimental analysis of wave interaction with fixed porous cylinder structures

Porous structures have been widely applied in the coastal and ocean engineering due to their wave energy dissipation mechanism. The macroscopic computational fluid dynamics (CFD) approach where the quadratic pressure drop condition of porous surface is introduced to model the wave interaction with porous cylinders. A series of CFD simulations of waves interacting with a single porous cylinder and the combined structure of a porous cylinder with a concentric inner solid column are performed, with corresponding tank tests conducted. The CFD method is compared with experiments, linear potential model, and the quadratic BEM (boundary element method) model. The effects of porosity and porous cylinder radius on wave force and wave heights inside porous cylinder are analyzed to evaluate the performance of porous shell reducing wave loads and wave surface elevation, and the wave force variation with incident wave amplitudes are also investigated. The results demonstrate that the established CFD model is reliable for engineering analysis and thereby being of great significance for reference purpose in the CFD simulations of waves interacting with porous structures.

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.

Numerical analysis on wave load reduction effect of a solid wall with porous plate by macroscopic CFD approach

The porous structures are widely designed in coastal and ocean engineering to reduce the wave load. A macroscopic CFD approach is established to study the wave interaction with a porous plate placed in front of a solid wall. The established numerical wave tank was firstly validated by the analytical results of a vertical wall, and the macroscopic CFD model was validated by the experimental results of a single vertical porous plate. Then the wave load reduction effect of a solid wall with a vertical porous plate in front is investigated, the influences of porosity and relative gap width are compared, and the effects of wave height are also analyzed. The results demonstrate that the porosity and relative gap width are the main effect factors to the wave force reduction effect, and the wave forces on structure increase almost linearly with the increase of relative wave height, while the wave load reduction coefficient and refection coefficient are not linearly. A porosity of 0.2 and relative gap width of 0.2–0.3 are deemed to be optimal geometry parameters at which the wave load reduction effect is optimal. The total horizontal wave force increases nonlinearly with the increase of wave heights, and the quadratic pressure drop condition is essential when studying the wave force on thin porous structures.

Iterative dual boundary element analysis of a wavy porous plate near an inclined seawall

In this work, wave trapping characteristics of a permeable composite wavy barrier located near an inclined seawall are examined under the framework of linear potential flow theory. For the waves past the thin barrier, a semi-empirical discharge equation having a quadratic pressure drop along with the velocity continuum is considered. The boundary value problem is solved using the dual boundary element method (DBEM). The accuracy of the numerical results is ascertained with a numerical model based on an independently developed multi-domain boundary element method (MBEM). Various numerical results such as wave reflection, wave run-up, the vertical force on the barrier and horizontal force on the wall are discussed considering various values of barrier porosity, number of ripples, ripple amplitude, relative spacing between the wall and barrier, relative submergence depth, seawall inclination, and seawall porosity. The numerical results suggest that the performance of the wavy barrier within sight of partial reflection leeward inclined wall is marginal for the design of the development of breakwaters as contrasted to the vertical and seaward inclined wall. The ripple amplitude minimizes the vertical force on the wavy barrier and the reduction is 63% for the non-dimensional ripple amplitude a/B=0.25 as compared to a flat plate where ‘a’ and ‘B’ refers to ripple amplitude and plate width respectively. The parametric study is expected to serve as an incentive for the design and development of composite wavy porous plates to secure the available seawalls.

Surface gravity wave scattering by multiple slatted screens placed near a caisson porous breakwater in the presence of seabed undulations

In the present study, the effectiveness of using multiple slatted screens placed in front of a caisson porous breakwater to dissipate the incident wave energy is analyzed. To model the water flow through the slatted screens, a quadratic (nonlinear) pressure drop condition which involves both the inertial and drag effects is considered. Further, for thick porous structure, the well known Sollitt and Cross (1972) model is used. To handle the nonlinear boundary conditions, an iterative multi-domain boundary element method (BEM) is used. The numerical convergence of the multi-domain BEM based solutions is presented. Energy identities for flow past a single slatted screen, multiple slatted screens and for the present problem are derived. The study shows that the minimum reflection coefficient (<1%) and maximum wave energy dissipation (>98%) can be obtained by using four slatted screens. Further, for moderate values of perforation-effect Keulegan-Carpenter number (KC), the reflection coefficient attains maximum for b1λ≈n2,n=1,2,3,⋯ (b1 is the distance between the rigid caisson and the front slatted screen, λ is the incident wavelength) irrespective of the variations in the number of slatted screens. Further, the transmission coefficient, horizontal and vertical wave forces acting on the rigid caisson attain maximum for b1/λ≈0.55n, and minimum occurs for 2n+14<b1λ<3n+26,n=0,1,2,3,⋯. The Bragg resonance in the reflection coefficient occurs at Bragg value ≈1.6 irrespective of the variations of KC number. Moreover, the reflection coefficient increases around the Bragg value for higher values of KC number without altering the peak value in the reflection coefficient.

Analytical study for oblique wave interaction with a submerged horizontal perforated plate near a partially reflecting vertical wall

A novel iterative analytical solution is developed to study the oblique wave interaction with a horizontally submerged perforated plate near a partially reflecting vertical wall. The boundary condition on the perforated plate is considered as a quadratic pressure drop to incorporate the effect of wave height on the wave energy dissipation. The boundary value problem is analyzed analytically under the assumptions of small amplitude wave theory. To validate the analytical model, an iterative solution based on MBEM (multi-domain boundary element method) is independently developed, and the comparison between the analytical solution and the experimental data are given. The performance characteristics of a submerged perforated plate before a partially reflecting vertical wall is studied by analyzing the hydrodynamic coefficients of the system for the effects of plate porosity, relative plate width, spacing, depth of submergence, and the direction of wave attack. The proposed model is expected to act as an effective breakwater in the attenuation of wave energy for the creation of a tranquillity zone in the marine environment. Moreover, the analytical model developed in the present study can be effective and reliable for practical preliminary design.

Hydrodynamic analysis of a seaside quarter-circular breakwater with an array of porous cages using DBEM

ABSTRACT The gravity wave interaction with an array of porous cages in the presence of a seaside submerged quarter-circular porous breakwater (QCPBW) is analyzed within the framework of linearized water wave theory. A quadratic pressure drop is adopted for the wave past the porous structures to capture the effect of wave height on the energy dissipation. The boundary value problem is solved using the iterative Dual Boundary Element Method (DBEM). The numerical code is validated with known results in the literature. One or upto four units of submerged and bottom fixed porous cages are considered. The effect of porosity on QCPBW and porous cages are investigated pragmatically to study the scattering and wave forces. It is found that introducing a single porous cage with 10% porosity at the rear side of a QCPBW with a porosity of 25% can reduce the wave transmission coefficient by about 50% for relative water depth . Increasing the porosity of the single cage from 10% to 40% has a significant reduction on the horizontal and vertical force coefficients on the cage, whereas the change is insignificant on QCPBW. The horizontal wave force is three to four times higher when compared to the vertical wave force for both QCPBW as well as on the porous cage. For a system comprising four units of cages, both the horizontal and vertical wave forces on cage 1, 2, 3, and 4 reduce progressively (cage 2 is at the lee side of cage 1 and so on). The results of the present study can be used for the optimal design of open sea porous systems.

A BEM model for wave forces on structures with thin porous elements

A boundary element method (BEM) model is presented for wave forces on structures composed of solid and porous surfaces, where the porous surface can be subject to either a linear or quadratic pressure–velocity relation. In the case of the quadratic relation, the solutions to the radiation and diffraction problems cannot be superimposed to obtain a solution for body motions in waves. Instead, a solution method is proposed which solves for the motion response and wave forces on the body simultaneously. Solutions for the radiation and diffraction problems are then obtained as special cases. Hydrodynamic identities and expressions for the mean drift force for combined solid-porous bodies are also derived. It is shown that in the case of a quadratic pressure drop, the hydrodynamic coefficients are no longer symmetric and the Haskind relation must be modified to account for the pressure drop across the porous surface. The BEM solution is verified against an analytical calculations and results for the excitation and mean drift forces are shown to agree well. A case study is presented for a floating truncated cylinder, with a concentric porous outer cylinder. It is shown that the porous outer cylinder significantly increases the damping at low frequencies, where wave radiation damping is low, leading to a lower motion response.

Effect of pressure, concentration and temperature on the oscillatory rheology of guar gum dispersions: Response surface methodology approach

Guar gum dispersions were subjected to high-pressure treatment at ambient temperature, and thereafter, selected samples were in situ thermally treated in a rheometer. The aim of the present work was to investigate the influence of the gum concentration (0.5–1.25%), pressure (300–600 MPa for 15 min), and heating temperature (25–70 °C) on the oscillatory rheology of guar gum dispersions. Guar gum dispersions with and without pressure exhibited liquid-like (G″>G′) behavior in the lower frequency range, which shifted to solid-like (G′>G″) behavior after the gel point. A quadratic polynomial regression model was employed for dynamic moduli and it was observed that the linear (concentration and temperature), quadratic (pressure), and interaction (concentration and temperature) model terms were significant (P < 0.05) for both G′ and G″ with higher R2 values (>0.97) and insignificant lack-of-fit. The temperature effect of the HP-treated guar gum was evident at >55 °C indicating a disruption of the guar gum network. While the time-temperature superposition principle applied to viscoelastic data, it was found that the G″-ω fitted the model adequately.

New analytical solutions of oblique wave scattering by submerged horizontal perforated plates using quadratic pressure drop condition

This study develops new analytical solutions for oblique wave scattering by horizontal perforated plates submerged below the water free surface using a quadratic pressure drop boundary condition. Both kinds of submerged plates are considered: a) A horizontal perforated plate as an offshore breakwater; b) A horizontal perforated plate attached to a vertical wall as a wave absorber. The velocity potential decomposition method combined with an efficient iterative algorithm is adopted to obtain the analytical solutions, and the hydrodynamic parameters of engineering interest for the reflection, transmission and energy-loss coefficients as well as the vertical wave force are considered. A new simple formula for predicting the discharge coefficient in the quadratic pressure drop condition, which is critical to make the analytical solutions to be practically applicable, is developed by fitting the analytical results and an amount of experimental data in literature. The present analytical solutions are in good agreement with previous analytical solution for special case and the numerical results of an independently developed multi-domain boundary element method solution. The reflection and transmission coefficients calculated by the present analytical solutions are also in good agreement with experimental data when adopting a well-predicted discharge coefficient. The present analytical solutions are valuable for fast assessing the hydrodynamic performance of horizontal perforated plates in the initial stage of engineering design. The present analytical model can also be readily extended to the multi-layer horizontal perforated plates and similar structures.

A Semi-Analytical Potential Solution for Wave Resonance in Gap Between Floating Box and Vertical Wall

Based on potential flow theory, a dissipative semi-analytical solution is developed for the wave resonance in the narrow gap between a fixed floating box and a vertical wall by using velocity potential decompositions and matched eigenfunction expansions. The energy dissipation near the box is modelled in the potential flow solution by introducing a quadratic pressure loss condition on the gap entrance. Such a treatment is inspired by the classical local head loss formula for the sudden change of cross section in channel flow, where the energy dissipation is assumed to be proportional to the square of local velocity for high Reynolds number flows. The dimensionless energy loss coefficient is calibrated based on experimental data. And it is found to be insensitive to the incident wave height and wave frequency. With the calibrated energy loss coefficient, the resonant wave height in gap and the reflection coefficient are calculated by the present dissipative semi-analytical solution. The predictions are in good agreement with experimental data. Case studies suggest that the maximum relative energy dissipation occurs near the resonant frequency, which leads to the minimum reflection coefficient. The horizontal wave forces on the box and the vertical wall attain also maximum values near the resonant frequency, while the vertical wave force on the box decreases abruptly there to a small value.

Wave scattering by inverted trapezoidal porous boxes using dual boundary element method

A numerical model is developed based on the dual boundary element method (DBEM) to solve the problem of surface gravity waves past multiple floating inverted trapezoidal porous boxes under the framework of linear potential flow theory. The continuity of velocity and semi-empirical quadratic pressure drop equation is adopted on the porous boundary to incorporate the effect of incident wave amplitude on the energy loss coefficient. Various results by an array of inverted trapezoidal porous boxes are discussed in the form of scattering coefficients (wave reflection, transmission, and energy loss coefficients), and force coefficients (horizontal force, vertical force, and moment). The verification of the numerical model is carried out by an independently developed multi-domain boundary element method (MBEM) code, and also with results available in the literature. The study reveals that, in most cases, the long waves exhibit insignificant hindrance whereas intermediate waves delineate the harmonic oscillations and deep water waves feel maximal hindrance by inverted trapezoidal porous boxes. The wave damping of KL≥0.95 is obtained for structural width B/λ≥0.66 by a pair of identical inverted trapezoidal porous boxes. The angle of porous wings is helpful to alter the scattering coefficients and also helps in the fine-tuning of force coefficients. The angle of partially open wings within 300≤θ≤600 is suggested along with structural porosity μ=0.25 for effective constriction of incident waves. The parametric analysis will be useful in the meticulous design of cost-effective breakwaters.

Gravity wave interaction with multiple submerged artificial reefs

In the present study, gravity wave interaction with a series of submerged artificial permeable reefs is analysed within the framework of linearised water wave theory. For wave past porous walls of the artificial reefs, a quadratic pressure drop is assumed to account for the wave energy dissipation due to the changes in wave height. The physical problem is handled for a solution using a numerical model based on the iterative multi-domain boundary element method and the developed numerical model is validated with known results in the literature. The iterative model is compared with the numerical model based on linear pressure drop boundary condition (i.e., Darcy law). The study reveals that the wave transmission reduces with the increase in the number of reef units. It is demonstrated that the transmission coefficient can be reduced to less than 0.5 when the number of reef units is greater than or equal to three for a relative height greater than 0.7, reef porosities less than 20% and for 0.4&lt;k0h&lt;3.0. Bragg reflection is observed when the porosity is in the range of 0 to 10% and above which the recurrent nature of wave reflection gradually fades away due to dissipation effects. The number of peaks occurring in the shallow and intermediate water depths is equal to the number of reef units wherein the maxima occur at the first frequency. A relative increase in the base width improves the energy losses but the rate of increase in energy loss decreases. The scattering coefficient pattern is oscillatory when the relative spacing between the barriers is varied and their hydrodynamic performance is invariant for higher relative base width.

On the selection of measure-valued solutions for the isentropic Euler system

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measure-valued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact, these measure-valued solutions also do not arise from a sequence of weak solutions of the Euler equations, in contrast to the incompressible case. Such a phenomenon has already been observed by Chiodaroli, Feireisl, Kreml, and Wiedemann using an A-free rigidity argument, but only for non-deterministic initial datum. We develop their rigidity result to the case of non-constant states and combine this with a compression wave solution evolving into infinitely many weak solutions, constructed by Chiodaroli, De Lellis, and Kreml. Hereby, we show that there exist infinitely many generalized measure-valued solutions to the two-dimensional isentropic Euler system with quadratic pressure law, which behave deterministically up to a certain time and which cannot be generated by weak solutions with bounded energy or by vanishing viscosity sequences.