Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy's law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equa- tions with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be sta- bly numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy＇s flow problem, the exact analytical solution for the case of Darcy＇s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclu- sion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equa- tions; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pres- sure gradient increasing for the one-dimensional problem.

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.

Iterative analytical solution for wave reflection by a multichamber partially perforated caisson breakwater

This study examines wave reflection by a multi-chamber partially perforated caisson breakwater based on potential theory.A quadratic pressure drop boundary condition at perforated walls is adopted,which can well consider the effect of wave height on the wave dissipation by perforated walls.The matched eigenfunction expansions with iterative calculations are applied to develop an analytical solution for the present problem.The convergences of both the iterative calculations and the series solution itself are confirmed to be satisfactory.The calculation results of the present analytical solution are in excellent agreement with the numerical results of a multi-domain boundary element solution.Also,the predictions by the present solution are in reasonable agreement with experimental data in literature.Major factors that affect the reflection coefficient of the perforated caisson breakwater are examined by calculation examples.The analysis results show that the multi-chamber perforated caisson breakwater has a better wave energy dissipation function(lower reflection coefficient)than the single-chamber type over a broad range of wave frequency and may perform better if the perforated walls have larger porosities.When the porosities of the perforated walls decrease along the incident wave direction,the perforated caisson breakwater can achieve a lower reflection coefficient.The present analytical solution is simple and reliable,and it can be used as an efficient tool for analyzing the hydrodynamic performance of perforated breakwaters in preliminary engineering design.

Analytical and Experimental Studies on Wave Scattering by a Horizontal Perforated Plate at the Still Water Level

This research investigates water-wave scattering via a horizontal perforated plate fixed at the still water level through analytical studies and physical model tests.The velocity potential decomposition method is combined with an efficient iterative algorithm to develop an analytical solution in which the quadratic pressure drop condition is imposed on the horizontal perforated plate.The analytical results are in good agreement with the results of an independently developed iterative boundary element method(BEM)solution.Experimental tests are carried out in a wave flume to measure the reflection coefficient and transmission coefficient of the horizontal perforated plate,and the analytical results agree reasonably well with the experimental data.The influence of various structural parameters of the horizontal perforated plate on the hydrodynamic parameters of reflection coefficient,transmission coefficient,energy-loss coefficient,and wave force are analyzed on the basis of the analytical solution.Useful results for the practical engineering application of horizontal perforated plates are also presented.