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The flow pattern and the hydrodynamic characteristics of coarse particles in deep-sea hydraulic lifting pipes are simulated using a numerical approach developed by combining the computational fluid dynamics (CFD) method with the discrete element method (DEM) in the Euler-Lagrange framework. This paper examines the effects of feed concentration, two-phase flow initial mixture velocity, and particle gradation on the dynamic characteristics of particles and flow pattern in the pipe by validating the rationality of numerical simulation. The results demonstrate that particles are distributed from the pipe center to the pipe wall, while the lift force causes more particles to be distributed in the pipe center. Moreover, greater inertia makes large particles more concentrated in the center. Particle-induced turbulence modifies the lift force and causes particles to move from the pipe's center to its wall. Due to the increasing trend of lift force, particles gather toward the center of the pipe at an increasing pace as initial velocity increases. The greater the feed concentration, the more particles disperse toward the pipe wall due to the violent momentum exchange caused by the high concentration and the significantly altered lift force caused by the high turbulent kinetic energy resulting from a high concentration. From the particle gradation 1:1:1 to 1:6:1, the pressure drop decreases gradually as the reduction of small particles decreases the number of particles near the wall, and the frictional energy loss between the particles and the pipe wall decreases.

The mechanics of offshore mooring lines are described by a set of nonlinear equations of motion which have typically been solved through a numerical finite element or finite difference method (FEM or FDM), and through the lumped mass method (LMM). The mooring line nonlinearities are associated with the distributed drag forces depending on the relative velocities of the environmental flow and the structure, as well as the axial dynamic strain-displacement relationship given by the geometric compatibility condition of the flexible mooring line. In this study, a semi analytical-numerical novel approach based on the power series method (PSM) is presented and applied to the analysis of offshore mooring lines for renewable energy and oil and gas applications. This PSM enables the construction of analytical solutions for ordinary and partial differential equations (ODEs and PDEs) by using series of polynomials whose coefficients are determined, depending on initial and boundary conditions. We introduce the mooring spatial response as a vector in the Lagrangian coordinate, whose components are infinite bivariate polynomials. For case studies, a two-dimensional mooring line with fixed-fixed ends and subject to nonlinear drag, buoyancy and gravity forces is considered. The introduced boundary and initial conditions enable the analysis of an equilibrium or steady-state of a catenary-like mooring line configuration with variable slenderness and flexibility. Polynomials’ coefficients computation is performed with the aid of a MATLAB package. Numerical results of mooring line configurations and resultant tensions are presented for deep-water applications, and compared with those obtained from a semi-analytical and finite element model. The PSM applied to the mooring line in the present study is efficient and more computationally robust than traditional numerical methods. The PSM can be directly applied to the dynamic analysis of mooring lines.

The present study introduces a novel algorithm based on the homotopy analysis method (HAM) to efficiently solve the equation of motion of simply supported transversely and axially loaded double-beam systems. The original HAM was developed for single partial differential equations (PDEs); the current formulation applies to systems of PDEs. The system of PDEs is derived by modeling two prismatic beams interconnected by a nonlinear inner layer as Euler–Bernoulli beams. We employ the Bubnov–Galerkin technique to turn the PDEs’ system into a system of ordinary differential equations that is further solved with the HAM. The flexibility and straightforwardness of the HAM in computing time-dependent components of the system’s transverse deflection and natural frequencies, in conjunction with the observed fast convergence, offer a robust semi-analytical method for analyzing such systems. Finally, the transverse deflection is built through the modal superposition principle. Thanks to a judicious and high-flexibility selection of initial guesses and convergence control parameters, numerical examples confirm that at most six iterations are needed to achieve convergence, and the results are consistent with the selected benchmark cases.