Constrained Interpolation Profile Research Articles

Effects of nonlinearities on the gap resonances between two free-heaving barges

The hydrodynamic characteristics of the gap resonances between two identical side-by-side barges are investigated in a constrained interpolation profile method-based numerical wave tank. Each barge can heave freely under the excitation of the incident waves. This paper mainly concentrates on the influences of the nonlinearities during the gap resonances on their hydrodynamic performance, via the parameter study of the incident-wave height. Numerical results demonstrated that the nonlinear gap resonances' magnitudes may reach close to or exceed that of the linear gap resonance and naturally contain strong nonlinearities when considering the heave responses of the side-by-side barges. Meanwhile, the effects of the heave responses on the key hydrodynamic parameters cannot be ignored as well. Therefore, the wave run-ups of the barges are of significant importance for the investigation. On the one hand, the wave run-ups directly reflect the coupling effects between the gap resonances and the twin barges' heave responses. On the other hand, the nonlinearities of the wave run-ups perform stronger than the wave elevation at the gap. Moreover, the linear wave run-ups are proposed via the linear formulas to qualitatively and quantitively investigate the effects of the nonlinearities on the inner process of the wave run-ups by comparison. Based on the harmonic analysis, the features of the distributions of the first four order harmonics of the wave elevation at the gap, the wave run-ups, and the wave loads are illustrated.

Effects of multi motion responses and incident-wave height on the gap resonances in a moonpool

The hydrodynamic characteristics of a free-floating moonpool encountering the gap resonances are investigated based on the constrained interpolation profile method in numerical wave tank. This paper mainly concentrates on the influences of the moonpool's motion responses and the incident-wave height on the gap resonances in the free-floating moonpool. Numerical results demonstrated that the heave response significantly changes the frequency and the magnitude of the linear gap resonance, while the roll motion influences more on the vertical wave loads and the wave responses in the fluid field. The heave and roll response of the free-floating moonpool are generally independent. Moreover, the magnitudes of the nonlinear gap resonances have the tendency of catching up and exceeding the linear gap resonance as the incident-wave height increasing for the free-floating moonpool, which is the consequence of the higher-order harmonics driven by the nonlinear processes and the linear secondary resonant region induced by the heave response. Based on the wavelet transform, it could be observed that the amplified harmonic component usually takes more time to be fully developed than other harmonic components during the development of its corresponding nonlinear gap resonance.

An accurate and efficient HOS-meshfree CFD coupling method for simulating strong nonlinear wave–body interactions

The study on strong nonlinear wave–structure interaction (WSI) is crucial in ocean engineering. The Smoothed Particle Hydrodynamics (SPH) method is popular in this field because of its advantages in simulating free surface flow. However, to predict the details of WSI, a high resolution of particles is usually required, which results in a huge computational cost. In this paper, a potential-meshfree CFD weak coupling method based on the High Order Spectral (HOS) and SPH method is developed, which combines the advantages of the former in generating nonlinear waves quickly and accurately and the latter in adapting to strong nonlinear free surface problems. The accuracy of the coupling method was verified through the simulations of regular and irregular waves compared with experiment data. It shows the coupling method can effectively improve computational efficiency and extensively reduce the numerical dissipation. Furthermore, a simulation of the interaction between the focus wave and the floating body was conducted by using the coupling method, the original SPH method and the Constrained Interpolation Profile (CIP) method. Among all numerical methods, the coupling method is the best one for predicting the impact pressure of the floating body and is the only one that can capture the second impacting peak.

Effects of roll motion on linear and nonlinear gap resonances in a moonpool excited by various incident-wave heights

The linear and nonlinear gap resonances in a free-rolling moonpool are investigated using a two-dimensional in-house code based on the constrained interpolation profile method. This study primarily focuses on the effects of roll motion on hydrodynamic characteristics, taking into account both the linear gap resonance and the nonlinear process-driven gap resonances. The analysis of the incident-wave height is conducted to illustrate the relationship between higher-order harmonics and nonlinear gap resonances. Numerical results demonstrate that the resonant frequency is insensitive to the roll motion. However, the roll response plays an important role in influencing the vertical wave loads on the barges and affecting the energy conversion pattern in the fluid field. According to the analysis of the incident-wave height parameter, the magnitudes of resonance at the linear and nonlinear resonant points are approximately in sub-linear and superlinear relationships with the incident-wave height, respectively. Moreover, the vortex shedding modes around the entrance of the free-rolling moonpool gradually transit from being dominated by the pairs of vortices to being dominated by the single vortex as the gap resonance changes from linear to quartic.

Nonlinearities of the gap resonance for a free-heaving moonpool: Higher-order harmonics and natural frequency component

The piston-mode gap resonance of a free-heaving moonpool was investigated in a two-dimensional numerical wave tank based on a Constrained Interpolation Profile (CIP) method. The wave elevation at the narrow gap, the moonpool's heave response, and wave loads on the free-heaving moonpool were all specifically examined in the current study, which was primarily focused on the higher-order harmonics of the gap resonance for the free-heaving moonpool against various gap widths and body drafts. The flow pattern around the entrance of the gap of the free-heaving moonpool was identified at the typical resonant frequencies. The frequency range of the beat pattern for the wave elevation at the gap was initially suggested. Furthermore, the natural frequency component (NFC) was observed from the interesting beat patterns, and the NFC-based approach was proposed as a straightforward way to forecast the resonant frequency of the free-heaving moonpool. Numerical studies showed that the quadratic wave excitation (QWE) driven gap resonances were recorded for both the fixed and the free-heaving moonpools versus regular waves. The resonance frequency of the free-heaving moonpool can be directly, precisely, and rapidly estimated using the suggested NFC-based approach. Finally, four frequency zones were proposed to differentiate the various gap resonances for the free-heaving moonpool based on distinct hydrodynamic characteristics of the wave elevation at gap.

Transient responses of the linear and nonlinear gap resonances in the fixed and free-heaving moonpools

The transient responses in the fixed and free-heaving moonpools under the conditions of linear and nonlinear gap resonance are investigated. A two-dimensional numerical model was established using a constrained interpolation profile method-based in-house code. This paper mainly concentrates on the prediction and quantification of the developing and diminishing processes of the free-surface elevation at gap when encountering both the linear gap resonance and quadratic wave excitation (QWE)-driven gap resonance. The parameter analyses with respect to the distance between the barges and barge draft are considered to derive the variation in the response time and damping time. The continuous wavelet transform (CWT) is introduced to investigate the gap resonances for the first time to obtain the transient characteristics' distribution of the free-surface elevation at gap along time and frequency domains. Numerical results demonstrate that the heave motion has changed the developing patterns of both the linear gap resonance and QWE gap resonance in the free-heaving moonpool rather than those in the fixed moonpool. Nevertheless, the impact caused by heave is little as for the diminishment. By introducing a new hyperbolic-function-type amplification function, the development of the gap resonances for the free-heaving moonpool is well predicted. In addition, the opposite tendency is observed for both the response time and damping time between the fixed and free-heaving moonpools when considering from the linear to QWE gap resonances. According to the transient characteristics obtained from the CWT, it is attributed to the different proportions of the free-surface elevation's second-order component between the fixed and free-heaving moonpools at the QWE resonant point. Moreover, the duration of the damping time is found to be greatly influenced by the phase relation between the free-surface elevation at gap and the heave motion of the moonpool from the linear resonant point to the QWE resonant point.

Simulations of interaction between flow and a thin elastic beam using CIP-based model with adaptive mesh

The simulation of interaction between flow and the elastic structure is a complicated fluid-solid interaction problem, involving the structure's large deformation and fluid interface variation. In this paper, an adaptive CIP (constrained interpolation profile)-based numerical model has been presented to simulate this problem. In this CIP-based model, the constrained interpolation profile method, a high-order finite difference method, is employed as the base flow solver and THINC/SW (Tangent of Hyperbola for Interface Capturing with Slope Weighting) method, a volume of fluid method, is applied to capture the flow interface. IBM (immersed boundary method) method is used to treat the fluid-solid interface. The body deformation is calculated by coupled FDM-FEM (finite difference method and finite element method) method. Adaptive mesh is generated by Afivo (Adaptive Finite Volume quadtree/Octree mesh) framework. The cases of flow past a square and water entry of a symmetric solid wedge are carried out to validate the reliability of the flow solver in this present model. Cantilever beam deformation case is used to validate the accuracy of the solid solver in the present model. Then cases of a thin elastic beam in sloshing tank and dam-break flow are carried out by using this model. After that, the case of regular wave past a membrane breakwater is studied. Our numerical results are compared with previous experimental and numerical studies, and good agreements are obtained. It is demonstrated that this present numerical model has good applicability to simulate the interaction between flow and a thin elastic beam.

Anisotropic electron heating in an electron cyclotron resonance thruster with magnetic nozzle

In a grid-less electron cyclotron resonance plasma thruster with a diverging magnetic nozzle, the magnitude of the ambipolar field accelerating the positive ions depends on the perpendicular energy gained by the electrons. This work investigates the heating of the electrons by electromagnetic waves, taking their bouncing motion into account in a confining well formed by the magnetic mirror force and the electrostatic potential of the thruster. An electromagnetic particle-in-cell code is used to simulate the plasma in a magnetic field tube. The code's Maxwell solver is based on a semi-Lagrangian scheme known as the constrained interpolation profile which enables larger time steps. The results show that anisotropic plasma heating takes place exclusively inside the coaxial chamber, along a Doppler-broadened zone. It is also shown that a trapped population of electrons with a larger perpendicular energy exists in the plume.

Solitary wave interaction with an impervious house in the coastal community using a parallel three-dimensional incompressible CIP-based model

Three-dimensional numerical investigations on solitary wave interaction with an impervious house in the coastal community using a parallel incompressible CIP-based model are carried out. Simple and effective parallel computation will be realized based on algorithm of domain decomposition method and hybrid OpenMPI–OpenMP computational framework. The convection term is discretized by a constrained interpolation profile (CIP) method and an accurate interface capturing scheme, the Tangent of Hyperbola for Interface Capturing with Slope Weighting (THINC/SW), is applied to reconstruct the free surface. The reliability of the present model is validated by a series of benchmarks, including regular wave interaction with a single cylinder, regular wave reflection and diffraction on a vertical structure and interactions between a solitary wave and an impervious structure. The present results are in good agreement with the published results. The present model is applied to investigate the solitary wave reflection and diffraction on the house with different roof types. The variations of wave force, wave elevation and free surface are investigated, which considers the effect of natural disasters such as tsunamis on the safety of the house with different roof types in the coastal community. These results provide engineering references for the design of tsunami-resilient houses.

Numerical Study of Flat Plate Impact on Water Using a Compressible CIP–IBM–Based Model

Due to the entrapment of compressible air, the process of a flat plate impact on water is complicated, which cannot be reproduced using incompressible simulations. To investigate such a slamming process, an accuracy compressible fluid–structure interaction numerical model has been proposed. The solution of this model is based on the constrained interpolation profile (CIP) method to solve the Navier–Stokes equations for the computation of fluid, and an implicit immersed boundary method (IBM) is used to calculate the fluid–structure interaction. Firstly, the present (CIP–IBM–based) model is validated against the problem of the flow past a stationary cylinder. Then it is implemented to simulate the problem of a rigid flat plate impact on water. The predicated impact pressure is compared with reference experiments and other simulations. The CIP–IBM–based model shows a better performance in dealing with sub-atmospheric pressure and reloading. From the numerical view, it is shown that the oscillation of the slamming pressure is significantly affected by the compression and expansion of the entrapped air cushion, and under the influence of the air cushion, the slamming pressure distribution along the bottom is not constant, which also varies greatly with time.

Improvement of the surface condition of laser-drilled holes via a dual-wavelength double-pulse train

Laser drilling is widely used in a variety of applications due to its advantages such as environmental protection, expandability, and non-contact, but still has some drawbacks. One of the most annoying problems is how to improve the surface condition of laser-drilled holes. In this study, we used two nanosecond double-pulse lasers with the wavelengths of 355 and 532 nm to form a combined dual-wavelength double-pulse train for the hole drilling and obtained a series of holes on stainless steel sheets. By using such a dual-wavelength double-pulse train, the area ratio of the front surface to back surface is smaller than those of using both a single-wavelength double-pulse train and a single-pulse laser, and the surface state of drilled holes has not changed significantly. At the same time, the different power components of two pulses have been systematically studied for drilling with a dual-wavelength double-pulse laser. The optimum power ratios of two pulses have been found through a series of experiments. In addition, based on the computation by using an algorithm named as constrained interpolation profile, we theoretically analyzed the changes in density around the processing area for three kinds of lasers during the laser processing. The simulation results show that the dual-wavelength double-pulse approach is the most effective of the hole drilling with a small front-to-back area ratio among three cases. Our results have great potential in the field of laser precision processing.

A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses

A 1D1V hybrid Vlasov-fluid model was developed for this study to elucidate discharge current oscillations of Hall thrusters (HTs). The Vlasov equation for ions velocity distribution function with ionization source term is solved using a constrained interpolation profile conservative semi-Lagrangian method. The fourth-order weighted essentially non-oscillatory (4th WENO) limiter is applied to the first derivative value to minimize numerical oscillation in the discharge oscillation analyses. The fourth-order accuracy is verified through a 1D scalar test case. Nonoscillatory and high-resolution features of the Vlasov model are confirmed by simulating the test cases of the Vlasov–Poisson system and by comparing the results with a particle-in-cell (PIC) method. A 1D1V HT simulation is performed through the hybrid Vlasov model. The ionization oscillation is analyzed. The oscillation amplitude and plasma density are compared with those obtained from a hybrid PIC method. The comparison indicates that the hybrid Vlasov-fluid model yields noiseless results and that the steady-state waveform is calculable in a short time period.

Multiphase flow simulation of water entry of a structure with complex geometry using a three-dimensional parallel compressible model

Three-dimensional water entry of a structure with complex geometry is investigated by a parallel compressible model. The projection method is adopted for the present high-order difference model. The convection term is discretized by a constrained interpolation profile method. The free surface is captured by a tangent of hyperbola for interface capturing with slope weighting scheme. The compressible multiphase model is realized by supplementing the equation of state. It can automatically handle compressible and incompressible areas, which considers the mixing compressibility effect. The benchmarks are the water entry of a sphere, water entry of a horizontal cylinder, and water entry of the single skeg and twin skeg. Special attention of this study is focused on the parallelism algorithm, compressibility effect, and the application of the asymmetric water entry of the skeg. Furthermore, the displacement, velocity, and acceleration of the structure can be accurately predicted. The highly nonlinear free surface can be simulated with fairly good accuracy. Air pocket and secondary impact can be reproduced well. The accurate numerical prediction of the impact load with the compressibility of asymmetric water entry of the structure with complex geometry is vital for the design of the structure and the safety of the passengers. These are the further development of the present model and help us understand the short and destructive slamming process.

Boundary variation diminished conservative semi-Lagrangian method for both compressible and incompressible flows

Motivated by the enlightenment that diminishing the jump at the cell boundary can effectively reduce numerical dissipation near the critical region, and a novel constrained interpolation profile conservative semi-Lagrangian method is proposed based on a newly designed boundary variation diminishing algorithm. First, a constrained interpolation profile conservative semi-Lagrangian scheme with the piecewise tangent of hyperbola for interface capturing scheme is proposed as one candidate to represent jump-like discontinuities. Second, the constrained interpolation profile conservative semi-Lagrangian scheme with a fourth-order weighted essentially non-oscillatory limiter is used as another candidate to keep the high-order and non-oscillatory reconstruction for smooth solutions. The selection criterion of these two candidates is designed by minimizing the total variations of the first derivative at cell boundaries. A unified pressure-based projection formulation with a fractional step procedure is implemented with the proposed scheme to simulate both compressible and incompressible flows. A variety of numerical tests are studied, including linear and nonlinear scalar wave transport problems and compressible and incompressible flow problems. Results show that the proposed method can effectively eliminate numerical oscillation and diffusion, suggesting it has great potential to be applied to various types of engineering problems including both compressible and incompressible flows.

Improvement of the Constrained Interpolation Profile Method for LEMP Simulations

The constrained interpolation profile (CIP) method is one of the finite-difference methods. In principle, the CIP method can suppress numerical dispersion and instability even when a relatively coarse grid is used. However, the M-type and C-type CIP methods, proposed as of today, can deal with neither conductor in the air nor imperfectly conducting ground. In this article, the CIP method is improved and extended to be capable of considering the presence of conductors in air and imperfectly conducting ground. This extended CIP (X-CIP) method does not use the direction-splitting technique, which is used in the conventional CIP methods and makes it difficult to represent conductors in air. The X-CIP method is applied to the simulations of lightning electromagnetic pulses over a perfectly or imperfectly conducting ground and associated surges on conductors in air, and the computed results are compared with those computed theoretically or using the finite-difference time-domain method.

An improved CIP-based numerical model for simulating free-surface flow with adaptive mesh

In this paper, an improved CIP (constrained interpolation profile)-based numerical model, which combines with Afivo (Adaptive Finite Volume quadtree/Octree mesh, an adaptive mesh framework), has been presented to simulate free-surface flow. In the CIP-based model, the constrained interpolation profile (CIP) method, a high-order finite difference method, is employed as the base flow solver and THINC/SW (Tangent of Hyperbola for Interface Capturing with Slope Weighting) method, a volume of fluid method, is applied to capture the flow interface. In this present model, an adaptive mesh technique is introduced to reduce the computational cost and multigrid technique is used to improve the solution efficiency. In the adaptive mesh system, new spatial interpolation schemes are provided, which have better performance in mass conservation compared with schemes provided in Afivo. Lid-driven cavity flow test was carried out to validate the accuracy of the flow solver with adaptive mesh and two/three-dimensional interface benchmark tests were performed to test the reliability of the interface capturing scheme and spatial interpolation schemes. Then two/three-dimensional applications (dam-break flow case, tank sloshing case and dam-break flow over a rectangular obstacle case) were investigated to demonstrate the model accuracy. The numerical results were compared with the previous experimental and numerical results, and good agreements were observed. It is demonstrated that the present numerical model has good applicability in modeling the free-surface flow.

Two-dimensional numerical study on fluid resonance in the narrow gap between two rigid-connected heave boxes in waves

This paper considers the motion response and hydrodynamic characteristics of two rigid-connected boxes freely heaving in waves. Two numerical models were established for the numerical simulation, namely an in-house Constrained Interpolation Profile (CIP) method-based code and a commercial Computational Fluid Dynamics (CFD) tool STAR-CCM+ combined with the overset mesh approach. Based on the developed numerical models, the motion response and hydrodynamic forces of the rigid-connected boxes were simulated under incident waves of different frequency. The effects of the body draft and the distance between the connected boxes were also discussed. Numerical results demonstrated that the developed two numerical models can accurately predict the motion response and hydrodynamic resonance characteristics of the rigid-connected boxes in waves. A systematic analysis proved that the hydrodynamic characteristics of the rigid-connected boxes are quite different under free heave condition and under motion constrained condition.

Numerical Simulation of Water Entry of Wedges in Waves Using A CIP-Based Model

In this study, the water entry of wedges in regular waves is numerically investigated by a two-dimensional in-house numerical code. The numerical model based on the viscous Navier-Stokes (N-S) equations employs a high-order different method—the constrained interpolation profile (CIP) method to discretize the convection term. A Volume of Fluid (VOF)-type method, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) is employed to capture the free surface/interface, and an immersed boundary method is adopted to treat the motion of wedges. The momentum source function derived from the Boussinesq equation is applied as an internal wavemaker to generate regular waves. The accuracy of the numerical model is validated in comparison with experimental results in the literature. The results of water entry in waves are provided in terms of the impact force of wedge, velocity and pressure distributions of fluid. Considerable attention is paid to the effects of wave parameters and the position of wedge impacting the water surface. It is found that the existence of waves significantly influences the velocity and pressure field of fluid and impact force on the wedges.

Axisymmetric Electromagnetic Wave Propagation Computation Using the Constrained Interpolation Profile Scheme With Large Time Steps

Conventional explicit time-domain methods used to solve Maxwell's equations are reliable and robust but are conditionally stable and often require the fulfillment of the condition Courant-Friedrichs-Lewy (CFL) ≤ 1. While this is acceptable for many applications, in some instances, where Maxwell's equations are solved alongside systems with slower propagation velocities, explicit methods prove costly. This is the case for nonrelativistic electromagnetic Particle-In-Cell methods that are required to study plasma thrusters. Several algorithms have been proposed to retain a nearly explicit formulation using large time steps to achieve higher CFL values. Among these is the semi-Lagrangian constrained interpolation profile (CIP) method. While the ability of this method to handle CFL > 1 has been demonstrated for planar 2-D-3-D cases, this has not been done for 2-D cases with cylindrical symmetry. In this article, a procedure is presented to compute the electromagnetic wave propagation in 2-D domains with cylindrical symmetry using the CIP method. The CIP scheme is extended for CFL ≥ 1 cases, and a ghost node method is proposed to deal with the axis singularity and with the wall boundary condition. The results are compared with the fields of a Hertzian dipole and with a coaxial cable, and they show a good agreement.

Implementation of the RCIP scheme and its performance for 1-D age computations in ice-sheet models

Abstract. Ice-sheet age computations are formulated using an Eulerian advection equation, and there are many schemes that can be used to solve them numerically. Typically, these differ in numerical characteristics such as stability, accuracy, and diffusivity. Furthermore, although various methods have been presented for ice-sheet age computations, the constrained interpolation profile method and its variants have not been examined in this context. The present study introduces one of its variants, a rational function-based constrained interpolation profile (RCIP) scheme, to one-dimensional ice age computation and demonstrates its performance levels via comparisons with those obtained from first- and second-order upwind schemes. Our results show that the RCIP scheme preserves the pattern of input surface mass balance histories in terms of the vertical profile of internal annual layer thickness better than the other schemes.