Constrained Interpolation Profile Method Research Articles

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.

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.

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.

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.

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.

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.

Numerical simulation of water entry of two-dimensional structures with complex geometry using a CIP-based model

The present study extends the CIP-based model for the treatment of water entry of two-dimensional structures with complex geometry. The high-order difference model, where the modified ghost-cell immersed boundary method is taken to deal with the moving body with complex geometry more accurately, and the convection term is discretized by the CIP (Constrained Interpolation Profile) method and the free surface is captured by the THINC/SW (Tangent of Hyperbola for Interface Capturing with Slope Weighting) method. The benchmarks are the symmetric water entry of a two-dimensional ship section in single degree of freedom and asymmetric water entry of a two-dimensional bionic dolphin section in three degrees of freedom. The impact peaks of local pressure and local force can be accurately predicted. The phenomena of the secondary impact and the air entrapment can be reproduced with fairly good accuracy. A further observation is that diving two-dimensional bionic dolphin section can use the horizontal velocity to increase the maximum penetration depth while maintaining relative stability in the engineering applications. These numerical investigations can help us to understand the influence of complex geometric parameters on hydrodynamic behavior.

Numerical investigation of solitary waves interaction with an emerged composite structure

The present study describes a numerical investigation on the hydrodynamic characteristics of the impinging and overtopping by solitary waves on an emerged impermeable trapezoidal seawall on a sloping beach. A flow solver of Navier-Stokes equations is employed with the constrained interpolation profile method and a volume-of-fluid type method to capture the strong nonlinear free surface. Three typical solitary waves are simulated to validate the capability of the numerical model. A favorable agreement between experimental and numerical results reveals that this model can reproduce the nonlinear hydrodynamic features of solitary wave impinging and overtopping involving vortex, breaking and impacting. Furthermore, parametric studies are conducted by numerical modeling to examine the effects of wave parameters including water level and wave nonlinearity and leeward slope ratio of the seawall on hydrodynamic stability including sliding and overturning stability and erosion potential. Vorticity and pressure fields are exhibited to show the dynamic properties of flow separation, air entrainment and breaking. It is found that the wave parameters and leeward slope ration influence the hydrodynamic stability of the seawall and erosion of the leeward slope in a different way.

Animating turbulent fluid with a robust and efficient high‐order advection method

AbstractThe accuracy of advection has a great influence on the visual effect of fluid simulation. Constrained interpolation profile (CIP) method has been an important advection scheme because of its third‐order accuracy and the fact that it only needs to be performed over a compact stencil, but extending it to high‐dimensional advection equations is not easy, because it involves complex calculations and large memory overheads, and is usually unstable. In this article, we propose a stable and efficient three‐dimensional (3D) CIP scheme which can maintain high accuracy but requires low computation and memory cost. We first construct an efficient two‐dimensional (2D) CIP scheme based on dimensional splitting and local Taylor expansions, and then propose an effective way to extend it for 3D applications without decreasing the computational accuracy or affecting the stability. The experimental results show the advantages of our method over the state‐of‐the‐art advection schemes.

Simulation of advection\u2013diffusion\u2013dispersion equations based on a composite time discretization scheme

In this work, we develop a high-order composite time discretization scheme based on classical collocation and integral deferred correction methods in a backward semi-Lagrangian framework (BSL) to simulate nonlinear advection–diffusion–dispersion problems. The third-order backward differentiation formula and fourth-order finite difference schemes are used in temporal and spatial discretizations, respectively. Additionally, to evaluate function values at non-grid points in BSL, the constrained interpolation profile method is used. Several numerical experiments demonstrate the efficiency of the proposed techniques in terms of accuracy and computation costs, compare with existing departure traceback schemes.

물 표면 시뮬레이션을 위한 보존적 USCIP법

We propose a physical simulation method based on the shallow water equation(SWE) to represent water surface effectively. In this paper, the water which can be represented has a much larger width compared to the depth does not have a large vertical direction flow. In order to calculate the water flow efficiently, we start with the shallow water equation as the governing equation, which is a simplified version of the Navier-Stokes equation. In order to numerically calculate the advection term of the SWE, we introduce a new conservtive USCIP(CUSCIP) method which improves the Constrained Interpolation Profile (CIP) method to preserve the physical quantity while increasing the numerical accuracy. The proposed method is based on Kim et. al.’s Unsplit Semi-lagrangian CIP[<xref ref-type="bibr" rid="B9">9</xref>], and calculates advection term with additional constraints on term that consider integral values. The experimental results show that the CUSCIP method is robust to the loss of physical quantity due to numerical dissipation, which improves wave detail and persistence.

Numerical Study of Solitary Wave Interaction with a Submerged Semicircular Cylinder

The propagation on submerged structures of solitary wave, as a typical nonlinear wave, has guiding significance for the design and operation of coastal engineering. This paper presents a numerical model based on Navier-Stokes equations to study the interaction of the solitary wave with a submerged semicircular cylinder. A multiphase method is utilized to deal with water and air phase. The model uses the CIP (Constrained Interpolation Profile) method to solve the convection term of the Navier-Stokes equations and the THINC (Tangent of Hyperbola for Interface Capturing) scheme to capture the free surface. Three representative cases different in relative solitary wave height and structure size are simulated and analyzed by this model. By comparing the surface elevations at wave gauges with the experimental data and the documented numerical results, the present model is verified. Then, the wave pressure field around the submerged semicircular cylinder is presented and analyzed. At last, the velocity and vorticity fields are demonstrated to elucidate the characteristics of wave breaking, flow separation, and vortex generation and evolution during the wave-structure interaction. This work presents the fact that this numerical model combining the CIP and THINC methods has the ability to give a comprehensive comprehension of the flow around the structure during the nonlinear interaction of the solitary wave with a submerged structure.

Experimental and numerical study of regular waves past a submerged breakwater

Laboratory experiments are performed to investigate the hydrodynamics around a submerged breakwater due to regular incident waves. With an in-house code, a constrained interpolation profile (CIP)-based model is employed to simulate this process. The model is built on a Cartesian grid system with the Navier-Stokes equations using a CIP method for the flow solver and an immersed boundary method (IBM) is used for the treatment of the solid body boundary. A more accurate interface capturing scheme, the tangent of hyperbola for interface capturing/slope weighting (THINC/SW) scheme, is used to track the free surface. The numerical results are compared with experimental data. Reasonably good agreement is achieved in terms of the wave profiles at six measuring stations, the flow velocities at three different space locations and the pressures of eight points on the surface of the submerged breakwater. Moreover, the water mass transfer over the breakwater is discussed using a two-phase VOF model and the wave spectrum is also presented for analysis. It is indicated that the present model can accurately predict the hydrodynamic characteristics of the wave over a submerged bar. Furthermore, the experimental data in the present work can provide reliable basic data, including the wave transformations, the velocities and the dynamic pressures, for the validation of other CFD models.

Improved prediction of 3-D water entry with a CIP method and parallel computing

This paper presents the further development of a constrained interpolation profile (CIP) based method to solve highly non-linear 3-D slamming problems with a parallel computing algorithm and an improved interface capturing scheme. The water entry problems, governed by the Navier-Stokes (N-S) equations, were treated as multi-phase problems (solid, water and air) and solved by the fractional step method, in which the 3-D CIP method was employed for the advection phase and a pressure-based algorithm was applied to the non-advection phase. The interfaces between different phases were captured by using density functions. The advection of the water phase was solved by applying the THINC/WLIC scheme, i.e., tangent of hyperbola for interface capturing with weighted line interface calculation. A six-degree-of-freedom (6DOF) motion solver was developed to simulate the free fall water entry. A Message Passing Interface (MPI) parallel computing scheme was implemented by decomposing the computational domain. The speed-up performance was investigated. Validation studies were carried out for the water entry of a 3-D wedge with prescribed velocities. The free fall water entry of a modified Wigley hull and an inclined circular cylinder was also studied. The computed slamming forces and motions were compared with experimental results and other published numerical results.