In this article, the overset grid assembly method is improved to efficiently solve several critical problems that occur when applying overset grids to the complicated geometries and moving body. First, instead of usin...In this article, the overset grid assembly method is improved to efficiently solve several critical problems that occur when applying overset grids to the complicated geometries and moving body. First, instead of using the two-step searching (i.e. cut and paste), a single-step searching method based on the grid cell size is proposed to modify holes and optimize the grid overlapping automatically. Second, discrepancies between the wall surface representations, where the grids overlap, are handled by introducing the wall's normal directions to the hole-map and projecting the interpolated points on the recipient mesh into the donor mesh. Finally, the dynamic overset method is modified to address the complex moving body problem. At every time step, the initial hole surface of the previous time step is dynamically adjusted to accomplish hole cutting and avoid the time consuming hole-map procedure. Numerical experiments show that the enhanced overset grid assembly method obtains satisfactory results.展开更多
Overset or Chimera grid approach is one of methods to cope with complex geometries. A spline based overset grid assembling system has been developed. The system is based on structured grid approach and covers comprehe...Overset or Chimera grid approach is one of methods to cope with complex geometries. A spline based overset grid assembling system has been developed. The system is based on structured grid approach and covers comprehensive features for overset assembling, i.e., grid generation, grid modification, and computing Domain Connectivity Information （DCI） for overset interpolation. Ferguson spline curve is used to compute curves through grid points and cell centers for trimming grids and computing DCI robustly and accurately. Flow simulation around Kriso Container Ship （KCS） with jointed grids shows good continuity of flow field between the grids. The overset grid assembling is enhanced to unsteady problem as dynamic overset approach coupled with a solver which also has been developed in National Maritime Research Institute, Japan. Computed results for pitchup spheroid are compared with measured data and show good agreement in unsteady force acting on the spheroid. It is confirmed that the system has capability to simulate flow field around jointed grids and unsteady flow with dynamic overset assembling approach practically.展开更多
A numerical method based on solutions of Euler/Navier-Stokes （N-S） equations is developed for calculating the flow field over a rotor in hover. Jameson central scheme, van Leer flux-vector splitting scheme, advectio...A numerical method based on solutions of Euler/Navier-Stokes （N-S） equations is developed for calculating the flow field over a rotor in hover. Jameson central scheme, van Leer flux-vector splitting scheme, advection upwind splitting method （AUSM） scheme, upwind AUSM/van Leer scheme, AUSM＋ scheme and AUSMDV scheme are implemented for spatial discretization, and van Albada limiter is also applied. For temporal discretization, both explicit Runge-Kutta method and implicit lower-upper symmetric Gauss-Seidel （LU-SGS） method are attempted. Simultaneously, overset grid technique is adopted. In detail, hole-map method is utilized to identify intergrid boundary points （IGBPs）. Furthermore, aimed at identification issue of donor elements, inverse-map method is implemented. Eventually, blade surface pressure distributions derived from numerical simulation are validated compared with experimental data, showing that all the schemes mentioned above have the capability to predict the rotor flow field accurately. At the same time, vorticity contours are illustrated for analysis, and other characteristics are also analyzed.展开更多
In the present study,an efficient overset grid method by means of parallel implicit hole-cutting is proposed for the sake of simulating unsteady flows in aerospace engineering involving multiple bodies in relative mov...In the present study,an efficient overset grid method by means of parallel implicit hole-cutting is proposed for the sake of simulating unsteady flows in aerospace engineering involving multiple bodies in relative movement.In view of the degraded computational efficiency and robustness for conventional overset grid assembly,several innovative techniques are developed within the overset grid assembly process,viz.,a bookkeeping alternative digital tree method to speed up the donor-cell searching,a fast parallel advancing front algorithm to accelerate the wall-distance calculation and a message-passing strategy with efficient information communication and lower storage expenditure within distributed computational architecture.The contribution of the developed techniques is evidenced by comparison with the existing alternative ways in terms of computing efficiency.Subsequently,the overset grid method is embedded into an inhouse programed URANS solver to examine its capability in predicting the flow field of complex applications such as helicopter,store separation and component deploying.Results show that the developed overset grid methodology is,in practice,able to resolve the aerodynamic characteristics of complex aerospace engineering with a high-fidelity flow topology and accuracy.展开更多
A cell-centred overset unstructured grids approach is developed.In this approach,the intergrid boundary is initially established based on the wall distance from the cell centre,and is then optimized.To accelerate the ...A cell-centred overset unstructured grids approach is developed.In this approach,the intergrid boundary is initially established based on the wall distance from the cell centre,and is then optimized.To accelerate the intergrid-boundary definition much more,a neighbor-toneighbor donor search algorithm based on advancing-front method is modified with the help of minimum cuboid boxes.To simplify the communications between different grid cell types and to obtain second-order spatial accuracy,a new interpolation method is constructed based on linear reconstruction,which employs only one layer of fringe cells along the intergrid boundary.For unsteady flows with relative motion,the intergrid boundary can be redefined fast and automatically.Several numerical results show that the present dynamic overset unstructured grids approach is accurate and reliable.展开更多
An overset grid methodology is developed for the fully coupled analysis of fluid-structure interaction (FSI) problems. The overset grid approach alleviates some of the computational geometry difficulties traditionally...An overset grid methodology is developed for the fully coupled analysis of fluid-structure interaction (FSI) problems. The overset grid approach alleviates some of the computational geometry difficulties traditionally associated with Arbitrary-Lagrangian-Eulerian (ALE) based, moving mesh methods for FSI. Our partitioned solution algorithm uses separate solvers for the fluid (finite volume method) and the structure (finite element method), with mesh motion computed only on a subset of component grids of our overset grid assembly. Our results indicate a significant reduction in computational cost for the mesh motion, and element quality is improved. Numerical studies of the benchmark test demonstrate the benefits of our overset mesh method over traditional approaches.展开更多
With an increase in model resolution,compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor co...With an increase in model resolution,compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor computers.To provide high-quality numerical ocean simulation on overset grids,we tried a novel formulation of the fourth-order multi-moment constrained finite volume scheme to simulate continuous and discontinuous problems in the Cartesian coordinate.Utilizing some degrees of freedom over each cell and derivatives at the cell center,we obtained a two-dimensional(2D)cubic polynomial from which point values on the extended overlap can achieve fourth-order accuracy.However,this interpolation causes a lack of conservation because the flux between the regions are no longer equal;thus,a flux correction is implemented to ensure conservation.A couple of numerical experiments are presented to evaluate the numerical scheme,which confirms its approximately fourth-order accuracy in conservative transportation on overset grid.The test cases reveal that the scheme is effective to suppress numerical oscillation in discontinuous problems,which may be powerful for salinity advection computing with a sharp gradient.展开更多
We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use t...We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method.Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability.By numerical experiments we demonstrate the stability,accuracy,efficiency,and the applicability of the methods to forward and inverse problems.展开更多
A method of unstructured dynamic overset grids is developed for the numerical simulation of helicopter unsteady rotorairframe aerodynamic interaction.For the effective treatment of the relative motion between the roto...A method of unstructured dynamic overset grids is developed for the numerical simulation of helicopter unsteady rotorairframe aerodynamic interaction.For the effective treatment of the relative motion between the rotor and the airframe,the domain of flowfield is divided into two overset subzones,namely,a rotational subzone containing the blades and a stationary subzone containing the airframe.The overset part of two subzones is used to convect the flow variables of the two zones.The Taylor series expansion is used to obtain a second-order spatial accuracy,and dual-time stepping is adopted to improve the solution accuracy.Mesh deformation from the blade motion in forward flight is treated by using a spring analogy.Validation is made by numerically simulating the flows around a wind tunnel configuration and comparing the predicted time-averaged and instantaneous inflow and airframe surface pressure distributions with the experimental data.It shows that the present method is efficient and robust for the prediction of complicated unsteady rotor-airframe aerodynamic interaction phenomena.展开更多
This paper proposes an automatic structure overset grid method, which utilizes the hole-surface optimization with one-step searching, wall-surface grid oversetting, and dynamic overset grid approaches to achieve the h...This paper proposes an automatic structure overset grid method, which utilizes the hole-surface optimization with one-step searching, wall-surface grid oversetting, and dynamic overset grid approaches to achieve the high adaptability of overset grids for complex multi-body aircrafts. Specifically, based on the automatic structure overset grids, the method first solves the coupling of Navier-Stokes(N-S) unsteady flow equation and 6DOF motion equation, and establishes the multi-body collision model. Then, the numerical simulation of unsteady flow for complex aircrafts' multi-body separation, the simulation of multi-body separating trajectory and the separation safety analysis are accomplished. Thus, the method can properly handle practical engineering problems including the wing/drop tank separation, aircraft/mount separation, and cluster bomb projection. Experiments show that our numerical results match well with experimental results, which demonstrates the effectiveness of our methods in solving the multi-body separation problem for aircrafts with complex shapes.展开更多
In the present study, the performance of the NTNU Blind Test 1 wind turbine is analyzed in the computational fluid dynamics (CFD) simulations by using the CFD code FANS with structured overset grids. First, the numeri...In the present study, the performance of the NTNU Blind Test 1 wind turbine is analyzed in the computational fluid dynamics (CFD) simulations by using the CFD code FANS with structured overset grids. First, the numerical methods including the governing equations, the turbulence closure model, and the flow solver are introduced. In addition, the NTNU BT1 wind tunnel experiment is described. Then, structured overset grid blocks are generated in the computational domain with fully resolved wind turbine geometry, including the blades, hub, nacelle, and tower. Afterward, unsteady Reynolds averaged Navier-Stokes (RANS) simulations with the two-layer k - ε turbulence model are performed with an inlet velocity of 10 m/s and a tip-speed ratio (TSR) of 6. The overset-grid capability of FANS is leveraged to handle the rotation of the rotor. Finally, simulations are performed for a range of TSRs and a comparison is made among the present CFD results, other numerical results obtained from representative methods, and the experimental data. It is observed that the CFD-predicted thrust coefficients match the experimental measurement at low TSRs while under-predicting the values at high TSRs, and potential reasons for this deviation are discussed.展开更多
Ice-going ships play a crucial role in polar transportation and resource extraction.Different from the existing modeling approach which assumes that ships remain stationary,dynamic overset grid technology and DFBI(Dyn...Ice-going ships play a crucial role in polar transportation and resource extraction.Different from the existing modeling approach which assumes that ships remain stationary,dynamic overset grid technology and DFBI(Dynamic Fluid-Body Interaction)method are employed in this paper to enable the free-running motion of the ship in modeling.A numerical model capable of simulating a ship navigating through pack ice area is proposed,which uses Computational Fluid Dynamics(CFD)method to solve the flow field and applies the Discrete Element Method(DEM)to simulate ship-ice and ice-ice interactions.Besides,the proposed high-precision method for generating pack ice area can be used in conjunction with the proposed numerical model.By comparing the numerical results with the available model test data and experimental observations,the effectiveness of the numerical model is validated,demonstrating its strong capability of predicting resistance and simulating ship navigation in pack ice,as well as its significant potential and applicability for further studies.展开更多
The hydrodynamic performance of high-speed planing hulls has gained considerable interest,with recent advancements in computational fluid dynamics and hull design techniques enhancing the understanding of planing hull...The hydrodynamic performance of high-speed planing hulls has gained considerable interest,with recent advancements in computational fluid dynamics and hull design techniques enhancing the understanding of planing hull hydrodynamics.In this study,we conducted a numerical investigation using the Reynolds-averaged Navier-Stokes approach with overset grids to capture large motions at high speeds.This study aims to improve the hydrodynamic performances of planing hulls,specifically focusing on total resistance,trim,and sinkage.The initial Fridsma hull with a deadrise angle of 20°has been used for validation,demonstrating good agreement with measurements at different Froude numbers.Subsequently,new configurations based on the Fridsma hull have been designed by varying the deadrise angle,number of chines,and transverse steps.Our findings reveal a correlation between the deadrise angle,the number of chines,and the Froude number.As the deadrise angle increases,total resistance also increases.Additionally,a single chine yields superior results at higher Froude numbers,while multiple chines offer advantages at lower values.The introduction of transverse steps consistently increases total resistance,highlighting their role in improving planing hull performance.This research not only offers valuable insights into planing hull design but also leverages state-of-the-art numerical methods to advance the understanding of hydrodynamic behaviors at high ship speeds.展开更多
基金National Natural Science Foundation of China (90716010)
文摘In this article, the overset grid assembly method is improved to efficiently solve several critical problems that occur when applying overset grids to the complicated geometries and moving body. First, instead of using the two-step searching (i.e. cut and paste), a single-step searching method based on the grid cell size is proposed to modify holes and optimize the grid overlapping automatically. Second, discrepancies between the wall surface representations, where the grids overlap, are handled by introducing the wall's normal directions to the hole-map and projecting the interpolated points on the recipient mesh into the donor mesh. Finally, the dynamic overset method is modified to address the complex moving body problem. At every time step, the initial hole surface of the previous time step is dynamically adjusted to accomplish hole cutting and avoid the time consuming hole-map procedure. Numerical experiments show that the enhanced overset grid assembly method obtains satisfactory results.
文摘Overset or Chimera grid approach is one of methods to cope with complex geometries. A spline based overset grid assembling system has been developed. The system is based on structured grid approach and covers comprehensive features for overset assembling, i.e., grid generation, grid modification, and computing Domain Connectivity Information （DCI） for overset interpolation. Ferguson spline curve is used to compute curves through grid points and cell centers for trimming grids and computing DCI robustly and accurately. Flow simulation around Kriso Container Ship （KCS） with jointed grids shows good continuity of flow field between the grids. The overset grid assembling is enhanced to unsteady problem as dynamic overset approach coupled with a solver which also has been developed in National Maritime Research Institute, Japan. Computed results for pitchup spheroid are compared with measured data and show good agreement in unsteady force acting on the spheroid. It is confirmed that the system has capability to simulate flow field around jointed grids and unsteady flow with dynamic overset assembling approach practically.
基金Innovation Foundation of BUAA for PhD Graduates (300522)
文摘A numerical method based on solutions of Euler/Navier-Stokes （N-S） equations is developed for calculating the flow field over a rotor in hover. Jameson central scheme, van Leer flux-vector splitting scheme, advection upwind splitting method （AUSM） scheme, upwind AUSM/van Leer scheme, AUSM＋ scheme and AUSMDV scheme are implemented for spatial discretization, and van Albada limiter is also applied. For temporal discretization, both explicit Runge-Kutta method and implicit lower-upper symmetric Gauss-Seidel （LU-SGS） method are attempted. Simultaneously, overset grid technique is adopted. In detail, hole-map method is utilized to identify intergrid boundary points （IGBPs）. Furthermore, aimed at identification issue of donor elements, inverse-map method is implemented. Eventually, blade surface pressure distributions derived from numerical simulation are validated compared with experimental data, showing that all the schemes mentioned above have the capability to predict the rotor flow field accurately. At the same time, vorticity contours are illustrated for analysis, and other characteristics are also analyzed.
基金supported by the National Natural Science Foundation of China(Nos.11672133,12002161)Open Foundations of EDL Laboratory,China(No.EDL19092111)+2 种基金supports from National Science Foundation of Shaanxi Province,China(No.2021JQ-078)Fundamental Research Fund of Zhuhai,China(No.ZH22017003210011PWC)Aeronautical Science Foundation of China(No.F2021110)are acknowledged as well。
文摘In the present study,an efficient overset grid method by means of parallel implicit hole-cutting is proposed for the sake of simulating unsteady flows in aerospace engineering involving multiple bodies in relative movement.In view of the degraded computational efficiency and robustness for conventional overset grid assembly,several innovative techniques are developed within the overset grid assembly process,viz.,a bookkeeping alternative digital tree method to speed up the donor-cell searching,a fast parallel advancing front algorithm to accelerate the wall-distance calculation and a message-passing strategy with efficient information communication and lower storage expenditure within distributed computational architecture.The contribution of the developed techniques is evidenced by comparison with the existing alternative ways in terms of computing efficiency.Subsequently,the overset grid method is embedded into an inhouse programed URANS solver to examine its capability in predicting the flow field of complex applications such as helicopter,store separation and component deploying.Results show that the developed overset grid methodology is,in practice,able to resolve the aerodynamic characteristics of complex aerospace engineering with a high-fidelity flow topology and accuracy.
基金supported by the National Basic Research Program of China (2009CB724104)
文摘A cell-centred overset unstructured grids approach is developed.In this approach,the intergrid boundary is initially established based on the wall distance from the cell centre,and is then optimized.To accelerate the intergrid-boundary definition much more,a neighbor-toneighbor donor search algorithm based on advancing-front method is modified with the help of minimum cuboid boxes.To simplify the communications between different grid cell types and to obtain second-order spatial accuracy,a new interpolation method is constructed based on linear reconstruction,which employs only one layer of fringe cells along the intergrid boundary.For unsteady flows with relative motion,the intergrid boundary can be redefined fast and automatically.Several numerical results show that the present dynamic overset unstructured grids approach is accurate and reliable.
文摘An overset grid methodology is developed for the fully coupled analysis of fluid-structure interaction (FSI) problems. The overset grid approach alleviates some of the computational geometry difficulties traditionally associated with Arbitrary-Lagrangian-Eulerian (ALE) based, moving mesh methods for FSI. Our partitioned solution algorithm uses separate solvers for the fluid (finite volume method) and the structure (finite element method), with mesh motion computed only on a subset of component grids of our overset grid assembly. Our results indicate a significant reduction in computational cost for the mesh motion, and element quality is improved. Numerical studies of the benchmark test demonstrate the benefits of our overset mesh method over traditional approaches.
基金Dr.X.L.Li at the China Meteorological Administration.This study was supported by grants from the National Natural Science Foundation of China(Nos.41575103 and 91637210).
文摘With an increase in model resolution,compact high-order numerical advection scheme can improve its effectiveness and competitiveness in oceanic modeling due to its high accuracy and scalability on massive-processor computers.To provide high-quality numerical ocean simulation on overset grids,we tried a novel formulation of the fourth-order multi-moment constrained finite volume scheme to simulate continuous and discontinuous problems in the Cartesian coordinate.Utilizing some degrees of freedom over each cell and derivatives at the cell center,we obtained a two-dimensional(2D)cubic polynomial from which point values on the extended overlap can achieve fourth-order accuracy.However,this interpolation causes a lack of conservation because the flux between the regions are no longer equal;thus,a flux correction is implemented to ensure conservation.A couple of numerical experiments are presented to evaluate the numerical scheme,which confirms its approximately fourth-order accuracy in conservative transportation on overset grid.The test cases reveal that the scheme is effective to suppress numerical oscillation in discontinuous problems,which may be powerful for salinity advection computing with a sharp gradient.
基金This work was supported in part by the National Science Foundation under Grant NSF-1913076.Any opinions,findings,and conclusions or recommendations expressed in this material are those of the author(s)and do not necessarily reflect the views of the National Science Foundation.
文摘We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method.Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability.By numerical experiments we demonstrate the stability,accuracy,efficiency,and the applicability of the methods to forward and inverse problems.
基金supported by the Basic Research Program of Northwestern Polytechnical University (Grant No. JC201219)the Postdoctoral Science Foundation of China (Grant No. 20100481368)
文摘A method of unstructured dynamic overset grids is developed for the numerical simulation of helicopter unsteady rotorairframe aerodynamic interaction.For the effective treatment of the relative motion between the rotor and the airframe,the domain of flowfield is divided into two overset subzones,namely,a rotational subzone containing the blades and a stationary subzone containing the airframe.The overset part of two subzones is used to convect the flow variables of the two zones.The Taylor series expansion is used to obtain a second-order spatial accuracy,and dual-time stepping is adopted to improve the solution accuracy.Mesh deformation from the blade motion in forward flight is treated by using a spring analogy.Validation is made by numerically simulating the flows around a wind tunnel configuration and comparing the predicted time-averaged and instantaneous inflow and airframe surface pressure distributions with the experimental data.It shows that the present method is efficient and robust for the prediction of complicated unsteady rotor-airframe aerodynamic interaction phenomena.
文摘This paper proposes an automatic structure overset grid method, which utilizes the hole-surface optimization with one-step searching, wall-surface grid oversetting, and dynamic overset grid approaches to achieve the high adaptability of overset grids for complex multi-body aircrafts. Specifically, based on the automatic structure overset grids, the method first solves the coupling of Navier-Stokes(N-S) unsteady flow equation and 6DOF motion equation, and establishes the multi-body collision model. Then, the numerical simulation of unsteady flow for complex aircrafts' multi-body separation, the simulation of multi-body separating trajectory and the separation safety analysis are accomplished. Thus, the method can properly handle practical engineering problems including the wing/drop tank separation, aircraft/mount separation, and cluster bomb projection. Experiments show that our numerical results match well with experimental results, which demonstrates the effectiveness of our methods in solving the multi-body separation problem for aircrafts with complex shapes.
基金Project supported by the National Natural Science Foundation of China(Grant No.52131102).
文摘In the present study, the performance of the NTNU Blind Test 1 wind turbine is analyzed in the computational fluid dynamics (CFD) simulations by using the CFD code FANS with structured overset grids. First, the numerical methods including the governing equations, the turbulence closure model, and the flow solver are introduced. In addition, the NTNU BT1 wind tunnel experiment is described. Then, structured overset grid blocks are generated in the computational domain with fully resolved wind turbine geometry, including the blades, hub, nacelle, and tower. Afterward, unsteady Reynolds averaged Navier-Stokes (RANS) simulations with the two-layer k - ε turbulence model are performed with an inlet velocity of 10 m/s and a tip-speed ratio (TSR) of 6. The overset-grid capability of FANS is leveraged to handle the rotation of the rotor. Finally, simulations are performed for a range of TSRs and a comparison is made among the present CFD results, other numerical results obtained from representative methods, and the experimental data. It is observed that the CFD-predicted thrust coefficients match the experimental measurement at low TSRs while under-predicting the values at high TSRs, and potential reasons for this deviation are discussed.
文摘Ice-going ships play a crucial role in polar transportation and resource extraction.Different from the existing modeling approach which assumes that ships remain stationary,dynamic overset grid technology and DFBI(Dynamic Fluid-Body Interaction)method are employed in this paper to enable the free-running motion of the ship in modeling.A numerical model capable of simulating a ship navigating through pack ice area is proposed,which uses Computational Fluid Dynamics(CFD)method to solve the flow field and applies the Discrete Element Method(DEM)to simulate ship-ice and ice-ice interactions.Besides,the proposed high-precision method for generating pack ice area can be used in conjunction with the proposed numerical model.By comparing the numerical results with the available model test data and experimental observations,the effectiveness of the numerical model is validated,demonstrating its strong capability of predicting resistance and simulating ship navigation in pack ice,as well as its significant potential and applicability for further studies.
基金Supported by the UK Department for Transport,as part of the UK Shipping Office for Reducing Emissions(UK SHORE)Programme and the UK Engineering and Physical Sciences Research Council(EPSRC)[grant number EP/Y024605/1].
文摘The hydrodynamic performance of high-speed planing hulls has gained considerable interest,with recent advancements in computational fluid dynamics and hull design techniques enhancing the understanding of planing hull hydrodynamics.In this study,we conducted a numerical investigation using the Reynolds-averaged Navier-Stokes approach with overset grids to capture large motions at high speeds.This study aims to improve the hydrodynamic performances of planing hulls,specifically focusing on total resistance,trim,and sinkage.The initial Fridsma hull with a deadrise angle of 20°has been used for validation,demonstrating good agreement with measurements at different Froude numbers.Subsequently,new configurations based on the Fridsma hull have been designed by varying the deadrise angle,number of chines,and transverse steps.Our findings reveal a correlation between the deadrise angle,the number of chines,and the Froude number.As the deadrise angle increases,total resistance also increases.Additionally,a single chine yields superior results at higher Froude numbers,while multiple chines offer advantages at lower values.The introduction of transverse steps consistently increases total resistance,highlighting their role in improving planing hull performance.This research not only offers valuable insights into planing hull design but also leverages state-of-the-art numerical methods to advance the understanding of hydrodynamic behaviors at high ship speeds.
