Cartesian Cut Cell Mesh Research Articles

Novel numerical method to simulate hydrodynamic characteristic of moving body through free surface and stratified-fluid interface

In order to solve the difficult physical problem of moving structures through free surface and stratified fluid interface, the paper presents and develops a new method based on surface capturing method and Cartesian cut cell mesh to handle transient density interface and moving boundary. Therein, quad-tree data storage structure, adaptive grid techniques, and high-resolution reconstruction limiter are introduced to improve computational efficiency and precision. Hybrid mesh systems of structural and Cartesian cut cell mesh are extended to be applied in the laminar flow problem at low Reynolds number. On this basis, a series of experimental tests of cylindrical models falling through air-water, air-oil and oil-water interfaces are carried out respectively, then several cases are taken to validate the numerical method. Sequentially, the characteristics of interface variation and closure shape versus different density ratios are compared, and the effect of stratified fluid interface on the hydrodynamic force of moving body is mainly discussed. In summary, this paper unifies free surface and stratified fluid interface together as density interface, which can be handled and studied by the same numerical method. Thus, by combining water entry with stratified fluid problem, the influence of various density interfaces on moving body can be better analyzed.

Numerical investigation on free surface effect on the supercavitating flow over a low aspect ratio wedge-shaped hydrofoil

In this study, the effect of the free surface on the supercavitating flow that surrounds a low aspect ratio wedge-shaped hydrofoil is studied based on computational fluid dynamics (CFD), with Cartesian cut-cell mesh method. The volume of fraction (VOF) method and the k - e turbulence model with the Schnerr Sauer cavitation model are used for simulating the supercavitation in turbulent flow. Both quasi-3-D and full-3-D cases are considered. The quasi-3-D simulation results are compared with the previous water tunnel experimental data and analytical data, and the results agree well with each other. The results are presented for a wide range of submerged distance in terms of the free surface effect on the cavity shape, wave elevation and force coefficients. The range of the free surface effect is found. The simulation results show that the cavitating flow around the hydrofoil become shorter and thicker as the submerge depth decreases. As the aspect ratio of the hydrofoil studied in the full-3-D cases is low, the 3-D effect on the supercavitating flow is strong. The relationship between the flow patterns and vortex structures is also revealed from the numerical results.

Accuracy considerations of mixed explicit implicit schemes for embedded boundary meshes

AbstractWe consider a mixed explicit implicit finite volume scheme for solving the advection equation on a Cartesian cut cell mesh: the scheme couples an implicit time stepping scheme on cut cells with an explicit time stepping scheme on cells away from the cut cells by means of flux bounding. We examine the accuracy of the resulting mixed scheme to see if a second‐order explicit and a second‐order implicit scheme result in a second‐order mixed scheme.

Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence

• The tip-leakage cavitating flow is simulated by LES with Cartesian cut-cell mesh. • Three types of cavitating vortical flows, including TLV, TSV and IV, are analyzed from Euler/Lagrangian viewpoints. • Three stages for the spatial–temporal evolution of tip-leakage cavitating flow are discussed. • The physical mechanism for cavitation-turbulence interaction is clarified. Cavitation within a tip leakage flow remains a challenging issue in a variety of axial hydraulic machines. It is still not possible nowadays to predict cavitation occurrence in such a flow with acceptable accuracy. In the present study, we have carried out numerical simulations of a tip leakage cavitating flow, generated by a straight NACA0009 hydrofoil. We have used the LES method combined with the Schnerr–Sauer cavitation model. The numerical results agree well with experimental data. The evolution of the tip leakage cavitating flow, involving tip-leakage vortex (TLV), tip-separation vortex (TSV) and induced vortex (IV), is analyzed from Eulerian and Lagrangian viewpoints. The results show that the spatial evolution of the tip leakage cavitating flow can be divided into three stages: Stage Ⅰ, Independent development of the TLV and TSV; Stage Ⅱ, Fusion of the TLV and TSV; and Stage Ⅲ, Development of the IV and dissipation of the TLV. The Lagrangian coherent structures (LCSs) obtained from the numerical results indicate that the TLV cavitation significantly influence the local flow patterns. The vorticity transport equation was then used to further analyze the influence of the cavitation on the vortices. The results demonstrate that the stretching term dominates the TLV evolution and the dilatation term is responsible for the vorticity reduction inside the TLV cavity. The results also show how the cavitation influences the local turbulence and that the transport term in the turbulent kinetic energy equation influences the turbulence distribution near the TLV cavity.

Large eddy simulation of tip leakage cavitating flow focusing on cavitation-vortex interaction with Cartesian cut-cell mesh method

In the present paper, flow configurations of cavitating flow around a straight NACA0009 foil with a gap between the foil tip and sidewall are investigated numerically by large eddy simulation (LES) coupled with Zwart-Gerber-Belamri (ZGB) cavitation model. A Cartesian cut-cell method is used for mesh generation, which is of good orthogonality and high quality. A good agreement is obtained between simulation and experiment. Two influencing factors on vorticity distributions, the interaction between different vortices and the occurrence of cavitation, are discussed in detail based on the numerical results. A series of Ω- shaped loops are observed during the development of the induced vortex, which is a result of the instability of vortex pair. This finding may provide a new viewpoint to control the evolution of tip-leakage vortex (TLV) cavitation. Moreover, it is found that the dilatation term plays a much more important role in the evolution of TLV cavitation compared with that in sheet cavitation.

Analysis of Near-wall effect on cloud cavitating flow that surrounds an axisymmetric projectile using large eddy simulation with Cartesian cut-cell mesh method

Near-wall effect is important for cavitation of flow and vortex structures. These structures are commonly investigated in cavitation of tip-vortex leakage, but are rarely discussed in cloud cavitating flow. In this study, typical experiments and numerical simulation of cloud cavitating flow were conducted near a wall that surrounds an underwater axisymmetric projectile. The experimental observations of cavity development are consistent with numerical results and validate the method’s accuracy. Changes in the cavity of the distal and near wall side differ throughout the entire evolution process. The cavity grows faster on the near wall side than on the distal side, whereas the re-entry jet inside the cavity moves slowly toward the shoulder of the model. The strong vortex around the projectile is non-axisymmetric because of the collapsing cavity, which may also affect the cruising stability.

Cloud Cavitating Flow That Surrounds a Vertical Hydrofoil Near the Free Surface

Unstable cavitation presents an important speed barrier for underwater vehicles such as hydrofoil craft. In this paper, the authors concern about the physical problem about the cloud cavitating flow that surrounds an underwater-launched hydrofoil near the free surface at relatively high-Froude number, which has not been discussed in the previous research. A water tank experiment and computational fluid dynamics (CFD) simulation are conducted in this paper. The results agree well with each other. The cavity evolution process in the experiment involves three stages, namely, cavity growth, shedding, and collapse. Numerical methods adopt large eddy simulation (LES) with Cartesian cut-cell mesh. Given that the speed of the model changes during the experiment, this paper examines cases with varying constant speeds. The free surface effects on the cavity, re-entry jet location, and vortex structures are analyzed based on the numerical results.

Numerical study of periodic long wave run-up on a rigid vegetation sloping beach

Coastal vegetation can reduce long wave run-up on beaches and inland propagation distances and thus mitigate these hazards. This paper investigates periodic long wave run-up on coastal rigid vegetation sloping beaches via a numerical study. Rigid vegetation is approximated as rigid sticks, and the numerical model is based on an implementation of Morison's formulation [21] for rigid structures induced inertia and drag stresses in the nonlinear shallow water equations. The numerical model is solved via a finite volume method on a Cartesian cut cell mesh. The accuracy of the numerical model is validated by comparison with experimental results. The model is then applied to simulate various hypothetical cases of long periodic wave run-up on a sloping vegetated beach with different plant diameters and densities, and incident long waves with different periods. The sensitivity of long wave run-up to plant diameter, stem density and wave period is investigated by comparison of the numerical results for different vegetation characteristics and different wave periods. The numerical results show that rigid vegetation can effectively reduce long wave run-up and that wave run-up is decreased with increase of plant diameter and stem density. Moreover, the attenuation of long periodic wave run-up due to vegetation is sensitive to the variation of the incident wave period, and the attenuation of wave run-up is not increased or decreased monotonically with incident wave period.

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem--that standard schemes are not stable on the arbitrarily small cut cells--while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in $$L^1$$L1 and between first- and second-order accurate along the embedded boundary in two and three dimensions.

Further Application of Surface Capturing Method and Cartesian Cut Cell Mesh on Hydroelastic Water-Entry Problems of Free-Falling Elastic Wedge

In order to study the interactions between fluid and elastic structure (such as marine lifeboat falling down and ship), this paper presents a new CFD method on hydroelastic water-entry problem of free-falling elastic wedge, which can more conveniently handle moving solid boundaries. In the CFD solver, a surface capturing method and the Cartesian cut cell mesh are employed to deal with the moving free surface and solid boundaries, respectively. On the other hand, in structural analysis, the finite element method and lath-beam structural model are introduced to calculate the elastic response. Furthermore, based on the current CFD and structural solver, a particular data transfer method and coupling strategy are presented for the fluid-structure interaction. Finally, by comparing numerical results with experimental data, the present method is validated to be available and feasible for hydroelastic water-entry problem and further successfully adopted to analyze the motion characteristics of free-falling elastic wedge.

Numerical study of vegetation damping effects on solitary wave run-up using the nonlinear shallow water equations

Vegetation damping effects on propagating water waves have been investigated by many researchers. This paper investigates the effects of damping due to vegetation on solitary water wave run-up via numerical simulation. The numerical model is based on an implementation of Morison's formulation for vegetation induced inertia and drag stresses in the nonlinear shallow water equations. The numerical model is solved via a finite volume method on a Cartesian cut cell mesh. The accuracy of the numerical scheme and the effects of the vegetation terms in the present model are validated by comparison with experiment results. The model is then applied to simulate a solitary wave propagating on a plane slope with vegetation. The sensitivity of solitary wave run-up to plant height, diameter and stem density is investigated by comparison of the numerical results for different patterns of vegetation. The numerical results show that vegetation can effectively reduce solitary wave propagation velocity and that solitary wave run-up is decreased with increase of plant height in water and also diameter and stem density.

Numerical Simulation of Water Impact on a Wave Energy Converter in Free Fall Motion

Results are presented for the 3D numerical simulation of the water impact of a wave energy converter in free fall and subsequent heave motion. The solver, AMAZON-3D, employs a Riemann-based finite volume method on a Cartesian cut cell mesh. The computational domain includes both air and water regions with the air/water boundary captured automatically as a discontinuity in the density field thereby admitting break up and recombination of the free surface. Temporal discretisation uses the artificial compressibility method and a dual time stepping strategy. Cartesian cut cells are used to provide a boundary-fitted grid at all times. The code is validated by experimental data including the free fall of a cone and free decay of a single Manchester Bobber component.

A Cartesian cut cell free surface capturing method for 3D water impact problems

SUMMARYA Cartesian cut cell mesh generation procedure is developed together with a finite volume Euler solver for a two‐fluid system with a free surface. A fast and robust triangle to triangle overlap scheme is used to determine the intersection of a body‐surface with the background Cartesian mesh. Improvements to the cut cell routines include a new treatment for multiple cuts within a single cell and a surface trimming procedure to ensure a good quality mesh around solid boundaries. The formulae for calculating all necessary information about a cut cell are also presented. These are generic and can be used for arbitrarily irregular boundary elements. A collocated finite volume method with a high resolution Godunov‐type scheme in space is used for discretization of the governing flow equations. By computing in both the air and water regions simultaneously in a consistent manner, the free surface is automatically captured as a contact discontinuity in the density field without the need for any special free surface tracking method. The algorithm incorporates the artificial compressibility method with a dual time stepping strategy to maintain a divergence free velocity field. The mathematical formulation including its numerical implementation of the method is reviewed and results for a number of test cases are also presented. Copyright © 2012 John Wiley & Sons, Ltd.

Analysis of Mechanical Energy Transport on Free‐Falling Wedge during Water‐Entry Phase

For better discussing and understanding the physical phenomena and body‐fluid interaction of water‐entry problem, here mechanical‐energy transport (wedge, fluid, and each other) of water‐entry model for free falling wedge is studied by numerical method based on free surface capturing method and Cartesian cut cell mesh. In this method, incompressible Euler equations for a variable density fluid are numerically calculated by the finite volume method. Then artificial compressibility method, dual‐time stepping technique, and Roe′s approximate Riemann solver are applied in the numerical scheme. Furthermore, the projection method of momentum equations and exact Riemann solution are used to calculate the fluid pressure on solid boundary. On this basis, during water‐entry phase of the free‐falling wedge, macroscopic energy conversion of overall body‐fluid system and microscopic energy transformation in fluid field are analyzed and discussed. Finally, based on test cases, many useful conclusions about mechanical energy transport for water entry problem are made and presented.

Comments on ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’

In the paper entitled ‘An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems’ ( Wang & Wang 2009 Proc. R. Soc. A 465 , 1843–1868 ( doi:10.1098/rspa.2008.0410 )), the present authors' earlier work ( Qian et al . 2006 , Proc. R. Soc. A 462 , 21–42 ( doi:10.1098/rspa.2005.1528 )) has been specifically applied to the study of water-entry and -exit of solid objects. An extended boundary condition, retaining the term owing to acceleration of moving boundaries in the momentum equation, has been implemented for calculating the pressure gradient at solid surfaces and, based on their numerical experiments, it was concluded by Wang and Wang that without this term the calculation will substantially under-predict the impact forces and may even break down. Therefore, a more complex procedure based on the exact solution of a Riemann problem for moving boundaries was implemented. In this short comment, by applying the authors' free surface capturing code to the same flow problem of water-entry of a wedge, it can, however, be demonstrated that the results from implementing the new pressure boundary condition are nearly identical to that of employing the original boundary condition without the acceleration term, indicating that its effects on the simulation results are minimal. A further examination of the implementation details on the pressure boundary condition also supports this conclusion.

An essential solution of water entry problems and its engineering applications

For solving water entry problems, a numerical method is presented, which is a CFD method based on free surface capturing method and Cartesian cut cell mesh. In this approach, incompressible Euler equations for a variable density fluid are numerically calculated by the finite volume method. Then artificial compressibility method, dual time-stepping technique and Roe’s approximate Riemann solver are adopted in the numerical scheme. Finally, some application cases are designed to show the ability of the current method to cope with water entry problems in ocean engineering.

Numerical Simulation of Shallow-Water Flow Using a Modified Cartesian Cut-Cell Approach

The Cartesian cut-cell method can be used to represent irregular and complex computational domains with less computational efforts by cutting the grid cells on the boundary surfaces in a background uniform Cartesian mesh. In this study, a modified Cartesian cut-cell grid technique is proposed to better represent complex physical geometries. A point shifting treatment was employed to determine the start and end points of a line segment in cut-cell grids. This led to an improved representation of sharply-shaped corners in surface polygons. Numerical simulation to solve a set of shallow-water equations was performed by incorporating a finite volume approach into the Cartesian cut-cell mesh. The advective fluxes at intercells were first estimated by a Harten, Lax and van Leer for contact wave approximate Riemann solver. In order to improve the model accuracy to the second order, a total variation diminishing-weighted average flux method was applied to work adaptively with the cut-cell mesh. The numerical model was then employed to simulate dam-break flow propagation in a small channel with a rectangular obstacle or a 45° bend. The numerical results show good agreement with available laboratory measurements.

An improved free surface capturing method based on Cartesian cut cell mesh for water-entry and -exit problems

The free surface capturing method based on Cartesian cut cell mesh is extended to the water-entry and -exit fields with body–fluid interaction. The governing equations are the incompressible Euler equations for a variable density fluid system with a free surface, which is treated as a contact discontinuity. The solver is based on the artificial compressibility method with a dual time-stepping technique for time advancing and the finite-volume method with a high-resolution Godunov-type upwind scheme on spatial discretization. For solving the numerical problem caused by the extension, the acceleration term of body is introduced for the new pressure condition of moving solid boundary, and the exact Riemann solution is used to calculate the flux of the solid boundary. In addition, a new solution of gradients based on the least-squares idea is presented for simply calculating the gradients. Finally, test cases show that the present method is available and can be successfully applied in various water-entry and -exit phenomena.

Cartesian Cut-Cell Method for Axisymmetric Separating Body Flows

A method for the calculation of unsteady, axisymmetric compressible flows involving both fixed and separating bodies is presented. The method uses a Cartesian cut-cell mesh approach and high resolution upwind finite volume scheme. A stationary background Cartesian mesh is generated on the computational domain with complex solid geometries represented by different types of cut cell. Solid bodies are allowed to move across the mesh using a finite volume scheme modified to deal with moving boundary problems. The flow solver employed is a MUSCL-Hancock Godunov-type scheme In conjunction with an approximate Riemann solver of the Harten, Lax, and van Leer type (for flow interfaces) and an exact Riemann solver for a moving piston (for fixed or moving solid faces). A cell-merging technique is used to maintain numerical stability in the presence of arbitrarily small cut cells and to retain strict conservation at moving boundaries. The method is applied to a muzzle blast flow and muzzle break problems involving both fixed and separating bodies