Cut Cell Mesh Research Articles

Accuracy Analysis for Explicit-Implicit Finite Volume Schemes on Cut Cell Meshes

The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. In May and Berger (J Sci Comput 71: 919–943, 2017), the mixed explicit-implicit approach in general and MUSCL-Trap (monotonic upwind scheme for conservation laws and trapezoidal scheme) in particular have been introduced to solve this problem by using implicit time stepping on the cut cells. Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping. In this contribution, we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme. As this result is unlikely to hold in two dimensions, we also introduce two new versions of mixed explicit-implicit schemes based on exchanging the explicit scheme. We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.

The Study of the Lift Coefficient of NACA-0012 Airfoil Subjected to an Incompressible Media by Using the Cartesian Cut Cell Method

In order to shorten product design time, engineers need to create high quality meshes within a few days or hours. Newer automated techniques have been published to tackle this need, the Cartesian Cut Cell is one of them. This study is focused on the effect of different Cut Cell meshing strategies on the accuracy of aerodynamic performance predictions. The method can be described as a methodology in which Cut Cells are applied to the geometry utilizing a process involving rectangular/hexagonal cells on a regular lattice cutting through the geometry. The Cut Cell meshing is a general purpose designed for ANSYS FLUENT, making use of Workbench to construct the airfoil shape and the mesh. The results obtained for NACA-0012 are computed using two models available in ANSYS FLUENT, namely the Eq. 2 k-ω SST and the Eq. 3 k-kl-ω models. The three-dimensional numerical simulations were created for steady incompressible flow around NACA-0012 shape. Lift coefficient, Boundary layer thickness, mesh expansion ratio, and mesh density variation parameters were investigated. For this application both models produce good lift results. k-kl-ω produce better lift and the results are close to the measured data. The Cut Cell method showed a very good agreement between Computational Fluid Dynamics results and experimental data. This work illustrated that the Cartesian Cut Cell method has the ability to generate high quality mesh, which captures the details of the viscous boundary layer easily. The future work is to use more sophisticated turbulence models and mesh refinement for air craft wing with flap.

TopoCut

Given a complex three-dimensional domain delimited by a closed and non-degenerate input triangle mesh without any self-intersection, a common geometry processing task consists in cutting up the domain into cells through a set of planar cuts, creating a "cut-cell mesh", i.e., a volumetric decomposition of the domain amenable to visualization (e.g., exploded views), animation (e.g., virtual surgery), or simulation (finite volume computations). A large number of methods have proposed either efficient or robust solutions, sometimes restricting the cuts to form a regular or adaptive grid for simplicity; yet, none can guarantee both properties, severely limiting their usefulness in practice. At the core of the difficulty is the determination of topological relationships among large numbers of vertices, edges, faces and cells in order to assemble a proper cut-cell mesh: while exact geometric computations provide a robust solution to this issue, their high computational cost has prompted a number of faster solutions based on, e.g., local floating-point angle sorting to significantly accelerate the process --- but losing robustness in doing so. In this paper, we introduce a new approach to planar cutting of 3D domains that substitutes topological inference for numerical ordering through a novel mesh data structure, and revert to exact numerical evaluations only in the few rare cases where it is strictly necessary. We show that our novel concept of topological cuts exploits the inherent structure of cut-cell mesh generation to save computational time while still guaranteeing exactness for, and robustness to, arbitrary cuts and surface geometry. We demonstrate the superiority of our approach over state-of-the-art methods on almost 10,000 meshes with a wide range of geometric and topological complexity. We also provide an open source implementation.

VEMPIC

The comprehensive visual modeling of fluid motion has historically been a challenging task, due in no small part to the difficulties inherent in geometries that are non-manifold, open, or thin. Modern geometric cut-cell mesh generators have been shown to produce, both robustly and quickly, workable volumetric elements in the presence of these problematic geometries, and the resulting volumetric representation would seem to offer an ideal infrastructure with which to perform fluid simulations. However, cut-cell mesh elements are general polyhedra that often contain holes and are non-convex; it is therefore difficult to construct the explicit function spaces required to employ standard functional discretizations, such as the Finite Element Method. The Virtual Element Method (VEM) has recently emerged as a functional discretization that successfully operates with complex polyhedral elements through a weak formulation of its function spaces. We present a novel cut-cell fluid simulation framework that exactly represents boundary geometry during the simulation. Our approach enables, for the first time, detailed fluid simulation with "in-the-wild" obstacles, including ones that contain non-manifold parts, self-intersections, and extremely thin features. Our key technical contribution is the generalization of the Particle-In-Cell fluid simulation methodology to arbitrary polyhedra using VEM. Coupled with a robust cut-cell generation scheme, this produces a fluid simulation algorithm that can operate on previously infeasible geometries without requiring any additional mesh modification or repair.

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space and an explicit strong stability preserving Runge-Kutta scheme in time. When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues. The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way. For piecewise constant polynomials in space and explicit Euler in time, the stabilized scheme is monotone for scalar problems. For higher polynomial degrees p, our numerical experiments show convergence orders of p+1 for smooth flow and robust behavior in the presence of shocks.

Demonstration of a three-dimensional dynamically adaptive atmospheric dynamic framework for the simulation of mountain waves

In this paper, Fluidity-Atmosphere, representative of a three-dimensional (3D) non-hydrostatic Galerkin compressible atmospheric dynamic framework, is generated to resolve large-scale and small-scale phenomena simultaneously. This achievement is facilitated by the use of non-hydrostatic equations and the adoption of a flexible 3D dynamically adaptive mesh where the mesh is denser in areas with higher gradients of variable solutions and relatively sparser in the rest of the domain while maintaining promising accuracy and reducing computational resource requirements. The dynamic core is formulated based on anisotropic tetrahedral meshes in both the horizontal and vertical directions. The performance of the adaptive mesh techniques in Fluidity-Atmosphere is evaluated by simulating the formation and propagation of a non-hydrostatic mountain wave. The 2D anisotropic adaptive mesh shows that the numerical solution is in good agreement with the analytic solution. The variation in the horizontal and vertical resolutions has a strong impact on the smoothness of the results and maintains convergence even at high resolutions. When the simulation is extended to 3D, Fluidity-Atmosphere shows stable and symmetric results in the benchmark test cases. The flows over a bell-shaped mountain are resolved quite smoothly. For steep mountains, Fluidity-Atmosphere performs very well, which shows the potential of using 3D adaptive meshes in atmospheric modeling. Finally, as an alternative cut-cell mesh in Fluidity-Atmosphere, the anisotropic adaptive mesh coupled with the Galerkin method provides an alternative accurate representation of terrain-induced flow.

An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries

Abstract The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.

A state redistribution algorithm for finite volume schemes on cut cell meshes

In this paper we develop a new technique, called state redistribution, that allows the use of explicit time stepping when approximating solutions to hyperbolic conservation laws on embedded boundary grids. State redistribution is a postprocessing technique applied after each time step or stage of the base finite volume scheme, using a time step that is proportional to the volume of the full cells. The idea is to stabilize the cut cells by temporarily merging them into larger, possibly overlapping neighborhoods, then replacing the cut cell values with a stabilized value that maintains conservation and accuracy. We present examples of state redistribution using two base schemes: MUSCL and a second order Method of Lines finite volume scheme. State redistribution is used to compute solutions to several standard test problems in gas dynamics on cut cell meshes, with both smooth and discontinuous solutions. We show that our method does not reduce the accuracy of the base scheme and that it successfully captures shocks in a non-oscillatory manner.

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.

A full-field simulation methodology for sonic boom modeling on adaptive Cartesian cut-cell meshes

This paper develops a full-field direct numerical simulation methodology of sonic boom in a stratified atmosphere. The entire flow field, ranging from the near field around a supersonic body to the far field extending to the ground, is modeled by the three-dimensional Euler equations with a gravitational source term. Thus far, it has been solved using a structured grid, and an application of previous simulation to complex geometries has been limited. In this study, we realize a full-field simulation by employing the following four numerical approaches: (i) a hierarchical structured adaptive mesh refinement (AMR) method, (ii) a Cartesian cut cell method, (iii) a well-balanced finite volume method, and (iv) a segmentation method of the computational domain. A new well-balanced, MUSCL-Hancock scheme applied over Cartesian AMR and cut cell grids for a stratified atmosphere is formulated. The computational results of an oblique shock wave in a stratified atmosphere agree well with the exact solution. A full-field simulation successfully reproduces the Drop test for Simplified Evaluation of Non-symmetrically Distributed sonic boom (D-SEND) #1, conducted by the Japan Aerospace Exploration Agency (JAXA). The results of this simulation are in good agreement with those of the previous computational study, the waveform parameter method, and flight test measurements. The grid convergence study shows that the mesh size is fine enough to assess pressure signatures over the entire flow field. These results demonstrate that a full-field simulation with AMR and cut cell grids is a powerful tool for extensively analyzing three-dimensional shock wave propagation in a stratified atmosphere.

A Stabilized DG Cut Cell Method for Discretizing the Linear Transport Equation

We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG) method in two dimensions with piecewise linear polynomials. The goal is to allow for explicit time stepping schemes despite the presence of cut cells. Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh and add penalty terms that stabilize the solution on small cut cells in a conservative way. Then one can use explicit time stepping, even on cut cells, with a time step length that is appropriate for the background mesh. In one dimension, we show monotonicity of the proposed scheme with a constant basis and total variation diminishing in the means stability for piecewise linear polynomials. We also present numerical results in one and two dimensions that support our theoretical findings.

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.

Consistent discretization of higher-order interface models for thin layers and elastic material surfaces, enabled by isogeometric cut-cell methods

Many interface formulations, e.g. based on asymptotic thin interphase models or material surface theories, involve higher-order differential operators and discontinuous solution fields. In this article, we are taking first steps towards a variationally consistent discretization framework that naturally accommodates these two challenges by synergistically combining recent developments in isogeometric analysis and cut-cell finite element methods. Its basis is the mixed variational formulation of the elastic interface problem that provides access to jumps in displacements and stresses for incorporating general interface conditions. Upon discretization with smooth splines, derivatives of arbitrary order can be consistently evaluated, while cut-cell meshes enable discontinuous solutions at potentially complex interfaces. We demonstrate via numerical tests for three specific nontrivial interfaces (two regimes of the Benveniste–Miloh classification of thin layers and the Gurtin–Murdoch material surface model) that our framework is geometrically flexible and provides optimal higher-order accuracy in the bulk and at the interface.

Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

Development of 2D inundation model based on adaptive cut cell mesh (K-Flood)

Development of 2D inundation model based on adaptive cut cell mesh (K-Flood)

A dimensionally split Cartesian cut cell method for the compressible Navier–Stokes equations

We present a dimensionally split method for computing solutions to the compressible Navier–Stokes equations on Cartesian cut cell meshes. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a number of test problems ranging from the nearly incompressible to the highly compressible flow regimes. All the computed results show good agreement with reference results from theory, experiment and previous numerical studies. To the best of our knowledge, this is the first presentation of a dimensionally split cut cell method for the compressible Navier–Stokes equations in the literature.

Numerical investigation of hydraulic and thermal performance of a honeycomb heat sink

The thermal and hydraulic performance of a heat sink having aluminum honeycomb fins is numerically investigated by using a finite-volume-based solver. The fin height (H), the distance between the fins in the streamwise direction (Sy) and the Reynolds number (Re) are selected as design parameters while the thermal resistance of the heat sink (Rth) and the pressure drop (ΔP) are regarded as the performance criteria. Each design parameter is examined in three levels and twenty-seven three-dimensional simulations are carried out in total. A computational domain consisting of three parts as the inlet and outlet sections and the test section where the honeycomb heat sink is located is used in the simulations. The CutCell mesh is employed for the test section while hexahedral meshes are used to discretize the inlet and outlet sections. The Reynolds-Averaged Navier-Stokes (RANS) based Realizable k-ε turbulence model in combination with the enhanced wall function is employed in the simulations. The effects of each design parameter on the performance criteria are examined in detail.

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.