Cut-cell Method Research Articles

An efficient pressure-based multiphase finite volume method for interaction between compressible aerated water and moving bodies

Maritime structures in heavy seas can experience wave impact events with high loads. The loads can lead to structural failure and even loss of life. Wave breaking in said sea states causes air to be entrained in water as aeration cloads, remaining long enough to be transported and to play a role in the impulsive interaction with the structure. A small amount of air in water already forms a highly compressible mixture. Compressibility influences the magnitude of the impact loads.A new cartesian grid method for compressible multiphase flow is introduced to account for water, air and homogeneous mixtures of air and water. The method is designed to predict the hydrodynamic loads on moving bodies engaging with interfaces between fluids having large density ratios. An equation for conservation of energy is omitted by enforcing pressure-density relations. The interface between fluids is transported using a geometric Volume-of-Fluid method. The interface between fluids and structure is taken care of by a cut-cell method. An additional fraction field for the amount of air in water in combination with a new formulation for the multiphase speed of sound prevent overprediction of compressibility by artificial air entrainment.New experimental data of 2D wedge impacts with aerated water, made available as open data, are presented to demonstrate the validity of the numerical method. For low aeration levels, the simulation results in terms of the impact loads on the wedge and the frequencies of pressure waves generated upon impact are in good agreement with the experimental data. Increasing the level of aeration reduces the maximum impact load on the wedge. Reflected density waves lead to secondary loads on the wedge. The intensity of the secondary loads, relative to the primary load of impact, increases with the aeration level while the density wave frequency decreases.

A coupled VOF/embedded boundary method to model two-phase flows on arbitrary solid surfaces

We present an hybrid VOF/embedded boundary method allowing to model two-phase flows in presence of solids with arbitrary shapes. The method relies on the coupling of existing methods: a geometric Volume of fluid (VOF) method to tackle the two-phase flow and an embedded boundary method to sharply resolve arbitrary solid geometries. Coupling these approaches consistently is not trivial and we present in detail a quad/octree spatial discretization for solving the corresponding partial differential equations. Modelling contact angle dynamics is a complex physical and numerical problem. We present a Navier-slip boundary condition compatible with the present cut cell method, validated through a Taylor–Couette test case. To impose the boundary condition when the fluid–fluid interface intersects a solid surface, a geometrical contact angle approach is developed. Our method is validated for several test cases including the spreading of a droplet on a cylinder, and the equilibrium shape of a droplet on a flat or tilted plane in 2D and 3D. The temporal evolution and convergence of the droplet spreading on a flat plane is also discussed for the moving contact line given the boundary condition (Dirichlet or Navier) used. The ability of our numerical methodology to resolve contact line statics and dynamics for different solid geometries is thus demonstrated.

A high order cut-cell method for solving the shallow-shelf equations

In this paper we present a novel method for solving the shallow-shelf equations in the presence of grounding lines. The shallow-self equations are a two-dimensional system of nonlinear elliptic PDEs with variable coefficients that are discontinuous across the grounding line, which we treat as a sharp interface between grounded and floating ice. The grounding line is “reconstructed” from ice thickness and basal topography data to provide necessary geometric information for our cut-cell, finite volume discretization. Our discretization enforces jump conditions across the grounding line and achieves high-order accuracy using stencils constructed with a weighted least-squares method. We demonstrate second and fourth order convergence of the velocity field, driving stress, and reconstructed geometric information.

The Cut-Cell Method for the Conjugate Heat Transfer Topology Optimization of Turbulent Flows Using the “Think Discrete–Do Continuous” Adjoint

This paper presents a topology optimization (TopO) method for conjugate heat transfer (CHT), with turbulent flows. Topological changes are controlled by an artificial material distribution field (design variables), defined at the cells of a background grid and used to distinguish a fluid from a solid material. To effectively solve the CHT problem, it is crucial to impose exact boundary conditions at the computed fluid–solid interface (FSI); this is the purpose of introducing the cut-cell method. On the grid, including also cut cells, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model with wall functions, and the temperature equation are solved. The continuous adjoint method computes the derivatives of the objective function(s) and constraints with respect to the material distribution field, starting from the computation of derivatives with respect to the positions of nodes on the FSI and then applying the chain rule of differentiation. In this work, the continuous adjoint PDEs are discretized using schemes that are consistent with the primal discretization, and this will be referred to as the “Think Discrete–Do Continuous” (TDDC) adjoint. The accuracy of the gradient computed by the TDDC adjoint is verified and the proposed method is assessed in the optimization of two 2D cases, both in turbulent flow conditions. The performance of the TopO designs is investigated in terms of the number of required refinement steps per optimization cycle, the Reynolds number of the flow, and the maximum allowed power dissipation. To illustrate the benefits of the proposed method, the first case is also optimized using a density-based TopO that imposes Brinkman penalization terms in solid areas, and comparisons are made.

A cut-cell method for the numerical simulation of 3D multiphase flows with strong interfacial effects

We present a numerical model for the approximation of multiphase flows with free surfaces and strong interfacial effects. The model relies on the multiphase incompressible Navier-Stokes equations, and includes surface tension effects on the interfaces between phases, and contact angles. The volume-of-fluid approach is used to track the interfaces and the free surfaces between the various phases and the ambient air. The numerical method relies on an operator splitting strategy. The space discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. An adaptive mesh refinement algorithm is incorporated to better track the interfaces and free surfaces on the finite element mesh, and approximate more accurately surface forces. The model is validated through numerical experiments, in particular for emulsion problems.

A flow control strategy for a near-wall square cylinder using porous media: A direct numerical simulation study

The topic of flows around a near-wall square cylinder has garnered increasing attention in recent decades. However, there are a few publications that have focused on mitigating the occurrence of a substantial negative lift in near-wall flows. In light of this, the present study has developed a novel flow control strategy that covers porous media at inward corners of a near-wall square cylinder to address this problem. We achieve such a control strategy with the aid of a high-fidelity computational framework at Re = 1000. Direct numerical simulations are employed to account for accurate flow behaviors, and the Cartesian cut-cell method as well as an adaptive mesh refinement algorithm are advocated to simplify grid generation and reduce computational costs. Additionally, a quasi-microscopic flow model is introduced to model the porous medium pore structure, providing an intuitive and accurate description of internal flows within the porous medium. Six porous medium layouts are first designed, and their influences and mechanisms on flow control are assessed using the presented computational framework to identify an optimal strategy. The optimal strategy yields a notable reduction of 52.472% in the lift coefficient. The identified strategy is then applied to a case involving a near-wall square cylinder with a substantial negative lift, where a gap ratio of 0.6 is determined via parameterization. The capacity of the presented strategy in flow control of the near-wall square cylinder is fully explored and demonstrated via the consideration of different porosities. The results indicate that the most effective flow control is achieved when the porosity exceeds 90%, leading to a near-zero lift coefficient. Furthermore, the underlying mechanism contributing to the variations in flow control effectiveness due to different porosities is analyzed.

An accurate and scalable direction-splitting solver for flows laden with non-spherical rigid bodies—Part 2: Moving rigid bodies

The Direction Splitting algorithm is a computational method that facilitates the solution of the Navier–Stokes equation by decomposing a multi-dimensional Poisson equation into a series of one-dimensional equations. This approach allows for efficient and scalable numerical computations. The incorporation of rigid bodies in the computational domain is made possible by modifying the diffusion and divergence stencils near the fluid–solid interface. In our recent study (Goyal and Wachs, 2023), we implemented stencil corrections for fixed non-spherical rigid bodies and demonstrated the scalability and accuracy of the DS algorithm for a wide range of test cases. This study presents an extension of the Direction Splitting solver to effectively compute the dynamics of moving non-spherical rigid bodies in the presence of fluid flow. The interaction between rigid bodies and fluid can be categorized as either one-way or two-way coupled. The stencils resolving the fluid–solid interface are updated at every time-iteration to account for changes in particle position and orientation. We demonstrate that the adaptation of the diffusion and divergence stencil in the literature (Goyal and Wachs (2023); Keating and Minev (2013)) is inadequate for accurately predicting the dynamics of moving rigid bodies in a two-way coupled simulation, especially at O(102) Reynolds number. Hence, we implement a conservative cut-cell method for the discretization of the divergence term, and we also take into account the proximity of the fluid–solid interface in the advection discretization. The results of multiple test cases demonstrate that the suggested stencil modifications effectively capture the dynamics of fluid flow in the presence of freely moving rigid bodies. However, the cut-cell discretization of the divergence term constitutes a significant computing overhead that impacts the fast computing property of the Direction Splitting algorithm.

A Three-Dimensional Fully-Coupled Fluid-Structure Model for Tsunami Loading on Coastal Bridges

A three-dimensional (3D) fully-coupled fluid-structure model has been developed in this study to calculate the impact force of tsunamis on a flexible structure considering fluid-structure interactions. The propagation of a tsunami is simulated by solving the 3D Navier–Stokes equations using a finite volume method with the volume-of-fluid technique. The structure motion under the tsunami impact force is simulated by solving the motion equation using the generalized alpha method. The structure motion is fed back into the fluid solver via a technique that combines a sharp-interface immersed boundary method with the cut-cell method. The flow model predicts accurate impact forces of dam-break flows on rigid blocks in three experimental cases. The fully coupled 3D flow-structure model is tested with experiments on a large-scale (1:5) model bridge under nonbreaking and breaking solitary waves. The simulated wave propagation and structure restoring forces generally agree well with the measured data. Then, the fully-coupled fluid-structure model is compared with an uncoupled model and applied to assess the effect of flexibility on structure responses to tsunami loading, showing that the restoring force highly depends on the dynamic characteristics of the structure and the feedback coupling between fluid and structure. The maximum hydrodynamic and restoring forces decrease with increasing structure flexibility.

A conservative sharp interface method for two-dimensional incompressible two-phase flows with phase change

A conservative sharp interface method is proposed in this work to simulate two-dimensional/axisymmetric incompressible two-phase flows with phase change. In this method, we use the cut cell method to generate unstructured meshes near the interface, of which the cell edges overlap with the interface at each time step. On such mesh, the mass and heat transfer during phase change and all the jump conditions can be incorporated into the calculation of fluxes at the cell edges, to ensure that they are strictly satisfied at the interface in a sharp manner. The governing equations, including the incompressible Navier–Stokes equations, heat equation, and vapor mass fraction equation, are discretized by a second-order finite volume method in the arbitrary Lagrangian–Eulerian framework. To well couple the mass, heat, momentum, and interface evolution, the solution procedure is carefully designed and performed with several techniques. In such a way, the sharp discontinuity of the velocity, stress, temperature gradient, and vapor fraction, caused by the mass/heat transfer during phase change, can be simulated accurately and robustly. The performance of this method is systematically examined by cases of phase change at or below the saturated temperature, including vapor bubble in superheated liquid, film boiling, droplet evaporation at different relative humidity conditions, droplet evaporation under gravity, and droplet evaporation under forced convection. The applicability of the present method for incompressible two-phase flows with phase change is well demonstrated by comparing the numerical results with the benchmark, theoretical or experimental ones.

Direct numerical simulations of flows around an array of rough cylinders partially coated with porous media at Re = 3900

In recent years, interest has been growing in the stability control of cylindrical structures in a flow field using full/partial coatings of porous media. However, the partial application of porous coatings to arrays of cylinders with rough surfaces has rarely been studied. This paper numerically investigates the passive flow control of an array of rough cylinders partially coated with porous media on their leeward side in the subcritical flow regime (Re = 3900). This is achieved through two-dimensional direct numerical simulations and the Cartesian cut-cell method, which facilitate flow simulations around complex porous media and rough surfaces in an accurate and flexible manner. In contrast to macroscopic models, this study employs a quasi-microscopic model to simplify the porous structure for each cylinder, providing an accurate and intuitive means of describing the inflow details of porous media near the central cylinder. The porosities and pores per inch of the porous media are defined in terms of the alignment angles and radii. Similarly, the surface roughness of a cylinder is simulated by the attachment of small circular appendages. The accuracy and capacity of the presented numerical approach are demonstrated via a numerical case study with a single smooth cylinder. The influence of the porous media on the flows around a single rough cylinder and an array of rough cylinders is thoroughly investigated and discussed.

High-Order Cut-Cell Discontinuous Galerkin Difference Discretization

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations.

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.

Steady three-dimensional unbounded flow past an obstacle continuously deviating from a sphere to a cube

We perform three-dimensional particle resolved direct numerical simulations of the flow past a non-spherical obstacle by a Finite Volume cut-cell method, a sub-class of non-body-conforming methods that provides a sharp description of the boundary, which is strictly mass and momentum conservative and can be easily extended to adaptive grids. The present research work discusses the effect of corner rounding and the incidence angle for a range of Reynolds numbers for which the flow exhibits a steady-state behavior. The obstacle is placed in a large cubic domain that properly models an unbounded domain. Hierarchically refined Cartesian meshes are used where the obstacle resides at the finest level of the mesh hierarchy, thus ensuring that the resolution of the boundary layer and the wake of the obstacle is highly accurate, along with significantly reducing the number of grid cells and the computing time. Specifically, we characterize the drag force and the main features of the flow past a bluff obstacle transitioning in shape from spherical to cuboidal through a superquadric geometrical representation. A superquadric representation is suitable for our study since it preserves geometric isometry, and our analysis, thus, focusses on non-sphericity caused by the level of curvature. We investigate a range of Re from 10 to 150, which spans the flow from attached to symmetric and separated past five different obstacle shapes, with the corner radius of the curvature of r/a=2/ζi=1,2/2.5,2/4,2/8 and 0 placed at incidence angles of α=0°, 15°, 30°, and 45° with respect to the streamwise direction. In general, our results show that the obstacle bluffness increases with α and ζi and this increase is more prominent at higher Re. Higher drag forces are a consequence of either higher viscous forces for more streamlined bodies and in less inertial regimes or higher pressure forces for more bluff bodies and in highly inertial regimes, depending on how the corners are contributing to the frontal and lateral surface areas.

Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.

Improvement of the stability and accuracy of solid-wall immersed boundary schemes for the linearized Euler equations using boundary constraints

A Cartesian cut-cell method with explicit high-order finite-difference stencils for solid walls at internodal wall positions has been developed to solve the linearized Euler equations. For this, additional boundary constraints besides the physical impermeability condition have been incorporated into Hermite-based finite-difference stencils near the boundary. We demonstrate that our methodology and the ghost point formulations of Tam & Dong (1994) and Gloerfelt (2001) are equivalent when a wall intersects with a grid point. Other than Tam & Kurbatskii's (1997) extrapolation-based cut-cell treatment, our formulation can be regarded as a straightforward generalization of the above-mentioned ghost point formulations. A numerical reflection problem demonstrates improved accuracy in 1D for high-order schemes of sixth- and 12th-order. The scattering of a plane wave with 8 PPW for a 7-point-, and 5 PPW for an 11-point boundary scheme at a 2D-cylinder in fluid at rest demonstrates the extension of the proposed method to higher dimensions. For the 1D case, stability analysis has been carried out to show the significant impact of using boundary constraints to improve the numerical stability of a boundary scheme. In the multidimensional case, weak instabilities have to be suppressed by selective filtering. Our new formalism for boundary methods enables the assessment of the propagation characteristics of modified 1D-boundary stencils for first derivatives by Fourier analysis. It is shown that the incorporation of boundary constraints improves the modified wavenumber signature of the boundary stencils significantly.

An implicit Cartesian cut-cell method for incompressible viscous flows with complex geometries

A versatile conservative three-dimensional Cartesian cut-cell method for simulation of incompressible viscous flows over complex geometries is presented in this paper. The present method is based on the finite volume method on a non-uniform staggered grid together with a consistent mass and momentum flux computation. Contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which avoids numerical instability without any additional small cut-cell treatment. Strict conservation of the mass and momentum for both fluid and cut cells is enforced through the PISO algorithm for the pressure–velocity coupling. The versatility and robustness of the present cut-cell method are demonstrated by simulating various two- and three-dimensional canonical benchmarks (flow over a circular cylinder, airfoil, sphere, pipe, and heart sculpture) and the computed results agree well with previous experimental measurements and various numerical results obtained from the boundary-fitted, immersed boundary/interface, and other cut-cell methods, verifying the accuracy of the proposed method.

A conservative and consistent implicit Cartesian cut-cell method for moving geometries with reduced spurious pressure oscillations

A conservative and consistent three-dimensional Cartesian cut-cell method is presented for reducing the spurious pressure oscillations often observed in moving body simulations in sharp-interface Cartesian grid methods. By analysing the potential sources of the oscillation in the cut-cell framework, an improved moving body algorithm is proposed for the cut-cell method for the temporal discontinuity of the solid volume change. Strict conservation of mass and momentum for both fluid and cut cells is enforced through pressure-velocity coupling to reduce local mass conservation errors. A consistent mass and momentum flux computation is employed in the finite volume method. In contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which prevents numerical instability without any additional small cut-cell treatment. The effectiveness of the present cut-cell method for reducing spurious pressure oscillations is demonstrated by simulating various two- and three-dimensional benchmark cases (in-line and transversely oscillating cylinder, oscillating and free-falling sphere), with good agreement with previous experimental measurements and other numerical methods available in the literature.

A hybrid semi-Lagrangian cut cell method for advection-diffusion problems with Robin boundary conditions in moving domains

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L1, L2, and L∞ norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary.

A strong coupling prediction-correction immersed boundary method

In the traditional immersed boundary methods for solving compressible fluid-structure interaction problems, conservation is one of the problems that must be considered. When the coupling boundary moves on the fixed grid, the structure coverage will change, resulting in many dead elements and emerging elements on the fluid grid. In the ghost-cell immersed boundary method, the reconstructed grid can not maintain the strict mass conservation when the dead elements and emerging elements appear. In order to overcome the shortcomings of traditional methods, a strong coupling prediction-correction immersed boundary method for compressible fluid-structure interaction problems was proposed. Firstly, the matrix representation of a general fluid-structure coupling system was described, and a strong coupling Gauss-Seidel iterative scheme of fluid-structure coupling system was derived. Furthermore, a prediction-correction scheme was derived, and a prediction-correction immersed boundary method was proposed. The fluid-structure coupling boundary was regarded as a free surface, and the space originally occupied by the solid was initialized as zero mass elements, allowing the fluid to pass through the coupling boundary freely. For the calculation of fluid, the second-order MUSCL finite volume scheme with the minmod limiter and the AUSM+-up flux based on Zha-Bilgen splitting were used to advance the time step with the third-order Runge-Kutta scheme. In the correction step, the transport process was realized by a set of mass conservation transport rules. The transport algorithm could be summarized as marking the fluid inside the boundary, dividing and moving the fluid in a uniform way according to the marking order, generating a flow pointing to the outside of the boundary, and finally applying a velocity correction near the boundary to ensure the no-slip condition. The marking and transport algorithm avoided the tedious geometric treatment of the cut-cells, and ensured the easy implementation of the algorithm. For the calculation of solids, the first-order difference scheme and the implicit dynamic finite element scheme were used to solve the rigid body and linear elastic body respectively, and the Gauss quadrature was used to obtain the coupling force on the solid surface. The one-dimensional and two-dimensional problems were calculated by the prediction-correction immersed boundary method. In the one-dimensional piston problem, the accuracy, conservation and convergence of the method were investigated by comparing the results with those in the literature. In the two-dimensional shock wave impact problem, the experimental optical schlieren images were compared with those obtained by the numerical simulation, and the deflection history of the plate structure was investigated. The study shows that this method can accurately maintain the mass conservation of the flow field and has the advantage of easy implementation, which is different from the traditional ghost-cell method and the cut-cell method. This method has the first-order convergence accuracy, and can accurately predict the flow field after shock diffraction and the deflection of plate under shock waves. It provides a new idea for the development of fluid-structure coupling algorithms.

Topology optimization in fluid mechanics using continuous adjoint and the cut-cell method

A topology optimization method for steady-state flows of incompressible fluids which is capable of imposing accurate boundary conditions along the solid walls of the sought fluid paths is presented. In each topology optimization cycle, body-conforming Cartesian meshes are generated around shapes of any complexity by tracing the fluid-solid interfaces and a cut-cell flow solver is implemented together with its adjoint. Sensitivity derivatives are computed along the fluid-solid interfaces via the continuous adjoint method. Changes in topology are caused by expressing the computed sensitivity derivatives w.r.t. an auxiliary background material distribution, that helps updating the fluid-solid interfaces. The proposed method performance is assessed on three 2D benchmark examples and a 3D case. Two out of the three 2D examples are also solved using a porosity-based topology optimization approach in which impermeable regions are penalized by a Brinkman term and useful conclusions are drawn. For a fair comparison, designs optimized using the porosity-based method are re-evaluated after extracting fluid-solid interfaces from the computed porosity fields.