Sharp Interface Cartesian Grid Method Research Articles

A sharp-interface Cartesian grid method for time-domain acoustic scattering from complex geometries

This paper is concerned with the development of a Cartesian grid method for predicting the acoustic field scattered by complex geometries immersed in an infinite fluid. The sound wave in the fluid is formulated by the linearized Euler equations (LEEs), which are computed on a fixed Cartesian grid by using a fourth-order dispersion-relation-preserving (DRP) scheme. The solid object naturally cuts the underlying Cartesian grid with its geometrical boundary, which is modeled by the Lagrangian method. A sharp-interface Cartesian grid model based on the ghost-cell immersed boundary method is established to impose the non-penetrating conditions on the complex boundary of the solid. The values of the acoustic variables at the ghost points are constructed by a linear extrapolation in conjunction with a constrained moving least-squares (CMLS) interpolation method. The interpolation method is capable of eliminating the numerical instability encountered in the conventional moving least-squares (MLS) formulation. The perfectly matched layers are adopted to absorb the out-going sound waves without any reflections from the computational boundary to the interior domain. The proposed method is verified by several benchmark problems in computational aeroacoustics, including the initial value problem of a pressure perturbation scattered by a rigid cylinder, and Gaussian distributed acoustic sources scattered by a single cylinder or two cylinders. Application of the proposed method to the problem of acoustic waves scattered by solids with complex geometries is also presented to demonstrate the effectiveness and robustness of the method.

A sharp interface Cartesian grid method for viscous simulation of shocked particle-laden flows

ABSTRACTA Cartesian grid-based sharp interface method is presented for viscous simulations of shocked particle-laden flows. The moving solid–fluid interfaces are represented using level sets. A moving least-squares reconstruction is developed to apply the no-slip boundary condition at solid–fluid interfaces and to supply viscous stresses to the fluid. The algorithms developed in this paper are benchmarked against similarity solutions for the boundary layer over a fixed flat plate and against numerical solutions for moving interface problems such as shock-induced lift-off of a cylinder in a channel. The framework is extended to 3D and applied to calculate low Reynolds number steady supersonic flow over a sphere. Viscous simulation of the interaction of a particle cloud with an incident planar shock is demonstrated; the average drag on the particles and the vorticity field in the cloud are compared to the inviscid case to elucidate the effects of viscosity on momentum transfer between the particle and fluid phases. The methods developed will be useful for obtaining accurate momentum and heat transfer closure models for macro-scale shocked particulate flow applications such as blast waves and dust explosions.

A sharp-interface Cartesian grid method for viscoelastic fluid flow in complex geometry

Computational methods based on Cartesian mesh are popular in simulating fluid flow with moving boundaries of complex geometry. In this paper, a sharp-interface Cartesian grid method is proposed for simulating viscoelastic fluid flow. We implement a finite volume numerical scheme with an improved Rhie-Chow interpolation on the open-source toolbox OpenFOAM. In the benchmark test of flow past a stationary cylinder, the velocity for Newtonian fluid flow is found to be second-order accurate with linear/bi-linear fitting functions for local reconstruction and third-order accurate with quadratic fitting functions. Only first-order accuracy is achieved with current solver for the Oldroyd-B fluid flows due to the difficulty in handling the extra stress near the boundary. However, our sharp-interface Cartesian grid method has been verified to correctly predict the extra stress on the surface of the cylinder. Simulation results of confined Oldroyd-B fluid flow past a pair of cylinders are also reported. A comparison between the proposed sharp-interface Cartesian grid method and a smoothed-interface immersed boundary method is carried out with respect to accuracy and efficiency.

A three-dimensional sharp interface Cartesian grid method for solving high speed multi-material impact, penetration and fragmentation problems

This work presents a three-dimensional, Eulerian, sharp interface, Cartesian grid technique for simulating the response of elasto-plastic solid materials to hypervelocity impact, shocks and detonations. The mass, momentum and energy equations are solved along with evolution equations for deviatoric stress and plastic strain using a third-order finite difference scheme. Material deformation occurs with accompanying nonlinear stress wave propagation; in the Eulerian framework the boundaries of the deforming material are tracked in a sharp fashion using level-sets and the conditions on the immersed boundaries are applied by suitable modifications of a ghost fluid approach. The dilatational response of the material is modeled using the Mie-Gruneisen equation of state and the Johnson–Cook model is employed to characterize the material response due to rate-dependent plastic deformation. Details are provided on the treatment of the deviatoric stress ghost state so that physically correct boundary conditions can be applied at the material interfaces. An efficient parallel algorithm is used to handle computationally intensive three-dimensional problems. The results demonstrate the ability of the method to simulate high-speed impact, penetration and fragmentation phenomena in three dimensions.

Sharp interface immersed-boundary/level-set method for wave–body interactions

A sharp interface Cartesian grid method for the large-eddy simulation of two-phase turbulent flows interacting with moving bodies is presented. The overall approach uses a sharp interface immersed boundary formulation and a level-set/ghost–fluid method for solid–fluid and fluid–fluid interface treatments, respectively. A four-step fractional-step method is used for velocity–pressure coupling, and a Lagrangian dynamic Smagorinsky subgrid-scale model is adopted for large-eddy simulations. A simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed. A variety of test cases of different scales ranging from bubble dynamics, water entry and exit, landslide-generated waves, to ship hydrodynamics are performed for validation. Extensions for high Reynolds number ship flows using wall-layer models are also considered.

FLOW DYNAMIC COMPARISON BETWEEN RECESSED HINGE AND OPEN PIVOT BI-LEAFLET HEART VALVE DESIGNS

The flow dynamics through the peripheral and hinge regions of a bi-leaflet mechanical heart valve are complex and result in abnormally high shear stresses particularly during the closing phase of the valve function. It has been observed that, the late stages of closure is more significant in the dynamics of platelet activation; therefore, the later stages of closure is simulated by solving the two-dimensional Navier-Stokes equations using an Eulerian Levelset based sharp interface Cartesian grid method. Using a fixed Cartesian mesh incorporating local mesh refinement for solution accuracy and efficiency, the flow through and within a recessed hinge design and an open pivot hinge design is compared. Platelets are modelled as point particles by Lagrangian particle tracking algorithm with one way coupling. A dilute particle flow is assumed and particle-particle interactions are neglected. It was observed that the hinge region of the open pivot valve indicated a lower potential for activation of platelets compared to that in valves with a recessed hinge design.

Adaptively refined, parallelised sharp interface Cartesian grid method for three-dimensional moving boundary problems

Sharp interface Cartesian grid methods are capable of simulating complex moving boundary problems on fixed meshes while treating embedded interfaces accurately. This article further enhances the effectiveness of the sharp interface method by devising techniques for adaptive mesh resolution combined with parallel processing. These extensions enable dealing with problems involving disparate length scales encountered in many applications. A tree-based adaptive local mesh refinement scheme is developed to complement the sharp interface Cartesian grid method for efficient and optimised calculations. Detailed timing and accuracy data are presented for a variety of benchmark problems involving moving boundaries. Guidelines for selecting mesh refinement criteria for moving boundary calculations are developed. Issues associated with parallelisation of the overall framework are tackled. The capabilities of the method are demonstrated in a number of moving boundary problems, which require adequate resolution of a wide range of length scales and three-dimensional flows.

Sharp interface Cartesian grid method III: Solidification of pure materials and binary solutions

A numerical technique is presented for computing dendritic growth of crystals from pure melts and binary solutions. The governing equations are solved on a fixed Cartesian mesh and the immersed phase boundary is treated as a sharp solid–fluid interface. The interface is tracked using a level-set field. A finite-difference scheme is presented that incorporates the immersed phase boundary with only a small change to a standard Cartesian grid Poisson solver. The scheme is simple to implement in three-dimensions. The results from our calculations show excellent agreement with two-dimensional microscopic solvability theory for pure material solidification. It is shown that the method predicts dendrite tip characteristics in excellent agreement with the theory. The sharp interface treatment allows discontinuous material property variation at the solid–liquid interface. This facilitates sharp-interface simulations of dendritic solidification of binary solutions.

Sharp interface Cartesian grid method II: A technique for simulating droplet interactions with surfaces of arbitrary shape

A fixed-grid, sharp interface method is developed to simulate droplet impact and spreading on surfaces of arbitrary shape. A finite-difference technique is used to discretize the incompressible Navier–Stokes equations on a Cartesian grid. To compute flow around embedded solid boundaries, a previously developed sharp interface method for solid immersed boundaries is used. The ghost fluid method (GFM) is used for fluid–fluid interfaces. The model accounts for the effects of discontinuities such as density and viscosity jumps and singular sources such as surface tension in both bubble and droplet simulations. With a level-set representation of the propagating interface, large deformations of the boundary can be handled easily. The model successfully captures the essential features of interactions between fluid–fluid and solid–fluid phases during impact and spreading. Moving contact lines are modeled with contact angle hysteresis and contact line motion on non-planar surfaces is computed. Experimental observations and other simulation results are used to validate the calculations.

Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations

A Cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three-dimensional immersed boundaries. The embedded boundaries are represented using level-sets and treated in a sharp manner without the use of source terms to represent boundary effects. The narrow-band distance function field in the level-set boundary representation facilitates implementation of the finite-difference flow solver. The resulting algorithm is implemented in a straightforward manner in three-dimensions and retains global second-order accuracy. The accuracy of the finite-difference scheme is established and shown to be comparable to finite-volume schemes that are considerably more difficult to implement. Moving boundaries are handled naturally. The pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries. Benchmarking of the method is performed against available numerical as well as experimental results.

A particle-level set-based sharp interface cartesian grid method for impact, penetration, and void collapse

An Eulerian, sharp interface, Cartesian grid method is developed for the numerical simulation of the response of materials to impact, shocks and detonations. The mass, momentum, and energy equations are solved along with evolution equations for deviatoric stresses and equivalent plastic strain. These equations are cast in Eulerian conservation law form. The Mie–Grüneisen equation of state is used to obtain pressure and the material is modeled as a Johnson–Cook solid. The ENO scheme is employed to capture shocks in combination with a hybrid particle level set technique to evolve sharp immersed boundaries. The numerical technique is able to handle collisions between multiple materials and can accurately compute the dynamics of the immersed boundaries. Results of calculations for axisymmetric Taylor bar impact and penetration of a Tungsten rod into steel plate show good agreement with moving finite element solutions and experimental results. Qualitative agreement with theory is shown for the void collapse phenomenon in an impacted material containing a spherical void.

A Sharp Interface Cartesian Grid Method for Simulating Flows with Complex Moving Boundaries

A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian–Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier–Stokes equations are discretized using a second-order-accurate finite-volume technique, and a second-order- accurate fractional-step scheme is employed for time advancement. The fractional-step method and associated boundary conditions are formulated in a manner that properly accounts for the boundary motion. A unique problem with sharp interface methods is the temporal discretization of what are termed “freshly cleared” cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A simple and consistent remedy for this problem is also presented. The solution of the pressure Poisson equation is usually the most time-consuming step in a fractional step scheme and this is even more so for moving boundary problems where the flow domain changes constantly. A multigrid method is presented and is shown to accelerate the convergence significantly even in the presence of complex immersed boundaries. The methodology is validated by comparing it with experimental data on two cases: (1) the flow in a channel with a moving indentation on one wall and (2) vortex shedding from a cylinder oscillating in a uniform free-stream. Finally, the application of the current method to a more complicated moving boundary situation is also demonstrated by computing the flow inside a diaphragm-driven micropump with moving valves.