Fractional-step Projection Method Research Articles

DNS of Marangoni effects on a suspension of droplets in microgravity using the unstructured conservative level-set method

The multi-marker unstructured conservative level-set (UCLS) method for two-phase flow with variable surface tension is used for the Direct Numerical Simulation of thermocapillary-driven motion of a bi-dispersed suspension of droplets in microgravity. The so-called Marangoni stresses induced by surface tension gradients on the interface lead to a coupling of the momentum transport equation and the thermal energy transport equation. The finite-volume method discretizes the transport equations on three-dimensional collocated unstructured meshes. The UCLS method performs interface capturing with mass conservation of the fluid phases, whereas the multiple marker approach circumvents the numerical coalescence of fluid particles. The fractional-step projection method solves the pressure-velocity coupling. Unstructured flux-limiters schemes discretize the convective term in transport equations. Numerical and physical findings are reported.

Unstructured Conservative Level-Set (UCLS) method for reactive mass transfer in bubble swarms at high density ratio

This research presents a parallel Unstructured Conservative Level-Set (UCLS) method for reactive mass transfer in bubbles with a high-density ratio. This method uses the multiple-marker UCLS approach to circumvent the numerical coalescence of bubbles, combined with the finite-volume method to discretize transport equations on 3D collocated unstructured meshes. The fractional-step projection method solves the pressure-velocity coupling. The central difference scheme discretizes the diffusive term, whereas convective term within the momentum transport equation, level-set advection equations, and mass transfer equation are discretized by unstructured flux-limiters schemes. Such a combination of numerical techniques preserves the numerical stability in bubbly flows with a high Reynolds number and high-density ratio. Numerical and physical findings on the effect of physical properties ratios on the reactive mass transfer are reported, including their effect on the Reynolds number, interfacial area and Sherwood number.

Numerical Simulation and Characterization of Swirl Fluid Motion through Cylindrical Chambers

Axisymmetric solution of radial, axial and tangential components of velocity of flow through a cylindrical chamber with an inflow containing swirl and outflowis investigated numerically using Finite Volume Method. An in-house developed computer code for solving the three components of velocity and pressure is developed and validated using experimental data available from literature on the problem of axial vortex breakdown of confined flow in a cylinder due to rotation of one of the end walls. The code uses a staggered approach and fractional step projection method for decoupling the velocity and pressure. The convection and diffusion terms are approximated using a second-order accurate central difference scheme. This paper discusses the characteristics features of flow such as mixing streams, streams of fluid flowing close to the solid walls (cooling streams), swirl motion and development of flow structures.

Solid–liquid coupled material point method for simulation of ground collapse with fluidization

An improved version of the solid–liquid coupled material point method (MPM) is proposed to simulate ground collapses with fluidization involving transition processes from soil structures to flowing mixture. A water-saturated soil is assumed based on porous media theory, and the physical quantities of the soil and water phases are assigned to two separate sets of particles (material points). The main contribution of this study is the introduction of the fractional step projection method for the time discretization of the momentum equation of the water phase on the assumption of incompressibility. Thanks to this, the proposed solid–liquid coupled MPM is capable of suppressing the pressure oscillations caused by the weak incompressibility of water, which is commonly assumed in the previous studies, and of representing the wide range of behavior of the soil–water mixture at relatively low computational cost. Also, B-spline basis functions are utilized for the spatial discretization to suppress the cell-crossing errors caused by particles crossing element (cell) boundaries. Several numerical tests are conducted to examine the performance of the proposed method that inherits the beneficial features of MPM and demonstrate the capability of reproducing a model experiment of wave collision to sandpile that exhibits the water flow-induced fluidization process of soil involving scouring, transportation and sedimentation.

Numerical study of rising bubbles with path instability using conservative level-set and adaptive mesh refinement

This paper focuses on three-dimensional direct numerical simulations of rising bubbles in the wobbling regime, and the study of its dynamical behavior for Eötvös number 1 ≤ Eo ≤ 10 and Morton number 1e−11 ≤ M ≤ 1e−9. The computational methodology is based on a mass Conservative Level-Set method, whereas the spatial discretization of the computational domain employs an Adaptive Mesh Refinement strategy for the reduction of computational resources. The Navier–Stokes equations are discretized using the finite-volume approach on a collocated unstructured mesh; the pressure-velocity coupling is solved using a classical fractional-step projection method. This methodology is applied to a series of verification and validation tests, which are compared with experiments and numerical results from the literature. Finally, buoyancy bubbles rising in the wobbling regime are researched at moderate to high Reynolds numbers (100 < Re < 3000). Terminal Reynolds number, drag coefficient and frequency of path oscillations are compared with empirical correlations and numerical studies from the literature. Results show the discharge of alternate oppositely-oriented hairpin vortex structures. Moreover, depending on the characteristics numbers of the system, different path features, bubble shape, and vortical structures in the wake are reported.

An Euler–Lagrange model for simulating fine particle suspension in liquid flows

This article presents a two-way coupled Euler–Lagrange model to simulate the suspension of fine particles in liquid flows. The goal is to develop a three-dimensional numerical model that is capable of replicating the detailed features of particle-laden turbulent flow. A two-phase fractional-step projection method is developed to ensure mixture incompressibility by solving a modified Poisson equation for pressure, which in turn affects the particle motion through the pressure gradient. An efficient particle-moving algorithm that exchanges particles between Eulerian meshes is developed that automatically retains the necessary particle information at each time step and does not require any Lagrangian particle tracking. A soft-sphere particle collision model is employed to avoid excessive particle overlap and to achieve the random closed packing limit for depositing particles. Since particles are all inherently localized in Eulerian meshes, efficient particle searches can be achieved when calculating particle–particle collision. This model is then used to simulate the gravitational settling of particles, and the results confirm the effect of mixture incompressibility and demonstrate that the model is capable of reaching the random closed packing limit. Numerical examples for the flow problems of particle-induced stratification are conducted, and the model is able to reveal the detailed features of particle-laden flows. Sensitivity on the grid resolution and deviations from the existing single-phase model results are also discussed.

A multiple marker level-set method for simulation of deformable fluid particles

A novel multiple marker level-set method is introduced for Direct Numerical Simulation of deformable fluid particles (bubbles and droplets), which is integrated in a finite-volume framework on collocated unstructured grids. Each fluid particle is described by a separate level-set function, thus, different interfaces can be solved in the same control volume, avoiding artificial and potentially unphysical coalescence of fluid particles. Therefore, bubbles or droplets are able to approach each other closely, within the size of one grid cell, and can even collide. The proposed algorithm is developed in the context of the conservative level-set method, whereas, surface tension is modeled by the continuous surface force approach. The pressure–velocity coupling is solved by the fractional-step projection method. For validation of the proposed numerical method, the gravity-driven impact of a droplet on a liquid–liquid interface is studied; then, the binary droplet collision with bouncing outcome is examined, and finally, it is applied on simulation of gravity-driven bubbly flow in a vertical column.

Numerical analysis of the impact of two droplets with a liquid film using an incompressible SPH method

In this study, a truly incompressible smoothed particle hydrodynamics (SPH) algorithm combined with an effective surface tension model is extended to simulate the dynamic process of multiple droplets impacting on a liquid film in 2D and 3D. This approach uses a pressure Poisson equation to satisfy the incompressibility constraints, and the Navier–Stokes equations are solved in a Lagrangian form using a fractional-step projection method. The mathematical model is first validated by the simulations of several fluid impact phenomena in comparison with those obtained by other numerical methods. Then the interesting phenomena of two 2D droplets impacting successively on a rigid solid/liquid film are numerically predicted and compared with the corresponding experimental results. Next, the fluid mechanics of two 2D droplets impinging simultaneously on a thin liquid film are numerically investigated. The effects of the impact velocity and the two droplets’ horizontal spacing on the collision behavior are discussed in detail. Lastly, the splashing phenomenon of a 3D droplet impacting on a thin liquid film is simulated. All numerical results obtained are in agreement with the available data.

Spurious currents in finite element based level set methods for two‐phase flow

SUMMARYA study of spurious currents in continuous finite element based simulations of the incompressible Navier–Stokes equations for two‐phase flows is presented on the basis of computations on a circular drop in equilibrium. The conservative and the standard level set methods are used. It is shown that a sharp surface tension force, expressed as a line integral along the interface, can give rise to large spurious currents and oscillations in the pressure that do not decrease with mesh refinement. If instead a regularized surface tension representation is used, exact force balance at the interface is possible, both for a fully coupled discretization approach and for a fractional step projection method. However, the numerical curvature calculation introduces errors that cause spurious currents. Different ways to extend the curvature from the interface to the whole domain are discussed and investigated. The impact of using different finite element spaces and stabilization methods is also considered. Copyright © 2011 John Wiley &amp; Sons, Ltd.

An unstructured finite volume projection method for pulsatile flows through an asymmetric stenosis

An unstructured finite volume method coupled with a fractional-step projection method is developed for the numerical solution of the incompressible Navier–Stokes equations. Based on the use of the proposed method, numerical investigations are performed for a pulsatile flow through a stenotic channel. The geometry of the stenotic segment is assumed to be asymmetric, which can mimic the deformations in the diseased arteries. The numerical results reveal that for the pulsatile flow, the Womersley and Reynolds numbers have a great influence on the vortex generation and the wall shear stress distributions. The size and number of vortices depend largely on the Womersley number but less on the Reynolds.

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

AbstractA numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley &amp; Sons, Ltd.

A Cartesian ghost-cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost-cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional-step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier-Stokes equations. Point-wise Gauss-Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub-level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V-cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid-driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. © 2010 John Wiley & Sons, Ltd.

SVD–GFD scheme to simulate complex moving body problems in 3D space

The present paper presents a hybrid meshfree-and-Cartesian grid method for simulating moving body incompressible viscous flow problems in 3D space. The method combines the merits of cost-efficient and accurate conventional finite difference approximations on Cartesian grids with the geometric freedom of generalized finite difference (GFD) approximations on meshfree grids. Error minimization in GFD is carried out by singular value decomposition (SVD). The Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations on convecting nodes is integrated by a fractional-step projection method. The present hybrid grid method employs a relatively simple mode of nodal administration. Nevertheless, it has the geometrical flexibility of unstructured mesh-based finite-volume and finite element methods. Boundary conditions are precisely implemented on boundary nodes without interpolation. The present scheme is validated by a moving patch consistency test as well as against published results for 3D moving body problems. Finally, the method is applied on low-Reynolds number flapping wing applications, where large boundary motions are involved. The present study demonstrates the potential of the present hybrid meshfree-and-Cartesian grid scheme for solving complex moving body problems in 3D.

Numerical Investigation on Non-Newtonian Flows Through Double Constrictions by an Unstructured Finite Volume Method

This article presents a numerical investigation on a steady non-Newtonian flow through a two-dimensional channel with double constrictions. The power-law mode is employed in describing the non-Newtonian behavior of the flow. An unstructured finite volume method combined with a fractional-step projection method is developed for the discretization of incompressible equations governing the non-Newtonian flows. The important flow dynamics related with the arterial diseases, such as the wall shear stress and vortex generation, are also numerically studied in detail. Numerical results reveal that there are marked differences between Newtonian and non-Newtonian models.

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow (Power-Law model). The collocated (i.e. non-staggered) arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

Incompressible Turbulent Flow Simulation Using theκ-ɛModel and Upwind Schemes

In the computation of turbulent flows via turbulence modeling, the treatment of the convective terms is a key issue. In the present work, we present a numerical technique for simulating two-dimensional incompressible turbulent flows. In particular, the performance of the high Reynoldsκ-ɛmodel and a new high-order upwind scheme (adaptative QUICKEST by Kaibara et al. (2005)) is assessed for 2D confined and free-surface incompressible turbulent flows. The model equations are solved with the fractional-step projection method in primitive variables. Solutions are obtained by using an adaptation of the front tracking GENSMAC (Tomé and McKee (1994)) methodology for calculating fluid flows at high Reynolds numbers. The calculations are performed by using the 2D version of theFreeflowsimulation system (Castello et al. (2000)). A specific way of implementing wall functions is also tested and assessed. The numerical procedure is tested by solving three fluid flow problems, namely, turbulent flow over a backward-facing step, turbulent boundary layer over a flat plate under zero-pressure gradients, and a turbulent free jet impinging onto a flat surface. The numerical method is then applied to solve the flow of a horizontal jet penetrating a quiescent fluid from an entry port beneath the free surface.

Computation of free surface flows with a projection FEM in a moving mesh framework

A numerical model for the simulation of incompressible viscous free surface flows is presented. The equations governing the fluid flow are solved with the finite element fractional-step projection method. A moving mesh approach, similar to the arbitrary Lagrangian–Eulerian method is introduced to deal with the presence of a free surface. Validations are carried out with a two-dimensional sloshing problem with small deformations. Thus, results concerning, respectively, the damped oscillations of a gravity and a capillary wave are compared with the initial-value linear theory. Then, a three-dimensional computation concerning the behavior of a uniform flow field subjected on its free surface to an external applied traction is presented.

Incompressible Navier–Stokes solutions by a triangular spectral/ p element projection method

The application of the fractional step projection method recently proposed by Guermond and Quartapelle to the numerical approximation of unsteady Navier–Stokes solutions by means of a spectral/ p element method is considered. In particular we illustrate the second-order pressure correction technique and evaluate its accuracy properties in some test cases. Stability with respect to the compatibility condition between the approximation spaces for velocity and pressure is also addressed. The high (spectral) accuracy in space and the second-order accuracy in time are verified by two simple test cases with analytical solution. A more interesting problem is solved showing the ability of the method to produce very accurate results also for problems in complex geometries.

A Conservative Finite-Volume Second-Order-Accurate Projection Method on Hybrid Unstructured Grids

This paper describes a finite volume discretization method to compute steady, two-dimensional incompressible viscous recirculating flows using hybrid unstructured meshes, composed of triangles and quadrilaterals. However, the proposed formulation is not restricted to these topologies.The new method includes a second-order least squares scheme for convection discretization, and a fractional step projection method based on a staggered grid arrangement for pressure velocity coupling. Numerical results are reported to demonstrate the robustness, second-order accuracy, and flexibility of the proposed method.To the authors' knowledge, this paper represents the first general unstructured grid finite volume method to achieve full second-order accuracy for the steady incompressible 2D version of Navier–Stokes equations.

On the approximation of the unsteady Navier-Stokes equations by finite element projection methods

This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method.