Fractional Step Method Research Articles

Numerical Investigation of Vortex Shedding Modes of Flow past Two Circular Cylinders in Side-by-Side Using Immersed Boundary Method

The vortex shedding modes of flow past two circular cylinders in side-by-side arrangement are investigated numerically in this paper. The simulations are carried out using a ghost cell immersed boundary method which imposes the boundary condition through reconstruction of the local velocity field near the immersed boundary. The two-dimensional unsteady incompressible Navier-Stokes equations are solved using an implicit fractional step method based on cell-center, collocated arrangement of the primary variables. Vorticity contours of the flow around the cylinders and force time histories are presented. Anti-phase and in-phase vortex shedding modes were found to exist in the flow simulation. These results of simulations were in agreement with phenomena observed in experiment and numerical results of previous researchers.

Parallel numerical algorithm for simulation of a three-dimensional lid-driven cavity flow problem with the use of compact schemes

The parallel algorithm for simulation of the three-dimensional lid driven cavity flow problem with Reynolds numbers up to 10000 on a grid 128x128x128 was developed and implemented in the research. The problem is solved by numerical method where momentum equation is solved by a modified fractional step method with the use of compact scheme and pressure equation is solved by Fourier method in combination with the matrix sweep. The technique of the domain decomposition for parallelization of the proposed numerical method is described. In the process of computing and comparative experiment the dependency of the resulted acceleration and the coefficient of the calculation scalability on number of processors was found. Mathematics Subject Classication: 76F65

Ghost Cell Immersed Boundary Method for Simulating Flow over Two Circular Cylinders in Tandem Arrangement

In this work, a ghost cell immersed boundary method is applied to the numerical simulation of a uniform flows over a circular cylinder and two circular cylinders in tandem arrangement. The Navier-Stokes equations are solved using an implicit fractional step method employed on collocated arrangement variables. Immersed boundary method permit the use of structured Cartesian meshes to simulate flows involving complex boundaries. The shedding of vortices and flow interference between two circular cylinders in tandem arrangement are investigated numerically. The calculations are validated against the experimental and numerical results obtained by other researchers to prove the accuracy and effectiveness.

직교격자 기반 수치기법을 이용한 선박의 대변위 운동해석

In this study, a Cartesian-grid method based on finite volume approach is applied to simulate the ship motions in large amplitude waves. Fractional step method is applied for pressure-velocity coupling and TVD limiter is used to interpolate the cell face value for the discretization of convective term. Water, air, and solid phases are identified by using the concept of volume-fraction function for each phase. In order to capture the interface between air and water, the tangent of hyperbola for interface capturing (THINC) scheme is used with weighed line interface calculation (WLIC) method which considers multidimensional information. The volume fraction of solid body embedded in the Cartesian grid system is calculated using a level-set based algorithm, and the body boundary condition is imposed by a volume weighted formula. Numerical simulations for the two-dimensional barge type model and Wigley hull in linear waves have been carried out to validate the newly developed code. To demonstrate the applicability for highly nonlinear wave-body interactions such as green water on the deck, numerical analysis on the large-amplitude motion of S175 containership is conducted and all computational results are compared with experimental data.

Study on UV Curable Waterborne Polyurethane Modified by Epoxy Resin

In this paper, UV curable waterborne polyurethane emulsions were prepared by fractional step method using toluene diisocyanate (TDI), polyethylene glycol (PEG, MW=1000), α,α-dimethylol propionic acid (DMPA), 2-hydroxyethyl methacrylate (HEMA) as raw material, and UV-WPUs were modified with epoxy resin. The effect of different content of epoxy resin on the viscosity, the water resistance, and other properties was investigated. The synthesized polyurethane structure was confirmed by the Fourier transform infrared spectroscopy (FTIR). The FTIR analysis identified that the target product was expected epoxy modified waterborne UV polyurethane. And with –COOH content increasing, the emulsion viscosity of the WPU increased and the water absorption of the films increased; with epoxy content increasing, the viscosity of the WPU emulsion increased and the water resistance of the films was improved.

Validation process of ISIS CFD software for fire simulation

Fire propagation constitutes a major safety concern in nuclear facilities. In this context, IRSN is developing a CFD code, named ISIS, dedicated to fire simulations. This software is based on a coherent set of models that can be used to describe a fire in large, mechanically ventilated compartments. The system of balance equations obtained by combining these models is discretized in time using fractional step methods, including a pressure correction technique for solving hydrodynamic equations. Discretization in space combines two techniques, each proven in the relevant context: mixed finite elements for hydrodynamic equations and finite volumes for transport equations. ISIS is currently in an advanced stage of verification and validation. The results obtained for a full-scale fire test performed at IRSN are presented.

A low-dissipative, scale-selective discretization scheme for the Navier–Stokes equations

A new Scale-Selective Discretization (SSD) procedure for the Navier–Stokes equations is proposed. The aim is to reduce the numerical dissipation of already existing numerical schemes to make the SSD scheme easily implementable to the existing CFD codes. In particular, the new procedure is designed to decrease the dissipation errors arising from the discretization of the convection term using upwind-biased convection schemes. Such dissipative errors reduce the quality of high-fidelity simulation approaches in fluid dynamics such as Large-Eddy Simulations (LES). The new discretization procedure is based on separating small and large scales of the flow using a high-pass filter. As a first pre-processing step the convecting velocity field ui is decomposed into a rapidly fluctuating part ui′ using the high-pass filter and a smooth part ui-ui′. After this the derivatives involving ui-ui′ may be discretized with a centered scheme whereas the derivatives involving ui′ can be discretized using an upwind method. The new procedure is tested in Navier–Stokes simulations by implementing the method into a second order accurate incompressible finite volume code based on the fractional step method. The numerical tests on the 2D lid-driven cavity at laminar conditions Re=2500 imply that the new method clearly improves the quality of the simulations. At Re=10,000 the SSD scheme captures the post-critical state of the cavity flow. The advantages of the new method are quantitatively assessed by studying a 2D temporally evolving shear layer. The results imply that the SSD scheme significantly reduces the numerical diffusion in contrast to the conventional upwind-biased schemes. Results from marginally resolved turbulent channel flow at Reτ=590 imply that the new scheme can be used for 3D simulations.

Two‐phase dynamic analysis by material point method

SUMMARYThis paper extends the material point method to analyze coupled dynamic, two‐phase boundary‐valued problems via a velocity formulation, in which solid and fluid phase velocities are the variables. Key components of the proposed approach are the adoption of Verruijt's sequence of update steps when integrating over time and the enhancement of volumetric strains. The connection between fractional step method and the time‐stepping algorithm presented in this paper is addressed.Enhancement of volumetric strains allows lower order variations in pressure and mitigates spurious pressure fields and locking that plague low‐order finite‐element implementations.A stress averaging technique to smoothen stress variations is proposed, and the local damping procedure adopted by FLAC is extended to handle two‐phase problems. Special Kelvin‐Voigt boundaries are developed to suppress reflections at artificial boundaries. Idealized examples are presented to demonstrate the capability of the proposed framework to accurately capture the physics of wave propagation, consolidation and wave attack on a sea dike. Copyright © 2012 John Wiley & Sons, Ltd.

The second-order upwind finite difference fractional steps method for moving boundary value problem of nonlinear percolation coupled system

The second-order upwind finite difference fractional steps method for moving boundary value problem of nonlinear percolation coupled system

On the superiority of the mixed element free Galerkin method for solving the steady incompressible flow problems

Abstract The element free Galerkin (EFG) method is a promising method for solving flow problems, but it meets the difficulty of volumetric locking for solving the incompressible flow problems. In this paper, a mixed EFG method is proposed for solving the steady incompressible flow problems, which avoids the volumetric locking and inherits the meshfree properties. The method employs two sets of nodes, one for the velocity approximation and the other for the pressure approximation. Specially, the ratio between the velocity node number and the pressure node number is taken as the only indicator for the locking behavior of the mixed EFG method. And inf–sup tests are carried out to investigate the relationship between the ratio and locking behavior. By two numerical examples, the accuracy, rate of convergence and efficiency of the mixed EFG method are also carefully studied. The results show that the accuracy, convergence and efficiency of the mixed EFG method are superior to that of the time-related fractional step methods.

Large eddy simulations of turbulent Couette–Poiseuille and Couette flows inside a square duct

AbstractTurbulent Couette–Poiseuille and Couette flows at different mean strain rates (velocity ratio of Couette wall to bulk flow, $r= 0. 6$–$3. 15$), in a square duct at a bulk Reynolds number $\approx $10 000 are investigated by large eddy simulation. The numerical framework consists of a finite-volume method with a staggered-grid arrangement of dependent variables. Spatial derivatives are approximated using second-order centred differencing, and a fractional-step method is used for temporal integration. Simulations are conducted with $160\ensuremath{\times} 160\ensuremath{\times} 256$ grids. Secondary flow near the Couette wall shows a gradual change of vortex size and position as the moving wall velocity increased, where the two clockwise rotating vortices gradually merge in tandem with speed of the moving wall and form a large clockwise vortex. A linear relation is observed to exist between the angle of the two vortices and the parameter $r$, and the angle saturates beyond $r\ensuremath{\sim} 2. 06$. Also, at $0. 6\lt r\lt 1. 6$, together with a small counter-clockwise corner vortex, this vortex pattern is similar to that observed in the corner region of the duct flow with a free surface. The change of the vortex patterns also influences the dominant transport terms in the streamwise vorticity transport equation. Near the moving wall due to the reduction of the streamwise velocity fluctuation at the moving wall, turbulence structure gradually moves towards a rod-like axisymmetric turbulence, and as $r$ increases beyond $1. 2$, turbulence reverts to the disc-like structure.

Fixed Cartesian grid based numerical model for solidification process of semi-transparent materials I: Modelling and verification

An Eulerian–Lagrangian model was developed to numerically solve solidification process of semi-transparent, absorbing, emitting and anisotropically scattering materials. The model was based on the Fixed Cartesian Grid Front Tracking Algorithm combined with the Immersed Boundary Technique and for the first time was used for tracking highly curved, internal boundaries in participating medium with fluid flow taken into account. The convection in semitransparent melt was considered by applying the Fractional Step Method at a collocated grid. The Least Square Gradient Reconstruction Based Technique was used to find unknown temperature, velocities and pressure gradients at reshaped control volumes centres as well as to find unknown temperature, velocities and pressure values and gradients at control volume faces which were internally cut by the immersed boundary. The developed model enabled dealing accurately with different thermophysical and optical properties in the solid and the liquid phase of solidifying materials. The phenomena such as absorption, emission and diffuse or specular reflection of the thermal radiation at the opaque solid–liquid interface as well as transmission of the thermal radiation at the totally transparent solid–liquid interface were also introduced in proposed model. Several test cases, with fixed internal boundaries and with moving solid–liquid interface were solved to perform numerical verification of the model.

An artificial compressibility based fractional step method for solving time dependent incompressible flow equations. Temporal accuracy and similarity with a monolithic method

In this note, an artificial compressibility based fractional step method is analysed against a monolithic scheme for solving incompressible flow equations. The artificial compressibility (AC) procedure presented in this paper is stabilized via a characteristic based split (CBS), and thus it is referred to as the AC-CBS method. The monolithic method used for comparison in the present study is the pressure stabilized Petrov–Galerkin (PSPG) method. It is shown that the AC-CBS and PSPG procedures are identical in structure, except for the stabilization parameters. For unsteady problems, a dual time stepping algorithm is employed in the AC-CBS scheme. Unlike classical fractional step methods, this dual time stepping mechanism circumvents the temporal pressure splitting error, and thus provides the anticipated temporal accuracy. The temporal accuracy of the AC-CBS method is demonstrated via a standard benchmark problem. Up to fourth order time accurate schemes are introduced for a thorough analysis of the AC-CBS scheme.

Large Eddy Simulation of Gas-Liquid Two-Phase Flow for a Nested Type Fixed-Cone Valve

large eddy simulation cooperated with a physical fractional-step method is applied to simulate steady flow around a nested type fixed-cone valve; and the equations are solved with the finite volume method. The free fluid surface is simulated by the VOF method. The pressure contours and vorticity magnitude are obtained. The modeling results conform to physical law, and show that the large eddy simulation theory has powerful capacity in simulation of microstructures of turbulent flows, and the function of the nested type fixed-cone valve for energy dissipating is good.

비정수압 자유수면 모형의 3차원 점성 흐름에의 적용

본 연구에서는 3차원 점성 흐름에 적용될 수 있는 비정수압 자유수면 모형을 수평방향 직교 곡선좌표계에서 개발하였다. 개발된 수치모형은 엇갈린 격자를 사용함으로써 발생되는 자유수면에서의 경계조건 종결 문제를 수면층 방정식을 도입하여 해결하였으며, 난류의 유동 해석을 위한 폐합식으로 등방성의 k-<TEX>${\varepsilon}$</TEX> 난류모형을 이용하였다. 본 연구에서 운동량방정식은 이송-확산항만으로 중간단계의 유속을 예측하고, 압력 및 중력을 포함하는 생성항과 연속방정식을 결합하여 다음 시간단계의 유동장을 결정하는 계산 단계 분리법을 이용하였다. 수치모형의 적용성 평가를 위하여 폐쇄된 2차원 수조에서의 취송류, 급경사를 가지는 2차원 수로에서의 흐름, 원심력에 의한 이차류 흐름특성 분석을 위한 3차원 급변 만곡류에 대한 모의를 실시하였다. 수치모의 예측치는 수리모형 실험값과 수위, 유속, 난류특성 등에서 일치하는 양상을 보이는 것이 확인되었다. A horizontally curvilinear non-hydrostatic free surface model that was applicable to three-dimensional viscous flows was developed. The proposed model employed a top-layer equation to close kinematic free-surface boundary condition, and an isotropic k-<TEX>${\varepsilon}$</TEX> model to close turbulence viscosity in the Reynolds averaged Navier-Stokes equation. The model solved the governing equations with a fractional step method, which solved intermediate velocities in the advection-diffusion step, and corrects these provisional velocities by accounting for source terms including pressure gradient and gravity acceleration. Numerical applications were implemented to the wind-driven currents in a two-dimensional closed basin, the flow in a steep-sided trench, and the flow in a strongly-curved channel accounting for secondary current by the centrifugal force. Through the numerical simulations, the model showed its capability that were in good agreement with experimental data with respect to free surface elevation, velocity, and turbulence characteristics.

A simple and efficient direct forcing immersed boundary framework for fluid–structure interactions

A direct forcing immersed boundary framework is presented for the simple and efficient simulation of strongly coupled fluid–structure interactions. The immersed boundary method developed by Yang and Balaras [J. Yang, E. Balaras, An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys. 215 (1) (2006) 12–40] is greatly simplified by eliminating several complicated geometric procedures without sacrificing the overall accuracy. The fluid–structure coupling scheme of Yang et al. [J. Yang, S. Preidikman, E. Balaras, A strongly-coupled, embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies, J. Fluids Struct. 24 (2008) 167–182] is also significantly expedited by moving the fluid solver out of the predictor–corrector iterative loop without altering the strong coupling property. Central to these improvements are the reformulation of the field extension strategy and the evaluation of fluid force and moment exerted on the immersed bodies, by taking advantage of the direct forcing idea in a fractional-step method. Several cases with prescribed motions are examined first to validate the simplified field extension approach. Then, a variety of strongly coupled fluid–structure interaction problems, including vortex-induced vibrations of a circular cylinder, transverse and rotational galloping of rectangular bodies, and fluttering and tumbling of rectangular plates, are computed. The excellent agreement between the present results and the reference data from experiments and other simulations demonstrates the accuracy, simplicity, and efficiency of the new method and its applicability in a wide range of complicated fluid–structure interaction problems.

A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.

A time splitting algorithm for numerical solution of Richard’s equation

► Advection and diffusion forming the Richard’s equation are studied separately. ► A non-iterative numerical algorithm is proposed based on time splitting technique. ► A combined finite volume and characteristic methods are proposed for advection. ► To preserve the efficiency, an implicit procedure is proposed for characteristic part. ► A 4th order scheme is designed for diffusing without noticeable change in efficiency. The Richard’s equation mathematically expressing the infiltration problem is a transport type of equation describing two different phenomena namely advection and diffusion being hyperbolic and parabolic type of differential equation, respectively each having their own physical properties thus a single numerical scheme would not essentially be suitable for both. In this paper a numerical algorithm based on time splitting or fractional step method has been proposed and described. This has allowed designing and applying some different numerical schemes more compatible with the mathematical and physical properties of the corresponding phenomenon. In advection part an implicit characteristic based method has been employed to calculate the cell face fluxes and then the new unknowns have been obtained using finite volume method (FVM) explicitly. Two different speed type quantities namely wave celerity and mass velocity have been distinguished in the advection of infiltrated water to the soil, the magnitude of the first being several times larger than the other. The implicit characteristic based method has been designed specifically to cope with the mentioned high wave celerity. An implicit numerical scheme has been employed in which the space derivative term contributed in definition of cell face flux has been discretized in fourth order accurate manner. The extra cells getting involved due to the higher order discretization have been taken in account in a way not to change the tri-diagonal matrix coefficient shape preserving the efficiency of the algorithm. The performance, accuracy and efficiency of the out coming non-iterative numerical algorithm have been successfully examined by some test cases. The relative importance of the advection and diffusion terms in the concerned transport equation and the ration of their contribution in the overall infiltrated flux also have been discussed in detail.

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids. An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling. The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex. For the temporal integration of the momentum equations, an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term. The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM). The momentum interpolation is used to damp out the spurious pressure wiggles. The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure. The classic test cases, the lid-driven cavity flow, the skew cavity flow and the backward-facing step flow, show that numerical results are in good agreement with the published benchmark solutions.

SIMPLE Algorithm기반의 비압축성 Navier-Stokes Solver를 이용한 Immersed Boundary Method의 적용

Immersed boundary method(IBM) is a numerical scheme proposed to simulate flow field around complex objectives using simple Cartesian grid system. In the previous studies, the IBM has mostly been implemented to fractional step method based Navier-Stokes solvers. In this study, we implement the IBM to an incompressible Navier-Stokes solver which uses SIMPLE algorithm. The weight coefficients of the bi-linear and quadratic interpolation equations were formulated by using only geometric information of boundary to reconstruct velocities near IB. Flow around 2D circular cylinder at Re=40 and 100 was solved by using these formulations. It was found that the pressure buildup was not observed even when the bi-linear interpolation was adopted. The use of quadratic interpolation made the predicted aerodynamic forces in good agreement with those of previous studies. For an analysis of moving boundary, we smulated an oscillating circular cylinder with Re=100 and KC(Keulegan-Carpenter) number of 5. The predicted flow fields were compared with experimental data and they also showed good agreements.