Fractional Step Method Research Articles

An octree structured finite volume based solver

The present work describes the development of a parallel distributed-memory implementation, of an octree data structure, linked to an adaptive cartesian mesh to solve the Navier–Stokes equations. The finite volume method was used in the spatial discretization where the advective and diffusive terms were approximated by the central differences method. The temporal discretization was accomplished using the Adams–Bashforth method. The velocity-pressure coupling is done using the fractional-step method of two steps. Moreover, all simulated results were obtained using a external solver for the Poisson equation, from the pressure correction, in the fractional step method. Results are presented both for adaptive octree mesh and for a mesh without refinement. These were determined in the verification and validation processes for the present computational code. Finally, we consider the simulations for the problems of a laminar jet and the lid-driven cavity flow. Numerical results are compared with numerical and experimental data.

A formulation for high-fidelity simulations of pool boiling in low gravity

Direct numerical simulation of pool boiling under varying gravity environments was performed and compared with experimental data. The three dimensional solver is fully explicit in time and uses a fractional step method for Navier-Stokes, energy and mass transfer equations. A level-set technique is used to represent and advect the interface. Sharp jumps are applied to the diffusion terms using a ghost-fluid method and the advection terms are discretized using a third order weighted essentially non-oscillatory scheme. A multi-level, block-structured adaptive mesh refinement is used to selectively refine the grid around the interface and thermal boundary layer. The latter contributes towards accurate calculation of evaporative heat flux. The pressure Poisson solver uses Hypre to solve the linear system on the coarsest level and geometric multi-grid method to prolong-restrict the solution between successive refinements. Parallelization is achieved using MPI on distributed memory systems and a dynamic contact line model is implemented to account for inertial effects on vapor bubbles due to wall adhesion force. Validation cases are presented to test the accuracy of the solver in comparison to analytical and experimental results. Replicability is also demonstrated by comparing results with already published data. A simple mathematical model is used to account for nucleation site density and results of pool boiling simulations are compared with experimental results to illustrate the effects of subcooling and gravity.

A 2D vertical model for simulating surface and subsurface flows using finite element–finite volume methods

AbstractA numerical model is presented for simulation of hydrodynamics of a 2D vertical free surface domain consisting of an arbitrary partitioned porous and non-porous area. To this end, modified Navier–Stokes equations are considered which could be applied in surface water and in subsurface flows, simultaneously. A wide range of Reynolds number has been considered, from which non-Darcy effects have also been taken into account. A fractional step method has been used in the time discretization procedure, where the convection and diffusion terms are separated from the pressure term while solving the momentum equations. To include the variation of surface elevation in computation, the domain has been divided into two parts, namely, ‘interior subdomain’, which never gets dry during the simulation period, covered by fixed unstructured triangular grids and ‘top layer’ with only a one layer structured grid, the position of which varies with the water surface. The validation of the proposed model has been achieved by comparison of its results with both theoretical and experimental data reported in the literature.

Comparison of efficacy of two treatment regimens under ureteroscope in patients with calculi in incarcerated upper ureteral duct

Objective To compare the efficacy of two treatment regimens under ureteroscope in the treatment of calculi in incarcerated upper ureteral duct. Methods A retrospective analysis on January 2016 to February 2017 in our hospital for treatment and diagnosis of 86 patients with calculi in incarcerated upper ureteral duct, 55 cases of men and women in 31 cases, aged 29-69 years, the average age was (38.5±7.4) years old. One step method of 43 cases of observation group, 43 patients accepted the fractional step method for the control group, two groups of patients in such aspects as age, basic situation, condition, there was no statistically significant difference(P>0.05). Postoperative urine bacterial culture and drug sensitivity test were conducted on 2 groups of patients, and regular reexamination was conducted. Two groups of operation time, surgical success rate, stone clearance rate, postoperative complications, postoperative fever rate to 38.5℃temperature acuity, 2 groups of patients in postoperative 1 day the reaction levels of inflammatory markers measurement and stress related factors were compared. Results The average operation time of observation group was(37.45±3.40) min, and the operation time of control group was (55.45±4.80) min, the time of observation group was obviously less than control group(P 0.05). Two groups of patients in postoperative body temperature of 38.5℃or higher, observation group occurred in 1 case, 12 cases of control group. There was no statistical significance in inflammatory factor, tumor necrosis factor-a(TNF-a) and interleukin-10(IL-10) before surgery(P>0.05), but in postoperative 1, 3, 5 d, observation group was lower than the control group(P 0.05), and 1, 3, 5 d after surgery, the observation group and control group decreased, but the observation group was lower than the control group, with statistical significance(P 0.05). Conclusions One-step lithotripsy can shorten the operation time, has a good surgical field in operation, it can effectively guarantee the operation quality and the operation time, the low incidence of postoperative patients with fever and inflammation, reduce the patient's medical expenses, the fractional step method has obvious application advantages. Key words: Ureteral Calculi; Ureteroscopy; Lithotripsy

A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows

An unstructured overset method capable of performing direct numerical simulation (DNS) and large eddy simulation (LES) of many (O(105)) moving bodies, utilizing many computational cores (O(105)), in turbulent, incompressible fluid flow is presented. Unstructured meshes are attached to bodies and placed within a fixed background domain. Body meshes are allowed to arbitrarily overlap and move throughout the domain. Within each mesh a high resolution, unstructured, non-dissipative finite volume method is used to solve for the flow field. Boundary conditions for each mesh are provided by interpolation from flow solutions on overlapping meshes. When many unstructured meshes of different resolution overlap, care is required in the connection between the different flow solutions. An interpolant is created which seeks to preserve volume conservation of flow quantities between meshes regardless of overlapping mesh differences. An implicit fractional step method is used for time advancement, requiring the calculation of a predicted fluid velocity and corrector pressure field. For the predictor step, the resulting interpolation is directly introduced into the implicit equations for the predicted flow field. For the corrected pressure field, the continuity between meshes is weakly enforced using a penalty formulation. The pressure formulation is symmetric, positive-definite and non-singular resulting in a formulation which is readily solvable using traditional iterative matrix inversion techniques. An Arbitrary Euler-Lagrangian (ALE) method coupled to a 6 degrees of freedom rigid body equation system (6-DOF) is used for body motion. For rotation, a quaternion representation is used to solve Euler's equations of rigid body motion. A linear spring damper model, which uses geometry information readily available from the overset assembly process, is used for collisions. Validation of the method for canonical flow fields is presented including assessment of order of accuracy and kinetic energy conservation properties. Particle-resolved direct numerical simulation (PR-DNS) of single particles in various flow fields are presented for validation. PR-DNS results of 50,000 spherical particles freely moving within turbulent channel flow are shown as a demonstration of the method at full scale. LES results of a marine propeller under crashback conditions are shown to demonstrate the ability to simulate highly unsteady turbulent flows over complex, moving geometries.

Variable spaced particle in mesh‐free method to handle wave‐floating body interactions

SummaryIn this work, the motion of a two‐dimensional rectangular freely floating body under waves is simulated using Improved Meshless Local Petrov‐Galerkin method with Rankine Source function (IMLPG_R) with variable spacing resolutions. The IMLPG_R method is a particle method that solves Navier–Stokes equations using the fractional step method to capture the wave properties. However, many existing particle methods are computationally intensive to model the wave‐floating body due to the requirement of fine particles, needing uniform distribution throughout the domain. To improve the computational efficiency and capture the body response properly, variable spaced particle distribution with fine resolution near the floating body and coarse resolution far from the body is implemented. Numerical schemes to handle variable resolutions are reported. An iterative scheme to handle the wave‐floating body is implemented in the particle method. Two test cases, one with small wave and another with steep waves, are simulated for uniform particle distribution and the result shows good agreement with literature. Based on this, the performance of the variable spaced particle distribution is tested in coupling with floating body solver. The application of the method for wave impact load from the green water loading of the floating structure is also simulated.

Optimization of Process Conditions for Hydrotreating Biodiesel from Gutter Oil

In this paper, the catalyst Ni-Pt/γ-Al2O3 was prepared by method of fractional steps and impregnation with gutter oil as raw material. Ni and Pt were selected as active components and γ-Al2O3 as carrier under 400°C setting conditions. The results showed that the reaction space velocity was 0.5 h−1, reaction pressure was 8.0 Mpa, hydrogen oil ratio was 1200:1, under the condition of above-mentioned facts, the pour point of biodiesel obtained from gutter oil hydrogenation was reduced by 9°C, cetane number was 57, yield was 74.03%. The yield of light component was significantly promoted, below 160°C fraction yield above 3.5%, the yield of diesel fractions could be reached 55.2%, the yield increased by 15%. This process greatly increased the additional value for the gutter oil and provided experimental basis for producing diesel fraction from gutter oil.

Relaxation and simulation of a barotropic three-phase flow model

We focus here on a three-phase flow model in order to represent complex flows involving liquid metal droplets, liquid water, and its vapour. The governing equations and its main properties are given, and focus is given on the pressure–velocity relaxation process on the one hand, and on the structure of solutions of the one-dimensional Riemann problem associated with pure convective effects. A fractional step method that computes successively the convective part and the relaxation effects is used to obtain approximate solutions on unstructured meshes. Details of algorithms are provided, and it is shown that the numerical method preserves positive values of statistical fractions and partial masses. Verification and validation test cases are presented, and some perspectives are eventually drawn.

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.

Simulation of a macrosegregation benchmark in a cylindrical coordinate system with a meshless method

The main objective of the present paper is to define a new benchmark test for macrosegregation in axisymmetry and to verify a novel meshless method on it. The test case represents a solidification of Al4.5 wt%Cu alloy in two different types of geometries, a solid and a hollow cylinder, cooled at the vertical boundaries. The volume averaging method is used to formulate the coupled mass, energy, momentum, and species transport equations for solid-liquid flow. The lever rule is used for determination of liquid and solid fraction.The meshless numerical approach, verified in this paper, is called the diffuse approximate method. The method is formed by using the weighted least squares approximation, where the second-order polynomial basis and Gaussians are used as trial and weight functions, respectively. The method is localised with the use of subdomains, each containing thirteen computational nodes. The explicit Euler scheme is used to perform the temporal integration. The fractional step method is used to couple the pressure-velocity fields. The stability of the method is attained by an adaptive shift of the computational node and Gaussian weight in the upstream direction.Results are presented for three geometrically different simulations. The results are compared with the classical finite volume method. All results show a very good agreement with the finite volume method. The simulations are performed on a uniform equidistant node arrangements of different node densities and the convergence of the node spacing is evaluated and compared. The results can also be used as a benchmark for other numerical methods.

Solution of three-dimensional temperature and turbulent velocity field in continuously cast steel billets with electromagnetic stirring by a meshless method

This paper demonstrates the development of meshless Local Radial Basis Function Collocation Method (LRBFCM) for solution of three dimensional (3D) turbulent molten steel flow and solidification under the influence of electromagnetic stirring (EMS) and its application to continuous casting process of steel billets. A mixture continuum approach is used to formulate the coupled set of macroscopic equations for mass, momentum, energy, turbulent kinetic energy, and dissipation rate in Cartesian coordinates. The mushy zone is treated as a Darcy porous media. The explicit Chorin fractional step method is used to resolve the pressure-velocity coupling. The LRBFCM uses explicit time discretization and spatial discretization with shape-adaptive multiquadrics radial basis functions collocation on non-uniform seven-noded influence domains, and displacement-adaptive upwind scheme. The Lorentz force due to EMS is provided by a one-way coupling by Elmer software. The advantages of the meshless method are trivial adaptation of the nodal densities, simple upgrade to 3D from previous two-dimensional models, and high accuracy. A study on the influence of changing the EMS parameters on the calculated temperature and velocity fields is performed. The paper confirms the ability of LRBFCM meshless solution of a realistic 3D multiphysics industrial problem with complicated swirling flow pattern.

A low-dissipation finite element scheme for scale resolving simulations of turbulent flows

The present work extends the conservative convective scheme proposed by Charnyi et al. (2017) [13], originally formulated for mixed finite elements and tested in laminar flows, to equal order finite elements. A non-incremental fractional-step method is used to stabilise pressure, allowing the use of finite element pairs that do not satisfy the inf-sup conditions, such as equal order interpolation for the velocity and pressure used in this work. The final scheme preserves momentum and angular momentum at the discrete level; the error in the conservation of kinetic energy introduced by this stabilisation is of O(δt,h2) in the case of linear finite elements. The low dissipation strategy is tested on a set of relevant turbulent cases. First, by using direct numerical simulation on the inviscid and viscous Taylor-Green vortex problem at Re=1600 and later, coupled with the Vreman (2004) [25] sub-grid stress model for performing large-eddy simulations on a turbulent channel flow at Reτ=950, the flow past a sphere at ReD=104 and the flow around an Ahmed body at ReH=2×105. In all cases the performance of the presented formulation is fairly good and it has been capable of reproducing the reference results with good accuracy.

Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations

This paper provides an unconditional optimal convergence of a fractional-step method for solving the Boussinesq equations. In this method, the convection is treated by the Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated by the projection method. There are lots of authors who worked on this method, and some authors gave the error estimate of this method. But, to our best knowledge, the error estimate for this method is under certain time-step restrictions. In this paper, we prove that the methods are stable almost unconditionally, i.e., when τ and h are smaller than a given constant. The basic idea of our analysis is splitting the error function into three terms, one term between the finite element solution and the projection, the other term between the projection and the time-discrete solution, the third term between the time-discrete solution and the exact solution, and giving the error estimates for each term respectively. Then, we obtain the optimal error estimates in L2 and H1-norm for the velocity and L2-norm for the pressure. In order to show the efficiency of our method, some numerical results are presented.

Numerical simulation of mixed convection in a square cavity using Direct Meshless Local Petrov-Galerkin (DMLPG) method.

This paper present the extension of the Direct Meshless Petrov-Galerkin (DMLPG) for solving mixed convection heat transfer in a square cavity for the flow up to Reynolds number up to Re 400 and Grashof number Gr = 100. The semi algebraic forms of the governing equations are solved using fractional step method. The numerical results of the DMLPG method are in good agreement with the conventional numerical results in the literature. Based on the numerical results it is shown that The DMLPG is very promising in dealing with the fluid flow and heat transfer problem.

High-Schmidt-number dissolved oxygen transfer from turbulent flows to permeable microbial sediment bed

We investigated the high-Schmidt-number mass transfer of dissolved oxygen (DO) from turbulent flows to a permeable microbial sediment bed by large eddy simulations over the Schmidt number range of 100 ≤ Sc ≤ 690. The three-dimensional open channel flows were numerically obtained using a fractional-step finite difference method for a friction Reynolds number of Reτ=180. The bottom sediment layer populated with DO-absorbing bacteria was described by a generalized Darcy–Brinkman–Forcheimer model coupled with a biogeochemical model. The dependence of the mass transfer on the intrinsic variable of the sediment bed (permeability, porosity, and microbial biomass) were analyzed using the statistical turbulent quantities including the mean velocity, mean concentration, turbulent intensity, and instantaneous flow and scalar characteristics, such as the fluctuating velocity and DO concentration fields, especially near the sediment-water interface. This study revealed that the role of turbulence in the DO transfer process increases with the Darcy number and with increasing porosities of the permeable sediment bed. Concerning DO transport from turbulent flows to a highly permeable sediment bed, the turbulent diffusion by large-scale motion is more important than viscous diffusion in the mass transfer process near the sediment/water interface. This process can be further accelerated with an increase in the turbulence intensity. An increase in the microbial biomass and a reduction in the Schmidt number leads to an enhanced sediment oxygen demand.

Modeling of the continuous casting process of steel via phase-field transition system. Fractional steps method

Here we consider the phase field transition system (a nonlinear system of parabolic type), introduced by G. Caginalp to distinguish between the phases of the material that is involved in the solidification process. On the basis of the convergence of an iterative scheme of fractional steps type, a conceptual numerical algorithm is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such approach is that the new method simplifies the numerical computations due to its decoupling feature. The finite element method (fem) in 2D is used to deduce the discrete equations and numerical results regarding the physical aspects of solidification process are reported. In order to refer the continuous casting process, the adequate boundary conditions was considered.

Efficiency of Two Approaches to Computing the Flow around an Airfoil with Flaps in the Case of Flow Separation

Model equations describing convective transport are used to analyze the approximation errors of an explicit numerical scheme and various implicit schemes with the same approximation of spatial derivatives. It is shown that, under time step constraints determined by the Courant–Friedrichs–Lewy condition, the implicit scheme is inferior in accuracy to the explicit one and, with a further increase in the time step, the accuracy of simulated convective processes degrades substantially. Two methods for implementing the marching procedure in time are considered, namely, a fractional step in the case of an explicit scheme and a dual step in the case of an implicit scheme. It is shown that the fractional step method is efficient only on grids with a scatter of cell sizes of 100–1000. For the numerical solution of problems with no-slip conditions on solid walls (scatter of cell sizes of 104–105), two approaches are proposed: an implicit scheme with a dual step in all cells and an zonal approach, in which a dual step is used in a thin near-wall domain (about 3% of the thickness of a developed turbulent boundary layer), while a fractional step is applied in the rest of the domain. These two approaches are used to compute the flow over an airfoil with flaps. Numerical and experimental data are compared. The accuracy of the numerical results is estimated. The causes of error formation are examined. The domain of efficient application is determined for each of the indicated approaches.

2차원 비압축성 점성 열유동 해석을 위한 WMLS 기반 무격자 해법의 개발

In this study, a newly-developed Eulerian gridless solver for simulating two-dimensional unsteady incompressible viscous thermal flows in low Reynolds number is introduced. The Boussinesq approximation is applied to deal with heat transfer and spatial derivatives of governing equations are solved by WMLS(Weighted Moving Least Square) interpolations on a cloud of points. A fractional time step method is adopted and the Poission equation for pressure are solved by successively in a WMLS sense. Some heat transfer problems are solved by the developed solver for the validation and the results are compared with analytic solution and those from other CFD simulations.

Numerical investigation of flow and combined natural-forced convection from an isothermal square cylinder in cross flow

Unsteady flow and heat transfer from a horizontal isothermal square cylinder is studied numerically using a three-dimensional computational model to investigate the influence of buoyancy on the forced flow and heat transfer characteristics. The numerical model is based on a horizontal square cylinder subjected to laminar fluid flow in an unconfined channel. The governing equations in 3D form are solved using a fractional step method based on the finite difference discretization in addition to a Crank–Nicholson scheme employed to the convective and the viscous terms. Two working fluids–air (Pr = 0.7) and water (Pr = 7)–are considered, and the flow and heat transfer simulations were carried out for the Reynolds and Richardson numbers in the intervals 55 ≤ Re ≤ 250 and 0 ≤ Ri ≤ 2, respectively. The flow characteristics such as time-averaged drag/lift, rms drag/rms lift coefficients as well as Strouhal number were computed. The heat transfer from the cylinder is assessed by mean Nusselt number (and rms Nusselt number) over the total heated cylinder walls. As the buoyancy increases, the mass and the velocity of the fluid flowing underneath the cylinder increases. The fluid is injected into the near wake region with an upward motion which significantly alters the flow field in the downstream as well as upstream regions. The effects of Reynolds, Richardson and Prandtl numbers on the flow field and temperature distributions are discussed in detail. It is shown that the flow and heat transfer characteristics are influenced more for air than water. To fill the void in the literature, useful empirical correlations of practical importance are derived for pure forced and pure natural as well as mixed convection. The mixed convection correlations, in terms of the ratio of pure forced convection, are also developed, and their implications are discussed.

An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown.