Fractional Step Technique Research Articles

Virtual body-fitted grid-based immersed boundary method for simulation of thermal flows with Dirichlet and Neumann boundary conditions

In this work, a virtual body-fitted grid-based immersed boundary method (IBM) combined with the thermal lattice Boltzmann flux solver (TLBFS) is proposed to efficiently simulate thermal flows with Dirichlet and Neumann boundary conditions. The method involves generating a virtual body-fitted grid along the immersed boundary and implementing the boundary conditions using a fractional step technique. In the prediction step, the finite-volume TLBFS is used to obtain the predicted flow field on the Eulerian mesh without considering the immersed boundary, while in the correction step, the correction process of the flow field is conducted on the virtual grid. Thus, the effect of the boundary is transferred to the fluid domain through the virtual grid indirectly. For Dirichlet boundary conditions, such as the no-slip condition and the specified temperature condition, the velocity and temperature corrections are considered unknowns and solved through a matrix system formed from enforcing the boundary conditions. As for the Neumann boundary condition, such as the specified heat flux condition, the temperatures at virtual layers inside the immersed body are regarded as unknowns and computed from those at virtual layers outside the wall according to the quadratic distribution. Several benchmark cases including natural, forced, and mixed convection problems are well simulated to validate the method. The numerical results are in good agreement with reference data, demonstrating the capacity of the present method to simulate thermal flows with Dirichlet and Neumann boundary conditions while requiring fewer Lagrangian points and achieving higher computational efficiency.

An immersed boundary velocity correction method combined with virtual body-fitted grid for simulation of incompressible flows

In this work, a virtual body-fitted grid is introduced into the velocity correction-based immersed boundary method (IBM) to simulate incompressible flows. The impact of the immersed boundary is indirectly transmitted to the flow field via a virtual body-fitted grid. In this method, the fractional step technique consisting of the predictor and the corrector is adopted. The prediction step is executed on the Eulerian mesh, and the correction step is done on the virtual grid to fulfill the no-slip boundary condition. After the correction step, the corrected velocity field on the virtual grid is then assigned to that on the Eulerian mesh to update the flow field. Being able to adjust the grid spacing flexibly, the virtual body-fitted grid alleviates the shortcomings of the conventional IBM that uses the smooth Dirac delta function to associate Lagrangian points with their surrounding Eulerian points. As a result, the present method is easy to apply to non-uniform Cartesian grids, which is inapplicable to the conventional IBM with the smooth Dirac delta function. Numerical experiments concerning flow past a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the present method, i.e., fewer Lagrangian points are required to avoid the streamline penetration of boundary and the range of “diffuse interface” can be narrowed by reducing the normal grid spacing of the virtual body-fitted grid to improve numerical results on a coarse mesh. In addition, an accuracy assessment on the decaying vortex problem reveals that the present IBM has a second-order accuracy.

Acoustic multipole source–simplified lattice Boltzmann method for simulating acoustic propagation problems

AbstractCombining the advantages of the acoustic multipole source (AMS) method and the simplified lattice Boltzmann method (SLBM), a method called AMS–SLBM is proposed for simulating the propagation of acoustic multipole sources in fluids. A particle source term is introduced to the right‐hand side of the lattice Boltzmann equation using the AMS method, and the macroscopic equations with source terms are derived via the Chapman–Enskog expansion analysis. Employing the fractional step technique, the solving process of the macroscopic equations can be divided into three steps: the predictor, corrector, and supplement steps. In the predictor and corrector steps, macroscopic equations without source terms are solved by SLBM, and in the supplement step, the time advancement of the source terms is solved using the finite difference method. AMS–SLBM uses SLBM to simulate the propagation of sound waves by directly evolving the macroscopic variables, which evades the evolution and storage of the distribution function, and the computational process is simpler and memory can be reduced compared to the standard LBM. Moreover, since the acoustic source term is introduced to the right‐hand side of the lattice Boltzmann equation by the AMS method, AMS–SLBM avoids the disadvantage that the traditional forced equilibrium distribution function method will interfere and cover the original flow field during the calculations. Several cases including the propagation of a plane wave, a Gaussian pulse and acoustic monopole, dipole, and quadrupole sources are simulated to validate the robustness and accuracy of the present method. The results show that AMS–SLBM can well simulate acoustic multipole sources propagation, and it affords second‐order accuracy.

A VMS–based fractional step technique for the compressible Navier–Stokes equations using conservative variables

In this paper we address the compressible Navier–Stokes equations written in the so-called conservative formulation. In particular, we focus on the possibility of uncoupling the computation of the problem unknowns, namely, density, linear momentum and total energy, a technique usually labeled as fractional step method, which allows to reduce the associated computational cost. The proposed methodology is a finite-element solver supplemented with a stabilization technique within the Variational Multi-Scale framework. In this regard, we consider orthogonal and dynamic definitions for the subscales. This discretization in space shows an adequate stability, permitting in particular the use of equal interpolation for all variables in play. However, we complement it with a shock-capturing operator in order to solve problems involving shocks. Several representative benchmark flow simulations are performed, which demonstrate the suitability of the proposed algorithm for a vast range of regimes.

A unified immersed boundary-lattice Boltzmann flux solver (UIB-LBFS) for simulation of flows past porous bodies

A novel numerical method named the unified immersed boundary-lattice Boltzmann flux solver (UIB-LBFS) for simulating incompressible flows past homogeneous porous bodies is proposed in this paper. A diffuse layer through which the porosity is smoothly changed is introduced. As a consequence, the governing equations in the porous domain and the pure-fluid domain can be unified. The solutions to each domain can be smoothly transitioned from one to the other through the diffuse layer around the domain interface. A fractional-step technique is employed to split the computational procedure into the predictor step and the corrector step, respectively. In the predictor step, an intermediate flow field is first predicted without considering the domain interface by the unified lattice Boltzmann flux solver. Then, the physical conditions at the fluid–porous interface are implemented through the immersed boundary method to correct the flow field in the corrector step. All the flow quantities are evaluated at the cell centers, while the viscous and the inviscid numerical fluxes are locally reconstructed at each cell interface simultaneously. Numerical validations are carried out, and excellent agreements between the present and published results are achieved. The accuracy and the reliability of the UIB-LBFS are thus proven.

Improving accuracy of the moving grid particle finite element method via a scheme based on Strang splitting

Particle finite element method (PFEM) is a computational tool suitable for simulating fluid dynamics problems characterized by presence of moving boundaries. In this paper a new version of the method for incompressible flow problems is proposed aiming at accuracy improvement. This goal is achieved essentially by applying Strang operator splitting to Navier–Stokes equations and selecting adequate integration schemes for the resulting advective and Stokes sub-problems. For achieving efficient implementation, the pressure and the velocity in the Stokes part are decoupled via the fractional step technique as in the classical PFEM. However, at the first fractional step an explicit pressure prediction procedure for alleviating mass losses is introduced. Three test cases are solved, validating the methodology and estimating its accuracy. The numerical evidence proves that the proposed scheme improves the accuracy of the PFEM.

Immersed boundary–simplified thermal lattice Boltzmann method for incompressible thermal flows

An immersed boundary–simplified lattice Boltzmann method (IB-STLBM) is proposed in this paper for the simulation of incompressible thermal flows with immersed objects. The fractional step technique is adopted to resolve the problem in two successive steps. In the predictor step, the simplified thermal lattice Boltzmann method (STLBM) is utilized to resolve the intermediate flow variables without considering the immersed objects. The STLBM is advantageous over the conventional thermal lattice Boltzmann method (TLBM) in memory cost, boundary treatment, and numerical stability. In the corrector step, the boundary condition-enforced immersed boundary method (IBM) is used to give correction values of velocity and temperature for accurate interpretation of the Dirichlet boundary conditions on the surface of the immersed objects. Based on the present IBM, some novel strategies can be applied in the evaluation of hydrodynamic forces or thermal parameters of the immersed objects. Five numerical examples are presented for comprehensive validation of the accuracy and robustness of IB-STLBM in various two- and three-dimensional thermal flow problems.

A diffuse‐interface immersed boundary method for simulation of compressible viscous flows with stationary and moving boundaries

SummaryIn this paper, a diffuse‐interface immersed boundary method (IBM) is proposed for simulation of compressible viscous flows with stationary and moving boundaries. In the method, the solution of flow field and the implementation of boundary conditions are decoupled into two steps by applying the fractional step technique, ie, the predictor step and the corrector step. Firstly, in the predictor step, the intermediate flow field is resolved by a recently developed gas kinetic flux solver (GKFS) without consideration of the solid boundary. The GKFS is a finite volume approach that solves the Navier‐Stokes equations for the flow variables at cell centers. In GKFS, the inviscid and viscous fluxes are evaluated as a single entity by reconstructing the local solution of continuous Boltzmann equation. Secondly, in the corrector step, the intermediate flow field is corrected by the present diffuse‐interface IBM. During this process, the velocity field is firstly corrected by the implicit boundary condition–enforced IBM so that the no‐slip boundary condition can be accurately satisfied. After that, the density correction is made by an iterative approach with the help of the continuity equation. Finally, the correction of the temperature field is made in the same way as that of the velocity field. Good agreements between the present simulations and the reference data in literature demonstrate the reliability of the proposed method.

Implicit heat flux correction-based immersed boundary-finite volume method for thermal flows with Neumann boundary conditions

In the frameworks of immersed boundary method (IBM) and finite volume method (FVM), an implicit heat flux correction-based IB-FVM is proposed for thermal flows with Neumann boundary conditions. With the use of a fractional-step technique, the preconditioned Navier–Stokes (N–S) equations are solved by the FVM to obtain the intermediate solution in the prediction step and the heat flux is corrected by enforcing the Neumann condition in the correction step. Different from existing IBMs, the cell face centers are defined as the Eulerian points due to the heat flux computation at each face in the FVM. The Neumann condition is implemented in such a way that the interpolated temperature gradient is equal to the specified boundary value at the same point when the corrected gradient field is interpolated from the face centers to the Lagrangian points. To achieve an implicit algorithm, the temperature derivative corrections at the Lagrangian points are set as unknowns and a system of algebraic equations is established by constructing hybrid thin-plate splines (TPS) interpolation/delta function distribution. In the derivative interpolation process, the much more accurate TPS is introduced because the use of cosine delta function yields a less accurate solution. After the distribution process, the heat flux correction of a fluid cell is evaluated by using the solved temperature derivative corrections at the face centers, but that of a solid cell is calculated by using their additive inverses to supplement the same amount of heat flux into the solid domain as that flowing into the fluid domain across the boundary. Finally, the heat flux of a cell is corrected by adding the correction to the intermediate value and the corrected heat flux is utilized to solve the N–S equations in the prediction step. As compared with the available implicit IBMs for Neumann conditions, the present method avoids the introduction of auxiliary layers of Lagrangian points as well as the approximate conversion from the Neumann to Dirichlet condition and thus is suitable for an arbitrary geometry. The proposed method is verified by simulating several benchmark thermal flows with Neumann conditions, the natural convection in an annulus and the steady or unsteady forced convection over a stationary or oscillating cylinder. All the computed results agree well with the literature data.

Immersed boundary-simplified lattice Boltzmann method for incompressible viscous flows

An immersed boundary-simplified lattice Boltzmann method is developed in this paper for simulations of two-dimensional incompressible viscous flows with immersed objects. Assisted by the fractional step technique, the problem is resolved in a predictor-corrector scheme. The predictor step solves the flow field without considering immersed objects, and the corrector step imposes the effect of immersed boundaries on the velocity field. Different from the previous immersed boundary-lattice Boltzmann method which adopts the standard lattice Boltzmann method (LBM) as the flow solver in the predictor step, a recently developed simplified lattice Boltzmann method (SLBM) is applied in the present method to evaluate intermediate flow variables. Compared to the standard LBM, SLBM requires lower virtual memories, facilitates the implementation of physical boundary conditions, and shows better numerical stability. The boundary condition-enforced immersed boundary method, which accurately ensures no-slip boundary conditions, is implemented as the boundary solver in the corrector step. Four typical numerical examples are presented to demonstrate the stability, the flexibility, and the accuracy of the present method.

A Semi-Explicit Multi-Step Method for Solving Incompressible Navier-Stokes Equations

The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predictor-corrector semi-explicit methods. In the present work, we derive a scheme based on using the fractional step technique in conjunction with explicit multi-step time integration within the framework of Galerkin-type stabilized finite element methods. We show that under certain assumptions, a Runge–Kutta scheme equipped with the fractional step leads to an efficient semi-explicit method, where the pressure Poisson equation is solved only once per time step. Thus, the computational cost of the implicit step of the scheme is minimized. The numerical example solved validates the resulting scheme and provides the insights regarding its accuracy and computational efficiency.

A simplified thermal lattice Boltzmann method without evolution of distribution functions

In this paper, a simplified thermal lattice Boltzmann method (STLBM) without evolution of the distribution functions is developed for simulating incompressible thermal flows. With the assistance of the fractional step technique, the macroscopic governing equations recovered from Chapman–Enskog (C–E) expansion analysis are resolved through a predictor–corrector scheme. Then in both the predictor and corrector steps, using the isentropic properties of lattice tensors and relationships of C–E analysis, the macroscopic flow variables are explicitly calculated from the equilibrium and non-equilibrium distribution functions. In STLBM, the equilibrium distribution functions are calculated from the macroscopic variables, while the non-equilibrium distribution functions are evaluated from the differences between two equilibrium distribution functions at different locations and time levels. Therefore, STLBM directly updates the macroscopic variables during the computational process, which lowers the virtual memory cost and facilitates the implementation of physical boundary conditions. Through von Neumann stability analysis, the present method is proven to be unconditionally stable, which is further validated by numerical tests. Three representative examples are presented to demonstrate the robustness of STLBM in practical simulations and its flexibility on different types of meshes and boundaries.

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

AbstractIn this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

A Robust Immersed Boundary-Lattice Boltzmann Method for Simulation of Fluid-Structure Interaction Problems

AbstractA robust immersed boundary-lattice Boltzmann method (IB-LBM) is proposed to simulate fluid-structure interaction (FSI) problems in this work. Compared with the conventional IB-LBM, the current method employs the fractional step technique to solve the lattice Boltzmann equation (LBE) with a forcing term. Consequently, the non-physical oscillation of body force calculation, which is frequently encountered in the traditional IB-LBM, is suppressed greatly. It is of importance for the simulation of FSI problems. In the meanwhile, the no-slip boundary condition is strictly satisfied by using the velocity correction scheme. Moreover, based on the relationship between the velocity correction and forcing term, the boundary force can be calculated accurately and easily. A few test cases are first performed to validate the current method. Subsequently, a series of FSI problems, including the vortex-induced vibration of a circular cylinder, an elastic filament flapping in the wake of a fixed cylinder and sedimentation of particles, are simulated. Based on the good agreement between the current results and those in the literature, it is demonstrated that the proposed IB-LBM has the capability to handle various FSI problems effectively.

A Spectral Multidomain Penalty Method Solver for the Simulation of the Velocity Attenuation in Hyporheic Flows

A hydrodynamic model, based on a spectral multidomain penalty method (SMPM), is proposed to study the dissipation of the velocity fluctuations at the hyporheic zone. The model is based on the one dimensional Navier Stokes equations, assuming incompressibility and a hydrostatic approximation. The spatial discretization of the governing equations is done via the SMPM that is a multidomain collocation approach based on discontinuous non-overlapping subdomains that are connected by a penalty term that ensures stability of the solution by imposing weak continuity at the subdomain interfaces. The temporal discretization of the equations is handled with a high-order fractional time step technique. A spectral filter is used to stabilize the solution when spurious oscillations appear in the solution. A sinusoidal pulse is imposed at the water/sediment interface as a boundary condition, and the flow dynamics was evaluated under different conditions of kinematic viscosity. A progressive damping of the velocity fluctuations was observed in all cases, until full dissipation is reached. At that point, the flow can be governed by Darcy's law.

Incompressible SPH Model for Simulating Violent Free-Surface Fluid Flows

Abstract In this paper the problem of transient gravitational wave propagation in a viscous incompressible fluid is considered, with a focus on flows with fast-moving free surfaces. The governing equations of the problem are solved by the smoothed particle hydrodynamics method (SPH). In order to impose the incompressibility constraint on the fluid motion, the so-called projection method is applied in which the discrete SPH equations are integrated in time by using a fractional-step technique. Numerical performance of the proposed model has been assessed by comparing its results with experimental data and with results obtained by a standard (weakly compressible) version of the SPH approach. For this purpose, a plane dam-break flow problem is simulated, in order to investigate the formation and propagation of a wave generated by a sudden collapse of a water column initially contained in a rectangular tank, as well as the impact of such a wave on a rigid vertical wall. The results of simulations show the evolution of the free surface of water, the variation of velocity and pressure fields in the fluid, and the time history of pressures exerted by an impacting wave on a wall.

Efficient Scheme for Chemical Flooding Simulation

In this paper, we investigate an efficient implicit scheme for the numerical simulation of chemical enhanced oil recovery technique for oil fields. For the sake of brevity, we only focus on flows with polymer to describe the physical and numerical models. In this framework, we consider a black oil model upgraded with the polymer modeling. We assume the polymer only transported in the water phase or adsorbed on the rock following a Langmuir isotherm. The polymer reduces the water phase mobility which can change drastically the behavior of water oil interfaces. Then, we propose a fractional step technique to resolve implicitly the system. The first step is devoted to the resolution of the black oil subsystem and the second to the polymer mass conservation. In such a way, jacobian matrices coming from the implicit formulation have a moderate size and preserve solvers efficiency. Nevertheless, the coupling between the black-oil subsystem and the polymer is not fully resolved. For efficiency and accuracy comparison, we propose an explicit scheme for the polymer for which large time step is prohibited due to its CFL (Courant-Friedrichs-Levy) criterion and consequently approximates accurately the coupling. Numerical experiments with polymer are simulated : a core flood, a 5-spot reservoir with surfactant and ions and a 3D real case. Comparisons are performed between the polymer explicit and implicit scheme. They prove that our polymer implicit scheme is efficient, robust and resolves accurately the coupling physics. The development and the simulations have been performed with the software PumaFlow [PumaFlow (2013) Reference manual, release V600, Beicip Franlab].

HYBRID FINITE DIFFERENCE/FINITE VOLUME METHOD FOR 3-D CONDUCTING MEDIA PROBLEMS

A hybrid time-domain method combing flnite-difierence and cell-centered flnite-volume method is presented in this paper. This method is applied to solve three dimensional electromagnetic problems which involve media having flnite conductivity. The fractional-step technique (FST) for FVTD scheme is applied to solve these problems. Local time-step scheme is used to enhance the e-ciency of this method. Numerical results are given and compared with a reliable numerical method, which is used to show the validation of this method.

A Review of Full Eulerian Methods for Fluid Structure Interaction Problems

We have recently developed a novel numerical method for fluid–solid and fluid–membrane interaction problems. The method is based on a finite difference fractional step technique, corresponding to a standard numerical approach for simulating incompressible fluid flows, and applicable to treating nonlinear constitutive laws of solid/membrane and large deformations. The temporal change of the solid deformation is described in the Eulerian frame by updating the advection equations for a left Cauchy-Green deformation tensor, which is used to express the constitutive equations for materials and membranes. This method is reviewed in detail with some numerical results.

An $L^2$-Projection for the Galerkin-Characteristic Solution of Incompressible Flows

This work is devoted to the development of an accurate Galerkin-characteristic method for simulation of incompressible viscous flows. The Galerkin-characteristic formulation is obtained by using a semi-Lagrangian discretization of the total derivative. The spatial discretization is performed using the finite element method on unstructured meshes. It can be interpreted as a fractional step technique where the convective part and the generalized Stokes part are treated separately. The semi-Lagrangian method requires high-order interpolating procedures for accuracy. A class of $L^2$-projection methods is developed and two interpolating schemes are considered by tracking the feet of the characteristic lines from the integration nodes. To solve the generalized Stokes problem we used a conjugate gradient algorithm. This method does not require any special correction for the pressure. We illustrate the performance of the proposed approach for an advection-diffusion equation with known analytical solution and for the benchmark problem of flow past a circular cylinder. We also present numerical results for a problem of incompressible buoyancy flow. The Galerkin-characteristic method using the $L^2$-projection has been found to be feasible and satisfactory.