Fractional Step Technique Research Articles

A conjugate gradient algorithm for solving the Galerkin-characteristic approximation of interfacial flows

We present a conjugate gradient algorithm for solving the Galerkin-characteristic approximation of interfacial flows. The governing equations are the incompressible Navier–Stokes for two fluids separated with an interface in the computational domain. We consider a level set method to track the interface in these equations. The method combines advantages of the semi-Lagrangian method to accurately solve the convection-dominated flow problems with a finite element method for space discretization of the governing equations. It can be interpreted as a fractional-step technique where the transport part and the Stokes part are treated separately. A limiting procedure is implemented for the reconstruction of numerical solutions at the departure points. The implementation of the proposed Galerkin-characteristic method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. Therefore, due to the Lagrangian treatment of convection, the standard Courant–Friedrichs–Levy condition is relaxed and the time truncation errors are reduced in the Stokes part. To solve the generalized Stokes problem we implement a conjugate gradient algorithm. This method avoids projection techniques and does not require any special correction for the pressure. The focus is on constructing efficient algorithms with a large stability region to solve interfacial flow problems. We verify the method for a passive transport of a slotted cylinder and for the benchmark problem of rising bubbles. We also present numerical results for a problem of barotropic flow in the Strait of Gibraltar. The conjugate gradient algorithm has been found to be feasible and satisfactory.

Cell-Centered Finite-Volume Time-Domain Method for Conducting Media

This letter describes a cell-centered finite-volume time-domain method for solving electromagnetic problems which involve media having finite conductivity. Two different approaches, the additive induced source technique and the fractional-step technique (FST), are considered and compared with a reliable numerical method. Namely, the former treats the induced electric current density as an additive source term, whereas the latter is based on the FST.

Numerical modeling of two-phase transonic flow

Our work is aimed at the development of numerical method for the modeling of transonic flow of wet steam including condensation/evaporation phase change. We solve a system of PDE’s consisting of Euler or Navier-Stokes equations for the mixture of vapor and liquid droplets and transport equations for the integral parameters describing the droplet size spectra. Numerical method is based on a fractional step technique due to the stiff character of source terms, i.e. we solve separately the set of homogenous PDE’s by the finite volume method and the remaining set of ODE’s either by explicit Runge-Kutta or implicit Euler method. The finite volume method is based on the Lax-Wendroff scheme with conservative artificial dissipation terms for structured grid. We also note result achieved by recently developed finite volume method with VFFC scheme. We discuss numerical results of steady and unsteady two-phase transonic flow in 2D nozzle, 2D and 3D turbine cascade and 2D turbine stage with moving rotor cascade.

MHD channel flow control in 2D: Mixing enhancement by boundary feedback

A nonlinear Lyapunov-based boundary feedback control law is proposed for mixing enhancement in a 2D magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow, which is electrically conducting, incompressible, and subject to an external transverse magnetic field. The MHD model is a combination of the Navier–Stokes PDE and the Magnetic Induction PDE, which is derived from the Maxwell equations. Pressure sensors, magnetic field sensors, and micro-jets embedded into the walls of the flow domain are employed for mixing enhancement feedback. The proposed control law, designed using passivity ideas, is optimal in the sense that it maximizes a measure related to mixing (which incorporates stretching and folding of material elements), while at the same time minimizing the control and sensing efforts. A DNS code is developed, based on a hybrid Fourier pseudospectral-finite difference discretization and the fractional step technique, to numerically assess the controller.

A Galerkin-Characteristic Method for Large-Eddy Simulation of Turbulent Flow and Heat Transfer

This work aims at the development of a nonoscillatory Galerkin-characteristic method for large-eddy simulation of turbulent flow and heat transfer. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization of the incompressible Navier–Stokes/Boussinesq equations in primitive variables. It can be interpreted as a fractional step technique where the convective part and the Stokes/Boussinesq part are treated separately. A limiting procedure is implemented for the reconstruction of numerical solutions at the departure points. The main feature of the proposed Galerkin-characteristic method is that, due to the Lagrangian treatment of convection, the standard Courant–Friedrichs–Levy condition is relaxed, and the time truncation errors are reduced in the Stokes/Boussinesq part. To solve the generalized Stokes/Boussinesq problem we implement a conjugate gradient algorithm. This method avoids projection techniques and does not require any special correction for the pressure. We verify the method for an advection-diffusion equation with a known analytical solution and for the benchmark problem of mixed convection flow in a squared cavity. We also present numerical results for a problem of heat transport in the Strait of Gibraltar. The Galerkin-characteristic method has been found to be feasible and satisfactory.

Numerical simulation of natural and mixed convection flows by Galerkin‐characteristic method

AbstractA numerical investigation is performed to study the solution of natural and mixed convection flows by Galerkin‐characteristic method. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization in primitive variables. It can be interpreted as a fractional step technique where convective part and Stokes/Boussinesq part are treated separately. The main feature of the proposed method is that, due to the Lagrangian treatment of convection, the Courant–Friedrichs–Lewy (CFL) restriction is relaxed and the time truncation errors are reduced in the Stokes/Boussinesq part. Numerical simulations are carried out for a natural convection in squared cavity and for a mixed convection flow past a circular cylinder. The computed results are compared with those obtained using other Eulerian‐based Galerkin finite element solvers, which are used for solving many convective flow models. The Galerkin‐characteristic method has been found to be feasible and satisfactory. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical solution of the shallow water equations with a fractional step method

A fractional step technique for the numerical solution of the shallow water equations is applied to study the evolution of the potential vorticity field. The height and velocity field of the shallow water equations are discretized on a fixed Eulerian grid and time-stepped with a fractional step method recently reported in [M. Shoucri, Comput. Phys. Comm. 164 (2004) 396; M. Shoucri, A. Qaddouri, M. Tanguay, J. Côté, A Fractional Steps Method for the Numerical Solution of the Shallow Water Equations, International Workshop on Solution of Partial Differential Equations, The Fields Institute, Toronto, August 2002], where the Riemann invariants of the equations are interpolated at each time step along the characteristics using a cubic spline interpolation. The potential vorticity, which develops steep gradients and evolve into thin filaments during the evolution, is nicely calculated at every time-step from the solution of the code. The method is efficient and has lower numerical diffusion than other methods, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem, and without the iteration associated with the intermediate step of solving a Helmholtz equation, usually associated with other methods like the semi-Lagrangian method. The absence of iterative steps in the present technique makes it very suitable for problems in which small time steps and grid sizes are required, as for instance in the present problem where steepness of the gradients and small scale structures are the main features of the potential vorticity, and more generally for problems of regional climate modeling. The simplicity of the method makes it very suitable for parallel computer.

A fractional step ELLAM approach to high-dimensional convection–diffusion problems with forward particle tracking

In this paper, a fractional step method combined with an Eulerian–Lagrangian localized adjoint method (ELLAM) is proposed to solve high-dimensional convection–diffusion problems. The method reduces high-dimensional problems to a series of uncoupled one-dimensional problems in each time step interval, in which one-dimensional ELLAM is used to solve the one-dimensional splitting equations. The approach takes the attractive advantages of the ELLAM method and the fractional step technique. It reduces computational complexities, large memory requirements, and long computation durations due to the application of the splitting technique. It reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation. It effectively eliminates non-physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way. Numerical experiments show the efficient performance of the approach.

A fractional step lattice Boltzmann method for simulating high Reynolds number flows

A fractional step lattice Boltzmann scheme is presented to greatly improve the stability of the lattice Boltzmann method (LBM) in modelling incompressible flows at high Reynolds number. This method combines the good features of the conventional LBM and the fractional step technique. Through the fractional step, the flow at an extreme case of infinite Reynolds number (inviscid flow) can be effectively simulated. In addition, the non-slip boundary condition can be directly implemented.

Mathematical model and simulation of gas flow through a porous medium in high breaking capacity fuses

We present a one-dimensional model to describe the gas flow and the heat transfer in high breaking capacity (HBC) fuses. The compressible Euler equations for perfect gas have been used as the basic flow model coupled with a porous medium model taking into account the mechanical interaction between the gas and the silica sand and the heat transfer between hot gas and cold silica sand. In addition, to describe the solid temperature evolution, we introduce the heat equation for the solid. The governing equations are discretized following a finite volume scheme coupled with a fractional step technique and the fluxes are evaluated using the Roe method. The computational fluid dynamics model is used for the numerical investigations of a gas-flow in a porous medium for HBC fuses.

Numerical Simulation of Darcy and Forchheimer Force Distribution in a HBC Fuse

We present the breaking of a short-circuit current in a HBC fuse simulation based on an isentropic non-stationary model in a porous medium for a one dimensional geometry. The fluid flow is affected by the nature of the gas and by the morphology of the silica sand. To model the gas–silica sand interaction, we introduce two classical laws: the Darcy's law due to the viscous interaction and the Forchheimer's law due to the inertial force. Numerical simulations with realistic physical parameters have been performed using a finite volume scheme with a fractional step technique. We show the evolution of Darcy and Forchheimer forces during time and according to the position in the fuse. We place in prominent position the fact that either force is predominant in the fuse according to the time and the position which justifies a numerical treatment to cover all the situations.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.

Fractional-Step Method Solution of Diffusive Wave Equation

Flood routing modeling of one-dimensional unsteady flow in open channels is sometimes based on the numerical solution of the diffusive wave equation. This paper presents a computational method for the solution of the diffusive wave problem with lateral inflow, based on the fractional-step technique. This method converts the diffusive wave problem into two single problems by utilizing separate equations for convection and diffusion. This separation is well adapted for computerization in distributed hydrological models. The applicability and the accuracy of this method are studied by mathematical analysis. Results show that this method provides an efficient and accurate resolution of the diffusive wave equation under some conditions on space and time steps and on spatial and temporal distribution of lateral inflow.

Linear elements for metal-forming problems within the flow approach

Simulation of transient and stationary metal-forming processes is presented in the context of a mixed velocity–pressure formulation with a rigid-viscoplastic material model. A fractional step technique is used to obtain in a natural way a discrete Laplacian in the pressure–pressure entries of the system matrix which, as is known, introduces the stabilizing effect and allows for the use of equal order interpolating functions for both discretized variables, in spite of the Babuska–Brezzi condition. The method is suitable for both implicit and explicit time integration schemes. In particular, the explicit version (forward Euler with mass lumping) is developed. The discretization is carried out for linear triangles and tetrahedra which most easily can make use of automatic mesh generators and result in a simple code. Attention is focused on the stabilizing effect as a function of material and numerical parameters. The technique is illustrated with extrusion and upsetting simulation and compared with a known pressure stabilization method.

Direct numerical simulation of turbulent flow over a wavy wall

The Navier–Stokes equations have been solved, by a pseudospectral method, for pressure-driven flows between a no-slip wavy wall and a slip flat wall. Periodic boundary conditions were used in the streamwise and spanwise directions. The physical domain is mapped into a computational domain that is a rectangular parallelepiped using a nonorthogonal transformation. The pseudospectral solution procedure employed in previous studies, for example, Lam and Banerjee [Phys. Fluids A 4, 306 (1992)], eliminated the pressure and solved for the wall–normal velocity and vorticity. The other velocity components were calculated using the definition of vorticity, and the continuity equation. This procedure leads to oscillations in the pressure field when solutions were attempted in the mapped computational domain. To overcome the problem, the procedure had to be modified and the pressure solved for directly using a fractional time step technique. For the cases examined here, these modifications resulted in spectral accuracy being maintained. Flow over sinusoidal wave trains has been simulated and the results compare well with available experiments. The simulations show significant effects of the wavy boundary on the mean flow and the turbulence statistics. The mean velocity profile differs substantially from the profile for the flat-wall case, particularly in the buffer region where the fluid is under the influence of both the wavy wall and the slip boundary. The velocity fluctuations in the streamwise direction decrease in the buffer region. This effect becomes more pronounced when the wave amplitude increases. Most of the redistribution of energy, from the streamwise direction to the spanwise and wall–normal directions, occurs in a thin layer close to the boundary, downstream of the wave troughs. The energy primarily redistributes into spanwise fluctuations. High shear stress regions form downstream of the wave troughs, and streaky structures and quasi-streamwise vortices are also seen to initiate in these regions. The length of the streaks, and the extent of the quasi-streamwise vortices, scale with wave length for the two cases investigated.

Multiscalar mapping closure for mixing in homogeneous turbulence

A new methodology to use mapping closures for the mixing of several scalars in homogeneous turbulence is explained. The main idea is the unidimensional mapping for each scalar, with the cross-dissipation handled by a joint reference field. A restricted and a general closure are described. A Monte Carlo code based on a fractional step technique has been developed. As an example, the segregated double-delta two-scalar mixing is analytically and numerically solved and predictions are shown.

Monte Carlo implementation of a single-scalar mapping closure for diffusion in the presence of chemical reaction

It is shown how to implement, via a Monte Carlo method, a single-scalar mapping closure for diffusion when chemical reactions or other processes are present. The mixing term in the one-point probability density function (pdf) equation of the scalar is closed via a mapping. The chemical term, closed in the pdf formulation, is treated exactly. To isolate each process, a fractional step technique is used. The methodology is the same if more processes are present. As an example, the pdf equation of one scalar reacting with itself in a statistically homogeneous forced turbulent flow is solved. Comparisons with computed direct numerical simulation data are excellent.

On solving the stiff ODE's of the kinetics of chemically reacting gas flow

We study the efficiency of computational methods for the stiff ordinary differential equations of chemical kinetics that arise when the partial differential equations of chemically reacting gas flow are treated by a fractional step technique. In this application, the overhead work associated with evaluating partial derivatives and decomposing matrices for the Newton-; like corrector iterations used in most algorithms for stiff ODE's can be eliminated for the most part by keeping in store a small number of suitably- chosen copies of the Jacobian matrix, reduced to Hessenberg form to facilitate changes of stepsize and order. Computational results in the case of ignition and propagation of a one-dimensional, premixed laminar flame with different realistic chemical kinetic models are presented to show the reduction of computational work obtained by modifying a modern general-purpose ODE-code in this manner.

Improved stability techniques for the solution of Navier-Stokes equations

When solving the Navier-Stokes equations for transient incompressible viscous flow problems, one normally encounters a decrease in numerical stability of the time integration algorithm with an increase in Reynolds number. This instability cannot be easily overcome due to the non-linearities present. The present paper, using the finite element method to integrate the equations in the spacial dimension, incorporates a time-staggered semi-implicit fractional step technique to improve stability at the higher Reynolds numbers. Unlike the upwind or directional differencing schemes normally employed to increase numerical stability, the present scheme does not introduce numerical damping or artificial viscocity, and becomes more stable as the Reynolds number increases. Results for this scheme are compared with various explicit integration schemes for the case of flow around a circular cylinder at Reynolds numbers of 100 to 400. For comparable accuracy the time-staggered semi-implicit fractional step technique was found to be up to 25 times more efficient than the other explicit integration schemes.