Full Approximation Storage Research Articles

Simulation of 3D Gas-Liquid Two-Phase Flow Characteristics of Carrousel Oxidation Ditch

In this paper, the gas–liquid two-phase mixture model with the k-ε turbulence model was used to numerically simulate the characteristics of an oxidation ditch. The proposed model concerns with the drag force and the drift velocity. The numerical method is based on a pressure-correction algorithm of the SIMPLE-type. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence, while the κ-ε model with wall functions is used. The numerical results for velocity and turbulent kinetic energy in the oxidation ditch are obtained．

3D Simulation of Gas–Liquid Two-Phase Flow in an Aeration Tank

In this paper, a numerical two-phase flow model combining with the Realizable k–ε turbulent model for compressible viscous fluid is presented for the computation of flow characteristics in an aeration tank; and the equations are solved with the finite volume method. The free fluid surface is simulated by the VOF method. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence. The numerical results for velocity and turbulent kinetic energy distribution in the aeration tank are obtained. It is shown that the Computational Fluid Dynamics (CFD) is a valuable tool to analyze the interaction of flow field and aeration.

Numerical Simulation of Hydraulic Characteristic of Oxidation Ditch

A numerical simulation study of the hydrodynamics of oxidation ditch is presented. The numerical method is based on a pressure-correction algorithm of the SIMPLE-type. A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence, while as a turbulence model the RNG κ-ε model with wall functions is used. The numerical results for velocity, as well as for the pressure distribution in the oxidation ditch are obtained．

Numerical Study of the Flow over a Two-Dimensional Car Model

A numerical study of the flow over a two-dimensional car model in proximity to the ground is presented. The numerical method is based on a pressure-correction algorithm of the SIMPLE-type．A multigrid technique based on the full approximation storage (FAS) scheme is employed to accelerate the numerical convergence, while as a turbulence model the standard κ-ε model with wall functions is used. The numerical results for the u and v velocity ,as well as for the pressure distribution over the car surface are obtained．

Multi-level adaptive solution of anisotropic porous media flow with strong discontinuous jumps in permeability

A fast and robust multi-grid algorithm for the efficient solution of diffusion-like, elliptic problems which exhibit strong discontinuous jumps in diffusivity is presented. Although generally applicable to this class of problem, the focus for illustrative purposes is that of porous media flow; in particular, such flows for which accurate solutions can only be achieved if the full permeability tensor is taken into consideration. The merits of adopting one or the other of two different approaches to deriving a discrete analogue to the steady-state Darcy equation, namely a novel weighted average of permeability formulation and a continuity of flux preservation method, are explored. In addition, automatic mesh refinement is incorporated seamlessly via a multi-level adaptive technique, making full use of the local truncation error estimates available from the inclusive full approximation storage scheme. Adaptive cell- and patch-wise mesh refinement strategies are developed and investigated for this purpose and used to solve a sequence of benchmark problems of increasing complexity. The results obtained reveal: (a) the ease with which the overall approach deals with generating accurate solutions for flows involving both distributed anisotropy and strong discontinuous jumps in permeability; (b) that both discrete analogues produce equivalent results in comparable execution times; and (c) the significant reductions in computing resource, memory, and CPU, to accrue from employing automatic adaptive mesh refinement.

Inertial two- and three-dimensional thin film flow over topography

The effect of inertia on gravity-driven thin film free-surface flow over substrates containing topography is considered. Flow is modelled using a depth-averaged form of the governing Navier–Stokes equations and the discrete analogue of the coupled equation set solved accurately using an efficient full approximation storage (FAS) and full multigrid (FMG) technique. The free-surface disturbance induced by topographic features is illustrated by considering examples of gravity-driven flow over and around peak, trench and occlusion topography. Results are presented which demonstrate how increasing Reynolds number can significantly enhance the magnitude of free-surface disturbances, a feature which may have important consequences for the wide range of coating process that aim to maximise free-surface planarity.

Analysis of Non-Darcy Models for Mixed Convection in a Porous Cavity Using a Multigrid Approach

This article investigates the performance of two models; namely the Brinkman-Forchheimer Darcy model (BFDM) and the Brinkman-extended Darcy model (BDM), in a problem involving mixed convection in a square cavity filled with a porous medium using the multigrid method. The left and right walls, moving in opposite directions, are maintained at different constant temperatures, while the top and bottom walls are thermally insulated. The transport equations were solved numerically by the finite-volume method on a colocated grid arrangement using a quadratic upwind interpolation for convective kinematics (QUICK) scheme. The influence of the key parameters, namely the Darcy number (Da) and Grashof number (Gr) on the flow and heat transfer pattern is examined. Further, the issue of reliability of the results is addressed. The results demonstrate that BDM over-predicts the momentum and heat transfer rates compared with BFDM, which is in conformity with the fact that the additional term present in the BFDM hinders convective effects. The full approximation storage (FAS) multigrid method achieves considerable acceleration of convergence for the present relatively unexplored problem.

Inertial thin film flow on planar surfaces featuring topography

A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over an inclined planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier–Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient full approximation storage (FAS) algorithm and a full multigrid (FMG) technique; together with error-controlled automatic adaptive time-stepping and proper treatment of the associated nonlinear convective terms. An extensive set of results is presented for flow over both one- and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. In the case of one-dimensional (spanwise) topography, moderate Reynolds numbers and shallow/short topographical features, the results obtained are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography, it is shown that the free-surface disturbance is influenced significantly by the presence of inertia leading, as in the case of spanwise topography, to an increase in the magnitude and severity of the resulting capillary ridge and trough formations: the effect of inclination angle and topography aspect ratio are similarly explored.

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier–Stokes equations

The two-dimensional steady incompressible Navier–Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge–Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.

Thin film flow over flexible membranes containing surface texturing: Bio-inspired solutions

Gravity-driven continuous thin liquid film flow over a flexible membrane containing surface topography is modelled using lubrication theory. The associated coupled non-linear equation set, for the film thickness, pressure and membrane deflection, is solved using a state-of-the-art full approximation storage multi-grid algorithm with automatic mesh refinement and adaptive time-stepping, in order to maximize computational efficiency when fine-scale resolution is required while ensuring accurate mesh-independent solutions at the micro-scale. The robustness of the approach is demonstrated through the solution of a series of problems and comparisons drawn with the same flow on an equivalent completely rigid membrane. It is shown that the former differs considerably from the latter in that the film thickness affects the shape of the flexible membrane, the compliance of which in turn impinges on the profile of the resulting free-surface disturbance.

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Here we discuss the efficient solution of the resulting second order nonlinear elliptic equation. The equation is discretized by the finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. We develop a special implementation, that allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here as an essential part of an accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid-adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.

A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids

The convergence of high-order methods, such as recently developed spectral difference (SD) method, can be accelerated using both implicit temporal advancement and a p-multigrid ( p = polynomial degree) approach. A p-multigrid method is investigated in this paper for solving SD formulations of the scalar wave and Euler equations on unstructured grids. A fast preconditioned lower–upper symmetric Gauss–Seidel (LU-SGS) relaxation method is implemented as an iterative smoother. Meanwhile, a Runge–Kutta explicit method is employed for comparison. The multigrid method considered here is nonlinear and utilizes full approximation storage (FAS) [Ta’asan S. Multigrid one-shot methods and design strategy, Von Karman Institute Lecture Note, 1997 [28]] scheme. For some p-multigrid calculations, blending implicit and explicit smoothers for different p-levels is also studied. The p-multigrid method is firstly validated by solving both linear and nonlinear 2D wave equations. Then the same idea is extended to 2D nonlinear Euler equations. Generally speaking, we are able to achieve speedups of up to two orders using the p-multigrid method with the implicit smoother.

Development and application of a parallel multigrid solver for the simulation of spreading droplets

AbstractWe consider the parallel application of an efficient solver developed for the accurate solution of a range of droplet spreading flows modelled as a coupled set of nonlinear lubrication equations. The underlying numerical scheme is based upon a second‐order finite difference discretization in space and a second‐order, fully implicit, adaptive scheme in time. At each time step, this leads to the need to solve a large system of nonlinear algebraic equations, for which the full approximation storage multigrid algorithm is employed. The motion of the contact line between the three phases (liquid, air and the solid substrate) is based upon the assumption of a thin precursor film, with a corresponding disjoining pressure term in the governing equations. It is the inclusion of this precursor film in the model that motivates the need for a parallel solution method. This is because the thickness of such a film must be very small in order to yield realistic predictions, while the finite difference grid must be correspondingly fine in order to obtain accurate numerical solutions. Results are presented which demonstrate that the parallel implementation is sufficiently efficient and robust to allow reliable numerical solutions to be obtained for a level of mesh resolution that is an order of magnitude finer than is possible using a single processor. Copyright © 2008 John Wiley & Sons, Ltd.

Thin film flow on surfaces containing arbitrary occlusions

The flow of a continuous liquid film, only several microns in thickness, on a plane surface containing occlusions, is modelled using lubrication theory. The resultant time-dependent non-linear equation set is discretised using both finite difference and finite element analogues; the former is solved using a very efficient full approximation storage ( FAS ) multigrid algorithm employing both automatic spatial and temporal adaptivity, the latter exploits the COMSOL suite of software to solve a weak form of the governing equations. The efficiencies and accuracies of both approaches are compared and a variety of problems, of increasing complexity, explored. Although the adaptive multigrid approach is clearly the most efficient, with the two sets of solutions indistinguishable, COMSOL offers an attractive alternative to the non-specialist user. The results, the first of their kind, further demonstrate: (i) the power of using adaptive multigriding, in time as well as space, for the simultaneous control of both spatial and temporal errors; (ii) the significant reduction in memory requirements and CPU time that can be achieved. It is revealed that occlusions lead to many of the features inherent in the flow of thin liquid films over fully submerged micro-scale topographic features; namely, the presence of capillary ridges linked to “bow-wave” plus “comet-tail” free-surface disturbances.

Adaptive FAS‐multigrid method for nonlinear elliptic equations on unstructured grids

AbstractA two‐step, node‐wise assembly procedure is developed to enhance the classical assembly process that is repeatedly required for the nonlinear finite elements method. Using the proposed assembly method, an efficient nonlinear point‐wise Gauss–Seidel method is designed and used as a smoother in a full approximation storage multigrid (FAS_MG) that results in solving nonlinear problems on unstructured grids as efficient as the linear ones. Next, the FAS_MG solver is incorporated into an adaptive full multigrid cycle that solves the nonlinear problem on a coarse grid and uses a posteriori error estimator to adaptively and automatically construct a next finer grid. Numerical results show that the proposed method solves adaptively nonlinear finite elements elliptic problems within the textbook MG efficiency. Copyright © 2007 John Wiley & Sons, Ltd.

The efficient and accurate solution of continuous thin film flow over surface patterning and past occlusions

AbstractThe benefits of a spatially and time‐adaptive full approximation storage (FAS) multigrid method for the solution of the thin film lubrication equations, as used to model gravity‐driven flow over complex submerged topography and past small‐scale occlusions, are investigated. Results are presented which demonstrate how flow orientation has a significant effect on the magnitude of the resultant free‐surface disturbances and how reduction of the in‐plane dimensions of the associated topography increases the computational resources needed to achieve grid‐independent solutions. In addition, the ability to automatically restrict fine grid resolution only to those regions in and around small, localized topography is shown to lead to significantly increased computational efficiency over grid‐independent solutions obtained on globally fine meshes with the same degree of accuracy. Copyright © 2007 John Wiley & Sons, Ltd.

A heterogeneous space–time full approximation storage multilevel method for molecular dynamics simulations

AbstractA heterogeneous space–time full approximation storage (HFAS) multilevel formulation for molecular dynamics simulations is developed. The method consists of a waveform Newton smoothing that produces initial space–time iterates and a coarse model correction. The formulation is coined as heterogeneous since it permits different interatomic potentials to be applied at different physical scales. This results in a flexible framework for physics coupling. Time integration is performed in windows using the implicit Newmark predictor–corrector method that permits larger time integration steps than the explicit method. The size of the time steps is governed by accuracy rather than by stability considerations of the algorithm. We study three different variants of the method: the Picard iteration, constrained dynamics and force splitting. Numerical examples show that FAS based on force splitting provides significant time savings compared to standard explicit methods and alternative implicit space–time schemes. Parallel studies of the Picard iteration on harmonic problems illustrate the time parallelization effect that leads to a superior parallel performance compared to explicit methods. Copyright © 2007 John Wiley & Sons, Ltd.

A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

AbstractThis paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.

An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features

Gravity-driven continuous thin film flow over a plane, containing well-defined single and grouped topographic features, is modelled as Stokes flow using lubrication theory. The associated time-dependent, nonlinear, coupled set of governing equations is solved using a full approximation storage (FAS) multigrid algorithm by employing automatic mesh adaptivity, the power, efficiency and accuracy of which is demonstrated by comparing the results with corresponding global fine-mesh solutions. These show that automatic grid refinement effectively restricts the use of fine grids to regions of rapid flow development which, for the topographies considered, includes the topography itself, the upstream Capillary ridge, downstream surge region, and the characteristic bow wave. It is shown that for the accurate solution of the associated lubrication equations, adaptive multigrid offers increased flexibility together with a significant reduction in memory requirement. This is further demonstrated by solving the problem of transient flow over a trench topography, generated by a sinusoidally varying inlet condition.

A modified full multigrid algorithm for the Navier–Stokes equations

A modified full multigrid (FMG) method for the solution of the Navier–Stokes equations is presented. The method proposed is based on a V-cycle omitting the restriction procedure for dependent variables but retaining it for the residuals. This modification avoids possible mismatches between the mass fluxes and the restricted velocities as well as the turbulent viscosity and the turbulence quantities on the coarse grid. In addition, the pressure on the coarse grid can be constructed in the same way as the velocities. These features simplify the multigrid strategy and corresponding programming efforts. This algorithm is applied to accelerate the convergence of the solution of the Navier–Stokes equations for both laminar and high-Reynolds number turbulent flows. Numerical simulations of academic and practical engineering problems show that the modified algorithm is much more efficient than the FMG–FAS (Full Approximation Storage) method.