An adaptative augmented Lagrangian method for three-dimensional multimaterial flows

The direct numerical simulation of incompressible multimaterial flows, based on predictor/corrector and volume of fluid (VOF) approaches is presented. An original adaptative augmented Lagrangian method is proposed to solve the predictor solution, satisfying at the same time the conservation equations as well as the incompressibility constraint. This algorithm is based on an Uzawa optimisation technique. The corrector solution is obtained with a projection method on a divergence free subspace. Several examples of two- and three-dimensional flows are proposed to illustrate the ability of the method to deal with unsteady, multimaterial problems.     

On energy conservation for finite element approximation of flow-induced airfoil vibrations

In this paper the numerical approximation of a two-dimensional fluid–structure interaction problem is addressed. The fully coupled formulation of incompressible viscous fluid flow interacting with a flexibly supported airfoil is considered. The flow is described by the incompressible system of Navier–Stokes equations, where large values of the Reynolds number are considered. The Navier–Stokes equations are spatially discretized by the finite element method and stabilized with a modification of the Galerkin Least Squares (GLS) method; cf. [T. Gelhard, G. Lube, M.A. Olshanskii, J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows, Journal of Computational and Applied Mathematics 177 (2005) 243–267]. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method and the stabilizing terms are modified in a consistent way with the ALE formulation.     

Numerical simulation of Newtonian and non-Newtonian flows in bypass

This paper deals with the numerical solution of Newtonian and non-Newtonian flows with biomedical applications. The flows are supposed to be laminar, viscous, incompressible and steady or unsteady with prescribed pressure variation at the outlet. The model used for non-Newtonian fluids is a variant of power law. Governing equations in this model are incompressible Navier–Stokes equations. For numerical solution we use artificial compressibility method with three stage Runge–Kutta method and finite volume method in cell centered formulation for discretization of space derivatives. The following cases of flows are solved: steady Newtonian and non-Newtonian flow through a bypass connected to main channel in 2D, steady Newtonian flow in angular bypass in 3D and unsteady non-Newtonian flow through bypass in 2D. Some 2D and 3D results that could have application in the area of biomedicine are presented.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.     

Lagrangian particle model for multiphase flows

A Lagrangian particle model for multiphase multicomponent fluid flow, based on smoothed particle hydrodynamics (SPH), was developed and used to simulate the flow of an emulsion consisting of bubbles of a non-wetting liquid surrounded by a wetting liquid. In SPH simulations, fluids are represented by sets of particles that are used as discretization points to solve the Navier-Stokes fluid dynamics equations. In the multiphase multicomponent SPH model, a modified van der Waals equation of state is used to close the system of flow equations. The combination of the momentum conservation equation with the van der Waals equation of state results in a particle equation of motion in which the total force acting on each particle consists of many-body repulsive and viscous forces, two-body (particle-particle) attractive forces, and body forces such as gravitational forces. Similar to molecular dynamics, for a given fluid component the combination of repulsive and attractive forces causes phase separation. The surface tension at liquid-liquid interfaces is imposed through component dependent attractive forces. The wetting behavior of the fluids is controlled by phase dependent attractive interactions between the fluid particles and stationary particles that represent the solid phase. The dynamics of fluids away from the interface is governed by purely hydrodynamic forces. Comparison with analytical solutions for static conditions and relatively simple flows demonstrates the accuracy of the SPH model.     

Current status of computational methods for transonic unsteady aerodynamics and aeroelastic applications

The current status of computational methods for unsteady aerodynamics and aeroelasticity is reviewed. The key features of challenging aeroelastic applications are discussed in terms of the flowfield state—low-angle high speed flows and high-angle vortex-dominated flows. The critical role played by viscous effects in determining aeroelastic stability for conditions of incipient flow separation is stressed. The need for a variety of flow modeling tools, from linear formulations to implementations of the Navier-Stokes equations, is emphasized. Estimates of computer run-times for flutter calculations using several computational methods are given. Applications of these methods for unsteady aerodynamic and transonic flutter calculations for airfoils, wings and configurations are summarized. Finally, recommendations are made concerning future research directions.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

A numerical method for the multiphase viscous flow equations

A numerical technique is developed for the solution of the equations that govern multiphase viscous flow. We demonstrate that the equations can be written as a coupled system of Partial Differential Equations (PDEs) comprising: (i) first order hyperbolic PDEs for the volume fraction of each phase; (ii) a generalised Stokes flow for the velocity of each phase; and (iii) elliptic PDEs for the concentration of nutrients and messengers. Furthermore, the computational domain may vary with time for some applications. Appropriate numerical methods are identified for each of these subsystems.The numerical technique developed is then demonstrated using two exemplar applications: tissue engineering; and avascular tumour development. This allows verification that the technique is appropriate for many features of multiphase viscous flow modelling.     

Parallel-multigrid computation of unsteady incompressible viscous flows using a matrix-free implicit method and high-resolution characteristics-based scheme

A three-dimensional parallel unstructured non-nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated. The finite-volume Navier–Stokes solver is based on the artificial compressibility approach with a high-resolution method of characteristics-based scheme for handling convection terms. The unsteady flow is calculated with a matrix-free implicit dual time stepping scheme. The parallelization of the multigrid solver is achieved by multigrid domain decomposition approach (MG-DD), using single program multiple data (SPMD) and multiple instruction multiple data (MIMD) programming paradigm. There are two parallelization strategies proposed in this work, first strategy is a one-level parallelization strategy using geometric domain decomposition technique alone, second strategy is a two-level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques. Message-passing interface (MPI) and OpenMP standard are used to communicate data between processors and decompose loop iterations arrays, respectively. The parallel-multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid-driven cavity flow. A maximum speedup of 22.5 could be achieved on 32 processors, for instance, the lid-driven cavity flow of Re = 1000. The results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements. A detailed study of the time step size and number of pseudo-sub-iterations per time step required for simulating unsteady flow are presented in this paper.     

Numerical simulation of steady compressible flows using a zonal formulation. Part II: viscous flows

Viscous flow simulations are usually based on the Navier–Stokes equations representing the balance of mass, momentum, and energy. For many high Reynolds number flows, the viscous effects are only limited to small regions in the neighborhood of solid surfaces and in the wakes. We present here a zonal formulation with a reduced system of equations in the outer inviscid flow region. As in part I of this work, the velocity components are calculated from a generalized form of the Cauchy–Riemann equations with non-vanishing vorticity only in the inner region, where the governing equations including the viscous terms are solved. The viscous effects are transferred to the Cauchy–Riemann equations through a forcing function (vorticity), and the process is repeated until convergence. Preliminary results are presented and compared to standard Navier–Stokes calculations for two and three dimensional flow problems.     

Numerical studies in high Reynolds number aerodynamics

Two numerical approaches are presented for the computation of viscous compressible flows at high Reynolds' numbers. In the first approach, named global approach, the whole flow field, which includes viscous and inviscid regions, is determined as the solution of a single set of equations, which may be the full Navier-Stokes equations, or some approximate form of these equations. The second approach, named coupling approach, consists in solving two different sets of equations in their respective domains simultaneously; one of the two sets governs an inviscid flow whose boundary conditions are provided by the viscous effects, determined by the other set.The discussion of the global approach is centred on two particular features of the finite-difference method used: a discretization technique, directly in the physical plane with arbitrary meshes: and a mesh adaptation technique, which combines a coordinate transformation to fit the mesh system to particular lines in the flow, and a technique of dichotomy for mesh refinement. Numerical results are presented for an axisymmetric compression corner and a shock-boundary layer interaction on a flat plate, both in supersonic regime, and for a transonic nozzle flow.For the coupling approach, emphasis is given firstly to the improvement resulting from an interacting analysis where the viscous and inviscid computations are matched, and not only patched. It is shown that the parabolic problems associated with simple viscous theories are always replaced by elliptic problems, even for supersonic flows, and that “supercritical interactions” or “critical points”, as defined by Crocco-Lees, are removed. Secondly, a new coupling method, fully automatized and capable of solving directly a well-posed problem for supersonic flow, is illustrated by examples involving shock wave-boundary layer interactions and reverse flow bubbles; they concern flows over symmetrical transonic airfoils and supersonic compression ramps.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Simulating flows with moving rigid boundary using immersed-boundary method

The present study is to apply the immersed-boundary method to simulate 2- and 3-D viscous incompressible flows interacting with moving solid boundaries. Previous studies indicated that for stationary-boundary problems, different treatments inside the solid body did not affect the external flow. However, the relationship between internal treatment of the solid body and external flow for moving-boundary problems was not studied extensively and is investigated here. This is achieved via direct-momentum forcing on a Cartesian grid by combining “solid-body forcing” at solid nodes and interpolation on neighboring fluid nodes. The influence of the solid body forcing within the solid nodes is first examined by computing flow induced by an oscillating cylinder in a stationary square domain, where significantly lower amplitude oscillations in computed lift and drag coefficients are obtained compared with those without solid-body-forcing strategy. Grid-function convergence tests also indicate second-order accuracy of this implementation with respect to the L¹ norm in time and the L² norm in space. Further test problems are simulated to examine the validity of the present technique: 2-D flows over an asymmetrically-placed cylinder in a channel, in-line oscillating cylinder in a fluid at rest, in-line oscillating cylinder in a free stream, two cylinders moving with respect to one another, and 3-D simulation of a sphere settling under gravity in a static fluid. All computed results are in generally good agreement with various experimental measurements and with previous numerical simulations. This indicates the capability of the present simple implementation in solving complex-geometry flow problems and the importance of solid body forcing in computing flows with moving solid objects.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

Numerical simulation of the transient flow behaviour in chemical reactors using a penalisation method

We present a numerical scheme to study the transient flow behaviour in complex geometries. The Navier–Stokes equations are discretized with a high-resolution Fourier pseudo-spectral discretization with adaptive time-stepping. Using a penalisation technique solid boundaries of arbitrary shape can be easily taken into account. As application we present different simulations of unsteady flows, typically encountered in chemical reactors. We study transitional flows in tube bundles (arrays of cylinders and squares) for different Reynolds numbers and angles of incidence and a channel flow with an obstacle.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

On modelling angular momentum and vorticity in compressible fluid flow

The conservation of angular momentum and the preservation of vorticity are examined in particle-in cell and finite difference solutions to the equations for viscous, compressible flow. Both methods are found to conserve angular momentum in the solution of the Lagrangian equations of motion to O(Δt²). In the modeling of convection, however, the finite difference method has computational diffusion that is absent in the particle-in-cell method. In numerical experiments, the effect of computational diffusion is shown to be greater as the number of grid points is decreased.     

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier-Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29] and [12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.