Usability of explicit filtering in large eddy simulation with a low‐order numerical scheme and different subgrid‐scale models

Based on a priori tests, in large eddy simulation (LES) of turbulent fluid flow, the numerical error related to low‐order finite‐difference‐type methods can be large in comparison with the effect of subgrid‐scale (SGS) model. Explicit filtering has been suggested to reduce the error, and it has shown promising results in a priori studies and in some simulations with fourth‐order method. In this paper, the effect of explicit filtering on the total simulation error is studied together with a second‐order scheme, where the numerical error should be even larger. The fully developed turbulent channel flow between two parallel walls is used as a test case. Rather simple SGS models are applied, because these models are most likely used in practical applications of LES. Explicit filtering is here applied to the non‐linear convection term of the Navier–Stokes equations, four three‐dimensional filter functions are applied, and the effect of filtering is separated from the effect of SGS modelling. It is shown that the effect of filtering is rather large and smooth filters introduce an additional error component that increases the total simulation error. Finally, filtering via subfilter‐scale modelling is applied, and it is shown that this approach performs better. However, the large‐frequency components of the resolved flow field are not as effectively damped as when the non‐linear convection term is filtered. Copyright © 2007 John Wiley & Sons, Ltd.     

On the efficiency of linearization schemes and coupled multigrid methods in the simulation of a 3D flow around a cylinder

This paper studies the efficiency of two ways to treat the non‐linear convective term in the time‐dependent incompressible Navier–Stokes equations and of two multigrid approaches for solving the arising linear algebraic saddle point problems. The Navier–Stokes equations are discretized by a second‐order implicit time stepping scheme and by inf–sup stable, higher order finite elements in space. The numerical studies are performed at a 3D flow around a cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

A fast method for solving fluid–structure interaction problems numerically

The paper presents a semi‐implicit algorithm for solving an unsteady fluid–structure interaction problem. The algorithm for solving numerically the fluid–structure interaction problems was obtained by combining the backward Euler scheme with a semi‐implicit treatment of the convection term for the Navier–Stokes equations and an implicit centered scheme for the structure equations. The structure is governed either by the linear elasticity or by the non‐linear St Venant–Kirchhoff elasticity models. At each time step, the position of the interface is predicted in an explicit way. Then, an optimization problem must be solved, such that the continuity of the velocity as well as the continuity of the stress hold at the interface. During the Broyden, Fletcher, Goldforb, Shano (BFGS) iterations for solving the optimization problem, the fluid mesh does not move, which reduces the computational effort. The term ‘semi‐implicit’ used for the fully algorithm means that the interface position is computed explicitly, while the displacement of the structure, velocity and the pressure of the fluid are computed implicitly. Numerical results are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the time‐dependent Navier–Stokes equations

This article considers numerical implementation of the Crank–Nicolson/Adams–Bashforth scheme for the two‐dimensional non‐stationary Navier–Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of hybrid algorithms for the Navier–Stokes equations with respect to hydrodynamic stability theory

Hybrid three‐dimensional algorithms for the numerical integration of the incompressible Navier–Stokes equations are analyzed with respect to hydrodynamic stability in both linear and nonlinear fields. The computational schemes are mixed—spectral and finite differences—and are applied to the case of the channel flow driven by constant pressure gradient; time marching is handled with the fractional step method. Different formulations—fully explicit convective term, partially and fully implicit viscous term combined with uniform, stretched, staggered and non‐staggered meshes, x‐velocity splitted and non‐splitted in average and perturbation component – are analyzed by monitoring the evolution in time of both small and finite amplitude perturbations of the mean flow. The results in the linear field are compared with correspondent solutions of the Orr–Sommerfeld equation; in the nonlinear field, the comparison is made with results obtained by other authors. Copyright © 2002 John Wiley & Sons, Ltd.     

A finite volume method for viscous incompressible flows using a consistent flux reconstruction scheme

An incompressible Navier–Stokes solver using curvilinear body‐fitted collocated grid has been developed to solve unconfined flow past arbitrary two‐dimensional body geometries. In this solver, the full Navier–Stokes equations have been solved numerically in the physical plane itself without using any transformation to the computational plane. For the proper coupling of pressure and velocity field on collocated grid, a new scheme, designated ‘consistent flux reconstruction’ (CFR) scheme, has been developed. In this scheme, the cell face centre velocities are obtained explicitly by solving the momentum equations at the centre of the cell faces. The velocities at the cell centres are also updated explicitly by solving the momentum equations at the cell centres. By resorting to such a fully explicit treatment considerable simplification has been achieved compared to earlier approaches. In the present investigation the solver has been applied to unconfined flow past a square cylinder at zero and non‐zero incidence at low and moderate Reynolds numbers and reasonably good agreement has been obtained with results available from literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Comparison of Subgrid-scale Viscosity Models and Selective Filtering Strategy for Large-eddy Simulations

Explicitly filtered large-eddy simulations (LES), combining high-accuracy schemes with the use of a selective filtering without adding an explicit subgrid-scales (SGS) model, are carried out for the Taylor-Green-vortex and the supersonic-boundary-layer cases. First, the present approach is validated against direct numerical simulation (DNS) results. Subsequently, several SGS models are implemented in order to investigate if they can improve the initial filter-based methodology. It is shown that the most accurate results are obtained when the filtering is used alone as an implicit model, and for a minimal cost. Moreover, the tests for the Taylor-Green vortex indicate that the discretization error from the numerical methods, notably the dissipation error from the high-order filtering, can have a greater influence than the SGS models.     

Parallel solution of three‐dimensional Marangoni flow in liquid bridges

This paper describes the implementation and performances of a parallel solver for the direct numerical simulation of the three‐dimensional and time‐dependent Navier–Stokes equations on distributed‐memory, massively parallel computers. The feasibility of this approach to study Marangoni flow instability in half zone liquid bridges is examined. The results indicate that the incompressible, non‐linear Navier–Stokes problem, governing the Marangoni flows behavior, can effectively be parallelized on a distributed memory parallel machine by remapping the distributed data structure. The numerical code is based on a three‐dimensional Simplified Marker and Cell (SMAC) primitive variable method applied to a staggered finite difference grid. Using this method, the problem is split into two problems, one parabolic and the other elliptic A parallel algorithm, explicit in time, is utilized to solve the parabolic equations. A parallel multisplitting kernel is introduced for the solution of the pseudo pressure elliptic equation, representing the most time‐consuming part of the algorithm. A grid‐partition strategy is used in the parallel implementations of both the parabolic equations and the multisplitting elliptic kernel. A Message Passing Interface (MPI) is coded for the boundary conditions; this protocol is portable to different systems supporting this interface for interprocessor communications. Numerical experiments illustrate good numerical properties and parallel efficiency. In particular, good scalability on a large number of processors can be achieved as long as the granularity of the parallel application is not too small. However, increasing the number of processors, the Speed‐Up is ever smaller than the ideal linear Speed‐Up. The communication timings indicate that complex practical calculations, such as the solutions of the Navier–Stokes equations for the numerical simulation of the instability of Marangoni flows, can be expected to run on a massively parallel machine with good efficiency. Copyright © 1999 John Wiley & Sons, Ltd.     

Applications of the artificial compressibility method for turbulent open channel flows

A three‐dimensional (3‐D) numerical method for solving the Navier–Stokes equations with a standard k–ε turbulence model is presented. In order to couple pressure with velocity directly, the pressure is divided into hydrostatic and hydrodynamic parts and the artificial compressibility method (ACM) is employed for the hydrodynamic pressure. By introducing a pseudo‐time derivative of the hydrodynamic pressure into the continuity equation, the incompressible Navier–Stokes equations are changed from elliptic‐parabolic to hyperbolic‐parabolic equations. In this paper, a third‐order monotone upstream‐centred scheme for conservation laws (MUSCL) method is used for the hyperbolic equations. A system of discrete equations is solved implicitly using the lower–upper symmetric Gauss–Seidel (LU‐SGS) method. This newly developed numerical method is validated against experimental data with good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order upwind control‐volume method based on integrated RBFs for fluid‐flow problems

This paper is concerned with the development of a high‐order upwind conservative discretization method for the simulation of flows of a Newtonian fluid in two dimensions. The fluid‐flow domain is discretized using a Cartesian grid from which non‐overlapping rectangular control volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle‐point rule. One‐dimensional integrated radial basis function schemes using the multiquadric basis function are employed to represent the variations of the field variables along the grid lines. The convection term is effectively treated using an upwind scheme with the deferred‐correction strategy. Several highly non‐linear test problems governed by the Burgers and the Navier–Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well. Copyright © 2010 John Wiley & Sons, Ltd.     

An operator splitting numerical scheme for thermal/isothermal incompressible viscous flows

A numerical scheme for time‐dependent incompressible viscous fluid flow, thermally coupled under the Boussinesq approximation is presented. The scheme combines an operator splitting in the time discretization and linear finite elements in the space discretization, and is an extension of one previously applied for isothermal incompressible viscous flow governed by the Navier–Stokes equations. To show the efficiency of the scheme, numerical results are presented for mixed convection, and natural convection at high Rayleigh numbers. Restricting the scheme to the isothermal case, some numerical results at high Reynolds numbers are included, i.e. the scheme is tested for a small viscosity and a large force term, which are not trivial tasks to deal with. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite element thermally coupled flow formulation for phase‐change problems

A finite element, thermally coupled incompressible flow formulation considering phase‐change effects is presented. This formulation accounts for natural convection, temperature‐dependent material properties and isothermal and non‐isothermal phase‐change models. In this context, the full Navier–Stokes equations are solved using a generalized streamline operator (GSO) technique. The highly non‐linear phase‐change effects are treated with a temperature‐based algorithm, which provides stability and convergence of the numerical solution. The Boussinesq approximation is used in order to consider the temperature‐dependent density variation. Furthermore, the numerical solution of the coupled problem is approached with a staggered incremental‐iterative solution scheme, such that the convergence criteria are written in terms of the residual vectors. Finally, this formulation is used for the solutions of solidification and melting problems validating some numerical results with other existing solutions obtained with different methodologies. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element modified method of characteristics for the Navier–Stokes equations

An algorithm based on the finite element modified method of characteristics (FEMMC) is presented to solve convection–diffusion, Burgers and unsteady incompressible Navier–Stokes equations for laminar flow. Solutions for these progressively more involved problems are presented so as to give numerical evidence for the robustness, good error characteristics and accuracy of our method. To solve the Navier–Stokes equations, an approach that can be conceived as a fractional step method is used. The innovative first stage of our method is a backward search and interpolation at the foot of the characteristics, which we identify as the convective step. In this particular work, this step is followed by a conjugate gradient solution of the remaining Stokes problem. Numerical results are presented for:

• a Convection–diffusion equation. Gaussian hill in a uniform rotating field.
• b Burgers equations with viscosity.
• c Navier–Stokes solution of lid‐driven cavity flow at relatively high Reynolds numbers.
• d Navier–Stokes solution of flow around a circular cylinder at Re=100.

Copyright © 2000 John Wiley & Sons, Ltd.     

Fourth‐order finite difference scheme for the incompressible Navier–Stokes equations in a disk

We develop an efficient fourth‐order finite difference method for solving the incompressible Navier–Stokes equations in the vorticity‐stream function formulation on a disk. We use the fourth‐order Runge–Kutta method for the time integration and treat both the convection and diffusion terms explicitly. Using a uniform grid with shifting a half mesh away from the origin, we avoid placing the grid point directly at the origin; thus, no pole approximation is needed. Besides, on such grid, a fourth‐order fast direct method is used to solve the Poisson equation of the stream function. By Fourier filtering the vorticity in the azimuthal direction at each time stage, we are able to increase the time step to a reasonable size. The numerical results of the accuracy test and the simulation of a vortex dipole colliding with circular wall are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element and implicit Runge–Kutta implementation of an acoustics–convection upstream resolution algorithm for the time‐dependent two‐dimensional Euler equations

The second of a two‐paper series, this paper details a solver for the characteristics‐bias system from the acoustics–convection upstream resolution algorithm for the Euler and Navier–Stokes equations. An integral formulation leads to several surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi‐dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier–Stokes equations, an approximation that remains multi‐dimensional independently of the orientation of the reference axes and computational cells. The time‐dependent discrete equations are then integrated in time via an implicit Runge–Kutta procedure that in this paper is proven to remain absolutely non‐linearly stable for the spatially‐discrete Euler and Navier–Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics–convection upstream resolution algorithm generates essentially non‐oscillatory solutions for subsonic, transonic and supersonic flows, encompassing oblique‐ and interacting‐shock fields that converge within 40 time steps and reflect reference exact solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

Newton linearization of the incompressible Navier–Stokes equations

The present study aims to accelerate the convergence to incompressible Navier–Stokes solution. For the sake of computational efficiency, Newton linearization of equations is invoked on non‐staggered grids to shorten the sequence to the final solution of the non‐linear differential system of equations. For the sake of accuracy, the resulting convection–diffusion–reaction finite‐difference equation is solved line‐by‐line using the proposed nodally exact one‐dimensional scheme. The matrix size is reduced and, at the same time, the CPU time is considerably saved due to the decrease of stencil points. The effectiveness of the implemented Newton linearization is demonstrated through computational exercises. Copyright © 2004 John Wiley & Sons, Ltd.     

An L2‐stable approximation of the Navier–Stokes convection operator for low‐order non‐conforming finite elements

We develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low‐order non‐conforming finite element approximations (the so‐called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown to enjoy an L² stability property, which may be seen as a discrete counterpart of the kinetic energy conservation identity. In addition, numerical experiments confirm the robustness and the accuracy of this approximation; in particular, in L² norm, second‐order space convergence for the velocity and first‐order space convergence for the pressure are observed. Copyright © 2010 John Wiley & Sons, Ltd.     

Modelling and simulation of fires in vehicle tunnels

Applying a low‐Mach asymptotic for the compressible Navier–Stokes equations, we derive a new fluid dynamics model,which should be capable to model large temperature differences in combination with the low‐Mach number limit. The model is used to simulate fires in vehicle tunnels, where the standard Boussinesq‐approximation for the incompressible Navier–Stokes seems to be inappropriate due to the high temperatures developing in the tunnel. The model is implemented using a modified finite‐difference approach for the incompressible Navier–Stokes equations and tested in some realistic fire events. Copyright © 2004 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.