Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Unstable Displacement of Non-aqueous Phase Liquids with Surfactant and Polymer

In this paper, we study two-phase multicomponent displacement of two immiscible fluids in both homogeneous and heterogeneous porous media. In many applications such as enhanced oil recovery, fluid mixing and spreading can be detrimental to the efficacy of the process. Here, we show that when an initially immobile phase is being displaced by a finite-size slug of solvents (surfactant and polymer), viscous fingering significantly enhances mixing and spreading of solvents. These effects are similar to those caused by medium heterogeneity and lead to poor displacement efficiency. We first quantify the displacement efficiency subject to different mobility ratios, Peclet numbers, and levels of medium heterogeneity. We observe a non-monotonic behavior in displacement efficiency as a function of mobility ratio, indicating that although stable frontal interface is desirable, miscible viscous fingering on the rear interface will eventually disintegrate the solvents slugs and reduce the displacement efficiency. Then, we show that miscible viscous fingering developing on the rear interface of the chemical slug could be greatly suppressed when viscosity contrast is gradually decreased using exponential or linear functions, leading to 10% increase in displacement efficiency while using the same amount of chemicals. To elucidate this low displacement efficiency, we study the evolution of mixing, spreading, and interfacial length and show that while higher viscosity ratios are quite effective in mobilizing the initially immobile phase in 1D displacements, they are in fact detrimental in 2D unstable displacements since they enhance mixing and spreading of solvents.

     

Higher‐order‐accurate numerical method for temporal stability simulations of Rayleigh‐Bénard‐Poiseuille flows

For Rayleigh‐Bénard‐Poiseuille flows, thermal stratification resulting from a wall‐normal temperature gradient together with an opposing gravitational field can lead to buoyancy‐driven instability. Moreover, for sufficiently large Reynolds numbers, viscosity‐driven instability can occur. Two higher‐order‐accurate methods based on the full and linearized Navier‐Stokes equations were developed for investigating the temporal stability of such flows. The new methods employ a spectral discretization in the homogeneous directions. In the wall‐normal direction, the convective and viscous terms are discretized with fifth‐order‐accurate biased and fourth‐order‐accurate central compact finite differences. A fourth‐order‐accurate explicit Runge‐Kutta method is employed for time integration. To validate the methods, the primary instability was investigated for different combinations of the Reynolds and Rayleigh number. The results from these primary stability investigations are consistent with linear stability theory results from the literature with respect to both the onset of the instability and the dependence of the temporal growth rate on the wave angle. For the cases with buoyancy‐driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers. The largest growth rates are obtained for a wave angle of 90°. For the cases with viscosity‐driven instability, the linear growth rates are lower and the first mode to experience nonlinear growth is a higher harmonic with half the wavelength of the fundamental.     

Two‐dimensional implicit time‐dependent calculations on adaptive unstructured meshes with time evolving boundaries

An implicit multigrid‐driven algorithm for two‐dimensional incompressible laminar viscous flows has been coupled with a solution adaptation method and a mesh movement method for boundary movement. Time‐dependent calculations are performed implicitly by regarding each time step as a steady‐state problem in pseudo‐time. The method of artificial compressibility is used to solve the flow equations. The solution mesh adaptation method performs local mesh refinement using an incremental Delaunay algorithm and mesh coarsening by means of edge collapse. Mesh movement is achieved by modeling the computational domain as an elastic solid and solving the equilibrium equations for the stress field. The solution adaptation method has been validated by comparison with experimental results and other computational results for low Reynolds number flow over a shedding circular cylinder. Preliminary validation of the mesh movement method has been demonstrated by a comparison with experimental results of an oscillating airfoil and with computational results for an oscillating cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit pseudo‐spectral multi‐domain method for the simulation of incompressible flows

A pseudo‐spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi‐domain approach and it is able to cope with non‐constant fluid properties and non‐orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi‐implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure–correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite‐volume discretizations. Copyright © 2003 John Wiley & Sons, Ltd.     

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional transient Navier–Stokes solvers in cylindrical coordinate system based on a spectral collocation method using explicit treatment of the pressure

A spectral collocation method is developed for solving the three‐dimensional transient Navier–Stokes equations in cylindrical coordinate system. The Chebyshev–Fourier spectral collocation method is used for spatial approximation. A second‐order semi‐implicit scheme with explicit treatment of the pressure and implicit treatment of the viscous term is used for the time discretization. The pressure Poisson equation enforces the incompressibility constraint for the velocity field, and the pressure is solved through the pressure Poisson equation with a Neumann boundary condition. We demonstrate by numerical results that this scheme is stable under the standard Courant–Friedrichs–Lewy (CFL) condition, and is second‐order accurate in time for the velocity, pressure, and divergence. Further, we develop three accurate, stable, and efficient solvers based on this algorithm by selecting different collocation points in r‐, ? ‐, and z‐directions. Additionally, we compare two sets of collocation points used to avoid the axis, and the numerical results indicate that using the Chebyshev Gauss–Radau points in radial direction to avoid the axis is more practical for solving our problem, and its main advantage is to save the CPU time compared with using the Chebyshev Gauss–Lobatto points in radial direction to avoid the axis. Copyright © 2010 John Wiley & Sons, Ltd.     

Improving the CBS‐based partitioned semi‐implicit coupling algorithm for fluid‐structure interaction

We detail in this work 2 simple but effective alternatives to improve the characteristic‐based split–based partitioned semi‐implicit coupling algorithm for fluid‐structure interaction. The basic idea lies in introducing the end‐of‐step velocity into the implicit stages of the 2 algorithms integrating different splits. The algorithm built upon the second‐order pressure split is further stabilized via the pressure gradient projection with particular emphasis on the extremely low mass ratio. The smoothed finite element method is exploited for spatial discretization of fluid and solid equations. Even without any accelerators, both the semi‐implicit solvers incorporating fixed‐point iterations engender visible improvements versus the previously published data for several benchmarks.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

A boundary element method for the solution of finite mobility ratio immiscible displacement in a Hele‐Shaw cell

In this paper, the interaction between two immiscible fluids with a finite mobility ratio is investigated numerically within a Hele‐Shaw cell. Fingering instabilities initiated at the interface between a low‐viscosity fluid and a high‐viscosity fluid are analysed at varying capillary numbers and mobility ratios using a finite mobility ratio model. The present work is motivated by the possible development of interfacial instabilities that can occur in porous media during the process of CO₂ sequestration but does not pretend to analyse this complex problem. Instead, we present a detailed study of the analogous problem occurring in a Hele‐Shaw cell, giving indications of possible plume patterns that can develop during the CO₂ injection. The numerical scheme utilises a boundary element method in which the normal velocity at the interface of the two fluids is directly computed through the evaluation of a hypersingular integral. The boundary integral equation is solved using a Neumann convergent series with cubic B‐Spline boundary discretisation, exhibiting sixth‐order spatial convergence. The convergent series allows the long‐term nonlinear dynamics of growing viscous fingers to be explored accurately and efficiently. Simulations in low‐mobility ratio regimes reveal large differences in fingering patterns compared with those predicted by previous high‐mobility ratio models. Most significantly, classical finger shielding between competing fingers is inhibited. Secondary fingers can possess significant velocity, allowing greater interaction with primary fingers compared with high‐mobility ratio flows. Eventually, this interaction can lead to base thinning and the breaking of fingers into separate bubbles. Copyright © 2015 John Wiley & Sons, Ltd.     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.     

A fixed‐grid b‐spline finite element technique for fluid–structure interaction

We present a fixed‐grid finite element technique for fluid–structure interaction problems involving incompressible viscous flows and thin structures. The flow equations are discretised with isoparametric b‐spline basis functions defined on a logically Cartesian grid. In addition, the previously proposed subdivision‐stabilisation technique is used to ensure inf–sup stability. The beam equations are discretised with b‐splines and the shell equations with subdivision basis functions, both leading to a rotation‐free formulation. The interface conditions between the fluid and the structure are enforced with the Nitsche technique. The resulting coupled system of equations is solved with a Dirichlet–Robin partitioning scheme, and the fluid equations are solved with a pressure–correction method. Auxiliary techniques employed for improving numerical robustness include the level‐set based implicit representation of the structure interface on the fluid grid, a cut‐cell integration algorithm based on marching tetrahedra and the conservative data transfer between the fluid and structure discretisations. A number of verification and validation examples, primarily motivated by animal locomotion in air or water, demonstrate the robustness and efficiency of our approach. Copyright © 2013 John Wiley & Sons, Ltd.     

On the implicit time integration of semi‐discrete viscous fluxes on unstructured dynamic meshes

Numerical simulations of viscous flow problems with complex moving and/or deforming boundaries commonly require the solution of the corresponding fluid equations of motion on unstructured dynamic meshes. In this paper, a systematic investigation of the importance of the choice of the mesh configuration for evaluating the viscous fluxes is performed when the semi‐discrete Navier–Stokes equations are time‐integrated using the popular second‐order implicit backward difference algorithm. The findings are illustrated with the simulation of a laminar viscous flow problem around an oscillating airfoil. Copyright © 1999 John Wiley & Sons, Ltd.     

An assessment of a parallel,finite element method for three‐dimensional,moving‐boundary flows driven by capillarity for simulation of viscous sintering

A parallel, finite element method is presented for the computation of three‐dimensional, free‐surface flows where surface tension effects are significant. The method employs an unstructured tetrahedral mesh, a front‐tracking arbitrary Lagrangian–Eulerian formulation, and fully implicit time integration. Interior mesh motion is accomplished via pseudo‐solid mesh deformation. Surface tension effects are incorporated directly into the momentum equation boundary conditions using surface identities that circumvent the need to compute second derivatives of the surface shape, resulting in a robust representation of capillary phenomena. Sample results are shown for the viscous sintering of glassy ceramic particles. The most serious performance issue is error arising from mesh distortion when boundary motion is significant. This effect can be severe enough to stop the calculations; some simple strategies for improving performance are tested. Copyright © 2001 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.     

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.     

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.     

On the efficiency of semi‐implicit and semi‐Lagrangian spectral methods for the calculation of incompressible flows

Classical semi‐implicit backward Euler/Adams–Bashforth time discretizations of the Navier–Stokes equations induce, for high‐Reynolds number flows, severe restrictions on the time step. Such restrictions can be relaxed by using semi‐Lagrangian schemes essentially based on splitting the full problem into an explicit transport step and an implicit diffusion step. In comparison with the standard characteristics method, the semi‐Lagrangian method has the advantage of being much less CPU time consuming where spectral methods are concerned. This paper is devoted to the comparison of the ‘semi‐implicit’ and ‘semi‐Lagrangian’ approaches, in terms of stability, accuracy and computational efficiency. Numerical results on the advection equation, Burger's equation and finally two‐ and three‐dimensional Navier–Stokes equations, using spectral elements or a collocation method, are provided. Copyright © 2001 John Wiley & Sons, Ltd.