Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

The Multidimensional Optimal Order Detection method in the three‐dimensional case: very high‐order finite volume method for hyperbolic systems

The Multidimensional Optimal Order Detection (MOOD) method for two‐dimensional geometries has been introduced by the authors in two recent papers. We present here the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids, and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set of numerical tests for both the convection equation and the Euler system that the optimal high order of accuracy is reached on smooth solutions, whereas spurious oscillations near singularities are prevented. At last, the intrinsic positivity‐preserving property of the MOOD method is confirmed in 3D, and we provide simple optimizations to reduce the computational cost such that the MOOD method is very competitive compared with existing high‐order Finite Volume methods.Copyright © 2013 John Wiley & Sons, Ltd.     

Assessment of a high‐order MUSCL method for rotor flows

This work presents the implementation of a high‐order, finite‐volume scheme suitable for rotor flows. The formulation is based on the variable extrapolation MUSCL‐scheme, where high‐order spatial accuracy (up to fourth‐order) is achieved using correction terms obtained through successive differentiation. A variety of results are presented, including 2‐ and 3‐dimensional test cases. Results with the proposed scheme, showed better wake and higher resolution of vortical structures compared with the standard MUSCL, even when coarse meshes were employed. The method was also demonstrated for 3‐dimensional unsteady flows using overset and moving grids for the UH‐60A rotor in forward flight and the Enhanced Rotorcraft Innovative Concept Achievement tiltrotor in aeroplane mode. For medium grids, the present method adds reasonable CPU and memory overheads and offers good accuracy on relatively coarse grids.     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Computation of unsteady viscous incompressible flows in generalized non‐inertial co‐ordinate system using Godunov‐projection method and overlapping meshes

Time‐dependent incompressible Navier–Stokes equations are formulated in generalized non‐inertial co‐ordinate system and numerically solved by using a modified second‐order Godunov‐projection method on a system of overlapped body‐fitted structured grids. The projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The second‐order Godunov method is applied for numerically approximating the non‐linear convection terms in order to provide a robust discretization for simulating flows at high Reynolds number. In order to obtain the pressure field, the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain so that the moving‐boundary problem can be solved economically. Numerical results are then presented to demonstrate the performance of this projection method for a variety of unsteady two‐ and three‐dimensional flow problems formulated in the non‐inertial co‐ordinate systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

Theoretical analysis for achieving high‐order spatial accuracy in Lagrangian/Eulerian source terms

In a fully coupled Lagrangian/Eulerian two‐phase calculation, the source terms from computational particles must be agglomerated to nearby gas‐phase nodes. Existing methods are capable of accomplishing this particle‐to‐gas coupling with second‐order accuracy. However, higher‐order methods would be useful for applications such as two‐phase direct numerical simulation and large eddy simulation. A theoretical basis is provided for producing high spatial accuracy in particle‐to‐gas source terms with low computational cost. The present work derives fourth‐ and sixth‐order accurate methods, and the procedure for even higher accuracy is discussed. The theory is also expanded to include two‐ and three‐dimensional calculations. One‐ and two‐dimensional tests are used to demonstrate the convergence of this method and to highlight problems with statistical noise. Finally, the potential for application in computational fluid dynamics codes is discussed. It is concluded that high‐order kernels have practical benefits only under limited ranges of statistical and spatial resolution. Additionally, convergence demonstrations with full CFD codes will be extremely difficult due to the worsening of statistical errors with increasing mesh resolution. Copyright © 2006 John Wiley & Sons, Ltd.     

A comparative study of high‐order variable‐property segregated algorithms for unsteady low Mach number flows

In this paper, five different algorithms are presented for the simulation of low Mach flows with large temperature variations, based on second‐order central‐difference or fourth‐order compact spatial discretization and a pressure projection‐type method. A semi‐implicit three‐step Runge–Kutta/Crank–Nicolson or second‐order iterative scheme is used for time integration. The different algorithms solve the coupled set of governing scalar equations in a decoupled segregate manner. In the first algorithm, a temperature equation is solved and density is calculated from the equation of state, while the second algorithm advances the density using the differential form of the equation of state. The third algorithm solves the continuity equation and the fourth algorithm solves both the continuity and enthalpy equation in conservative form. An iterative decoupled algorithm is also proposed, which allows the computation of the fully coupled solution. All five algorithms solve the momentum equation in conservative form and use a constant‐ or variable‐coefficient Poisson equation for the pressure. The efficiency of the fourth‐order compact scheme and the performances of the decoupling algorithms are demonstrated in three flow problems with large temperature variations: non‐Boussinesq natural convection, channel flow instability, flame–vortex interaction. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

A higher‐order Boussinesq‐type model with moving bottom boundary: applications to submarine landslide tsunami waves

A two‐dimensional depth‐integrated numerical model is developed using a fourth‐order Boussinesq approximation for an arbitrary time‐variable bottom boundary and is applied for submarine‐landslide‐generated waves. The mathematical formulation of model is an extension of (4,4) Padé approximant for moving bottom boundary. The mathematical formulations are derived based on a higher‐order perturbation analysis using the expanded form of velocity components. A sixth‐order multi‐step finite difference method is applied for spatial discretization and a sixth‐order Runge–Kutta method is applied for temporal discretization of the higher‐order depth‐integrated governing equations and boundary conditions. The present model is validated using available three‐dimensional experimental data and a good agreement is obtained. Moreover, the present higher‐order model is compared with fully potential three‐dimensional models as well as Boussinesq‐type multi‐layer models in several cases and the differences are discussed. The high accuracy of the present numerical model in considering the nonlinearity effects and frequency dispersion of waves is proven particularly for waves generated in intermediate and deeper water area. Copyright © 2006 John Wiley & Sons, Ltd.     

A stable and accurate projection method on a locally refined staggered mesh

In this paper, a robust projection method on a locally refined mesh is proposed for two‐ and three‐dimensional viscous incompressible flows. The proposed method is robust not only when the interface between two meshes is located in a smooth flow region but also when the interface is located in a flow region with large gradients and/or strong unsteadiness. In numerical simulations, a locally refined mesh saves many grid points in regions of relatively small gradients compared with a uniform mesh. For efficiency and ease of implementation, we consider a two‐level blocked structure, for which both of the coarse and fine meshes are uniform Cartesian ones individually. Unfortunately, the introduction of the two‐level blocked mesh results in an important but difficult issue: coupling of the coarse and fine meshes. In this paper, by properly addressing the issue of the coupling, we propose a stable and accurate projection method on a locally refined staggered mesh for both two‐ and three‐dimensional viscous incompressible flows. The proposed projection method is based on two principles: the linear interpolation technique and the consistent discretization of both sides of the pressure Poisson equation. The proposed algorithm is straightforward owing to the linear interpolation technique, is stable and accurate, is easy to extend from two‐ to three‐dimensional flows, and is valid even when flows with large gradients cross the interface between the two meshes. The resulting pressure Poisson equation is non‐symmetric on a locally refined mesh. The numerical results for a series of exact solutions for 2D and 3D viscous incompressible flows verify the stability and accuracy of the proposed projection method. The method is also applied to some challenging problems, including turbulent flows around particles, flows induced by impulsively started/stopped particles, and flows induced by particles near solid walls, to test the stability and accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

The advantages of using high‐order finite differences schemes in laminar–turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two‐dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second‐order central differences, a fourth‐order central differences, a fourth‐order compact scheme and a sixth‐order compact scheme. A classic fourth‐order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small‐amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high‐order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Fourth‐ and tenth‐order compact finite difference solutions of perturbed circular vortex flows

In this study, high‐order compact finite difference calculations are reported for 2D unsteady incompressible circular vortex flow in primitive variable formulation. The fourth‐order Runge–Kutta temporal discretization is used together with fourth‐ or tenth‐order compact spatial discretization. Dependent on the perturbation initially imposed, the solutions display a tripole, triangular or square vortex. The comparison of the predictions with the detailed spectral calculations of Kloosterziel and Carnevale (J. Fluid Mech. 1999; 388 :217–257) shows that the vorticity fields are very well captured. The spectral resolution of the present method was quantified from the decomposition of the vorticity distribution in its azimuthal components and compared with reported spectral results. Using identical grid resolution to the reference results yields negligible differences in the main features of the flow. The perturbation amplitude and its first harmonic are virtually identical to the reference results for both fourth‐ or tenth‐order spatial discretization, as theoretically expected but seldom a posteriori verified. The differences between the two spatial discretizations appear only for coarser grids, favouring the tenth‐order discretization. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the shallow‐water equations using an adaptive moving mesh method

This paper extends an adaptive moving mesh method to multi‐dimensional shallow water equations (SWE) with source terms. The algorithm is composed of two independent parts: the SWEs evolution and the mesh redistribution. The first part is a high‐resolution kinetic flux‐vector splitting (KFVS) method combined with the surface gradient method for initial data reconstruction, and the second part is based on an iteration procedure. In each iteration, meshes are first redistributed by a variational principle and then the underlying numerical solutions are updated by a conservative‐interpolation formula on the resulting new mesh. Several test problems in one‐ and two‐dimensions with a general geometry are computed using the proposed moving mesh algorithm. The computations demonstrate that the algorithm is efficient for solving problems with bore waves and their interactions. The solutions with higher resolution can be obtained by using a KFVS scheme for the SWEs with a much smaller number of grid points than the uniform mesh approach, although we do not treat technically the bed slope source terms in order to balance the source terms and flux gradients. Copyright © 2004 John Wiley & Sons, Ltd.     

High resolution Godunov‐type schemes with small stencils

Higher‐order Godunov‐type schemes have to cope with the following two problems: (i) the increase in the size of the stencil that make the scheme computationally expensive, and (ii) the monotony‐preserving treatments (limiters) that must be implemented to avoid oscillations, leading to strong damping of the solution, in particular linear waves (e.g. acoustic waves). When too compressive, limiting procedures may also trigger the instability of oscillatory numerical solutions (e.g. in advection–dispersion phenomena) via the artificial amplification of the shorter modes. The present paper proposes a new approach to carry out the reconstruction. In this approach, the values of the flow variable at the edges of the computational cells are obtained directly from the reconstruction within these cells. This method is applied to the MUSCL and DPM schemes for the solution of the linear advection equation. The modified DPM scheme can capture contact discontinuities within one computational cell, even after millions of time steps at Courant numbers ranging from 1 to values as low as 10^‐4. Linear waves are subject to negligible damping. Application of the method to the DPM for one‐dimensional advection–dispersion problems shows that the numerical instability of oscillatory solutions caused by the over compressive, original DPM limiter is eliminated. One‐ and two‐dimensional shallow water simulations show an improvement over classical methods, in particular for two‐dimensional problems with strongly distorted meshes. The quality of the computational solution in the two‐dimensional case remains acceptable even for mesh aspect ratios Δx/Δy as large as 10. The method can be extend to the discretization of higher‐order PDEs, allowing third‐order space derivatives to be discretized using only two cells in space. Copyright © 2004 John Wiley & Sons, Ltd.     

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.