A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids

This paper is concerned with the development of a new high‐order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non‐uniform grids. Specifically, both a high‐order fluxes integration and the implicit deconvolution of the volume‐averaged field are considered. This way, the numerical solution effectively stands for a fourth‐order approximation of the point‐wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high‐order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large‐eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two‐dimensional time‐evolving mixing layer is simulated, with and without sub‐grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.     

Well‐balanced finite difference weighted essentially non‐oscillatory schemes for the blood flow model

The blood flow model maintains the steady‐state solutions, in which the flux gradients are non‐zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted essentially non‐oscillatory (WENO) schemes to this model with such well‐balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well‐balanced property, and keep good resolution for smooth and discontinuous solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Local maximum principle satisfying high‐order non‐oscillatory schemes

The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.     

Compact finite difference schemes on non‐uniform meshes. Application to direct numerical simulations of compressible flows

In this paper, the development of a fourth‐ (respectively third‐) order compact scheme for the approximation of first (respectively second) derivatives on non‐uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi‐ and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed. Copyright © 1999 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.     

Improved error and work estimates for high‐order elements

Work estimates for high‐order elements are derived. The comparison of error and work estimates shows that even for relative accuracy in the 0.1% range, which is one order below the typical accuracy of engineering interest (1% range), linear elements may outperform all higher‐order elements. As expected, the estimates also show that the optimal order of element in terms of work and storage demands depends on the desired relative accuracy. The comparison of work estimates for high‐order elements and their finite difference counterparts reveals a work‐ratio of several orders of magnitude. It thus becomes questionable if general geometric flexibility via micro‐unstructured grids is worth such a high cost. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

A recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Linear and non‐linear stability analysis for finite difference discretizations of high‐order Boussinesq equations

This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non‐linear and extremely dispersive water waves. The analysis demonstrates the near‐equivalence of classical linear Fourier (von Neumann) techniques with matrix‐based methods for formulations in both one and two horizontal dimensions. The matrix‐based method is also extended to show the local de‐stabilizing effects of the non‐linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep‐water non‐linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non‐normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non‐linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non‐linear PDEs. Copyright © 2004 John Wiley & Sons, Ltd.     

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered,non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

It is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

A higher order compact (HOC) finite difference solution procedure has been proposed for the steady two‐dimensional (2D) convection–diffusion equation on non‐uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. Effectiveness of the method is seen from the fact that for the first time, an HOC algorithm on non‐uniform grid has been extended to the Navier–Stokes (N–S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for difficult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non‐uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex flows such as that in the lid‐driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient discretization of pressure‐correction equations on non‐orthogonal grids

An alternative discretization of pressure‐correction equations within pressure‐correction schemes for the solution of the incompressible Navier–Stokes equations is introduced, which improves the convergence and robustness properties of such schemes for non‐orthogonal grids. As against standard approaches, where the non‐orthogonal terms usually are just neglected, the approach allows for a simplification of the pressure‐correction equation to correspond to 5‐point or 7‐point computational molecules in two or three dimensions, respectively, but still incorporates the effects of non‐orthogonality. As a result a wide range (including rather high values) of underrelaxation factors can be used, resulting in an increased overall performance of the underlying pressure‐correction schemes. Within this context, a second issue of the paper is the investigation of the accuracy to which the pressure‐correction equation should be solved in each pressure‐correction iteration. The scheme is investigated for standard test cases and, in order to show its applicability to practical flow problems, for a more complex configuration of a micro heat exchanger. Copyright © 2003 John Wiley & Sons, Ltd.     

High‐order finite difference schemes for incompressible flows

This paper presents a new high‐order approach to the numerical solution of the incompressible Stokes and Navier–Stokes equations. The class of schemes developed is based upon a velocity–pressure–pressure gradient formulation, which allows: (i) high‐order finite difference stencils to be applied on non‐staggered grids; (ii) high‐order pressure gradient approximations to be made using standard Padé schemes, and (iii) a variety of boundary conditions to be incorporated in a natural manner. Results are presented in detail for a selection of two‐dimensional steady‐state test problems, using the fourth‐order scheme to demonstrate the accuracy and the robustness of the proposed methods. Furthermore, extensions to higher orders and time‐dependent problems are illustrated, whereas the extension to three‐dimensional problems is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids

This paper presents a family of High‐order finite volume schemes applicable on unstructured grids. The k‐exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock‐capturing capability of the scheme. This procedure combines the simplicity of the k‐exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High‐order accurate shock capturing schemes. To further improve the efficiency, an efficient problem‐independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non‐oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High‐order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Methods based on exponential finite difference approximations of h⁴ accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h⁴ accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

A Mach‐uniform unstructured staggered grid method

A novel Mach‐uniform method to compute flows using unstructured staggered grids is discussed. The Mach‐uniform method is a generalization of the pressure‐correction approach for incompressible flows, and is valid for Mach numbers ranging from 0 (incompressible) to > 1 (supersonic). The primary variables (ρ u ,p and ρ) are updated sequentially. The grid consists of triangles. A staggered positioning of the variables is employed: the scalar variables are located at the centroids of the triangles, whereas the normal momentum components are positioned at the midpoints of the faces of the triangles. Discretization of the two‐dimensional flow equations on unstructured staggered grids is discussed. For the cell face fluxes there is a choice between first‐order upwind and central approximation. Flows around the NACA 0012 airfoil with freestream Mach numbers ranging from 0 to 1.2 are computed to demonstrate the Mach‐uniform accuracy and efficiency of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.     

High‐order compact finite difference schemes for the vorticity–divergence representation of the spherical shallow water equations

This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth‐order compact, the sixth‐order and eighth‐order SCFDM, and the sixth‐order and eighth‐order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi‐implicit Runge–Kutta method is used. In addition, to control the nonlinear instability, an eighth‐order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high‐order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid‐latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth‐order and eighth‐order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth‐order compact method. Copyright © 2015 John Wiley & Sons, Ltd.     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.