A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

An overlapping control volume method for the Navier–Stokes equations on non‐staggered grids

An algorithm, based on the overlapping control volume (OCV) method, for the solution of the steady and unsteady two‐dimensional incompressible Navier–Stokes equations in complex geometry is presented. The primitive variable formulation is solved on a non‐staggered grid arrangement. The problem of pressure–velocity decoupling is circumvented by using momentum interpolation. The accuracy and effectiveness of the method is established by solving five steady state and one unsteady test problems. The numerical solutions obtained using the technique are in good agreement with the analytical and benchmark solutions available in the literature. On uniform grids, the method gives second‐order accuracy for both diffusion‐ and convection‐dominated flows. There is little loss of accuracy on grids that are moderately non‐orthogonal. Copyright © 1999 John Wiley & Sons, Ltd.     

Towards a transparent boundary condition for compressible Navier–Stokes equations

A new artificial boundary condition for two‐dimensional subsonic flows governed by the compressible Navier–Stokes equations is derived. It is based on the hyperbolic part of the equations, according to the way of propagation of the characteristic waves. A reference flow, as well as a convection velocity, is used to properly discretize the terms corresponding to the entering waves. Numerical tests on various classical model problems, whose solutions are known, and comparisons with other boundary conditions (BCs), show the efficiency of the BC. Direct numerical simulations of more complex flows over a dihedral plate are simulated, without creation of acoustic waves going back in the flow. Copyright © 2001 John Wiley & Sons, Ltd.     

An efficient edge based data structure for the compressible Reynolds-averaged Navier–Stokes equations on hybrid unstructured meshes

An efficient edge based data structure has been developed in order to implement an unstructured vertex based finite volume algorithm for the Reynolds-averaged Navier–Stokes equations on hybrid meshes. In the present approach, the data structure is tailored to meet the requirements of the vertex based algorithm by considering data access patterns and cache efficiency. The required data are packed and allocated in a way that they are close to each other in the physical memory. Therefore, the proposed data structure increases cache performance and improves computation time. As a result, the explicit flow solver indicates a significant speed up compared to other open-source solvers in terms of CPU time. A fully implicit version has also been implemented based on the PETSc library in order to improve the robustness of the algorithm. The resulting algebraic equations due to the compressible Navier–Stokes and the one equation Spalart–Allmaras turbulence equations are solved in a monolithic manner using the restricted additive Schwarz preconditioner combined with the FGMRES Krylov subspace algorithm. In order to further improve the computational accuracy, the multiscale metric based anisotropic mesh refinement library PyAMG is used for mesh adaptation. The numerical algorithm is validated for the classical benchmark problems such as the transonic turbulent flow around a supercritical RAE2822 airfoil and DLR-F6 wing-body-nacelle-pylon configuration. The efficiency of the data structure is demonstrated by achieving up to an order of magnitude speed up in CPU times.     

A Taylor series‐based finite volume method for the Navier–Stokes equations

A Taylor series‐based finite volume formulation has been developed to solve the Navier–Stokes equations. Within each cell, velocity and pressure are obtained from the Taylor expansion at its centre. The derivatives in the expansion are found by applying the Gauss theorem over the cell. The resultant integration over the faces of the cell is calculated from the value at the middle point of the face and its derivatives, which are further obtained from a higher order interpolation based on the values at the centres of two cells sharing this face. The terms up to second order in the velocity and the terms up to first order in pressure in the Taylor expansion are retained throughout the derivation. The test cases for channel flow, flow past a circular cylinder and flow in a collapsible channel have shown that the method is quite accurate and flexible. Copyright © 2008 John Wiley & Sons, Ltd.     

Using staggered grids with characteristic boundary conditions when solving compressible reactive Navier–Stokes equations

A specific (hybrid) arrangement of variables is discussed to solve reactive compressible Navier–Stokes equations on staggered‐like grids with high‐order finite difference schemes. The objective is to improve the numerical flow solution at boundaries. Hybrid arrangement behaviour is compared with ‘pure’ colocated and staggered strategies. Classical Fourier analysis shows accuracy to be significantly improved in the hybrid case. One‐dimensional laminar flame test demonstrates increased robustness (in terms of mesh resolution), whereas computation of 1D exiting pressure wave propagation gives evidence that the method also improves accuracy in the prediction of non‐reflecting outflows, compared e.g. with the fully staggered scheme of (J. Comput. Phy. 2003). Multidimensional extension is illustrated through turbulent 2D planar and 3D expanding flames simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

A multi‐energy‐level lattice Boltzmann model for the compressible Navier–Stokes equations

In this paper, we propose a new lattice Boltzmann model for the compressible Navier–Stokes equations. The new model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. As the 25‐bit model, we obtained the equilibrium distribution functions and the compressible Navier–Stokes equations with the second accuracy of the truncation errors. The numerical examples show that the model can be used to simulate the shock waves, contact discontinuities and supersonic flows around circular cylinder. The numerical results are compared with those obtained by traditional method. Copyright © 2007 John Wiley & Sons, Ltd.     

On the discontinuous Galerkin method for the numerical solution of the Navier–Stokes equations

The paper deals with the use of the discontinuous Galerkin finite element method (DGFEM) for the numerical solution of viscous compressible flows. We start with a scalar convection–diffusion equation and present a discretization with the aid of the non‐symmetric variant of DGFEM with interior and boundary penalty terms. We also mention some theoretical results. Then we extend the scheme to the system of the Navier–Stokes equations and discuss the treatment of stabilization terms. Several numerical examples are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

An adaptive multigrid finite-volume scheme for incompressible Navier–Stokes equations

An algorithm for the solutions of the two-dimensional incompressible Navier–Stokes equations is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. It is based on an artificial compressibility method and uses higher-order upwind finite-volume techniques for the convective terms and a second-order finite-volume technique for the viscous terms. Three upwind schemes for discretizing convective terms are proposed here. An interesting result is that the solutions computed by one of them is not sensitive to the value of the artificial compressibility parameter. A second-order, two-step Runge–Kutta integration coupling with an implicit residual smoothing and with a multigrid method is used for achieving fast convergence for both steady- and unsteady-state problems. The numerical results agree well with experimental and other numerical data. A comparison with an analytically exact solution is performed to verify the space and time accuracy of the algorithm.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

Exact solutions of the generalized Navier– Stokes equations for benchmarking

The generalized Navier– Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier– Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright © 2002 John Wiley & Sons, Ltd.     

The fundamental solution method for incompressible Navier–Stokes equations

A complete boundary integral formulation for incompressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch show good agreement with available experimental data. © 1998 John Wiley & Sons, Ltd.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Two‐level finite element method with a stabilizing subgrid for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method with a stabilizing subgrid of a single node is described, and its application to the Navier–Stokes equation is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems. The results show that the proper choice of the subgrid node is crucial in obtaining stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Copyright © 2008 John Wiley & Sons, Ltd.     

A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis

Stabilized finite element methods have been shown to yield robust, accurate numerical solutions to both the compressible and incompressible Navier–Stokes equations for laminar and turbulent flows. The present work focuses on the application of higher‐order, hierarchical basis functions to the incompressible Navier–Stokes equations using a stabilized finite element method. It is shown on a variety of problems that the most cost‐effective simulations (in terms of CPU time, memory, and disk storage) can be obtained using higher‐order basis functions when compared with the traditional linear basis. In addition, algorithms will be presented for the efficient implementation of these methods within the traditional finite element data structures. Copyright © 2001 John Wiley & Sons, Ltd.     

Three-dimensional incompressible Navier–Stokes equations on non-orthogonal staggered grids using the velocity–vorticity formulation

This paper is concerned with the numerical resolution of the incompressible Navier–Stokes equations in the velocity–vorticity form on non-orthogonal structured grids. The discretization is performed in such a way, that the discrete operators mimic the properties of the continuous ones. This allows the discrete equivalence between the primitive and velocity–vorticity formulations to be proved. This last formulation can thus be seen as a particular technique for solving the primitive equations. The difficulty associated with non-simply connected computational domains and with the implementation of the boundary conditions are discussed. One of the main drawback of the velocity–vorticity formulation, relative to the additional computational work required for solving the additional unknowns, is alleviated. Two- and three-dimensional numerical test cases validate the proposed method. © 1998 John Wiley & Sons, Ltd.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 John Wiley & Sons, Ltd.     

A Chebyshev collocation method for the Navier–Stokes equations with application to double-diffusive convection

A Chebyshev collocation method for solving the unsteady two-dimensional Navier–Stokes equations in vorticity–streamfunction variables is presented and discussed. The discretization in time is obtained through a class of semi-implicit finite difference schemes. Thus at each time cycle the problem reduces to a Stokes-type problem which is solved by means of the influence matrix technique leading to the solution of Helmholtz-type equations with Dirichlet boundary conditions. Theoretical results on the stability of the method are given. Then a matrix diagonalization procedure for solving the algebraic system resulting from the Chebyshev collocation approximation of the Helmholtz equation is developed and its accuracy is tested. Numerical results are given for the Stokes and the Navier–Stokes equations. Finally the method is applied to a double-diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below.