Derivation and characteristics analysis of an acoustics–convection upstream resolution algorithm for the two‐dimensional Euler and Navier–Stokes equations

The first of a two‐paper series, this paper introduces a new decomposition not of the hyperbolic flux vector but of the flux vector Jacobian. The paper then details for the Euler and Navier–Stokes equations an intrinsically infinite directional upstream‐bias formulation that rests on the mathematics and physics of multi‐dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the differential equation level, before the spatial discretization, within a characteristics‐bias governing system. Through a decomposition of the Euler flux divergence into multi‐dimensional acoustics and convection–acoustics components, this characteristics‐bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable‐strength upstreaming that correlates with the spatial distribution of characteristic velocities. Copyright © 2005 John Wiley & Sons, Ltd.     

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

An exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

Least‐squares meshfree method for incompressible Navier–Stokes problems

A least‐squares meshfree method based on the first‐order velocity–pressure–vorticity formulation for two‐dimensional incompressible Navier–Stokes problem is presented. The convective term is linearized by successive substitution or Newton's method. The discretization of all governing equations is implemented by the least‐squares method. Equal‐order moving least‐squares approximation is employed with Gauss quadrature in the background cells. The boundary conditions are enforced by the penalty method. The matrix‐free element‐by‐element Jacobi preconditioned conjugate method is applied to solve the discretized linear systems. Cavity flow for steady Navier–Stokes problem and the flow over a square obstacle for time‐dependent Navier–Stokes problem are investigated for the presented least‐squares meshfree method. The effects of inaccurate integration on the accuracy of the solution are investigated. Copyright © 2004 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.     

Development of a convection–diffusion‐reaction magnetohydrodynamic solver on non‐staggered grids

This paper presents a convection–diffusion‐reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one‐dimensional constant‐coefficient CDR equation is employed. For purposes of extending this discrete formulation to two‐dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.     

The steady Navier–Stokes/energy system with temperature‐dependent viscosity—Part 1: Analysis of the continuous problem

In this first part we propose and analyse a model for the study of two‐dimensional incompressible Navier–Stokes equations with a temperature‐dependent viscosity. The flow is supposed in a mixed convection regime and considers an outflow region, leading to a strongly coupled problem between the Navier–Stokes and energy equations, which will be justified theoretically. The coupling in the continuous problem is treated by an outer temperature fixed point strategy. Existence results for a particular variational formulation follows from this study. Further, a particular uniqueness result for small data is also obtained. Copyright © 2007 John Wiley & Sons, Ltd.     

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

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

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.     

Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two-dimensional laminar internal flows up to high Reynolds numbers.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐intrusive reduced‐order modelling of the Navier–Stokes equations based on RBF interpolation

We present a new non‐intrusive model reduction method for the Navier–Stokes equations. The method replaces the traditional approach of projecting the equations onto the reduced space with a radial basis function (RBF) multi‐dimensional interpolation. The main point of this method is to construct a number of multi‐dimensional interpolation functions using the RBF scatter multi‐dimensional interpolation method. The interpolation functions are used to calculate POD coefficients at each time step from POD coefficients at earlier time steps. The advantage of this method is that it does not require modifications to the source code (which would otherwise be very cumbersome), as it is independent of the governing equations of the system. Another advantage of this method is that it avoids the stability problem of POD/Galerkin. The novelty of this work lies in the application of RBF interpolation and POD to construct the reduced‐order model for the Navier–Stokes equations. Another novelty is the verification and validation of numerical examples (a lock exchange problem and a flow past a cylinder problem) using unstructured adaptive finite element ocean model. The results obtained show that CPU times are reduced by several orders of magnitude whilst the accuracy is maintained in comparison with the corresponding high‐fidelity models. Copyright © 2015 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.     

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.     

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.     

Newton linearization of the incompressible Navier–Stokes equations

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

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

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.     

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.     

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two‐dimensional unsteady convection–diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5?μ?1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection–diffusion problem and three flows of varying complexities governed by the two‐dimensional incompressible Navier–Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd.