On the smallest scale for the incompressible Navier-Stokes equations

We prove that, for solutions to the two- and three-dimensional incompressible Navier-Stokes equations, the minimum scale is inversely proportional to the square root of the Reynolds number based on the kinematic viscosity and the maximum of the velocity gradients. The bounds on the velocity gradients can be obtained for two-dimensional flows, but have to be assumed in three dimensions. Numerical results in two dimensions are given which illustrate and substantiate the features of the proof. Implications of the minimum scale result, to the decay rate of the energy spectrum are discussed.Research was supported in part by the National Acronautics and Space Administration under NASA Contract No. NAS1-18107, while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support was provided by the National Science Foundation under Grant DMS-8312264 and the Office of Naval Research under Contract N-00014-83-K-0422.     

Preconditioned conjugate gradient methods for the incompressible Navier-Stokes equations

A robust technique for solving primitive variable formulations of the incompressible Navier-Stokes equations is to use Newton iteration for the fully implicit non-linear equations. A direct sparse matrix method can be used to solve the Jacobian but is costly for large problems; an alternative is to use an iterative matrix method. This paper investigates effective ways of using a conjugate-gradient-type method with an incomplete LU factorization preconditioner for two-dimensional incompressible viscous flow problems. Special attention is paid to the ordering of unknowns, with emphasis on a minimum updating matrix (MUM) ordering. Numerical results are given for several test problems.     

On pressure boundary conditions for the incompressible Navier-Stokes equations

The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time-varying divergence-free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well-defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.     

Implementation of the Lagrange-Galerkin method for the incompressible Navier-Stokes equations

This paper is concerned with the implementation of Lagrange-Galerkin finite element methods for the Navier-Stokes equations. A scheme is developed to efficiently handle unstructed meshes with local refinement, using a quad-tree-based algorithm for the geometric search. Several difficulties that arise in the construction of the right-hand side are discussed in detail and some useful tricks are proposed. The resulting method is tested on the lid-driven square cavity and the vortex shedding behind a rectangular cylinder and is found to give satisfactory agreement with previous works. A detailed analysis of the effect of time discretization is included.     

Linear and non-linear iterative methods for the incompressible Navier-Stokes equations

In this study, the discretized finite volume form of the two-dimensional, incompressible Navier-Stokes equations is solved using both a frozen coefficient and a full Newton non-linear iteration. The optimal method is a combination of these two techniques. The linearized equations are solved using a conjugate-gradient-like method (CGSTAB). Various types of preconditioning are developed. Completely general sparse matrix methods are used. Investigations are carried out to determine the effect of finite volume cell anisotropy on the preconditioner. Numerical results are given for several test problems.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

An ILU smoother for the incompressible Navier-Stokes equations in general co-ordinates

The solution of the incompressible Navier-Stokes equations in general two- and three-dimensional domains using a multigrid method is considered. Because a great variety of boundary-fitted grids may occur, robustness is at a premium. Therefore a new ILU smoother called CILU (collective ILU) is described, based on r-transformations. In CILU the matrix that is factorized is block-structured, with blocks corresponding to the set of physical variables. A multigrid algorithm using CILU as smoother is investigated. The performance of the algorithm in two and three dimensions is assessed by numerical experments. The results show that CILU is a good smoother for the incompressible Navier-Stokes equations discretized on general non-orthogonal curvilinear grids.     

Chebyshev collocation method and multidomain decomposition for the incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations in primitive variables have been solved by a pseudospectral Chebyshev method using a semi-implicit fractional step scheme. The latter has been adapted to the particular features of spectral collocation methods to develop the monodomain algorithm. In particular, pressure and velocity collocated on the same nodes are sought in a polynomial space of the same order; the cascade of scalar elliptic problems arising after the spatial collocation is solved using finite difference preconditioning. With the present procedure spurious pressure modes do not pollute the pressure field. As a natural development of the present work a multidomain extent was devised and tested. The original domain is divided into a union of patching sub-rectangles. Each scalar problem obtained after spatial collocation is solved by iterating by subdomains. For steady problems a C¹ solution is recovered at the interfaces upon convergence, ensuring a spectrally accurate solution. A number of test cases have been solved to validate the algorithm in both its single-block and multidomain configurations. The preliminary results achieved indicate that collocation methods in multidomain configurations might become a viable alternative to the spectral element technique for accurate flow prediction.     

A spectral method with staggered grid for incompressible Navier-Stokes equations

In order to solve the Navier-Stokes equations by spectral methods, we develop an algorithm using a staggered grid to compute the pressure. On this grid, an iterative process based on an artificial compressibility matrix associates the pressure with the continuity equation. This method is very accurate and avoids naturally most of the effects of parasite modes appearing in classical spectral methods with a velocity—pressure formulation.     

Compact finite difference-Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported. The project supported by the National Natural Science Foundation of China     

A divergence-conforming hybridized discontinuous Galerkin method for the incompressible Reynolds-averaged Navier-Stokes equations

We present a hybridized discontinuous Galerkin (HDG) method for the incompressible Reynolds-averaged Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The method extends upon an HDG method recently introduced by Rhebergen and Wells for the incompressible Navier-Stokes equations. With a special choice of velocity and pressure spaces for both element and trace degrees of freedom (DOFs), the method returns pointwise divergence-free mean velocity fields and properly balances momentum and energy. We further examine the use of different polynomial degrees and meshes to see how the order of the scalar eddy viscosity affects the convergence of the mean velocity and pressure fields, specifically for the method of manufactured solutions. As is standard with HDG methods, static condensation can be employed to remove the element DOFs and thus dramatically reduce the global number of DOFs. Numerical results illustrate the effectiveness of the proposed methodology.     

A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations

We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity form, can be approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low-to-medium Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re ? 7500 point-SOR iteration is used and the convergence is fast.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

Finite element stream function-vorticity solutions of the incompressible Navier-Stokes equations

The incompressible, two-dimensional Navier-Stokes equations are solved by the finite element method (FEM) using a novel stream function/vorticity formulation. The no-slip solid walls boundary condition is applied by taking advantage of the simple implementation of natural boundary conditions in the FEM, eliminating the need for an iterative evaluation of wall vorticity formulae. In addition, with the proper choice of elements, a stable scheme is constructed allowing convergence to be achieved for all Reynolds numbers, from creeping to inviscid flow, without the traditional need for upwinding and its associated false diffusion. Solutions are presented for a variety of geometries.     

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.     

A mixture differential quadrature method for solving two-dimensional incompressible Navier-Stokes equations

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TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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Solution of the discretized incompressible Navier-Stokes equations with the GMRES method

We describe some experiences using interative solution methods of GMRES type to solve the discretized Navier-Stokes equations. The discretization combined with a pressure correction scheme leads to two different systems of equations: the momentum equations and the pressure equation. It appears that a fast solution method for the pressure equation is obtained by applying the recently proposed GMRESR method, or GMRES combined with a MILU preconditioner. The diagonally scaled momentum equations are solved by GMRES(m), a restarted version of GMRES.     

A high-order discontinuous Galerkin method for the incompressible Navier-Stokes equations on arbitrary grids

A high-order discontinuous Galerkin (DG) method is proposed in this work for solving the two-dimensional steady and unsteady incompressible Navier-Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax-Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.     

Invariant discretization of the incompressible Navier-Stokes equations in boundary fitted co-ordinates

The discretization of the incompressible Navier-Stokes equation on boundary-fitted curvilinear grids is considered. The discretization is based on a staggered grid arrangement and the Navier-;Stokes equations in tensor formulation including Christoffel symbols. It is shown that discretization accuracy is much enhanced by choosing the velocity variables in a special way. The time-dependent equations are solved by a pressure-correction method in combination with a GMRES method. Special attention is paid to the discretization of several types of boundary conditions. It is shown that fairly non-smooth grids may be used using our approach.