Kinetic-based numerical schemes for incompressible Navier–Stokes equations

Numerical schemes for incompressible Navier–Stokes equations based on low Mach number limits of kinetic equations are presented. Discretizations of the incompressible Navier–Stokes equations are derived based on discretizations of the Boltzmann equation and consideration for the low Mach number limit. In the incompressible Navier–Stokes limit the discretizations reduce to explicit high-order numerical schemes. Numerical results for several test cases and comparisons with other well-known approaches are also presented.     

A Pseudospectral Multi-Domain Method for the Incompressible Navier–Stokes Equations

A pseudospectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method is designed for moderately complex geometries by means of a multi-domain approach. Key components are a Chebyshev collocation discretization, a special pressure-correction scheme and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the method with respect to the multi-domain functionality is investigated and compared to finite-volume approaches.     

A Conservative Isothermal Wall Boundary Condition for the Compressible Navier–Stokes Equations

We present a conservative isothermal wall boundary condition treatment for the compressible Navier-Stokes equations. The treatment is based on a manipulation of the Osher solver to predict the pressure and density at the wall, while specifying a zero boundary flux and a fixed temperature. With other solvers, a non-zero mass flux occurs through a wall boundary, which is significant at low resolutions in closed geometries. A simulation of a lid driven cavity flow with a multidomain spectral method illustrates the effect of the new boundary condition treatment.     

Algebraic Fractional-Step Schemes for Time-Dependent Incompressible Navier–Stokes Equations

The numerical investigation of a recent family of algebraic fractional-step methods (the so called Yosida methods) for the solution of the incompressible time-dependent Navier–Stokes equations is presented. A comparison with the Karniadakis–Israeli–Orszag method Karniadakis et al. (1991, J. Comput. Phys. 97, 414–443) is carried out. The high accuracy in time of these schemes well combines with the high accuracy in space of spectral methods.     

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.     

A Comparative Study of Time Advancement Methods for Solving Navier–Stokes Equations

A qualitative and quantitative study is made for choosing time advancement strategies for solving time dependent equations accurately. A single step, low order Euler time integration method is compared with Adams–Bashforth, a second order accurate time integration strategy for the solution of one dimensional wave equation. With the help of the exact solution, it is shown that the presence of the computational mode in Adams–Bashforth scheme leads to erroneous results, if the solution contains high frequency components. This is tested for the solution of incompressible Navier–Stokes equation for uniform flow past a rapidly rotating circular cylinder. This flow suffers intermittent temporal instabilities implying presence of high frequencies. Such instabilities have been noted earlier in experiments and high accuracy computations for similar flow parameters. This test problem shows that second order Adams– Bashforth time integration is not suitable for DNS.     

Numerical solution of the incompressible Navier–Stokes equations with exponential type schemes

A new exponential type finite-difference scheme of second-order accuracy for solving the unsteady incompressible Navier–Stokes equation is presented. The driven flow in a square cavity is used as the model problem. Numerical results for various Reynolds numbers are given, and are in good agreement with those presented by Ghia et al. (Ghia, U., Ghia, K.N. and Shin, C.T., 1982, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multi-grid method. Journal of Computational Physics, 48, 387–411.).     

A residual-based compact scheme for the unsteady compressible Navier–Stokes equations

A dissipative compact scheme is developed within a dual time stepping framework for the computation of unsteady compressible flows. The design of the scheme relies on the vanishing, at steady state with respect to the dual time, of a residual that includes the physical time-derivative. High-order accuracy and numerical dissipation are obtained in a simple way through the systematic use of derivatives of this residual. The accuracy and robustness of the approach are assessed on the simple advection of an inviscid vortex and compressible mixing layer problems involving shock waves.     

A nonconforming finite element method¶with anisotropic mesh grading¶for the incompressible Navier–Stokes equations¶in domains with edges

Any solution of the incompressible Navier–Stokes equations in three-dimensional domains with edges has anisotropic singular behaviour which is treated numerically by using anisotropic finite element meshes. The velocity is approximated by Crouzeix–Raviart (nonconforming 𝒫₁) elements and the pressure by piecewise constants. This method is stable for general meshes since the inf-sup condition is satisfied without minimal or maximal angle condition. The existence of solutions to the discrete problems follows. Consistency error estimates for the divergence equation are obtained for anisotropic tensor product meshes. As applications, the consistency error estimate for the Navier–Stokes solution and some discrete Sobolev inequalities are derived on such meshes. These last results provide optimal error estimates in the uniqueness case by the use of appropriately refined anisotropic tensor product meshes, namely, if N _e is the number of elements of the mesh, we prove that the optimal order of convergence h∼ N _e ^{− 1/3}. Received:July 2001 / Accepted: July 2002     

Global well-posedness of the incompressible fractional Navier–Stokes equations in Fourier–Besov spaces with variable exponents

We study the Cauchy problem of the fractional Navier–Stokes equations in critical variable exponent Fourier–Besov spaces $F{\stackrel{?}{B}}_{p\left(?\right),q}^{4?2\alpha ?\frac{3}{p\left(?\right)}}$. We discuss some properties of variable exponent Fourier–Besov spaces and prove a general global well-posedness result which covers some recent works about classical Navier–Stokes equations.     

A Very High-Order Accurate Staggered Finite Volume Scheme for the Stationary Incompressible Navier–Stokes and Euler Equations on Unstructured Meshes

We propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier–Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests are provided to assess the effectiveness of the method to achieve up to sixth-order convergence rates. Simulations for the benchmark lid-driven cavity problem are also provided to highlight the benefit of the proposed high-order scheme.     

An accelerated algorithm for Navier–Stokes equations

In this paper, the numerical solution of the Navier–Stokes equations by the Characteristic-Based-Split (CBS) scheme is accelerated with the Minimum Polynomial Extrapolation (MPE) method to obtain the steady state solution for evolution incompressible and compressible problems.The CBS is essentially a fractional time-stepping algorithm based on an original finite difference velocity-projection scheme where the convective terms are treated using the idea of the Characteristic-Galerkin method. In this work, the semi-implicit version of the CBS with global time-stepping is used for incompressible problems whereas the fully-explicit version is used for compressible flows.At the other end, the MPE is a vector extrapolation method that transforms the original sequence into another sequence converging to the same limit faster then the original one without the explicit knowledge of the sequence generator.The developed algorithm, tested on two-dimensional benchmark problems, demonstrates the new computational features arising from the introduction of the extrapolation procedure to the CBS scheme. In particular, the results show a remarkable reduction of the computational cost of the simulation.