Finite-difference schemes for steady incompressible Navier-Stokes equations in general curvilinear coordinates

Time-marching finite-difference schemes for solving the steady incompressible Navier-Stokes equations are proposed. In these schemes, the fractional-step method and the SMAC method are applied to a general curvilinear coordinates grid, so that the continuity condition can be satisfied identically. The momentum equations of contravariant velocities are newly derived for the accurate and easy treatment of the boundary conditions and are formulated concisely by introducing the contravariant vorticities. Spurious errors and numerical instabilities can be reasonably suppressed by employing the staggered grid and the upwind difference for moderate-to-high Reynolds number flows. Numerical results for two-dimensional laminar duct flow over a backward-facing step are shown, and compared with existing results to assess the reliability of the present schemes.     

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier-Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29] and [12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.     

Difference schemes on triangular and tetrahedral grids for Navier-Stokes equations of incompressible fluid

New approximations of Navier-Stokes equations are proposed for incompressible fluid on triangular and tetrahedral grids in the predictor-corrector method. Estimates are presented for the unconditional stability of difference schemes, and calculation results for a two-dimensional problem about a fluid flow in a cavity with a floating top cover; and they are compared with the solutions of other authors. The problems of grid equations monotonization are discussed.     

Finite element solution of the unsteady Navier-Stokes equations by a fractional step method

A finite element method is presented for the numerical simulation of time-dependent incompressible viscous flows. The method is based on a fractional step approach to the time integration of the Navier-Stokes equations in which only the incompressibility condition is treated implicitly. This leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost-effective solutions of large-scale problems. Numerical results indicate the versatility and effectiveness of the proposed method.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

A segregated-implicit scheme for solving the incompressible Navier-Stokes equations

The proposed segregated-implicit (SI) scheme, which is based on the artificial compressibility method, is discretized by the finite difference numerical scheme and verified by simulating a shear-driven cavity flow. The current results demonstrate that the SI scheme is a simple algorithm capable of fast solving the incompressible Navier-Stokes equations.     

Accelerated artificial compressibility method for steady-state incompressible flow calculations

A simple method is presented for accelerating the convergence of Chorin's artificial compressibility method for steady-state incompressible flow calculations. The acceleration is achieved by introducing an artificial bulk viscosity to dissipate the artificial sound waves more rapidly. The method is purely explicit and hence is well suited to vector and parallel computation. It is trivial to implement and does not alter the final steady solution. Use of the method in simple test calculations has resulted in computer time savings of 20–60%.     

Manifold method coupled velocity and pressure for Navier-Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier-Stokes (N-S) equations was discussed in this paper, and numerical manifold schemes for N-S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N-S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N-S equations.     

Null-space-free methods for the incompressible Navier-Stokes equations on non-staggered curvilinear grids

The discrete Poisson equation used to enforce incompressibility in the projection method of Chorin has a null space. The null space often manifests itself in producing solutions with checkerboard pressure fields. The staggering of variables by Harlow and Welch effectively eliminates the null space; however, when it is used in the context of curvilinear coordinates its consistent implementation is complicated because it requires the use of contravariant velocity components and variable coordinate base vectors. In this paper we present and analyze a null-space-free approximate projection method, which is based on cartesian vector components and non-staggered grids. The approximate projection method has been implemented in the computer code HEMO. Several examples of two-and three-dimensional flows using HEMO are presented.     

A higher order finite element algorithm for the unsteady Navier-Stokes equations

A higher order element, the Tocher 10 or C₀ Cubic on triangles, is the base for formulation of a finite element algorithm for numerical calculation of fluid flows governed by the unsteady Navier-Stokes equations. Results from the calculation of supersonic free shear layer flow are numerically accurate and in excellent agreement with finite difference solutions. Diverse characteristics for these two classes of methods emerge when the requirements of core storage and computer time are considered.     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

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.     

Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method

Our aim in this article is to study a new method for the approximation of the Navier-Stokes equations, and to present and discuss numerical results supporting the method. This method, called the nonlinear Galerkin method, uses nonlinear manifolds which are close to the attractor, while in the usual Galerkin method, we look for solutions in a linear space, i.e., whose components are independent. The equation of the manifold corresponds to an interaction law between small and large eddies and it is derived by asymptotic expansion from the exact equation. We consider here the two- and three-dimensional space periodic cases in the context of a pseudo-spectral discretization of the equation. We notice however that the method applies as well to more general flows, in particular nonhomogeneous flows.     

Chebyshev pseudospectral solution of the incompressible Navier-Stokes equations in curvilinear domains

A pseudospectral method for the calculation of 2-D flows of a viscous incompressible fluid in curvilinear domains is presented. The incompressible Navier-Stokes equations, expressed in terms of the primitive variables velocity and pressure, are solved in a non-orthogonal coordinate system. All the variables are expanded in double truncated series of Chebyshev polynomials. Time integration is performed by an implicit finite differences scheme for both the advective and diffusive terms. The pressure is calculated by the use of a truncated influence matrix involving all the collocation points in the field. A preconditioned iterative method is used to solve the system of linear equations resulting from the pseudospectral Chebyshev approximation. The algorithm is applied to the classical problem of the Green-Taylor vortices in order to check its accuracy; then 2 examples of viscous flow calculation are given in the case of a driven polar cavity and of a 2-D channel.     

Field equations for incompressible non-viscous fluids using artificial intelligence

In this article, we introduce new field equations for incompressible non-viscous fluids, which can be treated similarly to Maxwell’s electromagnetic equations based on artificial intelligence algorithms. Lagrangian and Hamiltonian formulations are used to arrive at field equations that are solved using convolutional neural networks. Four linear differential equations, which describe the two fields, namely, the dynamic pressure and the vortex fields, are derived, and these can be used in place of Euler’s equation. The only assumption while deriving this equation is that the dynamic pressure and vortex fields obey the superposition principle. The important finding to be noted is that Euler’s fluid equations can be converted into field equations analogous to Maxwell’s electromagnetic equations. We solve the flow problem for laminar flow past a cylinder, sphere, and cone in two dimensions similar to the conduction in a uniform electric field and arrive at closed-form expressions. These closed-form expressions, which are obtained for the potentials of fluid flow, are similar to the streamline potential functions in the case of fluid dynamics.

     

Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integro-differential formulation

A numerical method for the solution of the Navier-Stokes equations is developed using an integro-differential formulation of the equations. The method permits the actual computation to be confined to the viscous region of the flow and offers a drastic reduction in the number of data points required in the numerical procedure. The integro-differential formulation is presented along with discussion of the kinetic and kinematic aspects of the problem and the interplay between the two aspects. Results for several parallel flow problems and for the flow past a circular cyliner are presented. For the circular cylinder, it is shown that the introductions of a splitter plate behind the cylinder suppresses vortex shedding.     

Numerical solution of a phase field model for incompressible two-phase flows based on artificial compressibility

We present an artificial compressibility based numerical method for a phase field model for simulating two-phase incompressible viscous flows. The phase model was proposed by Liu and Shen [Physica D. 179 (2003) 211–228], in which the interface between two fluids is represented by a thin transition region of fluid mixture that stores certain amount of mixing energy. The model consists of the Navier–Stokes equations coupled with the Allen–Cahn equation (phase field equation) through an extra stress term and a transport term. The extra stress in the momentum equations represents the phase-induced capillary effect for the mixture due to the surface tension. The coupled equations are cast into a conservative form suitable for implementation with the artificial compressibility method. The resulting hyperbolic system of equations are then discretized with weighted essentially non-oscillatory (WENO) finite difference scheme. The dual-time stepping technique is applied for obtaining time accuracy at each physical time step, and the approximate factorization algorithm is used to solve the discretized equations. The effectiveness of the numerical method is demonstrated in several two-phase flow problems with topological changes. Numerical results show the present method can be used to simulate incompressible two-phase flows with small interfacial width parameters and topological changes.     

Laminar incompressible flow past a class of blunted wedges using the Navier-Stokes equations

The Navier-Stokes equations are solved, numerically, to determine the symmetric laminar incompressible flow past blunted wedges. Similarity-type variables are used in a coordinate system that comprises the optimal coordinates for the corresponding boundary-layer problem. This formulation leads to better numerical accuracy and produces the correct solution near the leading edge, far downstream and transversely far from the wedge surface. The flow over parabolas, the flat plate, and the verticval wall are obtained as particular cases of the present solutions and compare well with the available results for these problems. The numerical method that converges rapidly to accurate results.     

The numerical solution of the Navier-Stokes equations for laminar incompressible flow past a paraboloid of revolution

A numerical methods is presented for the solution of the Navier-Stokes equations for flow past a paraboloid of revolution. This method is based upon the ideas of van de Vooren and collaborators [1,2]. The flow field has been computed for a large range of Reynolds numbers. Results are presented for the skinfriction and the pressure together with their respective drag coefficients. The total drag has been checked by means of an application of the momentum theorem.     

Space-time adaptive simulations for unsteady Navier-Stokes problems

In this paper we apply to the unsteady Navier-Stokes problems some results concerning a posteriori error estimates and adaptive algorithms known for steady Navier-Stokes, unsteady heat and reaction-convection-diffusion equations and unsteady Stokes problems. Our target is to investigate the real viability of a fully combined space and time adaptivity for engineering problems. The comparison between our numerical simulations and the literature results demonstrates the accuracy and efficiency of this adaptive strategy.