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Numerical manifold method was applied to directly solve Navier-Stokes (N-S) equations for incompressible viscous flow in this paper, and numerical manifold schemes for N-S equations coupled velocity and pressure were derived based on Galerkin weighted residuals method as well. Mixed cover with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. As an application, mixed cover 4-node rectangular manifold element has been used to simulate the incompressible viscous flow around a square cylinder in a channel. Numerical tests illustrate that NMM is an effective and high-order accurate numerical method for incompressible viscous flow N-S equations.

In computational fluid dynamics (CFD), Navier-Stokes (N-S) equations for incompressible viscous flow can be solved by several numerical methods generally, such as finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) [

Numerical manifold method (NMM) also known as manifold method or finite cover method (FCM) is a generalized numerical method proposed by Shi in the early 1990s [

Flow around a square cylinder is a typical model to validate the performance of numerical methods for solution of incompressible viscous N-S equations. The flow structure has been investigated experimentally and numerically. Experimental investigations have shown that the flow characteristic is different at different Reynolds numbers [

In this paper, numerical manifold schemes of direct solutions coupled velocity and pressure for N-S equations were constructed and applied to analyze incompressible viscous flow around a square cylinder in a parallel channel. The validations of numerical schemes to steady and unsteady flow were completed.

For numerical solution of incompressible viscous flow, the integration expressions for the continuity equation and N-S equations can be obtained by Galerkin weighted residual method, and the weak solution form can be stated as

When NMM is applied into solution of N-S equations, suggested that there are

By substituting (

In Galerkin integration equations (

The partial derivatives of element cover functions can be obtained from (

For each element

Equations (

In FEM, when velocity and pressure field are discretized in the same way, the discrete element can not be ensured to satisfy the Ladyzanskya Babuska Brezzi stability condition (LBB condition), and these kinds of element cannot be applied to directly solve N-S equations coupled velocity and pressure for the spatial oscillation of pressure field. Mixed elements can be built to meet the LBB condition by increasing velocity interpolation node [

In finite element cover system of NMM, the element cover functions are composed of the weight functions and the cover functions of physical covers. Generally, the weight functions will adopt the element shape functions as in FEM, which are usually defined by the element shape and nodes. The cover functions of every physical cover for different physical variables can employ different order functions according to the solving physical equations. When NMM is applied to solve N-S equations for incompressible viscous flow, the velocity and pressure field can be discretized in the same way and the weight functions will employ the same interpolation functions for element velocity and pressure variables. But, the cover functions of every element physical cover can apply different order functions for velocity and pressure variables, so it will form a manifold element with mixed cover functions for velocity and pressure variables, which can meet the requirements of different order approximate functions for velocity and pressure in Galerkin integration expressions (

In theory, velocity components and pressure covers can employ very high-order functions, but it will cause a very complicated calculation process, so low-order polynomial functions are favorable. If pressure cover function of every physical cover is defined as constant function, velocity cover functions can adopt linear polynomial functions, and then the basic series in (

As to 2D steady N-S equations, element manifold equations (

The manifold equations (

As to 2D unsteady N-S equations, the first-order linear ordinary differential equation (

The manifold equations (

To illustrate the validity of the present numerical manifold method, low-Re incompressible viscous flow around a square cylinder in a channel and flow past a step are investigated in details through direct solution of N-S equations in dimensionless form in this paper as two numerical examples. In numerical manifold analysis, standard rectangular finite element cover system with 4-node manifold element as show in Figure

4-node rectangular manifold element.

In numerical analysis of flow past a step, finite element cover system and boundary conditions are shown in Figure

The streamline patterns and pressure distributions for flow field past a step.

Flow field configuration and boundary conditions

NNM solutions (

FVM solutions (

FVM solutions (

The streamline and pressure distribution in flow field past a step at Re = 200 are shown in Figure

The flow field configuration and boundary conditions of flow around a square cylinder in a channel are shown in Figure

Flow field configuration and boundary conditions of flow around a square cylinder.

The flow field is meshed by three mixed grids, one is that

The numerical analyses of steady flow at different Re numbers (

The streamline patterns and pressure distributions at different Re numbers are shown in Figure

Streamline pattern and pressure distribution for flow field at different Re numbers.

In NMM analysis unlike FVM and FEM analysis, the pressure distributions of flow field are obtained from direct numerical solutions of N-S equations and continuity equation, so it can improve the solution accuracy of pressure field. The pressure distributions of flow field show that high-pressure area is formed in frontage of the square cylinder for flow blockage, low pressure areas are formed near behind two front corner points for flow separating, and large pressure gradient is produced near the corner points. The pressure distributions are symmetrical, and the pressure of flow field will decrease as Re number increasing.

The velocity

Comparison of velocity

Comparison of pressure along horizontal lines through geometric center of flow field.

The flow field configuration and boundary conditions of unsteady flow around a square cylinder in a channel are the same as these of steady flow, and the flow field is meshed by the third mixed grids. The unsteady flows at different Re numbers (

When

The streamline patterns for flow field from

T/4

T/2

3T/4

The pressure contours for flow field at

The pressure contours for flow field from

T/4

T/2

3T/4

Numerical manifold method for direct coupled solution of incompressible viscous flow N-S equations has been developed in this paper. Numerical manifold schemes integrated velocity and pressure were derived based on Galerkin-weighted residuals method as well. Mixed cover with linear polynomial function for velocity and constant function for pressure was adopted in finite element cover system. Compared with FVM and FEM, in NMM for incompressible viscous flow, the accuracy of velocity variables approximation can be improved by adopting high-order cover function, direct numerical solution of N-S equations, and continuity equation coupled velocity and pressure variables can be implemented by adopting finite cover system with mixed cover manifold element, so it can improve the solution accuracy of velocity and pressure variables.

As an application, mixed cover 4-node rectangular manifold element has been used to simulate flow around a square cylinder in a channel and past a step at low Re numbers. As to flow around a square cylinder in a channel, accurate numerical results have been presented for steady flow at

Financial support for this project was provided by the National Natural Science Foundation of China (no. 50975050).