LOCAL METHOD OF APPROXIMATE PARTICULAR SOLUTIONS FOR TWO-DIMENSIONAL UNSTEADY INCOMPRESSIBLE FLOW

The isothermal incompressible viscous flow is used to describe flows taking advantage of simplified flow equations that describe fluid motion, which is known as the incompressible Navier-Stokes equations. There are numerous approaches developed to solve the incompressible Navier-Stokes equations; the algorithm used in this article represents the Chorin’s projection method (fractional step) that belongs to the class of the decoupled methods which are more commonly used in computational fluid dynamics (CFD). Other way to classify the algorithms is to determine the version of the underlying governing equations, whether they are expressed using the primitive variables, velocity component and pressure, or not [1]. The mentioned fractional step method uses primitive variable formulation of the Navier-Stokes equations. The fractional step algorithm is chosen to approximate the solutions of the incompressible Navier-Stokes equations in conjunction with the localized method of approximated particular solutions (LMAPS) that represents meshless collocation method [2]. The fractional step method solves the incompressible NavierStokes equations in three steps. In the first step the simple form of the momentum equations is obtained by dropping the pressure source term; this results in the intermediate velocities. In the next step the intermediate velocities are projected on the divergence-free plane using the solution of the Poisson’s pressure equation and, subsequently obtaining velocity corrections using the pressure gradients [1]. The main drawback of mesh-based numerical methods is the requirement of the mesh generation, which may be difficult to process during the numerical implementations, especially for multi-dimensional problems. In order to avoid mesh generation process many meshless numerical methods, such as the local boundary integral element method (LBIEM) [3 7], smoothed particle hydrodynamics (SPH) method [8], meshless strong (collocation) method [9 11] and meshless local Petrov-Galerkin method (MLPG) [12 and13] have been developed. The LMAPS used in this study represents a stable, accurate tool for simulating the two-dimensional incompressible viscous flow field with the projection method. Numerical experiment of the two-dimensional lid-driven cavity flow problem is presented in the article with the comparison against literature data [14], which verifies the performance of the LMAPS.


Introduction
The isothermal incompressible viscous flow is used to describe flows taking advantage of simplified flow equations that describe fluid motion, which is known as the incompressible Navier-Stokes equations. There are numerous approaches developed to solve the incompressible Navier-Stokes equations; the algorithm used in this article represents the Chorin's projection method (fractional step) that belongs to the class of the decoupled methods which are more commonly used in computational fluid dynamics (CFD). Other way to classify the algorithms is to determine the version of the underlying governing equations, whether they are expressed using the primitive variables, velocity component and pressure, or not [1]. The mentioned fractional step method uses primitive variable formulation of the Navier-Stokes equations.
The fractional step algorithm is chosen to approximate the solutions of the incompressible Navier-Stokes equations in conjunction with the localized method of approximated particular solutions (LMAPS) that represents meshless collocation method [2].
The fractional step method solves the incompressible Navier-Stokes equations in three steps. In the first step the simple form of the momentum equations is obtained by dropping the pressure source term; this results in the intermediate velocities.
In the next step the intermediate velocities are projected on the divergence-free plane using the solution of the Poisson's pressure equation and, subsequently obtaining velocity corrections using the pressure gradients [1].
The main drawback of mesh-based numerical methods is the requirement of the mesh generation, which may be difficult to process during the numerical implementations, especially for multi-dimensional problems. In order to avoid mesh generation process many meshless numerical methods, such as the local boundary integral element method (LBIEM) [3 -7], smoothed particle hydrodynamics (SPH) method [8], meshless strong (collocation) method [9 -11] and meshless local Petrov-Galerkin method (MLPG) [12 and13] have been developed. The LMAPS used in this study represents a stable, accurate tool for simulating the two-dimensional incompressible viscous flow field with the projection method. Numerical experiment of the two-dimensional lid-driven cavity flow problem is presented in the article with the comparison against literature data [14], which verifies the performance of the LMAPS.

Governing equations
An unsteady incompressible flow is governed by Navier-Stokes equations, which can be in a two dimensional case written in dimensionless form using the primitive variables as

Localized method of approximated particular solutions
In order to apply the LMAPS, the following procedure may beadopted to solve physical variable u with any given differential operator.Even though the implementation details of the LMAPS are quite similar to the meshless collocation methods, its origin follows the globally defined method of approximated particular solution (MAPS) (see [2]). In this study the LMAPS is treated as special case of the meshless collocation methodwith the support domain concept already included in the formulation. The area of interest Ω with the boundary 2Ω is covered by points within the area and also on the global boundary (see Fig. 1). Consider a local circular sub-domain Ω s centered at every point s. This sub-domain is regular around all the points, and it is called support domain. Then using the points in a particular support domain any function can be expressed using just nodal values as where NS is the total number of computational nodes in the support domain, F is the radial basis function, x is coordinate vector and α i are the local weighting coefficients to be determined. Because the values of u expressed using Eq. (12) at the computational nodes should be the same asu specified in these nodes, we can write following matrix expression u Fa = where where u, v is the velocity vector component in the direction x and y respectively, p is the pressure, and Re represents Reynolds number, assuming that density and characteristic velocity equals to unity. A fractional step algorithm is used to solve this problem (see [1, 15 and 16]). The time derivative of the velocity in momentum equations Eq.(1, 2) can be replaced with a difference and the following relation is obtained: where upper indexes n and n+1 indicate time level and Δt is the length of the timestep. The first order finite difference, used here to discretize unsteady term, is preffered beacuse the explicit nature of the momentum equations. Eq. (4, 5) arean explicit formula for the convection andviscous terms, and an implicit one for the pressure term. Eq. (4, 5) are simplified using the fractional time step approximation (e.g., [15 and 16] Comparing (4,5) and (6, 7) respectively gives The intermediate velocity components wherer depicts the distance between the collocation points and c is a shape parameter.

Handling derivative boundary conditions
Strong-form methods can produce accurate results for partial differential equations, when the boundary conditions are all of Dirichlet type. If there is any derivative boundary condition, the accuracy of the solution deteriorates drastically, and the solution can be unstable; small changes in the setup of the problem can lead to a large change in the solution. The discretized system equation behaves like an ill-posed problem in which errors introduced into the system are magnified in the output. A number of strategies can be used to impose the derivative boundary conditions in the strong form methods. The method using fictitious points is used in this article to enforce the pressure gradient on the boundary.
Along the derivative boundaries, a set of fictitious points is added outside the problem domain. In this case, two sets of equations are established at each derivative boundary node, one for the derivative boundary condition, and the other for the governing equation.

Discretization of Navier-Stokes equations
In this section, the numerical procedures for the approximation of the Navier-Stokes equations via the fractional step method and the LMAPS will be explained. The fractional step begins with Eq.
Since F is invertible, we will obtain the unknown weighting coefficients as Now, for operators concerned in solving Navier-Stokes equations, we firstly consider the operator with highest order, the Laplace operator ∇ 2 , and let where f is the radial basis function with the essential condition as f=∇ 2 F. Replacing α i with Eq. (17) we get where { is the vector of shape functions Laplacian. We apply similar procedures to the gradient operator ∇ and we get where γ is the vector of shape function gradients and subscript j represents the coordinate of the gradient.
The processed operators may now be applied directly into numerical implementations, which discretizes the governing equations and the essential conditions into formulation of linear systems. All boundary conditions in this paper have also been processed using the LMAPS. Numerical procedures for solving Navier-Stokes equations will be elaborated in subchapter 3.3, and more details of the LMAPS can be found in Ref. [2].

Integrated multiquadrics radial basis functions
The trial function basis used in the LMAPS incorporates multiquadrics RBF, but different approach than the standard meshless collocation methods is adopted. In the case of LMAPS the MQ-RBF [2 and 17] represents the form of basis function after differentiation using highest order operator. To obtain the basis function, MQ-RBF needs to be integrated. Re = 5000

Fig. 3 Lid-driven cavity, streamlines pattern for different Reynolds numbers
Three different meshes are tested here (61⨯61, 81⨯81, and101⨯101 points), with each having a non-uniform distribution similar to that used in [15]. For all calculations the uniform timestep Δt=10 -3 . The strategy of finding neighboring points for RBF interpolation is the main problem facing the use of anonuniform distribution of points.
The segmentation strategy (see [7]) gives the best results for this type of network. For higher Reynolds numbers the solution found using a coarse network does not converge and denser networks must be used. This problem probably arises because the point distribution in the coarse network is unable to capture the velocity gradient within a thin boundary layer for higher Reynolds numbers. The results of the LMAPS are compared with the solutions presented in [14] for the Reynolds number Re=100, 400, 1000 and 5000. The vertical velocity components along a horizontal line through the center ofthe cavity and the horizontal velocity components along a vertical line through the center of the cavity are plotted in Fig. 4.
These comparisons for the three types of mesh show that, as expected, the most dense mesh (101⨯101 points) leads to the results closest to those of [14]. Figures 3, 5 and 6 present streamlines, velocity magnitude and pressure contours for Reynolds numbers Re=100, 400, 1000 and 5000.

Conclusions
A possible use of the localized method of approximate particular solutions with integrated MQ-RBF interpolation the present study. Next, we obtain pressure term from Poisson's equation of the second stage, which can be written as Then, in the third stage we execute the velocity correction explicitly as The convergence criterion for steady state is given as ε =10 -7 for all numerical experiments in this paper. The fractional time step algorithm described above is now used to solve the Navier-Stokes equations at every time step [15]:

Numerical example
The solution of the Navier-Stokes equations using the LMAPS developed in the previous sections is validated here by solving lid-driven cavity flow examples. Lid-driven cavity flow is used as a standard test problem for the validation of numerical solutions of incompressible Navier-Stokes flow. The top wall of the cavity moves with a velocity u 0 =1, and no-slip impermeable boundary conditions are assumed along the other three walls. The geometry and boundary conditions arepresented in Fig. 2. The difficulty of this problem lies in the presence of singularities of pressure and velocity at the two upper corners of the cavity [15]. Therefore, the density of points used increases toward the corners of the cavity (see Fig. 2).