Numerical Simulation of the Couette Flow Using Meshless Weak-strong Method

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Procedia Engineering 91 ( 2014 ) 334 – 339

Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) doi: 10.1016/j.proeng.2014.12.070

ScienceDirect

XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP) (TFoCE 2014)

Numerical Simulation of the Couette Flow Using Meshless

Weak-strong

Method

Juraj Mužík

a

_*

a_{Department of geotechnics, Facultyof Civil Engineering, University of Žilina, Žilina 01026, Slovak Republic}

Abstract

The paper deals with use of the meshless method for incompressible fluid flow analysis. There are many formulations of the meshless methods. In the present article, the meshless weak-strong method (MWS) – local formulation of the Navier-Stokes equations is presented. The crucial part of the meshless numerical analysis in the construction of shape functions. The article shows the radial point interpolation method (RPIM) for the shape functions construction, which is able to reproduce the 2nd_order

derivatives with reasonable accuracy. At the end the results of the Couette flow numerical solution using MWS method are presented.

Peer-review under responsibility of organizing committee of the XXIII R-S-P seminar, Theoretical Foundation of Civil Engineering (23RSP).

Keywords:meshless analysis;meshless Petrov-Galerkin method; Navier-Stokes equation;

1. Introduction

Incompressible Navier-Stokes flow in two dimensions is one of the several major problems in fluid mechanics that have been extensively studied both theoretically and numerically. In general, the formulation of incompressible Navier-Stokes equations using primitive variables is often used, but it has limitation in approximating the velocity and pressure. The meshless weak-strong method (MWS) is truly meshless method, which requires no elements or global background mesh, for either interpolation or integration purposes. The first article applying MWS method to compute convection-diffusion and incompressible flow was published by Lin[1]. In his work, he used the MWS with the vorticity-stream function form of the N-S equations. The most attractive aspect of MWS is the lack of integration for the model nodes, because the strong form of governing equation is expressed in sense of shape functions and their direct derivatives. The present paper focuses on the MWS primitive variable method using fractional step method to achieve velocity-pressure decoupling to solve incompressible viscous flow [2].

* Corresponding author. Tel.: +421 41 5135762.

E-mail address:[email protected]

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2. Meshless Shape Function Formulation - Radial Point Interpolation method

Using radial basis functions (RBF) is one of the best solutions to construct the meshless shape function. Multi-quadratic RBF function is one of the most popular radial function [3] and is defined as

q

i

i r

R _X 2_H2

(1) where ri is the distance between the desired point (X) and the field node i (Xi) defined simply as 2D Euclidean

distance. Constants İ and q in the Eq.1 are constants that depend on the type of problem. The RPIM interpolation augmented with polynomials can be written as

_¦

n i m j j j i i a p b R u 1 1 ) ( ) (X X R (X)a p (X)b X T T (2) where Ri(X) is the radial basis function (RBF), n is the number of RBFs, pj(X) is polynomial basis function, m is

number of polynomial basis function, ai and bj are interpolation coefficients. In order to determine ai and bj a support

domain is formed for the point of interest at X, and n field nodes are included in the support domain. Interpolation coefficients can be determined by enforcing the Eq.2 to be satisfied at these n nodes surrounding the point of interest X. This leads to n linear equations, one for each node. The equation system in matrix form can be expressed as

b P Ra

U_S _m (3)

where US is the vector of nodal function values, the RBF moment matrix is and the Pm is polynomial moment

matrix. Eq.3 can be subsequently rewritten as

>

@

» ¼ º « ¬ ª b a X p X R (X)b p (X)a R X T T T T u (4)

and following set of equations is used to obtain unknown vectors of parameters a, b

» ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª » ¼ º « ¬ ª b a G b a 0 P P R 0 U U T m m S S ~ (5)

Using Eq.5 we can obtain

X

>

RT(X) pT(X)

@

G1U~S ĭ~T(X)U~S

u (6)

and finally the RPIM shape functions corresponding to the nodal vector ĭ(X) are obtained as

X

>

X

@

ĭT

I

1

I

2

I

n (7)

Then the unknown function approximation can be written using the RPIM shape functions and nodal displacements

_¦

n i i iu u 1 ) ( S I T _X _U ĭ X (8)

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1 I T I T G X X p X X R X ĭ » ¼ º « ¬ ª w w w wI ( ) I ( ) I (9)

Note that G-1_{usually exists for arbitrarily scattered nodes. Therefore, there is no singularity problem in the RPIM as}

a small number of nodes are used in the local support domain [4]. 3. Governing Equations and Fractional-Step Algorithm

The governing equations for unsteady incompressible viscous fluid flow are Navier-Stokes equations with the continuity equation in the convection term [2,5, 6]. This equation can be written as

x f x p y u v x u u y u x u t u w w w w w w ¸ ¸ ¹ · ¨ ¨ © § w w w w w w 2 2 2 2 Re 1 (10) y f y p y v v x v u y v x v t v w w w w w w ¸ ¸ ¹ · ¨ ¨ © § w w w w w w 2 2 2 2 Re 1 ₍₁₁₎ 0 w w w w y v x u ₍₁₂₎

where u and v are the velocities in x and y direction respectively, p is the pressure, fx and fy are the body forces, Re

is Reynolds number. Eq. 10 and Eq. 11 are the momentum equations and Eq.12 is the continuity equation. A fractional-step algorithm is used to solve this problem (see [2,3]). The time derivative of the velocity vector in a momentum Eq. 10 and Eq. 11 can be replaced with a difference approximation and following relation is obtained

n x n n n n n n n n n f x p y u v x u u y u x u t u u w w w w w w ¸¸ ¹ · ¨¨ © § w w w w ' 2 2 2 2 1 Re 1 (13) x p t f y u v x u u y u x u t u u n n x n n n n n n n n w w ' » ¼ º « ¬ ª w w w w ¸¸ ¹ · ¨¨ © § w w w w ' 2 2 2 2 1 Re 1 (14) and n y n n n n n n n n n f y p y v v x v u y v x v t v v w w w w w w ¸¸ ¹ · ¨¨ © § w w w w ' 2 2 2 2 1 Re 1 (15) y p t f y v v x v u y v x v t v v n n y n n n n n n n n w w ' » ¼ º « ¬ ª w w w w ¸¸ ¹ · ¨¨ © § w w w w ' 2 2 2 2 1 Re 1 (16)

where upper indexes n and n+1 indicate the time step. Eq.14 and Eq.16 are explicit formula for convection and viscous terms and the implicit one for a pressure term. To simplify Eq.14 and Eq.16 we used the fractional step approximation (see e.g. [2]). According this approximation the intermediate velocities Ǌn_{and ܢ}n_{are computed using}

simplified momentum equation

» ¼ º « ¬ ª w w w w ¸¸ ¹ · ¨¨ © § w w w w ' n x n n n n n n n n _f y u v x u u y u x u t u u 2 2 2 2 Re 1 ~ (17)

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» ¼ º « ¬ ª w w w w ¸¸ ¹ · ¨¨ © § w w w w ' yn n n n n n n n n _f y v v x v u y v x v t v v 2₂ 2₂ Re 1 ~ (18)

When we compare Eq.14, 16 and Eq.17, 18 we get

x p t u u n n n w w ' 1 ~ ₍₁₉₎ y p t v v n n n w w ' 1 ~ (20)

The intermediate velocities Ǌn_{and ܢ}n_{does not satisfy the continuity equation(Eq. 12). The velocities u}n+1_{and v}n+1

must satisfy the continuity equation which implies

¸¸ ¹ · ¨¨ © § w w w w ' w w w w y v x u t y p x pn n ₁ ~n ~n 2 2 2 2 (21)

Eq.21 is the Poisson’s equation with non-zero source term [2]. The Eq.19 and Eq.20 are solved explicitly by updating nodal values for velocities. The pressure Eq.21 is solved using MWS over problem domain with boundary conditions pn_|ī

u = pթn and pn/n = qթn.

4. The MWS method formulation

The meshless weak-strong form method (MWS) is truly meshless method which requires no elements or global background mesh, for either interpolation or integration purposes. In MWS the problem domain is represented by a set of arbitrarily distributed nodes. There is possibility to use two different sets of nodes, one set for interpolation of the primary function and second as collocation points at which the governing equation should be satisfied. In MWS method implementation the same nodes are used as interpolation and collocation nodes. The unknown function defined in terms of nodal values using the RBF shape functions have the form of Eq.8 and its derivatives in form of Eq.9. These terms used directly in strong form of the Poisson equation (Eq.21) so for each node the corresponding equation is formulated without any integration. The local weak form of Eq.21 is used only for points within global boundary where natural boundary condition is defined. This hybrid formulation makes the imposition of the natural boundary condition very easy. The equation for collocatable points can be written as:

¸¸ ¹ · ¨¨ © § w w w w ' » ¼ º « ¬ ª w w w w ĭT( , ) ĭT( , ) _p 1 ĭT( , )_u~n ĭT( , )_v~n 2 2 2 2 y y x x y x t y y x x y x n (22)

For points within the global boundary with natural boundary condition, the weighted residual method is used to create the discrete equation by integrating the governing equation over local quadrature domain. The quadrature domain can be arbitrary in theory, but very simple regularly shaped domain, such as rectangles for 2D problems are often used for ease of implementation [6]. A generalized local weak form of the pressure Poisson Eq. 21 defined over local sub-domain ȍs can be written as

_³

³

: : w * * : : ¸¸ ¹ · ¨¨ © § w w w w ' * ' * » » ¼ º « « ¬ ª * w w : ¸¸ ¹ · ¨¨ © § w w w w w w w w s s sq su s d y w y x x w y x t d w n y x w n y x t d w q wd n y x d y w y y x x w x y x n T n T n T n T n T T T v ĭ u ĭ v ĭ u ĭ p ĭ ĭ ĭ ~ ) , ( ~ ) , ( 1 ~ ) , ( ~ ) , ( 1 ) , ( ) , ( ) , ( 2 1 (23)

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The fractional step algorithm can be summarized as follows x Start with initial velocity and pressure field

x Compute intermediate velocity field using Eq.17 and Eq.18 for each node x Solve the Poisson equation (Eq.21) using MWS to obtain the pressure x Update velocities using Eq.19 and Eq.20

5. Numerical Example

In this section the MWS solution of Navier-Stokes equations developed in the previous sections is validated by solving the Couette flow example as a benchmark problem of fluid flow simulation [7, 9]. The problem domain (see Fig.1) is formed by regular nodal model with 41x81 nodes and nodal spacing of ns = 0.025.

Fig. 1. Network and boundary conditions for Couette flow.

The plain Couette flow is defined as viscous flow between two parallel plates with zero external pressure gradient. The bottom plate is stationary and the upper plate is moving with constant velocity and the distance between plates h = 1 (see Fig.1). The exact solution of the unsteady Couette flow is often reported as

_¦

f

1 2 2 sin 1 2 , k t k k e y k k y t y u S Q S S (24) where Ȟ is the kinematic viscosity of fluid [8]. Rectangular computational domain (see Fig. 1) has been used to solve Couette flow. Because the laminar Couette flow is valid only for very low Reynolds’ number, our computations have been carried out with Re = 1. The time step has been constant ǻt = 0.0001 [3].

Horizontal velocity distribution at various time intervals has been compared with the corresponding exact solution according Eq.24 and presented in Fig. 2. The results for time t = 0.01[s] and t = 0.1[s] shows that the solution using MWS moves slightly away from the exact solution with errors according to L2 norm err|t=0.01 = 4.789x10-1 and

err|t=0.1 = 3.745x10-1_{. Table 1 lists errors for all five time intervals.}

Table 1. L2 errors of Couette flow.

Time interval

t = 0.001 t = 0.01 t = 0.05 t = 0.1 t = 1.0

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The differential nature of the MWS method is responsible for the slight inaccuracy because differentiation is roughening operator unlike the integration. Methods based on strong formulation are unstable and needs some sort of tuning (support domain size, MQ function shape parameters, nodal distribution) which is mainly valid for specific type of problem or geometry.

Fig. 2. Comparison of results for Couette flow.

6. Conclusions

In this article, a numerical algorithm using the meshless weak-strong (MWS) method for the incompressible Navier-Stokes equations is demonstrated. To deal with convection term, the fractional step method was adopted and the set of recurrent equations was derived for time stepping procedure. The ability of the MWS code to solve fluid dynamics problems was presented by solving Couette flow problem with reasonable accuracy when compared to exact solution.

Acknowledgements

This contribution is the result of the project supported by Scientific Grant Agency of Slovak Republic (VEGA) No. 1-0789-12.

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