论文研究-求解二维非定常不可压缩Navier-Stokes方程的简化差分变换方法与微扰迭代算法的比较

158 浏览量 2020-05-24 10:13:53 上传评论收藏 1.92MB PDF 举报

资源详情

资源评论

Journal of Applied Mathematics and Physics, 2018, 6, 2518-2543

http://www.scirp.org/journal/jamp

ISSN Online: 2327-4379

ISSN Print: 2327-4352

DOI:

10.4236/jamp.2018.612211 Dec. 12, 2018 2518 Journal of Applied Mathematics and Physics

A Comparison between the Reduced

Differential Transform Method and

Perturbation-Iteration Algorithm for Solving

Two-Dimensional Unsteady Incompressible

Navier-Stokes Equations

Abdul-Sattar J. Al-Saif, Assma J. Harfash

Department of Mathematics, College of Education for Pure Science, University of Basrah, Basra, Iraq

Abstract

In this work, approximate analytical solutions to the lid-

driven square cavity

flow problem, which satisfied two-

dimensional unsteady incompressible

Navier-Stokes equations, are

presented using the kinetically reduced local

Navier-Stokes equations. Reduced differential transform method and pertur-

bation-iteration algorithm are applied to solve this problem. The convergence

analysis was discussed for both methods. The numerical results of both me-

thods are given at some Reynolds numbers and low Mach numbers, and

compared with results of earlier studies in the review of the literatures. These

two methods are easy and fast to implement, and the results are close to each

other and other numerical results, so it can be said that these methods are

useful in finding approximate analytical solutions to the unsteady incompressi-

ble flow problems at low Mach numbers.

Keywords

Unsteady Incompressible Viscous Flows, Reduced Differential Transform

Method, Perturbation-Iteration Algorithm

1. Introduction

Fluid flow is one of the most important engineering phenomena that have re-

ceived widespread attention in theoretical and practical scientific research. Many

of these studies focus on simulated mathematical models which represent these

phenomena. Therefore, the equations of Navier-Stokes, which are the basic

How to cite this paper:

Al-Saif, A.-S.J. and

Harfash

, A.J. (2018) A Comparison be-

tween the Reduced Differential Transform

Method and Perturbation

-Iteration Algo-

rithm for Solving Two

-Dimensional Un-

steady Incompressible Navier

-Stokes Equa-

tions

Journal of Applied Mathematics and

Physics

, 2518-2543.

https://doi.org/10.4236/jamp.2018.612211

Received:

November 12, 2018

Accepted:

December 9, 2018

Published:

December 12, 2018

8 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

A.-S. J. Al-Saif, A. J. Harfash

DOI:

10.4236/jamp.2018.612211 2519 Journal of Applied Mathematics and Physics

model for describing the movement of fluid, have received considerable atten-

tion from researchers to find their analytical and numerical solutions.

In this work, unsteady viscous incompressible flows characterized by

two-dimensional Navier-Stokes equations are studied. The non-dimensional

momentum and continuity equations have the following form

( ) ( )

t x y x xx yy

t x y y xx yy

u uu vu p u u

v uv vv p v v

=− +++ +

=− ++ + +

(1.1)

and

= 0,

uv+

(1.2)

where

is the physical time,

( )

,,u xyt

and

( )

,,v xyt

are the fluid velocity

components,

( )

,,p xyt

is the pressure, and

is the Reynolds number. Since,

the Navier-Stokes equations are nonlinear partial differential equations and

there is no explicit equation for calculating pressure, these equations are difficult

to solve, so many studies have suggested the alternative thermodynamic description

of incompressible fluid flows. One of these alternative formulas is the kinetically

reduced local Navier-Stokes (KRLNS) equations [1] [2] [3] [4] [5] which is

obtained by replacing the pressure by

= +

(1.3)

and the continuity equation by

( )

( ) ( )

t x y xx yy

g uv g g

=− ++ +

(1.4)

where

is the Mach number and

( )

,,g xyt

is the grand potential. The time

scale in INS equations is related to that of KRLNS equations;

( )

KRLNS NS

t Ma t

= ×

Then, the system of equations of KRLNS has the following form

( ) ( )

( )

t x x y x xx yy

t x y y y xx yy

t x y xx yy

u uu vv vu g u u

v uv uu vv g v v

g uv g g

=− ++++ +

=− ++ ++ +

=− ++ +

(1.5)

The KRlNS equations suggested in [1] of the reduced equations for the grand

potential and the fluid momentum were derived from the compressible Navi-

er-Stokes equations in order to present the thermodynamic description of in-

compressible fluid flows at low Mach numbers. The two-dimensional KRLNS

system is simplified and compared with a Chorin’s artificial compressibility me-

thod for steady state computation of flow in two-dimensional lid-driven cavity

and Taylor-Green vortex flow in [2]. In [3], KRLNS equations were applied to

two-dimensional simulation of doubly periodic shear layers and decaying ho-

A.-S. J. Al-Saif, A. J. Harfash

DOI:

10.4236/jamp.2018.612211 2520 Journal of Applied Mathematics and Physics

mogeneous isotropic turbulence, where the central difference scheme is used for

the spatial discrimination and four stage. Runge-Kutta method is utilized for the

time integration. High order approach of the KRLNS equations was applied to

two-dimensional numerical simulations of Womersley problem, doubly periodic

shear layers and three-dimensional decaying homogeneous isotropic turbulence

in [4] [5].

The lid-driven cavity problem refers to the flow in a box cavity with no-slip at

the walls, one or more which move at constant speed. It has been used exten-

sively as a benchmark case for the study of computational methods to solve

Navier-Stokes equations, because the simplicity of its geometry and boundary

conditions. Numerous literature studies have offered the solutions for this prob-

lem by using the different numerical methods in rectangular or square cavities.

For example, in [6], the implicit cell-vertex finite volume method was described

to solve the steady and unsteady two-dimensional lid-driven cavity problem at

high Reynolds numbers. In [7], Chebyshev-collocation method in space is in-

troduced with Adams-Bashforth backward-Euler scheme for the time integra-

tion to calculate the solution of three-dimensional lid-driven cavity flows. The

finite element scheme based on the Galerkin method of weighted residuals of

unsteady laminar mixed convection heat transfer in a lid driven cavity is per-

formed in [8]. The vorticity-stream formulation of the Navier-Stokes equation

with the strong-stability-preserving Runge-Kutta (SSPRK (5, 4)) scheme in very

fine grid mesh was used for solving lid driven cavity at high Reynolds number in

[9]. For the problem of flow inside a square cavity with constant velocity, the fi-

nite volume method with numerical approximations of second-order accuracy

and multiple Richardson extrapolations is utilized in [10]. The compact finite

difference approximation is developed for non-uniform orthogonal Cartesian

grids in [11] for solving the stream function-velocity formulation of the steady

two dimensional incompressible lid-driven square cavity flow problem. The

numerical simulations of two-dimensional fluid flow and heat transfer in a

four-sided lid-driven rectangular domain have been preformed in [12], where

the quadratic upstream interpolation for convective kinematics (QUICK)

scheme of finite volume methods was used and semi-implicit method for pres-

sure linked equations (SIMPLE algorithm) was adopted to compute the numeri-

cal solutions of the flow variables.

The main aim of this study is to obtain the approximate analytical solutions

for two-dimensional lid-driven square cavity flow problem, since most of the re-

search focused on the numerical solutions for this problem. Reduced differential

transform method (RDTM) and perturbation-iteration algorithm (PIA) are used

for this purpose for several reasons. The first reason is that both methods have

not previously been applied to resolve this problem. Secondly, these methods

can directly be applied to KRLNS equations. Moreover, these methods can re-

duce the size of the calculations and at the same time maintain the accuracy of

the numerical solution.

We have organized this paper into seven sections, of which this introduction

A.-S. J. Al-Saif, A. J. Harfash

DOI:

10.4236/jamp.2018.612211 2521 Journal of Applied Mathematics and Physics

is the first. In Section 2 and 3, we describe the reduced differential transform

method and perturbation-iteration algorithm, and applied them to KRLNS equ-

ations. We derived the condition of convergence for both methods (Section 4).

We then present the approximate analytical solutions for two-dimensional

lid-driven cavity flow, which are obtained by applying differential transform

method and perturbation-iteration algorithm (Section 5). Next, we introduce the

numerical results and compare these results with other works (Section 6). The

last Section summarizes the major findings of this study.

2. Reduced Differential Transform Method (RDTM)

The RDTM was first introduced by Keskin [13]. It is an iterative procedure

based on the use of the Taylor series solution of differential equations. It has

been successfully applied to solve various nonlinear partial differential equations

[13]-[27]. Since it does not require any parameter, discretization, linearization or

small perturbations, thus it reduces the size of computations and can be easily

used. The RDTM was used for solving the generalized Korteweg-de Vries equation

[14], the fractional Benney-Lin equation [15], the Wu-Zhang equation [16], the

equal width wave equation and the inviscid Burgers equation [17], the Sine-Gordon

equation [18], the Burgers and Huxley equations [19], the time-fractional telegraph

equation [20], the generalized Drinfeld-Sokolov equations and Kaup-Kupershmidt

equation [21], the Zakharov-Kuznetsov equations [22], the heat-like equations

[23], the coupled Ramani equations [25], two integral members of nonlinear

Kadomtsev-Petviashvili hierarchy equations [26], and the second order hyper-

bolic telegraph equation [27]. Few studies have been applied RDTM to solve the

Navier-Stocks equations, which is one of the reasons for choosing it as a method

for solving the lid-driven cavity flow.

In this section, we give some properties of the (2 + 1)-dimensional RDTM

[16] [18] [20] [22] [23] [24] [26] [27] which is used to find the approximate so-

lutions to two-dimensional Navier-Stokes equations. Consider

( )

,X xy=

be a

vector, if

( )

,u Xt

is analytic function and continuously differentiable with re-

spect to time

and space in the domain of interest. Then, let

(

) ( )

U X u Xt



∂



∂



(2.1)

is the

-dimensional spectrum function of

( )

,u Xt

which is the transformed

function. The reduced differential inverse transform of

( )

is defined as

( ) ( )

u Xt U X t

∞

∑

(2.2)

from Equation (2.1) and Equation (2.2), we can conclude that

( ) ( )

, ,.

u Xt u Xt t

∞



∂



∂



∑

(2.3)

The fundamental mathematical operations performed by RDTM are readily

obtained and listed in

Table 1.

剩余25页未读，继续阅读

评论收藏

内容反馈

论文研究 - 求解二维非定常不可压缩Navier-Stokes方程的简化差分变换方法与微扰迭代算法的比较

评论0

最新资源

论文研究 - 求解二维非定常不可压缩Navier-Stokes方程的简化差分变换方法与微扰迭代算法的比较

评论0

最新资源

相关推荐

【Matlab源码】二维非定常Navier-Stokes方程的求解.zip

解非定常不可压缩N_S方程的迭代压力Poisson方程

二维navier-Stokes (非定常)MATALAB 示例

R中不可压缩的Navier-Stokes方程的经典基本唯一解

隐式格式的MATLAB代码-cavity-NS:NavierStokes方程的二维求解器在腔体流动问题，有限差分方案和分数阶方法学中的应用

Navier_Stokes_2D:二维有限差分纳维斯托克斯码

12-steps-to-navier-stokes:该项目包含12个用于计算Navier Stokes方程的代码

黏性系数依赖于密度的一维可压Navier-Stokes方程的自模解 (2010年)

CFDPython：Jupyter笔记本的序列，具有“ Navier-Stokes的12个步骤” http://lorenabarba.com

2D_Cavity_Flow.py

matlab求解微分方程组代码-navier_stokes_multigrid:使用压力校正方案和并行多网格v周期求解3DNavierStok

涡流函数公式中的通道流求解器：Navier-Stokes 方程涡流函数公式中二维通道流问题的求解器-matlab开发

平板上超声速流动的数值解：求解层流状态下的 Navier-Stokes 方程以模拟平板上的超声速流动。-matlab开发

CFD-Navier-Stokes:在 MATLAB 中构建纳维斯托克斯方程的有限差分解决方案的集合。-matlab开发

2d流体模拟软件

顶盖驱动流cavity.cpp

有限差分matlab代码-PseudospectralTutorial:伪光谱教程

FluiDNS_Windows:Windows版本的2D DNS求解器-FluiDNS

用CT-VOF法在数值波试验槽中生成线性和非线性波 (2012年)

基于非结构化同位网格的SIMPLE算法 (2003年)

MATLAB有限元分析与应用源代码-flow_sims:二维盖驱动腔流模拟：FEM，FDM和SIMPLE

气液两相分层流动相界面迁移的数值模拟 (2009年)

双曲柱面matlab代码-CFD-Simulation:用MATLAB和基于Python的PyFR编写的流体求解器代码的数值模拟。方法包括有限

环形喷嘴气动谐振加热性能 (2010年)

港口内靠码头系泊船波浪力时域模型 (2003年)

Qt 5实现串口调试助手 （源工程文件、0积分下载）

【SystemVerilog】路科验证V2学习笔记（全600页）.pdf

AutoSAR标准协议4.2.2

光伏-储能并网系统仿真.rar

Qt 5实现串口调试助手（源工程文件、0积分下载）