1. Introduction

The governing equation for the fluid flow is known as Navier-Stokes equation, which is however difficult to solve analytically; and therefore, a lot of numerical techniques have been proposed and developed. Nevertheless various complex flow phenomena such as turbulent flow, multiphase flow, compressible flow, combustion, and phase change encountered in the fields of engineering would have still difficulties to circumvent even using both present computational resources and numerical techniques. The present chapter devotes not to elucidate such complex phenomena, but to introduce rather simplified fluid flow by using the finite difference method.

One focuses on incompressible flows, in which physical properties such as the viscosity, the thermal conductivity, the specific heat are constant and even the fluid density is not a thermodynamic variable. This simplified assumption makes the fluid flow phenomena much easier to

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

be handled and it is valid when the flow velocity is much slower than the sound velocity and/ or the temperature difference in the fluid is small enough to consider the thermal expansion coefficient is independent to the temperature. The former situation is known the low Mach number approximation, while the latter one the Boussinesq approximation.

y ¼ 0 : u ¼ uw y ! ∞ : u ! 0

In order to reduce the partial differential equation to an ordinary equation, the following

<sup>u</sup> <sup>¼</sup> uwUð Þ <sup>η</sup> , <sup>η</sup> <sup>¼</sup> <sup>y</sup>

dU

The boundary condition for the ordinary differential equation is as follows using the similar

η ¼ 0 : U ¼ 1 η ! ∞ : U ! 0

As a consequence, one needs to solve this boundary value problem. The theoretical solution

ffiffiffi <sup>π</sup> <sup>p</sup> ðη 0

<sup>¼</sup> <sup>1</sup> � erfð Þ¼ <sup>η</sup> <sup>1</sup> � <sup>2</sup>

2 ffiffiffiffi

<sup>ν</sup><sup>t</sup> <sup>p</sup> (3)

http://dx.doi.org/10.5772/intechopen.72263

exp �ξ<sup>2</sup> � �d<sup>ξ</sup> (6)

<sup>d</sup><sup>η</sup> <sup>¼</sup> <sup>0</sup> (4)

Numerical Analysis of the Incompressible Fluid Flow and Heat Transfer

(2)

245

(5)

�

dimensionless velocity U and the similar variable η are introduced

Then, the following ordinary differential equation can be obtained

can be easily obtained and expressed using the error function

<sup>U</sup> <sup>¼</sup> <sup>u</sup> uw

The velocity profile is shown in Figure 1.

variable η instead of y:

Figure 1. Velocity profile.

d2 U <sup>d</sup>η<sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>η</sup>

�

Another simplification on the incompressible flows is the reduction of dimension due to the characteristic of similarity and periodicity. For the boundary layer flows such as the Blasius flow, the stagnation-point flow, and the von Kármán rotating disk flow have the similar solution where the flow transition from laminar to turbulence does not occur. In those cases, a combined dimensionless variable (similar variable) η is introduced and the velocity distribution can be only a function of η. While for the onset of instability such as the Rayleigh-Bénard convection, the Bénard-Marangoni convection, and the Taylor-Couette flow, the periodic characteristic of flow structure is observed. At the stage of onset of instability, the non-linear term is negligible and therefore the function of flow field is separated into the amplitude part and periodic part, respectively. This makes the effort on numerical analysis to reduce significantly and also to contribute the augmentation of accuracy of the results.

This chapter consists of three main bodies. First, a numerical technique for solving the boundary value problem called the first Stokes problem or the Rayleigh problem [1] is introduced. The differential equation is transferred into an ordinary equation and it is solved by a finite difference method using the Jacobi method. Second, similar solution of natural convection heat transfer heated from a vertical plate with uniform heat flux is introduced together with the method how to obtain the system of ordinary differential equations. The obtained Nusselt numbers are compared with some previous studies. Third, for example, of the linear stability analysis, one shows that the HSMAC method can be applied to obtain the critical values for the onset of secondary flow such as the Taylor-Couette flow. The Eigen functions of flow and pressure fields are visualized.
