Local Nusselt number for various values of Prandtl number (the upper: present results, the lower: Churchill and Ozoe [4]).
Abstract
The present chapter introduces incompressible Newtonian fluid flow and heat transfer by using the finite difference method. Since the solution of the NavierStokes equation is not simple because of its unsteady and multidimensional characteristic, the present chapter focuses on the simplified flows owing to the similarity or periodicity. As a first section, the first Stoke problem is considered numerically by introducing the finite difference method. In the second section, 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. In the third example, linear stability analysis for the onset of secondary flow during the TaylorCouette flow is numerically treated using the HSMAC method.
Keywords
 finite difference method
 similar solution
 boundary layer
 linear stability analysis
 HSMAC method
1. Introduction
The governing equation for the fluid flow is known as NavierStokes 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 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
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
This chapter consists of three main bodies. First, a numerical technique for solving the boundary value problem called the
2. Unsteady flow due to sudden movement of the plate
2.1. Governing equations
An infinite length plate is set in a stationary fluid as an initial condition. Let us consider the situation that the infinite length plate suddenly moves along its parallel direction at a constant speed
Here,
In order to reduce the partial differential equation to an ordinary equation, the following dimensionless velocity
Then, the following ordinary differential equation can be obtained
The boundary condition for the ordinary differential equation is as follows using the similar variable
As a consequence, one needs to solve this boundary value problem. The theoretical solution can be easily obtained and expressed using the error function
The velocity profile is shown in Figure 1.
2.2. Numerical method for solving the ordinary differential equation using finite difference method
For numerical solution, it is necessary to define the range of
As illustrated in Figure 2, in which vertical and horizontal axes are exchanged from Figure 1, one needs to obtain each value of dimensionless velocity numerically. The approximated velocity profile is expressed by connecting these values smoothly. For simplicity, the intervals between neighboring two points are the same and it is noted as
Here,
Here,
In the following, the case of
This kind of tridiagonal matrix is often seen and can be solved by a direct numerical method, such as Tomas method. However, the rank of the matrix is usually extremely large and one introduces an iterative method for solving the kingsize matrix.
2.3. Iterative method for matrix solver
In general, the rank of the matrix appearing in computational fluid dynamics (CFD) is large and iterative methods such as Jacobi, GaussSeidel, or successive over relaxation (SOR) methodare employed. In this subsection, the Jacobi method is explained. The matrix can be divided into three parts of lower, diagonal, and upper as follows:
In the Jacobi method, only the diagonal part is put in the lefthand side (
Here,
This is equivalent to the following equation:
By using Eq. (9), Eq. (14) is computed repeatedly and then the value of each grid gradually converges to a certain solution. The GaussSeidel and SOR methods are known as the faster convergence method.
3. Similarity solution for natural convection heated from a vertical plate
3.1. Introduction
In this section, let us consider the natural convection heat transfer for a vertical plate heated with uniform heat flux in the wide range of Prandtl number from zero to infinity. In order to explain the numerical method as how to solve the governing equations, one assumes that the flow and temperature fields formed in the vicinity of the heated plate have a similarity and then one introduces the finite difference method to obtain numerical results.
3.2. Governing equations
One assumes that the flow is incompressible laminar and boundary layer equations are used in this analysis. The governing equations with presuming the Boussinesq approximation are shown in Eqs. (15)–(17) together with the boundary condition (18). Here, one defines that
Continuity of mass
Momentum equation
Energy equation
Boundary equation
Here,
3.3. Nondimensionalization
First, dimensionless variables, such as velocity and temperature, are set as follows using the unknown reference value denoted with subscripts
Equation (19) is substituted into Eqs. (15)–(18), and one gets
At the moment stage,
Putting [3] = 0, and one obtains
Putting [6] = 1, and one gets
Putting [5] = 1, and one gets
Putting [1] = 1,
Putting [4] = 1,
In the above process, finally one obtains the dimensionless equations as follows:
The dimensionless variables are summarized as follows:
Furthermore, one assumes that the velocity and temperature fields has a similarity along the direction of vertical plate, so one puts
Continuity of mass
Momentum equation
Energy equation
Boundary conditions
The dimensionless variables and nondimensional numbers are defined as follows:
The local Nusselt number can be obtained by the following derivation:
Therefore, the local Nusselt number can be obtained just from the dimensionless temperature at the wall using Eq. (36)
If the Prandtl number is higher than unity, the following equations are useful:
Continuity of mass
Momentum equation
Energy equation
Boundary conditions
The dimensionless variables and nondimensional numbers are defined as follows:
3.4. Numerical results
Figure 3 shows the numerical result for the various Prandtl number cases. The upper figures indicate the vertical velocity and lower ones the temperature. The lefthand side figures show the cases of Pr ≥ 1, while the righthand side ones the cases of Pr ≤ 1
Table 1 shows the summary of the local Nusselt number for various Prandtl number cases together with the reference of Churchill and Ozoe for comparison [4]. The agreement is quite good except for the extreme cases such as Pr → 0 and ∞. In such extreme cases, a small amount of discrepancy exists. In this study, the boundary condition for Pr → 0
and that for Pr → ∞
are used. Owing to this kind of special treatments for the boundary condition of such extreme cases, one can obtain accurate numerical results for the system of ordinary equations. The results between the solution of the present method and that of Le Fevre [5] for the case of constant temperature of heated wall are identical to each other. The value for Pr → ∞ is 0.5027 and that for Pr = 0 is 0.6004.
Pr  0  0.01  0.1  1  10  100  ∞ 


N/A  0.4564 0.456 
0.5234 0.524 
0.5495 0.550 
0.5631 0.5627 


0.7107 0.6922 
0.6694 0.670 
0.5970 0.597 
0.4564 0.456 
N/A 
4. Linear stability of TaylorCouette flow
4.1. Governing equations
In the text book of Chandrasekar [6], various examples of the linear stability analysis such as the
Here, it is indicated that
4.2. Basic state and linearization
The cylindrical enclosure is long enough to neglect the top and bottom ends. In that situation, the basic states for the azimuthal component of velocity and pressure are as follows:
Azimuthal velocity
Pressure
Here,
After neglecting the secondorder disturbance, the following linearized equations are obtained:
By considering the periodicity of the secondary flow which could be happened, each component of infinitesimal disturbance is assumed to be given in the following form. Here,
4.3. Linear stability analysis
The dimensionless simultaneous ordinary equations are summarized as follows:
Basic velocity
Disturbance equations for amplitude functions
Here, the dimensionless variables and nondimensional numbers are as follows. The outer radius
The boundary conditions are as follows:
After Chandrasekar [6], the following two nondimensional numbers are introduced to verify the computational results:
In this section, it is assumed that
Table 2 shows the computational results for various rotation speeds at
Present (201 grids)  Chandrasekar [6]  

Critical wave number  Critical 
Wavenumber  
0  1/4  6.286  15316  6.4  15332 
0.4  1/6  6.293  19518  6.4  19542 
0.6  2/17  6.299  22617  6.4  22644 
1.0  0  6.325  33062  6.4  33100 
4/3  −1/8  6.403  53210  6.4  53280 
1.6  −1/4  6.715  98520  6.4  99072 
1.8  −4/11  7.819  197715  7.8  199540 
1.9  −9/21  8.733  288761  8.6  293630 
2.0  −1/2  9.602  417734  9.6  428650 
The simultaneous ordinary equations from (56) to (59) were divided into the real and imaginary parts. However, only four equations among the eight equations are necessary to solve in this problem because of the symmetricity and antisymmetricity of the complex variables. In Figures 6 and 8, the real part of
Here, the subscripts
References
 1.
Schlichting H, Gersten K. Boundary Layer Theory. 8th Revised and Enlarged ed. Berlin: Springer; 1999  2.
Stokes GG. On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society. 1856; 9 (Part II):8106 or Collected Papers III  3.
Rayleigh L. On the motion of solid bodies through viscous liquids. Philosophical Magazine. 1911; 21 :697711  4.
Churchill SW, Ozoe H. A correlation for laminar free convection from a vertical plate. Journal of Heat Transfer. 1973; 95 (4):540541  5.
Le Fevre EJ. Laminar free convection from a vertical plane surface. In: Proceedings of the 9th International Congress of Applied Mechanics; Brussels; Vol. 4; 1956. pp. 168174  6.
Chandrasekar S. Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press; 1961  7.
Koschmieder EL. Bénard cells and Taylor vortices. In: Cambridge Monographs on Mechanics and Applied Mathematics. United State of America: Cambridge University Press; 1993  8.
Hirt CW, Nichols BD, Romero NC. SOLA: A Numerical Solution Algorithm for Transient Fluid Flows. Los Alamos, New Mexico, United State of America: Los Alamos Scientific Laboratory of University of California. 1975. p. LA5852  9.
Tagawa T, Egashira R. Fluid flow of a liquid metal in a cylinder driven by a rotating magnetic field. Transactions of the Japan Society of Mechanical Engineers, Part B. 2012; 78 (794):16801695  10.
Tagawa T. Numerical investigation of BénardMarangoni convection of paramagnetic liquid in annular layers. In: The 15th International Heat Transfer Conference; Kyoto, Japan; August 2014