7.1 Introduction

Finite volume method (FVM) is another widely used numerical technique [1,2]. The fundamental conservation property of the FVM makes it the preferred method compared to various existing methods viz. finite difference method (FDM), finite element method (FEM), etc. In this approach, similar to the known numerical methods like FDM (Chapter 5) or FEM (Chapter 6), the volumes (elements or cells) are evaluated at discrete places over a meshed geometry. Then, the involved volume integrals of respective differential equation containing divergence term are converted to surface integrals by using well‐known divergence theorem [3,4]. Further, the simulated differential equation gets transformed over differential volumes into discrete system of algebraic equations. In the final step, the system of algebraic equations is solved by standard methods to determine the dependent variables.

Aforementioned FVM has various other advantages in handling differential equations occurring in science and engineering problems. For instance, a key feature of the FVM is formulation of physical space (domain of the differential equation) on unregulated polygonal meshes. Another feature is that the FVM is quite easy to implement various boundary conditions in a noninvasive manner, because the involved unknown variables are calculated at the centroids of the volume elements, rather than at their boundary faces [5]. These characteristics of the FVM have made it suitable for the numerical simulations in variety of applications viz. fluid flow, heat and mass transfer, etc. [6,7]. As such, in this chapter a brief background of the FVM is explained with respect to a simple example problem.

In the next section, we present the discretization techniques of the FVM.

7.2 Discretization Techniques of FVM

In the FVM, the considered domain on which the conservation laws are applied is subdivided into a set of nonoverlapping cells [ 4 , 5 ]. The conservation laws are applied on each cell to regulate the variables at some discrete points named as nodes. Similar to the FEM [8], there is a considerable freedom in the choice of cells and nodes (i.e. nodes may be at various locations of the cells, such as cell‐centers, cell‐vertices, or mid‐sides). As such, the complete geometrical approach of the FEM can be used in the FVM. There exist different types of cells viz. triangular, quadrilateral, etc. In this regard, Figure 7.1 shows some typical examples of choices of nodes with the associated definition variables [ 1 , 2 ].

The following section presents the general solving procedure of a convection–diffusion problem using the FVM.

7.3 General Form of Finite Volume Method

The FVM depends on approximate solution of the integral form with respect to conservation equations 2,9,10]. In the FVM, the given domain of differential equation is divided into a set of nonoverlapping finite volumes and then the respective integrals of the conservation equations are evaluated by using nodal (function) values at computational nodes. For easy understanding of readers, we present the FVM procedure by solving the convection–diffusion problem. The control volume domain of the convection–diffusion problem is presented in Figure 7.2.

The general form of convection–diffusion problem [ 9 ,11] is represented as

(7.1)

where

• = Transient term (ρ‐Density, φ‐Conservative form of all fluid flow),
• ∇ · (ρuφ) = Convection term (u‐velocity),
• ∇ · (D∇φ) = Diffusion term (D‐Diffusion coefficient) and S_φ = Source term.

Integrating Eq. (7.1) over the given domain V, we obtain

(7.2)

For a fixed domain, and commute each other [11] , so we have

(7.3)

Then, using the Gauss‐divergence theorem [12], we change the volume integrals of Eq. (7.2) into surface integrals as

(7.4)

(7.5)

where n‐is the outward unit normal of the surface S (Fig. 7.2 ).

Now substituting Eqs. (7.3)–(7.5) in Eq. ( 7.2 ), the required integral form for the FVM is obtained as

(7.6)

7.4 One‐Dimensional Convection–Diffusion Problem

The steady‐state convection–diffusion equation 9, 11 ] can be derived from transport equation 7.1) by replacing transient term as

(7.7)

Then, the grid generation for solving Eq. (7.7) is illustrated in Section 7.4.1.

7.4.1 Grid Generation

In the initial step of grid generation, divide the considered domain into equal parts of small domain and place nodal points at the midway in between each small domain as given in Figure 7.3. Then, using these nodal points, create control volume near the edge in such a way that the physical boundaries coincide with control volume boundaries [ 5 , 6 ].

Further, place a general nodal point P for the control volume on the space between A and B in Figure 7.3 . The boundaries of control volumes are placed in the middle, between the adjacent nodes. Then, each node is enclosed by a control volume or cell. The nodes to the west and east are identified by W and E, respectively. The west‐side face and east‐side face of control volume are referred to as w and e, respectively. The distances between WP, wp, pe, and PE are denoted by δx_WP, δx_wp, δx_pe, and δx_PE, respectively.

7.4.2 Solution Procedure of Convection–Diffusion Problem

In the absence of source S_φ from Eq. ( 7.7 ), the steady‐state convection–diffusion in one‐dimensional flow field [ 11 ,13] is given by

(7.8)

Then, by integrating Eq. (7.8) as over control volume [ 11 , 13 ], we may obtain

(7.9)

where the gradient from east to west is calculated with the help of nodal points (Figure 7.3 ) as

(7.10)

Now, by substituting Eq. (7.10) in Eq. (7.9), we may get

(7.11)

Assume S_w = S_e = S, δx_PE = δx_WP = δx, F = ρu (convective mass flux per unit area), and (diffusion conductance) in order to obtain the discretized equation 9, 11 ]. Further, by rearranging Eq. (7.11), we obtain

(7.12)

where F_e = (ρu)_e and F_w = (ρu)_w.

The central difference approximation with linear interpolation [ 9 , 13 ,14] has been used to compute the cell face values φ_e and φ_w in a uniform grid

(7.13)

On substituting Eq. (7.13) in Eq. (7.12), we obtain

(7.14)

Rearranging Eq. (7.14), we get

7.15

After simplifying Eq. (7.15), we obtain the discretized equation as

(7.16)

where

(7.17)

(7.18)

(7.19)

and F_e = F_W = F ⇒ F_e − F_W = 0.

Now, by using Eqs. (7.17) and (7.18), one can rewrite Eq. (7.19) as

(7.20)

Finally, the linear system (discretized equations) (7.16) can be handled using any standard method to evaluate the unknown cell face values φ_P, φ_W, and φ_E.

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