Equations with first derivatives

Here the finite volume method will be illustrated for the general first-order equation

where appropriate choices of E, F and G represent the various equations of motion. For example, for and Eqn. (15.1) is the two-dimensional continuity equation and for , it is the inviscid momentum equation in the x-directions, and so on.

In a similar manner, for x direction viscous momentum equation,

(15.2)

Assuming the finite volume (quadrilateral) ABCD shown in Fig. 15.1 is the representative of the control volume we consider the area integral of (15.1) over :

(15.3)

Recall the Green's theorem

(15.4)

Applying Green's theorem, (15.3) becomes

H.n

(15.5)

where H and is the outward unit normal of segment (see Figure 14.5). For a segment On a counter-clockwise contour, the outward unit normal where For the continuity equation, and

Figure 15.1: Finite Volume Mesh system.

In Cartesian coordinates,

Equation (15.5) is just a statement of conservation. For the particular choice, , in Eqn. (15.5) concides with an integral statement of the conservation of mass. As mentioned, the finite volume method is a discretization of the governing equation in integral form, in contrast to the finite difference method, which is usually applied to the governing equation in differential form.