Notes on Computational Fluid Dynamics: General Principles

2.4 Conservation of mass

The law of conservation of mass can be written

(2.8)

where is the mass density of the fluid. The equation can be derived by considering a volume fixed in space (note !!), bounded by a surface . If the volume is filled by a fluid with density , its mass is

The rate of increase of mass inside the volume must match the rate of inflow of mass across the volume’s surface. The latter is calculated by integrating the mass flux over the surface, noting the negative sign due to pointing out of the volume.

Gauss’s Theorem relates surface and volume integrals by

(2.9)

Equating the rate of mass increase to rate of inflow and applying Eq. (2.9 ) gives

(2.10)

Since the integral is valid for any volume , it follows that the integrand (in ) must equal 0, resulting in Eq. (2.8).

Divergence

Divergence, denoted by , indicates the tendency of a vector field to point outward of a closed surface. For example, When the divergence of velocity is positive, the fluid is expanding; negative divergence indicates contraction. Imagine a volume of fluid with a complex distribution of at its bounding surface, below left. The calculation will isolate the diverging component (right) from uniform flow (centre).

The divergence of velocity is calculated by integrating — i.e. summing — fluxes over the closed surface. To define divergence at a point, we consider the limiting case where the volume tends to zero. For a surface that encloses a volume , divergence is the flux across the surface per unit volume, as , e.g.

(2.11)

The nabla operator

The nabla symbol can be considered a vector operator

(2.12)

Divergence is the inner product with , e.g.

(2.13)