2.3 Flow through a surface

The concept of flow through a surface appears in many areas of CFD, including fluid dynamics equations, numerical methods, boundary conditions and general flow calculations. When we talk about something that travels through a surface, the term flux is generally used.2

To quantify the flux of some property, we multiply the area of surface by the property at the surface. If the property is a vector, we take the component normal to the surface. For example, the flux eqn associated with velocity through a surface segment of area eqn would be eqn.

PICT\relax \special {t4ht=

As shown in the figure, eqn can be calculated from eqn by the inner product with normal vector eqn of unit length, expressed as

un = n u: \relax \special {t4ht=
The flux associated with eqn is
 d = n u dS = dS u: \relax \special {t4ht=
It is a good habit to write eqn first since the order matters with a tensor, e.g. stress eqn, introduced in Sec. 2.6 , i.e. eqn

Inner product of two vectors

The normal component of velocity eqn is described in Eq. (2.5 ) by the inner, or “dot”, product of eqn and eqn. It is calculated for vectors eqn and eqn as shown in the figure below, where eqn denotes the magnitude of the vector and eqn is the internal angle between the two vectors.

PICT\relax \special {t4ht=

The inner product of two vectors is a scalar invariant, since the magnitudes and angle are the same irrespective of the co-ordinate system used. It is calculated from vector components as follows:

a b = a b + a b + a b : x x y y z z \relax \special {t4ht=

The inner product of two vectors is commutative, i.e. eqn. It is distributive, i.e. with an additional vector eqn,

a (b + c) = a b + a c: \relax \special {t4ht=

Scalar multiplication and inner products are associative, i.e.

s(a b) = sa b = a sb = (a b)s: \relax \special {t4ht=

An inner product of a vector with itself is simply the square of the vector magnitude, i.e.

a a = jaj2: \relax \special {t4ht=

2Joseph Fourier used the term fluxion in relation to flow of heat in Théorie analytique de la chaleur, 1822; James Clerk Maxwell used the term flux in A Treatise on Electricity and Magnetism, 1873.

Notes on CFD: General Principles - 2.3 Flow through a surface