Notes on Computational Fluid Dynamics: General Principles

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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 $\text{[math]}$ associated with velocity through a surface segment of area $\text{[math]}$ would be $\text{[math]}$ .

$PICT\relax \special {t4ht=$

As shown in the ﬁgure, $\text{[math]}$ can be calculated from $\text{[math]}$ by the inner product with normal vector $\text{[math]}$ of unit length, expressed as

un = n u: \relax \special {t4ht=

(2.5)

The ﬂux associated with $\text{[math]}$ is

d = n u dS = dS u: \relax \special {t4ht=

(2.6)

It is a good habit to write $\text{[math]}$ ﬁrst since the order matters with a tensor, e.g. stress $\text{[math]}$ , introduced in Sec. 2.6 , i.e. $\text{[math]}$

Inner product of two vectors

The normal component of velocity $\text{[math]}$ is described in Eq. (2.5 ) by the inner, or “dot”, product of $\text{[math]}$ and $\text{[math]}$ . It is calculated for vectors $\text{[math]}$ and $\text{[math]}$ as shown in the ﬁgure below, where $\text{[math]}$ denotes the magnitude of the vector and $\text{[math]}$ 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. $\text{[math]}$ . It is distributive, i.e. with an additional vector $\text{[math]}$ ,

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=

(2.7)

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