Notes on CFD: General Principles

[chapter contents][previous topic][next topic][index]

2.2 Velocity

Like force, velocity $\text{[math]}$ is a vector with direction and magnitude, with SI units of $\text{[math]}$ . Using the vector $\text{[math]}$ to denote the position of a particle of ﬂuid, its velocity is

dx x u = ---= lim ---: dt t!0 t \relax \special {t4ht=

(2.4)

$PICT\relax \special {t4ht=$

Vector ﬁelds

While $\text{[math]}$ can be used to denote a single velocity with magnitude and direction, it can also denote a vector ﬁeld of velocity which varies from point to point across a spatial domain. A vector is represented by 3 numbers, relating to the co-ordinate system being used, e.g. $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , in the Cartesian system.

$PICT\relax \special {t4ht=$

While the magnitude and direction of a vector is ﬁxed, it is not invariant since the 3 values depend on the co-ordinate system used. We represent a vector without reference to the co-ordinate system by bold text, e.g. “ $\text{[math]}$ ” (compared to scalar “ $\text{[math]}$ ”).

Basic vector algebra

Addition and subtraction of 2 vectors is performed by operating on respective components. Subtraction of two vectors $\text{[math]}$ and $\text{[math]}$ is performed by

a b = (ax bx;ay by;az bz): \relax \special {t4ht=

Multiplication of any vector $\text{[math]}$ by a scalar $\text{[math]}$ is performed by multiplying all the components by the scalar, e.g.

sa = (sa ;sa ;sa ) x y z \relax \special {t4ht=

Addition and multiplication are commutative, i.e. variables can be in any order ( $\text{[math]}$ ). Subtraction is not commutative. Products between scalars and vectors are distributive, i.e.