1.6 Dimensionless Form of Equations
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1.6 Dimensionless Form of Equations
Experimental studies of flows are often carried out on models, and the results
are displayed in dimensionless form, thus allowing scaling to real flow conditions. The same approach can be undertaken in numerical studies as well.
The governing equations can be transformed to dimensionless form by using
appropriate normalization. For example, velocities can be normalized by a
reference velocity vo, spatial coordinates by a reference length Lo, time by
some reference time to, pressure by pvi, and temperature by some reference
temperature difference TI - To. The dimensionless variables are then:
If the fluid properties are constant, the continuity, momentum and temperature equations are, in dimensionless form:
dT* a(ujT*) -
-
1 d2T*
-at*
ax;
Re P r dxj2
The following dimensionless numbers appear in the equations:
which are called Strouhal, Reynolds, and Froude numbers, respectively. yi is
the component of the normalized gravitational acceleration vector in the xi
direction.
For natural convection flows, the Boussinesq approximation is often used,
in which case the last term in the momentum equations becomes:
where Ra is the Rayleigh number, defined as:
and p is the coefficient of thermal expansion.