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2.4 Reduced Forms of the Navier-Stokes Equations 57

2.4 Reduced Forms of the Navier-Stokes Equations

The conservation equations can be reduced to simpler forms by examining the
relative magnitudes of the terms in the equations. In the application of this
procedure, known as "order-of-magnitude" analysis, to two-dimensional steady
flows, it is common to introduce two length scales L and 8 (which are, respec-
tively, parallel and normal to the wall) to assume a typical velocity to be of
order u e, and to estimate the relative magnitudes of inertia, pressure, and vis-
cous and body force terms in the Navier-Stokes equations. For example, if we
assume that a typical viscous stress is at the form

du
dy
then the viscous forces are of order
u e

per unit area and, since a typical pressure force is of the order

2
QU e
per unit area, , the ratio of the two forces is
Pressure Force QU e QU eL _ u eL
2
= RL (2.4.1)
Viscous Force uu e/L a V
For an incompressible flow with small temperature differences, the Reynolds
number, i?£, is the principal parameter for determining the nature of the flow.
Two important nondimensional groups for heat flux, with equivalents for
other scalar properties, are the Prandtl number

k k/QC p a { }

which represents the ratio of diffusion coefficients, v and a, and the Grashof
number
Gr = Ri • R\ (2.4.3)
2
which represents the product of the Richardson number Ri [= (Ag/g)(gh/u ))
and the square of the Reynolds number.
Using order of magnitude arguments, the conservation equations of the pre-
vious section can be simplified by neglecting some of the viscous terms. For
example, in some three-dimensional flows the viscous terms da xz/dz, da yz/dz
and d(j zz/dz in Eqs. (2.2.2) to (2.2.4) and the heat transfer term dq z/dz in
Eq. (2.2.22) are omitted, and the resulting form of the Navier-Stokes equa-
tions, referred to as "parabolized Navier-Stokes equations," are solved together
with the continuity equation. In other flows the Navier-Stokes equations are