Nondimensionalization of the NavierStokes Equation
(Section 102, Çengel and Cimbala)
Nondimensionalization
:
We begin with the differential equation for conservation of linear momentum for a
Newtonian fluid, i.e., the
NavierStokes equation
. For incompressible flow,
Equation 102 is
dimensional
, and each variable or property (
ρ
,
V
r
,
t
,
µ
, etc.) is also
dimensional
.
What are the primary dimensions (in terms of {m}, {L}, {t}, {T}, etc) of
each term in this equation?
Answer:
{
}
To nondimensionalize Eq. 102, we choose
scaling parameters
as follows:
We define
nondimensional variables
, using the scaling parameters in Table 101:
To plug Eqs. 103 into Eq. 102, we need to first rearrange the equations in terms of the
dimensional variables, i.e.,
()
**
*
*
0
1
1
tt
x
L
x
V
V
V
f
PP P PP
gg
g
L
∞∞
==
=
=+
−
=
∇
=
∇
r
r
rr
r
r
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View Full DocumentNow we substitute all of the above into Eq. 102 to obtain
Every additive term in the above equation has primary dimensions {m
1
L
2
t
2
}. To
nondimensionalize the equation, we multiply every term by constant
L
/(
ρ
V
2
), which has
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 Fall '05
 CIMBALA
 Fluid Dynamics, primary dimensions, NavierStokes equation

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