Basics of Fluid Mechanics, 2014a

320 CHAPTER 9. DIMENSIONAL ANALYSIS
For the Continuity Equation (8.17) for non–compressible substance can be transformed
into
0
∂ρ
✁∂t ✁✁✕ + ∇·(˜ρ U) =0 (9.51)
For the N-S equation, every additive term has primary dimensions m 1 L −2 t −2 .
To non nondimensionalization, we multiply every term by L/(V 2 ), which has primary
dimensions m −1 L 2 t 2 , so that the dimensions cancel.
Using these definitions equation (8.111) results in
fh ∂ŨU
( )
(ŨU
U 0 ∂˜t + · ˜∇
Pmax − P ∞
)ŨU = −
˜∇ ˜P
ρŨU
+ 1 ⃗f
ŨU 2 g + 1 ˜∇ 2
ρŨU ŨU (9.52)
h
gh μ
Or after using the definition of the dimensionless parameters as
St ∂ŨU (ŨU
∂˜t + · ˜∇
)ŨU = −Eu ˜∇ ˜P + 1
Fr ⃗ 2 f g + 1 Re ˜∇ 2 ŨU (9.53)
The definition of Froude number is not consistent in the literature. In some places Fr
is defined as the square of Fr = U 2 /g h.
The Strouhal number is named after Vincenz Strouhal (1850 1922), who used
this parameter in his study of “singing wires.” This parameter is important in unsteady,
oscillating flow problems in which the frequency of the oscillation is important.
Example 9.23:
A device is accelerated linearly by a constant value B. Write a new N–S and continuity
equations for incompressible substance in the a coordinate system attached to the body.
Using these equations developed new dimensionless equations so the new “Froude number”
will contain or “swallow” by the new acceleration. Measurement has shown that
the acceleration to be constant with small sinusoidal on top the constant such away as
( ) f
a = B + ɛ sin
(9.XXIII.a)
2 π
Suggest a dimensionless parameter that will take this change into account.
Supplemental Problems
1. An airplane wing of chord length 3 [m] moves through still air at 15 ◦ Cand 1 [Bar]
and at at a speed of 15 [m/sec]. What is the air velocity for a 1:20 scale model
to achieve dynamic similarity between model and prototype? Assume that in the
model the air has the same pressure and temperature as that in prototype. If the
air is considered as compressible, what velocity is required for pressure is 1.5[bar]
and temperature 20 ◦ C? What is the required velocity of the air in the model test
when the medium is made of water to keep the dynamic similarity?