5 Experimental Fluid Mechanics
345
∂u
∂ x
+
∂v
∂ y
+
∂w
∂z
=0
If it is to become a dimensionless form, a dimensionless transformation is
performed on the quantities in the system of equations. It is illustrated by
the x-direction equation (the other two directional component equations are
similar) and the continuous equation. Introduce dimensionless variables into
the equations, there is
t
∗
=
t
T
, x
∗
=
x
L
, u
∗
=
u
V 0
, p
∗
=
p
p 0
, . . .
where L, T , V 0 , p 0 are characteristic length, time, speed, and pressure,
respectively.
The dimensionless continuous equation becomes
∂u ∗
∂ x ∗ +
∂v ∗
∂ y ∗ +
∂w ∗
∂z ∗
= 0
The equation in the x-direction is
V 0
T
∂u ∗
∂t ∗ +
V 2
0
L
u
∗ ∂u ∗
∂ x ∗ + v
∗ ∂u ∗
∂ y ∗ + w
∗ ∂u ∗
∂z ∗
= g −
p 0
ρ L
∂ p ∗
∂ x ∗ + v
V 0
L 2
∂ 2 u ∗
∂ x ∗2 +
∂ 2 u ∗
∂ y ∗2 +
∂ 2 u ∗
∂z ∗2
finished into an infinite formation
Sh
∂u ∗
∂t ∗ + u
∗ ∂u ∗
∂ x ∗ + v
∗ ∂u ∗
∂ y ∗ + w
∗ ∂u ∗
∂z ∗
=
1
Fr 2 − Eu
∂ p ∗
∂ x ∗ +
1
Re
∂ 2 u ∗
∂ x ∗2 +
∂ 2 u ∗
∂ y ∗2 +
∂ 2 u ∗
∂z ∗2
where Sh is the Strohal dimensionless number, i.e.
Sh =
L
V 0 T
Fr is the Froude dimensionless number, i.e.
Fr =
V 0
√
gL