1 Foundation of Fluid Mechanics
37
viscous stress cannot be neglected. It is the same order of magnitude as
the inertial force, and the fluid particles are rotating. The thickness of
the boundary layer can be estimated based on the assumption that the
viscous force and inertial force in the boundary layer are of the same
order of magnitude. Take the flow around a flat plate as an example. Let
the velocity of the incoming flow be U, the length in the x-direction be
L, and the thickness of the boundary layer be δ. In the boundary layer,
the inertia force of the fluid micro mass is zero.
F J = m
du
dt
∝ ρ L
2
δ
U
T
= ρ L
2
δ
U
L/U
= ρLU
2
δ
The viscous force of the fluid microelement is
F μ = ρν A
du
dy
∝ ρ L
2
ν
U
δ
= ρ L
2
ν
U
δ
Based on the assumption that the inertial force and the viscous force are
of the same order of magnitude,
F J ≈ F μ , ρ LδU
2
≈ ρ L
2
ν
U
δ
δ
L
≈
1
√
Re L
, Re L =
U L
ν
It is shown that the ratio of the thickness of the boundary layer to the
length of the plate L is inversely proportional to the square of the total Re L
number calculated by the previous flow velocity and the length of the plate. If
the inflow velocity U = 14.6 m/s, the plate length L = 1.0 m, the air moving
viscous coefficient ν = 1.46 × 10 −5 m 2 /s, Re L = 10 6 , the boundary layer
thickness is in the order of millimeter, which is equivalent to 1/1000 of the
plate length. Theoretical Solution δ ≈ 5.0 mm for Laminar Boundary Layer
of Plate.
Accordingly, the two-dimensional laminar boundary layer governing equations derived by Prandtl (a simplified form of N-S equations) are as follows:
∂u
∂ x
+
∂v
∂ y
= 0
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
= f x −
1
ρ
∂ p
∂ x
+ ν
∂ 2 u
∂ y 2
37
viscous stress cannot be neglected. It is the same order of magnitude as
the inertial force, and the fluid particles are rotating. The thickness of
the boundary layer can be estimated based on the assumption that the
viscous force and inertial force in the boundary layer are of the same
order of magnitude. Take the flow around a flat plate as an example. Let
the velocity of the incoming flow be U, the length in the x-direction be
L, and the thickness of the boundary layer be δ. In the boundary layer,
the inertia force of the fluid micro mass is zero.
F J = m
du
dt
∝ ρ L
2
δ
U
T
= ρ L
2
δ
U
L/U
= ρLU
2
δ
The viscous force of the fluid microelement is
F μ = ρν A
du
dy
∝ ρ L
2
ν
U
δ
= ρ L
2
ν
U
δ
Based on the assumption that the inertial force and the viscous force are
of the same order of magnitude,
F J ≈ F μ , ρ LδU
2
≈ ρ L
2
ν
U
δ
δ
L
≈
1
√
Re L
, Re L =
U L
ν
It is shown that the ratio of the thickness of the boundary layer to the
length of the plate L is inversely proportional to the square of the total Re L
number calculated by the previous flow velocity and the length of the plate. If
the inflow velocity U = 14.6 m/s, the plate length L = 1.0 m, the air moving
viscous coefficient ν = 1.46 × 10 −5 m 2 /s, Re L = 10 6 , the boundary layer
thickness is in the order of millimeter, which is equivalent to 1/1000 of the
plate length. Theoretical Solution δ ≈ 5.0 mm for Laminar Boundary Layer
of Plate.
Accordingly, the two-dimensional laminar boundary layer governing equations derived by Prandtl (a simplified form of N-S equations) are as follows:
∂u
∂ x
+
∂v
∂ y
= 0
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
= f x −
1
ρ
∂ p
∂ x
+ ν
∂ 2 u
∂ y 2