53
2.1 Aerofoils
We first consider a flow with a zero pressure gradient. Far from the wall, the
shear stress is zero. There is a certain value on the wall. So dτ/dy < 0. The momentum equation demonstrates that friction reduces the velocity within the
streamtube in the flow sense. The height of the streamtube thus increases in the
flow sense. The same applies to the entire zone close to the wall, where flow is
retarded by the wall friction. The zone affected by the wall friction is called the
boundary layer. Its thickness is denoted by δ δ in Fig. 2.6. Note that this thickness
cannot be determined precisely. The shear stress is partly of molecular nature
due to the relation
(2.3)
where μ is the dynamic viscosity coefficient. We recall from fluid mechanics that
friction is generated between two adjacent fluid layers with different velocity by the
chaotic motion of the microscopic fluid particles (molecules or atoms) around the
average macroscopic motion. Fluid particles leap from the fast layer to the slower
one, where they arrive with a higher average momentum and so push forward the
slower layer. Oppositely, fluid particles leaping from the slower layer to the faster
one produce a breaking effect. The resulting effect is described macroscopically by
Eq. (2.3), with the viscosity coefficient μ as a positive quantity. According to the
momentum Eq. (2.2), friction affects the velocity change with μ/ρ. We thus define
the kinematic viscosity coefficient
(2.4)
The dimension of kinematic viscosity is m
2
/s. The value for water at atmospheric
temperature is about 10
−6
m
2
/s. The value for air at atmospheric temperature and
pressure is about 15 10
−6
m
2
/s. So water and air are not highly viscous fluids.
This results in a very thin boundary layer. It is demonstrated in fluid mechanics
that, on a smooth wall without a pressure gradient, boundary layer thickness is
approximately
Here, v ∞ represents the velocity far away from the wall (outside the boundary layer
zone) and x the distance covered by the boundary layer. A boundary layer thickness (with laminar flow) maximally amounts to some thousandths of the covered
length. It is very important to keep in mind the small thickness of a boundary layer
in boundary layer analyses. When drawing a boundary layer, the thickness must
,
dv
dy
t m
=
.
m
r
=
ν
xv
/ x 5 /
.
d
∞
≈
ν
2.1 Aerofoils
We first consider a flow with a zero pressure gradient. Far from the wall, the
shear stress is zero. There is a certain value on the wall. So dτ/dy < 0. The momentum equation demonstrates that friction reduces the velocity within the
streamtube in the flow sense. The height of the streamtube thus increases in the
flow sense. The same applies to the entire zone close to the wall, where flow is
retarded by the wall friction. The zone affected by the wall friction is called the
boundary layer. Its thickness is denoted by δ δ in Fig. 2.6. Note that this thickness
cannot be determined precisely. The shear stress is partly of molecular nature
due to the relation
(2.3)
where μ is the dynamic viscosity coefficient. We recall from fluid mechanics that
friction is generated between two adjacent fluid layers with different velocity by the
chaotic motion of the microscopic fluid particles (molecules or atoms) around the
average macroscopic motion. Fluid particles leap from the fast layer to the slower
one, where they arrive with a higher average momentum and so push forward the
slower layer. Oppositely, fluid particles leaping from the slower layer to the faster
one produce a breaking effect. The resulting effect is described macroscopically by
Eq. (2.3), with the viscosity coefficient μ as a positive quantity. According to the
momentum Eq. (2.2), friction affects the velocity change with μ/ρ. We thus define
the kinematic viscosity coefficient
(2.4)
The dimension of kinematic viscosity is m
2
/s. The value for water at atmospheric
temperature is about 10
−6
m
2
/s. The value for air at atmospheric temperature and
pressure is about 15 10
−6
m
2
/s. So water and air are not highly viscous fluids.
This results in a very thin boundary layer. It is demonstrated in fluid mechanics
that, on a smooth wall without a pressure gradient, boundary layer thickness is
approximately
Here, v ∞ represents the velocity far away from the wall (outside the boundary layer
zone) and x the distance covered by the boundary layer. A boundary layer thickness (with laminar flow) maximally amounts to some thousandths of the covered
length. It is very important to keep in mind the small thickness of a boundary layer
in boundary layer analyses. When drawing a boundary layer, the thickness must
,
dv
dy
t m
=
.
m
r
=
ν
xv
/ x 5 /
.
d
∞
≈
ν