54
2 Basic Components
necessarily be exaggerated, which creates a false impression. Turbulent flow contains eddies (whirling flow patterns) as a result of the fragmentation due to stretching and bending of the vortices generated by shear zones. Eddies have a macroscopic size. Their motion is strongly chaotic around an average flow, similar to, but
on a far larger scale, than the microscopic fluid particles. The turbulent motion thus
also generates shear stress on the average flow. This stress is expressed by an eddy
viscosity μ t according to
(2.5)
The eddy viscosity coefficient μ t is no fluid characteristic and depends on the local
flow. It is important to realise that μ t is a positive quantity and that it may be up to
100 times bigger than the molecular viscosity coefficient. So, due to the turbulence,
the boundary layer thickness increases significantly. The thickness stays small however compared to the covered length, namely some hundredths. From the positive
values of μ and μ t and the negative value of dτ/dy, we understand that the velocity
within the streamtube decreases in flow sense, due to friction, but that inversion
of the flow sense is impossible. Separation thus cannot be caused by the viscosity
effect only.
In order to understand the role of the pressure gradient, we must also analyse the
pressure variation in the normal direction. The easiest way is with a relative frame
attached to a fluid particle, as used in Sect. 2.1.1, leading to a balance between the
normal pressure gradient and the centrifugal force caused by the curvature of the
streamline:
(2.6)
R is the radius of curvature. It is assumed in Eq. (2.6) that friction forces do not
contribute. As is clear in Fig. 2.6, there is also shear stress on the inlet and outlet
faces of the infinitesimal streamtube. A stress change in the flow direction thus contributes to the force in the normal direction. As boundary layers are very thin, with
changes in the normal direction being much bigger than in the flow direction, the
contribution of friction may be ignored.
A consequence of Eq. (2.6) is that the pressure distribution over an aerofoil, as
sketched in Fig. 2.5, is only slightly dependent on the fluid viscosity, except maybe
with very viscous fluids. Another consequence of Eq. (2.6) is an almost identical
pressure variation over the various streamlines in a boundary layer, as the radius of
curvature R is very large compared to the boundary layer thickness. The decrease
of momentum, arising on the various streamlines according to Eq. (2.2) with an
adverse pressure gradient, is thus about the same for all streamlines. Consequently,
the velocity decrease is relatively stronger as the streamline is nearer to the surface.
(
) .
t
dv
dy
t
m m
= +
2
1
.
dp v
dy R
r
=
2 Basic Components
necessarily be exaggerated, which creates a false impression. Turbulent flow contains eddies (whirling flow patterns) as a result of the fragmentation due to stretching and bending of the vortices generated by shear zones. Eddies have a macroscopic size. Their motion is strongly chaotic around an average flow, similar to, but
on a far larger scale, than the microscopic fluid particles. The turbulent motion thus
also generates shear stress on the average flow. This stress is expressed by an eddy
viscosity μ t according to
(2.5)
The eddy viscosity coefficient μ t is no fluid characteristic and depends on the local
flow. It is important to realise that μ t is a positive quantity and that it may be up to
100 times bigger than the molecular viscosity coefficient. So, due to the turbulence,
the boundary layer thickness increases significantly. The thickness stays small however compared to the covered length, namely some hundredths. From the positive
values of μ and μ t and the negative value of dτ/dy, we understand that the velocity
within the streamtube decreases in flow sense, due to friction, but that inversion
of the flow sense is impossible. Separation thus cannot be caused by the viscosity
effect only.
In order to understand the role of the pressure gradient, we must also analyse the
pressure variation in the normal direction. The easiest way is with a relative frame
attached to a fluid particle, as used in Sect. 2.1.1, leading to a balance between the
normal pressure gradient and the centrifugal force caused by the curvature of the
streamline:
(2.6)
R is the radius of curvature. It is assumed in Eq. (2.6) that friction forces do not
contribute. As is clear in Fig. 2.6, there is also shear stress on the inlet and outlet
faces of the infinitesimal streamtube. A stress change in the flow direction thus contributes to the force in the normal direction. As boundary layers are very thin, with
changes in the normal direction being much bigger than in the flow direction, the
contribution of friction may be ignored.
A consequence of Eq. (2.6) is that the pressure distribution over an aerofoil, as
sketched in Fig. 2.5, is only slightly dependent on the fluid viscosity, except maybe
with very viscous fluids. Another consequence of Eq. (2.6) is an almost identical
pressure variation over the various streamlines in a boundary layer, as the radius of
curvature R is very large compared to the boundary layer thickness. The decrease
of momentum, arising on the various streamlines according to Eq. (2.2) with an
adverse pressure gradient, is thus about the same for all streamlines. Consequently,
the velocity decrease is relatively stronger as the streamline is nearer to the surface.
(
) .
t
dv
dy
t
m m
= +
2
1
.
dp v
dy R
r
=