52
2 Basic Components
2.1.4 Boundary Layer Separation
We can obtain more detailed understanding of the flow around an aerofoil by deriving the momentum equations for an infinitesimal streamtube close to a wall.
Figure 2.6 sketches such a streamtube for the suction side. We consider a flow that
follows the surface, in other words, that does not separate.
The axis in flow direction is denoted by x. This direction approximately follows
the surface. The normal direction is denoted by y. Shear stress on a face of the control volume is denoted by τ. The flow between the streamtube considered and the
wall has a braking effect on the streamtube. The flow between the streamtube and
the free flow has a driving effect. The momentum theorem in the flow direction for
an infinitesimal part of the streamtube, on condition of small curvature, is
From this:
v( dy )dv
( dy )dp
( dy )dU ( dx )d .
r
r
t
= −
−
+
(2.1)
From now on, we assume constant density, but extension for variable density is
possible. With constant density, the effect of gravity onto the pressure may be expressed by considering pressure as relative to the hydrostatic pressure. This is relevant for liquid flow. The effect of gravity may simply be ignored for gas flow. We
thus simplify (2.1) to
(2.2)
dv
dp
dU d
v
.
dx
dx
dx dy
t
r
r
= − −
+
.
dv
dp d
v dx
dx dy
t
r
= −
+
Fig. 2.6 Infinitesimal streamtube close to a wall and velocity profile evolution with an adverse
pressure gradient
2 Basic Components
2.1.4 Boundary Layer Separation
We can obtain more detailed understanding of the flow around an aerofoil by deriving the momentum equations for an infinitesimal streamtube close to a wall.
Figure 2.6 sketches such a streamtube for the suction side. We consider a flow that
follows the surface, in other words, that does not separate.
The axis in flow direction is denoted by x. This direction approximately follows
the surface. The normal direction is denoted by y. Shear stress on a face of the control volume is denoted by τ. The flow between the streamtube considered and the
wall has a braking effect on the streamtube. The flow between the streamtube and
the free flow has a driving effect. The momentum theorem in the flow direction for
an infinitesimal part of the streamtube, on condition of small curvature, is
From this:
v( dy )dv
( dy )dp
( dy )dU ( dx )d .
r
r
t
= −
−
+
(2.1)
From now on, we assume constant density, but extension for variable density is
possible. With constant density, the effect of gravity onto the pressure may be expressed by considering pressure as relative to the hydrostatic pressure. This is relevant for liquid flow. The effect of gravity may simply be ignored for gas flow. We
thus simplify (2.1) to
(2.2)
dv
dp
dU d
v
.
dx
dx
dx dy
t
r
r
= − −
+
.
dv
dp d
v dx
dx dy
t
r
= −
+
Fig. 2.6 Infinitesimal streamtube close to a wall and velocity profile evolution with an adverse
pressure gradient