66
2 Basic Components
Equation (2.16) may be integrated along a streamline into
(2.18)
With the cosine rule on the velocity triangles follows
Thus:
(2.19)
This means that, for constant mechanical energy of the incoming flow in the absolute frame and uniform v u , the constant in Eq. (2.19) and thus in Eq. (2.18) is the
same on all streamlines. Eq. (2.18) then also implies
(2.20)
Combined with Eq. (2.17), this gives
(2.21)
The significance of Eq. (2.21) may be understood by calculating the circulation on
an infinitesimal contour consisting of two pieces of neighbouring streamlines connected with straight segments as shown in Fig. 2.13 (  x is streamline direction, y is
normal direction), and by applying Stokes’ circulation theorem, which is an integral
theorem for the rotor of a vector quantity:
where S is a surface spanned by the contour and
n is the unit normal vector on
the surface in the sense corresponding with the sense on the contour. On the
infinitesimal contour of Fig. 2.13, this gives, with R the radius of curvature of the
constant.
2
1 2
1
w
p
r
+
=
2
2
2
u
w
u v 2uv .
= + −
constant
2
2
1
1
u
2
2
1
v
p
u uv
.
r
+
+
−
=
dw 1 dp
w
0.
dy
dy
r
+
=
2
dw w
w
0.
dy R
+
=
S
w.dl
(
w ).ndS ,
= ∇×
∫
∫