Mechanics of Fluids

As δθ is very small, cos δθ → 1 and sin δθ → δθ. Hence
giving
since
(u + δu)δθ sin µ = (u + δu − u) cos µ
δθ = δu
δu δu
cot µ → cot µ =
u + δu u u (M2 − 1) 1/2
Supersonic flow round a corner 513
(11.38)
sin µ = 1/M (11.21)
Equation 11.38 shows that, with positive δθ measured from the original
direction in Fig. 11.16, δu is positive. In other words, the velocity
increases round a convex corner. As the component parallel to the Mach
wave remains unchanged the normal component must increase and, to satisfy
the continuity relation ρun = constant, the density must decrease.
If an initially uniform flow makes a succession of small turns, as in
Fig. 11.17, there will be a number of regions of uniform flow separated by
Mach waves emanating from the corners. If the straight portions between
the corners, and also the angles δθ1, δθ2 etc., are indefinitely decreased a
continuously curved surface is obtained, from which an infinite number of
Mach waves is generated.
Since opportunity for heat transfer at these high velocities is so slight the
process across each infinitesimal wave may be considered adiabatic. Then,
since changes of elevation are negligible, the energy equation (11.11) may
be written
� �
γ p 1
+
γ − 1 ρ 2 u2 = a2 1
+
γ − 1 2 u2 = constant C
whence
1 1 C
+ =
(γ − 1)M2 2 u2 SF, A, PG (11.39)
Fig. 11.17