Key to HW #7
Yang
04/21/06
1
Hint to Problem 1
From eqn(11.13) on page 591, we have the following
"µ ¶
µ ¶2 #
2
γ−1
∂φ
∂φ
2
2
+
a = a0 −
2
∂x
∂y
and since we are given with the velocity potential, then
as follows
ux
uy
=
∂φ
∂x
=
V∞ + 2π p
=
∂φ
∂y
=
−2π70e−2π
∂φ
∂x
and
∂φ
∂y
can be found
√
70
2
e−2π 1−M∞ y cos(2πx)
2
1 − M∞
√
2 y
1−M∞
sin(2πx)
Now substituting the potential derivatives into eqn(11.13), you’ll get the local
speed of sound a, local speed ux and uy . Then you could obtain the local Mach
number; Once you’ve got the Mach number, it’s easy to obtain T , p, and ρ from
the following equations
T0
T
p0
p
ρ0
ρ
=
=
=
γ−1 2
1+
M
2
µ
¶γ/(γ−1)
γ−1 2
1+
M
2
µ
¶1/(γ−1)
γ−1 2
1+
M
2
Now think about why we can use these isentropic relations? The reason is the
velocity has already been given to us in the potential form which implies the
flow is irrotational or isentropic.
1
2
Hint to Problem 2
To solve this problem, you’ve gotta understand what the compressibility correction means. It actually says if we know the pressure coefficient Cp,0 for a given
air foil under an incompressible flow, then we could get a rather accurate approximate for Cp through several rules, e.g. Prandtl-Glauert, eqn(11.51), KarmanTsien, eqn(11.54), and Laitone, eqn(11.55), for the corresponding compressible
flow over the same air foil.
For this problem, we know Cp,0 and we know Mach number. So just directly
substitute them into the aforementioned equations.
3
Hint to Problem 3
Again conceptual issue, what does critical Mach number mean? The critical Mach number is that freestream Mach number at which sonic flow is first
achieved on the airfoil surface.
With that on mind, we equate the following two equations(eqn.(11.60) and
eqn.(11.51)
"µ
#
¶γ/(γ−1)
2
2
1 + [(γ − 1)/2]Mcr
Cp,cr =
−1
2
γMcr
1 + (γ − 1)/2
Cp
=
Cp,0
2
1 − M∞
p
Since from the problem, we already know Cp,min , then solve this equation Cp =
Cp,cr for Mcr .
2