### AE 301 Aerodynamics I

```Supersonic Potential Flow
•
For supersonic flow, we write the small perturbation
potential equation as:
2xx   yy  0
  M 2  1
•
Writing the equation in this form highlights the
difference from the subsonic equation.
•
A PDE in this form is said to be hyperbolic - while the
subsonic equation is said to be elliptical.
•
The names come from the form of the equations for
ellipses and hyperbolas:
ellipse: ax2  by2  0
•
hyperbola: ax2  by2  0
However, the differences between the equations is more
fundamental then appearances.
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Supersonic Potential Flow 
•
Elliptical equations are characterized by the following:
– Smooth and continuous interior solutions; maximum and
minimum on boundaries.
– All points on the interior depend upon all points on the
boundary (if only very slightly!).
•
In contrast, hyperbolic equations are characterized by:
– The possible existence of discontinuities in interior.
– Wavelike propagation of information from boundaries into
interior.
– Regions of influence and regions of “silence”.
•
You might recognize some of these from the theories
for subsonic and supersonic flow you have learned.
•
These characteristics dictate how to approach solutions!
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Supersonic Wavy Wall
•
Once again, consider the wavy wall problem:
y
yw  h cos2x / l 
x
l
•
x
h
The flow tangency boundary condition is the same as
before:
dyw
2hV
 2x 
 y ( x,0)  vw  V
dx

l
sin 

 l 
•
However, this time we will be solving:
•
Since this equation is hyperbolic, we will not be using
separation of variable to solve this.
2xx   yy  0
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Supersonic Wavy Wall 
•
Instead, assume a solution in the form of:
•
Either of these two functions are solutions to the
governing equation as can be demonstrated by:
 x, y   f x  y   g x  y 

f x  y   f ' x  y 
x
2
f x  y   f x  y 
2
x

f x  y   f ' x  y 
y

g  x  y   g '  x  y 
x
2
g x  y   g x  y 
2
x

g x  y   g ' x  y 
y
2
2


f
x


y


f x  y 
2
y
2
2


g
x


y


g x  y 
2
y
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Supersonic Wavy Wall 
•
Combining these yields:
2 f 2 f
2
2






f


f   0
2
2
x
y
2
2 g
2  g
2
2






g


g   0
2
2
x
y
2
•
However, these two functions represent two different
types of solutions.
•
Since f is a function of x-y, then it will have constant
values along lines described by equation.
•
Similarly, g is a function of x+y, then it will have
constant values along these lines.
•
These lines thus form waves as seen below.
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Supersonic Wavy Wall 
Left running
waves
f = constant
x  y  constant

M
Right running
waves
•
•
g = constant
x  y  constant
Note that the angle shown, , is given by:


 1 
1
 dy 



  atan   atan
 asin
 M 2 1 
 dx 
 M 



Thus, these lines represent Mach waves.
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dy 1

dx 
dy  1

dx 
Supersonic Wavy Wall 
•
For our solution, we are obviously only interested in the
left running waves. Thus:
 x, y   f x  y 
•
With the velocity components:
u  x  f 
•
•
v   y  f 
To satisfy our flow tangency boundary condition, lets
use surface slopes:
dy
v  f 
 tan 

dx
V
V
Thus, the u perturbation velocity, assuming small
angles, is:
V tan
V
u  


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M 2  1
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Supersonic Wavy Wall 
•
So, finally, our pressure distribution along this wall is:
•
You may recall deriving the same equation as the limit
for weak shocks on a thin airfoil.
•
From this solution, we see that the supersonic pressure
coefficient varies directly with slope - rather than
curvature like the subsonic solution.
•
Also, we never applied a far field boundary condition
since these waves, in 2-D, will propagate to infinity.
•
Note that these waves don’t coalesce or fan-out like
shock or expansion waves - a result of our linearization.
 2u
2
Cp 

V
M 2  1
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Supersonic Wavy Wall 
•
The pressures predicted by our new relation are 90o out
of phase with the wall oscillations:
x
x
•
High pressures occur on the front face of each wave,
low pressures on the back face.
•
As a result, an integration of pressures results in a drag
force in the x direction - wave drag.
•
In application, there is also a slight drag due to the total
pressure loss in shock waves. But this is usually small
unless it is a normal shock - I.e. transonic flow!
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Supersonic airfoils
•
A nice thing about the supersonic solution is due to the
hyperbolic nature of the problem.
•
Due to the limited regions of influence, the solution
applies equally to a isolated segment of the wall as to
the whole wall.
•
Thus, for a parabolic arc airfoil (constant curvature, so
linear slope variation), the pressure distribution looks
like:
Cp 1
x/c
+
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Supersonic Similarity
•
Our solution for the supersonic pressure coefficient also
yields a supersonic similarity rule:
C p1 M 12  1
1

C p 2 M 22  1
2
•
Note the close similarity to the subsonic Prandlt-Glauert
rule.
•
Paradoxically, this rule states that, for the same
geometry, as the Mach number increases, the pressure
coefficient (and thus lift coefficient) decreases.
•
However, since dynamic pressure also increases with
Mach number, the actual pressures and forces increase.