Transcript AE 301 Aerodynamics I - Prescott Campus, Arizona

Prandlt-Glauert Similarity
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Consider the linearized potential equation for subsonic
 2xx  yy  0
flow:
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where:
•
•
  1  M 2
Consider a coordinate transformation in the x direction
dx  dx
y  y
x  x
No changes occur in the y direction, but the x derivative
become:
 dx   x
x 


x dx x 
   dx   2 xx
xx  2   
 2
2
x

 dx  x
2
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Prandlt-Glauert Similarity [2]
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With this transformation, the governing equation
becomes:
 2xx   yy  xx   yy  0
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Thus, the solution in this transformed space is simply a
solution for incompressible flow.
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To proceed from this point, we must consider how to
apply the boundary conditions.
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If two solutions are obtained for the same geometry
but using the two equations above:
 2xx  yy  0
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The solutions will differ by:
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o xx  o yy  0
o


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Prandlt-Glauert Similarity [3]
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The pressure coefficients, found from the x derivatives,
will be related by the same factor:
 2x  2 x  2o x C po
Cp 



V
V
V

•
Cp 
C po
1  M 2
Similarly, the forces found by integrating pressures will
differ by the same factor.
CL 
CLo
1  M 2
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This idea of correcting incompressible solutions for
compressibility is know as Prandlt-Glauert similarity.
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Generally, this approach works to speed of around Mach
= 0.6 – but breaks down soon after that.
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Prandlt-Glauert Similarity [4]
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Another approach is to find the changes in geometry to
achieve the same potential solution.
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Since the solution is dictated by the boundary condition,
this requirement, in 2-D, becomes:
1 dy
dy

 dx surface dx surface
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We can handle this at least two ways:
– fix the y axis and stretch the x axis, as we did when we started.
– or, in order to keep the planform the same, fix the x axis and
stretch the y axis:
y

 y
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y  y 1  M 2
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Prandlt-Glauert Similarity [5]
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Thus, to have the same solution (including lift), the
compressible flow geometry would be thinner then the
incompressible geometry by the factor .
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This agrees with what you may have already observed
for aircraft.
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To capture both these effects, a generalization of the
Prandlt-Glauert similarity rule is:
C p1 1  M 12
1
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
C p 2 1  M 22
2
Which relates the thickness, pressure coefficient and
Mach number at one condition to that at another.
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Subsonic Wavy Wall
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The wavy wall problem is a classical solution which
demonstrates differences between flow types.
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Consider the infinite wall given by:
y
yw  h cos2x / l 
x
l
•
x
h
The flow along this wall will satisfy our small
perturbation potential:
2
 xx  yy  0
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With the flow tangency boundary condition:
 y ( x, yw )  vw  V
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dyw
2hV  2x 

sin 

dx
l
 l 
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Subsonic Wavy Wall [2]
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However, rather than apply this BC on the wall, it is
consistent with our small perturbation assumptions to
apply it on the y=0 axis:
2hV
 2x 
 y ( x,0)  
sin 

l
 l 
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To solve this system, use the classical separation of
variables technique.
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First assume that the solution can be expressed by the
product of functions, each of which are dependant upon
only one variable:
 x, y   F xG y 
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The derivatives of our potential function are then:
xx  F G
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 yy  FG
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Subsonic Wavy Wall [3]
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Thus, our governing equation becomes:
 2 F G  FG  0
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Or:
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This last equation states that the some relation which
depends only on x must be equal to another relation
which depends only on y.
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The only way for this to be true for all x and y is if the
two relations equal a common constant, k2:
F 
G 
 2
F
 G
F 
G
  2  k 2
F
 G
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Subsonic Wavy Wall [4]
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Thus, our assumption of separation of variables leads to
two independent 2nd order ODE’s:
d 2G
2 2

k
 G0
2
dy
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d 2F
2

k
F 0
2
dx
Assuming that k is real, then these two ODE’s have
solutions:
G y   A1eky  A2eky
F x  B1 sin(kx)  B2 cos(kx)
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The first BC that we can apply is the fact that in
subsonic flow, the disturbances created by the wall
should not propagate to infinity.
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This BC can only be satisfied if:
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A2  0
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Subsonic Wavy Wall [5]
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The remaining terms, combined, become:
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And the perturbation vertical velocity:
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Or, on the axis:
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Comparing with our flow tangency, BC:
 ( x, y)  F ( x)G y   A1eky B1 sin(kx)  B2 cos(kx)
y ( x, y)  kA1eky B1 sin(kx)  B2 cos(kx)
 y ( x,0)  kA1 B1 sin(kx)  B2 cos(kx)
2hV
 2x 
 x ( x,0)  
sin 

l
 l 
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shows that:
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B2  0
2
k
l
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A1 B1 
hV
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
Subsonic Wavy Wall [6]
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Our solution to this problem is thus:
 2y 1  M 2

 ( x, y) 
exp 

l
1  M 2

V h
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To get the pressure coefficient, we need the x velocity
perturbation:
2

2

y
1

M
2V h


x  u 
exp 

l
l 1  M 2

•
  2x 
 sin

  l 

  2x 
 cos

  l 

And then:
 2y 1  M 2
2u
 4h

Cp  

exp 

V l 1  M 2
l

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  2x 
 cos

  l 

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Subsonic Wavy Wall [7]
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The first thing we note about this solution is that the
pressure distribution is 180o out of phase with the wall.
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Thus high pressures occur a the low spots, low
pressures on the peaks.
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More importantly, the pressure distribution is symmetric
about the peaks (or troughs)
x
x
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Thus, an integration of the pressures would produce no
net force in the x-direction - no drag!
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A closer evaluation would show that the pressures are
related to the 2nd derivative – or curvature – of the wall.
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Subsonic Wavy Wall [8]
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Next, the rate of attenuation of the disturbance away
from the wall decreases with the Mach number.
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Thus, the higher the Mach number, the disturbances are
felt further away from the wall – or stronger at the
same location:
M = 0.6
M=0
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Improved Compressibility Corrections
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In the wavy wall solution, the Prandlt-Glauert factor
once again appears as a similarity parameter.
•
An improvement on the Prandlt-Glauert rule was
suggested by Laitone using the local Mach number
rather than the freestream.
Cp 
•
C p0
1 M 2
Bu using issentropic relations to relate the local mach
and pressure coefficient, Laitone obtained:
Cp 
C p0
 2   1 2 
2 
1  M  M  1 
M   / 2 1  M  C p 0
2

 

2

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Improved Compressibility Corrections [2]
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Laitone’s correction is an improvement over PrandltGlauert, although with less mathematical basis.
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However, Laitone’s correction does tend to overpredict
the compressible effect.
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Another correction based upon some simplified solution
methods was suggested by Karman-Tsien:
Cp 
•
C p0
2

 C p0
M
2


1 M   
1 1 M 2  2
 

This correction, while of comparable accuracy to
Laitone’s, has greater acceptance.
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