Airfoils in Supersonic Flow
Linear theory
Recall that for isentropic flow over a corner (Prandtl-Meyer Expansion), we obtained:
dp 
Md
M2 1
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Narayanan Komerath
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Now we can argue that for small angles of compression, only weak oblique shocks will be
formed, and the entropy increase is minimal. In fact, it can be shown that
,
so that
is small for small
for weak compressions as well.
. Therefore, the relations for isentropic flow should hold
Using the expression for dp, we can write, for small changes,
or,
or
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Sign of cp
Note that
by definition should be positive when
, i.e. the pressure is
higher than the freestream static pressure (compression). This is easier to keep in mind
than any sign convention for
.
Lift Coefficient
For small
,
where
is the equation to the airfoil
surface.
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Lift per unit span,
where
Note that
respect to
is pressure on the lower surface and
where
is pressure on the upper surface.
is the distance along the chord and
is the slope with
.
Given the equations to the upper and lower surfaces
can find
.
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and
we
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Example 1
Flat plate at angle of attack  .
Upper surface:
Lower surface:
((How do we figure
g out whether it should be +  or -  ? Think about the sign
g of cP.
The upper surface is an expansion surface -- negative cP while the lower surface is a compression
surface … positive cp)
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This is a very useful result.
result In fact,
fact this is valid for relatively large values of  as well.
well
Note that
attack.
Cl 
2
1 M
1
for an ideal thin airfoil in subsonic flow at small angle of
2
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Example 2
Airfoil with thickness at angle
g of attack.
Upper surface is described by
Lower surface is described by
At angle of attack, upper surface slope,
At angle
l off attack
tt k llower surface
f
slope,
l
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dZl dZ u  x 
Cl 
2   
d  
 



 dx
dx  c 
M 2 1 0 
2
At
=0 and 1,
1 
(leading edge and trailing edge)
So,
Astonishing result! Neither thickness nor camber produced any lift in supersonic flow.
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Drag Coefficient
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Drag per unit span
For small
,
(in radians)
ie
i.e.
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Drag due to angle of attack, thickness and camber
Consider a cambered airfoil at angle of attack.
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Note:
1.Everything causes drag.
2.We can split up the airfoil at the angle of attack into
(a): flat plate at AOA
(b): camber line at zero AOA
(c): thickness distribution at zero AOA
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The slope
of the airfoil surface is the sum of the slopes due to the angle of
attack,
tt k th
the camber
b andd the
th thi
thickness
k
di
distribution.
t ib ti
Similarly,
l 
2
  tl  cl     2 tl  cl 
2
2
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Now,
Giving,
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Pitching Moment
Pitching moment due to the net force at a distance x from the leading edge on an area (dx times 1)
is
Moment coefficient is
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If we split  as before into AOA, camber and thickness, we will find that
a.the contribution of thickness is zero
b.both AOA and camber cause pitching moment
Aerodynamic Center
It is the ppoint about which the ppitchingg moment is independent
p
of angle
g of attack. The above
equation shows that the aerodynamic center is at the mid chord.
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Shock -Expansion technique
The pressures on the airfoil surface can also be calculated using oblique shocks and P-M expansions.
Note: at the trailing edge, there are a few conditions to be met.
1. same flow direction
2. same static pressure on both sides of the slip line
This is the most exact way of calculating forces and moments. It is also the most tedious, but it is
necessary ffor angles
l > 5° .
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Second order theory (Busemann's theory)
If we retain terms of the order of
This gives better results for
results for
,
in our expression for
,
,
than the linearized theory, but worse
.
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Supersonic Drag From the Performance Point of View
Drag at supersonic speeds consists of
•friction drag & pressure drag,
•wave drag due to volume,
•wave drag due to lift,
•induced drag due to finite aspect ratio
•and
d other
th miscellaneous
i
ll
d
drag it
items (i
(interference
t f
d
drag, which
hi h may b
be partly
tl
included in pressure drag).
For low drag,
drag supersonic configurations tend to be long
long, thin
thin,slender.
slender
Drag at supersonic speeds is very dependent on configuration shape, relative
size and locations of the components.
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Courtesy:
Dr. B.Komerath
Kulfan, BOEING Co.
Narayanan
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Source of Drag: Fluid Mechanics Point of View
(1) Boundary Layer drag: laminar and turbulent
Fluid molecules hitting the surface, and thus losing their momentum, then bouncing off and
hitting other molecules zipping along in the flow. This is called "viscous
viscous drag
drag".. Clearly this is a
bigger deal if this "momentum transfer" is a big part of the momentum of the flow. This
momentum transfer is confined to a region near the surface, called the "boundary layer". This
is whyy the "Reynolds
y
Number" is important:
p
it ggives a measure of how large
g the inertial forces
of the flow are, with respect to the viscous forces. When the Reynolds number is low (< ~
100,000? ), viscous drag becomes very important. Generally, the Reynolds number for the
flow over an airplane wing is quite large (20 million based on wing chord?), so viscous drag is
not very significant in subsonic flight. It does become significant when other kinds of drag are
minimized, and at high speeds every source of drag becomes important. We generally speak
of "skin friction drag" in high speed flight, because the high temperatures created at the skin
worry us. The temperature becomes high because the molecules which were going smoothly
along in the flow now start getting bounced all over the place when they collide with the
molecules bouncing off the surface, and their motion becomes chaotic: the kinetic energy is
converted
t d to
t heat,
h t andd the
th flflow near the
th surface
f
becomes
b
hot,
h t andd transfers
t
f heat
h t tto th
the
surface.
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Note:
E
Even
if the
th external
t
l flow
fl is
i supersonic,
i inside
i id the
th boundary
b d layer
l
the
th velocity
l it decreases
d
to
t
zero.
Potential flow theory looks at the flow outside the boundary layer
(remember that we assumed "irrotational"? The boundary layer is certainly rotational), so
viscous drag is not captured by potential flow theory.
We have to use "common sense" tricks like the "Kutta condition" to figure out the
"right"amount of rotation in the boundary layer to solve an aerodynamics problem.
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(2) Pressure drag.
drag
The flow becomes distorted so that regions of low pressure occur on the downstream
side, pulling the aircraft backwards. We try our best in aircraft design to avoid such regions.
Examples of such regions are flow separation zones or "recirculation
recirculation bubbles"
bubbles , and wakes
wakes. We try
to make wakes as thin as we can, by keeping the low-momentum fluid in the boundary layer as
close to the surface as possible.
Pressure drag,
drag in many cases
cases, arises from viscosity,
viscosity but its cure may be related to making the
boundary layer turbulent, so that the flow remains attached. So, what you do to reduce skin
friction drag (i.e., making the boundary layer less turbulent) may in fact increase your pressure
dragg because the flow separates
p
over the aft pportions of the airfoil. For this reason,, it is a ggood
idea to consider these types of drag separately.
A flow separation region may be carried on a supersonic configuration, though of course the
velocityy inside this region
g mayy be low relative to the surface. An example
p is the recirculation
region formed over the wing of a winged re-entry vehicle when its control surfaces are deflected
(Space Shuttle, X-33, or the X-38 Crew Return Vehicle)
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(3) Drag due to shocks:
Shocks are regions where the entropy goes up, and the stagnation pressure drops. Thus they
cause drag.
Shocks cause large amounts of drag: the entropy increases suddenly across them. This is
because inside the sharply discontinuous shock, dissipative effects such as viscosity and heat
conduction take away some of the energy of organized motion and convert it to the energy of the
random,
d
chaotic
h ti motion
ti off gas molecules.
l l
Entropy rise shows up as a DROP in stagnation pressure.
The entropy rise across a shock is related to the third power of the static pressure change, or the
turning angle, whereas the drag coefficient in wave drag is proportional to the square of the
turning angle.
angle
The drag by a shock of given static pressure ratio is thus considerably greater than the total
wave drag due to several waves,
waves each of small pressure ratio
ratio.
Moral:
If you have to make waves in supersonic flow, make a series of small waves rather than
one big one. This is used in designing wings, fuselages, and engine inlets.
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(4) Wave Drag:
This is a strange phenomenon which is very much felt on supersonic aircraft, but would not be
anticipated without thinking of the differerence between subsonic and supersonic flows
.
Calculating Wave Drag:
We calculated drag coefficients in supersonic flow, from the pressure coefficient distribution over an
airfoil in supersonic
p
flow,, after assumingg that the flow was ppotential,, and neglecting
g
g boundaryy layer
y
effects. This is called "wave drag". It is a feature of supersonic flow. Disturbances in ideal supersonic
flow propagate out to infinity, unchanged in amplitude. The energy of these disturbances must come
from the kinetic energy of the flow
flow. This is the source of the "wave
wave drag
drag".
Wave drag increases as angle of attack (lift) increases. It also increases as thickness or camber or
any other disturbance increases.
increases
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In subsonic potential flow, on the other hand, the passage of an object leaves no trace: conditions
return to what theyy were before the object
j came by.
y This is because the disturbances from
different portions of the airfoil cancel out as one goes away from the airfoil.
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(5) Lift-Induced Drag Due to Finite Aspect Ratio
”Lift-induced
Lift-induced drag
drag", or "induced
induced drag"
drag . This is,
is literally,
literally drag induced by our efforts to generate
lift. As we know, lift is simply any aerodynamic force perpendicular to the freestream, and any lift
generation is accompanied by induced drag. We have seen how to calculate induced drag of
wings in low
low-speed
speed flow.
flow
Is there lift-induced drag in supersonic flow? Well, any time you generate lift, you turn the flow,
aandd tthere
e e must
ust be so
somee pe
penalty
a ty assoc
associated
ated with
t tthis.
s Thiss will sshow
o up in you
your wave
a e ddrag
ag
calculation.
At the wingtips,
g p the ppressure difference between the freestream and the upper
pp and lower
surfaces, causes the flow to turn in on the upper surface and out on the lower surface, just like
in subsonic flow. This redirection of momentum is not recovered - it is left behind, and after the
aircraft leaves, it forms tip vortices in the atmosphere, spinning away like little tornadoes. It was
the burning of fuel that generated the energy now used to make air go round and round - that is
drag, and it happened because of lift.
Although the wing cannot “feel” the effect of these vortices, you can account for the lift loss and
drag increase, by looking at how much energy is in the spinning vortices.
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Drag Mechanisms: Comparison Between Subsonic and Supersonic Flow
Type
Mechanism
Subsonic flow Supersonic Flow
Viscosity in the boundary
layer: Molecules bouncing off
surface transfer their
momentum change to the
molecules in the stream,
slowing them down.
down
Yes.
Yes (boundary layer is
still subsonic!!). Also,
compressibility leads to
substantial density
change and temperature
change: worsens drag.
drag
1 b) Skin friction: Momentum transfer
turbulent
becomes chaotic as entire
packets of fluid start moving
across the boundary layer:
High-speed flow comes
closer
l
tto surface,
f
lleading
di tto
greater shear, hence greater
drag.
Yes
Yes (b.l. still subsonic).
Also, compressibility
leads to substantial
density change and
temperature change:
worsens drag.
d
1 a) Skin friction:
laminar
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Type
2. Pressure
drag:
Mechanism
Subsonic
flow
Supersonic Flow
Wakes and recirculation regions of
low pressure
Yes
Yes (subsonic inside these
regions)
3. Lift-induced Energy formerly contained in
Yes
drag.
streamwise momentum, now goes
into forming rotating regions of flow
(tip vortices). Lift is lost as well,
near the wing tips.
b)
4. Wave drag
Pressure disturbances not
recovered by cancellation
No
Yes.
5. Shocks
Entropy rise due to irreversible
No
sudden compression s is roughly
proportional to the cube of the
static
t ti pressure change,
h
or th
the cube
b
of turning angle for oblique shocks:
3
Yes
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a)
Wave drag increases
with lift coefficient.
Momentum redirection
at the wing tips.
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