In 
the last class, we have mentioned that the
oblique shock and expansion theory can be
used to construct many important aerodynamical
problems, particularly for the geometries
which have straight segments. The most important
aerodynamical problem is of course, the flow
past an airfoil and wing. Since, the airfoil
and wings are very thin, it is customary to
assume the wing and airfoil to be flat plate.
Flat plate airfoils are very widely analyzed
and always taken as the first problem in aerodynamic
of wings and airfoils.
Let us consider a supersonic flow. At first,
a flat plate, say at an incidence of alpha.
Let us consider a flat plate of chord C and
at an angle of attack alpha. The chord is
C. Let us consider a supersonic stream is
approaching this airfoil. We will consider
this angle of attack is such that even after
the disturbance you created on this free stream,
the flow remains supersonic always. Now, since
there is no upstream influence in supersonic
stream, the streamlines ahead of this plate
remain straight and the upper surface flow
is independent of the lower surface flow.
Now, the flow on the upper surface turns and
since it is an expansive turn; it turns through
a centered expansion fan, and the 
Mach number increases, pressure decreases
downstream or on the surface of the plate.
The plate being straight there is no further
change in the properties. So, if we consider
a particular streamline ahead of these, which
is straight, now turns and become parallel
to the flat plate surface. Now, coming to
the trailing edge, this flow needs to turn,
so that it is parallel to the free stream
and; obviously, these turn then in opposite
direction and will be achieved through a shock
and the streamline will now become parallel
to the free stream.
On the lower surface the reverse happens.
The flow undergoes again a turn of angle of
attack alpha, but through a shock, and at
the trailing edge again it turns by an centered
expansion and a lower surface streamline can
be Now, knowing the amount of turn alpha and
knowing the Prandtl- Meyer function nu upstream
corresponding to M 1 and alpha, we can see
what is the Prandtl-Meyer function here.
So, by knowing nu 1, we can find nu 2 here,
where nu 2 is simply nu 1 plus alpha. Alpha
is the magnitude of the angle of attack. Now,
knowing nu 2, this of course, gives us the
mach number M 2, and using isentropic relation,
p 2, T 2 from isentropic relation and again
knowing the flow turning angle alpha here,
and the mach number of ahead of these shock
M 2, using the M theta beta relation or the
M theta beta chart, we can find the wave angle
corresponding to the shock. Once the beta
for this shock is known, we can calculate
the pressure, temperature, density; all other
properties at downstream. The pressure of
course, should be the upstream pressure.
Similarly, on the lower surface, knowing M
1 and the flow turning angle theta, we can
find out the wave angle beta and the downstream
mach number here. Let us call that to be M
3 and p 3. The pressure will be higher here,
and again knowing M 3 and the turning angle,
we can also calculate the downstream flow
properties. Now, looking to the pressure distribution
over the airfoil, we see that ahead of the
flat plate, the pressure is p 1 and then on
the upper surface pressure is reduced to certain
value p 2, while on the lower surface; the
pressure is increased to value p 3.
So, what we see is that there is higher pressure
on the lower surface and lower pressure on
the upper surface. However, the flat plate
being straight over this, the pressure is
uniform over the entire plate. Consequently,
there is a lift force acting on this airfoil.
The lift force L is now given by p 3 minus
p 2 into c cos alpha, of course, per unit
span, and this can be approximately written
as p 3 minus p 2 into c, for small alpha.
We can also see that there is a drag force
given by p 3 minus p 2 into c sin alpha, which
again is, approximately p 3 minus p 2 into
c alpha, for small angle of attack.
Coming back to this flow problem again, you
see that the lowest upper surface flow, first
encounters an expansion wave, expansion fan,
through which this mach number increases and
the shock is at upstream mach number of M
2, which is higher than M 1. While on the
lower surface, the flow encounters the shock
first at a mach number M 1 and isentropic
expansion fan at the trailing edge, which
increases the mach number again back to M
1.
Now, the upper surface flow and then lower
surface flow experiences the shock at different
mach number. The upper surface shock being
at higher mach number and consequently for
same amount of turning, this will be a stronger
shock. So, on the upper surface we have a
stronger shock compared to the shock on the
lower surface, as a consequence, the entropy
change experienced by the upper surface flow
and the lower surface flow are not equal.
Hence, there is a difference in the entropy
level on the upper and lower surface flow
and consequently we will have a slip stream.
As you have seen, in earlier cases that on
the two sides of the slip stream other than
the pressure and flow direction, the other
flow parameters can have different value.
The entropy values are obviously different
as you have seen. Similarly, the density and
the velocity, they will also have different
values or as if this works as a tangential
vorticity, which creates a difference in tangential
velocity across it.
Now, this is the flow we are near the flat
plate; however, if we try to look to the far
field then you can see that these expansion
fans may interact with the shock. That is
as you have seen earlier that there might
be shock expansion interaction, consequently
the shock may become curve and there will
be deflected surface or deflected flow pattern
and the far field flow pattern is likely to
be and consequently the shock is likely to
be curved, which shows the slip line and this
there can reflect back and again this reflector
wave.
See that in the far field and particularly
flow downstream is quite complex with having
shock expansion interaction and then deflection
and again interaction of those deflected waves
and so on. However, in these cases, these
reflections are will not be incident on the
flat plate itself and consequently the flow
over the flat plate or the in the immediate
vicinity of the flat plate will remain undisturbed
of these shock expansion interaction. The
lift and drag for inviscid flow are quite
accurate and exact.
One very important thing that we notice here
is that in a supersonic flow, even where there
is in two dimensional cases, there is a finite
amount of drag force. You have seen that in
subsonic two dimensional inviscid flows, there
is no drag force; the drag force in inviscid
flow can come only in three dimensions, which
is lift induced drag. However, in case of
supersonic flow we see that there is a drag
in two dimensional inviscid flow as well.
This drag is known as the wave drag and this
drag, which is in this case as p 3 minus p
2 into c sin alpha. This drag is called wave
drag inviscid supersonic flow. So, this is
a special feature of supersonic flow that
there is a drag even in two dimensional inviscid
flows and this drag is called the wave drag.
This is due to the presence of waves in supersonic
flow this drag comes in and this will always
be present in a supersonic flow.
Now, let us consider another very important
supersonic airfoil which is the diamond section
airfoil.. Let us consider, we have a symmetric
Consider a symmetric section of symmetric
airfoil at zero incidence; however, the construction
of the flow will be same even if the airfoil
is not symmetric and the angle of attack is
nonzero. Let us consider this angle to be
epsilon and epsilon; that is total two epsilon
angles and same here is also two epsilon.
Let us consider again a supersonic free stream,
now see that as you have discussed earlier
that here the flow will turn and become parallel
to the wall, through a shock and on the both
surface this will be a shock in this case.
However, whether it is a shock and expansion
fan that will depend on the amount of turn
the flow will undergo. So, in case this airfoil
at an angle of attack and that alpha is such
that this becomes an expansive turn, in that
case, there will be an expansion fan here.
So, subsequently of course, here the flow
will turn again through an expansion fan,
and then again it will turn here by an oblique
shock. So, remember that we are considering
the flows where even occur decrease in mach
number through a shock, the flow still remains
supersonic.
We are always considering only weak solutions.
So, if we consider a streamline upstream of
the mach numbers. Upstream of the airfoil,
the streamline is straight at a mach number
M 1 with pressure p 1. At the first shock
it turns and become parallel to this straight
segment. Through the expansion fan, the flow
again smoothly turns and become parallel to
this downstream surface, again it turns through
the oblique shock and become parallel to the
free stream and exactly identical happens
at the lower surface.
Now, if the flow is symmetric or if the geometry
perfectly symmetric, then the mach number
at the corresponding segment on the upper
and the lower surface are exactly identical.
All the shocks occur at same mach number and
the entropy change undergone by this flow
on the upper surface, either on the lower
surface, are identical. Hence, in the wedge,
the flows have same entropy and there is no
slip stream here. However, if there is symmetry
in the geometry, which will cause symmetry
in the flow, then there will be a slip stream
from the trailing edge, which deduction will
be determined by the total term that the flow
undergoes the upper surface and on the lower
surface.
Now, let us say the mach number here is M
2, and the pressure is p 2, and here it is
M 3 and p 3, and here it is M 4 and p 4, which
should be p 1. This p 4 should be p 1. Now,
the pressure distribution, if we plot on the
airfoil surface of the upstream of the geometry,
the pressure is uniform at p 1 at the shock
pressure jumps to p 2 and on the surface pressure
changes to pressure remain constant at p 2.
On the downstream face the pressure is again
uniform at p 3 at which is much lower, remember
the turn here is the double of the turn here.
Now, if the geometry is symmetric and at zero
angle of attack the flow is also symmetric
on the upper and lower surface, as we have
discussed, and consequently the pressure on
this face is also equals p 2 and pressure
on this face is also equals p 3. Consequently,
there is no lift force in this case, the pressure
distribution having upper and lower surface
symmetry, no lift force, pressure distribution
has;
However, since there is an over pressure on
this forward face and lower pressure, under
pressure on the downstream face or their rearward
face; over pressure on the forward face, and
under pressure on the rearward face, this
results in a drag force. If this maximum thickness
is denoted by t, and the chord by c, then
the drag force can be written as drag force
D.
This drag force can be written as p 2 minus
p 3 into a component of e in the drag direction
into t per unit span and that is equal to
p 2 minus p 3 into t for small epsilon. Now,
let us consider a curved airfoil section.
Consider a curved airfoil section 
for supersonic flow the airfoils are usually
having sharp leading edge so that the shock
remain attached and there is no detached bow
shock and hence no large pressure loss. So,
considering a curved airfoil and in this case
there is a shock attached at the nose; however,
subsequently there is a continuous expansion
occurs along this surface and consequently
in this case, these expansions are very close
to this shock and they will interact and 
consequently the shock will become curved.
At each interaction, the shock strength will
decrease and due to this attenuation the wave
angle will also change and the shock will
continuously bend. The pressure distribution
here, we can see that 
at 
there is no upstream influence as in the case
of supersonic flow. So, the pressure up to
the leading edge remains at p 1 and at the
leading edge there is a increase in pressure
due to the shock; however, the pressure continuously
falls and again it increases to the free stream
pressure.
The shock at the trailing edge will also be
curved because these interaction of these
expansion fan and shock, expansion waves and
shocks, there will a reflection and these
reflections will definitely hit or intersect,
interact with the trailing edge shock. Not
only that, even these reflected waves may
interact from the airfoil surface and through
these interactions, there will be continuous
change in entropy and also of vorticity and
the flow will be rotational even though this
is in inviscid flow.
Now, in this case, of course, because of this
continuous change in pressure, it is not straightforward
to apply this shock expansion theory and using
this exact shock expansion results, you cannot
compute the properties on this surface of
these airfoils. To do this, we need to simplify
the shock expansion theory or results of the
shock expansion theory.
So, we will and which gives us a supersonic
thin airfoil theory. As you have seen that
if we apply this shock expansion theory we
get exact result; however, this shock expansion
theory is not very convenient for a curved
airfoil section and that is the shock expansion
theory result cannot be expressed in a concise
analytical form. Hence, it is not suitable
for various application particular having
curved surfaces.
Now, we can linearize this shock expansion
theory for a thin airfoil, because when you
have a thin airfoil theory at small angle
of attack; the flow deflections are always
small. Consequently, we can use that basic
pressure rise relationship, approximate pressure
rise relationship. Now, when the turn is very
small and the pressure rise is also small,
the flow deflection is also small and the
difference in pressure ahead and behind the
wave is also not large. Consequently, p 2
and M 2 are, that is the pressure and the
mach number behind the wave is not much different
from p 1 and M 1 that is pressure and mach
number ahead of the wave. For small deflection,
shock is weak and M 2, p 2 are close to M
1, p 1.
Using this approximation, we can even write
delta p by p 1 is approximately equal to gamma
M 1 square by. Now, theta is now relative
to free stream and now the pressure coefficient
is defined as p minus p 1 by half rho 1 u
1 square and that is delta p by gamma y 2
p 1 M 1 square. that is pressure coefficient
at any point is expressed in terms of the
slope of the local position.
So, this is in a sense local inclination theory,
which says that the pressure coefficient at
any point on a thin airfoil is simply given
by the free stream mach number and the local
flow inclination. Now, considering, if we
applying this to a flat plate, at angle of
attack alpha, the pressure coefficient on
the upper surface is minus 2 alpha by and
pressure coefficient on the lower surface
is
Consequently, the lift curve, lift coefficient
C l is p l minus p u into c cos alpha by.
So, for flat plate at small angle of attack
alpha, we can see that lift coefficient according
to this linearized theory or thin airfoil
theory in a supersonic flow is 4 alpha by
root M 1 square minus 1, or the lift curve
slope is, while the same result for incompressible
thin airfoil theory; incompressible thin airfoil
theory is 2 pi. We 
also see 
the drag coefficient. So, that the wave drag
coefficient from thin airfoil theory; we can
clearly also see that the aerodynamics enter
in this case is at the mid chord wing.
Similarly, if we consider the diamond section
airfoil for the diamond section airfoil; this
C p is given as plus minus 2 epsilon by root
over M 1 square minus 1; plus is on the front
face and similarly, minus is for the rear
face, and this gives the pressure difference
on the front face and the rear face and that
had to be front face minus rear face into
half. Using the formula that we have derived
earlier the drag is approximately p 2 minus
p 3 into t, which is p 2 minus p 3, and this
t can be expressed in terms of epsilon into
c. Substituting these pressure coefficients,
this becomes 4 epsilon square by root over
M 1 square minus 1 into half rho 1 u 1 square
into c and this simply is C D equal to.
Now, for a curved general curved airfoil for
a general curved airfoil similarly, we can
express pressure coefficient in terms of the
local surface inclination and in terms of
the local surface inclinations, pressure coefficient
on the upper surface is and on the lower surface
it is. If x axis is aligned with chord then
C p u.
The lift force can be obtained as half rho
1 u 1 square into 0 to C, and drag force can
be obtained as half rho 1 u 1 square into
0 to c and similarly, the lift coefficient
can be 1 by c, 0 to c, C p L minus C p u of
d x and 
the wave drag coefficient similarly, will
become 0 to c, C p L minus d y l d x plus
p u d y u d x. In this case, we have assumed
that the x. These all horizontal reactions
not aligned with the chord and the appropriate
relations it can be used if we align the x
with the chord
Since, the average camber, if the average
camber over the entire airfoil is zero, the
lift coefficient airfoil with average camber
zero. In that case, this lift coefficient
will again come to 4 alpha by root over M
1 square minus one; however, C d will become
as 4 by root over M 1 square minus 1; alpha
square plus square of average camber plus
average thickness. So, to summarize we have
first constructed flow problems for some simple
two dimensional geometrical configuration,
which have state geometry and we have seen
that for these cases we can apply our exact
shock expansion theory to find the forces
acting on these airfoils.
However, since exact shock and expansion theory
results are not amenable to concise analytical
form, we have seen that these exact relationship
are not straightforward or not conveniently
used for geometry with curved section. For
such situation, we have linearized or made
approximation of thin airfoil or thin geometry.
We set a thin and small angle of attack and
we have approximated for shock expansion theory
results and seen that the pressure coefficient
at any point on geometry is a function of
the local surface inclination. Using that
theory, you have again computed the lift and
the coefficient of some simple geometries;
however, this must be remembered that these
are based on the assumptions that the flow
patterns remains simple at least on the surface
and very close vicinity of the geometry, that
is the reflected waves they do not; they are
not incident upon the geometry and if the
reflected wave is incident of the geometry,
then of course, the simple relation simple
analysis is invalid and more complex analysis
or in general numerical simulation of the
complete equations are essential.
