Deck 9: Hypersonic Vehicles

Question 1

Consider a laminar boundary layer on a flat plate. At the trailing edge of the plate, with a free-stream Mach number of 2, the boundary layer thickness is 0.3 in. Assuming that the Reynolds number is held constant, calculate the boundary layer thickness for a Mach number of 20.

Answer 1

The relation between laminar boundary layer thickness
and Mach number for a compressible flow is expressed as,
…… (1)
Here,
is the Mach number of the flow, and
is the Reynold's of the flow.
For Mach number equivalent to
, solve for Reynolds number from equation (1) as follows,
And,
…… (2)
Similarly, for Mach number equivalent to
, solve for Reynolds number from equation (1) as follows,
…… (3)
Compare equation (2) and (3) and solve for boundary layer thickness
at Mach number
as,
…… (4)
Substitute
for
on the left part of the equation (4) and
for
on the right part of the equation (4),
for
and solve,
Hence, the boundary layer thickness for Mach number of 20 is
.

Question 2

Consider a hypersonic vehicle flying at Mach 20 at a standard altitude of 59 km. Calculate the air temperature at a stagnation point on this vehicle. Comment on the accuracy of your answer.

Answer 2

The relation between temperature ratio and Mach number for a subsonic compressible flow is expressed as,
…… (1)
Here
is the temperature at the stagnation point,
is the temperature of the free-stream flow,
is the Mach number of the flow,
is specific heat ratio, equivalent to
;
being specific heat at constant pressure and constant volume, respectively.
At a standard altitude of
, consider the temperature
of the free-stream as,
Consider the specific heat ratio
for air equivalent to
.
Now, calculate the temperature
at the stagnation point from equation (1) as,
Substitute
for
,
for
,
for
, in the above equation and solve for
as,
Hence, the temperature at the stagnation point is
.
Comment:
The temperature at the stagnation point calculated above is extremely high temperature. Air becomes highly chemically reacting at such high temperatures, and the ratio of specific heats no longer remains constant; moreover, the above equation, that assumes constant
, no longer remains valid.
It is thus implied that hypersonic flows can be very high temperature flows. However, the gas temperature at the stagnation point will be much lower than that calculated above, as the dissociation of the air requires energy; it will be approximately
, which is still quite high, but is sufficient to cause massive dissociation of the air.

Question 3

Assume that the nose of the Space Shuttle is spherical, with a nose radius of 1 ft. At Mach 18. calculate ( a ) the pressure coefficient at the stagnation point and ( b ) the pressure coefficient at a distance of 6 in away from the stagnation point measured along the surface.

Answer 3

Rayleigh Pitot tube formula is expressed as,
…… (1)
Here
is the total pressure behind the shock wave,
is the pressure of the free-stream,
is the specific heat ratio,
is the Mach number of the free-stream.
Pressure coefficient
at stagnation point is expressed as,
…… (2)
Here
is the total pressure behind the shock wave,
is the pressure of the free-stream,
is the specific heat ratio,
is the Mach number of the free-stream.
Modified Newtonian law is expressed as,
…… (3)
Here
is the pressure coefficient at stagnation point,
is the pressure coefficient, and
is the free-stream direction.
Diagram representing the flow over a sphere.
Since,
Thus, angle
is expressed in radians as,
In the above diagram, angle
is calculated in degrees
as,
…… (4)
Here
is the distance, and
is the radius of the sphere.
Now, calculate the angle
as,
Substitute
for
, and
for
in equation (4) and solve,
Now angle
is calculated as,
Substitute
for
in the above equation and solve,
…… (5)
(a)
Consider the specific heat ratio
for air equivalent to
.
Solve for pressure ratio
from equation (1) as,
Substitute
for
, and
for
in the above equation and solve,
…… (6)
Now calculate the magnitude of
from equation (2) as,
Substitute
for
,
for
, and
for
in the above equation and solve,
…… (7)
Hence, pressure coefficient at stagnation point is
.
(b)
Calculate pressure coefficient
from equation (3) as,
Substitute
for
, and
for
, in the above equation and solve,
Hence, pressure coefficient at a distance of
away from the stagnation point measured along the surface is
.

Question 4

Consider an infinitely thin, flat plate. Using Newtonian theory, show that C L , max = 0.77 and that it occurs at ? = 54.7°.

Question 5

Consider hypersonic flow over an infinitely thin, flat plate. The zero-lift drag coefficient is denoted by C D ,0. (Note that the zero-lift drag for a flat plate is entirely due to skin friction.) Consider that the wave drag coefficient is given by the Newtonian result for drag coefficient-that is, by Eq. (10.12). Also assume that the lift coefficient is given by the Newtonian result in Eq. (10.11). We wish to examine some results associated with ( L / D ) max for this flat plate. Because ( L / D ) max occurs at a small angle of attack, make the assumption of small ? in Eqs. (10.11) and (10.12). Under these conditions, show that at maximum L/D , ( a )
and occurs at
and ( b ) the wave drag coefficient = 2 C D ,0.