An analysis of supersonic aerodynamic heating with continuous fluid injection Page: 1 of 10
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AN ANALYSIS OF SUPERSONIC AERODYNAMIC HEATING
WITH CONTINUOUS FLUID INJECTION
By E. B. KLUNKER and H. REESE IVEY
An analysis of the steady-state aerodynamic heating problem
at high-supersonic speeds is madefor two-dimensional flows with
laminar boundary layers. The aerodynamic heating is shown
to be reduced substantially by injecting a small amount of coolant
through a porous surface into the boundary layer. The coolant
injection acts in two ways to decrease the aerodynamic heating:
First, and most important, the velocity profile is altered such
that the rate at which heat is conducted to the surface is reduced
and, second, the coolant absorbs an amount of heat which is a
function of the difference in temperature between the surface
and the coolant. The first effect prcrides the advantage of cooling
by injection orer that of simply using a coolant to absorb heat
from the surface. Calculations of the stability of the laminar
boundary layer show that for a wide range of high-speed flight
conditions the boundary layer would remain laminar at all
Reynolds numbers according to the stability-theory consideration.
The analysis includes calculations of the cooling requirements
and equilibrium surface temperatures for fiat plates and for
flat porous surfaces with several rates of fuid injection at 1ach
numbers from 5 to 15 and altitudes from sea lerel to 200,000
feet. Some calculations of the skin friction are also included.
The aerodynamic heating problem assumes considerable
importance at high-supersonic speeds. Singer and Bredt
(reference 1) have calculated the high-speed aerodynamic
forces and equilibrium surface temperatures at extremely high
altitudes where the molecular mean free path is large (free-
molecule-flow region) compared with a characteristic body
dimension. Although under these conditions the surface
temperatures are low, the maximum lift-drag ratios become
very small. Consequently, flight at these altitudes may be
satisfactory for nonlifting missiles but would be unsatisfactory
for steady level flight. On the other hand in the lower atmos-
phere (say, sea level to 200,000 ft) the lift-drag ratios are
considerably larger, but the aerodynamic heating is most
acute and some means of surface cooling must be employed
at very high speeds. The scope of the present report is
limited to a consideration of the steady-state aerodynamic
heating problem at altitudes where the air may be considered
a continuum, and particular emphasis is placed on a means of
cooling the surface by continuous injection of a fluid in the
Both the heat transfer and the drag coefficients are known
to be lower for laminar than for turbulent flows. At super-
sonic speeds it may be possible to maintain a laminar bound-
ary layer and thus alleviate the aerodynamic heating
somewhat. The theoretical investigation of Lees (refer-
ence 2) on the stability of the laminar boundary layer in
compressible flow indicates that the laminar layer is com-
pletely stable at all Reynolds numbers at supersonic speeds
for a sufficiently low ratio of surface temperature to stream
temperature. Thus, for flow over smooth surfaces small
disturbances are damped and a turbulent boundary layer
does not develop even at high Reynolds numbers. The
possibility of maintaining a laminar flow under these condi-
tions with finite but small disturbances in the boundary
layer-that is, maintaining a laminar boundary layer over
surfaces that may be employed on a high-speed aircraft-has
not been verified experimentally. Experimental verification
of this question as well as the general conclusions of Lees'
work is therefore desirable. Nevertheless, in view of the
conclusion of laminar stability for infinitesimal disturbances,
it is desirable to investigate theoretically methods of decreas-
ing the aerodynamic heating for laminar flows.
The aerodynamic heating is alleviated somewhat with
increasing altitude, but even with laminar flow at altitudes
of 200,000 feet the surface temperatures may be excessive at
high-supersonic speeds. An investigation of means of cooling
the surface is therefore desirable. One possible method of
cooling is that of injection of a cool fluid through a porous
surface into the boundary layer (reference 3). This coolant
injection acts in two ways to decrease the aerodynamic
heating: First, it alters the velocity profile such that the
heat-transfer rate from the fluid to the surface is reduced
and, second, the coolant absorbs an amount of heat which is
a function of the difference in temperature between the
surface and the coolant. The first effect provides the
advantage of cooling by fluid injection over that of simply
using a coolant to absorb heat from the surface.
Injection of the coolant through a porous surface affects the
boundary-layer stability in two ways. The direct effect of
injecting a fluid in the boundary layer is to alter the velocity
profile such that the flow is less stable. The indirect effect
is to give a lower surface temperature which in turn tends
to make the flow more stable. Actual calculations must
determine whether the flow would be stable for given
conditions. This question will be discussed in more detail.
The blowing rate should be kept as low as possible, consistent
with adequate cooling, since the coolant will probably be
carried by the aircraft. Consideration of storage would
probably dictate the use of a liquid, and a coolant which has
a low temperature and a high heat capacity would be most
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Klunker, E. G. & Ivey, H. Reese. An analysis of supersonic aerodynamic heating with continuous fluid injection, report, September 29, 1949; (digital.library.unt.edu/ark:/67531/metadc60328/m1/1/: accessed September 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.