# An analysis of supersonic aerodynamic heating with continuous fluid injection Page: 1 of 10

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REPORT 990

AN ANALYSIS OF SUPERSONIC AERODYNAMIC HEATING

WITH CONTINUOUS FLUID INJECTION

By E. B. KLUNKER and H. REESE IVEYSUMMARY

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.

INTRODUCTION

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

boundary layer.

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 heatingsomewhat. 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

efficient.

701

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### Reference the current page of this Report.

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.