Skin temperatures and heat transfer over wedge wings at extreme speeds

CoA Report No. 105

Kluyverweg 1 iS DELFT *^°"°

THE COLLEGE OF AERONAUTICS

CRANFIELD

SKIN TEMPERATURES AND HEAT TRANSFER

OVER WEDGE WINGS AT EXTREME SPEEDS

by

- 2 m. 1957

REPORT NO. 105 AUGUST. 1956.

T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D

Skin. Teniperatures and Heat T r a n s f e r o v e r Vfedge ïïings a t Extreme Speeds of F l i g h t h y -T. N o n w e i l e r , B . S c . , A . P . I . A . S . S U Li 11 A R Y By p a y i n g s p e c i a l a t t e n t i o n t o t h e vang d e s i g n and a l t i t u d e of f l i g h t , i t i s p o s s i b l e t o e n s u r e t h a t t h e ' h i g h e s t t e m p e r a t u r e r e a c h e d a t t h e l e a d i n g edge of t h e vdng of an a i r c r a f t , i n l e v e l f l i g h t a t s p e e d s of t h e o r d e r of t h e

c i r c l i n g v e l o c i t y , need be no more t h a n about 1000 C, Formulp.e and c h a r t s a r e p r e s e n t e d t o e n a b l e t h e a c t u a l

s k i n t e m p e r a t u r e c l o s e t o t h e nose t o be p r e d i c t e d , f o r a wedge-shaped vving, i n t e r n s of s k i n t h i c k n e s s and c o n d u c t i v i ' j y ,

Contents

I

Notation used in Main Text.

1. Introduction

2. Deduction of a Limit to Skin Temperature

3. Measuj^es for the Reduction of the Maximum Skin Temperature

4. Some Numerical Values

List of References

Appendix I: Conditions in the Boundary Layer of an Inclined

Piano sTorface producing an Intense Attached Shockwave

Appendix II: Conditions behind an Intense Inclined Shockwave in Air

Notation used in Appendices

L wing loading

L measTore of L in kg/sq.m (1 kg/sq.m = 0.205 l b / s q , f t )

n

Q constant chosen so that

Q Z / I T

approximates to rate of heat

transfer per unit surface area from boundary layer

close to leading edge

Q maximum

x>f

Q vd.th respect to flight speed (achieved for

max

z r,

T)\

q-, = 2 g R) ^ 3

Q (5) value of Q relative to that for flat plate moving parallel

to itself, at same speed, and having same surface pressure

R earth's radiiis

T surface temperature

TT.„ 'iTalue of T at wing leading-edge

T maximum of T^.^ with, respect to flight speed

" ^ (achieveS^f or q^ = 2 g R )

3

d skin thickness

d measure of d in cms.

m

f

non-dimensional temperature defined by equation (l), and

evaluated in fig, 2

g gravitational acceleration

k thermal conductivity of skin

k measure of k in cals/cm, sec. deg.C. (1 cal/cm. sec.deg.C

ra

_{= 242 B. Th, U/ft. hr. deg.}

_{F )}

X charaj3teristic length of surface affected by conductivity,

and defined by equation (2),

p. STJcrface pressure on underside of wedge a e r o f o i l , taken a^

constant, (measured in atmospheres)

q. speed of flight (measured i n Km./sec.)

X d i s t a n c e from leading edge

X measure of x i n metres

m

5 i n c l i n a t i o n of undersurface of wedge t o d i r e c t i o n of

f l i g h t ( p o s i t i v e if forward facing)

e e m i s s i v i t y of e x t e r n a l surface of vmderside of wing

s . r a t e of n e t t heat l o s s by r a d i a t i o n pejr u n i t area of i n t e r i o r

surface of underside of wing 4- crï^'

_ Q 1

cr S t e f a n ' s C o n s t a n t ( = 5 . 7 x 10~ w a t t s / s q . m . ( d è g . c ) ,

1• Introduction

At speeds of flight usual for present-day aircraft it is weèl-knovm that due to aerodynamic heating, the v/ing surface of an

aircraft - at least neajr its nose - reaches the "thermometer"

teinperature whose value depends only on the speed of flight. If the same v/ere true at such speeds as v/ould, for example, be involved in flight of satellite vehicles, the ccnsequences v/ould be profound, because the corresponding therraometer temperatiires are ct the order

of 10,000 C and more. Fortunately, it is not true, and the present report discusses one limit to the maximum teinperature reached close to the wing leading-edge, in sustained level flight at such speeds, as imposed by the action of conduction of heat within and along the skin. There may VTCII be other and indeed lov/er limits, but the one

considered can be established v/ith a fair amount of certainty, even if it represents an unnecessarily pessimistic view of the intensity of kinetic heating,

The results v/e deduce are relevant only to a wing of v/edge section (constructed as a hollov/ shell v/ith a thin skin), not because of any particular merit that this section may have, but because the boundary layer flow about it may be most easily deduced for flight at such extreme speeds as these v/e have in mind. In the same "vvay, the aircraft miglit v/ell not be designed as a "fljmig-wing" so that the temperature reached by the fuselage v/ould also present a problem of some importance. However, as we shall shew that the wing loading

p

_s

2

-raust be low, and the v/ing thickness (or v/edge angle) can be without penalty very large, it is likely that an all-wing design might after all be involved.

We first of all establish (in section 2) how we may determine the limit to the skin temperature, and also the rate of heat transfer at extreme speeds. Details of this latter determination are given in Appendix I, and as the flov/ in the boundary layer is aXfected by

the intense shock v/ave developed by a wedge at incidence, it is necessary also to discuss the relevant shock wave conditions: this is dcane in Appendix II. \7e next suggest (in section 3) measures which may be adopted to reduce the maximum skin temperature fotind by these means; and finally, by v/ay of illustration, we calculate numerical values of thP temperature reached by an aircraft, designed to include these measures, and flying at the particular speed at which kinetic heating

effects are m«st severe. This speed is incidentally shown to be equal olrcling

to juat over 80% of the velocity (namely, 6.5 Kn/sec,), It is the intention to show that particular attention to the flight plan and overall design can greatly simplify the problems of kinetic heating. Nose teanperatures of no more than aroxond 1000 C are all that need be involved. Dovmstream of the nose temperatures will of course become progressively smaller still, but it is not our object to determine

anything other than a limit to the maximum values.

2. The Deduction of a Limit to Skin Temperature.

Supposing that there are no regions of turbulent flow, ejid that steady conditions prevail, the highest temperature will be reached at the nose of the body or v/ing surface, where the heat transfer is greatest. As mentioned in the introduction, in what follows v/e shall be dealing expressly with a v/edge shaped wing (fig.la) constructed as a hollow shell. The s\irface pressures in inviscid flow over such a wing are constant on the upper, lower and base surfaces, and the usual approach of boundüry layer theory shows that at the nose, the rate of heat transfer due to kinetic heating is infinite, irnless the local surface temperature has the particular "thermometer" value. If this approach were indeed valid, then it would follov/ that a non-conducting skin woiild instantaneously heat up to the thermometer temperat\jre characteristic of the flight condition.

Hov/ever', the argument which leads to this deduction can be faiiLted on tv/o counts. In the first place, the rate of heat transfer is not likely to be infinite at the nose, and the theory vrfiich predicts such a singularity breaics dov/n in this vicinity. For a start it is ivell-knov/n -chat the simplified form of the Prandtl equations of motion and of energy, usually invoked in boundary layer analysis, are inapplicable in this region, and the full Navier Stokes equations are to be preferred, though they have proved intractable. Quite apart from this, there are modifications to the usual solutions to account

-4-for the self-induced pressure field of the boundary layer, and the proximity of the leading edge shock wave, and also to acco\ant for the existance of a velocity of slip and a temperature jump at the surface. All of these factors are known to produce effects of increasing iirportance towards the leading-edge, vrfiich might v/ell determine a maximum rate of heat transfer. Ihilst orders of magnitiide of this maximum rate can be estimated by existing analyses, it is not considered that sufficient is known as yet to predict the value with any certainty.

In what folloY/s we shall treat the usual ajjproximations to the rate of kinetic heating as valid right up to the leading-edge, and although this is plainly v/rong, it does err on the pessimistic side in providing a local over-estimate of hea.t fliox. But even then it does not follov/ tliat the nose v/ill heat up to the "thermometer temperatvire", because the second assumption v/e mentiohed in connection with this deduction can also be faulted. This is that the skin is a non-conductor of heat. IWiilst this may be an adequate assumption in dealing with the surface temperatixre distribution downstream, it can certainly lead to important errors in the vicinity of the nose, v/here - if the skin v/ere really non-conducting, there would exist severe longitudinal teraperatiore gradients; even in a medium of small conductivity these gradients v/ould irnply an appreciable longitudinal rate of heat flux along the skin, v/hich v/oiold tend to "cool" the nose at the expense of the structure dovmstream.

The problem of the action of conductivity in limiting the nose temperatTore has been discussed elsewhere . The assumptions of this analysis are that:

(i) The skin is of uniform thickness d, which is small cocipared with the characteristic length t of the region affected by conductivity (to be defined later)

(ii) The rate of kinetic heating per unit area to the external surface i/2

is Q/x , where x is the distance from the nose, and Q may be

f .

taken as a constant over the length ", (The value of Q may be different on upper and lower wing surfaces).

(iii) The exterior surface loses by radiation a nett heat fltix equal to e OÏ v/here cr is Stefan's constant, and T the surface temperature; and the interior surface suffers a nett loss of heat by radiation equal to e.oT^

(iv) The interior of the shell formed by the wing skin is non-conducting. Then it follows that

T =

(s 4^.)Vkd U J

' 0 l' J

(1)

where u is the characteristic length and

i

.

k^d^

J

-é-These relations can be deduced on the basis of dimensional homogeneity, and the function f satisfies a non-linear second-order differential equation. The fuiiction f is sho^vn in fig. 2, which represents the

result of a more refined method of solution of the differential equation involve! than that given in the reference v/ork. The evaluation of the maximum temperatxxre v/hich - as will be seen from fig. 2 - is attained at the nose (x = O ) , is possible once we have allotted a s-uitable value to the constant, Q, As the heat transfer at, or close to, the nose depends only on the local value of the sxorface temperature, and not on its dovmstream variation, it is possible to

evaluate Q by reference to calculations of the heat transfer to sijrfaoes with constant surface temperature, v/hich fortunately is the condition most usually studied. This temperature is placed equal to T.j^ and then the relation

^LE = •'•IS

.V,3

(3)

(e +e .) cr kd

. o i'

obtained by noting that f(o) = 1.15, is an implicit equation for T

At extreme flight Mach numbers it was shovm in ref,2 that Q is nearly independent of the sxirface temperature - vmless it has improbably high values. It is only flight at extreme speeds T/riiich interests us in the present discussion, so that the results of this reference work will be invoked. They ore relevant to the lajninar boun(3£iry layer on a surface at uniform pressure p^ moving at velocity

q., and inclined a t only a small angle of incidence t o the flow, and

t h e y show t h a t , xl/2

g ,, 2 / % \ ., /

1/2 = "•''• 'i^ {

ra

v/here q,^ is in Kir/sec, p^ in atmospheres and :: in metres. This dimensional form of the result demands no assiimption about the

atmospheric conditions prevailing as will be clear from the analysis of Appendix I, which generalises the previous resiilts so as to make them applicable to forv/ard facing siorfaces inclined at a finite angle 5 to the direction of motion. We find in fact in this Appendix that if dissociation effects are ignored

Q = ^ 4 Q ( 5 ) q y / 2 : ^ ^ 0 B . ) 3 / 2

^ ° (4)

where the function Q ( 6 ) is shown in fig. 3 for all angles 6 below that for shock wave detachment at high speeds. This is taken to be about 50 , which is higher than the figure ustmlly quoted: however the figure used is cased on an allowance for the change in the

specific heats cf aii' at the elevated temperatures existing behind an intense shock wave. The precise form of these allowances is

detailed in Appendix II, and a comparison of the variation of Q ( 6 ) obtained by assigning the specific heats constant is provided in fig.3. In this matter It wxy also be pointed out that some investigators liave suggested that the effects of radiation Proa "behindan intense shockv/ave itself are such that it can be treated as virtually an isothermal process, in which event the corresponding

-8-foïïn of the function Q ( 6 ) approximates simply to Q ( 6 ) » oos 6, also shown in fig.3.

It will be noticed that all these assumptions lead to the döduotion that the heat transfer reduces with increasing surface deflection, 6, The adopted assumptions of the present

•voxk siiggest in fact a reduction intermediate between the others. This reduction over a forward facing storface may appear unreasonable unless it is recalled that the coniparison is based on surfaces

subjected to equal pressiore (pg) . A n inclined surface will only preserve this same pressure if the eiltitude of flight is hi^er, and the speed of flow outside the boundary layer is lower. If the altitude were kept the same, then of course the inclined surface would be subjected to a iiiiach higher rate of heat transfer. For all

rearward facing surfaces, the arguments of ref. 2 remain unaffected and the value of Q ( 5 ) can be consequently taken as unity for 6 ^ 0 .

The above discussion v/as prefixed by the statements that

there were to be no regions of turbulent flow, and that steady conditions prevail, "Whether or not steady conditions prevail, will largely

depend on the flight plan; in all consequent assessments we shall take the wing to be on an aircraft in steady level flight. But it m\ist be pointed out that in most other conditions of interest, the rate of heat transfer at the nose is so large that steady conditions would at least be quickly reached in the region affected by

Again, in regard to the implied possibility that higher rates of kinetic heating - and vidth them, higher surface temperatures, might be reached in regions of turbulent flow, if present, it should be noted that over a forward facing inclined surface the flow outside the boundary layer has a. relatively small Mach Number (of order unity) and a very high temperat\Are (of order M. times that of the ambient air, if M is the flight Mach Number). Thus, the ratio of siirface to local static temperature is in all cases of interest small (and much less than unity). Almost certainly this v/ould suggest that the rate of heat transfer is sufficiently large to stabilise the boundary layer in all disturbances; research in this field has already deduced this to be true at least for tv/o-dimensional wave type disturbances in such conditions, v/hatever the Rejmolds Number of the flov/.

3, Meas"ures for the Reduction of the Maximum Skin Temperature,

Equations (3) and (4) above provide a basis for the caloiiLation of the maximum skin temperature. Y/ithout introducing any numerical values, they permit us to deduce qualitatively those measures which vsrill reduce its value.

For instance, increase of skin conductivity evidently lowers the maximum temperature reached and this might conceivably affect the choice of material used, tho'ugh the author is not competent to discuss all the issues that such a measure might involve. Moreover it must be ronembered that the conductivity of materials at elevated teanperatures can be very different from its value at ordinary temperatures: for most

-10-piire metals it is lower, and for alloys higher. For example all steels or ferrous substances at high temperatures tend to have about the same conductivity, though the high alloys are poor conductors in other circttmstances. On the other hand, it must also be recalled that the use of special conducting material v/ould be necessary only

over the length of the order t (the characteristic length defined by equation (2) ) , In most examples of interest the numerical value of I is not more than a few centimetres;thus the opportxxnity for choice of material is probably v/ider than might at first sight appear. This important fact must also be remembered in many other connections.

Evidently it also pays to make the skin thickness as large as possible, again over the length affected significantly by

conductivity. Hov/ever, if the skin thickness d becomes appreciable compared with t we cannot expect the analysis leading to the results under consideration to remain valid, as this violates one of the siraplifjring assumptions of the method. This point will be returned to at a later stage, but the general conclusion is always valid that the thicker the skin, the smaller will be the temperatiore.

Again the emissivity of the outer surface should be made as high as possible, to radiate as much heat away as possible. This is Y/ell-knov/n as a desirable feaXvice of design. It eq-ually pays to radiate heat av/aj"- from the inner siorface, provided it is not reflected or radiated back to it again. Now in relation to the particiiLar

Kanaalstraat 10 - DELFT

1 1

-is not so Irrge that the upper s-orface -is forward facing, the prerr,iare on this surface vd.ll be very small, and as a result so also will be the heat input to it by kinetic heating. Thus one can afford to radiate heat to it from the underside, which is suffering severe kinetic heating. All surfaces being "black" half of the heat emitted by the inner surface of the underside is returned from the facing surface, while half is radiated away from the outer surface cf the upper side. The temperature of the upper

side becomes (l/t/2 ) times that of the xander side, and the value of e.. = O.5.- (if any of the surfaces are not "black", then

s. <, 1/2. ) Yfe see from (3) that allcv/ing a transfer of heat by radiation of this kind is equivalent in effect to over a threefold increase ^Jl thickness in alleviating the maximum temperature. In case it is undesirable to heat any partic\ilar part of the wing interior in this way, it woxild be a simple matter to provide it with a

reflecting surface by v/ay of insulation.

Even greater reductions can be obtained by projecting the tinderside of the wing forward of the other so that radiation from both its sides can be obtained, ijnihhibited by any re-emission and absorption (fig, 1b); in this way v/ith black svirfaces it would be appropriate to take e - e s i . This may not appear too

o 1

undesirable a modification if it is again remembered that the

projection need be only of short length, to achieve the desired effect. The only other measvtre open for us to adopt, to decrease

-12-the temperature, is to decrease -12-the rate of heat transfer from -12-the boundary layer, so far as is possible. Now equation (4) shows that at a fixed speed of flight this means decreasing the surface pressure, or increasing the lower surfaxse inclination (6). For a wing of the type considered, in which the i^per surface pressure will be small compared with that of the underside at extreme Mach Numbers, the lift loading vdll be ( Pc cos 6 ) , whilst if the wing loading is L, say, the lift loading required for level flight is

R being the radius of the earth. Hence we see that reduction of surface pressioi-e, and so of temperature, is really a matter of reducing the v/ing loading.

Equations (4) and (5) together show that the angle of inclination (or incidence) of the underside may be increased with advantage provided that Q ( 6 ) / , / C O S 6 is thereby decreased.

The least value of this function is probably attained for a value in excess of the angle 5 required for Shockwave detachment; but as the analysis quoted is not applicable in this range, we can only say with assurance that increasing the altitude of flight, up to the highest value reconcilable vd.th the maintainance of an attached

shock at the wedge leading edge v/jJl reduce the rate of heat transfer, and so be of advantage. Such large wing incidences as are thereby involved have another advantage, in that it is possible to use quite

-13-large wedge angles for the v/ing section, v/ithout making any

appreciable difference to the heat flux to the upper surface,

Tïhich v/ill be to all intents and purposes exposed to a

near-vacuum.

To sum up, the meas\ires to be adopted to reduce the

maximum temperature are:

(i) To provide an adeqiJate skin thickness; ^ over a Dength of the

/..\ „

j . - n o u - ' u - i j - . - ^ underside of the order

(,11; To use a material of high conductivity; I

I of the characteristic

(iii) To project the underside for\7ard of the upper; , ,, 0

(iv) To ensiire that all surfaces have a good emissivity, even if

this implies allov/ing the upper side of the v/ing to be heated

by the ijnderside;

(v) To ensure that the wedge angle of the wing is not large enough

to cause the upper surface pressure to be appreciable compared

with th^.t existing over the londerside;

(vi) To allow flight to taJrce place at an altitude at least as high

as that compatible v/ith the existence of an attached shock wave

at the leading-edge;

and (vii) To design the aircraft v/ith a small Tiving loading.

4. Some Nimerical Vrlues

It is instructive to see how the application of the measures

just discussed lead to relatively modest values of temperature rise

even in the most extreme conditions of flight.

We see in fact from relations (4) and (5) tha.t the heat

transfer v/ill be higliest at the speed at, which

/ q2V/2

-14-is ma:cimum, i.e. at q. = 2/3 g R . Th-14-is corresponds to

6,3

Knv/sec., and might typically imply a flight Mach Number of

about 20. The relative reduction in nose tein[veratxire at other

speeds, supposing that the altitijde is adjusted to give a constant

lift coefficient, is shown in fig.4.

Y/e take the incidence as being tliat giving a maximum

deflection shock at these extreme speeds, so that from (4) and (5),

in dimensional terms,

. 1/2 , 3/2 . , ,

'^mBjr = ^•''^^h,

-V»-'^ (L^ inK^g/sq.m.) ^^^

The altitude giving this rate of heat transfer Td.ll be such that

the lift coefficient equal to 1.23 and the dynamic head is 0.271

times the v/ing loading. Thus if the v/ing loading v/ere 100 Kg/sq.m

(20.5 Ib/sq.ft), the indicated air speed v/ould be only 20 metres/sec.

(45 m.p.h.). The relotive air density would be about 10 , iniplying

a height in the region of 80 Km (5O miles). The structural advantages

resulting from a flight path restrained to employ such low indicated

air speeds, need no amplification,

Assuming the underside surface projects forward of the

upper one, and all surfaces are bla.ck (so that e = e. = 1 ) , v/e

have from (6), in (3), that at this critical speed the nose

temperature reaches its highest value of

/ ' L 2

\ ^ ^ 3

/

L ;

^ma.c=

''^H'^t]

'Ik d i

deg.C.absolute (7)

ly^ m m y

d in cms. J\n alinement chart for a rapid interpretation of this resul.t is given in fig.5. Fraii this one may quickly be

persuaded that nose temperatures in this, the most extreme condition of flight likely to be encountered, can still be of reasonable

proportions. Figures of aroxmd 1000 C are all that need be involved. Remember, too, that these figures are probably based on a pessimistic

estimate of the heat transfer in the neighbourhood of the nose, One reservation must be made at this stage, however. It will be recalled that the above result has been obtained on the assumption that ( d/i) is small. A few specimen exact calculations

(for a projecting lip bevelled at the nose to form a v/edge, as shown in fig. 1c) have indicated that v/here the value of ö./t is about unity, the actual nose temperature is higher, by nearly 15/5, than tha.t ^vhich would be predicted by (7). This may not sound too serious an error until it is realised tliat, to obtain the correct answer by equation (7), we should have to substitute in place of d .a figure for the "effective" skin thickness v/hich is only one fifth of the actual thickness,

Greater skin thicknesses than Ii appear to produce little or no further reduction in teiriperaturc, so that the effective skin thickness is never likely to exceed i/5 ^ , no matter hov/ thick the skin is made,

In view of this is is sajfer in practice to use the relation obtained from equations (2) (3) and (6),

I 0'

. 1/5 ,2

T ,, = 1.151 - » » i s a 2 L _ —

-16-1/5

deg,C.absolute

v/hich gives T in terms of an effective value of d/-6 rather than d, max

A nomogram to interpret maximum temperattires on this basis is given in fig, 6. The chart of fig,5 may then subsequently be used to

determine d, and so also t, relevant to the initial choice of ( d/-^ ) , But it v/ill be a.ppreciated that the value of d so found is an

"effective" thickness, and the actual skin thickness required will be considerably H.arger if the initial value of ( d/-^ ) has been chosen as around 0,1.

In many structural considerations, the teraperatiore gradient and the duration of tine during which severe heating occurs are of interest. From equiitions (l) and (2) and by references to fig. 2, it is possible to show that the maximum. temperati.urc gradient is

( f )

=

0.11

3 s

maorz ^

and those measures which reduce the nose temperatvire v/ill also reduce the tempcratiire gradient.So far as the dura-tion of the period of high temperature is concerned, it all depends on the flight plan. A time scale is suggested in fig.4 for a glide pa.th at constant path at ccaistant lift ooefficient. Once again -m a.dvanta.ge of using a high incidence appears, this time in that the l/D ra.tio being low, a relatively rapid decelleration is achieved.

Conclusions

(i) The actual rate of heat transfer to the sxirface at the nose of a wedge v/ing may v/ell be finite; but even if it is assumed to go to infinity (like 1 / ^ 7 ) as predicted by boundary layer theories, then the actual surface temperature is considerably below the thermometer temperature at the nose, due to conduction of heat along and within the skin. (ii) Equo-tions (3) and (4) of the main text together enable

the temperature a.t the nose of the wing to be estimated at high flight speeds

(iii) The temperat\ire at the nose is the maximum reached by the surface; higher values might be reached if turbulent flov/ existed, but this seems vinlikely due to the stabilising effect of the high rates of heat transfer on the Ifminar flov/.

(iv) An estimate is made (in Appendix l) of the rate of heat transfer to an inclined wedge aerofoil at extreme speeds, taking account of the conditions of flov/ behind the intense nose shock wave

(Appendix II). If the surface pressure and speed of flight are kept constant, increasing the v/ing incidence (i.e. increasing the altitude of flight) serves to decrease heat transfer,

(v) This is true at least up to incidences v/hich v/ould cause the shock v/ave to become detached from the nose of the v/ing. Due to the change in specific heats of air at elevated temperatures the maximum deflection through a shock wave may be greater than that usually suggested (Appendix II).

-18-(vi) Certain measures v/hich reduce the maxiraum temperature reached by the surface (at the nose) are detailed in section 3, and svimmarised at the end of this section.

(vii) The nose temperature will reach its highest value at

[^

cdxoling

a speed equal to -^ the ... , velocity (i,e, at 6,5 Kn/sec). The minimum temperature at other speeds of flight is shown in fig.4.

(viii) For flight at this speed, charts are given in figs, 5 and 6 to enable the maximum temperature to bo determined. The

temperatures involved need only be around about 1000 C if appropriate . but not unreasonable measures are taken to ensure this.

(ix) In section 4 it is noted that there is a limit to the reduction which can be hiade to the nose temperature by increasing the skin thickness.

(x) The above conclusions are based on a neglect of dissociation, and its presence v/ill affect both the shock v/ave conditions and the boundary layer heat transfer properties. In at least the latter connection there appears at the moment to be no adequate way of taking it into account.

REFEREIMCES 1. Nonweiler, T. 2. Nonweiler, T, 3. Shapiro, N.M.

4.

Young A„D. 5. Crocco, L.

Surface Conduction of the Heat transferred from a Boundary Layer

C, of A. Report No.59. (1952)

The Two-Dimensional Laoinar Boundary Layer a.t Hypersonic Speeds

C. of A. Report No.67, (l953) E f f e c t s of Pressure Gradient and Heat

Transfer on the s t a b i l i t y of t h e Compressible Lrminar Boundary Layer

J . Ae, S c i . Vol.23, No,1 p . 8 l . (l956) Skin F r i c t i o n i.n Laminar Bounda.ry Layer

i n Cc.iipressible Plov/

Aero. Quart. Vol, I, Part II, (l949) Lo Strato Limi.te Laminar e nei Gas

Monografie scientifiche di Aeronautica, No, 3. (1946)

-20-Notation used in Appendices

C Sutherland's constant (|i«: TV 7T+C) F shear stress at v/all

F ( 6 ) shear stress rela,tive to that of uninclined plate v/ith some surface pressure and moving at same speed H enthalpy, or total heat

M = q/a, local Hach number Q heat flux to wall

Q ( 5 ) heat flux relative to that of uninclined plate with same surface pressure and moving at same speed

\ = (R

) =

^ o

T

a

c =

°f

=

c =

P

c =

p

k

^h

=

m

P

q

u

V

r

Y

6

(pqx/iJ,), local Rej/nolds number

PoV/^^o temperature speed of sound C/T

^/^P" % > local skin friction coefficient dH/dT, specific heat at constant pressure (H-H^)/'(T-T^)

thermal conductivity

Q/ipcQ.!; } local heat transfer coefficient molecular weight

pressiire gas speed

component of gas velocity perpendicular to shock ' ' ' ' parallel to shock mc /m.c

P -l P^

ratio of specific heats.

max

P

cr

f

max (Ü

maxiraum value of 5 v/ith respect to f

viscosity density

kc /p, , Erandtl number

angle between shock wave and free stream direction value of f corresponding to maximum deflection power index in relation ti» [1:

Subscripts 'w' '6' •b' »1' '2»

denotes v/all conditions

denotes conditions at the outside of the boundary layer

denotes some standard atmospheric datum, say

I.C.A.N, sea-level,

denotes conditions upstream of shock-v/a.ve denotes conditions dov/nstream of shock-wave, ( N , B , In example considered of v/edge shaped aerofoil, subscripts '6' and '2' are interchangeable),

-22-/J^HÏ3\T)IX I

Conditions in Boundary La.yer of an Inclined Plane Surface producing an Intense Attached Shoclcv/ave

Supposing that the air in the boundary layer behaves as a perfect gas and has constant molecular v/eights, Prandtl nimber and specific heats, any non-dimensional expression of a siirface property (such as skin friction or heat transfer coefficient) can be expressed as a function of

R £ 5 ^

v/here '6' denotes conditions outside the boundary layer, and also in terms of the parameters

qg c gkg T / C

p

v/here T is the (constant) v/all teniperature, and w is the poT/er index in the viscosity temperatiore variation (ii«:T )» The pressure gradient is taken to be zero,

In the boundary layer condition vmder consideration the temperature outside the boundary layer vdll be a quantity such that from (2,12) of Appendix II

T T

^ = ^ = 0 ( M / ) ^1 ^1 ^

As the shockv/ave is 'intense', M ^ is indefinitely large J v/e shall suppose hov/ever that T remains bounded and of

order T, as if.^ •• » , - v/hich is plainly an assimiption of reality, because surface temperatures of the order of the large stagnation values are of little practical interest, Consequently

-23-and may,, in the presence of intense Shockwaves, be taken as zero o

n

Further r— vdll likewise be small which implies that wc may take w = ^ (i,e, jjiocT^ as is characteristic cf high temperatures). Also in general, the temperature within the boundary layer will be of the large magnitude, Tr

(except, in the limit, at infinitesimal distances from the surface). Thus the specific heats and Prandtl number, which have an asymptotic value at elevated temperatures, can Indeed be treated as constant across the boundary layer,

Hence it follows that the asisumptions implicit in the expression of the boundary layer properties at the beginning of "chis Appendix are applicable in the limiting condition iCf''-* oo, if we take

^ = 0, 0) = 1 (or Cg = 0 ) , Mg = Mg , (1.1) 6

together with the numerical values relevant to elevated temperatjLu-'es given in ref, 2 as

Yg = |- = '..286, o-g=-l| =0,782 (1.2)

Solutions ier- the 'flat plate' boundary layer, wi-ch

precisely the valies of T/Tg , w and cr given by (1,1) and (l,2)„ do not appear to have been vrorked, although closely cca-respondj-ag solutions are known. It would not be inapprcTpriate therefore to use one of the various inter-polating expressions which have been rervdsed,. Par instance, Young suggests

c^. = - - ^ = 0.664 0...5-.0.55 T 7 •^•09(Y5-1)IVI5 o^ / - - ; ^

(i.3)

-24-^IM.

IL

-%

^(1.4)

S i m i l a r l y Crocco"^ has suggested t h a t , i f v/e define

k, =

±

then 2P5% 20-;^ 1 + -T o^(Y5-l)Hg o e t « * * \ < « 5 /

which simplifies in the present instance to

Ji = 0.52 (l -H ï ^ j M* or

^

/lvI,(R^) V 1.4 // V l,e

6 Li2 ..(1.6)

It is interesting to note that for large values of Mp (i,e, smaj.l surface inclinations - a condition for which of course the simplifications made above are actually inapplicable), equations (1,4) and (1,6) yi-Qld

4^4

> \ 0j_89

v^ng(R^) 5 " ^ 5

v/hereas the v/ork of reference 2, v/hich is strictly relevant to this particular condition of the uninclined flat plate at hypersonic speeds, yields

e - 1«70 V -

M L

-Such a measure of agreement must be largely fortuitous} hov/ever it does imply that, as (lo4) end (1,6) are relevant to plates at finite inclinations^but also give nearly

angles of inclination, provided only that 11. is large, can be justified,

The values from (1,4) and (1.6) involve a kncrwledge of the local Reynolds nurabcr (R ) , based on conditions

out-X c

side the boundary layer. It is more convenient in- general

to use a Reynolds nuaaber based on free stream conditions, or

better still

p a x

•^o o

= (R )

^ X''

v/here suffix zero denotes some datum condition - say I,C.A,N, standard sea-level. New this parameter can only be intro-duced if v/e con relate the change of [X from its value (|ir) at the large temperature Tr. to its value at normal temper-atures, To do this we assiime, a.s in ref, 1, that there is no change in the molecular v/eight of the air, - i.e. no dissociation. Then we see that

^

(K)^

p a ro C

(R

)i

— X

o

1 + c

c

1 -: c,

or from (1.1) and (1.2)

P536

-12 P

yi = °-^^« I t ) L'^' °°^ -y ^ >

(1.7)

Proa (io7) in (lo4) and (I.6)

P = 1,67 -- 1 + l i ^ V ' ^ ' Po%1l(^-^°o>

14

_(K)''

_o

'h

_(1.8)

-'A

-"•KD^t") ^^

P a- q./! (1 +c )

(R )"^

^ x'

o

©!

— £ . 0

-Relative to vn equal distance beldnd the loa.ding edge of an uninclined plc.te, moving at the saane speed, but having the same surface pressvire

F

P flat plate

A

£_> =(?) ( l . ^ ) ( w ^ ) =Q(6),say^

lat plate \ "^1 / \ J.'L / \ ïC / ''flat plate ^2

The variation of — and M- is deduced in the Appendix II, and the functions F(S) and Q(6) are plotted in fig, 6,

A.'HJ]NDIX II

Conditions behind an Intense Inclined Shocl-cwave in Air Let us consider the conditions behind an inclined shoclcv/ave, talcing into account the variation of gas specific heats, an.d molecular v/eiglib. Denoting conditions upstream -dovmstrean of the shock by suffices '1' and '2' respectively, and using otrier symbols as explained in the list of syLibols a.t the end of tlils Appendix and in figure 7j 'VJe have from continuity,

P2 ^1

, J » 3 0 » 3 0 » # » » \ ^ « 1 / 0 O I

C o n s e r v a t i o n of nor:ient\:im a c r o s s the shock r e q u i r e s t h a t

^2 . L ^2V^1

2

„ = ^ + rA^ - ~ J

-^ . . . - (2.2)

•Pi I \ a^ / ^2

and normal t o i t , tha^t r^ = V2

Conservation of energy nay be expressed a^s

H^ - H^ = i (u^^ - U2) (2,4)

The p e r f e c t ga.s lav/ s t a t e s thsit

( 2 . 5 )

VJ, = V o . a . . . . (2.3)

P . P2 ^2 ^1 P2 Pi ^^1 ^2

and i s t a k e n a s b e i n g obeyed b y t h e edx\. I f v/e d e f i n e H-H, m c

- _ 1 r _ R

p - T-T. ' m.c ^ 1 1 p^ t h e n ( 2 , 4 ) c a n b e v a d t t e n , from (2,5)> ^s

!= £1. :!i , iijil 4 ( i 4 ) (2.6)

2 1 1

-28-whereas from (2,1) and (2.2)

^2 P1 ' ^1 A ^2\i^2 ,, ,v

Equation (2,6) and (2,7) together yield a quadratic for — v/hich can be solved to give

^1

^ 2 iA-1 , . ; i 1^1 u^^ r 2 y u^« v^^2 V v ^ 2 r ;

{. li-I-l^ r2 8)

2

I f (J- i s th3 i n c l i n a t i o n of the shockv/ave v/e have

u^ = q^ sin f , v^ = q^ cos f

where q i s the spsedj sim.ilarly i f 6 i s the r n g l u l a r

d e f l e c t i o n of the stream tliro-agh tho shock,

^2

tan (f"b) = —=•

^ ' Vg

or using (2o3)

t, an 5 = (l - --j tan ifr / (l + ^ ten'' i^ J (2.9)

TiTe note t h a t

2

^1 - 2

—^- = l i ; Kin V

a 1

ond i n the l i m i t as t h i s becomes large ( i „ e . M. "* «> , ^ / O),

the shoclcv/ave becomes intense and

'i (2r^«i).v,+i

2

From tills equation and (2,3) i t follows t h a t

[TT ) ~ \ TT I ^^^- ^ + °°2 f = { • ~ smi^ +cos

V1/ V^'l/ Li (2r -1-

i ( 2 r 2 - i ) Y ^ + i

(2.11)

1^ -'

Again, from (2.6) and (2,10)

Il £2 Pi_

2Y^(Y^-1)

Ij^s'^^^l-^ll

(2r2-i)Y^+iJ

, 8

. ! ^'i Pi P2

1.1^ s m f 1+0 .(2.12)

so t h a t

2Y2(YI-1)

li^z'^h^^W

VM;

*'

^ " r(2r2-i)Y.+iT ^

I t a3.so follov/s from (2.10) t h a t

(YI"')^+ L ( 2 V ) Y - , + I J , oof-i^

Hn

2 Y 2 ( Y I - 1 ) [ ( I V : ~ O Y ^ + 1 J

1+0

^ I ^ ,^^

,(2.14)

Also from (2.10) and (2.2)

P2 = 2

(ypY^f-i

(212-1 )Y.,+1

Pl^

ain*"!^

1+0

i— «1-2 .,2 .«...8a.*..«V'^*'5/

The relation (2,9) becomes

tan 6 -

2 | ( r 2 - O Y i + 1 j t o ^ l ^

r

1+0 1

'^1 '^

(2r2-l)Y^+1 + (Y^-l)tar=,^

The maximum d e f l e c t i o n tlorough the shockv/ave i s evidently

/ ( Y i - l ) [ ( 2 V ) f i + l l L. Vilt li

. , 0 . 0 . . 0 . » . 3 V ^ . l / /

0 = tan

max

v/hich i s a.chieved v/hen

tan f = tan^

_{If = f - f — 1+0( — )}

(2VOY1+1 r /-, N'

( V ^ L Vii?Z

-30-For the a i r upstream of the sliockv/ave we nay take

Y^ = 7/5

Noting from (2.12) that the temperature behind the shoclcwave is high if L. ^ is large, it is often appropriate to use the values

Yo = r_ = 9/7

which are appropriate to air at elevated tenperatiures. If v/e define an 'intense shockv/ave' as being tine limiting forrn given by taking Ï.C V" •*«>, and adopting the above numerical values, \V5 find tliat

and J . 5 | = = coJ'^f 1 + 6h. 9 1 ^ 64 '^ i 2 > COX f a±r?f P2 = 0 Pl*^1 ^^'^ tan 6 6 max = = — 7 tar,-^ 8+tan^ fr

tan-^

I

tan"'' -^d / 2 = . 51 approx, ) \ /(2.19) max

J

We note thab if 8 = 6 , Ï.L = 1, so that in general max ' 2 '

STREA FREE STREAM >

<a>

b '

f c J 1-2 * I _IO 0 8 0-6 0.4 0-2 O 1 2 3 4 5 r-|_ ^ VARIATION OF (NON-DIMENSIONAL) TEMPERATURE F I G . 2 . WITH DISTANCE ALONG SURFACE.

I O

Qf8l 0 8

0-6

0 4

0 2 CONSTANT SPECIFIC HEAT

ISOTHERMAL SHOCKWAVE

IO* 2 0 ' 3 0 ' 4 0 * SO* 8 ANGLE OF SURFACE

' INCLINATION

6 0 SPEED K M ,

NOSE TEMPERATURE AT VARIOUS SPEEDS

FIG. 3. ^^rJl^Y^IN^D^^^^a'^E^^WFE'^E^D^^ ^^^ ^ '^^^^^^^^ ^° T M... ^^v«d meSKnj^)

s

k ° ó ? CALf O

CM.SEC.4««.C

IN THE REGION SO MARKED, THE SKIN IS NOT'THERMALLY THIN"

>« QV^ IS NOT SMALL. SEE FIG. 6 .

T MAX dc9 Cobs 2 0 0 0 T ^ ^ ^ - -1900 I 8 0 0 -•I700 I 6 0 0 -• -• I 5 0 0 WOO-• WOO-• I 3 0 0 I 2 0 0 -•MOO lOOO-• lOOO-• 9 0 0 8 0 0 ' "TOO '/CM.SE&4C9.C

FIG. .5. ALIGNMENT TO DETERMMi TMAX

FROM K. L.AND S TMAX dcg C abs 2 0 0 0 i I600 • • * ^ ' * ° ° : 1300 I 2 0 0 -•IIOO l O O O ' . 9 0 0 8 0 0 - > J-700 9 o

FIG. 6 . ALIGNMENT TO DETERMINE ^ / t )

STREAM SHOCK WAVE FIG. 7 "2" EMERGENT STREAM

NOTATION USED IN APPENDIX H

I o

CO Q(^)

0 - 8 0 6 0 - 4 0 - 2 ' ^ \ \ F

- \ °(0

^

(S)X

\ ^ ^ \ I O " 2 0 ° 3 0 ° 4 0 ° ^ DEFLECTION ANGLE 5 0 " FIG. a

RELATIVE REDUCTION IN SKIN FRICTION AND HEAT TRANSFER ON INCLINED WEDGES RELATIVE TO THOSE

ON UNINCLINED PLATE MOVING AT SAME SPEED AND WITH SAME SURFACE PRESSURE.