Supersonic flow over flaps with uniform heat transfer

H

lA) o

von

KAR:M:AN

INSTITUTE

-

8

cC. 1 66

FOR FLUID DYNAMICS

TECHNICAL NOTE 30

SUPERSONIC FLOW OVER FLAPS WITH

UNIFORM HEAT TRANSFER

by

JEAN Jo GINOUX

,

RHODE-SAINT-GENESE, BELGIUM

"TH

I

teru\:.

. ltermijn: i.

r

TECHNICAL NOTE 30

SUPERSONIC FLOW OVER FLAPS WITH

UNIFORM HEAT TRANSFER

by

JEAN J. GINOUX

Page

Surnrnary . . . . . . . . . . . . . . . . . . . . . i

List of figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

Syrnbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Equiprnent and modeIs. 2 Wind tunnel. . . . . . . . . 2

Pressure rneasurernents. 2 Therrnal rneasurernents. . 2

Sublirnation technique. 3 ModeIs. . . . . . . . . . . 4

Results and discussion. 6 Pre s sure rneasurernents. . . . . . . . . . . . . . . . 7

Recovery ternperatures (natural transition) . . 8

Recovery ternperatures (artificial transition.). 11 Heat transfer rneasurern ents. 12 Conclusions .. 14

SUMMARY

Statie pres sure, recovery temperature and heat transfer distributions have been measured on flat plates with flaps at a Mach number of 2.02. A uniform amount of heat was dissipated at the surface of the modeis. The angle of the flaps was varied between L.

'5

0

::a:É.d 21 0

• Laminar, transitional, and turbulent flows were considered.

Well defined plateau pressures existed when reattachment was transiti:ónal . and not in fully laminar flows. A large peak of recovery temperature existed, when transitiört was in the re-attachment zone, followed by fairly low recovery temperatures. Heat transfer results generally agreed with other measurements made with

isothermal walls. No peak of heat transfer was detected in the reattache-ment region .

LIST OF FIGURES

Figure 1.

a, b. Shadowgraphs of the flow around a 5-deg. flap

6

-1

6

-1

a: R =1.5x10:m b: R =2.2x10 :m

e e

c, d. Shadowgraphs of the flow around a 7 -deg. flap

6

-1

6

-1

c: R=1.5x10:m d: R=3.0x10:m

e e

e, f. Shadowgraphs of the flow around a 10 deg. flap

6

-1

6

-1

e: R =1. 5x10:m f: R =3.0x10 :m

e e

g, h, i. Shadowgraphs of the flow arou11d a 15 -deg. flap

6

-1

6

-1

g: R =lx10 :m h: R =1. 5x10 :m

e

6

-1 e

i : R =3. Ox1 0 :m e

j, k. Sehlieren photographs of the flow around a 21-deg. flap

6

-1 6 -1

j: R =1. 5x10:m k: R =3.0x10 :m

e e

1, m, 11. Sehlieren photographs of turbulent flows around flaF-s at

6

-1

R =3.0x10 :m

e

n:6=21°

Figure 2

a. Heat transfer distribution over a 1. 5-deg. ra:mp ( :model CCA4) b. Pressure and heat transfer distributions on a 3-deg. ra:mp

(:models CC4 and CCA3)

c. Pressure and heat transfer distributions on a 5-deg. ra:mp (m.odels CC5 and CCA5)

d. Statie pressure distribution on a 7-deg. ra:mp (:model CC7) e. Pressure and heat transfer distributions on a lO-deg. ra:mp

(m.odels

cc6

and CCAl)

f. Pressure and heat transfer distributions on a lS-deg ra:mp ( m.odels CC9 and CCA6)- Natural transition

g. Pressure and heat transfer distributions on a l5-deg. ra:mp (m.odels CC9 and CCA6 ) - Turbulent flow

Figure 3 - Surnrnary of the static pres sure distributions for various flap angles 6 -1 a- R

=

1. 5 10 rn e

6 - 1

b- R =3· 0 10 rn

Figure 4 - Ternperafure distributions over the flap models a-

a

= v!. 5 ° b-

e

= 3° c-

e

= 5° d-

a

= 10° e-

a

15,0 f-

e

-

2:1°

Figure 5- Surnrnary of the recovery ternperatures for various flap angles 6 -1

a- R =1.5xl0 rn

e 6

b- R =3.0xl0 rn-1 e

Figure

6 -

Effect of unit Reynolds number on recovery ternperature distribution for a flap angle of 10 degrees.

Figure 7- Sublimation pictures of the flow around a 10-deg. flap

6

-1 a- R =1. 5xl0 rn

e 6 -1

b- R =3 .. Oxl () rn e

Figure 8 - Effect of transition location on the recovery ternperature distribution for a flap angle of 15 degrees.

Figure 9 - ~ecovery ternperature distributions for fully turbulent flows at various flap angle s.

Figure 10 - Surnmary of the heat transfer results for larninar and transitio-nal flow5!.

6

-1 a- R=1.5xl0 rn e

6 - 1

b- R =3.0xl0 rn e

Figure 11 - Surnrnary of the heat transfer re sults for turbulent flows over flap model s.

SYMBOLS

c Specific heat at constant pressure

p

h Heat transfer coefficient; h=q/(T -T )

w r

k Heat conductivity coefficient

e

L Length of flap

M Mach number upstream of interaction region (M =2.02)

e e

M

f Mach number at infinity downstream on the flap

N Nusselt number = hk /x

u e

p L o c a l statie pressure

Statie pre s sure upstream of interaction region

_•

Po Stagnation pressure

q heat flux per unit area and unit time, dissipated uniformly at the

2

surface of the models (cal/m sec)

R

e Unit Reynolds number in the flow parallel to the flat plate upstream

_U

p

R

=

e e (m-l )

e lLe

of interaction region ;

Reynolds number based on local distance x; R =xR

ex e

R ex

R Reynolds number based on distance x ; R

ex , 0 ex o 0 S Model span T Stagnation temperature o T Recovery temperature r

T Wall temperature with uniform heat flux

w

=x R o e

x Streamwise coordinate measured from the leading edge along the

surfacesof the plate and of the flap.

x Distance between the leading edge of the model and the point whe:re

o

the interaction is first felt x

t Distance between the leading edge of the model and transition

II e Viscosity coefficient upstream of interaction

p Density upstream of interaction

e

In the course of previous investigations on separated flows

made by the author (1,2,3), it was shown that regular and strong streamwise

vortices developed systematically in laminar reattaching boundary layers at

supersonic and hypersonic speeds. Although the main body of the research

was made with backward facing steps, the phenomenon was also observed in

other types of reattaching flows and, in particular, in separated flows over ramps or control surfaces. Such vortices were shown to cause spanwise variations of the heat transfer rate at and downstream of reattachement with local peaks higher than the known turbulent flow values. However the ~an heat transfer rate, i. e. the average over a portion of the span large

compa-red to the spacing of the vortices, was not appreciably affected by the vortices.

Wh en this phenomenon was qualitatively observed in the reattaching flow over a 10 -deg. flap, with a sublimation technique, regular

striations patterns appeared as usually observed by the author in other types

of separated flows. However the average rate of sublimation was surprisingly

high at reattachment. Although this could have been explained by the existence

of a peak

ei.

heat transfer in that region, as observed by other investigators

in reattaching flows, it was in order to make a direct verification of this facto Preliminary tests, made on such a flap model, showed that there was

no such peak (4) but rather that a maximum of the recovery temperature

existed in the reattachment region which had not been reported before. It was then decided to study this phenomenon in a more systematic manner.

This report is related to the results of such an investigation.

The research reported in this document has been sponsored

by the Air Force Office of Scientific Research, O. A. R., through the European office, Aerospace Research, United States Air Force, under Grants AF. EOAR

EQUIPMENT AND MODELS

Wind Tunnel

The tests were made in the VKI 16" x 16" continuous supersonic wind tunnel at a Mach number of 2.21, over a range of stagnation pres sures from 100 mm Hg (i. e . I. 8 psi) to 200mm Hg (3.6 psi) absolute which

corres-6

6

ponds to free stream Reynolds numbers of 1.5 10 and 3xl0 per metre re specti vely.

The stagnation temperature is of about 15°C to 30°C (59°F to 86°F) depending upon the stagnation pressure. At a given pressure level, it increa-sed gradually during a test at a rate of 1 to 2 ° C per hour.

The tunnel is equipped with shadow and single pass schlieren systems with parabolic mirrors. Pictures are taken with a spark light source of a few micro-seconds duration time.

Pressure Measurements

The static pressure distributions were measured with 0.5 and 1.0 psi strain gage transducers and scanning valves located inside the tunnel circuit. The output of the transducers was recorded by a four digit digital voltmeter. Calibrations were made before and af ter each test to within

±

0. 2 mm Hg.

Thermal Measurements

The recovery temperatures and heat transfer rates were measured with a steady state technique developped by the author (5).

Models are made of an insulating material (araldite) and covered with thin film of silver of nearly constant thickness (one micron thick plus or minus ten percent). Heat can thus be uniformly dis sipated at the surface of the models by the Joule effect in the silver films and the heat flux per unit area (9) can be computed from the ratio of the product,

current times voltage, to the area covered by the silver film. Heat losses

to the interior of the model are minimized by a symmetric design of the model. Thermocouples, installed flush with the surface along the center-line of the model, are used to measure the recovery temperature T

r

(i.

e. when q=O) and the wall temperature T with uniform heat dissipation

w

(qlO ) . The heat transfer coefficient (h), the Stanton number (St) and the Nus selt number (N )

u q

h=""T -T

w

r

are then calculated from: hk N u e

=

---'---x h

where subscript (e) is related to the flow parallel to the model surface, upstream of the interaction region.

Thermal equilibrium is generally achieved af ter one to one and a half hour of running time.

Thermocouples were made of copper-constantan wires O. I mm In diameter and calibrated before and af ter the tests to within

+

0.20

C. The total amount of heat dissipated at both the upper and lower surfaces of the models were generally small , about 50 watts, thus giving values of q of the order of 0.1 cal/m2. sec. The driving temperatures T -T were accordingly small , with maximum values of about 30 to 500

C

w

r

and minimum values (downstream of reattachment at large flap angles) of SoC. However, the accuracy of the measurements is believed to be good, because error s are eliminated by taking the difference between T a n d

w

T measured by the same thermocouples during the same test. With small

r

values of q , quasi-adiabatic conditions were reached which means that the flow was not affected by the transfer of heat. This was verified by measuring the pressure distribution on model CCA2 with and without heat dissipation at the model surface. No difference was found.

Sublimation T echnigue

The flow near the surface of one of the models was visualized by using a sublimation technique with acenaphthene as an indicator. The model was fir st sprayed with tracing blue ink , to increase the contrast, and then

with the indicator whose thickness was such that a running time of one to two hours was needed to obtain the surface pattern. On the photographs (fig. 7) included in this report, darker regions are representative of larger sublimation rates than brighter ones.

Models

Two types of models were used ; metallic ones (CC -series) for

pressure measurements and those made of araldite (CCA- series) for

thermal measurements. A different model was made for each flap angle.

The designation of the models and their main dimensions are

given in Table 1, where eis the flap angle in degrees, S the model span

and L the length of the flap in millimetres. On all the modeIs, the flat

plate upstream of the flap was 120 mm long and was inclined by 5 degrees

in the test section. The effecti ve Mach number (M ) was thus 2.02. This

e

inclination was necessary to ensure complete symmetrie flow condition,

when both the upper and lower surfaces of the araldite models were heated up at the same rate.

TABLE 1 Model e (degrees) S (mm) L(mm) CCAI 10 250 120 CCA2* 10 250 120 CCA3 3 250 120 CCA4 1.5 250 120 (+50)

**

CCA5 5 250 120 (+50)** CCA6 15 386 100 CCA7 21 386 70 (+25)

**

CC4 3 250 120 CC5 5 250 120

CC7 CC6 CC9

*

7.

10 15

with static pressure orifices 250 250 386 120 120 125

**

additional af ter body with surface parallel to tunnel free stream Araldite models were cast af ter in stalling the thermocouples and the electric wires in the mould. They were made of araldite type D except model CCA 7. Araldite D is simpler to use but the maximum wall tempelëa:~ ture is limited to 40 to 60°C (100 to 150°F). For large flap angles the

maximum temperature is reached , with uniform heat flux, in the separated flow region where St is smaller, while T

w remains rather low in the reattachment region of the flow. The driving temperature (T -T ) can

w r

thus be rather small with a corresponding increase in the inaccuracy of the heat transfer measureroents. For thi s reason, one of the models

(CCA7) was made of araldite F which can be used at higher temperatures before warping.

A special araldite model (CCA2) was made with a 10-deg. flap which contained a limited number of both pressure taps and thermocouples,

to verify the effect of heat transfer on the pressure distribution as men-tioned in describing the technique of heat transfer measur:ement.

To check the effect of heat conduction inside the model, model CCA6 was remade with a hollow flap such that only a thin skin of araldite , 0.3 mm thick and 25 mm wide, existed all along the centerline of the flap. Thermocouples were embedded in th at thin skin. No heat conduction effect

was found.

The araldite models were machined af ter casting, but their

leading edges were not as nice as on the metallic ones. To check the effect of leading edge irregularities, a comparison was made of the pressure distributions on models CCA2 and CC6. In another test, artificial

and heat transfer distributions were measured . There was no indication within the accuracy of measurements of a leading edge effect.

RESULTS AND DISCUSSION

The re sult of stat ic pres sure and heat transfer me asurements are presented and discussed in the following sections. The flap angle was vari'ed from 1. 5 to 21 degrees and two stagnation pressure levels were used, i. e.

100 and 200 millimetres of mercury corresponding to unit Reynolds number

6

6

of 1.5 10 and 3 10 per metre in the flow parallel to the flat plate.

Flow Visualization

Shadowgraphs or schlieren photographs of laminar. transitional and turbulent flows over flaps of various angle s are shown in figures 1 - ( a to n). Turbulent flow is obtained artificially by tripping the boun-dary layer upstream of the flap with a paper strip, 10 mm wide and 1.4 mm thick, covering the whole span of models and located 20 mm downstream of the nose of the modeis.

Pictures of the flow over 1.5 and 3 deg. ramps are not shown because they are similar to those given in figures la and 1 b for the 5 -deg. flap.

For a 5-deg. ramp, the flow is laminar over the whole model. The flow is on the verge of separation as shown by the pressure distribution.

For e=7degrees, the flow is separated and transition appears, at the highest free stream Reynolds number , far downstream of reattachment.

For e = 10 and 15 degrees, transition has moved closer to or in the reattachment zone and finally at e = 21 degrees, transition is located upstream of reattachment.

Table II gives the distance x

t of transition from the model leading edge as measured on photographs of figure 1, for e = 10, 15 and 21 degrees, for the two stagnation pressure levels (100 and 200 millimetres of mercury. ).

TABLE II

e

degrees p mmHg x t mm 0 10 100 170-200 200 140-150 15 100 150 200 135 21 100 105-110 200 95-100

With the roughne s s element, the flow remained attached on the 10

and 15 deg. flaps. For

e

=21 degrees, oil flow visualization showed a region

of reversed flow starting approximatively 10 mm upstream of the nose of the ramp.

Pressure Measurements

The ratio pi p is plotted versus x in figures 2b to 2g for two

e

stagnation pressure levels (100 and 200 mm Hg) corresponding to unit

6

Reynolds number s , in the flow parallel to the flat plate, of 105 10 and

3.0 10

6

respectively. p is the local static pressure, p the static pres"6ure

e

upstream of the interaction region and x the distance , in m illimetres, from

the leading edge of the model, measured along the flat plate and the flap

surfaces. These results are summarized by the curves of figures 3a and 3b.

Flow separation occurs at a flap angle of about 5 degrees as

indica-ted by the presence of a kink in the pressure distribution for

e

=5 degrees.

A plateau of nearly constant pressure develops for a flap angle larger than

7 degrees. From the flow pictures previously discussed, transition was

observed in the reattachrnent region in these conditions. Therefore, there

laminar separated flow with a well defined plateau pressure as is generally assumed in the literature (6,7,8) . The sensitivity of the present flows to the transition location will appear more fully when discussing the results of the thermal measurements.

The plateau pressures for the'transitional flows" are compared with Chapman's formulae (9) established for laminar flows:

=

where R is the Reynolds· number based on conditions upstream of the ex

interaction ~one and x the distance between the leading edge of the model

·0

and the point where the interaction is first feIt. A good agreement is obtained as indicated in figures 3a and 3b, showing that the presence of transition in the reattachment zone does not influence, as expected, the free interaction region located near separation.

The pressure distribution region for turbulent flow is shown in

figure 2g for a flap angle of 10 degrees and a stagnation pressure of 200 mmHg. The upstream influence of the flap is very small and there is no indication of flow separation.

The total pressure rise. is in good agreement with the inviscid theory with a tendency for overshooting in all three types of flows.

Recovery Temperatures (Natural Transition)

The distributions of recovery temperature (adiabatic wall) measured along the centerline of the models are plotted in figures 4a to 4f in degrees centigrade, for various flap angles and for the two free stream unit Reynolds

6

6

numbers 1.5xlO and 3.0xlO per metre. Temperatures are shown rather than recovery factors, because of the uneertainty in thè ealeulation of the

tempera--tures have been multiplied by the ratio of a reference stagnation

ternpe-rature of 3000

K

(i.

e. 27 ° C) over the actual stagnation ternperature rnea-sured in each test. This reference level is indicated in the figures as weU

as the corresponding theoretical values of the recovery ternperatures on a flat plate based on the initial Mach nurnber (M ) of 2.02 and also on the

e

final Mach nurnber (M

f), function of the flap angle. The larninar values are based on a recovery factor of 0.85 while the turbulent ones are based.

on 0.90.

These results are surnrnarized in figures 5a and 5b for flap angles between 5 and 21 degrees, by srnooth curves drawn through the experirnental points.

For

e

~ 5

0,

the recovery ternperature increases gradually through the interaction region, frorn the flat plate laminar flow value at the initial

Mach number to the corresponding value at the final Mach number. Thi s is in agreement with the flow pictures which show fully laminar flows.

For a flap angle of 10 degrees, a very interesting result is

obtained. At . the lowe~t u·n i t R nurnber , for which transition is located

e

downstream of the reattachment region, the recovery temperature

increa-ses smoothly , as expected, from the larninar flat plate value to the turbulent

one, possibly followed by a slight decrease. However at the highest

Reynolds nurnber, which corresponds to a transitional reattachment, the recovery ternperatureïncreases more rapidly , reaches a peak at

x

=

140 mm and then decreases rapidly down to a value which is roughly

the turbulent value on a flat plate at the initial Mach number rather than

the final one.

Figure 6 shows the gradual rnodification of the recovery ternpera -ture distribution with increasing Reynolds number. The temperature

peak increases and moves upstrearn, while the downstream temperature

decreases as the Reynolds number increases. At the same time, the

reattachment point as well as transition moves upstream. It seems from the flow pictures that transition was approximatively located at the

temperature peak. It iS. also observed that the temperature distribution did not vary upstream"'Ûf the nose of the flap as Rvaried.

e

For a flap angle of 15 degrees, a peak existed in the recovery tem-perature distribution over the whole range of Reynolds number. Figure 5b shows that this peak is higher than on the 1.0 rdeg.::'.pap while the~ecovery tem-perature drops, downstream of reattachment, to a lower value (even lower than the laminar flat plate value based on the final Mach number). Compa-rison with the flow pictures seems to indicate that transition is located slightly downstream of the temperature-peak at both unit Reynolds number s.

For a flap angle of 21 degrees, the temperature peak is still present although flatter and located this time, like transition, upstream of the nose of the ramp and also, of course, of reattachment. The final temperature downstream of reattachment is again fairly low compared to the theoritical turbulent flow value at the final Mach number of 1. 17. It is also observed from the figures that a second , although weaker, peak existed in the reatta-chment region at the 10west Reynolds number.

Figures 7a and 7b give the sublimation pictures of a 10-deg. flap model at unit Reynolds number of 1. 5xl 0 6 and 3. Oxl 06. At the lowest R ,

e a striation pattern exists which is characteri:stic of laminar reattaching

flows and their associated streamwise vortices (1). At the .:mghest R ,

e

striations are still present but there is a narrow region that runs along the span which is darker ,i. e. where the sublimation rate is higher than

el:s:ewhere. This phenomenon which was observed on araldite models in an

earlier part of the investigation can now be explained by:the existence of a

peak in the recovery temperature distribution. Indeed, such sublimation

patterns being obtained af ter a long running time of the tunnel, the model has ample time to reach thermal equilibrium. Furthermore with araldite models the internal heat conduction is not sufficient to smooth out a

tempe-rature peak. In the "peak region" , the concentration of chemical increases near the surface and therefore the sublimation rate is accordingly higher.

followed by fairly low recovery temperatures in transitional reattaching flows on flaps has not been reported yet, as far as the autoor is aware. This is probably due to the fact that recovery temperatures are not very of ten measured directly because their rneasureme·ht requi-re.s

facilities with long running times. It is of ten believed that the recovery factor is re1atively constant throughout the separated region and approxi-mati vely equal to the attached flow value (10, 11, 12). Indirect measurements obtained by extrapo1ating experimental data to zero heat flux have been

made for instance by Larson (13) and Naysmith (14) who did not observe temperature peaks. Seban, Emeny, and Levy (1S) who have used a tech-nique similar to the present one , to investigate subsonic turbulent reat-taching flows behind a backward facing step, have observed systeinatically that the recovery factors were substantially below the typical value for turbulent flows of air ( O. 80 instead of 0.89) which indicated an unba1anced energy distribution in tl).e flow. The authors do not mention the existence :

of a peak in the recovery temperature distribution although there seems to be a very slight maximum in the figures of their report. In transition

studies made with attached flows on hollow cylinders, Brinich (16) reports a peak in recovery factor followed by a drop to the turbulent va1ue ,attribu-ted to a violent mixing in the transition region. Brinich also reports (17) peaks and _.troughs in the recovery factor distributions in turbulent flows near two-dimensiona1 roughness but not in laminar flows.

Recovery Temperatures (Artificial Transition )

Because of the very limited range of Reynolds number available in the present investigation, it was necessary as a view to investigating further types of flows to trigger artificially the transition. This was done by tripping the boundary layer upstream of separation. By selecting a proper size of the roughness and by varying the Re.nuinber in_the .avä.ilable

range, it was possib1e to move transition on the lS-deg. flap model from

6

its natura1 1ocation of x

giving various types of transitional reattaching flows, as weIl as a fully turbulent unseparated flow.

The recovery te:mperature distributions are shown in figure 8 by s:mooth curves passing through the experi:mental points. x

t indicates the

location of transition. One can see that the te:mperature peak :moves

up-strea:m with transition until the flow becorries attached (i. e. for a unit

6

Reynolds nu:mber larger than 2. 1 x 10 ): Then the te:mperature peaks

re:main fixed at the nos.e of the ra:mp. Foll-owing the peak, the te:mperature

drops down to a value which is approxi:matively; the theoritical turbulent

flat plate value based on the initial Mach nu:mber, i. e. so:mewhat lower

than the one based on the final Mach nu:mber which one could expect. Further

downstrea:m, the te:mperature rises very slowly and it see:ms that '.a long

downstrea:m distance is needed for the boundary layer to readjust itself to

a fully turbulent f!-ow.

The te:mperature peak observed in fully turbulent flow is obviously associated with the rapid pressure rise at the nose of the ra:mp (see fig. 2g) and no longer with the transition process as it was probably the case in

transitional separated flows. This is further de:monstrated by the recovery

teFlperature di stributions :measured in turbulent flows over the various flap

:models and shown in fig.

9

.

A te:mperature peak. exists in each case at the

nose of the flap, which increases with fl:ap angle. For

e

=

210

the flow se-parated slightly upstrea:m of the flap and the te:mperature peak is flatter .

All these results indicate that, when transition is located either

downstrea:m or upstrea:m of the reattach:ment region, the te:mperature

peak is flatter than when transition is located in th at region. In the latter

case, the te:mperature peak is possibly due to a co:mbined effect of

transi-tion and of a steep pressure rise.

Heat Transfer Measurè:ments

in figures 4a to 4f, where the wall te:mperature T In degrees centigrade is w

plotted versus the strea:mwise coordinate x in :milli:metres, for various flap angles, for a stagnation te:mperature of 270

C (3000

K) and for unit Reynolds

6

-1

nu:mbers of 1. 5 and 3. OxlO :m . These were obtained by dissipating uniforrn-ly a certain a:mount of heat over the whole surface of the :modeIs. This a:mount was not nece s sarily the sa:me for each flap angle and it was taken into account in reducing the data in the for:m of N

I

./

R ,where the Nusselt nu:mber and

u

V

ex

Reynolds nu:mber are based on conditions upstrea:m of the interaction region,

i. e. : N u hk e x R e

u

p x e e q h T -T

w

r

The quantity N _u

I

\

_V

~

_~~ex

,

which is theoretically constant and equal to 0.37 for a flat plate with unifor:m heat flux (5);is plotted versus x in figures 2a to 2h. The heat transfer coeffi'Cient h, is inversely proportional to the driving tern-perature (T -T ) which is the difference between the curves given in fig. 4.

w wa

A su:m:mary of the heat transfer results is given in figures 10a and lOb, where s:mooth curves drawn through the experi:mental points are shown.

Figures 2 show that the value of Nul \

_V

r;;:-

_{.n. ex} is constant upstrea:m of the interaction region but lower than theoretically predicted by about 10%. Si:milar results were previously observed in flat plate studies :made by the author (5).

The effect of the ra:mp is to , first, decrease

N

_u

I

·

_V

r;;;:-.

_...

_ex

.

below the flat plate value for all flap angles ( even for 8=1.50

and 3.00

for which the flow was attached to the :model surface). N _u

I

\

_{V ...}

rp:-

_ex passes through a :mini:mu:m and then increases further downstrea:m. This :mini:mu:m decreases with increasing flap angle until 8 =150

and then rises again. The lowest value of N _u

I

,JR

_{V ...}

_ex which was :measured was of 0.15,i. e. approxi:matively one third of the plate value.

For unseparated flows, N / .

ç-

increases gradually over the surface u

V

n.ex

of the flap, at a rate which is independent from the unit Reynold number. For separated flows, the N / \

IR

begins to increalse ahead of the flap,

u

V

ex

i. e. in the separated flow region where the pressure is constant. The rate of increase over the flap depends upon

e

and also strongly on the Reynolds number. A maximum value of N /. _{u VKex}

r;:-

of 1.8 was measured which is about 5 times the flat plate value. For comparison the heat transfer distri-bution was measured in the case of fully turbulent flow. The results are shown in fig. 11. There is a steep increase of N / \ _{u V~~ex}

fR

near the nose of the

flap, followed by a rather constant value. These constant values are also shown in fig. lOb for compari son with the laminar and transitional iie.slalts. Similar values were obtained in turbulent flow by Sayano, Bausch and Donnelly at Douglas Cy (10).

As already mentioned, for

e

< 50

the effect of R on the curve e

N

/.,---p:-

ver sus x is small. For

e

=

100

, the Reynolds number has a

u

V

.L'-ex

large effect on the heat transfer rise over the flap and no influence on the heat transfer ne ar separation and in the separated region of the flow. For

e

> 100

, the effect of R is felt over the entire interaction region. The

e

influence of the location of transition is obviously important.

No rise in heat transfer was observed at separation like Miller et al found (19), which was attributed to a separation vortex. No peak in heat transfer was measured in the reattachment region as is typically observed in cavity flows; see for instance Nicoll's results at hypersonic speed (20).

CONCLUSIONS

Laminar and transitional reattaching flows were obtained in the range of unit Reynolds number of 1.5 to 3. Oxl0

6

per metre and flap angles up to 21 0 at a Mach number of 2. 02. It was néces sary to trip the boundary layer to obtain fully turbulent flows.

the flow for flap ang1es equa1 or larger than 7 degrees. In these cases, tran-sition was in the reattachment zone. Plateau pressures were in agreement with Chapman's formulae for laminar flows. The pressure distribution is rather sensitive to the location of transition. For turbulent flows, flap angles of about 21 0

are needed to separate the boundary layer.

A marked peak of recovery temperature exists near reattachment for transitional flows, followed by substantially low temperatures downstream

of reattachment. This peak increases, while the downstream temperature

decreases when the unit Reynolds or the flap angle are increased. Transition seems to be located in the temperature-peak region. However, the existence of the peak can also be caused by the large pressure rise at reattachment as shown by the unseparated turbulent flow results. Long downstream distances are necessary for the boundary layer to readjust itself to a fully turbulent flow.

Heat transfer distributions are even more sensitive to the location of transition than the pre s sure distributions. The effect of the flap is gener'-·

ally, to first lower the heat transfer to a minimum, which can reach in certain cases one third of the flat plate value, and then to increase it at a rate which increases with Reynolds number and flap angle. Heat transfer results are generally in agreement with measurements made by other inves-tigatbrs with isothermal models.

1. GINOUX, J. J. - The existence of three -dimensional perturbations in the reattachment of a two-dimensional super sonic

boundary layer af ter separation. Agard Report 272, 1960 2. GINOUX, J. J. - Leading edge effect on separated supersonic flows.

Proceedings of ICAS lIl. Stockholm 1962

3. GINOUX, J. J. - Streamwise vortice s in Laminar Reattaching Flows. Recent developm.ents in Boundary Layer Research. AGARDograph 97, Part 1. Naples, May 1965

4. GINOUX, J. J.- Laminar Separation in Supersonic and Hypersonic Flows.

Grant AF EOAR 65 -11. Final Report October 1965 5. GINOUX, J. J. - A Steady State Tecnique for local heat-transfel':, mea,"'.

surement and its application to the flat plate. - J. of Fluid Mechanics, Vol. 19, part 1, pp 21-29, 1964 6. LEES, L. and REEVES, B. L. - Supersonic Separated and reattaching

laminar flows. General theory and application to

adiabatic boundary layer . Shock wave interactions .

Galcit Techn. Rep. n03, oct. 1963 . .

7. NIELSEN,J.N. , LYNES, L.L. and GOODWIN,F.K. - Calculation of

laminar separation with free interaction by the method of integral relations. Tech. Rep. AFFDL-TR-65-l07 October 1965

8. HAKKINEN, R. J., GREBER, 1. ,TRILLING, L. , ABARBANEL, S. S. -The interaction of an oblique shockwave with a laminar boundary layer. NASA TM Memo 2 -18 - 59W. March1959 9 . CHAPMAN, D. R. , KUEHN, D. M. , and LARSON, H. K. - Investigation

of separated flows in supersonic and subsonic streams with emphasis on the effect of transition- NACA

TN 3869 , 1957.

10. WUERER, J. E. and CLAYTON, F . 1. - Flow Separation in high speed flight. Areview of the state-of-the-art. Douglas Report S. M. - 46429 - April 1965.

nal backward facing step from the high enthalpy super sonic flowinthe shock tube. AIAAJournal- Vol 2 - n02 - Feb. 64 12. BE CKER, J. V. and KORYCINSKI, P. F . - Heat transfer and pressure

distribution at M=6. 8 on bodies with conical flares and

exten-sive flow separation- NASA TN D - 1260 - April 1962 13. LARSON, H. K. - Heat transfer in separated flows -JAS Vol 26, p731,

Nov 59

14. NAYSMITH, A. - Heat transfer and boundary layer me asurements in a

region of supersonic flow separation and reattachment RAE • TN AERO 2558 - May 1958

15. SEBAN, R. A. , EMER Y, A. and LEVY, A. - Heat transfer to separated and reattached subsonic turbulent flows obtained downstream of a surface step - JAS - December 1959

16. BRLNICH, P. F. - A study of boundary layer transition and surface temperature distributions at M=3. 12 - NAGA, TN 3509 July 1955

17. BRINICH, P. F . . - Recovery ternperatures and heat transfer near two dirnensional roughness elements at M=3. 1

NAGA TN 4213 - Feb. 1958

18. GHAPMAN, D. R. - A theoritical analysis of heat transfer in regions of separated flow. NACA TN 3792 (1956)

19. MILLER, D. S . • HYMAN, R. and CHILDS, M. E. - Mach 8 to 22 studies of flow separations due to deflected control surfaces.

AI AA Jou rnal , Vol 2 , n° 2 , Feb. 1964

20. NICOLL, K. M. - A study of larninar hyper sonic cavity flows.

b) R

=

2.2 X 106 m-1

...J u.. dl CII '"0

S2

4 0 Z :::J or-I 0

E

0:: CD 4 0

....

_3:

) ( 0 0 ....J M u.. 11 CII UJ 0::

::r::

_I

--

_-

u.. 0 ti)

::r::

a.

4 0:: (.!)

3:

0 0 4

::r::

ti) dl u..

Fig.1 m - SCHLIEREN PHOTOGRAPH OF THE TURBULENT FLOW OVER A 15 deg. FLAP.

~

"Rex .75 o Re: 1.5 106

m'

à Re: 3 10' m-1 .50

~ Flat plate theory M : 2.02 _{@~ ~~è~bà}

~ è~~

0Gl

{j

ê

à ~

_ê

~~~

_~~~~

àGlàà AA

0

.25

Nose of the flap

/

o

50 100 150 200

xmm

250

.

1.5

.---,-1---.1

---,,---

---.---,

:, Model CC-4 Inviscid theory

1.510'mX.Re<310'm-l \ ... I.M ... M ...

~

'"='Uo 00

I

inDol

0 " 1.0~

_o

_ _ _ _

sa ...

1!te_i~~'tft'I"'""'" ... L . . . - - - - l - - - L - - - l 200 Xmm 100 150 250 1 . o . - - - , - - - . , . - - - . - - - r - - - , Model CCA-3 o Re: 1.5 10' m-1 IJ:, Re: 3 10' mI .75~----4_---_+---~---+_---~ .50

-.25

o

Aat plate theory Me: 2.02

t

_;

: i

i i !

l.&l

₁

à ~!à à

Nose of the flap

/ 50 100 150 0

,

~ à à 0 à

A

0 0 à

*

à

2QO

Xmm 250

Fig.2b-PRE55URE AND HEAT TRANSFER DISTRIBUTIONS OVER A 3- fLAP

I

·

I

Inv;,,;d .ho.,.,

.E.

Model CC-5 /

P,

1.5 "''''''''.''

""1

0 0 0 0 ' u -~ ~~LJ IUUOOOOo 0 ° 0 .0 ..,0

o

50 100 150 200

Xmm

250 1.0 Model CCA-5

Nu

-===

Re: 1.5 10' m'l

"Rex

Re: 3 10'

mI

.75 .50 ₀ A 0 A A

-

Flat plate theory Me: 2.02

~

ê

~ ~ à àà llè~ .25 .S!? Cl B J:,.8

Nose of the flap

/

o

50 100 150 200

xmm

250 Fig.2c-PRESSURE AND HEAT TRANSFER DISTRIBUTIONS OVER A 5- FLAP

P

P,

2.0

1.5

1.0

o

Chapman's formula ~ À bil.

.-50

<:> Re: 1.5 10'm-' ARe: 3 l)'m-'

~~

~.~~~~~~~

'~~~~~~AA

(!)

~&

bA ~!ä 100

150

200

X

mm

250

P

P,

Model CC-6 Re: 1.5 10· m-l Re: 3 10· m-1

/

1.5t-- - -- t - - - -t--- l--f---F1 - - - - ---1r---1 1.010~ ----'"""I!t'i!rt!t~--~~~~--1+-50---L.---I1l~Xrn-m-2.J50 1.5.---.,---.---,---:~=r--=---, Model CCA-l Re: 1.5 10'

m'

Re: 3 10'

m'

1.01--- - - + - - - f - - - - ---lJ-- - - --+F--- ---1 .50r---4---4---~-,~---~---~

N ose of the flap

/

o

50 100 150 20Q Xmm 250

Fig.2 e-PRESSURE AND HEAT TRANSFER DISTRIBUTIONS OVER A 10· FLAP

.E.

P,

2.0 1.5 1.0

o

Chapman's lormula 0 - -A-.~ Model CC-g c:> Re: 1.5 106rrï' A Re: 3 106 m-'

~=

50

I

Inviscid pressure rise.f.l.

..

/

'.~'; A --e

(!~

-liJ

: ~~ lf 100 150 200 Xmm 250 1.5.---.---.---.---.---'-, Model CCA-6 o Re: 1.5 106

m'

A Re: 3 106

m-'

1.0t--- - - t - - - t - - - ---t:>r..-+-I'---if---I

Open sym bols: upper surface

Filled symbols : lower surface

.'

CD

.50~---~---+----~-F~r---+---~

o

Nose of Ihe flap

/

50 100 150 200 Xmm 250

Ag.2f-PRESSURE AND HEAT TRANS,FER DISTRIBUTIONS OVER A 15- FLAP

P

P.

2.0 1.5 1.0 0 50 Model CC-9 Re: 3 10'

m'

~

..

)

nO

Model CCA-6

"

. .

"-

...

Ir

I Inviscid theory ~ 150 200 x mm 25 0 1.01--- - - 1 - - -- -- 4 - - - 1 - - - 4 - - - -- - 1 - - - 1 .51---~~----4---4---I---1

o

50 100 150 200 Xmm 250

Fig.2g- PRESSURE_ AND HEAT TRANSFER DISTRIBUTIONS CNER A 15- FLAP TURBULENT Fl.OW- Models CC-9 and CCA-6.

(1) Re: 1.5 10'm" A Re: 3 10'ml 2.01---+---+----.-...!ii!ll----\----+---~ 1.51---+---+----+---...-.----lH----+---~ 1. Ol---+---+----cl---+---+---+---~ 0.51---- - - - l - - - - ---I!I + - - + - - - - + - - - + - - - 1

o

50 100 150 200 Xmm 250

P P.

,

2.

n 1. 5 0 1. 0

----

5' 7'

...

--

-_

..

-

...

--

"...."

---

10' if"

.I

_.-

15'

_i

!

I

I

_-'-

_.-..--: / '

.-'

I

I ' : ! Chapman's

/

/

.,-formula

_

_..

_-

if ---

---.--"-

._

..

../.

""."

....

_.-~V/;·Z

1-" .. ... : •••• ; .,,'

...

Hose

..

.

...

of

.

..

flap

..

...

_._

...

..

...

..•.•. I 50 100 150 200

xmm

250

Fig.3a_ SUMMARY OF THE STATIC PRESSURE DISTRIBUTIONS FOR VARIOUS

2.5 P P, 2.0 1.5

lO

o

Chapman'~ formula

=.0'=

50 . 6 -1 FLAP ANGLES- Re =1.5 10

m

...

3'

----

5'

- -

7' _{1--" " "}

-

-"-'-_.-

10'

f

·

· ..

_

..

-

_15' i

!

I

I

I

,.-'-'-

.---.--I)~

---~

t-

...

~~.~

.~.==

.~.==~

Z

" .0-

...

/., _,'" .... ,,'" _{••••••} ... / / Hose of flap _I 100 150 200

Xmm

250

Fig.3b-SUMMARY OF JHE STATIC PRESSURE DISTRIBUTIONS FOR VARIOUS

lw,Tree

50'~---+---~~~~~~---~---~ 40,~---~--~~---+---~---4---~----~ 30r---~~~---~~---~---~---~ Heat~ wal! e R.: 1.5 10' m" A Re: 3 10' m'" 20r---~~--+_---4_---~---~---~

o

Turbulent Turbulent recovery

lemperalure Adiabalic wall Mt : 1.97

C;>---<i>- -<:>--0--0- ~~0 ~c;r-G"0~' ~'liT..I.

er--/1r- --ftr-A--A- A-~~bAA& -Ir'lr-é-GrA-k-, ~-E:>

laminar recovery lemperalure laminar

(Iheory: Me: 2.02) Mt= 1. 97

Nose of flap

"

50

100

150

·

200

Xmm

250

50~---~---+---+---~---~ 4~---+_---~~---_+---4_--~----~ 30~---~~L---_+---~---~---~ o Re: 1.5 10' mol A Rf.': 3 106 m"' 20j---~---t_~~--_t----~~~~~--~ Turbulent 10 M : 2.02

o

laminar Me: 2.02 c;>--- . ~--50 Turbulent Adiabatic wall Mt=1.91 _0--0-<;>---~. :-.

rl-A-

----/)p./l..er8-:

_k ..A-8--- "" laminar Mp 1.91

Nose of flap

,

100 150 200 Xmm 250

50~---~---~+---~~~---~---~ 40~---4---+~~~--~~+_---~---_1 30~---~---~~---~---~---~ Stagnation temperature o Re: 1.5 10' m-I A Re: 3 10' m-1 20r---~---+---~---,---~ Laminar Me: 2.02 50 e Nose of flap

"

o

1 0 200 Turbulent Mf:1.84 A Xmm 250

50r---~---_+---r_---~---~ 401~---_+---~---_+---~---_4 o Re: 1.5 106 mol A Re: 3 10' mol 30r---+---r~~---_+---~---~ Stagnation temperature 20r---1---~---+---~r_---~~---~

o

Laminar Me: 2.02 Nose of flap

'"

50 100 150 --0 __ -0 Turbulent M_f:1.66 _l>---A L.àminar M_{f :} 1.66 200 Xmm 250

50r---~---_+---~---~---~ 40r---~----~--~~---~---~---~ G Re= 1.5 10'm-I IJ:. Re= 3 10'm-1

30

Stagnation temperature

20

Turbulent Mf=1.46 Turbulent Laminar

10

Me: 2.02 Mt=1.46 0----' Sr,1!!

A--_+

Laminar Me= 2.02 Nose of flap

/

0

50

100

150

200

Xmm

250

50~---~T---~~---~~---+---~ 40~---r~---+--~----~~---J---~ C!> Re: 1.5 106 m O ' IJ!. Re: 3 106 mO ' 30~----.r--~---~~----~~--~---.---~ 20~---7----r---~-&~~~~r---~---1----~T·u-rb~u~le-n~t 1P'0''''t5 Mp 1.17

~"l/fr:f

«" p_ l!) Laminar " Mp 1.17

,Co!

t!f

Turbulent , / , ' M 2 02 ' Adiabatic wal! 10 e:' ,

rt, ....

Nose of flap

/

50 100 150 200 Xmm 250

5'

T r ·C Stagnation temperature level 10'

IS'

21'

Po: 100mm Hg

20~---+---~----~~~---r---~

o

Laminar flat plate Me' 2.02

50

Nose ol flap

/

Theory tor laminar flat

Plate~

based on final Mf

T: tran ition (tabIe 1)

100 150 Xmm 250

Fig.5a- SUMMARV OF JHE RECOVERV TEMPERATURES FOR VARIOUS FLAP ANGlES -Re=1.5 106

m-

I

30.---~---~---~---~---~

5'

Tr.C Stagnation temperature level 10'

IS' 21'

Po = 200mm Hg

20~---~---+--/-7~~--~~~~----~

.' T bas~ on final Mf _ _

~

TheorY for turbulent flat plate

o

Laminar flat plate

Me: 2.02 50 100

.

:T\

_{• T} Nose of flap

I

'

-Theory for laminar flat plate / based on final Mf

T : transition (tabIe I)

150 200 Xmm

Fig. 5 b -SUMMARV OF .THE RECOVERY TEMPERATURES FOR VARIOUS FLAP ANGLES _ Re=3 106

m-

I

20

10

o

~

1

- -

1.43

Stagnation temperature level

_{_}

_.

_-

_1.82

----

2.16

-_

.

_

.

_2.59

- -

2.96

;Z;

<::,~--~ ::.,,~ _{F---. ___} ' / /

!;ij:-

~

_.

_-

_.

_{-_.-}

----

_.

_{_}

_.

---~/ I-._._._-=:; V

~

~ Nose of flap

I

50 100 150 200 Xmm 250

Fig.6 _ EFFECT OF UNIT REYNOLDS NUMBER ON RECOVERY TEMPERATURE DISTRIBUTION FOR A FLAP ANGLE OF 10 DEGREES.

)

aRe: 1. 5 x 10 m

~M

.

.

.

.

~~~l~ogm~}

With

T ·C

_{r Stagnation temperature level}

20~---+---4-~-#~~~~

o

Laminar M e =2.02 ",'-:--' 50 Nose af flap

/

100 150 125 - - - 1 7!i h 130 _ .. - Ó5 roug ness 130 --1.05 135 -_.- 150 a ur:l. 132 _._.- 3.00 } N t I 150 --- lOS transItion Laminar Mf = 1.46 200 Xmm 250

Fig.8- EFFECT OF TRANSITION LOCA"(ION ON THE RECOVERV TEMPERATURE

DISTRIBUTION FOR A FLAP ANGLE OF 15 DEGREES.

30 T

_{r oe}

Stagnatien temperature 20 Turbulent 10 Me= 2.02

o

Laminar Me= 2.02 50

e

5' 10' 15' 21'

"----

.

-100 150 200 Xmm 250

Fig.9-RECOVERV TEMPERATURE DISTRIBUTIONS FOR FULLV TURBULENT

N _U

VrC

_~ 2 .5 2 .0 .5 .0 .5 0

o

Flap angles

----

l.S·

_

....

.

-

3---

5·

_.-

10·

_

..

-

15·

--

21· Po: lOOmm Hg

1\

\

I

/ /

I

V

/.

/

/.

j

//.

....

V ./ .. '

.'

~ Flat pl~~ the~ry Me: 2.02

~::.---~~

~~-_.--~~

#

--~

...

~~

...

~~

..

.

...

~

...

~.::-

...

.

...

...

_-, Nose of flap

I

50 100 150 200

Xmm

250

Fig.l0a - SUMMARY OF THE HEAT .TRANSFER RESULTS FOR LAMINAR AND TRANSITIONAL FLOWS - Re=1.5 10

6 m-l

Flap an~es 1.5· 3· 2.5 5· 10· 15· 21· 2.0 Po :200mm Hg Turbulent data. ~

....

1.5 1.01---+---+---,f---:--tl'---+---~ .50r---+----~+--~~'---+---+______,~·-····-~ a:-":;::':'

-o

Aat Nose of flap I 50 100 ~~:~

_

.

-_.

--150 200

Xmm

250

Fig.l0 b - SUMMARY OF JHE HEAT TRANSFER RESULTS FOR LAMINAR AND TRANSITIONAL FLOWS - Re=3 10

6 m-1

NU

-====

_VR

ex 2.0

lO

o

e

----

5·

_.-

10·

-.-

15·

--

21·

j/=

~.-

.

_-~:.

,'"

---

----_&

_--"

-

..

--

~

...

~

Nose of flap

I

50 100 150 200

xmm

250

Fig.ll- SUMMARY OF THE HEAT TRANSFER RESULTS FOR TURBULENT FLOWS OVER FLAP MODELS.