Wall temperature effects on subsonic gas flow

ARCH:EF

Irk

y.

AERODYNAMISCHE VERSUCI

GOTTINGEN

REPORT NR 67 A 26

WALL TEMPERATURE EFFECTS

ON SUBSONIC GAS

FLOW

by

Wm. J. Keinhofer

DEPARTMENT OF MECHANICAL ENGINEERING THE CATHOLIC UNIVERSITY OF AMERICA

WASHINGTON, D.C. 20017

FINAL REPORT

OC-T .11967

PREPARED UNDER CONTRACT Nonr. 0014-67-C-0149 FOR OFFICE OF NAVAL RESEARCH

ACKNOWLEDGMENT

The author gratefully acknowledges the financial support of the Fluid Dynamics Branch of. the Office of

Aerodynamisde Versuthsonstolt

Göttingen

BIaft.Nr.

i

WALL TE1PERATURE EFFECTS ON SUBSONIC GAS FLOW

Abstract

A theoretical study is made for calculating various boundary layer thick-nesses at a point on a heated or cooled wafl assuming n-power velocity and rn-power temperature profiles in subsonic gas flow. Also1 boundary layer measurements are made for air flowing over a flat plate heated to 250 °C

with velocities up to 30 rn/s . The experirnentál results are compared to previous theoretical results assuming

n m and the velocity and thermal

boundary Layer thicknesses equal in fully developed flow.

Contents

Forward Nomenclature Introduction

Theory-calculation of t and Pr effects

w Description of experiments Experimental.results Summary G. References Appendix

This Report includes: 77 pages inch

4 tabLes and

45 figures

AERODYNAMISCHE VERSi.1 CHSANSTALT GOETTINGIN The Director

*. f&'4t4

(Prof. Dr. H. Schlichting)

1

The Author (Dr. ni. J. Keinhofer) Associate Professor f Datum; _{17. 7. 1967} Bearbeitet: Abtsilwg: i Dr. F. W.. JUetels 8,i&t: (i7 A 2G

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Forward

The investigations reported herin were performed by the writer while a guest

at the AVA, during, the period 1 September 1966 through 31 July 1967. 1 wish

thank Professor Doctor H. Schlicht ing for accepting me at the AVA1

and Dr. F V. R i e g e t s, head of the low speed aerodynamics department,

for rende iing to me his departmental services necessary to carry out this work. 1\ly ;pecial th4nks go to Mr. J. Rotta who contributed much of his

time to the design of the experimental appazatus. offered much advice

regard-ing the runnregard-ing of the experiment, and, with Mrs. H. Senkbeil and I'Jrs. I. Wo Ike, helped reduce the databy computer in a most useful way. The

technican for the experiment, Mr. A. Sperber; is also thanked in a

spe-cial way for putting up daily with my requests, and carrying out the long

series of measurerents. I wish to thank the many other people, ali of whose

names I cannot mention here, who made my visit and work at the Institute most pleasant and worthwhile.

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Aerodynamische Versucksanstak Göttingen BJàtt-Nr 3 Nomenclature A _{parameter t} B parameter (1 - A) 1/rn C

parameter A/B

J)

parameter A/B

F3 finite series given by Equation (A-14)

G3 finite series given by Equation (A-9)

li boundary layer shape factor

boundary layer shape factor

/

boundary tayer shape factor

13 finite serìes given by Equation (A-8)

J3 finite series given by Equation (A-11)

Pr

Prandtl number

R radius of rotationaLly symmetric body

X intermediate variable

Cf 'ocal skin coefficient

C.

speàific heat at constant pressure

d _{dissipation function in Equation. (7)}

p

staticpressure

q rate of heat transfer per unit area

t temperature absolute

u velocity component in x-direction

y velocity component in y-direction

x _{streamwisecoordjñate.} y normal to x-coordinató Datum: Bearbeitet -Gesehen: I 8eritht: I 67A26

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4

A1, A2, A3, _{4 a} A5 boufldary layer th.tcknes s give n by Equations (17_21)

(S total boundary. layer thickness

mechanical energy loss thickness

*

6 dispLacement thickness

ratio _y/6

e

momentum Loss thickness

thermal energy convection thickness absolute viscosity ratio

_2'1

¼ mass density

-shear stress.

intermediate variable S ' b i. c ript s

o _{for the condition of o-constant}

i . condition at edge of veLócity boundary Layer

2 condition at. edge of thermal boundary Layer

condition at the wait

w . condition out8ide. the boundary layer

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1. Introduction

B)a,.Nr

Knowledge of how the characteristics of turbulent boundary Layer gas flow are affected by heat transfer at the wall is Oftén a problem of interest for

subsonic or supersonic, internat or external flow. Turbulent boundary layer calculations without heat transfer are difficult to perform, and this difficulty increases when the temperature of the wall, is not the saine as the free stream gas temperature With Large temperature differences between the wáll and free stream, Large density gradients exist which can affect skin friction and energy.dissipation in the boundary Làyer. With adverse pressure gradients, one would expect that flow separation, if It exists, would also be Ifected. Investigations into the effects of compressibility and the resulting heat trans-fer on the turbulent boundary layer for supersonic flow have been made both theoretically and experimentally. Only a Ñlatively few Invéstigations are

fôund fur the case of subsonic flow wIth heat transfer [1,2,2,4, 5,61'. For

these investigations, the density in the boundary layer is considered constant. The present writer made some theoretical investigations of the effects of the

ratio of waU temperature to freé stream tórnperaturé. Ç/t2

. and Prandt L

number, Pr , on turbulent boundry. Layer flow with variable density in the

boundary layer 17]. For these investigations n-power velocity and

tempera-ture profiles in the boundary layer were assumed, as wil as the velocity

boundary layer thickness, 61 , approzimnatty equal to the thermal boundary

Layer thickness, _c2 The results showed that aU. boundary Layer

characte-ristics, such as; displacement thickness, znómentum loss thickness,

e, mechanical energy toss thickness, thermal energy connection

thick-ness,

, various shape factors, and separation; are alt affected by t/t2

and Pr . The effectsare greater for laminar., flow, than for turbulent flow,

but all trends are the same. The accuracy of. these calculations were not

corn-paréd with experiments.

This report shows an extensión;of the theoretical procedure for calculation

Dotum:

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Numbers in brackets identify references listed on page 19.

Abeihjng: . . ..

.

. Bendit.

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r

of the and Pr effects at a point on a body where n-power velocity

profiles and rn-power temperature profiles are asawned. Also, _tSr ta not

considered equal to . The iffects of variable density in the boundary

layer can be computed separatly Liz closed form for certain combinations of n and _{in , and interpolated for other combinations of interest. However,}

new computations have not been carried out.

An experiment of subsonic, turbulent air flow over a flat plate with various

surface temperatures up to 250 °C was carried out, and the resuLts are also

given in this report. Boundary layer velocity and temperature profiles were measured, and various boundary layer characteristics computed. These results are compared with the calculatlona given in Reference 17].

2. Theory-calculation of t_w and Pr effects

The flow model is assumed to be that of a perfect gas flowing over a rota-tionally symmetric body as shown iii Figure 1. After a certain distance on the body, the flow is assumed fully turbulent. The body wall temperature is i.onsidered not equal to the free stream temperature and so a thermal

bound-ary layer 52(x) exists, as welt as.a velocity boundary layer £.1(x) . Since

O Pr 1. 0 is assumed,

'

when the thermal boundary layer is

fully developed. The objective is to obtáin a procedure for calculating the t and Pr effects on the turbulent boundary Layer characteristics. The equations of motion for the boundary layer are the continuity equation

a(uR)

+ a(QvR)t

ax _ay

Prandtt 's boundary Layer momentum equation

au QU + QV

r-8x + Datum: Bearbeise$: / Abtsøung: 1sricht: 67 A 26

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Bla?t..Nr.

and, the conservation of

enerr equation

at at a aq.

(3)

' a

'

ax ay a)

At the edge of the boundary layers one can write du d

..

-

i clx

Wildx

dt u2 du2 R dx e dx p The boundary conditions are

At y

uRvO

At y , u

u(x)

44t

= .0 = u2(x) =

(inc may eliminate v _{from the momentum equation by use of the continuity}

equation, and integrating the result once from y O to y ri

6,

the

inte-gral momentum equation is obtained. This procedure Is repeated after first niultiplying the. momentum equation through by. u, and the integral rnecha-nical energy equation is obtained. Finally, y i

eliminated from the ener'

equation and the result integrated from y O to y a6 to obtain the

in-tegral energy equation. Theseresults, respectively, are

de.f(H+2)

+1aî8!

dx

Lui

dx

RdxJ

2 d 3 du1 .1 dR 7

- + (

-. + -

o a dx u dx

Rdx'.

. 3 1 . glu1. (x)

, tt

w w.

O.. tt1

I t2 Datum: Bearbeitet; Abteilung: . 61.Ä26

Aerodynomische Versuchsonstolt Göttingen where de _{I i} du2

.1 dR

i1ç

q ±d

(8)

The effect of t t2 on the boundary Layer characteristics

e,

and

at a point on a body as given by Equation (9) can, in a general way, already be seen. 1f

t, ) t2 , the ratios

is always leas than unity inside the boundary Layer. Therefore1 the integranda of E) and at a

point O < y <S, are Less than they would be for t The integrand of

6', however,

would be greater than foi' the condition t t2 The

trends reverse, of course, for t< t2

Then, for t/t2

i. 0, E) and

decrease, arid 6*

_{increases; for}

_{t/t2ici.O, e and}

_increase, and 5* decreases; all other, conditions beinE equal. The effect on is

not so easy to discuss, since for

t

t2, e - o...

To show how these density variation effects can be calculated, Equations (9) are expanded after a concept cf A. Watz [81. Since

_{ap/ayo, anda}

per-fect gas is assumed, one can write

t1/t

and

t2/t

. Like

terms are added and subtracted to Equations (a) and they can be expressed as Datum: Bearbeitet: / Abteikmg: JGuebei,:

j

.1.._{_U.. T}

:87*26

Aerodynamische Versuchsanstolt Göttingen Datum Bearbeitet ¡ 0 , where *

.

6

=6+6

₍₁₀₎ o

i

®*e -

+4

(11) o 1 2

-

a

-636 -4+3

(12) o (13)

(1-.a)dy

6 Fi J

e -i

_o I u1 u1 o. 6

r'

6*

_o o 6 2 iciJ

(-)(i-

-i

o o u u'

_r'.

fu

'u-2] 6 J.1

L(i)dy

o -t dy -(14) (20) ß7 A 26 'ç ¿1

Aerodynamische Versuchsonstcilt Göttingen .10 (1 _- ) dy . (21) i

The terms

6, e

, and given by Equations (14), (15) and (16) respec-tiveLy are the displacement thickness, momentum Loss thickness, and mechá-nical energy loss thickness as they appear in the integral momentum and mechanical energy equations for the case

t

t1 t2 . The additional terms

in Equations (io), (li), and (12), i.e.,

A1. ₂

and £31 must then

re-present the effects of the temperature fIeld on d1, O, and for the case t t1 t2 . The term appears only in.the integral energy equation.

Clearly, for t = t1 = t2, e1

as well as Li

ói, A

and £4 go to zero.

One should also note that for Pi' 1. 0, and , since then

5=O and

r

Equations (14) through (21) can be integrated under the assumption of n-power velocity profiles and rn-power static temperature profiles In the boundary

layer, where n is not necessarily equal to ru Tbua

n a 1,2;3,...,io

t

-t

1/rn w = a 1/rn

t

_w

-t

_{2 2} m a 1,2,3k... .110

A further assumption is made timt the ratio of the Velocity to therTnal bound-(23) Datum Bearbeitet: jAbteilung: 87 A 26 u1 _di 1/ri _1/n (22 ),

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ary rayer thickness is a function only of Prandtt number, or

6 ₂

(24)

2 1

(ther than for the case ap/ax O , the assumption expressed by Equation

(24) is not yalid. For ap/ax O and Pr 1.0,

/62 =(Pr,x) .

How-ever1 this function is not known, and one might consider ₆₁₁₆₂ a weak

function of x

W ith the substitution of Equation (22) into Equations (14) (15), and (16),

6 and in nondixnensional form are

e

_n

6

(1+n)(2+n)

o Zn

(1 +n)(3+)

With the substitutions of Equations (22), (23),. and (24),. and a subsequent change of variable, Equations (17) through (21) as shown in appendix A of

this report, integrate respectively to

Z"(i+n)

_B;lIm.

.n.

m(A+HBI1/m) BS1ìm .G3 Datum: Bearbeitet: t Abteilung: 61 A 26

Aerodynomische Versuch sonstclt Göttingen 3 n

m(ABt1/m)

B /rn J . (30) i 4 n in - (1 + n.)

_B1/m

13 (31)

=L(l

-fl-F3]

(32) where: twit2 - A) Alan-Nr. 12

The.quaritities _'

G, J3,

and F3 are finite series given in appendix A,

respectively, as 1quations (A-6), (\-9), (A-11), and (A_14). Equations (23)

tlwc)ugh (32) are valid only Ior certain combinations of n and ¡n . 1low

-evel-, Uo otite i than these certain combinations, results can be obtained by

sinipie intctpolation. To understand fully the meaning of these last state.-meats, appendix A should be studied.

Under t!ìe assumptions stated above, the effects of Prandtt number and

variable density in the boundary layer on the boundary layer characteristics at a point on a body can be computed. One must first choose a value o.f n

and rit for a given Pr and

t/t,,

. The value of n might be taken the

sanie as for the case of

t/t2 = 1.0

. 1-lowever, very little insight is

avail-able as to what value of rn to choose. This knowledge might be obtained b experimentation or a closer took at the equations of motion in integral

furto. No computations for assumed combinations of n

and m have been

carried. However, Reference 17] gives calculations for the case n

m and

6,

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V ith regard to the effect of Pr and

t/t2

on boundary layer separation no extension of thé procedure given in Reference 171 has been made.

Evalua-tion of EquaEvalua-tions (9). however, is basic to the use of that procedure, and so one might say that the application of that procedure has been expanded.

3. 1)e.icription of experiments

The objective of this experiment was to measure boundary [ayer velocity and

temperature profiles along a hot, flat plate for various free stream velocities

and plate temperatures. A flat copper pLate 300 mm wide and 1 500 mm

long with associated heating apparatus was built into the test section of the low turbulence windtunnet at the Aerodynamische Versuchsanstalt, Göttingen.

A sketch of L L.pen-type windtunnei is shown in Figure 2. A sketch of the hot

plate and its installation in the windtuiinel is shown in Figure 3.

Installed iii the plaie were 1G static pressure taps of

0.4 mm bore, and

U5 copper eon.stantan thermocoupLes made of _{0. 2 mm diameter wwe. One}

extra tliernocouple not shown in Figure 3 was instaUed 24 rum fruni the [runt end of the plate making a total of 66 thermocoupLes. The wiring diagram fur the thernucoupLes is shown in Figure 4.

heating of the copper pLate was provided by 66, 220 Voit, .200 Watt, fLat, bar-type heating elements Lying cross,wie on the copper plate. Various methods uf wiring the heating elements were tried in an effort to obtain flat tempera-turc distributions along the plate. Figure 5 shows the most-satisfactory wiring fôund. The heating elements are divided into four groupallowingrnore heat-ing on the forward portion than ön the backward portion of the plate.

During the tests at 200 °C or more, it was found that the sheet of Insulation between the copper plate and the steel plate tying on the plexiglas window was

not thick enough, and the plexiglas became soft for a thickness of about

15 mm _{. A strip of insulation 35mm thick was pLaced between the steel}

plate and the plexiglas.

Dc?um: Abteilung: Gesehen: Bericht:

¡

,

Measurements in the boundary Layer were taken on the underside of the plate. This assured against any influence of natural convection currents. The

bound-ary layer probe, shown in Figure 6, consisted of a calibrated Chromel-Atu-mel wire thermocouple and a circular total pressure probe built together as

a unit. The thermocouple wire was 0. 07 mn-i thick. The total pressure

probe had an inside bore of 0. 4 mm and an outside diameter of 0. 5 mm

Connection of the thermocouple with the instrumentation is shown in Figure 4. The probe carne up through the floor of the windtunriel as shown in Figure 3. Tuis made the probe stern quite long, and so a profited windshield was pro-vided to reduce vibrations caused by wind flowing over the probe stern. The

probe, as a unit, could be moved in a longitudinal slot in the fluor of the wind-tunnel, thus all owing temperature and pressure profiles to be taken at any position along the centerline of the flat plate. Positioning of the probe in the vertical, direction was done with a fine threaded screw apparatus with posi-tionirig indications for every 0.01 inni . Contact of the probe with the

sur-face uf the flat plate was indicated by a Light energized volt meter using the

probe and the flat plate as terminals. Pressure readings were obtained by use of a Prand ti manometer. The niiUivolt signal from the thermocouple

was first amplified and then read-out on a digital voltmeter.

The test program consisted of taking temperature profiles (in terms of milli-volts) and dynamic pressure profiles in the boundary layer along the plate

at free stream veLocities of 10, 20, and 30 rn/s , and approximate plate

temperatures of room, 70, 110, 150, 200, and 250 °C . These

measure-iìients were taken at seven stations along the centerline of the plate as shown in Figure 3. For each profile, data were taken at no less than 50 and no more

than 59 vertical positions. More readings were taken close to the plate than nearer the edge of the boundary layer.

At all velocities and plate temperatures turbulent flow was maintained with no difficulty.Tests showed.that transition occured very near the front end of

the plate. A group of typical free stream velocity distributions relative to the average free stream velocity is shown.in Figure 7. Maximum deviations

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from the average were no more than t 2. 5 percent.

Approximately one and a half hours running time were required for the tern -perature of the plate to stabilize for a particularvetocity and tem-perature

setting. However, the average temperature would possibly increase or decrease a degree or two over a running time of 3 to 4 hours due to slight

changes in the room temperature or local atmosphere conditions. Tempera-ture distributions along the plate for a free stream velocity of 20 rn/s are shown in Figure 8. These temperature distributions are also typical for free

stream velocities of 10 and 30 rn/s.. Along the test section of the pLate the tern perature varied no more than ± 2 percent of the average for all tests. The current and voltage input to each group of heating elements on the plate for each velocity and approximate average plate temperature along the test section aru listed in Táble I . What was iiziportant for proper tern perature distributions, of course, was the power input to each heating element. The equations for calculating the power to the individual heating elements are shown in Figure 5, where the currents and voltages correspond to those of Table 1.

Approxinìatly i to i _{hours were required to measure together the} tem-perature and pressure profites at a given position átong the.ptate. This relatively long measuring time depended on the Length of time required for the pressure reading to stabilize.. The 0. 4 mm bore of the, total pressure

probe was, perhaps, too small. A flattened tube with a larger inside area

would probably reduce considerably the measuring time.

At 200 _{or higher a thin, dark skin would form rather quickly on the}

sur-face of the plate. This insulating skin made contact between the probe and the pLate difficult to find. Also, after a few hours of testing at the higher

tem-peratures, the total pressure probe had a tendency to stop-up, which was

probabLy due to very smalL flakes comming off the plate surface and finding their way to the probe. Cleaning of the plate surface with a fine polishing paper was not difficult, and this was done regularly.

Blott.Nr. ₁₅ Datum Bearbeitet Abteilung: JGesehen:. Bericht: 67 A 26

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4. Experimental results

The experimental data was reduced so that the effects of wäll temperature on the various boundary layer thickness could be seen. Typical velocity and temperature profiles along the plate for an average value of A _{1.43 and} u 20 in/s _{are shown in Figures 9 and 10 respectively. Velocity and}

tem-perature profites at x = 219 mm and x _{1 363 mm along the plate fur}

various values of _{A are shown In Figures 11 through 14. It appears that} the velocity boundary layer developes slightly faster than the thermal bound-ary layer atungthe plate. At a given position on the plate wall temperature has little effect on the general shape of the velocity profiles, but the thickness is slightly increased with increasing temperature. _{The same is true for the} therual boundary tayer. A comparison between the shapes of temperature and velocity profiles along the plate for _{A a 1.430 is shown in Figure 15.}

Figures 16 and 17 show this same comparison at .x

219mm and

x = 1 363 nim along the plate for various values of A . No significant

difference can be seen between the shapes of the temperature and velocity profiles except near the wall.

The boundary layer thicknesses, discussed in Section 2 of this Report, i.

_e.,

A A2, and A3 _{were computeci along the plate and the}

results for tree stream velocities of approximatly' _{10, 20, and 30 rn/s}

are sho;vn in Figures 18 through 38. The effects _{of A}

_{on Q and}

_{Ô are} slightly difficult to analyse, since the data is somewhat scattered. One _can

see, however, that the wall temperature effects on cS, , and ¿ become

greater with increasing boundary layer thickness. _{As velocity increases, the} effects of A on 6* _{decrease, but increase slightly on}

_{Q and}

₆

. The

effect of increasing A on is to make _{larger. The effect of}

in-creasing A

on O and

6 _{is to make them smaller. These trends are}

the saine as those found theoretically in _{Reference 171.}

The effects of A

and velocity on e. A1,. A2,

_{and A3 are very regular}

for all measurements on the plate. As A increases,

' A1,. A2', and

Datum

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,

A3 _{increase, and the., effect of increasing velocity is to decrease these}

quan-tities. Again, these effects are the same as those found in Reference [71. The fact that the shapes of the temperature and velocity profiles are similar except very close to the wall (see Figures 16 and 17), and that the trends of the wail temperature and velocity effects are in the same direction as those found in Reference [7, prompted a further comparison between the two results. The results of Reference [71are for ruuy developed velocity and thermal

bound-ary layers with

n = m and

. Therefore, this further comparison

i-; ruade only with the experimental results at x

i 33 mm

To obtain the ptoper experimental value of n , H was computed and then the val.ue f

.i obtained fronu JI = ((2/n) +1) . The assumptions were made that _a

.i.: ¿, for the experimental results. Then, the co!nparisons were made

6/S.

11/li ,

ii/ii

_, Ht , and H/IT, andare shown in

Figures 39 through 45 respectively.

The experinuentai values of n were between 5 and 6 . These were also

the theoretical values of n for the comparison of , IT/IT , and

il/IT . For c-/E , Il/il , and 11 , values of n between 5 and 9

were used, arid, for (S/s , values of n between 5 and 7 were used.

Theoretically, _11/110. is for all practical purposes independent of n

All comparisons between the experimental results and the theory are reasion-ably good. Sume scatter dependent on n exists, but all trends of the effects

of A are in the proper direction. Of the two effects compared here, i.e.,

n' and A , the effect of A is greater, and so the scatter effect of n. is not significant.

5. Summary

A theoretical analysis was performed for the computation of various bound-ary layer thickness at a point on a body whose temperature is different from

Datum: Bearbeitet:

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that of the free stream flow under the assumptions that n m,

and _1/62 (Pr) , where n _{indicates a power velôcity profite arid m}

indicates a power temperature profile. The velocity boundary layer

thick-ness is _{and the thermal boundary layer thickness is For certain}

integer combinations of n and rn , _{alt boundary [ayer thicknesses, e. g.,}

,

6, 6, etc.

can be computed in closed form. For other nr-n-combinatjons of

interest, resu Its can be obtained by interpolation. _{This analysis represents} an extension of the theory of Reference [71 where n nì and

were assunied. No new computations were carried _{out. These results can} be applied, however, to the method of boundary layer computation along a body as outlined in Reference Iii.

Experit:lerAts of turbulent, subsonic air flow _{over a f [at plate with surfaces}

tenperatuies of approxiniatly room, 70, 110, 150, 200, and 250 °C

were carried out, and show that A(m t/t2) has an effect on the various boundary laver thickness that are used (or can be used) for theoretical bound-arv laver analysis. Close to the wall the shape of the temperature _and

velo-city _{rofiIes arc different, but further away from the wal.t this difference}

dininishes. As _{the boundary tayer thickness increases aH wall temperature}

effects increase. The effect of A on is to increase but this

effect decreases as u1 increases. The effect of _{A on and is}

tO decrease these thickness, and this effect _{increases as} u increases.

A1, A), and A3 all increase with _{A and this effect decreases with}

ïncrcaing

u These trends are similar to those found _{tìeoreticatly in}

References 171.

By assun.ing that n m and

_{the experimental results in terms}

, H/H ,

ff/

_{, Ht and H/ at x 1}

were compared with the theoretical results of Reference [71. _{Outside of minor} scattering (lue to _{n ,. all comparisons are reasionally good, and show}

clearly the effects ;f A

Due to lack of time other flow characteristics, such as s!d.n friction

coeffi-Datum 8earbeitet Abteilung: J Geiehen: J Bericht: 67 A 26 Btofl.Nr. ₁₈

Aerodynom sche Versuchsanstat Göttingen 1.11 Heynnlds, W. C. Nays, Kline, S.J. 121 Dvorak, F. A. head, M. lt. [31 Perry, A. E. Belt, J.. B. Joubert, P.N. [41 McCarthy, T.F. Hartnett, S. P. BIen-Nr.' 19

cient, dissipation function, and heat transfer coefficient are not included in this Report. These quantities are presently being computed and will be included in a later and more complete Report.

6. References

heat transfer in the turbulent incompressible

boundary táyer

Constant wall temperature

Step wall temperature distribution

Arbitrary wall temperature and heat flux Effect of location of transition and predic-tion of heat transfer in a. known transipredic-tion region

NACA Memos (respectively), 12-1-58 V; 12-2-58 W; 1,2-3-58 W; 12-4-58 W

Heat transfer in the constant Property turbulent boundary Láyer

International Journal of Heat and Mass Transfer,

Vol. !2.(967), p. 61-81

Velocity and temperature profiles in adverse pressure gradient turbulent boundary layers

Journ'. Fluid Mech., Vol. 25, Part 2,'(1986),

p. 299-320

Heat transfer, to turbulent boundary layers with

a pressure gradient

Departm ent of M e chan ica t Engineering, Univer

-sity of Delaware, Techn. Rep. No. 2', Nov.

1963 '

[51 Kytateladze, S.S. _{Calculation of a turbulent boundary Layer for}

Le ont'ev, A. I.' _{substantial positive pressure gradients}

(Translation from Russian to English), Defense Documentation Center.No. AD-605877, 1964

[61 Rubesin, M. W. _{The effect of an arbitrary surface temperature}

variation along a flat plate on the convecture heat transfer in an incompressible turbulent boundary layer

NACA TN 2345, 1951

Datum: _{Abtlung: Gesehen:}

Bearbeitet

. I ' '

Aerodynamische Versuchsonstalt Göttingen jBIo*-Nr. 20 [7] Nelnhofer, W. J. [8J WaIz, A.

Analysis of effects of wall, temperature and Pran.dtl number on subsonic boundary Layer thicknesses for rotationally symmetric flow and application to turbulent flow separation Doctorat Dissertation, the Catholic University of America, Washington,' D. C., 1966

Beitrag zur Näherungstheorie kompressibler turbulenter Grénzschjchten

DVL-Bericht 84, 1959

Datum:

Bearbeitet: H.ie ser

/

Abteilung: Geiehen:

Ae rodyn orn ische Versuchsonstalt

Göttingen

Appendix

The procedure for the integration of Equations (1.7) _{through (21) is shown in} this appendix. In reading this appendix_{one ahould keep in mind that}

6

and n _{'u , and so subscripts and superscripts of terms are important.}

Equation 23 is expressed in. the frm

where

t

w

A=-2

with the use of Equation (24), Eqüation _{(A-1).becomes}

using the condition, t = t1. at

i

ç- ABh/m

and with use of Equation (A-2)one

_{ca write}

ti

_A+Bh/m

r .Aß1?mh/m

Appendix

A+B

I 1/rn 1/rn Ti' A) Stats-N,. 21 (A-1) (A-2) Datum: Bearbeitet , JAbteilung: JGesehen: . Bericht: 67 A 26 t., = t

l:civatiu (22) and (A-4) are substituted into Equation (17), and in nondirnen-sional forni 1 / (A

ß1/m

1 ¡ml - A +

B1/rn V1'1

r re

j

i 1/ri

i/m

I

Bhhu11 2 C +

'l'o CvalUrìte

.itt,al

₁₂ the fol iowin change of variabte is made

=

'ti 1/rn \rn/fl

(i nL\1 dX

= d; .\ = U

= I; X =

tIcn, liv ub.titutiuu,

J ,\ +

1/nt1/1u

1

Blott-Nr. 22

Aerodynomische

Versuchsansfalt Appendix

Aero d y nom s che Ve rsu ch so n sta I t Göttingen Appendix Blatf-Nr. 23 C P1-1 P1-1 P P 3 P1 - (P1 - 1) + (P1 - 2) .

(- i)

C + (- 1) 1 tn (C+ 1) f. (A-G)

['his series can be evaluated by computer. However, as

A.l. J, BO

and

Cw

, and the computation becomes difficult. At _{A = 1.0, 13} = _O

and so Iur values of A-1 t)

_{interpolation can be employed. For the}

coni-binatiuii of n and ru resulting in P1 _integer, 13 may be evaluated also by interpolation.

Datum Abteilung: Gesehen: Bericht:

Bearbeitet: _{G7 A 2G}

x(1h/1)

-1

C+\

dX ¡et: 13 =

_12 and

l' z m(1 - I I P 13 oef

C\

o

Vn n 1,2,3,..

10 and

ru = 1,2,3,.

.,1),

_{P1 can be (see Table A-I}

the wid nl this uppewIi)

1. integer (P1 i)

2 non-integer

1' 1

P1<1

IThe definite lutegi-al can _{be evaluated in closed forni by a finite series}

Aerod yna mische

Versuthsanstolt Göttingen

ThUS, by substitution into Equation (A-5)

A1 n ni(A +

Bh/m)

5 ₍ + n)

B1/m

3 G c 3 P2 (12 - 1) (P2 - 2) Datum: Bearbeitet: Abteilung: n nt(A B 2 ;-;i3 -

ß1/m

Appendix BIott-Nr. ₂₄ 2 + (-

l)2C

(C+1.)

n

fll(AB1/m)

j

S1 (34-n) -

ß1/n1

3 Geiehen: Bericht: (A-7) (A-3)

'v\Jiere P2 = tn(1 + 2/ti) - i = (integer). (Values of P2 for combinations

of ti

and m

are given in Table A-IL.)

c

P, -1

_{c3 +(_l)3c3ln(c1)}

P -1

P P

where P3 = n(1 ₊

3/n) - i

_{(integer) (Values of P3 for combinations}

ui n

and m

are given in TabLe A-UI)

The last two evaluations are of Equations (20) and (21). Equations (22) and (;\_2) are substituted into Equation (20), and in nondimensionai form is

67 A 26

and are evaluated. The results are

Aerodyno mische Versuchsanstofr Göttingen Appendix.

1 /

1/n

U

,'fl 'L1 ;-=RPI1

ABhfm'L

which integuate sin ply tu

4_

n

i

¿n

B/

13

1ìcre 13 is given by Equation (A-G).

Equation (21). by use of Equations (A1) and (24) is written i

i r .1

-A + B4

['he new liiiit.s of the inLegral are based on the boundary conditions

G Y

_1'2

' _{2 = 82/2} I Thci, )

d,2

Blatt-Nr. 2 5 (A-12)

where D

A/B The remaining integral is evaluated after making_the

follow-ing changé of variable

m-1 ;

dr2rnqc

_dV Daturr.: Beorbeitt: F Abteilung _Gesehen: 1.Bencht: 67 A 26 let:

l/m

Aerodynamische

Versuchsanstalt

Göttingen

and, for the limits of integration

®

2'

Thus, where 13 = I D (ni - 1)

- (ru -2)

(m

-3) +

(ni-1) (n-1)

i) In (D+ ( 1)(mn_1)D(n1_1)In(D +

or, combining the two series

Appendix

rn-I F3

f

.4d

For (rn - 1) an integer or zero1 this definite integral, can be evaluated

in cLoied form by two finite series. The result is

i _-

)-

-F3]

. (A13) rn-1)-1 (rn-i) 1) (ru-3)

D2(/m)

+ (-

D(m11

(m-2) D(.h/m) (rn - 2)

1/m)

Blatt.Nr. ' ' 26 Datum: Bearbeitet: Abteilung: Gesehen: .. J Bencht:. 67 A 26 ; V 1/rn

Aerodynomisdie Versuthsanstafi Göttingen Datum 8earbeitet:, r

F=

- 1 3

(m-1)'

S

'm2)

_-. 2 4- ₍₁ (in . 3) Appendix In - 1)D(IT -1)

(_P t i

'SL) _{+ g} (in 2)/rn)) + (- 1)(m_2)D(1n2)(1 1/m

l'ur the special case of Pr = 1.0, ₆

2 and . 1.0 then as

velT as

_5'i

_{identically to zero.}

Bloø-Nr ₂₇

+

Abteilung: _{Gesehen: 8erucht:}

67 A 26

Aerodyna mische

Versuchsan sta It

Göttingen

List of tabLes and figures

Table I Electrical inputs or heatig of flat plate

Figurer i Theoretical flow model

Figure 2 Low turbulence windtunnet

Figure 3 Sketch of hot plate, installation iii indtunnel1 and location of

data stations

l."igure 4 Tlieriitocouple wiring diagram

ligure 5 \\iring diagram for heating elements

l'i:tItc

6 13widarv layer probe

Viguru 7 l'y pical plate velocity distributions

Figure 8 Typical plate temperature distributions

Figure 9 Velocity profiles along the flat plate,

A 1.430, u1 20 rn/s.

Figure 10 Temperature profiles along the flat plate,

A = 1.430, u1 = 20 rn/s

Figure 11 Velocity profites at x . 219mm for various values of A

Figure 12 Velocity profiles at x 13ti3 mm for various values of A

Figure 13 Temperature profiles at x 219 nm for various values of A

Figure 14 Teniperature profiles at x = 1363 min for various values of A

Figure 15 (t

- t)/(t

- t2) vs. u/u1 along the flat plate

Figure 16 (t - t)/(t - t2) vs. u/u1

at. x

219 mm for various

of.A

Figure 17 (t - t)/(t - t ) vs. u/u1 at x 1363 mm for various

w w 2

of A

Figure 18 S along the flat plate for various A, u1 - 10 rn/s

Figure 19 along the flat plate for various A, u1 20 rn/s

Dcum

Bearbeite,..

Abteilung: Gesehen: - Beritht:

Blott-Nr. ₂₈

Aerodynomisde Versudnanstalt Gâttingen ____ 29 A3 o 6/6 o H/H o

atong the flat plate for vrious A, u1 3D rn/s

along the flat plateiorvarious A, u1 10 rn/a.

along the flat plate for various A u1 20 rn/a

along the fiat plate for Various A, u1 30 rn/a along the flat plate for various A,.. u1 10 rn/s along the flat plate for various A. u1 20 rn/a

along the flat plate for varloùa A, u1 30 rn/s along the flat plate for various A, u1 - 10 rn/s along the flatplate for various A, u1 - 20 rn/s along the flat plate for various A, _{u1 -} 30 rn/s along the flat plate for various

A, u1

_{10 rn/s} atòng the flat plate for various A. u1 - 20 tn/s along the flat plate før Various A, u1 s 30 rn/s along the flat plate for various A, u1 10 rn/s along the flat plate for various. A, u1 a 20 rn/s

along the flat plate for various A, u1 30 rn/s along the flat platè for various _{A, u1} 10 rn/a along the flat plate for various A, u1 - 20 rn/s

along the flat plate for various A, u1 30 m/8

vs. _{A compared with theory}

vs. A compared with theory

vs. A compared with theory

vs. A compared with theory

VB. A compared with theory

YB. A compared with theory

vs. A compared with theory

Datum: Bearbeitet: 67 A 26 Figure 20 Figure 21 Figure 22

e

Figure 23 Q Figure 24 Figure 25 Figure 26 ' Figure 27 Figure 28 Figure 29 Figure 30 A1 Figure 31 A1 Figure 32 A1 Figure 33 A2 Figure 34 A2 Fig:ire 35 A2 Figure 36 A3 Figure 37 A3 Figure 38 Figure 39 Figure 40 Figure 41 Figure 42 H/H Figure 43 Figure 44 _Ht Figure 45 H/H

A erod y no m ische

Versuchsanstolt

Göttingen

30

t-

I

Table I: Electrical Inp%its for Heating of Flat Plate

Datum: 8eorbeitet ¡ F Free Stream Velocity rn/s Approx. -Avg. Plate Temp. °C Groúp 1.

-Group 2 Group 3 Group 4

70 Volt 95 64 113 60 Amp 4.9 0.75 0.4 5.2 . i Volt Amp 130 6.65 871.03 175 0.6 83 7.1 10 150 Volt 1.0 106 204 105 Amp . 8 20 1.28 0.73 8.9 200 VoLt Amp 190 9 5 127

150

2420.85 127 10 5 250 Volt Amp 229 . 11.45 150 1.80 328 1.15 311 12.8 70 Volt Amp 124 6.'2 80 0.96 158 0.59 76 6.6 110 .. Volt Amp 168 8.5 108 . 1.3 223 0.8 .1078.8 20 150 Volt Amp 203 10.1 139 .1.6 270 0.95 127 10.9 200 Volt Amp 241 12 160 1.92 325 1.15 157 13.1 250 _AmpVolt 272 13.6 .184 2.2 401'1.4 282 15.1 70 Volt Amp 132 6.6 861.025 176 . 0.614 83 7.15 Volt 192 127 247 123 Amp. 9.7 . 1.54

091

10.5 30 150 Volt Amp 227 153 . -.

1.85.

314 1.17 155 12.43 200 . Volt . Amp 273 13.7 180 . 2.2 410 1.45 182 15.15 250 Volt Amp 310 15.7. 2062.45 460 1.60 420 17.3

Aerodynamisdie Versuthsanstalt Göttingen m(1

+1) - i

n Tabis A-1 Table A-U

.ma.

31 A-I; A-U 17/3 5 23/5 13/3

2/7

.4

35/9 19/5 22/3 13/2.

.6

17/3 38 f7 21/4 46/9 5 9.

.8

37/5 47/7 13/2. f 3.115 .31/4. 6819 37/5 9 79/9 43/5 io Oa um: Bearbeitet; Betidtts

97*26

2 3 4 5 1/2 1/3 1/4 1/5 2 5/3 6/4 1/5 1/2 3 11/4 13/5 5 13/3 4 19/5 13/2 17/3 .

3/4

5 8 7 28/4 31/5 19/2 5/.3 31/4 37/5 11 29/3 .9. 4315 25/2 11 41/4 49/5 14 37/3. 46/4 11 6 1/6 8/6 15/6. 22/6 29/6 . 6 43/6 50/6 57/6 64/6 7 1/7 9/7 17/7 25/7 33/7 41/7 7

57/7. 65/7

73/7 8 1/8 10/8 19/B 28/8 87/8

46/8: 55/8

8 73/8 82/8 9 1/9 11/9 21/9 31/9 41/9 51/9. . 81/9 71/9 9 91/9 10

i/io 12/lo 23/10 34/10 45/10 56/10 67/10 78/10 89/10

10 m(l + -m n

i

2 3 i 2 5 8 2 i 3 5 3 2/3 .7 / 3 4 4 1/2 2 7/2 5 2/5 9/5 16/5 6 1/3 5/3 3. 7 2/7 11/7 _20/1 8 1/4 3/2 11/4 9 2/9 13/9 24/9 10 1/5 7/5 13/5 25/2 58 / 5 14 13 11 37/3 74/7 83/7 41/4 23/2 10. 101/9 49/5 ..11 4 5 6

il

7 14 9 17

il

20 13 23 15 3213 37/3 19/2. ..11 44/5 51 / 5 .25/3. 29/3 8 65/7 26 29 17 19 13 _47/3

Ae ro d y n a m is ch e Versuchsanstalt Göttingen Bon-Nr. 32 Table. A-Ill Table A-III Datum Bearbeitet 7 JAbteitung Bsr$cI*:--. _67A26 i 2 3 4 5 6 7 8 9 10 - ,..-.- - - -

..--1

---1 3 7 11 15 19 23 27 . 31 35 39 2 3/2 4 13/2 9 - 23/2 14 33/2 19 43/2 24 3 1 3 - 5 7 9

ii

-. 13 15 17 19 4 3/4 10/4 17/4 6 31/4 38/4 45/4 13 59/4 66/4 5 3/5 11/5 19/5 2.7/5 7 43/5. 51/5 59/5 67/5 15 6 172 2 21/6 5 39/6 8 57/6 11 75/6. 84/6 7 3/7 13/7 23/7 33/7 43/7 53/7 9 73/7 . 83/7 93/7 8 3/8 14/8 25/8 36/8 47/8 _58/8 69/8 10 91/8 i02/8 9 1/3 15/9 27/9 39/9 51/9 7 75/9 87/9

ii

. 111/9 lo 3/10 16/10 29/10 42/10. -55/10 68/10 81/10 94/10 -107/10 12

Aerodynamische

Versuchsanstatt

Göttingen

Theoretical flow mOdel

IP1r.

Figure 1

Datum:

Aerodynom is che Versuchsanstalt

Göttingen

Low turbulence windtunnet

pi

34 II hWI'I iI:IiIi'III 1111:11 Datum: Bearbeitet: Abteihmg:. _{67 A 26.}

Steel Co ver 46 2 1S5 66 Heating Elements

'Y,

4:I

J-rFHFht

4 di irfi rh i r' i di i

rht4+4f+__t

Direktion of Air Flow

ø®®@®®®

-

373.

-

571---769

-967 . 1165 16 '68 1408

Static Pressure Stations

Aluminium Foil i1opper Plate 65 x 22 = 1430 Thermocouple Stations 265 1500 .

Section C-û

. No te: 411 dimensions are 1363

Positions of Boundary Layér Measuring

Stations Along the Plate

t rn:fl:

¡;r:-L

. -' H -' 'H "V'..

91q ,

pi

1ovi : '.' . ;& . h:

i-j va

w FJ#I_ Z#'#di7'

u__

, ;;. , 4 F 4 4 J 88 Insulation i: 7

M1:2

ir, millimeters G--Steel Plate s Plexiglas Window * versuchsanstalt I

Section 4-8

M1:1O; M 7:2

IDIatI-N,. 3.5 Aerodynaniische I

Sketch of hot plate. installation

in

wind-Göttingen

I

tunnel, and location of data stations

1Ftgure 3 Heating Elenien t .9 ' 17*370 22 C Volt; :200 Wa ft ODIum: 8.arbeUa: 1%t,.4s.i Abt.ulung: Gai,hen:

Boundary Layer Probe Wind Shield For Probe Section of Wind Tunnel showing /lotplate and Probe Insta lla'tion

Bericht:

67 A

26

4

f

j

Group Switch I Thermocouples 1-12

Copper Con stantan

11 13-24 Ice-Junction' Copper Thermocouple ¡n tw Chromel

Bounduy Layer Probe. :A,um.Il

tce-Junction

66 Thermocouples. in the !lotplate

25-36

I

I

i

I

I t

I.

!.

!,;

MalA Switch Constantan Copper Copper

IV'

37-48

Coppe, Amplifier s Copper

Y

49-60

Consta ntan

V

Thermocouple in the Nozzle of the Windtunnel

>SWitrh

Wgital Voltmeter

-c

C,

r

2 7 8 9 10 11 12 13 14 15 16 17 6roup i 72kW 1 C. Generator 6roup4. 70kW D.C. 6enerator

Group 2, Group 3,. Group 4;

20

0

u... al

1819.20

212223242526

_I J vJ L 6roup2l I Group3

.1

D.C. i I

i!oiiu-j

3Phase

Supply

Power per heating element

6roup 1;. For elements;

1'-9, 11, 14, 1?; Ph,E 12,5

VI.'1-For elements; 8,10, 12,13,15,16; PilE 12,5' 4 V2 12 NE V3h13 2 27

6364 6566

28

2930

Aerodynam ¡sthe Versuthsanstolt Göttingen Bdiidkry ayerprthQ : A

lrqO5

[J 402 V2A Thermocouple

Ids

Note: All dimensions are in millimeters I Datum: /Bearbeitet:

o-. - C 1,05 _UI UI,*vg 1,00

0'g5

4S0.

200

400

800

800

Plate Length (mml

1h00

i

w. m a

fi

-.3

-A.

w 1,000

29,58

xl,295

30,06 1/eSO 20,11

al,598

.9,71

1.

25O

T [°CJ

200

150

₁₀₀

o

o

500

o Aerodynamische Versuchsansta It Göttingen

po

All curves for 20 rn/sec

Test Section

Delum: earb&I.I

Typi ca L

I tate tern e rature (U stt'ibutioti

S

r

H

w_:1rflf,hI.I iehen: BIaIt.Nr. 41) Figure 8

1000

Plate Lngth [mml

1500 Brichi: (i? A 2G

Aerodynomsthe

Versuchsonstoh

Göttingen

Velocity profiles along the flat pla?e,

A 1.430, u1 20 rn/s 41 FigUre 9

u1120m/s (Approximate)

10

20

30 yfmmJ0

Datum: Bearbe;tet: 11 Abteilung: JGesehen: J Bericht: 67 A 26

Aerodynamische Versuchsanstolt

Göttingen

Temperature prafUel along the flat plate,

Aml.430, u12Qm/s

_{Figure. 10}

10

20

30 y[mm,

40

Datum:

Bearbeitet: 6.

Abteikrng 1GusIi.n; I Beridit:

I 67A26

42 BIaff.Nr.

1,0 Q UI

18

0,6

0,4

a2

10 y [thin]

15

Datum: Bearbeitet Ho. Abteflung: _Bericht: 67 A 2C V/V.

7;

r

x A 1,000 1,170 1,427 605

:736

::' o u

x4lßmm

-2Cm/s (Approximate) V

-î

Aerodynamische Versuchsanstalt Göttingen Velocity profiles at x 219 mm for various values of A

BIoI.Nr. 43

Aerodynom ische Versuchscjn stalt Göttingen 7,0

26

0,4

22

4 =1,000

t' ,168 u =1,2S4 =1,432 s, =7,55e 'i =7,763

u =2Cm/s (Appi"oximate)

x =1363mm

k

10

₂₀

_{30 y(rnm)}

₄₀

Datum: Bearbeitet:

Abteilung: _{Geiehen Betht:}

67 A 26 Velocity profiles at

various values

of A

x 1363 mm for

Blott-Nr. ₄₄

tw-t

th,t;

0,8

0,6

0,4

0,2

0

.5

_{10; yfmrnJ}

_/5

A o 1,170

o 1309

D 7,h27 1,608 1,736 u, =2Orn/s !Approximate) z -2 19mm II Datum: Bearbeitet: #4. Ableilung: 67 A 26 Aerodynamische Versuchsanstolt Göttingen Temperature profiles at

for various vatues of A

x 219 mm

BIott.Nr. ₄₅

i;

a8

o,'

0,2 Datum: Bearbeitet: 'SS Abt&lun: _Gesehen: 1

30 y(mm)

0

47 A 26

tw-tt

o

t,-t2

o C .

r'

A-1,168

"-1,294

1432

D-I,JJU

a*7,753

u -20 rn/s (Approximate)

x -7353mm

aa

--Aerodynamische Versuchsanstolt Göttingen Temperature profilès at

for various vaLues of A z - 1363 mm

Baft-Nr. ₄₆

Aerodynomische Versuchsanstah Göttingen 1,2

trt

twt2

.1,0

û,b

0,2

Datum: Bearbeitet: Ho.

t

- t)/(t

t2) vs. u/u1 along the

fiat plate

0,2

Abteilung: .0,4 Gesehen:

4A

Blatt.Nr. ₄₇ Figure 15 0,8 Bericht: 67 A 26 1,0 x[rnm] o ₂₁₉ ., A ₃₇₃ ° ₅₇₁ . .

;/

u1 =20 rn/s (A pproxima te)

:

:;

A =7,430 0 1165

'1363

..

I,.

41'

I.,..

0,6

O,'

0,2

Datum:

Bearbeitet: H0.

Abteilung: _{Gesehen:' Bericht;}

67 A 26 1,0

AI

o _1,170

AÎ,305

' . . u1 =2Cm/s

x 219mm

'.

. ' , . (Appròximate)

"

.

I

7

6Cê

1736

/

Ar

u Aerodynomisdie

Versuchsanstalt (t.

- t)/(t

. t'a). s'è. u/u1: at

Slatt.Nr. ₄₈

Göttingen x

219mm for various of A

_{Figure 16}

t'ct

tWt?

Aerodynomische Versuchsanstalt Göttingen

(-tW442).

X 13e3j 1,0 0,8 0,6 0, 0,2 rte)

tw-.t

twt2

f

- lv--J, A-1.170

Jfl9

N ₁₅₉₉ -1,763 -'

.

a u1 - 20m/s (4

proxii

- 1363mm

'1$. u/U1 at BIotsNr 49

étouaof A

Figur 17

02

a'

06

1,0

Datum: Abteilung: Gush.n: I Bericht:

[mml

5

2

i

7

200

b 110

600

800

Pia te Length

[mm]

1h00 mm

f

-' -' n O 0* C Q.. _ca O O 2-9- 01 o

Al

_1,000 0 163 (Average) 'J s' i' L u1 = 1Cm/s (A pproxima te)

ó1297

o1452

a-.

'C

m

î

o

r

j

I

i

7

6

6*

¡mml

b

3

2

i

200

400

600

800

Plate Length [mm]

1400 mm o, C O.. _{ca O} Q' A WOO (Average) u 's o7,167

a 29?

u1=2Orn/s (Approximate) O 1,430 1F4j599

743

[mm]

0

200

400

600

800

i/ate Length ¡mml

1h00

*

?

WOO (Average)

:

:

o 1,1.44

u1-30/ms (Approximate)

.

--.1 b.)

3

_o

[mm)

0

200

400

600

800

Plate L engt/i [mm]

1400 (Dm Os C (D o

A:

?

î

A

o )163

-X 1,000 fAveragefl 11 I> u,=ltlrn/s -4 1.292

°1,452

(Approximate)

4-'

4--

f594 1,774

_- 4-A

-8

-4-a

.4 4- '*4- - 4-U - 4-1

--4 -4

A1 BIoø-Nr. 54 Figure 22 CO Datum: Bearbeitet: Ho / Abteilung: Bericht: _{67 A 26}

\

\

\

\

\

\

\

\

\

\

\

\\

\

\

\.,\

_\.\

\

'-a

____A___

Q, a . Ci

ø s

,c o Aerodynomisdie Versuchsanstolt Göttingen :

e a'ong the flat piate

for various u1 3O ui/e

3

_a

fmrn]

O

200

bOO

600

800

Plate Length [mm]

1h00 mm ,% -' u O Ot C O (O O

I

À 11000

ol,gQ

(Average) a

Al298

7444

u,30rn/s (Approximate)

Z60b

1.743

Aerodynom ¡sthe

Versuchsanstolt

Göttingen

along the flat plate for various A,

.U1.á _IO:in/s 56

rè

24

°

Datum: Bearbeitet: Ho. Abteilu.,ng: Beridit: 67 A 2Í

\

\

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\

\

\.

\

H

\

\

\

Q)

v

\QSOø

\

\

A

\040*

s O X O

6

[mm]

o

200

400

600

800

Plate Length

[mm]

1h00

w

c»i A WOO (Average)

v

s A ¿297 1,599

.7743

01430

u1 =2Cm/s i'ApproÑnate)

-e

-0

Aerodynomisthe

Versuchsonstalt

Göttingen

along the flat plate for variou8 A,..

30 rn/s Datum: Bearbeitet: Ho. Abteilung: Geiehen: JBericht: 67A26

XOD4

\

- 't.-.

\

\

\

)COØ4\ 't 't 't 't . Q)

\

't

't.

oras 't 't. 't

'

't'.

_____k

II '.4. 4 0 ".4v .

ß

-X

o. C

3

_of

(mm]

A o 1,163 (Average)

a 1,292

ii

o1,452

Il . 1,594 1,7?4 's u1 = 1cm/s (Approximae)

200

400

600

800

Plate Length

frnmj

1400

200

400

600

800

Plate Length ¡mm]

1h00

'1 I-. o w r -' -'

. v

O O: C Q _ca (D 3 !4

09

A' o Z167 (Average) u u s' u

61297

o 1,t30 u, =2Orn/s fApproximateJ 1,599

ol.?42

A

.1

0

200

400

600

800

Plate Length[mmJ

1h00

-I In o O: C Q J1 L

ci

A

o 1,1O (Average)

'S « " I

61295

° Zb114 1,6O

u1-3Orn/s(Approxirnate)

.1S743.

3

[mm]

2

A o 1,163 (Average)

ó1292

o 1,t52

v.1,594.

n

fi,774

n n u =1Cm/s (Approximate)

200

¿00

600

800

Plate Length /mmJ

1 00

mm -I UI O: C O... CO O UI OE . .

Nr

_:0E

1

C

î

LI1

[mm]

2

3

0

200

400

600

800

Plate Length [mm]

11.00 p-. A o

1,167 (Average)

A 1.297

°

1430 u1=2Orn/s (Approximate) 1,599 7743 t :

-3

[mini

0

200

400

600

800

Plate Length [mm]

1400 o

g:

1

.1

A _{1,140 (Average)} 'i 444 n A 1,298

.1,743

o o u, =30 rn/s (A pproxirn ate)

dld,hhhhh1!hh1PP

J 1000

Plate length (mm) 1500

i

Ci 0) V.., Ci

2

(mm

2,0

t

u.

I 10mb I

.1

(Approzimatej

®4-U63

«-z2

7,fl4

(Averagel.

1J

¡s

s

..

io.

wo g-a m Q. C o) C) 4.

500

1000 Plate length

t'iim)

1500 112 (ll7ffl) I u1 - 2Cm/s I (Approximate) I 2,0 A

-Z167

a.iZ743.

:,.::WF

(Average)

10:.

.

-4

500

1000

Plate length (mm)

1500

f

2

m'nJ

=130,J/s MppJoximatej

2,0

4

A

-1,140

-U98

iI-7,444

"-i604

'"-1,743

AverageJ u ..H 8 o. . o o . o o C,

Aerodynom ische Versuchsanstalt

Göttingen

along the flat plate.fò various. A,

u1 10 mfs. 1Bl.ft.Nr;. 68

iFtu.

36

I

Datum: /Bearbeitet: . y ., ç 1

.

u G

De.

"i

Abteilung: Geiehen: Bericht:

o g-3 s

00

1000

Plate length (rom)

1500

<>

o

o Q: C Q.. D :.

ç

¿13 (mm)

2,0

- 20m/s (Approximate)

ou

A 1167 1,297 1,430 1,599 1,743 (Average) u 's

.

lo

T.

T

G)

I

g' 9. o -J

a-. - C g-3 C) -J C)

500

1000

Plate length (mm) 1500

C3 -C) o

A3

(rn/n)

u1 - 30m('s (Ptrimate)

-.

o A

1140 2g8 u 7,604

.«1.743

(Average) .

1,0

U

C

zo.

i; 1.

7cf

o

Experimental point at r

=

1363 mm,

u1 - 10 rn/s (Approximate)

g,

20

D ,, u n

30"

u

io

I i Theory Ref. (7)

15

20

r

u,

_LW.

_a7

1,0

1.5

20

-

Theory Rei (7) Experimental point at z

-1363mm, u1

-lOm/s (Approximate) A

tir

t2

o ap O

ri

g, g, N 20 N I"

I

30

'i

u

8

G a _I,

wo E

¿0-

t9

0,7-¡/6

io

o =5,35 n =5,63 = 45 = 5,75 =5,5 =5,67 _5,03 = 5,59 =5,6 =5.9' flg 7 =5

Theory Ref. (7) Experimental point at

x =1363mm, u1 10 rn/s (Appioximate)

I

20N

11 I

I

¿5

20

mm

r

-i In o 01 c a. (D o,' o,' o w

j.

C13

w

1,0

a

Theory Ret (7)

Experimental point at

II 'I I

15

IS N

30,'

i'

n A

20

5.--9

t2

am IjJ (n _{,O:C Q} a ca a

In .

o

'I

11' u = 1363 mm, u = 10 rn/s Approxi mate) ii II

2011

I 'I

t

_('z

Datum: Bearbeitet: vS Abteilung: Gesehen: JSericht: 67 A 26

Aerodynom ¡sche ßlo$i.Nr. 75.

Versuchsanstolt

Göttingen

H/ri-o va. coined with theory _Figure

- C o. !!. ; C 3 o G 3.

1,5

1,0

¿u

e -' Yb O O: C O.. _{(O O} O Yb Og_ (D o z Theory

Experimental

Ret (7/

point at

g, I x = 1363mm) g. ,, u1 = iO rn/s "

20

3Ø ' _t. (Approximate) 1 II fl=5 L -536 i=5,39

n5,33

= =5,03 67

-H=58 50 :5,35 I

L

-H 545 =5,63 iM

=

L2 1.0

Datum: Bearbeitet:

Abteilung: Ge.ehen: _I Beritht:

67 A 26

Aerodyna mische BIo$t-Nr. 77

Versuchsonstalt H/IT vs. A compared with theory

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