RESEARCH MEMORANDUM - NASA · the corrections for the pressure gradient due to the wake are given with the correct compressibility factors. All the final correction formulas together.

Copy No. 78

RM No. A7B28

- ____

RESEARCH MEMORANDUM BLOCKAGE CORRECTIONS FOR ThREE-DIMENSIONAL-FLOW CLOSED-

ThROAT WIND TUNNELS, WITh CCNS]DERATION

OF ThE EFFECT OF COMPRESSIBILITY

By John G. Herriot

Ames Aeronautical Laboratory Moffett Field, Calif.

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NATIONAL ADViSORY COMMITTEE - FOR AERONAUTICS

WASHINGTON

July 17, 1947

https://ntrs.nasa.gov/search.jsp?R=19930093811 2020-04-05T20:06:28+00:00Z

NACARMNO. A7B28.,

NATIONPJJ ADVISORY COMMITTEE FOR AERONAUTICS

RESEARCH MEMOPAI'fl)UM

BLOCKAGE CORRECTIONS FOR TEREE—DINENSIONAL.-FLOW CLOSED-

TBPOAT WIND' TUNNELS, WITH CONSIDERATION

OF THE EFEECT OF COMPRESSIBILITY

By John G. 'Herriot

SL1NMARY

Theoretical blockage corrections are presented for a body of revolution and for a three—dimensional unswept wing in a circular or rectangular wind tunnel. The theory takes account of the effects of the , wake and. of the compressibility of the fluid, and is based on. the assumption that the dimensions of the model are small in comparison with those of the tunnel throat. Formulas are given for correcting a number of the quantiticé, such as dynamic pressure and Mach number, measured in wind—tunneltests. The report presents'a summary and. unification of the existing literature on the subject.

INODUCTION

• When a model is placed. in a closed—throat wind tunnel there is an' effective constriction or blockage of the flow at the throat of the tunnel. The effect of this blockage is to increase the velocity of the fluid. flowing past the model; if the model is not to,o large relative to the tunnel throat, this velocity increment is a pproxi--mately the same at all points of the model so that th model is' effectively working in a uniform stream of , fluid the velocity of which is, however, 'eater than the .free-steam velocity observed. at some distance upstream of the model. It is therefore necessary to correct the observed velocity, dynamic pressure, Mach pimiber, and other measured quantities for the etfect 'of this contriction. This correction is frequently called. the correction for "solid blockage." In addition to thIs solid—blockage correction, a cor rection for "wage blockage" is also necessary if the true dynamic pressure and Mach number at the mode.l are to be determined.. This wake blockage arises because the fluid is slowed down in the wake and. consequently must be speeded up outside the wake. A further effect of the wake is to 'produce a pressure gradient which must be

2 _____ NACA RM No. A7B28

considered in correcting the drag coefficient.

Formulas for the solid-blockage correction for a model mounted in a two.-dimensional-flow wind, tunnel are given in references 1, 2, 3, and l i. . The first order effects of the compressibility of the fluid on these correctiOns are given in reference 5, as well as in references 3 and 14. References2, 3, á.nd 14.. consider also the wake-blockage correction, and reference 3 also considers the drag cor-rection due to the pressure gradient caused by the wake. The formulas given in reference 3 f Or the solid- and, wake-blockage cor-rections will usually be found most convenient whenever the engineer is confronted by a practical problem of determining the corrections for any configuration met in his experimental work.

The solid-blockage correction for a model mounted in a three-dimensional-flow wind tuniel has been given in a number of different forms by different authors. Not oily do different authors give the correction for the same configuration in different forms but no one author gives formulas which are applicable to both fuselages and wings in tunnels of' various shapes; for thisreason, the engineer confronted with a correction problem may have to refer to several reports to get the complete solution of his problem. Moreover the modifications of the formulas for the first order effects of fluid compressibility are given incorrectly In some cases. References 1 and 2 give a formula for the solid-blockage correction for a body of revolution in a circular or rectangular tunnel for the case of incomprdssible flow. In references 5 and. 6 the effect of the com-preesibility of the fluid on this correction is discussed, but the result given is ncorrect Reference ii- gives a forniu]a for the solid-blockage correction for a body of revolution and for a three-dimensional wing in the 7-. by 10-foot wind tunnel. The modification for compressibility is correct for the wing but' wrong f or the body of revolution. Reference 7 gives a formula for the solid-blockage correction for a body of revolution in a circular tunnel, correctly taking account of the effect of the comDrdssibility of the fluid. eferences 8, 9, and 10 give a formula for the solid-blockage cor-rection for any body in a circular wind tunnel together with the appropriate constants for a body of revolution and for a rectarg- - lar wing having various span-to_diameter ratios; the mOdification of this formula to take account of compressibility is correctly giver.

It Is clear that no one report gives all the necessary for-. mulas together with the appropriate constants and compressibility modifications to enable the engineer to calu1ate the solid--blockage correction for any case with which he may be confronted. Moreover, when the reeu1ts;of two or more reports overlap, the forms are frequently different so that it is not obvious whether the

NACA RMNo. A7B28 3

'results are In agreement. It is the purpose 'of the present report to summarize arid, extend theresu1ts of thepreviously mentioned reports. Formulas are given for the calculation of the solid-blockage correction for a body of revolution or a three-dimensional unswept wing in a ircular or rectangular tunnel. These foriflu1a' contain two constants, one depending on. the shape of the body and the other on the shape of the tunnel and the ratio of wing span to tunnel breadth. (This ratio may be taken to be zero for a body of revolution.) Values of the first constant for various bodies of revolution and for a number of frequently encountered wing-profile sections are given. Valuo of the second constant for a circular tunnel and for rectangular tunnels of a. number of coñmionly encoun-tered bx'ead.th-to--height ratios are given for variOus wing-span-to--tunnel-breadth ratios. Some of these values have been taken from the previously mentioned reports; whereas others appear for the first time in the present report. The discussion is limited to bodies centrally located in the wind tunnel.

The wake-blockage crrection for a model mounted ma three-dimensional-flow tunnel is given in references , 8, 9, and 10. For the case of incompressible flow the formulas.' are in agroethent, but the modification to .take account of compressibility given in reference 1 differs from that given in references 8, 9, and 10. This matter is diCcuseed in the present repor±. In reference '11 the corrections for the pressure gradient due to the wake are given with the correct compressibility factors.

All the final correction formulas together. with directions for their use are given in the final section ontitld "Concluding Remarks.' t Mathematical symbols are defined asintrod.uced in the text. For reference, a list of the more important symbols and their definitions is given in appendix B.

SOLID BLOCKAGE IN :INCO p SSIBI$ FLOW

In studying the flow over a thin airfoil of ámall camber at a small angle of attack it has been shown that the effects of camber and thickness may be considered independently. In treating the problem of wall interference, it is again convenient to consider the thickness and camber effects separately. ' The camber effect, as pointed out in reference 3, contribtes nothing to the blockage correction. Consequently 'it si,ff1ces to determine the blockage correction for synmetrica1 bodies at zer p 'angle of attack. This means that for wings it is necessry to consider only th base pro-file of the airfoil, the base profile being defined as the profile the airfoil would have if the camber were removed and the resulting syetrical airfoil placed at zero angle of attack; bodies of

NACA PM No. A7B28

revolution need; be considered. only at zero angle of attack.

.Rôctangular Tunnel

Three—dimensional_wing.— The blockage correction for a three—dimensional wing in a rectangular tunnel is considered in refer-ence ii. Numerical values are given ox4y 'or a wing of 6—foot span in a 7— by 10—foot wind, tunnel. The method may; however, be applied to any rectangular tunnel and. any span—to—breadth ratio.

Consider a rectangular tunnel of height H and. breadth B. Suppose that the wing span 2s is.iii the B—direction so that 2s/ is the span—to—breadth ratio. As in reference 1, let the wing be represented by a series of finite Unes of ources and sinks. (See fig. 1.) The strengths of these sources and sInks are assumed to depend on the airfoil profile and the determination of these strengths, which is a two—dimensional problem, is explained. presently. If each line retains.the same strength from one end to the other, the wing section cannot be exactly constant. It will thin down at the extreme tip and the plan form will not be exactly rectangular, but neither of. these features Is such as to detract from its usefulness for the present purpose; which Is to represent an actual wing sufficiently well to enable a calculation to be made of the velocity along the tunnel axis "induced" by images of the wing. It is this velocity induced by the images which represents the effect of the tunnel walls on the velocity at the modl, and which is the solid-blocka€e correction.

Consider the image line source CD of strength Q pex unit length. The velocity potential of a three--dimensional source of strength Q5y (volume per unit time) is —Q.5y/l'itr. it follows that the component velocity along the tunnel center line induced at A by the source element .QEy is

811A = 6y

where g and r are defined in figure 1. Putting 2 2 22 2

r = (mB—.y) + n + g and Integrating from -s to s give

- Qg. . mB+s 1UA t1((n2H2+) . /n2H2+2+(mB_s)2 J.

NACA PM No. A7B28

5

for the single line ource CD. Now CD is one Image of one finite line source used. to represent the wing in the tunnel. In order to find the velocity induced by all the. images of this particular line source, it is necessary to add the results obtaThed from equation (1) by giving m and n all positive and negative Integral values ,except m n = 0. Thus for a single line source,

.i r + . ' - 1(2) An2H2+g2 L/n2H2++(+s)2 J

where the prime denotes that the term m = n = 0 Ic to be omitted from the suation. Equation (2) may be rewritten

UA94 2it113 \, H' B H

where

1 r m+s/B H 2s n2+(g/H)2[y2(/)2(/)2 (B/H)2

- rn—s/B 1

n2+(g/H)2+(s/B} (B'/H) 2 J (3)

It Is convenient to define

s B"_l(B" 3/2 (g s B . V

'Ji' ' )YL' \' ,

so . thatV

UA = Qs - l/2 ¶I".g ! (Bu) 3/2 \\E' B' H)

Now and T depend only very slightly on g/E and so it usually suI'fices to take g/H = 0 in the evaluation of these quantities. Any wing profile can be reprsentod by a suitable distribution of sources and sinks along the chord. It follows that the total induced velocity due to the wing images is obtained. y sunning over this distribution and is approximately V /

6 NACA PM No. A7B28

=/2 1/2

<, .,(5)

It nay be noted here that setting g/H = 0 in the evaluation of a and. T is equivalent to representing the wing by a line doublet of strength EQg (analogous to references 1 and 2) instead of 'by a distribution of sources and. sinks.

The quantity EQg of equation () can be dotorrnined'approxl-mately from the wing profile and this determination is a two-diensiona1 problem. From reference 2, but with the notation' of reference 3, there Is obtained.

EQg = Ac2U'

(6)

where

c airfoil chord

U' apparent 'freestream. velocity at airfoil as dothrmined from easureent's" taken at. a point far ahead. of model

a factor dependent on shape of base profile

Substitution of equatiOn.(6) , into equation (5) yields

LU' = AT O ! . (BH) 3/2 16 ' B' E

(7) c 3/2 16 t/c \. B H,

where

t maximum thickness of airfoil,'

C cross—sectional area of tunnel

If V denotes the volume of the wing, then V = 2sctc 1 where , /

1 depends. on the shape of the base profile. Equation (7) may be rewritten '

NACA Rtr No. A7D8

7

____ - K1,-V(8) UIC3/2

or

____-

1C2i-2sct(9) U' C3/2

where

T = T(O, -

(10)

___ 1 11 1167

(12) 16 tIc

It is clear that'r depend.s only on the tunnel shape and. the win —span—ttmme1_breadth ratio; whereas K, K2 depend only on the shape of the base profile. -

The factor A can be detei!ned for any base profile from the relation (references 2 and 3)

= i f (13)

where

ordinate of base profile at chordwise station x:

dyt/d.x slope of surface of base profile at x

P base—rofne pressum coefficient at x in an incompressible flow

Values of A for a number of base profiles are given in reference 3. Thus the value of K2 can be calculated from equation (l2) The value of is iediate1y found.from the area of the base profile which ay be calculated, for example, by a numerical integration from the

.xt (1 ' .J_'+_,

8 NACA RN No. A7B28

ordinates of the base profile. As soon as is known, K1 can then be calculatpd from equation (i). The evaluation of T

requires the summation of the infinite series in equation (3). The summation of this series is explained in appendix A.

Values of r for rectangular tunnels of various breadth-to .-height ratios and for various wing.-span--to-tunnel-breadth ratios

are given in table I and figure 2. Values of K1 and. -K2 f or various base profiles are given in table.s II end III. The?chcice between the two formulas (8)- and (9) is entirely a matter of con ..

-venience and. should be decided in the light of the available data.

Body of revolution.- The blockage correction for a body of revolution in a rectangular tunnel is considered in references 1, 2, and. ii. In reference 14 numerical values are given for some average streamline body of revolution in a 7- by 10-foot wind tunnel; whereas in references 1 and 2 -numerical values are given for prolate spheroids and Rankine Ovolds in square and duplex (breadth equal to twice the height) wind tunnels. Either the method of reference 1i in which the body of revolution is represented b a suitable dis-tribution of sources and sinks along its chord or the method, of references land 2 in which the body is represented by a.doublet of suitable strength at its center may be used t2 obtain results for any streamline body of revolution in any rectangular tunnel. Since the method of reference was used for the three-dimensional wing, it is instructive to use the doublet method of references 1 and. 2 for the body of revolution case, although both methods give the same results.

Conaider a body of revolution of max:Lmum thickness t and. length c centrally located. in a rectangular tunnel of breadth B and. height H. As in reference. 2 the body may be represented, by a doublet of strength i given by the equation

where X is a constant depending only on the shape and fineness ratio of the body. The velocity induced at the model by the tunnel walls is the same as that induced br a doubly infinito array of imags of the doublet and is given by -

1U' = 1 (15) #1t L 2ff2+N2.B2)3/2

NACA RM No A7B28

9

the summation being taken over all positive and negative integral ia1ues of m and. n except vi n = 0. If g/H = 0 equation (3) may be rewritten .

a B B B i [(m^s)F - (s)E

- \ B 111 28 L n2 L -

((+s)F+(n-s/fl)E](16)

where the quantitIes E and. F, which are introduced, for con-venience, are defined, by the equations V

= /fl2+(vi+/3) 2 (B/H)2

mn = /fl2+($IB)2 (B/n)2

It follows that

5/

B 1(17)

- .

H) L [n2+ m2(B/H)2].3/2

Substitution of equation (17.) into equation (15) yields

= a0, 0, , H; S

If j.i is replaced. by its value froni equation ( 1I ) and. equation (Ii.) Is used., there is obtained.

= 3/2T(O, B " ct2 Xt T(o 0,

U' 8 ' (BH) 3/2 ' L' C312 8

NACA PM No. A7B28

If V denotes the volume of the body of revolut4.on, then V K2ct2 where K2 ^ depends on the shape of the meridian section of the body. (K2 = for a right circular cylinder whOse meridian section is . clearly a rectangle.) Equation (18) may be rewritten

____ - .KrV

U C/2

or -

z 1U' Kict2 U I C/2

where

T T (o, 0, (21)

K (22) 3. 8 c2

K4 = 3/2(23)

8 c

It should be noted that T for the case of the body of rev-lution is the saie as for the lImiting case of a wIng when the span approaches zero. As pointed-out n reference 2, X may be calculated for any body of revolution whose pressure distribution is known. The necessary formula is

= ()3f1 -) 2 l—P l+(dy/ax)2d() -- (2k)

where

y radius of body at chordwise station x

dy/d.x slope of meridian section at x

10

(19)

NACA PM No. A7B28 11

P pressure coefficient at x in incompressible flow

Values of X. for prolate epheroids and. Rankine Ovoids are given in references 1 and 2. As soon as X is Imown for a body K 4 can be calculated at once from e .uation (23).. The value of K2 marbe found. from the volume of the body of revolution which may be calculated f or exairple by a numerical integration, and. K 3 can. be calculated.from equation (22) as soon as tc2 is known.

Values of T for rectangular tunnels of various breadth..-to---height ratios are given in table I and. figure 2. Values of K3 and K for various bodies of revolution are given in tables IV and. V. Some of these bodies are drawn in figure 3. Unfortunately these tables are rather incomplete and. moreover fuselage shapes used in practice are extremely varied. However, in table IV it is observed. that the values of K3 do not depend very strongly on the shape of the body, although they do depend. on the thickness ratio. For this reason it appears that for most fuselages it will be sufficiently accurate to use the values of K 3 given for the NACA ill bodies. As the values of K 4 given in table V are more dependent on the body shape it is recommended that equation (19) and table IV be used in preference to equation (20) and table V whenever possible.

Circular. Tunnel.

Body of revolution.— It is convenient to consider the case of a body of revolution before considering the case of the three--d.imensional wing because more attention has been given to the former by other authors. (See references 1, 2,. 7 8, 9,. and 10.) It will now be shown that the blockage correction is again given by equations (19) and (20) where T has a value appropriate to a cir-cular tunnel. Since K and K 4 depend. on the model and not on the tunnel, they are still given by equations (22) and (23).

The most convenient starting point is the formula of. references 1 and 2, namely,

___ - (\\ 3/2 - .t .T\) (25)

where 5m Is the maximum cross—ectiona1 area of the model. It is only necessary to note, that

12

NACA RN No. A7:B28

Sm ,tt2/ll.

V=2ct2

It follows at once that -

g3/2 t° 3/2 ct2(26)

8 72r 8 C372

Substitution from equations (22) and (23) reduces this to equations (19) and (20), respectively. It should be noted that , of references 1 and 2 is identical' with , T, of the present report.

The value of T is given in table I and figure 2. Values of K3 and K4 are given in tables IV and V. Again equation (19) is preferable to equation (20).

The result of refereilce 7 fails to 'take into consideration the body shape and so it 'is not very useful except. for' less exact cal--culations. The result of references 8, 9, and. 10 is presented in a different form, namely,,

___ V ____ =

XIJT - . 27

where . - .

'factor depending on model shape .

TV factor depending on tunnel shape

V volunie of model '

B diameter of wind tunnel -

It is of interest to show that equations (19) and (20) can be deduced also from equation (27) showing the latterto be equivalent to equation (25). From reference 8, there is obtained

3

TV =T . -

NACA RM No. A7B28 13

Sine also C = tB2/i. equation (27) yields at once

____ - (ir t3 '\ ( (3/2- T

3/2 ct2 t )k ,) 8 c°/2,) :8 c3/2

This is the same as equation (26) which yields equations (19) and (20) directly.

Three—dimensional wing.— The blockage correction for a three—dimensional wing in a circular tunnel 18 given in references 8, 9, and. 10, the formula being of the same form as for a body of revo-' lution in a circular tunnel, namely, equatIon (27) where and v iave values approDriate to the threô—dimensional wing. Fom

refei'ence 8 (using the notation of reference 3 instead of refer-.-ence 2) there Is obtained

(28)

T_TS (29)

Since also C = itB2/li. equation (27) yields at once

L1Tj' (it 2sc2 T"\v(it/ 1 2sct A ,t3/2 U t \8 IT ,,\it' j \ 8 C 3/21 C3/2t/ 16

This is the same as equation (7) which leads directly to equations (8) and (9) with the same definitions of K 1 and K2 given in equations (ii) and (12); r is, however, given by equation (29) where TV is obtained, from referencea 8, 9, or 10.

Thus equations (8) and (9) may also be used for circular tunnels provided only that the appropriate values of r are used. Values of r are given in table I and 'igure 2. Values of K1 and. K2 are given in tables II and III.

l ii. NACA tM No. A7B28

SOLID BLOCKACE IN .COEESSIBLE FLOW

In the nreceding section the solid-blockage corrections have been 'determined under the assumpti'n that the fluid is incompressi-ble. It is now necessary to determine the inodificatDns required in these formulas to take account of the effects of th compressi-bility of the fluid. The methods of references 12'and 13 are very convenient for this urosè. As the required modifications are given incorrectly in references 5 and 6 partially incorrectiy in reference 1 , and correctly in reference s T. 8, 9 and. 10, it appears worth while to ivo some discussion of the matter.

For the purpose of deducing, the oroperties of a comDressible flow from those of a coesponding incompressible flow the s.c-called ttExtnsion of the Prandtl Rule which was first given in reference 12 and. repeated as Method IV in reference 13, is probably of most general application; but the other methodC of reference 13 are sometimes more convenient for certain problems. The Extension of the Prandtl Rule may he expressed in . the following manner:

The streamline pattern of a compressible flow to be calculated can be compared with the streamline pattern of an incompressible flow which results from the contraction of the y- and. zaxes includ-ing the profile contour by the factor (M = free.-streeia

Mach number) (x-axis in the direction of. the free stream). In th compressible flow the pressure coefficient as well as the increase in the longitudinal velocity are eator in the rati9l/(1-M) and the streamline slopes grater in the ratio l/,Jl_M2 than those at the corresponding points of the equivaleit incompressible flow.

Since formulas (8) and (19) for the three-dimensional wing and the body of revolution have the same form and also apply to both rectangular and circular tunnels, it SuffiCeS t.o determine the modification due to compressibility for them. Let the subse'ipt c refer to compressible flow and. the so.bscri.pt 1. to the correspond-lag incompressible flow. If V is the volume of the model in the compressible flow, then Vj, the volume of the model in the corresponding incomoressi-ble flow : is gl.ven by

= [l.-(M') 2 Vc

where Mt is the apparent free-stream Mach number at the model as determined from measurements taken at a poInt far ahead of the

NACA EMNo, A7B28

15

model. Also if C is the cross—sectional area of the tunnel in the compressible flow, then C 1 the cross—sectional area of the tunnel in the incompressible flow is given by

C1 = [1—(M') 2] Cc

T is unaffected by the transformation and. the effect on K 1 and. K3 is sufficiently smafl that it may be neglected. From the Extens ion of the Prandtl Rule it follows that

( LiU !) = 1 (1UT 1 K-V1 - 1 KjTVc U c l_(Mt) 2 u' ! - 1_(M') 2 c . 312 {1_.(M1)2)3/2 c3/2

where j denotes the numbers 1 or 3 . Since equations (8) and (9) are equivalent as are also equations (19) and (20), it follows that in all cases it is only necessary to multiply the blockage coriec-tions given by these formulas by fl—(M') 2j -3/2 in order to take account of the compressibility of the fluid.

As a check it is useful to determine the compressibility mod---Ification for the case of a body of revolution in a circular tunnel by Method II of rCference 13, since the derivation is so simple by thIs method. Both the body shape and the longitudinal velocities are the same in the corresponding compressible and incompressible flows; only the tunnel dimensions are altered by the factor ..Jl—(M') ' so that Cj = [1..(Mt 2 j Cc. There is obtained. -

= = KTV 1 K3Wc \. u 1 c 3/ tl_(w) 2 1 3/2 c0 3/2

as before.

WAI BLOCKAGE IN COMPRESSIBLE FLOW

The blockage due to the wake of a model in a tw-dimensional_ flow tunnel is discussed in some detail in reference 3 . Much of

16 NACA RN No. A7B28

this discussion is applicable without change to the case of a three-. d1mensiona—f1ow tunnel. The fundamental idea of replacing the model and Its wake by a source (in this case a three—dimensional source) of suitable strength iocated at the position of the model can be used again and in fact the determination of the source strength Q can be carried out in exactly the same ñianner. The result is Identical with that of reference 3, namely,

Q P'U'CD'S[ + (y_1)(M1)21 (30) 2

where

Q. mass flow of source rather than volume flow as used previously

p' mass denIty of fluid .at point far upstream

CDT uncorrected drag coefficient referred to apparent dynamic pressure q'

S area on which drag coefficient is based

y ratio of specific heat of gas at constant pressure to specifi.c (C heat at constan volumecv...

In equation (30) powers of Mt higher than (N') 2 have been neglected.

Consider now arectanXlar tunnel of height H and breadth B. The tunnel walls are replaced by a doubly infinIte array of. sources of. strength Q at distances n to the side and n.H above and below the position of the model..

By making use of method II. ofreforence 13, It is readily shown that a three—dimensional . source of strength Q (mass pex' unit time) in a uniform flow of compressible fluid will induce at the oInt the coordinates of which are x, y, z relaIvo to the source a stream..-wise velocity .

x 1ltp [x2 + ( 1-M2)(y2.tz2))3/2

NACA RM No. A7B28 17

where the uniform flow is in the x—direction and. p and. M are the density and. Mach number, respectively, of the undisturbed stream.

It follows that the streamwiso velocity Ut induced at a point onthe centerline of the tuflnel by the entire system of images is

I =

x2+fl_(W)2] ( + 2)}3/2 14.7tp

where p t and MT are the density and Mach number of the undis-turbed flow in the tunnel and the stunmatiôn Is taken over all positive and. negative inteal values of m and n except m = n = 0. The velocity induced by the image sources at an infi-nite distance upstream is

( 2u' )_ = + {l— ( ) 2 ] (m2+n2H2) }3/2

But conditions far upstream must remain unchanged and to achieve this it is necessary to counterbalance this velocity by super-position of a uniform flow of equal magnitude but opposite sign. The addition of this flow at all points in the field will result in a speeding up of the general flow at the position of the airfoil by the amount

x 3U =lim

I tp' (xS+{l_(Mt)2] (m2B2+n2H2)}3/2(31)

The suimation of this series caii be obtained by the'1ollowing artifice. Substitution of equation (30) into equation (31) and setting M' 0 gives

I Cjy'SUt X (32) ( 3u) = lim i (x2+ni2B2+n2H2) 3/2

NACA RM No. A7B28

But according to references ), 8, 9, and 10

(J' ' - 1 CD'S (33) u' -4 - BE

Comparison of' equations (32) and. (33) shows t1at

urn ' -___________ = 2 (4) x L (x?+m2B2+n2H2 )S/2 BE

If in equation (3 1i.) B apd H are replaced by bA7j1T3 B and

H, respectively, there is obtaiied

urn'

= 25Y

X- X2+l_(Mt)2}(m2B2+fl2H2)}3/2 [L_(M?)2JBH

Substitution of equations (35) and (30) into equation (31) yIelds

U' ^(y_l)(Mt)2 CD' S - 1--o.(M') 2 CD'S (36) - l—(M')2 4BH - T(M T )2 I1C

on setting ' = l.1. The preceding discussion is for a rectangular tunnel, but In references 8, 9, and. 10 the same formula is given for a tunnel of apy shape for the case M' = 0. Consequently, equation (36) may be taken to hold • Senerally for a tunnel of anf shape.

In reference the wake—blockage correction is given correctly for incompressible flow but the modification to take account of compressibility is given incor'e6t1y, as it is determined from ref-erence 5which,ae pointed out ir reference 13, i incorrect. The effect of' compressibility on the wake--blockage correction is deter-mined in references 8, 9, and 10 by means of' the correct form of the Extension of th Prandtl Ru1but there is so question whether this rule is applicable to this problem. It appears that in using this method to determine the effect of compressibility on the blockage

18

NACA M 'No. A7B2819

correcfion, the effect of compressibility on the source strength Q given by equation (30) . is overlooked. This exDlaiflS the discrepancy between equation (36) an the formula of references 8, 9,' and 10.'

WAKE PRESSIJPE GRADNT IN CO V1PRESSIBLE FLOW

The effect of the pressure gradient caused by the wake Is dis-cussed in reference 3 for compressible flow in a two—dimensional wind tunnel and in reference 11 for compressible flow in a circular wind, tunnel. The method, of reference 11 is equally applicable to the case of a rectangular wind tunnel if the appropriate value of T (table I) is used.

The longitudin1 velocity increment due to the effect of the tunnel boundaries on the source used. to simulate the wake has an approxiate1y linear gradient in the stream direction at the model location. This linear gradient in the velocity, is equivalent to a linear gradient in the pressure as is, easily seen from the approxi-mate relation ,

= - 2q' 42U

In reference 11 it is shown that the gradient in for a source in a wind tunnel is identically equal' to the value of U' for a doublet in a wind tunnel. In the notation of the present' report there is obtained

For compressible flow this becomes

- t 1 + (y-1),M' 2T' C'DS V

dx {1—(M')]

1 + O.4. (Mt)2 JiT 7 CD'S . V

[1— (M')2' 72 i VC_s7' V ' . V

The increase in the drag resu1tng from this pressure gradient is equal to the product of the pressure gradieiit 'by the sum 'of the actual model volume and the vrtua1 volume (reference 2). It

20 NACA RM No. A7B28

should be noted that the constants K1 and. K3 of tables II and IV are equal to r( /. ) times the ratio of the sum of the actual model

volume and the virtual volume to the actual model volume, these quantities being calculated und±the a , sumpion of incompressible flow. Thus the increase in drag cOff1cient caused by the pressure

gradient is given by

- o(M') KtC

LCDW - [1(MT)213/2c3'2

for the wing, and.

1 + o.1.(') K3TVbCD' ACDb' [l_(MT)233/2 c3"2

for the body of revolution.

It is pointed. out in reference 11 that the virtual volume is altered. by the compressibility of the fluid. Thus equations (37)

and (38) can be slightly improved if K1 and. .K3 appearing therein are corrected, to take account of this effect. The ;necessary modif i-cation is made by replacIng K1 and. K3. in these equations by

and. K3p, respectively, where

Kjp =[l ^ h(Mt ) j = 1,3 (39)

where h(M') is given by figure 1 which is reproduced. from reference 11. It should. be noted, that the improvement in modifying

K1 and. K3 for compressibility will be small especially in the case of K 3 for the 'body of revolution. It is important only for quite high Mach numbers. This additional ref inement, was not made in reference 3.

It apDears that a similar compressibility modification of K1

and. K3 in the formulas for the solid blockage should be mad.e, but the exact modification required is not known. However, it is believed. to be small.

(37)

(38)

21 NACA BM Io. A7B28

CORRECTION OF MEASURED QUANTITS

The true velocity U at a model consisting of a body of revolU-tion and. wing can be obtain6d. from the apparent velocity U' by applying the solid—blockage and. wake—blockage corrections. The true

velocity may be written in the form

U = u'(l +K)(Lo)

where

K=Kw+Kb+Kwk(1i)

In. equatiOn (38) K is the 8o11dblockage correction due to the

wing and. is given by

1 K1TVw (ii.2) = [1_(MI) 2 j 3 '2 03/2

or

1 K2T2sct (13)

= [i—CM' )213/2 C3!2

Kb is the 6oiid_blockage correction due to the body of revolution,

being given by

1 KrV 1411-

b =. (M' )2 i7. ( )

or

1 K4Tcbtb2 (14)

Kb - tl_(W)21/'2 C3/'2 -

is the wake—blockage correction and is given by

22 NACA RM No. .A7B28

1 + O.I.(M') CD'S. ('i.6) Kwk

l—(M')2 j.0

It is evident that a correctIon to the apparent velocity in a compressible flow implies corrections also to the apparent density, dynamic pressure, Reynolds number, and Mach number. These correc-tions are readily obtained on the basis of the usual assumption that the flow is adiabatic. It is assumed that the correction terms are small compared with unity, so that squares and. products of these terms may be neglected. The analysis follows the lines of'reference 3, anditis not necessary to repeat the details here. The following equations are obtained:

p ='ti - (M')2 (7)

q = q '{ i + C2—(M')21K }

(8)

R =R'{l + tlO.7(M'K} / ()

M = M t {l +[i+0.2(Mt)2JK} (50)

The drag coefficient must be corrected for the effect of the pressure 'adient due to the wake as well, as to refer it to the correct dynamic pressure. There is thus obtained

CD = cD t{ 1 [2—(M' ) 2 )K - LCD' - AC' }

(51)

where LCD' and. LCDb' are givexi by 'equations (37) and. (38).

Numerical values of the functions of M' which appear in these equations are given in table VI.

CONCLUDING REAR3

Data obtained from tests of three—dimensional models, which are small relative to the wind—tunnel dimensions, can be corrected for

NACA BM NO. A7B28

23

solid and. wake blockage and' for the pro ssuré gradient due to the wake by means of the following equations:

u = ut( 1+K ) (1O)

= + [2_(MT) 2 K} . (8)

= R' { +[1_O.7(M1)2]K} ('9)

M M!{-I + [l^O.2(Mt)2)K}. (50)

CD CD T{ i - [2 .(M')] K— LCD' (51)

In the preceding equations K is obtained front the following:

(ui)

where . . .

• . - ___________ K1 rV. . . (12) 'ST u1(MT) 2 ] 3/2 C 3/2 .

or . •

1 K2T2SCtw . (J ) K - [ 1_ (Mt ) 2 1 8/2 C 72 • .. .

K' - 1 K&rVb .()

0 - [. (Mt) 2 1 3/2 C3/2

or ,., .,

- 1 K4Tcbtb2

f1(M ? ) 2 ) 3/2 C3/2 •

'I

Kwk 1^O. 1i. (M) 2 CD'S - 6) 1_(M t )2 1•i•C

2i- NACA PM No. A7B28

1 + 0.1.(M 2 KjTVvD! (37) cDW' - [i _.(M T )2PY2 3/2

= 1 + o.I.(M') 2 KrVbCD' (38) [1 -(Mt)?)3/2 .C3/2

In these equations T is a factor which for.a body of revolu-tion depends only on the shape of the tunnel; whereas for a three--dimensional wing ¶ depends on the rtio'ofthe wing span to the tunnel breadth as well as on the tunnel shape. (The wing span is in the direction of the tunnel breadth.) Valuds of r are given in table I and. figure 2. The constants K1 and K2 for the three-dimensi6nal wing de pend only on the wing base profile shape and can be calculated. by means of equations (ii), (12), and (13). Values of K1 and. K2 for a number of wing profiles are given ix tables II arid. III. The constants K 3 and K for the body of revolution depend only on the shape of the body and can be calculated by means of equations (22), (23), and (2 1i). Values of these constants fpr a number of body shapes are given in tables IV and V; some of these shapes are drawn in figure 3. Since these tables are incomplete and fuselage forms are not standardized as are wing sections, itis recommended. that the values of K 3 given in table IV for the NACA 111 series of shapes be used for any fuselage shape which does not differ too 'eatly from an NACA 111 shape. This implies that equatiorr (19) and table IV should be used in prefe'ence to equation (20) and. table Vwhenever possible. Numerical values of the functions of W which aear in the corriction equations are given in table VI.

The constants K1 aiid K. 3 appearing in equations (37) and (38) may be modified for com-pressibility"by means of figure 1.. The modified. values of K 1 and K3 are to be used in equations (37) and (38) only and not in equations (I.2) and. (14i.).

Ames Aeronautical Laboratory, National Advisory Coiittee for Aeronautics,

Moffett Field, Calif.

NACA RM No. A7B28

25

APPENDIX A

STJ4ATION OF INFINI SERIES FOR a(0, s/B, B/H)

From equations (3) and. (16) it is seen that (o, s/B, /E) may be wi'itten in the a1terntive ferms

( )' (m+s LB - s LB.) (Al)

'B'E 2s .'fl \Emn 'mn

or

( sB"\ V 2m 0, , A2 • EF [ (m+s /B ) F-1+ (rn—s /B) Ep1

where the summation is taken for all positive 'and. negative integral values of m 'and. n except m = n = 0 'and. the quantities E and. Fma, which are introduced for conve'nience are define. by the equations

r Ema •/ n2+ (rn+s /3) (B

= / n + (s /3)2 (B/H)2

It is possible to sum the series for' e(O, s/B, B/H) exactly only when s/B = . In this case equations (Al) and. (A2) yield.

C') I -I

-a)

(0

I--14:\j

r-41?H H

+I C.) __

Ici

It-

a)

+ I

C')

cuLa)

1% __

lcI—,. '-

+ H. 0 +)

cci IL +

ri - a) + rO1 jCtJ +I_- —a C)

+ _ C) a)

Cu ---.. IcC I I_+ II

a)

L_-J \C')

8-.

^ 0)

I uRu

HId 0

II0)

H

Cu CU0)

ii II

rl

0)

HC'i 0

c

b

IC"

a)

H r -I

CU

+

p:H;i1

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0 \

cr +

C\J

a)

C')- -

+

çq$

a)

b

II

26

NACA Ii4 No. A7B28

NACA PM No. A7B28

—'---

4-)

4-)

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"-I

a)

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U)

U) 'ri

1

27

a)

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ccl

.' o a)

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4r3

c'J

cui

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^

c'J—.

+

H

+ ('jed — —'-

+

4-

. -1C') Ui +

-

ki —...

I-,--,

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ii (1

ri I.i1

lcd

rx

+

p

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+ ('I LJ

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IN

8.

L

cci 4)

cci

NACA PM No, A7B28

provided a, b >0. Now RN(0, s/B, B/H) is approximately equal to R(0, 0, B/H) and it is easily found that

N(°' 0, B/H) = + ) [2+2(B/H)2]3/2

n=l m=N-i-I n=N+l m=l

Co • 00

' •1• 2 T 1 +2 L (B/H) L

n=N+l ni=N+l

These sunm1ations may be approximated by suitable inte'als so that approximately

N\ BH

=( j;Co

+ d dYIN ) {y2+(B/H)2X2 ]3/2

( 1 \\ dx /

^2 1+ A7 \ (B/H)3JJN+1 x

If the 1nte'als of equation (A7) are evaluated by means of equations (A5) and. (A6), then equation (Al ) is obtained.

NACA RM No. A7B28

29

APPED( B

LIST OF IMPORTANT SYMBOlS

B tunnel breadth or diameter

H tunnel height

C tunnel cross—sectional area

c chord of airfoIl or body of evoiution

maximum thickness of airfoil or body of revolution

V volume f wing or body of revolution

Sm maximum cross—sectional .rea of body of revolution

a half span of wing

CD drag coefficient

S model area on which drag coefficient is based

U stream velocity

M Mach number•

R Reynolds number

7 ratio of specific heat of gas a,t constant pressure to specific heat at constant volume (Cp/C)

p mass density

q dynamic pressure

A,K1 ,K2 factors depending on shape of a.irf oil base profile (See equations (ii), (12), and (13) and tab'es II and. .)

X,K3 ,I 4 factors depending on shape of body of revolution (See equations (22), (23) and. ( 2 11. ) and tables IV and V.)

h(M') linear compressibility correction factor for virtual volume (reference 11)

30 NACA PM No. A7B28

factor depending. on tunnel shape and. wing—span—to-tunnel—breadth ratio (See equations (10) and (21) and. table I.).

K total blockage correction (See equation (lii).)

- K wing—blockage correction (See equations (l2) and (14.3).)

Kb body—blockage correction (See equations ( l )and (Is.5).)

Kwk wake—blockage correction (See equation (14.6).)

• Superscript

(') when pertaining to fluid properties, denotes values existing in tunnel far ups tream from model; when pertaining to airfoil characteristics, denotes values In tunnel, coefficients being referred to aDparent dynamic pressure q'

Subscripts (used only when necessary to avoid, ambiguity)

c denotes values in compressible fluid

I denotes values in incompressible fluid

w • denotes values for wing

b denotes values for body of revolution

wk denotes values for wake.

NCA RM No. A7B28

31

PEFENCES

1. Lock, C.N.H.: The Interference of a Wind. nne1 on a Symmetrical Body. R. M. No. 1275, British A.R.C., 1929.

2. Glauert, H.: Wind Tunnel Interfrenóe bn Wings, Bodies and Air-screws. R. & M.. No. 1566, British A.P.C. 19.33.

3. Allen, H. Julian, and 1lincenti, Walter G.: Wall Interference in a Twimensional1ow Wind TunneI; with Consideration of the fect of Compressibility. NCA ARR No. 14K03, i91.4.

-. Thom, A.: Blockage Corrections andChoking in the P.A.E. Hgii Speed Tunnel. Rep. No..Aero. 1891, R.A.E. (British Restricted! U.S. Restricted), Nov. 191 1.3 . .

5. Goldstein, S., and Young, A.D.: - The Linear Perturbation Theory fd of Compressible Flow, with Apiications to Wind-Tunnel nter- / ference. R. & M. No. 1909, British A.R.C., 1911-3.

6. Tsien, Hsue-shn, and Ies, Lester: The Glauert-Prandti Approximation for Subsonic Flows of a Compress:Lble Fluid. Jour. Aero. Sd., vol. 12, no. 2, April 1945, pp. 173-187 and 202.

7. von Baxanoff, A.: Zur Frage d.er imaikorrektur bei kompressi'bler tJnterscha.l1stroning. Deutsche Luftfahrtforschung Forschungsbericht Nr. 1272, 1911.0.

8. Gthert, B.: Windkanalkorrekturen bei hohan Unterscha11eschwindigkeiten unter besonderer berUcksichtigung des geschlossenen Kreiskanals. Deutsche Luftfabrtforschung Forschungsbericht Nr. 1216, 1911.0.

9. Gthert, B.: Windkanalkorrekturen bei hohen Unterscha11geschwindigkeiten. Lilienthal-Gesellschaft fr Luftfahrtforschung, Bericht 127, Sept. 1911-0, p. 1l4-120.

10. Wieseleberger, C.: Windkanalkorrekturen bel kompressibler Stroinung. LII ienthal-Ge sellschaft fir Luf tfa.hrtf orschung, Bericht 127, Sept. 1911.0, pp. 3-7.

32

NACA RiM No. A7B28

11. Luthdeg, H.: Wider$tandskorrektur in HochgeewindJkeitskan.1en. Doutche Luftfahrtforechung Forschungsbericht Ni'. 1955, 19.

12. G3thert, B.: Ebene und. r.um1iche Strômung bel hohen Unterschallgeschwindigkeiteri (Erweiterung dci' Prand.tlschen Regel). Lilientbal-Gesellscha.ft fiir Luftfairtforschung, Bericht 127, Sept. 1910, pp. 97-101.

13. Hrriot, John G.: The Linear Perturbation Theory of Axially Synmietric Coipressib1e Flow, with Application to the Effect of Compressibility on the Pressure Coefficient at the Surface of a Body of Revolution. NACA RiM No. A6H19, l91i7.

iii. . Young, A.D. and. Owen, P.3.: ASimplified. Theory for Streani.line Bodies of Revolution, and. its Application to the Development of High Speed Shapes. Rep. No. Aero. 1837, R.A.E., British Restricted/U.S. Restricted), July l93.

NACA RM No. A7B28

33

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