Research on Hydrofoil Profiles Suitable for a Straight Cascade of Accelerated Flow. Part 2 (Experimental study)

R e s e a r c h o n H y d r o f o i l P r o f i l e s S u i t a b l e f o r S t r a i g l i t C a s c a d e o f A c c e l e r a t e d F l o w / R e p o r t 2

(Expei-iniental S t u d y ) '

CcMt UnivGrsUy cf Technology

Sliip KviromecEisnics Laboratory

s

Library

Mekelweg 2 - 2628 CD Delft

The Netherlands

Phone: 31 15 786873 - Fax: 3115 781K8 Synopsis

In tfiis report the wind tunnel tests have been carried out on the profile specially designed for the accelerated cascade flow by our theoretical method and arranged iti cascade condition, so as to examine the reliability of our theoretical method. The pressure center of our profile has been proved to be almost immovable throughout the range of Ca = 0.3 ~ 1.5. Further, it is worthy to note that, the surface pressure distribution at the aimed lift coefficient Ca = 0.6, has been proved to coincide fairly well with the prescribed theoretical one.

1. Introduction

In the design of axial flow water turbines, the f o r m of the profile should be made suited to the accelerated cascade, as previously proposed by Prof. F. Numachi C1 ] . His research has been, however, along the line of a cascade of decelerated flow, and as to the cascade of axial turbines namely of accelerated flow, there seems some room left yet to be filled by substantiated research. And so f a r as the writer is aware, there has been felt the scarcity of reliable reports on experi-ments i n this field.

In the report 1 C 2 ) of the same title a theoretical method was described to figure out the profile which has ( i ) a given surface pressure distribution and ( i i ) the comparatively immovable pressure center i n a given cascade arrangement.

1. Report No. 88 (in European language) of the Institute of High Speed Mechanics, Töhoku University. This report was published in Japanese in the Memoirs of the Institute of High Speed Mechanics, Töhoku University, Vol. 8 (1952), No. 72, p. 9. Read at the 28th General Meeting of the Japan Society of Mechanical Engineers, on April 2, 1952.

2. Late Assistant Professor of the Institute of High Speed Mechanics, Tohoku University, Sendai, Japan.

120 Rep. Inst. High Sp. Mech., Japan, V o l . 9 (1958), No. 88

In the present report, the result of the experiment carried out i n the wind-tunnel is to be described by way of corroboration of the theory f o r calculating out the profile suitable for a cascade arrangement stated i n the Report 1.

The result of the experiment agreed w i t h the numerical value calculated by the author (which shows the applicability of his theory to the cascade profile w i t h comparatively small camber and thickness f o r axial flow water-turbines). And the excellent characteristics of the cascade profiles calculated theoretically seems to prove appropriateness of the author's idea in which he sought after the suitable cascad profiles for blade elements of Kaplan turbines.

2. Theory and Equipment of the Experiment

The whole apparatus is shown in the F i g . 1. I t is the wind-tunnel at the Institute of High Speed Mechanics [ 3 ) affixed w i t h a newly devised measurh g

Fig. 1. Experimental Arrangement ® 5 ff t w o stage axial blower

(D Honeycomb

© Adjustment wire-netting-strainer f o r attaining u n i f o r m distribution of w i n d velocity

® Cascade tested (^ = / = 150mm, d = 150°) @ Movable wall

apparatus. Based upon the experimental theory by Prof. F. Numachi [ 4 j , the upper and lower-walls at the rear of the cascade were made movable and the aspect of the flow near by the measuring apparatus was made to the possible l i m i t of approximity to the flow through the infinite number of hydrofoils arranged in a straight cascade. The outline of the measuring apparatus is shown i n Figs. 2 and 3. The arrangement of the cascade was made at the measuring apparatus w i t h seven profiles in which 6 (the angle between the flow velocity at the infinite

S. A B E : Research on Hydrofoil Profiles Suitable f o r Cascade / Rep. 2 121

Fig. 2. Photograph of Measuring Part and Test Profiles

F i g . 3. Measuring Part

f r o n t and the axis of the cascade) = 150°, pitch-chord ratio tjl = 1.0, chord length / = 150 m m , and span B = 150 mm. The bending-point of the movable walls was situated f r o m 40 96 f r o m the f r o n t edge of the profile and the distance between the farthest profile and the walls was 1/21. Details of the measuring apparatus was designed based upon the author's theory of experimental apparatus { 5 J.

-

Measurements of the l i f t and drag of the profile were made according to pressure distribution method and the pressure distribution was measured at 30 points on the profile surface as shown i n Figs. 7 and 9.

Fig. 4. Testpiece of Profile

Fig. 5. Wind Velocity Distribution at Measuring Part

B = the distance measured across the channel w i d t h f r o m the face of the rear wall

T = the distance measured along the cascade axis f r o m the lower w a l l Pitot tube used: 2.5 ^ Prandtl type \ ^ . . ,

S. A B E : Research on Hydrofoil Profiles Suitable f o r Cascade / Rep. 2 123

3. Preliminary Experiment

( 1 ) W i n d - V e l o c i t y a t t h e M e a s u r i n g A p p a r a t u s a n d t h e D e c r e a s e o f W i n d - P r es s u r e C a u s e d b y t h e F r i c t i o n L o s s a t t h e W a l l .

As shown in Fig. 5, the u n i f o r m distribution of the wind velocity was as-certained by the combined use of a pitot tube of Prandtl type w i t h the diameter of 2.5 m m and a N. P. L. type improved pitot tube w i t h the diameter of 6.45 m m . The difference of static pressures before and after the bellmouthed inlet of the wind-tunnel and their relation w i t h the w i n d velocity at the w o r k i n g section, and also the relation between the above mentioned difference of the static pressures and the decrease of static pressure caused by f r i c t i o n loss at the wall were meas-ured and summarized as shown in Fig. 6. A t the experiment of the cascade profiles, the difference of static pressures before and after the bell-mouthed inlet of the wind-tunnel was first measured, and then the wind velocity was calculated and the adjustment to the pressure lowering caused by f r i c t i o n at the wall was made according to F i g . 6.

Fig. 6. Relations between Static Pressure Difference A ^ , before and after Nozzle and A / J J , , Dynamic

Pressure of W i n d Velocity at Measuring P a r t ; ^^^pn, Pressure Loss f r o m Exit of Nozzle to Measuring Part and A / ) , ^ , Pressure

Loss f r o m Measuring Part to Exit of Wind Tunnel (Angle of

Mova-ble Wall 8 = 0 ° )

( 2 ) D e t e r m i n a t i o n o f t h e Z e r o I n c i d e n c e A n g l e .

As shown i n Fig. 7 , the symmetrical profile N. A. C. A. 0 0 0 8 was used to

obtain the zero incidence angle w i t h accuracy and the angle where the pressure distributions on the upper and lower surface of the profile conicide each other was determined as the zero point.

124 Rep. Inst. High Sp. Mech., Japan, V o l . 9 (1958), No. 88

Fig. 7. Above: - Symmetrical Aerofoil N.A.C.A. 0008 f o r Testing Zero Incidence Angle, and Below Pressure Distribution at Zero Incidence Angle (• > x denote

pressure measurement points)

( 3 ) A c c u r a c y o f t h e E x p e r i m e n t i n t h e C o n d i t i o n o f a S i n g l e A e r o f o i l P r o f i l e .

For the purpose of examining the accuracy of the present apparatus of experi-ment, an experiment was carried out on a single profile of Clark Y 8 ^ , one of

the existing profiles and its result was

1.2 1,0 0,8 OM 0,2 1 4 1 1 1 1 i

J

\

/ /

t

compared w i t h the results of the past experiments C 6 ] . As shown in F i g . 8, the l i f t curve obtained in this experiment coincides w i t h the experimental result of N . A . C. A. in the case of higher Rey-nolds numbers on condition that the l i f t is not too large. And the value of m i n i m u m resistance in the N . A . C . A . experiment w i t h adjusted Reynolds num-ber [ 7 3 agrees f a i r l y well w i t h the value of the minimum resistance i n the present experiment plus the calculated value of frictional resistance.

4. Result of Experiment ( 1 ) T e s t P r o f i l e s .

Fig. 8. Characteristics of Clark Y 8 % in Isolated Condition

Test profiles are, as shown i n F i g . 9, the cascade profile No. 1 which was cal-culated i n Report 1 and Clark Y 8 ^ , the detailed dimensions of which are shown in Tables 1 and 2. The cascade profile No. 1 is the one w i t h thickness ratio of 8 % calculated out so as to let i t have the pressure distribution shown i n F i g . 10 at the l i f t coefficient 0.6 and the small s h i f t i n g of the pressure center i n the neighborhood of C£, = 0.6, while Clark Y %% was selected out of existing profiles f o r the purpose of comparison.

S. ABE : Research on Hydrofoil Profiles Suitable f o r Cascade / Rep. 2 125

F i g . 9. Test Profiles Denoting Pressure Measurement Points w i t h the Dark Circle

Above : Cascade Profile No.l Below : Clark X %96

Table 1. Pressure Distribution f o r Calculation of Cascade Profile No. and its Dimension

pa = Pressure on the vacuum side

pu = Pressure on the pressure side

/>oo= Pressure correspond to V

X = Length measured f r o m the f r o n t edge of chord

P^= pV-l2, X = a : / /

X % _{(Pa-Pec.)/go:.} (Pu-P^^/Q^ X mm j^omm yu mm

0 1.000 1.000 0 0 0 0.25 -0.280 0.900 0.38 0.95 -0.36 Ü.50 -0.450 0.820 0.75 1.23 -0.48 1.25 -0.571 0.675 1.88 1.89 -0.81 2.50 -0.648 0.567 3.75 2.85 -1.14 5.00 -0.699 0.471 7.50 4.25 -1.56 7.50 -0.713 0.419 11.25 5.25 -1.83 10.00 -0.718 0.381 15.00 6.05 -2.03 15.00 -0.716 0.323 22.50 7.29 -2.34 20.00 -0.706 0.276 30.00 8.16 -2.60 25.00 -0.688 0.235 .37.50 8.69 -2.79 30.00 -0.667 0.197 45.Ü0 8.93 -2.93 40.00 -0.610 0.127 60.00 8.72 -3.12 50.00 -0.549 0.064 75.00 8.06 -3.30 60.00 -0.480 0.003 90.00 6.99 -3.32 70.00 -0.408 -0.053 105.00 5.66 -3.20 80.00 -0.326 -0.107 120.00 4.13 -2.66 85.00 -0.283 -0.128 127.50 3.18 -2.15 90.00 -0.239 -0.147 135.00 2.24 -1.64 92.50 -0.216 -0.152 138.75 1.73 -1.28 95.00 -0.192 -0.154 142.50 1.22 -0.87 98.00 -0.158 -0.146 147.00 0.51 -Ü.38 100.00 1.000 1.000 150.00 0 0

Minimum Pressure iPmin—Pa)/Q<^ -0.719

Position of Minimum y

Pressure 11 %

Theoretical Incidence

Angle m i '

Thickness Chord Ratio o„

Position of Maximum Thickness Xs Camber Ratio Position of Maximum 8.00 % 37 % 2.00 % no n/.

126 _{Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 88}

Table 2. Dimension of Clark Y 8 96 Profile Test-Piece X mm Jo mm yu mm 0 3.58 3.58 0.38 4.32 2.81 0.75 4.72 2.51 1.88 5.59 1.96 3.75 6.66 1.50 7.50 8.10 0.96 11.25 9.09 0.64 15.00 9.84 0.43 22.50 10.97 0.15 30.00 11.66 0.03 37.50 11.95 0 45.00 12.00 0 60.00 11.70 0 75.00 10.80 0 90.00 9.39 0 105.00 7.54 0 120.00 5.36 0 127.50 4.13 0 135.00 2.87 0 138.75 2.19 0 142.50 1.52 0 147.00 0.72 0 150.00 0.12 0

Thickness Chord Ratio o^o^ 8.00 %

Position of Maximum ^

Thickness " 30 o/o

Camber Ratio (u„,^ 2.43 %

Position of Maximum y

Camber 43 o/o

Fig. 10. Surface-Pressure Distribution of Cascade Profile No. 1 at Normal L i f t Coefficient

S. A B E : Research on Hydrofoil Profiles Suitable f o r Cascade / Rep. 2 127

( 2 ) L i f t a n d D r a g C u r v e .

The l i f t and drag curve of the Cascade Profile No. 1 is shown in Fig. 11 and that of Clark Y 8 ^ is in Fig. 12. I n both profiles, the l i f t increases suddenly w i t h the increase of incidence angle. The stall was not perceived i n the range of the present experiment. The minimum resistance coefficient (only of pressure drag) was comparatively small and was about 0.005 in both cases.

!,9 if

a

to

o.v

1 1 1

1 1 1

.1

.1

j

/

/

/

! /

/

/

/

/

/

A

1

1

1

/

f

/

' /

1

/

¥

/

^ )

Fig. 11. Camparison of Characteristics between Cascade Profile No. 1 i n Cascade and in Isolated Condition

Fig. 12. Comparison of Characteristics between Clark X. %% i n Cascade

and in Isolated Condition

( 3 ) C o m p a r i s o n w i t h t h e L i f t a n d D r a g C u r v e o f t h e S i n g l e P r o f i l e .

For comparison's sake, the characteristics of the single profiles were sought and shown in Figs. 11 and 12. I n both Cascade Profile No. 1 and Clark Y 8 96, the cascade profile shows much greater l i f t at the same incidence angle. The

128 Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 88

f.s

1,0

0,6

minimum resistance coefficient does not show much difference i n the cascade and the single profile, but the cascade profile shows the s h i f t of minimum resistance to the point where the incidence angle is smaller.

( 4 ) C o m p a r i s o n b e t w e e n t h e T h e o r e t i c a l V a l u e B a s e d o n P o t e n t i a l T h e o r y a n d t h e E x p e r i m e n t a l

V a l u e .

The l i f t curve [ 8 ] calculated on potential theory is shown i n Figs. 11 and 12, as corn-pared w i t h the experimental value. I n Cascade Profile No. 1 the curve coincides well w i t h the experimental value. And in case of Clark Y, the l i f t i n the experiment is shown as a little greater than the theoretical value.

The measured value of pressure distribution i n Cascade Profile No. 1 at the prescribed l i f t coefficient is shown i n F i g . 10 in dots and i t coincides f a i r l y well w i t h a given pressure distribution in calculation. The fact shows, i t may be admitted, that the theory stated in Report

1 was appropriate.

( 5 ) I n t e r f e r e n c e C o e f f i c i e n t o f t h e C a s c a d e .

For the sake of calculating the charac-teristics of profile in cascade arrangement out of the characteristics of the isolated profile, the following two questions must be considered : ( i ) change of no-lift angle of the cascade and the single profile and ( i i ) change of the slope of the (Ca - a)

curve of the cascade and the single profile. I n the present experiment, the theoretical value and the experimental value coincides well each other i n both cascade and single profile i n the former case; hence the ap-proximate value can be obtained by using the experimental value of the single profile w i t h theoretical adjustment. On the con-t r a r y , i n case of con-the slope of con-the (Ca — a)

curve, i t is true that theoretical value and experimental value show a f a i r coincidence i n case of cascade, but, in case of single profile, the theoretical value presents itself a little larger than the experimental value; so in this case i t is safe to adopt the usual

Cascade Profi/e) Clarf: Y8% J " ion in Cascat/e ^(t/i-/,0, Oi'/SO") 7 ' Ciscadt Prefik Ciar>:YS% bot/i/n miated condition

0.02

— Cu> 006 aos

F i g . 13. Comparison of Polar Dia¬ grammes between Cascade Profile

S. A B E : Research on Hydrofoil Profiles Suitable f o r Cascade / Rep. 2

calculation method in which the experimental value of the single profile is multiplied by the interference coefficient.

( 6 ) C o m p a r i s o n o f P o l a r D i a g r a m s o f C a s c a d e P r o f i l e N o . 1 a n d C l a r k Y 8 %.

The polar diagrams of Cascade Profile No. 1 and Clark Y 8 % are shown in Fig. 1.3, in which both profiles show excellent characteristics. I n both cases, the minimum d r a g / l i f t ratio is smaller in cascade. And i n the prescribed l i f t coefficient Ca = 0.6, the cascade profile excels the other in characteristics.

( 7 ) C h a n g e o f t h e C e n t e r o f P r e s s u r e b y L i f t C o e f f i c i e n t .

The change of the center of pressure by l i f t coefficient in Cascade Profile No. 1 and Clark Y 8 is calculated as tjl = 1.0, 6 = 156° as shown in Fig. 14. Clark Y 8 96 shows considerable change w i t h the change of l i f t , while Cascade Profile No. 1 shows little change. Both profiles are compared w i t h the numerical value derived by potential theory: in Cascade Profile No. 1 the difference between the theoretical value and experimental value is comparatively small, while in Clark Y 8 96 the theoretical value of the pressure center is drawn to the f r o n t compared w i t h the experimental value.

60 fO 20 X

. 1 .

lurk Yi

. 1 .

lurk Yi

. 1 .

Casa

Yoflle Mt

- }

eriment

Pot. neory

-Ca

Fig. 14. Comparison of X ' , Moment Center on the Chord between Cascade Profile No. 1 and Clark Y 8 ^ both i n Cascade Arrangement

itil = 1.0, 0 = 156°, i? = 3 X 10")

5. Inquiry into the Experiment

On examining the result of the present experiment the questions of ( 1 ) critical Reynolds number of the wind-tunnel and ( 2 ) aspect ratio at the w o r k i n g section should be considered. For ascertaining those points, simple inquiry was made into the experimental apparatus.

Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 88

( 1 ) . T u r b u l e n c e i n t h e W i n d - T u n n e l .

A cylinder w i t h the diameter d - 150mm, the span B — 150 m m was put into the working section (hight h = 525 m m , span B = 150 m m ) of the wind-tunnel, and its surface pressure distribution was measured w i t h i n the range of the wind velocity F = 13 ~ 35 m/sec. When the result of the measurement is arranged into the relation of Reynolds number R to drag coefRcient C„, i t w i l l present itself as shown in F i g . 15. Since djh being large, the value of drag coefficient in the present experiment where merely the infiuence of the upper- and lower-walls was adjusted on the assumption of potential velocity field, may be regarded as quan-titatively inaccurate. But so f a r as the critical Reynolds number is concerned, the result can be said f a i r l y reliable.

0,8

0.6

0,f

0

V

V

\

\

\

\

N A

—o— Our WM TumeJ ( d/A=OJff/)

N.RL WindTa/iM/ (Fage)

N.P.L. Wind Tumt/(/?e/f)

N.RL WindTa/iM/ (Fage)

N.P.L. Wind Tumt/(/?e/f)

06 Ofi 10

Fig. 15. Change Owing to Reynolds Number of Drag Coefficient of Circular Cylinder i n the W i n d Tunnel

When the present experimental result is compared w i t h the Göttingen wind-tunnel ( w i t h degree of turbulence 1.2) C9^ and N . P . L . wind-wind-tunnel CIO], the degree of turbulence of the present wind-tunnel is found to be i n the middle of the two, as shown in F i g . 15. Therefore, the characteristics of the profile used

Fig. 16. Change of Total Pressure in Span Direction at Immediate Front (Upper Row) and Immediate W

Rear (Lower Row) of the Cascade, Test Profile No. 1 ( / / / = 1.0, 01 = 150°). V Cylindrical Pitot tube of 4.5 mm diameter was used f o r measurement.

Sign X denotes the position where measurement was made

132 Rep. Inst. High Sp. Mech., Japan, V o l . 9 (1958), No. 88

i n this experiment can be regarded as appropriate, provided that the vicinity of the stalling point is not involved. In Fig. 15, the critical Reynolds number sphere in the wind-tunnel is considered to be in the neighborhood of 2.2 X 10'. And VW/V, the degree of turbulence, is calculated out of the above according to G. I . Taylor's theory (11}, which turns out to be y/ti^/V = 0.009.

( 2 ) A s p e c t R a t i o o f t h e W i n d - T u n n e l .

In order to ascertain if the flow at the main working section be two-dimensional i n the measurement condition, cylindrical pitot tube w i t h the diameter of 4.5 m m was used at several places immediately f r o n t and after the cascade to measure the distribution of total pressures in the direction of span.

Total pressure distribution in span direction at various incidence angles is shown i n F i g . 16, in which not only in an interval between profiles but also in the range of wake in the rear of the profile, total pressure distribution curve in span direction is comparatively u n i f o r m except in the vicinity of the wall. I n the experiment total pressure distribution is f a i r l y satisfactory at Ca = 0 and Ca = 0.6. Consequently, the result of this experiment carried out according to pressure dis-t r i b u dis-t i o n medis-thod in acceleradis-ted cascade arrangemendis-t is dis-to be regarded as com-paratively free f r o m the influence caused by small aspect ratio of the profile, provided that resistance coefficient is not too large.

6. Conclusion

( 1 ) I n order to inquire into the profile f o r m suitable f o r blade section of axial flow water turbines, the wind-tunnel tests have been carried out to examine the characteristics of a profile (a profile suitable f o r a cascade arrangement) specially designed by our theoretical method in accelerated cascade arrangement. And the characteristics of Clark Y S 96 i n accelerated cascade arrangement was also measured f o r comparison w i t h the above.

( 2 ) Though the polar diagrams in accelerated cascade are excellent in both two profiles, that of our profile seems to be slightly superior to that of Clark Y 8 96 in the neighborhood of Ca = 0.6. I n Clark Y 8 ^ , the center of pressure changes perceptively influenced by the incidence angle, while in our profile i t is almost immovable throughout the range of Ca = 0.3 -~ 1.5, which proves our aim f o r designing a suitable profile has been f a i r l y attained.

( 3 ) The pressure distribution actually measured i n the condition of the prescribed l i f t coefficient Ca = 0.6 and the given pressure distribution at the start of designing agrees comparatively well w i t h each other. That shows the theoretical calculation method stated in Report 1 was appropriate together w i t h the fact that a profile w i t h suitable characteristics was obtained.

S. A B E : Research on Hydrofoil Profiles Suitable for Cascade / Rep. 2 133

( 4 ) When the cascade characteristics of both profiles is compared w i t h that of an isolated profile, the l i f t of the former is found to be larger than the latter in the same incidence angle, and that much larger l i f t than the maximum l i f t of the latter is obtained.

( 5 ) The difference between the theoretical l i f t coefficient curves calculated according to potential theory and experimental ones is comparatively small and diminution of the slope of the l i f t coefficient curve under the influence of viscosity is not observed as is the case w i t h isolated or retarded cascade flow. Consequent-ly, as is commonly practiced, the calculation in which theoretical adjustment is given to the characteristics of an isolated profile as to cascade interference means

to estimate the characteristics of accelerated cascade on the side of safety.

In conclusion the w r i t e r wishes to express his cordial gratitude to Prof. F . Numachi for his idea which initiated the present research and also f o r his gene-rosity which has given the w r i t e r free use of various equipments. And especially he is thankful f o r the professor's pertinent guidance throughout his study. I t is his pleasant duty to acknowledge the valuable assistance given by M r . C . Mura-kami in the Mechanical Engineering Dep't and M r . H . Sakai in the Institute of High Speed Mechanics in manufacturing test pieces of the profile, and also his thanks are due to Assist. Prof. H . Éükuchi who gave h i m advice d u r i n g his experiment.

Bibliography

C1) F. Numachi and others. Cavitation Tests on Hydrofoil Profile Suitable for Arrangement in Cascade (Report 1 ~ 4 ) , Rep. Inst. High Sp. Mech., Japan, Vol. 2 (1952), p. 1, 21 and Vol. 3 (1953), p. 99, 139.

C 2 ) S. Abe, Research on Hydrofoil Profiles Suitable for a Straight Cascade of Accelerated Flow (Report 1), Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), p. 107.

[ 3 ] H. Kifiuchi, Experiments on Row of Blade Elements used in Axial Flow Blowers, Mem. Inst. High Sp. Mech., Japan, "Vol. 5 (1951), p. 71.

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C 6 ] B. M. Finlierton and H. Greenberg, Aerodynamic Characteristics of a Large Number of A i r f o i l s Tested in the Variable Density Wind Tunnel, N. A . C. A. Tech. Rep., No. 628 (1938).

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C 7 } A. Silverstein, Scale Effect on Clark Y A i r f o i l Characterisitics f r o m N . A . C. A. Full Scale Wind Tunnel, N . A . C . A . Tech. Rep., No. 502 (1934), p. 509.

[ 8 ] S. Abe, Theory f o r Characteristics of Straight Lattice Composed of Aero- or Hydro-foil Profiles of A r b i t r a r y Form (Supplement), Rep. Inst. High Sp. Mech., Japan, Vol. 6 (1956), p. 11.

[ 9 J L. Prandtl, Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttin-gen, 11 Lief., S. 23.

CIO) A. Page and J. H. Warsap, The Effects of Turbulence and Surface Rough-ness on a Drag of a Circular Cylinder, Tech. Rep. Aero. Res. Com., 1929, (R. and M . 1283).

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Cll] H. L. Dryden, G. B. Schubauer, W. C. Mock and H. K. Skramstad, Meas-urement of Intensity and Scale of Wind- tunnel Turbulence and Their Relation to the Critical Reynolds Number of Spheres, N . A . C. A . Tech. Rep., No. 581 (1937).