Drag and turbulent boundary layer of flat plates at low Reynolds numbers

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Ow I-z w -J DI-4 C, 4 0 DAVID WITAYLOR

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Maryland 20034

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c Scheepvaartkune

Te hi--e Hcqechcol, D&It

DATUM: DL.C..NT.TE

DRAG AND TURBULENT BOUNDARY LAYER OF FLAT PLATES AT LOW REYNOLDS NUMBERS

by

Paul S. Granville

27 FEB1976

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0i5-m-I December 1975 Report 4682

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I. REPORTNUMBER

4682

2. GOVT ACCESSION NO. 3. RECIPIENTS CATALOG NUMBER

4. TITLE (end Subtitle)

.DRAG AND TURBULENT BOUNDARY LAYER OF

FLAT PLATES AT LOW REYNOLDS NUMBERS

5. TYPE OF REPORT 6 PERIOD COVERED

6. PERFORMING ORG. REPORT NUMBER

-7. AUTHOR(a) .

Paul S. Granville

8. CONTRACT OR GRAM TNUMBER(.)

9. PERFORMING ORGANIZATION NAME AND ADDRESS

David W.. Taylor Naval Ship Research and

Development Center Bethesda, Maryland 20084

10. PROGRAM ELEMENT. PROJECT. TASK AREA 6 WORK UNIT NUMBERS

SRM231O1

Work Unit 1-i541-OO1

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December 1975

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37

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IS. SUPPLEMENTARY NOTES

19. KEY WORDS. (Continuó on rave ide if ,iiciiiaty end identify by block number)

Drag

Flat Plates

Turbulent Boundary Layer

20. ABSTRACT (ContifluO on iOviraa aide If neceeeary and identif* by block number)

A new formula is derived for flat plates which gives higher values of drag at low Reynolds

numbers than does the Schoenherr formula. This is due to anomalous effects at low Reynolds

numbers which are accounted fOr by adding viscosity to the outer law in the velocity similarity law analysis.

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UNCLASSIFIED

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TABLE OF CONTENTS

ABSTRACT

ADMINISTRATIVE INFORMATiON. INTRODUCTION

VELOCITY SIMILARITY LAWS ...

INNER LAW OR LAW OF THE WALL

OUTER LAW OR VELOCITY-DEFECT LAW LOGARITHMIC LAW

LOCAL SKIN FRICTION

LAW OF THE WAKE

BOUNDARY-LAYER PARAMETERS DRAG

HIGH REYNOLDS NUMBERS VERY LOW REYNOLDS NUMBERS

NUMERICAL EVALUATIONS . 14

SIMILARITY-LAW CONSTANTS 14

VELOCITY-DEFECT FACTOR AT LOW REYNOLDS

NUMBERS 14

COEFFICIENT OF LOCAL SKIN FRICTION 15

DRAG COEFFICIENT 18

PROPOSED FORMULA FOR DRAG COEFFICIENT 20

APPENDIX A - DERIVATION OF SQUIRE VELOCITY LAW

FOR BUFFER LAYER 25

APPENDIX B - DERIVATION OF FORMULA FOR LOCAL SKIN FRICTION FROM SCHOENHERR

FORMULA . . . 27

REFERENCES

LIST OF FIGURES

1 - Velocity Similarity Laws for Turbulent Boundary Layers 6 2 - Variation of Velocity-Defect Factor B2 with Coefficient of

Local Skin Friction 16

3 - Variation of Local Skin Friction Coefficient with Local

Reynolds Number 17 'U TECIPfl$CHE UNfl1ER$rrir Laboratorlum voor Scheepshydromechanlca Archlef Mekelweg 2, 2628 CD Deift TeL 015 786873- Fax: 015 781J38 Page 2 3 3 4 5 7 8 10 12 12 28

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iv

Page

VariationT0ç 2 _{with R0}

19

S - Fit of Proposed Formula for Drag Coefficient 21

6 - Variation of Drag Coefficient with Reynolds Number at

Low Reynolds Numbers 23

7 - Variation of Drag Coefficient with Reynolds Number at

High Reynolds Numbers 24

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NOTATION

A _{Slope of logarithmic velocity law}

a _{Factor defined in Equation (48)}

a1 _{Factor defined in Equation} ₍₅₇₎ B1 _{Law-of-the-wall factor}

B2 _{Velocity-defect factor}

B20 Constant value of B2:

b _{Factor defined in Equation (49)}

b1 _{Factor defined in Equation} ₍₅₈₎

CF Drag coefficient, Equation (43)

Cf _{Local skin-friction coefficient, Equation (60)}

D Drag

H _{Shape parameter, Equation (24)} Integral parameter, EqUation (20)

12 Integral parameter, Equation (21)

J _{Factor in Equation (5 1)} q Wake-modification function R Reynolds number, Equation (44)

Displacement thickness Reynolds number, Equation (34) R9 _{Momentum thickness Reynolds number, Equation (37)}

U Velocity of flat plate

u _{Streamwise velocity component}

u. Shear velocity, Equation (3)

w Wake function

x _{Streamwise distance or length of flat plate}

y Normal distance from wall

y* _{Inner-law Reynolds number, Equation (4)}

y0* _{Inner limit of log law}

(8)

y'

Nondimensional thickness of laminar sublayer

Iy\

_{Outer limit of log law}

5 Boundary-layer thickness Displacement thickness

Similarity-law Reynolds number, Equation (9)

O Momentum thickness

A1, A2 Factors in Equations (32) and (33)

v Kinematic viscosity of fluid p Density of fluid

o Local skin-friction coefficient, Equation (14)

r

Shear stress

Wall shear stress

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ABSTRACT

A new formtilä is derived for flat plates which gives higher values of drag

at low Reynolds numbers than does the Schoenherr formula This is due to anomalous effects at low Reynolds numbers which are accounted for by

add-ing viscosity to the outer law in the velocity similarity law analysis.

ADMINISTRATIVE INFORMATION

The work described in this report was funded by the Naval Sea Systems Command (Code

035) under the General Hydromechanics Research Program, SR-023-0 101, Work Unit

1-1541-001.

INTRODUCTION

Analytical prediction of development of the turbulent boundary layer and, the associated dragof flat plateszero pressure gradientis well-established by the use of classical velocity

similarity laws.1,2,3

Anomalies have been found in the velocity profile, however, at low Reynolds numbers

even well past transition from laminar flow. Most importantly the local skin friction is significantly higher than that predicted by the usual application of the similarity laws:

Coles4 observed from plots of measured velocity data that the outer similarity law is out of equilibrium ,at the low Reynolds numbers. Coles managed to correlate the velocity data by varying the velocity-defect factOr of the outer law with a, local Reynolds number. The

velocity-defect factor is then only constant at high Reynolds numbers instead of being con-stant at all Reynolds numbers, which is the usual consideration.

Simpson5 on the other hand correlated the velocity data of the outer similarity law by varying the von Krmán "constant" with Reynolds number instead of the velocity-defect

factor. Since the von Kármn constant also appears in the inner similarity law, this implies a variation of the inner law at low Reynolds numbers.

tIndwer L., "The Frictional Resistance of Flat Plates in Zero Pressure Gradient," Transactions of the Society of Naval Architects and Marine Engineers, Vol. 61, pp. 5-32 (1953). A complete listing of references is given on page 28.

D., "The Problem of the Turbulent Boundary Layer," Zeitschrift fir angewandtè Mathematik 'üñd Phyiik, Vol. 5, pp. 181-203 (1954).

3Coles, D., "The Law of the Wall in Turbulent Shear How;" in"50 Jahre Grenzschichtforschung," Edited by H. G&tler and W. Toilmien, Friedr. Vieweg and Sohn, Brauschweig (195).

4coles, D. E., "The Turbulent Boundary Layer in a Compressible fluid," RAND Corp.', Santa Monica, Calif., Report R-403-PR (Sep 1962).

5Simpson, R. L., "Characteristics of Turbulent Boundary Layers at Low Reynolds Numbers with and without Transpira-tion," Journal of Fluid Mechanics, Vol. 42, Part 4, pp. 769-802 (30 Jul 1970).

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Cebeci and Mosinskis,6 following the Simpson lead, modified the eddy-viscosity formula

fOr the inner part of the boundary layer for direct solutions of the turbulent boundary layer from the equations of motiOn. The modification involves the variation of the von Kármn "constant" with local Reynolds numbers.

However, McDonald7 varied the eddy-viscosity foirt ülatiOn, particularly fôt the outer

part of the boundary layer to allow for the effect of low Reynolds numbers.

Finally to settle the controversy, Huffman and Bradshaw8 critically examined the effect

of low Reynolds number and reached the conclusion that the outer law varied and not the inner law. Hence, the Coles analysis should be considered basically correct. It is stated8 that viscous effects arising from the turbulent irrotational interface affect the outer law at low

Reynolds numbers. As additional evidence, it is further asserted that there is no effect on the outer law at low Reynolds number for fully developed turbulent-duct flow which lacks such an interface.

It is now proposed to reanalyze development of the turbulent boundary layer and the drag of flat plates to include the effect at low Reynolds numbers. Only the outerlaw is

con-sidered affected in aàcordance with the conclusion of Huffman and Bradshaw.8 Viscosity is

added to the usual factors controlling the velocity defect of the outer law. This is the key to the analysis. The result is that the velocity-defect factor becomes a function of a similarity law Reynolds number. ihe non-logaiithmic part of the outer law is analytically stated by the usual law of the wake and a newly derived wake modification function.9

By use of these altered similarity laws, the usual boundary-layer parameters of displace-ment thickness, modisplace-mentum thickness, and shape parameter are derived. Finally the total drag

of the flat plate is obtained from these parameters.

From a fit to numerical evaluations, a new formula is proposód for the drag of flat

plates which agrees with the Schoenherr formUla at high Rçynolds numbers. The proposed formula gives higher values of drag at low Reynolds numbers than does the Schoenherr formula.

VELOCITY SIMILARITY LAWS

The velocity, similarity laws provide an analytical basis for describing the mean velocity

profiles of turbulent flow in boundary layers on flat plates in zero pressure gradient. For

6Cebeci, T. and G. J. Mosmskis, "Computation of Incompressible Turbulent Boundary Layers at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics Journal, Vol. 9, No. 8, pp. 1632-1634 (Aug 1971).

7McDonald H Mixing Length and Kinematic Eddy Viscosity in a Low Reynolds Number Boundaxy Layer Umted Aircraft Research Laboratories Report J214453-1 (Sep 1970).

8 - -

.,

. - - -- .

Huffman, G. D. and P. Bradshaw, A Note on von Karman s Constant in Low Reynolds Number Turbulent Flows, Journal of Fluid Mechanics, Vol. 53, Part 1, pp. 45-60 (9 May 1972).

..

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low Reynolds numbers, the outer law only is to be suitably modified, while the inner law is

to remain the same. This is in accOrd with the concepts ofColes4 and of Huffman and

Bradshaw.8

INNER LAW OR LAW OF THE WALL

The mean velocity component u parallel to the smooth wallroughness and drag-redUcing effects may also be includedis considered.dependent on the normal distance from the wall

y, the shearing stress at the wall;, and the fluid propertiesof density p and kinematic

viscosity v or

u=f[y,r,p,PJ

(1) Nondimensionally then = f[y*J ₍₂₎ UT where

r/p, the shear velocity (3)

uTy

y*

=. --

, the inner law Reynolds nUmber V

This is the usual statement of the inner law.

OUTER LAW OR VELOCITY-DEFECT LAW

The usual consideration of the velocity profile inward from the inviscid boundaryis that of a velocity-defect U - u as a function of relative position as well as the wail-shearing

stress; and fluid density p. Hçre U is the value of u at boundary-layer thickness &. Thus,

the outer law is usually stated as

UuF[y,6,r,p]

(5)

(12)

or nondinwnsionally as

Uu

ui _L5 (6)

It is to be noticed that viscosity does not explicitly appear in the outer law as it does in the inner law.

To account now for the effect óilow Reynolds number, it is:próposed to add visobsity to the factors affecting the outer law; This is justified physically by the large eddies of the outer boundary-layer flow being affected by the Viscous effects from the turbulent irrotátiönal

interface at low Reynolds numbers.8

Accordingly with the addition of kinematic viscosity v

U_.u=F[y,,r,p,v]

(7.) or nOndimensionally

U_u

-F'

' u Lo where ut&

fl -

is a similarity law Reynolds number

The effect of r on the outer law occurs then only at low Reynolds numbers.

LOGARITHMIC LAW

As demonstrated by Millikan,'. the ranges of validity of the inner and outer laws are considered to overlap, which mathematically implies loganthmic functions for both laws in the

common region.

To demonstrate this, the derivatives of u with respect to y fOr both laws are equated. Then it follows that

4

]

'°Mffljkan C. B., "A Ciiticai Discussion of Turbulent Flows in Chaññels and Circular Tubes," Proceedings of Fifth International Congress for Applied Mechanics (1938), John Wiley and Sons, New York, pp. 386-392 (1939).

(13)

afiy]

(y\ aF[y/o,]

ay*

_\oJ

_a(y/s)

By this arrangement, wherein the left side is a function of y* and the iight side of y/&, the simplest possible equality is to a constant A. Then both sides may be integrated to

give in turn

=A In y + B1

UT

for the inner law, and

Uu

_-

_A1n--+B,[]

(12)

UT

for the outer law

where A = 1/ where K is the Von Krmn constant

BL the wall factor is a constant for smooth surfaces

B2 the velocity-defect factor varies with i at low Reynolds numbers

The region of overlap is between y

and (j') ; see Figure 1.

It is to be noted that the

effect of low Reynolds number has been concentrated in B2 [fl]; 82 [ri] is obtained empiri-cally from experimental results. At high values of , B2 is constant for flat plates at value

B20.

LOCAL SKIN FRICTION

Equating the logarithmic laws, Equations (11) and (12) for the region of overlap produces

a very significant result, namely

o=Alni+B1+B[1

(13)

(10)

where

u.

_\pti/

(14)

is a convenient coefficient of local skin friction. 5

(14)

INNER LAW LOG LAW OUTER LAW OUTER LAW VELOCITY PROFILES

Oy77 (+),

U V

- =AIfly*+B1

-UT

(

6

wryl

rY

-

II +Aq

I-2L6J

L

Figure 1 - Velocity Similarity Laws for TurbulentBoundary Layers

S.) V* _TI U Ut = A In y + B1 + B2 U = fLy] UT INNER LAW

LAW OF THE WAKE

I

(15)

Since B1 is constant for smooth surfaces and A is a constant, in general it follows that with TI = fbi

B2 = f[a] (15)

This is a very useful result since in further analyses, a is used as the independent variable.

LAW OF THE WAKE

For the region oUtside the logarithmic region of the overlap, (-i) i, Coles has promulgated the law of the wake for the outer law, which in subse4uent modifications9

may be stated as

Uu

-

A,ln++B2

(1_[])_A[f]

(16)

where w wake function, and q = wake modification function. *

'0 y

The statement may be extended to the whole region of the outer law,

-

1, by

7? 6

suitable definitions of w and q. The outer law may also be stated in inner law form as

=Aly*+Bi+B2[] +A[--]

(11)

For convenience w and q are usually defined for the whole boundary layer, 0 1..

Polynomial expressions are given by Moses9 for the wake function

(y)2(y)3

and by Granville9 for the wake modification function

( \ 2

(16)

and

The outer law suggests the folloWing integral parameters

and shape parameter H where

BOUNDARYLAYER PARAMETERS

6*

6

the momentum thickness 0 where

0

6

Hence from the previous definitions 0

0

12= J01

(U_u)

The usual boundaiy-layer parameters are the displacement thickness 6* where

L'

(l.)d[]

U U 6* Ii

6. o

o 6 a 6*/a H 0/6

d [f]

(20)

fy

[a

(21) (22) (23) (24) I

(u

u

(17)

and

- 1 12/Il

H a

and 12 may be evaluated from the analytical description of the similarity laws

U-u

UI. *

_a_f(y*l,

o;- 1_;_.

X2-[(Alny0+B1)-2Ainy0+B1)+2A10

_J f2[y*Jcjy* 0 9 (27) (28) (30) 4819 ₂ 213. 13 2X1 a - X2 2520

A +

j

AB2:+ -- B +

(31) where * '0 A1 = (A In y + B1 - A) y0*

-

J f[y*J dy 0

U-u

_kin

I

wIyl\

lyl

Yo

(29)

U1-w and q being given by Equations (18) and (19).

Consequently, carrying out the integrations in Equations (20) and (21) yields

B2 X1

I

= - A+

(18)

and and R0 =

-10 2

In

_0/

ru

B2 1 4819 213 . 13 1 R0 = .A A2 + AB2 .+ ---

B2)j

l'his equation relates the local skin-friction coefficient a to local Reynolds number R0.

DRAG

The drag of a flat plate P is obtained from an integration of the wall.shearing stress, which for a plate of unit width is

(38) (39) With U6* R&. _-;;---

=°

(34) then R5 = Ii 7? (3.5)

/11

B2\ R5 = A

+ 7)

(36) Also, since Uo R0 (37) it follows that

(19)

where x is the streamwise distance along the flat plate, measured from the leading edge.

From the von Krmn momentum equation

dO

dx _pU2

D=pU2O (42)

For drag coefficient CF, defined as

CFE1_

D

20

_x

_-

2R0_R (43) where Ux (40) (41) (44)

Since R0 is given as a function of a in Equation (39), it is necessary to obtain R as a function of a in order to evaluate CF.

From a restatement of the von Kármn momentum equation

dR0

d R

-

(45')

Then as indicated by Landweber'

=

Jo2

_{d R0 = a2 R0 - 2}

_fR9

_{a da + const} (46)

(20)

The integral f R a do may be evaluated from the relation for R0 as a function of a in Equation (39). For low Reynolds numbers where B2 < B20, B2 isa function of a, and a

numerical integration is required for J R0 a do.

HIGH REYNOLDS NUMBERS

For high Reynolds where B2 is a constant, 82 = B20, a closed form integral results. If

Equation (39) is written as

/

b'

A2

R0= (a__)n_X1+_

₀ ₀ where

B0

a= -j--

A+--and Then

R =

[ao - (b

+ 2aA) a 2A (b + aA)] - X2 + const (50) VERY LOW REYNOLDS NUMBERS

There is a lower limit to the value of a for the vanishing of the logarithmic velocity laws

owing to the disappearance of the overlapping region y

to (f) .

Here the limiting value

of i is

y/(y/6)1,

and the corresponding value of a is obtained from Equation (13).

At stifi lower values of a, the flow in the boundary layer may be still turbulent, depenth

ing on external conditions. The boundary layer may be now considered to consist of a

viscous sublayer, 0 y* y*, starting from the walls and a transitional or buffer layer

which extends to the outer edge of the boundary layer. The buffer layer has both

laminar and turbulent contributiOns to the shearing stress.

4819 213

b=

A2+AB

+

2520 140

12

(21)

A velocity law is given by Squire11 for the buffer layer which is rederived in Appendix A

as

U

=Aln(y*_J)+B1

UT

which asymptotically joins the logarithmic velocity law, Equation (11). Here

J=B1+AInAA

and

= B1 + A In A (53)

Performing the necessary integrations with the Squire velocity law and with B2 = 0

results in the same basic formula for momentum thickness Reynolds number R0 as in Equation (39)

Iii

1

/4819

\1

x2 R0 = A 2520 A2)] 77 - A1 + with the difference that

aB1

A ₍₅₅₎

Then for R from Equation (46)

R = e

A _[a1a2 _{- (b1} ₊ _{2 a1 A) a + 2 A (b1} ₊ _{a1 A)]}

(56)

+ (a1 a2 - a) J - A2a + const

13

(54)

11Squire, H. B.,"Reconsideration of the Theory of Free Turbulence," Philosophical Magazine, Seventh Series, Vol. 36

(22)

where a₁

=4A

12 and 4819 b = A2 NUMERICAL EVALUATIONS SIMILARITY-LAW CONSTANTS

From plots of experimental data for the velocity similarity laws, Landweber1 arrives at values of A = 2.606, B1 = 4.0, and B20 = 2.0, which result in values of drag coefficient very

close to those .of the Schoenherr formula for the drag of flat plates. Since Landwéber uses a graphical representation for the non-logarithmic or wake part of the outer law, mstead of the

analytical representations used here, there is a Small difference in the resulting numerical

values of the boundary-layer parametets which involves integration of the velocity profile. Accordingly it is found that a small change to values of B1 3.88 and B20 2.19,

which still agree with the measured velocity profiles, with the ame A 2.606 for the

analyt-ical representations used here gives drag coefficients still close to the Schoenherr formula.

This will be apparent later. Of course with the inclusion of the effect of low Reynolds num-bérs, the agreement with the Schoenherr formula is then only at high Reynolds numbers.

VELOCITY-DEFECT FACTOR AT LOW REYNOLDS

NUMBERS

As previously shown, the velocity-defect factor B2 is a function of a. at lOw values of Reynolds numbers. Coles4 provides numerical values from a Study of a very extensive collec-tion of experimental velocity profiles Since Coles uses different values of similarity-law

con-stants of A = 2.439, B1 = 5.0,, and B2,0 = 2.68, the values of B2 have been adjusted to accom-modate the value of B20 2.19 used here, instead of 2.68; used by Coles., The difference is

subtracted. Also, B2 = 0 is taken at a = 19.7 which is the lowest value of a for the

over-lapping of the inner and outer laws and the, presence of a logarithmic relation. The results may be fitted by a polynomial

B2 =

_9.7)

(-0.344 a2 2.589 a'+ 297.6)

(73)3

(23)

for 19.1 a 27. A plot is shown in Figure 2. B2 increases with a until the constant

value B2 is reached.

COEFFICIENT OF LOCAL SKIN FRICTION

The coefficient of local skin friction Cf is usually expressed as a function Of local Reynolds number R0, where

rw ₂

Cf

a2

(60)

Values of Cf as a function of R0 may be obtained from Equation (39), using the value of B2

obtained from. Equation (59). All factors are now known, except A1 and X2 Lñdweber1

obtains A1 = 32.72, and A2 = 434.9, from an evaluation of experimental data. The results are given in Table 1 and in Figure 3.

TABLE 1 - CALCULATED VALUES OF DRAG COEFFICIENT

15 ci R0 x i0 CF io 20 805 2.515 6.402 21 1.075 3.653 5.886 22 1.443 5.358 5.386 23 1.962 7.993 4.909 24 2.708 12.124 4.467 25 3.806 18.734 4.063 26 5464 29546 3.699 27 8.029 47.611 3.373 28 11,932 77.200 3.091 29 17.712 124.26 2.851 30 26.267 198.88 2.641 31 38.922 316.86 2.457 32 57,631 502.90 2.292 33 85,279 795.54 .2.144 34 126,121 1254.80 2.010 35 186.428 1974.01 1.889 36' 275,449 . 3098.01 1 .778 37 406813 4851.30 1.677 38 600,602 7582.38 1.584

(24)

2.8 2.6 2:4 22 20 08 - 0.6 0.4 02 i8 19

/

I I 20 21 I I 22 23 24

COEFFICIENT OF LOCAL SKIN FRICTION o =

-UT

25 f

pU)

Figure 2 - Variation of Velocity-Defect Factor B2 with Coefficient of Local Skin Friction ó

(25)

5.0 4.5 C I-'C _4.0 C.) 2 0 C..) II. ₃₅ 2 U, -j 4 U

-

0 -0 I-₂ 'U U 'U 0 C., C, I I I I' I I I I I o LUDWIEG AND

A KLEBANOFF AND DIEHL

EXPER IMENTAL DATA (REF. 1) I I I- I I I I 2.0 1.5 I I I I 102 10' io io

MOMENTUM THICKNESS REYNOLDS NUMBER. R,

(26)

In Figure 3 comparison of the calculated results with the local skin-friction coefficient drived from the Schoenherr formula in Appendix B, Equation (76), shows the expected

agreement at high Reynolds numbers. There is also good agreement between the new formula and the experimental data which are plotted in Figure 3.

DRAG COEFFICIENT

Determination of the drag coefficient CF from Equation (43) requires calculation of a

local parameter R0 as a function of a in Euation (39) and an integrated factor R in Equa-tion (46) over the range from a 0 to the a in question. The variation ofa2 with R0 is

shown in Figure 4. Various regiOns may be delineated for the integration:

LaminarFlow,Oaa1

Here the classical Blasius solution is

o2 _{= 4.536 R0} (61)

Very Low Reynolds Numbers, a a 02

This is the region from laminar flow to the limit of overlap of the logarithmic

velocity laws; B2 = 0.

The change in R is obtained from EquatiOn (56).

Low Reynolds Numbers, 02 0 B2 = f [a] given by Equation (59).

The change in R is obtained from Equation (46) and involves a numerical integration

using Equation (39).

High Reynolds Numbers, a 03

B2 B20, a constant

The change in R is obtained from Equation (50). For the calculations here a = 10, 02 = 19.7, and 03 = 27.

The a Is taken from Figure 4 where the laminar line is tangent tothe line of very low

Reynolds numbers; 02 as shown in Figure 2 is the lower limit of the overlapping of the inner and outer similarity laws. 03 as shown in Figure 2 is the value of a, where B2 becomes constant.

The results of the calculations are presented in Table 1 and in part in Figure 5..

(27)

S

I I- I I

600 800 1000 1200 1400 1600

MOMENTUM THICKNESS REYNOLDS NUMBER. R0 =

Figure 4 - Variation of a2 with R0

1800 2000 600 500

.(III

.1Iii

_I0

400 -,

--

*0

.3_

\. 300

t14_

0'

.4

-/

/

200

/

10

:1

I I 200 400

(28)

PROPOSED FORMULA FOR DRAG COEFFICIENT

The results of the calculations for CF as a function of R through the medium Of the implicit parameter a as given in Table 1 may be fitted by an explicit formula. Hamat2 introduced the form

const CF =

(log R - const)2

to fit the Schoenherr formula

0.242 = log10 R CF It is found that 0.0776 60

c=

. + (64) F (log10 R - 1.88)2 R :

gives a close fit to the value displayed in Table I as shown in Figure 5. The added term accommodates the effect at low Reynolds number. There is excellent. agreement with the 'experimental data plotted in Figure 5.

Dissatisfaction with the Schoenherr formula at low Reynolds numbers led the 1957

International Towing Tank Conference (ITTC)'3 to adopt a formula, showing higher values

of CF at low Reynolds numbers

0.075 CF =

(log10 R -

2)2

This formula is based on the correlation of the drag of ships from model-scale Reynolds numbers to full-scale Reynolds numbers and represents the consensus of a committee.

Although officially termed a correlation line, this ITI'C-1957 formula is for the most part a flat-plate drag relation.

(65)

1 _{F R} Boundary Layer Characteristics for Smooth and Rough Surfaces Transactions of Socity of Naval

Architects and Marine Engineers, Vol. 62, pp. 333-358 (1954).

t3Acevedo M L and L Mazarredo Editors Proceedings of Eighth International Towing Tank Conference (1957) Canal dc Experieñcias Hidrodinamicas, El Pardo, Spain (1959).

(29)

6

5

SCHULTZ-GRUNOW ( EXPERIMENTAL DATA

WIEGHARDT

J (REF. 1)

PROPOSED FORMULA, EQ. (64)

- - - -CALCULATED. TABLE

.

106 iO

FLAT-PLATE REYNOLDS NUMBER., R

(30)

Comparison of the ITTC-1 957 formula with the proposed formula Equation (64) shows surprisingly good agreement in Figure 6. At high Reynolds numbers, both formulas agree

with the Schoenherr formula as shown in Figure 7.

Also shown in Figure 7 are some wind tunnel test data by Winter and Gaudet, assembled by Wiëghardt'4 that substantiate the Schoenherr formula at high Reynolds numbers.

'4Wieghaidt, K., "Uber den Reibungswiderstand der Platte," Institut für Schiffbau der Universität Haniburg, Report 291 (Apr 1973).

(31)

6

4

a I

PROPOSED FORMULA. EQ. (64) - - ITTC-1957 FORMULA. EQ (65)

- - - - SCHOENHERR FORMULA. EQ. (63)

4'

voo

%% 4! % -1 '4. '44 4% 4% 4 I I I I I

[II

-- I. 4% 4% .4. .4. 1o 1 6

FLAT-PLATE REYNOLDS NUMBER. R,1

(32)

3

WIND TUNNEL DATA

WINTER AND GAUDET (REF. 14) - PROPOSED FORMULA. EQ. (64) - ITTC-1957 FORMULA EQ. (65),

- SCHOENHERR FORMULA. EQ (63)

0

1 o

I I I I I -I I I I i I I I I I. I I 1 1 I I I I I I

108 io

FLAT-PLATE REYNOLDS NUMBER. R

(33)

or

APPENDIX A

DERIVATION OF SQUIRE VELOCITY LAW FOR BUFFER LAYER

For convenient reference the Squire velocity law is rederived for the transitional or

buffer layer which lies between the laminar sublayer and the logarithmic law layer,

y* _{y0* Here there are both laminar and turbulent contributions to shearing stress r}

du

(v + ) dy

where e is the turbulent eddy viscosity. Since the layer is thin, r

r,. Then

du

u = (p + e)

-Squire1' by dimensional reasoning and by considering turbulence to start at the outer

edge of the laminar sublayer _L uses a linear variation of e across the buffer layer or

= ,cu. (y - _YL)

Substituting e into Equation (67) gives the differential equation

u2

dy - v+KU,.(Y_yL)

with the initial boundary condition at the outer edge of the laminar sublayer at y = YL*,

given by

U

*

-

_UT

-Integrating Equation (69) produces

UT U

= A In (y* J) + B,

(34)

with

J=B1+AIn AA

(72)

and

yB1+A1nA

(73)

At increasing values of y the Squire velocity law becomes asymptotic to the usual

logarithmic law; see Equation (11).

(35)

DERIVATION OF FORMULA FOR LOCAL SKIN FRICTION

FROM SCHOENHERR FORMULA

The Schoenherr formula, Equation (63), fOr the total drag of a flat plate may be used to

supply a formula for local skin friction. The original derivation is not very accessible1 and

hence will be repeated here.

Substituting the relation for CF, Equation (43), into the Schoenherr formula, Equation

(63), produces

I

dR0

-.

(RO\ 1/2

dR

-

RXJ

R,/

_{4.l3/1og,0 (2 R0)}

Differentiation with respect to R gives

APPENDIX B

27

- _{log10 (2 R0) + log1} _e

/R0\

1/2

Substituting for -

from Equation (74) and using Equation (45) yields finally the formula for local skin friction

c

_0.01466

a2 pU2 2 _iog _{(2 R9)}

_[4

_{log10 (2 R0) +}_0.4343]

'5Granville, P. S., 'A Method for the Calculation of the Turbulent Boundary Layer in a Pressure Gradient," David Taylor Model Basin Report 752 (May 195!).

(74)

(75)

(36)

REFERENCES

Landweber, L., "The Frictional Resistance of Flat Plates in Zero Pressure Gradient," Transactions of the Society of Naval Architects and Marine Engineers, Vol. 61, pp. 5-32

(1953).

Coles D., "The Problem of the Turbulent Boundary Layer," Zeitschrift für angewandte Mathematik und Physik, Vol. 5, pp. 181-203 (1954).

Coles, D., "The Law of the Wall in Turbulent Shear Flow," in "50 Jahre

Grenzschichtforschung," edited by H. G&tler and W. Toilmien, Friedr. Vieweg and Sohn, Braunschweig (1955).

Coles, D. E., "The Turbulent Boundary Layer in a Compressible Fluid," RAND

Corp., Santa Monica, Calif., Report R-403-PR (Sep 1962).

Simpson, R. L., "Characteristics of Turbulent Boundary Layers at Low Reynolds Numbers with and without Transpiration," Journal of Fluid Mechanics, Vol. 42, Part 4,

pp. 769-802 (30 Jul 1970).

Cebeci, T. and G. J. Mosinskis, "Computation of Incompressible Turbulent Boundary

Layers at Low Reynolds Numbers," American Institute of Aeronautics and Astronautics Journal, Vol. 9, No. 8, pp. 1632--l634 (Aug 1971).

McDonald, H., "Mixing Length and Kinematic Eddy Viscosity in a Low Reynolds

Number Boundary Layer," United Aircraft Research Laboratories Report J214453-1 (Sep

1970).

Huffman, G. D. and P. Bradshaw, "A Note on von Karman's Constant in Low

Reynolds Number Turbulent Flows," Journal of Fluid Mechanics, Vol. 53, Part 1, pp. 45-60

(9 May 1972).

Granville, P. S., "A Modified Law of the Wake for Turbulent Shear Flows," NSRDC

Report 4639 (May 1975).

Millikan, C. B., "A Critical Discussion of Turbulent Flows in Channels and Circular Tubes," Proceedings of Fifth International Congress for Applied Mechanics (1938), John

Wiley and Sons, New York, pp. 386-392 (1939).

Squire, H. B., "Reconsideration of the Theory of Free Turbulence," Philosophical

Magazine, Seventh Series, Vol. 36 (1948).

Hama, F. R., "Boundary-Layer Characteristics for Smooth and Rough Surfaces,"

Transactions of Society of Naval Architects and Marine Engineers, Vol. 62, pp. 333-358

(1954).

(37)

Acevedo, M. L. and L. Mazarredo, Editors, "Proceedings of Eighth International To'wing Tank Conference (1957)," Canal de Experiencias Hidrodinamicas, El Pardo, Spain (1959).

Wieghardt, K., "Uber den Reibungswiderstand der Platte," Institut für Schiffbau der Universi1t Hamburg, Report 291 (Apr 1973).

Granville, P. S., "A Method for the Calculation of the Turbulent Boundary Layer in a

Pressure Gradient," David Taylor Model Basin Report 752 (May 1951).

(38)

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