Skin friction A contrubtion to the discussion of subject 2

RCHIEF

SKIN FRICTION

A Contribution to the Discussion of Subject 2

Sixth International Conference of

Ship Tank Superintendents [Me

September -

1951

Washington, D.C.

by

Louis Landweber

David. Taylor Model Basin

Lab. v.

Scheepshouwkendet

Technische Hogeschool

Sixth International Conference of Ship Tank Superintendents Subject 2 - Skin Friction

Contribution to the Discussion by: L. Landweber

Int rcrduc ti on,

It 'appears that the contributions on this subject at the present conference have not only increased the uncertainty as to the magnitude of the friction coefficient of smooth flat plates, but also the validity of some of the assumptions made in scaling resistance data has been questioned.

An attempt to clarify the former problem will be made by comparing the coefficients of shear stress derived from the fundamental shear-stress and boundary -layer measurements of Schultz-Grunow, with the corresponding values from several of the old and new friction formulations.

As a contribution to the latter question a modification of the present practice of scaling the viscous pressure drag (form resistance) will be proposed.

Scaling of Viscous Pre§sure Drag

In 1946 and 1947, under the stimulus of the Friction Committee of the American Towing Tank Conference, the towing tanks of Newport News and Stevens Institute jointly carried out a vigorous program to correlate the resistances of 4-foot models tested in their

tanks with those of 20-foot models tested at the Taylor Model Basin. These correlations were based on various friction formulations,

and employed the customary assumption that the form drag coef-ficient was independent of the Reynolds number. It was found that there was a trend towards higher residuary resistance coefficients for the smaller models. To correct for this trend, Mr. Comstock, then chairman of the ATTC Friction Committee, stated in a letter

to Dr. Davidson on 1 May

1947:

"An empirical factor applied to Schoenherr, varying from about 1010 for very lean models (high transverse curvature) to unity for models of about one foot or more in mean girth greatly improves this correlation".

In an attempt to establish a physical basis for the foregoing proposal, the present discussor carried through an analysis of the effect of tranverse curvature on frictional resistance

(TMB Report 689) in which it was concluded that the correction to the frictional resistance of a flat plate for a lean 4-foot model was only about 3 percent. It is now suggested that the remainder of the desired correction could be eliminated by means of a more rational assumption concerning the viscous pressure drag (form resistance).

The usual assumption of the constancy of the coefficient of form resistance is probably a good approximation for bodies with blunt sterns whose viscous pressure drag is principally due to turbulent separation. For streamlined bodies, however, from which the boundary layer shears away without separation, there is

evi-dence that the form resistance scales in _{a constant ratio to the}

frictional resistance as the Reynolds number is varied. In the

intermediate case, where a part of the form drag is _attributable

to separation, the form resistance would also _{be expected to}

decrease with increasing Reynolds number, but in an increasing

ratio to the frictional resistance. _{A rough symbolic} representa-tion of the aformenrepresenta-tioned _{cases could be as follows:}

Turbulent separation predominant: CDp = a

Wake partially due to turbulent separation: CDp = a 4 bCDF

Streamlined body: CDp = bCDF ,where

CDP is a viscous pressure drag coefficient

CDF is

a

friction coefficient a, b are constants.

For example consider a hypothetical fine form for which

the flat plate friction coefficients at _{corresponding Reynolds}

numbers are Cf = 0.0050 for a 4-foot model and Cf = 0.0030 for a 20-foot model, and suppose that the measured viscous drag

coef-ficients-(obtained at low Froude numbers _{so that wave resistance}

is negligible) are 0.0061 and 0.0035 respectively. _{These numbers}

as proposed by

_Comstock,

_{and adding C00005 as an assumed constant}

form resistance

coefficient-to both models. Now consider how the correlation would proceed on the assumption that the form resistance is proportional to the frictional. _{First, because of} the transverse curvature effect, the friction drag coefficient for the smaller model should be increased 3 percent over the flat

plate value, giving _CDF 0.00515. Hence the form drag coefficient

would be CDp = .00610 _{- .00515 = .00095. The form drag coefficient}

for thei 20-foot model would now be scaled as CDp = _{x .00095 =} .00055. _{Hence the predicted viscous drag coefficient for the} 20-foot model would be CDF 4 CDp _{= .00300 * .00055 = .00355, as}

compared with the assumed value of .0035.

The law of variation of the form drag of streamlined bodies

with Reynolds number Appears to have first been _{proposed by Young}

in 1939*. _{He derived this result semi-empirically from an analysis} of the computed resistances of several bodies of revolution at

various Reynolds numbers. Tulin, of the Taylor Model Basin, has shown that the proportionality of the _{pressure and firction drag} is a good theoretical approximation for 2-dimensional forms and is now attempting to derive the same result for bodies of revolu-tion. _{Tulin has also proposed a method of measuring the viscous/} part of the resistance of a body by means of a wake survey (TMB RepOrt

772)

_{and is now developing an experimental procedure for}

* (British) Reports and Memoranda No. 18740

carrying out this work. It is hoped that by means of this

tech-nique together with the devclopment

of

improved methods of computing

-the shearing and .1)ressure 'drags in preaSure gradients we will soon-

--.-know more certainly how to scale the,..7iscousdrag of a Ship MOdel6

-MC-tido:al

RettstanCe79f

a

Smooth Flat Plate with Turbulent

Boundary Layer n Zero Prep sure Gradient

In a recent report, (TMB Report 726), prepared for the

₁₉₅₀

meeting, at Ottowa, of the American Towing Tank Conference, the various theories that have been used to obtain formulas for the frictional resistance of a ,Smooth, flat plate with a turbulent :boundary layer in zero pressure gradient were reviewed and a new

and

'less approximate procedUi!e. for obtaining the frictional ,resistance was suggested. It is proposed, now to present the -results obtained by the new procedure and to compare

them:graph-ically with other theoretical and experimental; friction lines. The methods that have been used to obtain "universal" for-mUlas for the turbulent friction - drag of a smooth-

flat-

plate

begin

with

the VonKarMan moment-UM

equation

4 dx:

;where

Is the shear stress

coeffiCie4t,:.e

the momentum ' thick7 neSs. of the boundary layer, defined by

is the shearing stress at the wall p is the density of the fluid

is the distance along the plate, measured from the leading edge

II is the free stream velocity

U is the velocity of the stream at the distance y from the

-plate.7..

is the thickness, of :the :boundary layer..

Theory

and experiment have furnished two laws of the velocity

,..

-di. stribUtion

in.,

the :Theee

a The-LaW-::'Of the Wall

("Inter"

'Leif). In the neighborhood. .

ELF]

where

_[5]

b. The "Outer Law. At a sufficient distance from the

wall for the viscous stresses to be negligible in comparison with the Reynolds stresses

=

UT

= A .1) k

u

_ir

-v

au,

=

F(y/s)

_{[ 6]}

u,r

It has been found experimentally for both pipes and Plates that there is a range in the boundary layer where the .inner and outer laws overlap in which the velocity profile follows the so-called

universal logarithmic velocity distribution law. From this can

be detited the relation

With the aid of these lawsaf the boundary layer profile the

momentum thickness

e in [3]

has been expressed in various ways,

each of whiCh, when substituted

into

the momentum equation [1]

has given a differential equation relatingthe shearing stress and the Beynolds.number.

Prior to the experiments of Schultz-Grunaw in 1940 it had

been assumed, on the basis of pipe experiments, that the Logar-ithmic distribution law extended to the outer limit of the boundary layer. This assumptionlramongothers;.is incorporated in the

theory of the Prandtl-Schlichting and Von-karman friction

formu-lationS Schultz-Grunow measured the shearing_ stresses and velocity profiles on a plate and discovered marked deviations from the

velocity distributions in a pipe. He then applied the boundary

- .

layer laws in the form given by his data to obtain a new friction formula. It should be noted, however, that the form of the

analy-sis employed by Schultz-GrunOw4s- identical with that ofVon-Kar;ilan, the only difference lying..in the values chosen for four constants appearing in the ,analysis. Nevertheless, since

SChultz-.

dtlinow

chose his Constants on the basis of his plate measurements,

his formulation must be considered a great advance over previous

,

ones-.

_ .

The SchultzGrunow analysis

employed

Several assumptions

Which are either unnecessary or incorrect These will now be

considered.

1. It is assumed that the logarithmic distribution law

be seen,.,

this

appreciable errors at low Reynolds (Rx< 106). As will approximation can .easily be avoided.

-Schultz-Grunow did not use the Iogarithimic dis-tribution law given by his own boundary- layer

measure-,

ments. He determined the constants A and k in [ so as to make hits theoretical formula agree, in two points with

his experimental curve of C versus RX., He used this

procedure because with the measured values of A and k, the theoretical and experimental curves were not in

agreement. The great inconsistency between the values of ,A and k used in the friction formula and the values

given by his boundary layer measurements is shown by

'

the following comparisOn:.

From boundary layer measurements :Values used in frictionfori6la

It is believed that the foregoing

-to the next assumption.

inconsistency is due

30 It is assumed

that

a constant of integration

occur-ring in a relation betWeen'R and C, can be .evaluated by

employing conditions

at:

Rx

crux of the entire analysis is the existence- of a region in which the inner and outer laws overlap and the logar-ithmic distribution law obtains. But it can easily be shown that the overlapping. range diminishes with

O 38

2.58

3.62

2.89

,

-is -is incorrect. The

decreasing Reynolds numbers corresponding approximately to Cr,? 0.005. Hence the analytical -formula is valid only above a limiting Reynolds number. Since the con-stant of integration must be evaluated within the range of validity of the formula, Schultz-Grunows's procedure is incorrect. This restriction and criticism applies to all friction formulations based on the logarithic distribution law. Such formulae cannot be analytically extended to zero Reynolds number.

The present writer has carried through an analysis, based on Schultz-Grunows measurements in which the above assumptions have been avoided. The integration across the boundary layer was per-formed using the proper functional law in each regime, all the constants in the resulting differential equation between C-rand Rx were taken from Schultz-Grunow's functional laws of the

boundary layer, and the constant of integration obtained in

solving the differential equation was evaluated by comparison with

the experimental curve of CT versus R. It was found that this

constant could be chosen so as to bring the theoretical curve into excellent agreement with the experimental curve over the entire range of the latter. This fact is a partial confirmation of the validity of the theory..

The resulting equation for Rx as a function of

C-ris

0.388g

Rx 250,000 - 561g (0.286g2-3.45g4.14.0)e

[8]

where

An alternate form is obtained by expressing Re as a function of CI where Re =

Ue/e.

The relation between Re and CT is con-sidered more fundamental than that between Rx and gr since both Re and qy may be determined from local measurements and their connection may reasonably be assumed to be independent of the manner of transition from laminar to turbulent flow, whereas this

may not be the case for the relation between Rx and CT. The corresponding result is

500 ligg. (00286 242)ce0.388g [10]

where

g =

V2/CT

Figure 1 shows graphs of C versus Re derived from

Schultz-GrunoWs original data, his interpolation formula, and the new formula [10]. Curves derived from Schoehherr's friction formula and from the new NPL curve for a plank of length/draft = 5.0 are also shown. The values of Re near 103 correspond roughly to the small model range, the other end of the graph to the full scale ship range. It is seen that the new formula exceeds Schoenherr's curve in the small model range but is in very close agreement with it for Re,>

The large discrepancy between the Schultz,Grunow data and

the NPL curve indicatesthat the problem is far from being solved.

It does not appear plausible that the difference can be accounted

for

by the aspect ratio of. the NPL plank since it is generally

supposed that a deeper

plank

would have even lower resistance.

10

It is possible that the -free stream turbulence of the channel in which Schultz-Grthow- made -his measurements may have had an import-ant effect on the boundary layer -profiles. New measurements

similar to Schultz-Grunow's should be made under conditions of

low

free stream turbulence; it is planned to make such

measure-ments when a low turbulence wind tunnel becomes available at TMB.

There is some doubt also as to the continued validity

of

the

"outer law" of the boundary layer at very large Reynolds numbers. It is hoped that the shear stress and boundary layer measurements on the long cylinder to be towed at TMB will shed some light on this question as well.

0.005

0.004

_,0.0,03" 4

,

, . 'Figure -I Stress at W011 of a,

Flat:- Plate- with Tu-rbylent Boundary

Layer'',in'',2ie'ro..-?..r*ssure Gradient . -. . . . , .

Derived from Schultz-Grunow Interpolation Formula for Total Friction Coefficjeri.V. New Formula Based on Schultz- Grunow Curves for Inner and ..Outer Boundary Laye.r.°-,Low-s-._

-..,-.,3-.; y . -(' . .. .

Derived from Schoenherr Formula for Total Friction Coefficient

-,

..

.

,

Derived from new N.P.L CUri/e.Of'.Frictiorr,tbefficientsfor Length

/

Draft = 5;0. . . , J,.44 , A:...., 1. ..' ,...t ' ' t-. . .:.,,,,. 1-:.- , ' t ,-,-..--.. _ -,.--.. .

--Computed from Shear Stress- and Boundary Layer 'Profile Measurements of Schultz- GrUnoW

..iL-...--(Table --Rat'.) i-. ..,-,.,.-. . 4 ,,, ..._ . , . .:-.. -7'.., 0 .,. .. . - . -2 . ..0-. ..0-. . . - .,-' . ' , . 1:: , - .-. -: ., ... . --- ..__________ .,..., ) , t 4 ' t ---._--.-.-_ . . --, . . ,. ., . -4 'T? .6. 7.. ',Atiiii-: ' ' i, , , . . . .,-.- ^ 11, P . .-. :I-. . : ., ' : --r..,, . .. . '"P. t_{,, .16,} "g 1 : : --, I ...,..- ) dy .:.,, .":.1, -. .... 4 , _{_} . . .: ; t.,,-).;,--, 1.1

'

,-, F , ,._ ...

,

.4-. , 7 . . : i-- . z , 7 ' . _,...,.. , . ,-1. . . . ,. -. ' c'' --.,:4 ,

,

_V . .-V - . 1..."4, ,.... _ , f -. , " -,,i. .1-. v-fi': 4c _4,, ,:i'': . 341 34-S4 .. .

...

. . u , . . . ,v; ° z-: . ' .1 ' ,,.1. -, .m.7.-.g.c" . -.., . .; . ;,,b I' . ., . t 0