New velocity transfer theory

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ARCH1EF

New Vorticity Transfer Theory

By Hideo SASAJIMA

tab.

v. Scheepsboirvi),

Technische HogescLui

Delft

Reprinted from

TECHNOLOGY REPORTS OF THE OSAKA UNIVERSITY

Vol. 1 No. 2

Faculty of Engineering Osaka University

Osaka. Japan

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- No. 2

(Received 1, March. 1951)

New Vorticity Transfer Theory

-'By.

Hide() 'SASAJIMA (Department of Naval Architecture)

1

ap d2u

ax dY2

Preface

It. is not possible to distinguish between the momentum and the vorticiti7

transfer theories of turbulent flow by experimental verifications, and this suggest

us they are both not So reasonable. The author proposes here a: new vorticity transfer theory which contains the size of turbulent flukUparticles and the influ-ences of wall and viscosity for them. The theory agrees with the several

experi-mental results without any discrepancy.

1. New Vorticity Transfer Theory

Let us consider a steady turbulent flow, its mean velocity u is parallel to the

axis of x and is a function of y only.

G. I. 'Taylor" in his vorticity transfer

theory showed the following equation,

(v,at:tz2)2

_{( 1)}

where, L1, L2, L3 and a', w' are the mean lengths', of path and the velocities of turbulent fluid particles in the directions of x, y and z respectively, and the

bar indicates the time mean.

,

The above equation obtained from the consideration of three dirnentional

turbulence, and we may perceive it as a mosi reliable expression. But, since for

pratical purposes it is difficult to apply it as it is, we, have nothing for it but to

simplify into two dimensions. When we assume the turbulent motion is confined

to the plane xy, i. e. the fluid particles move as a long strand parallel to the z

axis, we have the two dirnentional vorticity transfer equation,

d2u ( 2;),

p ax dy2

(3)

22 New Vorticity Transfer Theory

To the contrary, if the turbulence is .confmed to yz plane, i. e. the long strand

i3 parallel to x axis

jj_p d (I-, du)

,p dylv clY71

this is the equation of the rhontentiiiii- transfer, theory.

Therefore, when an experiment coincides better tvith one of them, say (2), we may conceive the shape of particles shall be longer in z direction. The

difficulty of experimental decision between (2), and (3) suggest us that the

simpli-fications of (I) were not sound, and that we have to take the size or shape of

fluid particles into considerations. The author 'fixing the attention on these points derived the following new vorticity transfer theory.

Suppose the two dimentional motion as before. If we .assume the particles

move"1 with the iieloCity v' in yi direction coriserVing their mean vorticity,

d 2u

pv'l -0 represents the rate of transfer of momentum per unit volume. This is the case of the ordinary vorticity transfer theory , and may be true in an open space far:from a solid wall, but it is doubtful] in a closely attached .boundary

layer in which the constraint from the wall shall be the pain actor of fluid motion: From the hydrodynamics for inviecid fluid the velocity of an isolated vortex near

r'

a wall is-

_27rD' where r' is its circulation and D is the distance between the

vortex and its image. Therefore if we assume the turbulent particle were

,analo-gous with a vortex of same circulation, its velocity would be the same as before, and the direction is as u' in the momentum transfer theory. Since the number of fluctuating particles per unit time is proportional to v', let n be the proportional coefficient, the rate of transfer of momentum in y direction becomes pnv'r' 122-0, which must be identical with the Reynolds stress TR. Let ra2 be the Sectional

area of particle,

"

pn _{. 1(42vf} d2u_i2

This is the equation of new vortidity transfer theory:

. At a glance, this seems to be the second term of Momentum transfer

equation,

du p

dy2

but the physical meanings are quite different.

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2. The Effect of Viscosity

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H. SASAJIMA 23:

resistance; ;Then the rate of transfer of; momentum in the y direction is pnv2,7W;

which is identical with pnv'2, v' being the velocity in the presence of resistance.,

pnv'2 pnv2--W

The resistance W should be proportional to the cdefficient of viscosity it and

to the Velocity gradient between the particle and its surrounding fluid:

It is.

natural to think the gradient to be proportional to v and inversely. proportional

to an unknown length., On the ;other hand, the mixture length 1 maybe considered,

as a first approximation, to be inversely proportional to W. Therefore 1 stand for the above unknown length.

where is a constant depends on the size or shape of the particles: Then the

rate of transfer of momentum is

A similar relation holds for x direction also, but need not be considered,

,

because the Reynolds stress is the reaction of the resistance in x direction.

Suppose as usual

---- By

du du

v lc-Ty= By

where B is an universal constant so long as 1 is small, compared withthe dimen-sion of wall. Then

Jdu.tplu dy

1u-ry-k

---nB2

d k

represents the influence of viscosity, and' v' becomes zero at u so that the turbulent particles cannot intrude beyond this limit, i. e. the thickness- of laminar sublayei-. We shall come back to thiS point SomeWhat later.

The equation (6)(Seerias to give a' good ekplanation in the neighbourhood of

wall, but rather a less one in the region far from the wall. Vol.; in the latter

-du _{and --d7, approach to be proportional to --}

d 2u1 1

_{respectively, and therefore} v' becomes nearly constant, which does' not .agree with experiments.' -But, since

we have no theoretical argument to verify it, we put v' c/0 f(yl8.),. where f is an

( 5 )

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24 New \tonicity Transfer Theory

unknown function to be determined from the experimental results, and ô is the

thickness of boundary layer. Then (6) becomes

v' = By I cli4 IL) (1-7 ) ( 7 )

dy. dy y2 6

3. D and a

A) D, the distance between a particle and its image, is often difficult to calculate even for a simple problem. We show few examples.

Plane wall: D = 2y

Channel : D tan (71)

2 Al (12

7r 2b b

2b = distance between the Walls

Circular pipe : In the two dimentional consideration, the line of particles is a form of vortex ring which has its own velocity. Perhaps this does not fitted

for our purpose inspite of a hard calculation, and we have rather to avail the

result for channel.

From these examples, we put D =2Y.-(.Y16)

and g(y18)== 1 for plane wall

.3)

1/

Ti for channel

--=:-1/11----r-s for circular pipe

( 8 )

B) a is a characteristic size of turbulent particles. Following after Prandtl

we can apply dimensional arguments to determine a form of a. Consider a flow

along an infinite long plane wall in the .absence of pressure gradient, then 1, D, a and y are the lengths which enter directly into the problem. Since 1=By and D=2y, there remain two lengths a and y, and we have a dimensionless

combina-tion When the influence of viscosity is neglected must be constant or

a.'-Cy.

But in the neighbourhood of wall where the viscosity can not be neglected this form is not adequate, because the particles ought not to have their size in a laminar sublayer. The author assumed as follows.

a: C(Y-y1) ( 9 )

where yi, is the thickness of sublayer and C is a universal constant as B was so in (5).

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'10

SASAJIMA 25

An analogous consideration seems to be. used also for 1, but this leads to an absurd result in (6), that the resistance becomes infinitely large at

Putting (5)(9) into (4), we have

K2

PKy(y-

du (du,

k)

_da.2

d2u gym)

_.-.(y/a)

aY aY 6 ₍₁₀₎

K2= 1±B2C2

4

where K is an universal constant independent on the shape of wall so long as 1

and a are both small compared with the dimension of wall.

4. Coefficient of Correlation

The coefficient of correlation in our Case is defined as Rule- u'v' ₂ The

numerator means ritlp, and in the denominator' ;,ve may assume from (7)

f(y18)

K2 (y

57)2 dzu /du

.B2 y-f.,g; 'dy2/ dy

d2u

(y y)2and

converge to zero rapidly at y= yi. Where y becomes large,

aY

du 1 d2u 1

.,c/D-T, and dy, Cip

Al g

For a flow in channel, as an example, we have though very roughly, from the

experiment by Wattendorff2) EXP. WAT I I BY TEN DORFF ..._. 1 f(Y1°).-1 y ₍₁₂₎ 1(y11))2

=

1 1--2(y lb) If we neglect the variation of f

A{1(y11021

These are compared in Fig. 2 with the

result of calculation by KarmAn who

- based on the 'same experiment.

Agree-Ment is not goOd but may be said to be

better than the one from ordinary

theo-ries, where we can only expect constant

0 _'4

₆

_'8

Fig. 1

'04

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26 New Vorticity Transfer Theory

Hee even if we could to assume the correct value of v'.

Rtkvi

Fig. 2

. Properties of k and Structure of Boundary Layer

kly is a coefficient of the vorticity loss due to viscosity and depend on the property of turbulent particles. Therefore. k increases inversely with the growth

of turbulence, but is independent on the shape of wall. From the facts that the coefficient of resistance for a turbulent flow lies on a single curve, and that the

transition from laminar to turbulent takes place suddenly we have to conceive klv

be a function of Reynolds number (say` R), and for an actual turbulent flow it approaches very quickly to their final value. Thus kjv is an universal constant for fully developed turbulent motion. We will turn back to this point latter.

In the case v' 0, we have from (6) or (7)

duk

dy yf

This gives the thickness of laminar sublayer. Since yi is very thin we can-neglect

- du du

the Variatioh of

_{in this layer and put it equal to}

_{, the value on the}

ay dy 0 surface of wall:. ( du) k kdY10 (13) E.21/ To k ₍₁₄₎ P Yi

to is the shearing stress on the wall._ ur sometimes is called friction velocity oWing to its dimension, however, it-3 physical meaning was rather ambiguous.

(14) shows clearly that ur is proportional to the velocity at the outer end of

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k

From (7), > for turbulent region, and yi is given from (13). Therefore

dy y2

du

the distribution of vorticity becomes as shown in Fig. 3. The curve of cuts

kly2 at yi and Y2, the latter is the outer limit of the turbulent region. But for

practical purposes yz has little effect and we can neglect it generally.

H. SASAHMA 27

Fig. 3

6. Equation of Boundary Layer and its Solution

If the shearing stress distribution i known, we can Wright it aS follows

0.h(I(f) kelyil h(Yo-)

and the equation for turbulent region is

k2y(y -y1)2.44(ddyit ; ddy2112 gf(4/1:)) vegii+ viJy.72 ?lova) ₍₁₅₎ Since (15) can not be solved generally, as an approximation, divide it into two parts. At the first, in the neighbourhood of the wall each of f, g and h is

nearly equal to unity and (15) becomes independent on the shape of wall.

d u(du k d2u

duii

K2y(y -y1)2 Viri, 176

%+yyl- u

To solve this equation, put

+a2(L'iy)2 +as(b.)S (16)

coefficient al, az, az etc. are determined as the functions of K and klv. All of these except a/ represent the inatience of viscosity.

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28. New Vorti6eY .Transfer Theory

For the remaining part, the viscosity effect can be ignored arid ,(15) is much

simplified.

,du d2u Ayia)

k

₌

dy dy2 g(y f 8) yp The solution is Vkv k dy

Kyly14.

yiy

F

2y2f1" dv

Y3- f ,(17)

(17) is equal to the product of F4 and the first term of (16). Since the mutual influences between the viscosity and f are small we have an approximate

solution of (15) as follows.

du

it

- t 12 Y

) as 1)3+

The velocity in the turbulent region is

u=

fYikody +13, dudy

j 0 Yi J YidY

___kri+aiFi log +a2(1b-)+-la S1(511.12

+-la4t \(n--) +

Yi 2 3Z Y 3 (19)

In the above results, we are required only. to find the values of K and klv

from some conventional experiments. The- author obtained them from the results

of pipe flow. (see Table II)

7. Critical Reynolds Number

We have mentioned in 5 that kf v is a functiOn of the degree of turbulence and may be considered to be almost constant for turbulent flow. We will now to see a special case when -the turbulent motion just ceased,

i. e. the

motion at _the- critical Reynolds number. .

Fig. 4- shows the vorticity distribution in this

case. The, curve is laminar flow type and is tangent

du k

to dy==.372 at y--- ye, the value of k, say lee, is not

the same as for a turbulent motion but is much greater.: For, the turbulence in this case is the one existed in the laminar flow and is much weaker

than in the .tdrbulent one.

c

(18)

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H. SASAJIMA: 29

diameter; and suppose the turbulence existed, in each flow be constant. Then. the

du

curve- of lee Yz' is clraWn uniquely and all the (7. curve's become tangent to it, in other words, ieclY2 curve becomes an. of - curves.

' On the other hand, the envelope- for laminar flOws thronkh pipes under a

constant Reynolds number Can he obtained as follows

For the critical condition formerly : stated- this must be identical with the curve of

kely2. Therefore, wrighting Re instead of R lee 8 R

v 27 c

This shows the well known fact. that Re changes not with the size of pipe

but with the degree of turbulence.

The same relations hold also fok Other kind's Of Wail's. For a pressure ilow in channel of breadth 2b,

du 2 v R R:_2bU,Th ₍₂₂₎

dy y2 v

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For a uniform flow along a flat plate the solution was given by Blasius.3) Let

U be the uniform velocity and measuring x from the leading edge of the plate,

au

? 2, \

vie

₍₂₄₎

where,

₌

_1LE-V , 1,-= a function of

Ux

(722f "),na. 17 1.455 from graphical treatment:

-.". -= 1'455Ro/ . . . ... . . . ... . . (25) We can deduce the following general- relations from, these examples.

du v -a-3; x(Function of R) (26) Feely Function of Rc (27) ( 2 1 ) du 8

'dy27 y2

₍₂₀₎ R

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30 New VortiaitY Tranifer Theory

Equation (26) 'shoWS that the words ReynoldS number is constant ' fot lathinar

flows can be replaced- by the. words 'envelope to vorticity distribution curves is

constant '. And (27) shows that, if the turbulence existed in the stream is

con-stant the right hand eicie of thiS equatioil has a definite value independent On the shape of walls. Therefore the relations. between Re for different walls are given by (27). Table I shows R from experiments4' and by this relation (based on circular pipe). Result for plate is rather poor, .but, there are no reasons to

be-lieve the experiments were done under the same degree of turbulence as for pipe.

Table 1 wall exp. ' circular pipe charnel fiat plate . _critical Rey. No .2100 2,800 1-3 x 105

8. Example; Turbulent Flow in Circular Pipe

Experiments for circular pipe have been done by several authorities and are most reliable, so that it is Convenient. to utilize them for the determination of coefficients in §6..

For cireulai, pipe,

f

_{(f. )2}}

and

F=

2y1-

_dy1

y3(1+ylr)(1--2-y

Therefore the velocity distiibution is independent on the dimension of pipe.

Integrating the velocity (19) over the cross section, we have 1/k7v[1 ai lo

A/7) 3

2v-2_ g wiz' +az 1 +

)_11,

RAk

41/y

/7

1ial+za2 log avy

-k)+2a3+a4+

L,

Comparing this With the exPei-imen- t, w know is cOnstant-fof largei R, but for smaller R it grows slowly with the decrease of R. This tendency has already

been anticipated it §5 and is also seen in an experiment. Fig. 5 shows the

sesult by Nikuracise5) who obtained 1 iron experiments and momentum transfer

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du refered to the equivalent

aY

of which the breadth of turbulent region i's zero. 8 R 7 27 °

'20

ofs

0

*2

-6

--Fig. '5

therefore in this case-- . This does not actual turbulent flow of about =- <8, therefore_ur

the neighbourhood of Ro

The author assumed for it as follows,

A

at y Yi

Then, from (21).

agree with the experiments for A 7, must fall do quickly in

(29)

where k is a value when R becomes very large, and yilr is a function of R.

(See equation (33)). (29) satisfies the tendency in the neighbourhopd of Ro and becomes almost constant for actual floVv. Putting (29) into (28) they Coefficients

74-V.I 3.3 e 100 105_{-_,} ---1 -- -.

-/I

d ,/ -H. SASAJIMA 31

theory. tendency of ifi Fig. 5 i$ equivalent to increaSe rndie in the central

portion as R becomes smaller. The grows of / k in our case is similar to it

1 17

where ko is the linfitting value of k, and .72.0 is the corresPonding Reynolds

number. Distinction of Ro from Re is clear in Fig. 6. Putting Ro 1150, we have _{1 y} 185. But, from (19) we have

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32:: 1\revir'Voi.ticity Transfer Theory

are determined as shown in Table II As mentioned above V//iv may be used instead of 00) for ordinary purposes:

Table II

SeVeral results using (29) and values in Table If are,as follows.

2036 logioRAi 0947(RA) (666 .oRAi 223) (30) This coincides With the well knoWn formula

". At =-- 2 logioRAi 08 (30')

Both are shown in- Fig. 7.

du fidu`

_{ai(1--.0.+(a2-0.16.n)(n)2}

dy/ \dY b 7 .r y r y

+(a3+ 0'028n)(Y1)3+ (a4 + 0'063:n0)4+i/5(Y-1)5+ (31)

r

7-y .

ri yi

\y/

where

Fig. 8 shows (31) at ydr = 0

1.du\

k,Ili

\dY )0

ur = v . 1 [1+-1Klog y'1..-4-(a 0'16Y1\(2

f\

f+ as+ 0'028-YJ

x1.1--.(Y-1)21+- (1a4+0063.171)/1 -1(Y)3} + /1--(Y)41

y 3

r

7i 4

Where _y --,_yur

Fig. 9 shows (32) at y1/r=0 and y1Jr=1/20, and also the formula

yur

5.5+5.75 logic

/Jr

(32)

(321)

(32) agrees Keiji', well with the experiments of Nikuradse,6) but (32') is poor in

the neighbourhood of wall.

0.400 v 6.33

0-395 as 0.378

as 0.270 a4 0.148

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12

10

8

6

4

From (33) we can find the thickness of laMinar sublayer as a function of R.

Fig. 10 shows this:

io9 Fklck R 804-..[091 logior.+0'99-:11(0'83 jog/01:-Yi r Yi

4

Fig. 7 H. SASAJIMA 33

6 '8

'Fig. 9

4

4 1

_du

omattl

Cc/ -fi 5)] (33)

(15)

Fig. 10

Conclusion

The author proposed a new vorticity transfer theory containing the influence of viscosity, and obtained the following results.

Variation of the coefficient of correlation was explained.

Thickness of laminar sublayer and construction of boundary layer were

known.

General solution of boundary layer was obtained.

Physical meaning of Reynolds number especially of critical Reynolds

number was known.

Numerical calculation for pipe was quite satisfactory. Reference

Taylor, Proc. Roy. Soc. A, 135, 697-700 (1932). K4rman, Jour. Roy. Aero. Soc., 41 (1937).

Blasius, Zeitschr. f. Math. u. Physik, 56, 4-13 (1908).

Goldstein, "Modern Development in Fluid Dynamics" I, pp. 319, 367.

Nikuradse, V. D. I., Forschungsheft 356, 21 (1932).

5) " Ergebnisse der Aerodynamischen Versuchsanstalt zu Goettingen " IV, p. 20 (1932).

84

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New *eorticy 'transfer Theory

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