Diffusive Properties of Interfacial Layers

Technische lh~oschool De.ft 1:fd. Weg- en Waterhouwkl!:l'e Lab.

v

.

Vloeisto' e(,..u.!ll:..~a.

22-C-13

DIFFlrSIV E PROPERTIES OF INTER -FACIAL LATER

Cont:dbution to the XiIth lAHR congr ess by

Anders Sjö berg D.;vision of Hvd rau.lics Cha lrn er s Iri s t.it ute of Technology

Göteborg, Swcden 1967

1 NTERNATI ONAL ASSOCIATI ON FOR HYDRAULI C RESEARCH

Dl FFUSIVE PROPERTI ES OF I NTERFACI AL LAYERS By Mr. Anders Sjöberg, Civ.eng.

Divi sion of Hydr a ul.ics, Chalmers I nstitute of Technology, Göteborg Sweden

SYNOPSIS

The vertical stability of submerged s ewage fields is a matter of great im-por tance in Sweden where the coastal waters are strongly stratified. 'I'he

diffusive pl'operties of interfacial Iayers are there{ore di s cus sed and the

expe rimental results of different authors are compared.

SOMMAIRE

La stabili

té

verticale des eaux polluées suhmergées est diune grande im-portance en Suède ou les eaux cotiè r e s s ont sournises à une importante

stratification. C"est pourquoi 1"auteur a étudié les propriétés de diffusion des zones de transition et comparé les résultats expérirnentaux de diffé-rent s auteurs.

;. (

,", '; :

L INTRODUCTION

The Swedish coastal waters - particularly on the West Coast - have a pro-nounce d density stratification. This g.ives an opportunity to design a subma-rine outfall in s uch a way that a. s ubrne r ge d sewage field is established.

One of the questions whi.ch ha s to be answered is how the vertical stability of such a subrne rged sewage field is affe cted by surface cu r rent s, wind gene-rated waves, turbulence pr od uc ed at the bottom, etc.

The knowledge of the properties of interfacial layers is not satisfactory. No

feaaible theoretical a nalys is axi sts , and one must resort to model studies of density currents. Howe ve r, there art> few experimental investigations of str a« tified flow upon which a theoz-y c a n be ba.sed. It is the refor-e ne ce ss a ry to ex -tract as rnuch inform.ation as po.ssibl.e from these investigations.

The purpose of the following d.iscu ssiori is to present experimcntal data in a manner suitable te the de scripri on of turbulent transfer of matt er and rnorn err-turn acr o s s the interface.

H. DIMENSIONAL ANALYSIS

Let us corrs ider- an interfa.c ia l Ia ver between two pa r-a.IIel strearn s of different de nsit ies. As a fi r st app r oxi.mat ion this layer is supp o s ed not 1:0 be affected by rhe outer strearn s, 'I'h e properties of the la/er ca n then be described by its own independent var i.able s , which h e re are taken as

(1)

wh e r e U is a char a c t.e r is t.ic veIo c ity, ]:1 is a cha r a ct e r is t.ic elngth, p is tb e rn e a n

de ns ity of tbe two stre arns , .6p<'< p is th e de ne i.ty differ-enc e, g is the a.cc e l era> tion of gravity and

v

is the rne a n vaIu e of the kinematic viscosity. A dimension-al a na.Iys is then shows t.hat a.lI dirnensionless r atios cf interfacial data are

fUIlC-tiori s of two par am etc r s i fo r e xa.rnp Le

or (Za)

(Zb)

whe re

Re

_{::: "1'}

Uh

_;::;

Rc yno Id Ïs nurn b e r

).)

ê.e

G -

-P_

!. -

_

1-

_

::

sta.bil ity parameter p r opo s ed by Keule gan [1) .

- U3 - Re F2

A

The r e a scn for th€: introduction h e r e of the pa r arn e ter @ was that plots of experi-mental data against FA and Q seem tO be the mo st c onveni ent on~s for extrapola-tion. On tbe basis of stability analyses by Rou s e and Ma cagno [2] one ca n a150 ex-peet a neutral stability cur ve to have the sarne f orrn as the function Re F'c.2::,;constant.

[.ij

Kelllegan, G. H: Interfacial in stability and mixing in str a tiIi ed flow. Journal of Research of NBS, Vol. 43, 1949.

[2] Ma c agno, E. O. , Rou se, H: Interfacia.l mixing in str at ified flow. Transaction of the ASCE. VoL 127, 1962.

T'he selection of the characteristic quantities U and hare very important. Dif-ferent systems claim different quantities. Rous e a nd Macagno used

Dx

and h

as indicated in Fig. 2. In a [ree two-layer system thc corresponding variabiês

are Usand hs (Fig. 4).

Iriste ad of defining the geometrie variabie h from the velocity distribution one

can u s e density distribution. Such a length is lp:::lip/(dp /dz)max first introdu ced

by LöfquisL [3J . From the practical point of VIew, howe ver , 1 does not seem

to be a convenient. variabie . p

An alternati\~ cha ra ct.eristic velocity, used by Larsen

[

4] ,

is the s hea r

velo-city UI'_x

=

,

rr:-

/

p, wherc

T

.

is the maximum value of the stress in the interfacial

~-i' 1

layer.

rn.

CHOICE OF REFERENCE PLANE IN THE lNTERFACIAL LAYER.

'I'he idea of rn a ki ng the reference plane the plane of maximum s hea r stre s s

se ern s to be va r ifi e d aeproximately by th e experimental investigations of Mi chon ,

Goddet and .Bonnefille

L

S

]

.

A reworking of their experimental data, now car rie d

out by Sjöber g [6J . shows good agreement within experimental error between the

levels of rnaxi rn um s he a r stress. maximurn velocity g r a dienr. and maximum den

-sity gradient, although the turbulence is not syrnmetric, see Fig. 1. The plane

of maximum s h e a r stress (z=O) is thu s chosen as the reference plane and is here

called the interface. All quantities assigned to the interface are given the index "I".

When treating experimental data, (dujdz)i =(du!dz)max and (dp/dz)i =(dp jdz)max'

IV. INT ERF ACIAL SHEAR STRESS

lnstead of directly consiclering the interfacial shear stress

T

i'

Rouse and Macag

-no analyzed the ratio

T-

u.2. 1 IX (3.

I

dÛI

=

j/

I~

I

=

1

j/

P dz . _{dz .} 1 1

•

of the total stress to its laminar component. The experiment values showed de

fi-nite c or r e Iat ion with Re and F.a. . In Fig. 2. the plot has been transferred to th e

form of (2b) .

If Löfqui st"s results are treated in an analogous way, we obtain Fig. 3, which shows a significant dependenee on the quantity .1p /p. F.ó is approximately con

-stant throu gh out the experiments. This indicates that the me chanism of the i

n-terfacial layer is governed by th re e parameters F.a.' Q and .:lp/p. but it se e m s

difficult to give this observation a theoretical justilication. Itmust a150 be

poin-ted out that the density di fferences in Löfqui st"s experiment are extremely high. As Rouse and Macagno do not give the absolute values for their experimental data the corresponding c ornpar is o n is not p oss ibl.e.

[3J Löfq ui st,K: Flow and stress ne ar an interface be tw e e n stratified liquids.

The Physics of F'Iuid s , Vol. 3, No. 2, 1960.

[4] La r sen, I: Om tolagsströrnninger Il , Lab. for Havnebygning. Da ni s h

Insti-tute of Technology, Kopenhagen, 1962 (in Da nish]

[5] Michon, X.. Goddet,J. ,Bonnefille, R: Etude théorique et expérimentale des

c ou r a nt s de densité, T'orn e Ir. Laboratoire National d"Hyd r auIique ,

Chatou 1955.

[

6J

Sj

ö

b

e

rg,A: lnterfacial s h e a r stress and rate of mixing. Report to be

comple-ted during 1967. Divi sion of Hydraulies. Chalmers Institute of

Sjöberg [6] ha s d e ter mincd

13

i from the experimental data given by Michon,

Goddet arid Bonricf'il le, Fig. 4. T'he spread is large, but the trend is similar to that in Fig. 2.

Fig. 2 -4 ind ica te , at least app r oxi rna.tc Iy , that.F~ =cons ta nt along curve s(.3•• 9= ~ constant. Possiblv (3. in c r e a ses faster with dc c r e as in g Q than these cu/ves show. At ve r y high 'delsity di ffe r e n ces,(.3. mayalso depend on the absolute va l-ue of the dcnsity diffe rence. No significari.t infl ue nc e of neighboring so Ii d boun

d-ar ies ha s been found.

A quanti ta ti vo c orripa ris on between the fi gu r es is not p os siblc due to the different

bourida ry conditions. Mo reo ver, salt: water was used by Löf'qui st and by Rou se

and Maca gno while Michon, Goddet and Bonnefille used a clay suspension.

If the cha ra c.te r istic U<u . and h=l quantities are s elect cd , the plots show -the s a m c trend as indicated h~ove, bu~ the spread is not so large. F'u r tbe rrn o re,

the dependenee on .1p

l

p

in Fig.

3

disappears

[6J

V. DENSITY AND VELOCITY DISTRIBUTIONS

Löfquist gives the ernp ir ical e xpr es s ion

p

(z) -p A 1 [ z

;1

..1p

=

"2

1 - tanh

(\72}J ;

(3 )

for the density di str ibuti on in the interfacial layer. PA is the density of the l ower

current. This equ at ion is roughly satisfied by the data of Michon, Goddet and

Bonnefille [6J .

The practical UBC of the exprcssions 1.3.=f(FA19) requi r e s a connection between

(.3. a nd the velocity distribution. In the1present instance no feasible expression

exlsts to des c r ibe the velocity distribution in turbulent and strongly stratifïed

flows. When the density variations are s rn a l l, as in the lower layers of an adia

-batic atmosphere, the profile

?

I

d-

I

df [ ] 1/2

=-

==

1

2 ~ ...E. 1 -

v'

R.(z) .

p (z) dz dz 1 • (4)

ha s been proposed (see for exa rnp le Syono a nd Harnu r o

['IJ).

1 is a "mixing

length", (J is a function of "mixing lengths" and Ri(z) is the Ri chàr d s onIs num -ber

Ri(z) (5 )

Eq.

(4)

is essentially

th

e equation for homogenous flow given by Prandtl and

transferred to stratified flow. It cannot be integrated as long as we do not know

the length 1.

Thc expres sion

(6)

[7J Syono, S., Harnu r o ,M: Notes on the wind profile in t.h e Iowe r Iayers of a n adi abat ic atmosphere. J ourn. of the rneteorologi cal society of Japan, Series lI, Vol. 1, 1962.

p,iveI~by von Karman s e e m s rT10J:t' USCfllJ. as a w o rxing hvpothv sis. T'h e assurnp -(:jon't == constant [;i"l,(' 5, afL'!' double int c grat ion

dü 11

-

n.

1

I

}

f

~

d~)

i

I 1

₊

ln (

zl

I \

--

_. )(

+

) u.

-

u. Ut Ut

(

7

)

Drawing all jIUa..i()gy to s e dirrie nt laden s tr e a rn s

[B]

,

}[

C,Ul be expected to he 11'58

th a n 0,4 (X :-~O,4 for a b orn oge nou s flow) a nd to bc- a funct ion of

•

(Sf!.)

dz .

._

_R

.

--

·

2..·

}J 1-" {~)é. dz' i

Bc low the interface .-q. (7) fits the expe rirne nt.al r esul ts of Löfq ui st q ui te well with X. = 0.3. As exp e cte d , a bo vc the interface tbe correlation decreases with

th« tu rbuIe nce level, i. c . the vei.ccity of the hot.torn current. It must be pointe d out, howc v er, that th e expc r irn e nt.aI points ar ..' pr esented as a srn o ot.hed curve,

so syste m atic var iations in .l{ c o uld h~LV~'di.sapp eared. Ri(O) fal ls within th e lim

-its 1<. Ri(0)<- 10.

Ri (0) ₍₈₎

When cornp a red with expe rirne nta I data of Miehon, Goddet and BonnefilJe, cq,(7)

fails. The c ocffi cie nt Xs e ern s to vary in an i.rre gu Lar way. Several different vel o-city distribut.ions have been tried, for exarriple a profile a nalo gou s to the distri -bution between mixing strearn s (see Rousc [9J ). How o ver , no significant c o rr c«

lation betwo on the pararn et e r s in volvo d h as yet be c n Iound. VI. INTERF'ACIAL MIXING

Rous c and Ma cag no [2J a ssurrie d thr: volurnetr-I.cflux a cr o s s th« interface to be

Zero a nd defi ned a. rnass transfer velocity W by

<1

x

[(

P

(Z)ü(Z)dzJ + W 6p :-;;0;

A

Löfquist introduced a veloc:ity of entrairirnent , W , which does not seem to be derived from a complete salt o r rnass balance. I~owever, we here as s urne W_==W,

The dimensionless quantities

w

j

u

and

w

j

u

s show properties similar to t3.t:t?J,[6]

This could be expected , sinc e W a~d W d~pend on diffusion of rriat.ter , whil~

(3. depends on diffusion of mornent.um. eA cornpar is on between a c oe ffi ci e nt Io r tu.Èbulent tr a.ns fcr of rnat ter , é , and a coefficient for turbulent transfer of

mo-rnentum , ê_m, se ems then natur~L The former cocfficient is defined by (9)

(10

)

where qs is the resultant vertical transport of matter and D is the coefficient of rn ole cu lar diffusion. é:, is given by the relation m

T

d-=---

(

)=(v+ e )~. p z m d z • (11) 'I'hu s , at thc interface the ratio of ê. to

c

is s rn

[8J Progress report: Sediment transpo rta.tion rnecha.ni cs. Suspension of sediment.

Proceedings of th e ASCE. J'ou rnaLof the Hy dra ul ic Division, Vol. 89,

No. 45, Septernber 1963.

( 12)

On theoretica! basis Ellison and T'urne r [10] pos tulated

Rf 1,4 ( 1 - Rf)

c

_ Rf) 2 ( 13)

where Rf=Ri(z) é

I

e.

(the flux Rich ard s onïs nu m her ) arid Rf ha s a va lu o

about 0, 15. Munk. abd AWderson [1ijgive the e xp res sio n C

( 14)

which shows a s orn e wha t slow e r va riari on of ésl êrn wi th Ri( z).

Löfqu ist ~s exp c r i.rn e nta ldata rn ako it pos si bl(~ to c alc.u.late the ratio ê

,I

ê ,

ac c or ding to eq. (12) if we a ssurne W=We, Fig. 5 shows that the cxper~trll~nHH

points are low er than the t.he oretic al cur':..~ given by Ellison a nd Turner, but th at

.the tre nd is sirniIar. D is ch ose n 1,4·10 m2/ s, and experiments which give

ési <:D and ê .< V are not included. Thus, very s mall vaIues of the ratio ê /é

1 rni, 1 .fi d 1 s: rn c an oe e xpe ct ~d In strong y str ati H' a y ers . VI I. CONCLUSIONS A suitable r ep rese ntation of the interfacial s h ear stress

r.

is 1

T

v~g _1

=

i3

-

f(9

=

-~P-3 -P

J/!~

~

I

.

i U 1 F_À

The experimental data indicate that

}"'.c.

=: constant aIon g curves

(3

.

.

I:) -= constant. At v ery high d e n s ity differences, /3. mayalso depend1 on thc

absolute value of thc density d iff ercnc e. 1

'I'h e rate of rnixing acr o ss the interface shows an analogous correlation with G and F4 . T'h e ratio of the coefficient of turbulent transfer of matter to the

coe ffi.cient of turbulent transfer of momentum, h owe ver, dc cr eas es with

in-creasing Richa r d so n nurnbe r.

~oJ

Ellison. T. H., Turner, 1. S: Mixing of dense Huid in a turbulent pipo flow,

Journalof Fluid Mecha.nics. Vol. 8, 1960.

G

1J Munk , W. H., Anderson, E. R: Not e s on a theory of t.h.e thermocline.

Journalof Marine Research, Vol. VII. 1948.

1,5---..---I . ~ 1,01-...;:1---1

-•

Sa tet 10 0,1 0,5 1,0 1,5 1,5

E

1,0

.

....

x

ro 0,5 _...

S

_... 0 ..-: H .~ Il>"d .... (Ij Il> Jo<

E

eo

•

•

•

Level in meter of maximum s hea r stress. Level in meter of rnaxirnurn s he a r stress Fig. 1 Correlation between the level of rnaxirnurn velocity gradient and thc levels

of maximum shear stress and maximum density gradient in the interfacial • layer. Cal cul.ated from data give n by Miehon, Goddet and Bonnefille [5] .

T.

10

(3

.

1

=

1 dil lJP(dz). 1 ₅

Fig.2 Variation of interfacial she ar with Q and FA

by Rou s e and Macagno [Z] , 15

..

_Q.yz 11. 21

29

40 47

56

74

•

7-

10 K 0

•

0

•

4

"

(3.

=

1 dtt 1 Jlp{dz)i 5 _s

---_.

,2.J x 1~---~----4-~G4~

O,5~'_---

----

~--

--

~--

---

----~~~~~--10-4 5.10-

4

10-3 5'10-3 10-2 1,5' 10-2

Variation of interfacial she a r with Q and FA' Calculated from data

gi ven by Löfquist [3] Fig.3

•

.~.

I1

~:C

104 _I Lake Mead . ! and

_d

_

l

~

,.

L(~§_a_:.ll.ç_~

l"

·

lJ,)t

f'l ~

• ').]1

_t:

a

_~)

_.

~

1

,,7.6

103

I

. _ 5· 10- 7:1û -6. (i ~ ~~~lt __

~~~---b----~---

·

---

~----

--

'_----~

'2~

.

10-6

:;Ï.

10 - (; 10- 5 5 . 10 - 4 10 -3 2· 10 ~3 j.I~g Q::;____e_::: "<T3 V s

Va riat.ion of inte r Ia ci a l s h e ar wii.h 0 and FA' Ca.lcu lat.e d f1'o1"1'1data given by Mich.on , Go dd et and Bonnefille

U

>J.

Ll

p/p

<11'10-3. Fig. 4

6

_

---.-

-4·

t.

2 51 1U

._-6

mi

c

c

10

(

9.ë

)

dz . Ri(0) == _ g_ • 1 P (dfi) dz . 1

V~.~iat.ion of

êj?/

e,

i with Ri(O). Calculated hom data given by

Löt quist [3

J

.

Sym

hms.

sec Flg. 3.

1 .5

Fig.5

•

•

DIFFUSIVE PROP~R Tl~S OF INTERFACIAL LA YER

by

Mr

.

Ander" Sjöberg, cl v. eng. Pi,cusston 'qy the author.

In fig, 2.3 and

4-

te ntatfve \o'\lrves ~i' Q

=

constant with FA ::a constant

have been drawn. The cu r ve s

eau

a l s o be gaven by the equation

lJ

i

Re

=

constant; FA

=

constant

..

,

.

(15)

•

Thls relation is easily found by an {nspectional analysis of the equation of motion for Inhomogeueoua and fully turbulent flow.

The equatton of motion may be written in dimensionless farm by int r o>.

,ducing constant f'eference quantities .. These quantities form dimensionless groups. According to the theory of rnodeIa , similitude is achieved if the dimensionless rorrn ot the ditferential equation governing the phenomenon is identical in both model and prototype systems. This requires that Froude' s number

F

and a characteriatic concentration care nurn erica Il.y equal in model and prototype (influence of vi scosdty is neglected). In most cas e s the concentra-tion

c

may

be

replaced by a characteristic density difference

A

pip. The requirement may then be stated as

Fm

=

Fp;

(.Ê..e..)

= (~)

P ril P P

when the subscript m refers to the model and p to the prototype.

(16)

If we now introduce time average s , turbulent fluctuations and eddy viscosity

e

we wiU find that similtude requires that

(17)

• where

D

=

uL/e

is a Reynolds nutnber in which kinematic viscosity j)

is replaced by eddy viscosity é, . As the introduction of D is a pure con-struction it follows from equatron

(16)

and (17)that

D

=

f₁(F_. ~)._p' (18)

aearrangement and introduction of the densimetric Froude number F.A gives

~ - A - Re . f (F

A.e.) .

J) - fJ i - Z À' P •

( 19

)

FA

=

constant and

!e

=

constant now gives

13i

=