Liquid Spills on Water - The Problem of Oil Pollution

149

CHAPTER 7 LIQUID SPILLS O N W A T E R - T H E P R O B L E M O F OIL POLLUTION

T h e types of spills of present interest are those of liquids lighter t h a n water, w h i c h , w h e n spilled o n the surface, t e n d to s p r e a d

out

u n d e r t h e a c t i o n o f g r a v i t y or s u r f a c e t e n s i o n . M a n y l i q u i d s i n t h i s category are transported in large quantities b y ships, not only crude oil a n d r e f i n e d p r o d u c t s , b u t a l s o c h e m i c a l s s u c h a s a m m o n i a . I n a special category w e h a v e the cryogenic fluids, m a i n l y L N G (liquefied natural gas) or L P G (liquefied p e t r o l e u m g a s ) . Spills of such fluids o n w a t e r give rise to v i g o r o u s boiling a n d

to o t h e r p h y s i c o - c h e m i c a l

reactions. The spreading process and the physico-chemical processes often i n t e r a c t , a n d it m a y b e n e c e s s a r y to c o n s i d e r a c o u p l e d a n a l y s i s . E v e n for u n s t a b i l i z e d c r u d e o i l w e h a v e a c o n s i d e r a b l e m a s s l o s s , perhaps

5 0 % i n t h e first

12 h o u r s for o i l f l o w i n g d i r e c t l y f r o m

an

underwater well ("blowout"). T h e spreading theory, without correction for m a s s l o s s , g i v e s r e a s o n a b l e

estimates

and

ad-hoc

corrections

i m p r o v e t h e r e s u l t s f u r t h e r w i t h o u t m u c h effort. L a t e - t i m e e s t i m a t e s of oil spills h a v e n e v e r t h e l e s s p r o v e d to b e o f d u b i o u s v a l u e ,

not

b e c a u s e t h e t h e o r y i s w r o n g , b u t b e c a u s e c r u d e o i l is a c o m p o u n d o f m a n y substances, a n d the drifting a n d spreading slick c h a n g e s properties

substantially

over

time. W h e r e a s the

late-time

its

theory

predicts a very large slick of thickness only a few microns, in reality tar b a l l s or e m u l s i o n , t h e s o - c a l l e d " c h o c o l a t e m o u s s e " , a r e o b s e r v e d . A theory which assumes constant

and

uniform properties, clearly

will b e of limited value. A t e a r l y t i m e s , t h e t h e o r y d e v e l o p e d for o i l s l i c k s h a s p r o v e d v e r y useful, n o t o n l y for o i l , b u t a l s o for o t h e r a n d l e s s v i s c o u s s u b s t a n c e s s u c h as L N G . ( T h e t i m e scales, early a n d late, will d e p e n d o n spill size, a n d w i l l b e d i s c u s s e d l a t e r . ) T h e theory of oil spills also p r e d a t e s a n d m a y well h a v e inspired many

of the

atmosphere. solutions

theories

developed

"Box-models",

for m o v i n g

"slab

sources

or

for h e a v y - g a s d i s p e r s i o n models"

and

sources

in

certain

streams,

in

the

similarity were

first

d e v e l o p e d for t h e s p r e a d o f o i l o n w a t e r . T h e s e t h e o r i e s w e r e i n t u r n inspired b y classical solutions in g a s d y n a m i c s [ G . I . T a y l o r ' s "strong explosion"

(1950)]

mathematical

or

from

connections

are

"shallow-water" perhaps

clearer

wave than

theory. the

The

physical

s i m i l a r i t i e s , b u t it is w o r t h w h i l e t o a t t e m p t a u n i f y i n g v i e w o f t h e s e methods and results.

Chapter 7

150

7.1 7.1.1

Physical Processes and Spreading

Blokker

Forces

(1964)

and

Simplifications Parameters

proposed

what

amounts

to

a

semi-empirical

spreading l a w of limited range, a n d although incorrect (in terms of t i m e d e p e n d e n c e ) it s e r v e d t o d r a w a t t e n t i o n t o i m p o r t a n t

parameters.

F a y (1969) derived the correct spreading laws b y the m e t h o d discussed i n C h a p t e r 6. H e i d e n t i f i e d t h r e e r e g i m e s c h a r a c t e r i z e d r e s p e c t i v e l y b y the balance between gravity a n d inertial forces, gravity and viscous f o r c e s a n d finally s u r f a c e t e n s i o n a n d v i s c o u s f o r c e s . T h e importance o f gravity, as a spreading force, is illustrated

in

Fig. 7-1. A bucket of water si!bmerged in water will experience no p r e s s u r e o n t h e b u c k e t w a l l w h e n t h e free s u r f a c e s i n s i d e a n d o u t s i d e are o n the s a m e level. If the w a t e r in the b u c k e t is r e p l a c e d b y the s a m e w e i g h t o f a fluid o f lesser d e n s i t y ( e . g . o i l ) , the free

surface

i n s i d e w i l l b e e l e v a t e d a n d t h e b u c k e t w a l l is s u b j e c t e d t o a n o u t w a r d pressure.

O n denoting oil and

w a t e r w i t h i n d i c e s o a n d w,

the

condition of equal weight can be expressed Po9 o S

=

Pw9$

w

a n d t h e h e i g h t Ay a b o v e t h e w a t e r is A

z

*

& \Pw~Po)

water Fig.

7-1;

Ap

£

water Unbalanced hydrostatic

lateral pressure equilibrium.

Ap for

AP oil

volume

in

Liquid Spills on Water - The Problem of Oil Pollution The

parameter

Ap

= p

- p

w

will

0

occur

151

frequently

in

later

developments. A s t h e o i l , i n h y d r o s t a t i c e q u i l i b r i u m v e r t i c a l l y , is f o r c e d o u t w a r d d u e to t h e u n b a l a n c e d

lateral pressure,

the water layers in

contact

w i t h t h e oil a r e f o r c e d t o f o l l o w d u e t o t h e a c t i o n o f v i s c o u s s t r e s s e s (Fig. 7 - 2 ) . T h e s t r e s s o n t h e t o p s u r f a c e c a n b e n e g l e c t e d , a l t h o u g h w e w i l l l a t e r c o n s i d e r t h e effect o f w i n d s t r e s s o n t h e drift o f oil s l i c k s . O n t h e w a t e r s i d e , t h e "sudden" m o t i o n o f t h e s p r e a d i n g oil g i v e s r i s e to a v i s c o u s layer a n a l o g o u s to that d i s c u s s e d

in S e c t i o n 6.3,

the

Stokes' P r o b l e m . T h e v i s c o s i t y o f oil will m o s t often b e o n e to t w o orders of magnitude stress,

[i (du /dy) 0

0

higher than that of water. T h e continuity of

= ii (du /dy), w

w

s h o w s that the velocity

gradient

inside the slick will b e negligible in c o m p a r i s o n w i t h that in adjacent

the

w a t e r layer. Relative to the water, the slick m o v e s as

homogeneous

slab.

For

substances

less

viscous

than

oil,

a

this

conclusion can be questioned, but experimental observations indicate that

the

results

obtained,

using

this condition,

give

reasonable

a g r e e m e n t a l s o for l i q u i d s o n l y s l i g h t l y m o r e v i s c o u s t h a n w a t e r ( e . g . Diesel oil). T h e initial m o t i o n of the

oil is d r i v e n b y g r a v i t y a n d

resisted

primarily b y inertia in the usual case of negligible initial velocity. A s t h e s l i c k s p r e a d s , t h e r e is a r a p i d g r o w t h i n t h e w e t t e d a r e a , o i l - t o water, a n d the v i s c o u s stress at the oil-water interface b e c o m e s the p r i m a r y r e t a r d i n g f o r c e . T h e c h a n g e o v e r b e t w e e n t h e t w o r e g i m e s is gradual, but w e can estimate the characteristic time scales (the early a n d late times) from the equations o f motion. A s the slick is r e d u c e d ,

the gravity force will b e c o m e negligible in

thickness

comparison

w i t h the intermolecular forces, a n d surface tension t a k e s over as the

Fig.

7-2;

Water boundary spreading slick.

layer

and

velocity

distribution

for

Chapter 7

152

driving force. In w h a t

follows w e will s h o w h o w the

derived from the equations proportionality

(as

in

spreading

laws can

be

of motion, not only as expressions of

Fay's

similarity analysis), but

as

definite

solutions. T h e analysis will b e p u r s u e d in rather detailed form, as the effort

expended

will

lead

to

a

better

understanding

of

mathematically related problems of heavy-gas dispersion and

the

strong

explosions. We

will

follow

Waldman (1971,

here

the

original

solution

of Fannelop

and

1972) rather than the elaboration (with a changed

n o m e n c l a t u r e ) a p p e a r i n g i n H o u l t ' s r e v i e w ( 1 9 7 2 ) . A s a p r e l u d e to t h e somewhat

complicated

exact

solution,

w e will

first

derive

the

a p p r o x i m a t e "flat-slick" s o l u t i o n , w h i c h i n effect w a s t h e first " b o x model"

solution.

T h e flat-slick results

are

remarkably

accurate,

w h i c h i s e a s y to u n d e r s t a n d , b e c a u s e t h e s l i c k c r o s s - s e c t i o n p r e d i c t e d i n t h e e x a c t a n a l y s i s i s n e a r l y flat. A n early, but frequently forgotten p a p e r b y A b b o t t ( 1 9 6 1 )

should

a l s o b e m e n t i o n e d . It c o n s i d e r s t h e s p r e a d i n g p h e n o m e n o n f r o m t h e viewpoint of w a v e theory and discusses also the important role of the front

and

the

relevant boundary

condition. W e will discuss

this

p r o b l e m in detail later.

7.1.2

The Flat-Slick

Approximation

or "Box

Model"

W e c o n s i d e r a g i v e n oil v o l u m e V s p r e a d i n g e i t h e r i n a c h a n n e l (j = 0 ) o r r a d i a l l y (j = 1 ) . O n d e n o t i n g t h e l e a d i n g - e d g e p o s i t i o n x

L E

,

the

slick thickness will b e given b y

where

L defines the

corresponds

to a

length

channel

scale

such

that V = L

2

of unit width. Our

+

J and j = 0

slick has

a

finite

t h i c k n e s s at its l e a d i n g e d g e , a n d it w i l l i n t r u d e f o r w a r d i n t h e w a t e r in accord w i t h the l a w s discussed in Chapter 4

^ - ( k

a 5

\

l

/

w h e r e k\ is a n e m p i r i c a l c o n s t a n t , a = g Ap/p

2

w

a n d 6IE i s t h e l e a d i n g -

Liquid Spills on Water - The Problem of Oil Pollution edge

thickness.

On

substituting

8LE = & f r o m

153

(7-2), w e obtain

a

d i f f e r e n t i a l e q u a t i o n for xi£ w h i c h c a n b e i n t e g r a t e d d i r e c t l y

dx LE dt

k

a(l

l

1/2

+j)L J 2+

W i t h t h e u s u a l "similarity" c o n d i t i o n x integration _f

LE ~

X

which

3

+

A2/(3 j) +

fcj

(l+j)/2

LE

(2^

o(l

(t = 0 ) = 0, w e o b t a i n u p o n

L E

2 + j 11/(3 +J)

+J)L

2/(3 +J)

[-2-J

corresponds

very closely to the

result

from

the

(7-3)

complete

analysis. It m a y b e w o r t h w h i l e t o w r i t e o u t t h e r e s u l t s for t h e t w o c a s e s o f interest: f A \ 1/4. 1/4 ( r a d i a l , J = 1) :

x

(planar, J = 0) :

x

LE

LE

The empirical constant discussed.

It

consideration

is

= (±)

(7-3a,b)

[aL*) \"* l/

=

(aL ) 2

1 / 3

t ' 2

3

k\ i s s e t e q u a l to u n i t y for r e a s o n s y e t to b e

evident

from

the

analysis

that,

of both geometries simultaneously

although

leads to a

the more

c o m p a c t a n a l y s i s , t h e r e s u l t i n g e x p r e s s i o n i s a l s o m o r e d i f f i c u l t to assess a n d interpret. The

f l a t - s l i c k s o l u t i o n i s s o s i m p l e t h a t a d - h o c c o r r e c t i o n s for

m a s s loss ( e v a p o r a t i o n ) or c h a n g i n g oil properties (emulsification) easily can be incorporated. W e will discuss analogous i m p r o v e m e n t s i n c o n n e c t i o n w i t h t h e d e r i v a t i o n o f b o x - m o d e l s o l u t i o n s for h e a v y gases w h e r e m a s s addition ( e n t r a i n m e n t ) is of p r i m a r y interest.

7.2 7.2.1

T h e Mathematical T h e o r y of Spreading Oil Equations

of Motion

and Relation

of

Terms

In the flow r e g i m e s first c o n s i d e r e d , three nondimensional p a r a m e t e r s appear, the ratio o f oil a n d w a t e r densities or rather

154

Chapter 7

Aplp

, t h e o i l s l i c k d e p t h - t o - l e n g t h r a t i o < 5 / L , a n d (for t h e v i s c o u s

w

0

force) the characteristic

Reynolds number

layer. T h e nature of the

flow

0

for the w a t e r

boundary

(e.g., gravity-inertial or gravity-viscous)

m u s t be determined from the relationship b e t w e e n these

parameters.

T h e e q u a t i o n s o f m o t i o n , i n a f o r m v a l i d i n b o t h flow r e g i m e s , a r e given below. du Tdx ~

+

du

.u

Tdy "

-X

8U dU dU — : + u—- + u — = dt dx dy

^ =

(

0

1 dp Po d U -f- + FT p dx p dy

7

"

4

)

2

(7-5)

2

0

0

1 dp

dv dv dv — +u — +v — = dt dx dy

~ - g dy

p

7-6

y

0

H e r e j = 0 for t h e p l a n a r c a s e a n d j = 1 for t h e r a d i a l c a s e , w i t h x t a k e n to

be

the

radial

perpendicular

coordinate

in

that

to the w a t e r surface

instance.

The

coordinate

is y , a n d y = 0 represents

the

horizontal water surface prior to the spill. W e integrate E q u a t i o n s ( 7 4 ) t o ( 7 - 6 ) a c r o s s t h e o i l l a y e r o f t h i c k n e s s <5 f r o m t h e o i l - a i r i n t e r f a c e ( s u b s c r i p t e) to t h e o i l - w a t e r i n t e r f a c e ( s u b s c r i p t

(Fig. 7-2). In accord

w i t h the thin-layer a s s u m p t i o n w e will neglect vertical accelerations. Eq. (7-6) then yields the hydrostatic relation b e t w e e n the pressure the layer a n d the oil thickness: p - p

=g p

e

(y

Q

in

- y). B y d i f f e r e n t i a t i n g

e

the pressure relation w i t h respect to x a n d m a k i n g use of (7-1), i.e. o f t h e fact t h a t t h e w e i g h t o f t h e o i l e q u a l s t h e w e i g h t o f d i s p l a c e d w a t e r , we

derive

an

expression

for

the

pressure

gradient,

which

is

independent of y ^P^

Pw-Po^d

g

dx

b

h

G

p

ApdS dx

w

y

H

o

p

w

d x

It f o l l o w s f r o m E q . ( 7 - 5 ) , n e g l e c t i n g t h e s m a l l t e r m v du/dy v i s c o u s t e r m (\i ip ) d u/dy , 2

Q

that

even

if the

independent

2

Q

viscous

and the

that u is i n d e p e n d e n t o f y . W e note also term

is

retained,

u

is still v e r y

nearly

o f y . T h e r e a s o n for t h i s i s t h a t t h e v i s c o s i t y o f o i l i s

v e r y m u c h greater (by, say, o n e to t w o orders of m a g n i t u d e ) than that of water, so that the slick tends to m o v e locally as a h o m o g e n e o u s slab relative to the water, (Fig. 7-2). B y integrating the m o m e n t u m equation ( 7 - 5 ) across the layer, w e c a n express the v i s c o u s t e r m in a m o r e c o n v e n i e n t form

Liquid Spills on Water - The Problem of Oil Pollution

dU dU — + U dx j

dt

* Pw

dx

Po

[dy)

i

C o n t i n u i t y o f s t r e s s a t t h e i n t e r f a c e r e q u i r e s t h a t ii

Q

where r

= C u

w

2

Re

w

~

1

/

155

(du/8y)

= r ,

t

w

is the interfacial stress exerted b y the water

2

b o u n d a r y l a y e r , ( F i g . 7 - 2 ) . T h e ( w a t e r ) R e y n o l d s n u m b e r Re

w

on a suitably chosen reference velocity and

is b a s e d

the distance from

the

slick leading e d g e to the point considered. W e then h a v e

^ dt

(7-8)

u ^ =- g ^ - - ^ dx * Pw dx Po $

+

The normal velocity difference v

e

- v is obtained from the t

continuity

equation (7-4) [ ejdU

ll\

y

_

. U

JdU

T h e v a l u e s o f t h e n o r m a l v e l o c i t y at t h e t o p a n d b o t t o m o f t h e l a y e r , v

e

and v , are given b y {

d

y

(dd

Ap

dS

e —— | — + u — dX P \dt dX ^1 , „ f o _ ^ 83 UdX • e

dt

+ u

w

_ _ _

As 8 = y

e

- y, t

+

u

_

„

_____

( F i g . 7 - 2 ) , w e find

88 in88 u\ ^ — — Jdu fo —+ j— =0 dt dX \ dX X,

(7-9)

The integral m o m e n t u m and continuity equations (7-8) and (7-9) govern

the

motion

of an

oil slick

and

are

subject

to

suitable

conditions describing the m o d e of release of the oil. T h e y have

the

s a m e form as the e q u a t i o n s g o v e r n i n g tidal w a v e s of finite a m p l i t u d e (the shallow water theory, L a m b , 1945). T h e nonlinear terms u a n d u 88/8x a s du/dt a n d

8u/8x

cannot b e neglected, b e i n g of the s a m e order of magnitude 88/dt

To gain insight, w e shall simplify the problem b y assuming

that

Chapter 7

156

the

flow

is p l a n a r as

instantaneously,

o p p o s e d to radial,

that the

a n d t h a t it i n i t i a l l y h a s

oil is

a rectangular

released

shape. W e

take a =g

Ap — Pw

A t s o m e t i m e ti i n t h e d e v e l o p m e n t o f t h e s l i c k , let 8\ a n d L\ d e n o t e a characteristic depth and length of the slick s u c h that L cross-section

of

the

slick

in

the

plane

of

= 8\ L\ is t h e

2

motion.

We

nondimensionalize the terms in the equations b y setting i -

L

t =t t, x

x - L

x

x

, 8 - 88

, u = -—u

X

and r - L t ~ W ~ 1 1 T

W e then obtain

U

3 / 2

1

lo a \ yW "wJ

l

T W

,

2

°Z + u ^ +8 - ^ =0 dt dX dX du _ du — + dt dX

a8 t o Y

2 L

S

2

x

l

(7-10)

I

d

1

F~ \Pw Pw ^ l ) P8 \ >

dU

w

Q

X

w

/

2

^ -

(7-11)

l

d

F r o m E q . ( 7 - 1 0 ) , w e s e e t h a t all t e r m s i n t h e c o n t i n u i t y e q u a t i o n a r e o f e q u a l o r d e r . F r o m E q . ( 7 - 1 1 ) , w e f i n d t h a t t h e r a t i o o f v i s c o u s to inertial terms is proportional to L [PwPw]

t

1

/

2

——jPo ^

W e c a n c o n c l u d e t h a t i n t h e e a r l y s t a g e o f d e v e l o p m e n t o f t h e slick, the v i s c o u s t e r m in the m o m e n t u m e q u a t i o n is m u c h smaller t h a n t h e i n e r t i a l t e r m s ( b o t h L\ a n d t\ s m a l l ) a n d c a n t h e r e f o r e b e n e g l e c t e d , l e a v i n g a b a l a n c e b e t w e e n t h e i n e r t i a l a n d g r a v i t y (a d8/dx) t e r m s . In the later stages, h o w e v e r , the v i s c o u s t e r m c o m e s to p r e d o m i n a t e o v e r t h e i n e r t i a l t e r m s ( b o t h L\ a n d t\ l a r g e ) , s o t h a t t h e inertial terms o n the left-hand side o f Eq. ( 7 - 1 1 ) c a n b e neglected, a n d the v i s c o u s term is b a l a n c e d b y the g r a v i t y term. T h e crossover time b e t w e e n the gravlty-inertial and gravity-viscous regimes w o u l d be

Liquid Spills on Water - The Problem of Oil Pollution

157

e x p e c t e d to o c c u r w h e n t h e r a t i o o f v i s c o u s to i n e r t i a l f o r c e s b e c o m e s u n i t y , or w h e n L\ t\

=p

1/2

L /(p

fx ) .

2

0

If the inertial and gravity

l/2

w

w

t e r m s a r e t h e n still a p p r o x i m a t e l y i n b a l a n c e , w e w o u l d h a v e for t h e c r o s s o v e r t i m e t\ ~ a '

2

/

7

(p /p )

(PwlAW

6/7

0

for a c r u d e - o i l s l i c k w i t h L

3/7

w

2

^

8

/

7

» °

r

a b o u t 10 m i n

= 2 5 m . It is t h e r e f o r e s e e n t h a t 2

the

d u r a t i o n o f t h e g r a v i t y - i n e r t i a l r e g i m e is r e l a t i v e l y s h o r t .

7.2.2

One-Dimensional

Slick: Initial

Growth

W e r e t u r n to t h e d i m e n s i o n a l f o r m o f t h e i n t e g r a l e q u a t i o n s ( 7 - 9 ) a n d ( 7 - 8 ) , t a k e j = 0, a n d n e g l e c t t h e v i s c o u s t e r m . W e follow L a m b (1945) and

multiply Eq. (7-9) b y an

unknown

f u n c t i o n J' (8), w h e r e t h e p r i m e d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t to S. B y a d d i n g t h e e q u a t i o n w h i c h r e s u l t s to E q . ( 7 - 8 ) w e o b t a i n

provided

It f o l l o w s t h a t / = ± 2 c, w h e r e (7-12) is t h e w a v e v e l o c i t y . W e d e f i n e t h e R i e m a n n i n v a r i a n t s

ol

"

= C+U

(7

Q = 2 C-IL

13)

such that dP dt

i y

]

^^P dx

~

— + (u - c) — = 0 dt dx v

]

T h e n P is c o n s t a n t for a g e o m e t r i c a l p o i n t m o v i n g t o t h e r i g h t w i t h a v e l o c i t y u + c a n d Q is c o n s t a n t for a p o i n t m o v i n g to t h e left w i t h a velocity

u-c.

T h e p r o b l e m o f t h e i n i t i a l g r o w t h o f t h e s l i c k is a n a l o g o u s to t h e

Chapter 7

158

w a t e r c o n f i g u r a t i o n after a d a m s e p a r a t i n g t w o b o d i e s o f w a t e r

has

broken, Stoker (1957). T h e solution c a n b e d e t e r m i n e d b y the m e t h o d of characteristics.

T h e c o n d i t i o n s at t = 0 for a s l i c k w h i c h is i n i t i a l l y

flat a n d at r e s t a r e , u (x,0) = 0; 8 (x,0) = 8\ for 0 <; x <; x\, x > x\.

Near

characteristic

the

leading

edge

of the

slick,

8 (x,0) = 0 for

we have

from

the

equations P = 2c + u = 2 c

l

Q=2c-u=0 A d d i n g a n d s u b t r a c t i n g , it f o l l o w s t h a t t h e t h i c k n e s s a n d v e l o c i t y o f t h e e x p a n d i n g slick n e a r t h e l e a d i n g e d g e a r e g i v e n b y

(7-14a)

8 = -^ 4 u = c

l

(7-14b)

=
(7-14c)

l

A c c o r d i n g l y , t h e i n i t i a l s p r e a d o f t h e s l i c k is d i r e c t l y p r o p o r t i o n a l time: x

L E

= x\

+ c\ t. E q u a t i o n s

to

( 7 - 1 4 ) a p p l y to a finite region of

constant properties near the leading edge, see Fig. 7-3. T h e receding e d g e o f t h e d i s t u r b e d r e g i o n t r a v e l s w i t h t h e v e l o c i t y - ci ( s e e S t o k e r ( 1 9 5 7 ) for t h e d e t a i l s o f t h i s d e r i v a t i o n ) . B e t w e e n t h e r e c e d i n g e d g e a n d t h e r e g i o n o f c o n s t a n t t h i c k n e s s n e a r t h e l e a d i n g e d g e , t h e r e is a n expansion

r e g i o n i n w h i c h P = 2 c\.

The

straight

characteristics

within this region are given b y dx

^

x

~

=—

x

\

=

u-c = 2 c 3 c r

T h e r e g i o n is b o u n d e d o n t h e l e a d i n g - e d g e s i d e b y t h e l i n e dx/dt

= u-c

= 1 / 2 c i (from E q u a t i o n s ( 7 - 1 4 ) ) , x = x\ + 1 / 2 c\ t. F r o m t h e p r e c e d i n g c o n s i d e r a t i o n s , w e h a v e for t h e d i s t r i b u t i o n o f t h i c k n e s s

Liquid Spills on Water - The Problem of Oil Pollution

(x-

X l

159

) x

9

l

- c

l

t

< x <

X j + ^ C j t l

(7-15)

(see Fig. 7-3). T h e velocity of the leading e d g e in this solution is consistent

with

overall m a s s conservation: the m a s s contained in the region b o u n d e d by

the

advancing

and

receding edges equals

the

mass originally

c o n t a i n e d i n t h e r e g i o n s w e p t b y t h e r e c e d i n g e d g e . It is i n t e r e s t i n g to note that the solution obtained b y linearizing Equations ( 7 - 8 ) and (79) d o e s n o t satisfy this overall m a s s conservation r e q u i r e m e n t hence

must

be

rejected.

In the

corresponding

acoustic

and

problem,

linearization is possible, b u t in that case the signal v e l o c i t y greatly e x c e e d s the b u l k velocity in the fluid. A s a r e s u l t o f finite r e l e a s e r a t e s a n d o f r e f l e c t i o n s f r o m t h e c l o s e d end, the ideal w a v e s y s t e m d i s c u s s e d a b o v e will b e s o m e w h a t different from that o b s e r v e d in e x p e r i m e n t s . In addition, the slick is not thin initially a n d t h u s m a y not b e described adequately b y our differential

Fig.

7-3:

Initial

growth

of

slick.

Chapter 7

160

equations.

Since the initial m o d e of propagation (constant

e d g e v e l o c i t y ) h a s n o t b e e n o b s e r v e d i n e x p e r i m e n t s , it is to turn to the

p r o b l e m of slick spreading

at l a t e r

leading-

appropriate

times, w h e n

a

similarity solution can be derived. A numerical solution based on the method of characteristics

w h i c h a c c o u n t s for w a v e r e f l e c t i o n s

and

i n t e r a c t i o n s , w o u l d p r o d u c e t h e s a m e r e s u l t s , b u t o n l y after t h o u s a n d s of time-steps.

7.2.3

Similarity

Solution,

Gravity-Inertial

Regime

W e h a v e o b s e r v e d , for t h e s p r e a d i n g p r o b l e m s c o n s i d e r e d s o far, t h a t the l e a d i n g e d g e p r o p a g a t e s a s a p o w e r o f t i m e , x also

seen

that

the

front

velocity

is

controlled

condition, i.e. the Froude n u m b e r b a s e d

by

on frontal

~ t.

W e have

an

auxiliary

n

L E

thickness

and

d e n s i t y is c o n s t a n t . I n t h e a b s e n c e o f e v a p o r a t i o n or o t h e r l o s s e s , t h e total oil v o l u m e i s c o n s e r v e d ( g l o b a l c o n d i t i o n ) . F r o m s y m m e t r y it is a l s o o b v i o u s t h a t t h e v e l o c i t y at t h e s l i c k c e n t e r m u s t v a n i s h . W e can c o m b i n e these conditions w i t h the equations of m o t i o n in integral form, i.e. E q u a t i o n s ( 7 - 8 ) a n d (7-9), a n d attempt to find

an

appropriate

as

similarity solution. T h e outlook is hopeful i n a s m u c h

the mathematical

p r o b l e m is a close a n a l o g u e

to the

problem of

strong blast w a v e s considered b y G.I. Taylor (1950). W e introduce the n e w similarity variable

X =—

(7-16)

LE

X

where x

=At

L E

(7-17)

n

a n d A a n d n a r e c o n s t a n t s to b e d e t e r m i n e d . T h e equations of m o t i o n are transformed from the original system o f v a r i a b l e s (x,t) t o t h e s y s t e m (X,t) a s

a

nX

"di~

t

a ~di X

a ax t With this transformation,

follows

l LE

X

a ax

a ax

Equations (7-8) and (7-9) become

Liquid Spills on Water - The Problem of Oil Pollution

dS

( u

dt

\x du d

t

L E

X\

dd

8

t

dX

x \dX

( u

(du

' I le

tj

x

dX

X

a

' x

„

J

LE

X\ du n - — +

+I

. u\

161

dd ^ — = 0 dX

L E

The equation o f constancy o f m a s s o f the oil

r 8x dx LE

(2rif

J

=L

2

+

j

(7-18)

Jo where L J 2+

is the initial v o l u m e o f the spill, c a n o n l y b e satisfied in

terms of similarity if .7-19) LE

X

a n d w e find

f

L J 2+

D (X)X dX =

(7-20)

J

Jo

I n o r d e r t o satisfy s i m i l a r i t y , t h e v e l o c i t y m u s t h a v e t h e f o r m

u = _ ( X ) ^

(7-21)

and the continuity equation can b e written

(U' - n) D + (U - n X) D ' + j D | ^ - nj = 0

(7-22a)

or -£-[(U-nX)DX ] J

(U-nX)DX

J

=0

(7-22b)

= const.

(7-22c)

T h e o n l y s o l u t i o n t o t h i s e q u a t i o n s a t i s f y i n g t h e c o n d i t i o n U ( 0 ) = 0 is U=nX

(7-23)

Chapter 7

162

Substituting into the m o m e n t u m equation, integrating, a n d imposing E q . ( 7 - 1 8 ) , w e o b t a i n for t h e t h i c k n e s s d i s t r i b u t i o n o f t h e s l i c k

jf _{_ 2

1

+

D =

(l j)L J 2+

x

+

l+J

K

3

+

(7-24)

i

+

(3+j)

(2^

2

[3+jf

w h e r e K is an integration constant, a n d 2 3+j A

=

K

a

(7-25)

i / ( 3

+/ ) { 2 + /)/{3+/) L

The value of the constant K m u s t b e determined b y considering

the

rate o f p r o p a g a t i o n o f the b l u n t l e a d i n g e d g e o f the slick. Dimensional arguments lead us to the conclusion that the leadinge d g e v e l o c i t y m u s t b e p r o p o r t i o n a l t o t h e c h a r a c t e r i s t i c w a v e s p e e d for a s m a l l d i s t u r b a n c e c = y/ a SLE

u

Theoretical

^LE L

~

E

determination

dt of

= k

the

yJad

(7-26)

LE

constant

of proportionality

p r e s e n t s s o m e difficulty. A t late times, h o w e v e r , w h e n the

k

leading

e d g e is v e r y thin, the velocity w o u l d b e e x p e c t e d to e q u a l the

wave

s p e e d (k - 1 ) , b y a n a l o g y w i t h t h e a c o u s t i c l i m i t i n g a s d y n a m i c s . T h i s m a y b e contrasted w i t h the initial b e h a v i o r of the leading edge, w h e n according to E q u a t i o n s (7-14) u -

L E

= c\ = 2 c, g i v i n g k = 2 . T h e v a l u e s k

1 a n d 2 represent lower a n d u p p e r b o u n d s in the possible range of

v a r i a t i o n o f k. I f w e s u b s t i t u t e E q . ( 7 - 2 6 ) i n t o E q u a t i o n s ( 7 - 1 6 , 7 - 1 9 , 724

and

7-25)

integration

evaluated

at

the

leading

edge,

w e obtain

for

the

constant 1/(3 + / )

(3 + j ) ( l + j ) k" 3

K =2{2JV)J[2

(3 + j ) ~ ( l + j )

(7-27) k

2

w h i c h r a n g e s for p l a n a r flow (f = 0 ) f r o m ( 3 / 1 0 )

1 / 3

= 1.39 (k = 1) to 3 (fc

= 2 ) . F o r r a d i a l flow (J = 1), K r a n g e s f r o m 1.14 (k = 1) to oo (k = 2 ) .

Liquid Spills on Water - The Problem of Oil Pollution

163

S i n c e o u r s i m i l a r i t y s o l u t i o n i s v a l i d for l a t e t i m e s , it i s l o g i c a l to adopt

the late-time value k = 1 corresponding

to an

infinitesimal

d i s t u r b a n c e at the l e a d i n g e d g e . S u b s t a n t i a l empirical justification for t h i s c h o i c e h a s b e e n p r o v i d e d b y A b b o t t , w h o s e e x p e r i m e n t a l d a t a a r e m o s t s a t i s f a c t o r i l y c o r r e l a t e d b y t h e v a l u e k = 1. W e h a v e t h e n t h e f o l l o w i n g f o r m u l a for t h e s p r e a d o f a o n e - d i m e n s i o n a l s l i c k

(7-28)

T h e c o e f f i c i e n t 1.39 c o m p a r e s f a v o r a b l y w i t h t h e v a l u e o f 1.5 o b t a i n e d b y H o u l t et al. ( 1 9 7 0 ) , in their e x p e r i m e n t a l w o r k . For the radial case, we have (7-29)

This and

the previous solution do not give constant energy in

slick; i n d e e d , t h e e n e r g y c a n b e s e e n t o v a r y a s t ~ E n e r g y is p r e s u m e d

lost through

2

( 1

+

^

/

(

3+

the

A

turbulent dissipation near

the

blunt leading e d g e of the slick. S o m e features of this solution deserve c o m m e n t . Eq. (7-23) s h o w s a linear velocity distribution characteristic

property

inside

of this

the

slick. T h i s a p p e a r s to b e

type

of similarity

linearity prevails also in cases w i t h m a s s transfer clouds with entrainment).

solution.

(e.g. heavy gas

O n e could m a k e use of this property

develop simplified solutions; w e will d o so in the next subsection. ( 7 - 2 4 ) for t h e t h i c k n e s s

distribution

a

The

D , s h o w s this to b e

to Eq.

parabolic

w i t h the m a x i m u m v a l u e at the l e a d i n g e d g e . B u t the v a r i a t i o n

in

t h i c k n e s s is s m a l l , for p l a u s i b l e v a l u e s o f t h e i n t e g r a t i o n c o n s t a n t K . This is w h y the flat-slick a p p r o x i m a t i o n p r o d u c e s accurate results. Given

the

uncertainty

concerning

spill

external conditions of a real accident, a p p r o x i m a t e s o l u t i o n s u f f i c e s for m o s t

7.2.4

Gravity-Viscous

volume,

properties

one will conclude that

and this

purposes.

Regime

W h e n t h e s l i c k b e c o m e s v e r y l o n g c o m p a r e d to its t h i c k n e s s , it i s apparent from Eq. (7-11) that the acceleration terms can b e neglected in the m o m e n t u m equation, a n d Eq. ( 7 - 8 ) b e c o m e s

Chapter 7

164

a 6

Before

the

w

dx

simplified equations

(7-30)

Po of motion

can

be

s o l v e d , it

n e c e s s a r y to d e t e r m i n e the d r a g e x e r t e d b y the w a t e r o n the This

can

be

layer/water

accomplished

either

boundary-layer

by

equations

solving in

detail,

the

is

slick.

matched

oil

or b y d e r i v i n g a

s i m p l i f i e d m o d e l for t h e d r a g e x e r t e d b y t h e w a t e r b a s e d o n e x i s t i n g s o l u t i o n s . W e h a v e f o l l o w e d t h e latter c o u r s e . In our simplified drag m o d e l , w e can m a k e use of either the k n o w n Blasius boundary-layer

s o l u t i o n or s o l u t i o n s o f t h e r e l a t e d

Stokes'

p r o b l e m . Neither m e t h o d is e x a c t or u n i q u e in its application, various

approximations

accuracy

in

special

can

be

regions.

made

Unlike

with the

a

and

varying degree of

analogous

problem

of

boundary-layer g r o w t h in a shock tube, the present u n s t e a d y viscous problem cannot

be reduced

Galilean transformation

to o n e of steady flow b y m e a n s

because

of a

the leading-edge speed varies with

t i m e . A v a i l a b l e u n s t e a d y s o l u t i o n s a r e a l s o difficult t o a p p l y s i n c e t h e s p e e d v a r i e s f r o m p o i n t t o p o i n t (at f i x e d t i m e ) a l o n g t h e s l i c k . T h e s i m p l e s t d r a g l a w is t h a t for S t o k e s ' flow, d i s c u s s e d i n C h a p t e r 6.

T h e p r o b l e m n o w i s t o f i n d r e l e v a n t e x p r e s s i o n s for t h e velocity u

Q

and time t

R

in terms of slick variables.

T h e average velocity is

A n average time in m o t i o n t can b e expressed as

where U

dx LE

If w e b a s e the d r a g l a w o n a v e r a g e d p r o p e r t i e s , w e o b t a i n

reference

Liquid Spills on Water - The Problem of Oil Pollution

1/2

Pw

2

165

f \-l/2 - ' LE

Jt

Y X

(7-31)

LE

L

If w e a p p r o x i m a t e u b y a s t r a i g h t l i n e b e t w e e n x = 0 a n d x = x £ , u = L

" L E (1 + J'J/(2 + j j , w e h a v e 1/2

r

(2 j) +

-1/2

3/2 LE

(7-32)

LE

This is p r o b a b l y the simplest drag l a w that can be sensibly e m p l o y e d . We

can

reason,

as

an

improvement,

that

the

region near

the

l e a d i n g e d g e is m o s t a f f e c t e d b y v i s c o u s f o r c e s , a n d t a i l o r o u r t i m e in motion t

R

to b e as correct as possible in this region [ LE~ ) X

X

LE W e can also use the local velocity rather than an a v e r a g e velocity; i.e. u

Q

= u. W e t h e n o b t a i n

Pw PL

1/2

LE

JV

-1/2

1/2/

(7-33)

LE'

Eq. ( 7 - 3 3 ) c o r r e s p o n d s to local flat-plate similarity in b o u n d a r y - l a y e r t h e o r y . It h a s t h e a p p r o p r i a t e s i n g u l a r i t y at t h e l e a d i n g e d g e , a n d its f o r m is s u c h t h a t a s i m i l a r i t y - t y p e s o l u t i o n a p p e a r s p o s s i b l e . B e c a u s e Eq. (7-33) represents a superior description of the drag variation, w e w i l l u s e it i n t h e latter p a r t o f t h i s s e c t i o n a s o u r b e s t d r a g l a w . If w e e m p l o y t h e a v e r a g e d r a g l a w E q . ( 7 - 3 2 ) , t h e equation can b e integrated directly. W e will a s s u m e x £ L

a n d n a r e u n k n o w n c o n s t a n t s . S u b s t i t u t i n g for

x-momentum =A t

and u £ L

n

where A

-

dx /dt LE

in Eq. (7-32) and, o n introducing the result in (7-30), w e obtain

after

integration

8 (x,t)-S {x ,t) 2

2

LE

= (7-34)

Chapter 7

166

It is a p p a r e n t f r o m t h e e x p r e s s i o n d e r i v e d a n d f r o m t h e p h y s i c s o f the p r o b l e m that the m a x i m u m thickness

is at the center a n d

the

m i n i m u m t h i c k n e s s o c c u r s at t h e l e a d i n g e d g e . I n v i e w o f t h e fact t h a t t h e p r o b l e m c o n s i d e r e d h a s t w o u n k n o w n s ( i . e . , A a n d dig) a n d o n l y o n e c o n d i t i o n ( g l o b a l m a s s c o n s e r v a t i o n ) w h i c h c a n b e i m p o s e d , it b e c o m e s necessary to m a k e an additional assumption. W e will a s s u m e here that the slick thickness

v a n i s h e s a t t h e l e a d i n g e d g e . T h i s is

both

and

physically

reasonable

consistent

with

experimentally

observed behavior. Taking

<5

(x ,t)

= 0

LE

in

Eq.

(7-34)

and

invoking

the

mass

conservation condition Eq. (7-18), w e obtain

n =

3JZ (2 +

"2"

7j

jf

(7-35)

4(2 +j)

"(«P.)*V*A(Pvl>v)

4

< *J"2

17-36)

12

W e t h e n h a v e for t h e s p r e a d o f a o n e - d i m e n s i o n a l s l i c k LE

L

= j 2 ( 3 ^ ( a p s

L

1.87

[*W

0

)

1

/

4

( p

n x) --l /" 8 t3 / 8

(7-37a)

8 f

w

w

1 1 / 8

t

3 / 8

for

P

o

« p

(7-37b)

u

Pw

A c c o r d i n g to H o u l t e t al. ( 1 9 7 0 ) , t h e e x p e r i m e n t s i n d i c a t e a v a l u e for t h e coefficient o f 1.5, w h i c h c o m p a r e s r e a s o n a b l y w e l l w i t h o u r v a l u e o f 1.87. If w e h a d b a s e d o u r d r a g l a w o n t h e B l a s i u s

skin-friction

coefficient

value,

rather

than

theoretical coefficient

the

corresponding

aforementioned would be

ratio of the predicted v i s c o u s forces ( J _ ~

1 / 2

Stokes

the

1.74. A l t h o u g h t h e

/ 0 . 3 3 2 ) is a b o u t

1.7,

the

s p r e a d i n g l a w s a r e n e a r l y e q u a l d u e to t h e (1 / 8 ) - p o w e r in E q . ( 7 - 3 7 ) . For the radial case, w e h a v e

15 4TT

J

Is)

M

/12 l/4 t

[Pw Pw)

(7-38a)

Liquid Spills on Water - The Problem of Oil Pollution

167

1/12

t

1.18

for

1 / 4

p ~p 0

(7-38b)

u

Pw Let us n o w consider an i m p r o v e d analysis based on the best drag law, Eq. (7-33). Equations (7-9) a n d (7-30) are n o w coupled through the presence of the local velocity u on the right-hand side of Eq. (7-30). W e will look

for a s i m i l a r i t y s o l u t i o n s u c h

as

that developed in

the

previous section. W e introduce the similarity variable X as defined in Eq. (7-16), a n d observe that the treatment of the continuity equation leading

to Eq.

(7-23)

is

equally applicable here.

Hence we

can

substitute E q u a t i o n s (7-19), (7-21) a n d (7-23) in E q u a t i o n s (7-30) a n d (7-33),

obtaining

(7-35)

as

e q u a t i o n for t h e t h i c k n e s s

DD' =

3

/

2

before,

and

the

following

differential

distribution

f^ f

/

2

_ ! _

A

2 (2 J) +

x

(

x

_

x

)

- 1 /2

(

?

_

3

9

)

4 ( 2 +J)

O n integrating Eq. (7-39), imposing the condition D (1) = 0 discussed previously and

substituting the

result

in the

mass

conservation

condition Eq. (7-18), w e obtain

D = ± — - ( 2 + X ) {2jry i

1

/

2

( l - X )

1

/

(7-40)

4

where 1=

f X- (2+X) Jo 1

/

1

/

2

(l-X)

1

/

4

dX

(7-41)

and

(D" 2" (2 ) ' 4

2

3

+J

A =

{2rf

J

(2+J)

4

(a

P o

) 2 (2 +j) [p

w

pjf

4 (2 + / )

L

(7-42)

I

For the one-dimensional case, this gives LE

=

1/8*3/8

1.39(ap ) (p » )-"»t 1/4

0

w

w

(7-43a)

Chapter 7

168

1/8

t

1.39

3 / 8

for

(7-43b)

p ~p Q

w

Pw with the coefficient in g o o d a g r e e m e n t w i t h the experimental value of 1.5. F o r t h e r a d i a l c a s e , w e h a v e LE

0.98[ap ) {p

a)

l/6

o

w

1/12^1/4

(7-44a)

w

1/12

0.98

t

l / 4

for

(7-44b)

p ~p 0

w

Pw ( N o e x p e r i m e n t a l r e s u l t s a r e k n o w n for t h e r a d i a l c a s e . ) The

results

given

by

Equations

(7-43)

and

(7-44)

indicate

a

s o m e w h a t slower rate of spread t h a n that calculated w i t h the simpler drag law. This can be explained b y noting that the simpler drag law g i v e s c o n s t a n t d r a g f r o m t h e c e n t e r t o t h e e d g e at a n y i n s t a n t o f t i m e . T h e d r a g f r o m t h e b e s t d r a g l a w , E q . ( 7 - 3 3 ) , v a n i s h e s at t h e c e n t e r , is infinite at t h e l e a d i n g e d g e , a n d m a t c h e s the v a l u e o f t h e s i m p l e r d r a g l a w at t h e m i d p o i n t ; h e n c e m o r e d r a g i s c o n c e n t r a t e d n e a r t h e front. B o t h s o l u t i o n s g i v e a n infinite s l o p e at t h e l e a d i n g e d g e .

7.2.5

Surface

Tension-Viscous

Regime

A t v e r y l a t e t i m e s (or for v e r y t h i n s l i c k s ) t h e d o m i n a n t

spreading

force is t h e n e t d i f f e r e n c e o f t h e s u r f a c e t e n s i o n s b e t w e e n a ) t h e oil-air and oil-water interface a n d b ) the water-air interface. This difference Is d e n o t e d b y t h e s p r e a d i n g c o e f f i c i e n t a ^ a s

discussed in Chapter 2.

T h e d o m i n a n t retarding force is again the viscous drag of the water b e n e a t h t h e slick. A n exact analysis of this flow p r o b l e m a p p e a r s to be a c o m p l e x task.

In

what

follows

we

will

present

a

simple

approximation. W e note that the rate of spread in the d r i v e n r e g i m e is i n d e p e n d e n t

engineering

surface-tension

o f the v o l u m e of the spill (see Fay,

1969), and w e w o u l d therefore

expect the different

configurations

( p l a n a r or r a d i a l ) to s p r e a d at n e a r l y t h e s a m e r a t e . T h e f o r c e b a l a n c e for t h e p l a n a r a n d r a d i a l c a s e s c a n b e e x p r e s s e d as

follows

Liquid Spills on Water - The Problem of Oil Pollution

a =£

LE

[2jtx y

J

w

dx

(7-45)

e q u a t i o n for t h e u n k n o w n p o s i t i o n o f t h e l e a d i n g - e d g e XIE is

o b t a i n e d b y s u b s t i t u t i n g t h e e x p r e s s i o n for r Eq.

T

(2JTX)

N

LE

An

169

( 7 - 3 3 ) . B y l e t t i n g XIE = A t

from our best drag law,

w

w e find that dimensional

n

consistency

requires n = 3 / 4 , in a g r e e m e n t w i t h experimental observations (Lee, 1 9 7 1 ) . W e take u = u

(x/xi )

L E

( t h i s i s c o n s i s t e n t w i t h all t h e p r e v i o u s

E

solutions). Substituting and

integrating Eq. ( 7 - 4 5 ) ,

w e f i n d for

the

planar case

V, _ - ( g r The

(4JT/3)

coefficient

1

/

=

4

L'

3/4 17-46,

1 . 4 3 is in g o o d

agreement

with

the

experimental value 1 . 3 3 in L e e ( 1 9 7 1 ) . For the radial case w e obtain 1/2 LE

= 1-6

X

rn

t

3

/

C -

4

7

4

7

)

[Pw V ) w

a l m o s t t h e s a m e r e s u l t a s for t h e o n e - d i m e n s i o n a l s l i c k .

7.2.6

Summary

of

Results

T a b l e 7 - 1 s u m m a r i z e s t h e s p r e a d i n g l a w s for t h e t h r e e r e g i m e s . I n each case, the best

solution is given, involving the m o s t

accurate

description of v i s c o u s drag a n d the m o s t sophisticated a p p r o a c h to a similarity

solution.

In

those

cases

where

experimental

available, the experimental c o n s t a n t is a d d e d in parenthesis. 7 - 1 , w e have m a d e the approximation p Our

f i n d i n g s for the

planar

0

data

are

In Table

»PL_.

slick are

summarized

in Fig. 7 - 4 .

Initially a l a m i n a m o v e s out from the e d g e of the slick, j o i n e d to the contracting thickness

central of the

part

by

advancing

a

smooth

lamina

transitional

is

one

quarter

contour. the

The

original

t h i c k n e s s o f t h e s l i c k . T h e w a v e f o r m p r e d i c t e d a t t h e l e a d i n g e d g e is square,

but

the

retarding

action

of the

water

and

the

vertical

acceleration of the oil in the vicinity o f the w a v e , n e g l e c t e d in this analysis,

would

intermediate

tend

time, the

to

round

off

the

wave

front.

At

some

slick is e x p e c t e d to b e a p p r o x i m a t e l y flat.

Chapter 7

170

Table 7-1; Predicted spreading lawsfor the oil slicks. One-dimensional Gravllyinertial regime

/ 3 f

i

Radial

1/4

2/3

^

{PA,*-)

=

L

1 . 1 4 ( ^ ] [PW j

t

1 / 2

L

(1.5 from exp.) Gravityviscous regime

g

2

^=1.39

L

(4P) ' 2

1/8 t

3/8

^

PW V

= 0.98

1/12

\g M 2

2]

,1/4

PW V

W

W

(1.5 from exp.) Surface tensionviscous regime

X= LE

,3/4

1.43

X= LE

(Pu,^]

1 / 4

1.6

°N (PW

'

t

3/4

V)

1

W

(1.33 from exp.)

Later, the slick w o u l d tend to a p p r o a c h the parabolic form predicted by

t h e s i m i l a r s o l u t i o n for t h e g r a v i t y - i n e r t i a l r e g i m e , w i t h a l a r g e r

part of the m a s s in the outer region. A l t h o u g h this solution predicts a v e r t i c a l front for t h e s l i c k , t h e r e t a r d i n g a c t i o n o f t h e w a t e r a n d

the

v e r t i c a l a c c e l e r a t i o n a g a i n w o u l d t e n d t o r o u n d off t h e l e a d i n g e d g e . Still later, v i s c o u s d r a g c o m e s to d o m i n a t e o v e r t h e i n e r t i a l t e r m s

as

the slick slows d o w n , a n d m o s t of the slick tends to a s s u m e the form predicted b y the similarity solution for the g r a v i t y - v i s c o u s r e g i m e . The

l a r g e r p a r t o f t h e m a s s is n o w f o u n d i n t h e i n n e r r e g i o n o f t h e

s l i c k . S i n c e t h e d e n s i t y o f oil d i f f e r s t y p i c a l l y f r o m t h a t o f w a t e r b y 20%

or less; m o s t

of the

slick lies b e l o w the water level. A s

deceleration of the slick continues,

the

t h e effect o f g r a v i t y e v e n t u a l l y

falls off to z e r o a n d s u r f a c e t e n s i o n t a k e s o v e r a s t h e d r i v i n g f o r c e .

7.3

Additional Similarity

The

mathematical

Solutions

theories discussed

in the previous section

are

c o n c e r n e d w i t h a s o m e w h a t idealized case in w h i c h the oil, initially c o n s t r a i n e d b y a b a r r i e r , is s u d d e n l y free t o m o v e . T h e a c c u r a c y o f t h e similarity solution is n o t sensitive to the details of the release. fact, t h e i n i t i a l c o n d i t i o n s a r e s a t i s f i e d o n l y a p p r o x i m a t e l y (x

LE

In = 0

w h e n t = 0) a n d globally t h r o u g h the i n v a r i a n c e of the initial v o l u m e of oil. T h e r e are m a n y practical p r o b l e m s w h e r e these solutions

do

n o t a p p l y , e.g. t h e c a s e w h e n oil l e a k s i n t o a s t r e a m a n d w h e n

the

Liquid Spills on Water - The Problem of Oil Pollution

i

*±*r

i *\

initial

171

wave

"A "A "A • % •_% • % • % • \ • v^t?

flat

slick

gravity

- inertial

(

similar)

gravity

- viscous

(

similar)

(X (distorted Fig.

leak

7-4:

Evolution

rate varies with

of a

scales) slick.

can

be

obtained b y extensions of the similarity analyses. W e note that

time.

Solutions

to

such

problems

the

p r o b l e m s o f a m o v i n g s o u r c e o r a l e a k i n t o a s t r e a m a r e i n fact t h e s a m e as l o n g as the velocity is c o n s t a n t ( G a l i l e a n transformation) a n d for t h e s e c a s e s a s i m i l a r i t y s o l u t i o n h a s b e e n f o u n d . S i m i l a r i t y a l s o e x i s t s for t h e t i m e - d e p e n d e n t

source as long as the leak rate

v a r i e s a s p o w e r o f t i m e . A c o m p l e t e s e t o f s o l u t i o n s w a s first g i v e n b y W a l d m a n et al. ( 1 9 7 2 ) a n d t h e s e r e s u l t s a r e s u m m a r i z e d i n T a b l e s 7-2 a n d 7-3. A m o n g the solutions included w e find the equivalent of the "slab m o d e l " d e v e l o p e d m u c h l a t e r for a p p l i c a t i o n s r e l a t e d to h e a v y g a s d i s p e r s i o n , a s w e l l a s o t h e r r e s u l t s p r e s e n t e d a s "new" s o l u t i o n s i n more recent publications.

Chapter 7

172

Table 7-2; Summary of spreading laws for leaks onto calm water: m-Mt , £

One-dimensional

yregime inertial

=At . n

LE

Radial

In M An\ 1 / 3 ,„ .

G m V i t

x

,

(„ M An\ 1 / *

n

Surface tensionviscous regime

1.43 V

« (,„,,,„,)-»/« t ' «

r

3

LE

- 1.6 a „ > « ( p „ ^ J " " * t

(1.33fromexp.)

Gravity-inertial:

Gravity-viscous:

D=— n

aMD 2

3 +

P o

M-D MP

D=

2

—- , n A J'

+

, n=

' l0...u. \

l/4

~ . '

n

For both:

*J

—

l 2

?

+

, L

3 + 4^

"4(2+J)

V DX dX = (2jr)"A Jo J

In applications, numerical evaluation required to determine K j. t

Table 7-3: Summary of spreading laws for plumes with drift velocity U : m evaluated at t - z/U . T

T

L E

regime

rr regime Surface tensionviscous regime

U

t\2

2 P w

^.^|t*p(jt.]" (i)"',» ,

T [P j

U

2

w

[PwPwJ

C^{u o ^ 112

T

x

N

\ ) 2

J

3

/

4

Liquid Spills on Water - The Problem of Oil Pollution Note:

173

I n T a b l e 7-3 z i s t h e l e n g t h c o o r d i n a t e i n t h e d i r e c t i o n o f

drift a n d l /

t h e n e t drift v e l o c i t y , d u e t o w i n d s o r c u r r e n t s o r b o t h .

T

The equivalence of the

spreading

time

t of the

one-dimensional

s o l u t i o n ( T a b l e 7 - 2 ) , a n d the drift t i m e z / i 7 , is also n o t e d . A s the r

p l u m e d o w n s t r e a m of the continuous source, g r o w s in b o t h directions transverse to the plane o f s y m m e t r y , the equivalent source strength is only half of that of the one-dimensional solution derived in Section 7 . 2 . T h e o i l v o l u m e i n a s l a b o f p o s i t i o n z = If? t i s t h a t p r o d u c e d b y t h e s o u r c e a t t i m e t - z / If?. T h e e q u i v a l e n c e c a n b e s t a t e d a s m(t-z/U \

z

T

using the nomenclature in the tables.

7.4

T h e Containment and Collection of Oil Slicks

The only commonly accepted

method

for c l e a n i n g u p

oil spills

from the sea, is m e c h a n i c a l r e m o v a l ; in practice, c o n t a i n m e n t b y oil b o o m s (i.e. floating vertical barriers, stream) and removal by skimmers designed

to

straightforward inventive

remove

thin

problem, has

engineering

and

t o w e d b y ships or fixed in

(i.e. suction or a d h e s i o n

layers).

To

solve

this

the

devices

seemingly

p r o v e d e x t r e m e l y difficult. In spite o f large

expenditures

on

research

and

d e v e l o p m e n t , a r e l i a b l e s y s t e m for u s e o n t h e h i g h s e a s i s y e t n o t i n sight. T h e p r i m a r y element in the oil containment and collection system, is t h e oil b o o m . I t s p u r p o s e i s t o a r r e s t t h e s p r e a d o f t h e s l i c k a n d t o concentrate

the oil to a t h i c k n e s s sufficient to allow s k i m m e r s

and

p u m p s t o p i c k it u p a n d t r a n s f e r it t o h o l d i n g t a n k s w i t h o u t t o o m u c h w a t e r . T h e o r i e s for t h e r e q u i r e d d e p t h o f b o o m s a n d o f t h e i r e x p e c t e d performance w e r e developed in the early seventies. Most of these early theories are w r o n g as they ignore both the appearance of a h e a d w a v e a n d t h e a s s o c i a t e d e n t r a i n m e n t , w h i c h for a b o o m o f d e p t h

sufficient

to a v o i d d r a i n a g e , r e p r e s e n t s t h e m a j o r s o u r c e o f oil l o s s e s ( F i g . 7 - 6 ) . These early theories attempt to relate the hydrostatic pressure of the oil p o o l forced against

the barrier,

to the applied stress o n

the

pool surfaces, w i n d side, w a t e r side or both. D e n o t i n g the stress r a n d A = (p w

p )/Pw 0

w e have (Fig. 7-5)

Chapter 7

174

r=p Agh

—

0

W i t h T specified in terms of the turbulent stress, w e can write

where

i s t a n d s for w a t e r or air. O n a s s u m i n g

cj = c o n s t a n t

and

v a n i s h i n g t h i c k n e s s a t t h e l e a d i n g e d g e , t h e e q u a t i o n s a b o v e l e a d to a parabolic thickness

d i s t r i b u t i o n a n d a s l i c k l e n g t h far i n e x c e s s o f

t h a t o b s e r v e d , F i g . 7-5 ( H o u l t , 1 9 6 9 , a n d C r o s s a n d H o u l t , 1 9 7 0 ) . An

approach,

based

on

the

dynamic

equilibrium between

the

h e a d w a v e a n d the o p p o s i n g current, w o u l d c o m e closer to the truth. T h e t h i c k n e s s o f t h e h e a d w a v e w o u l d h a v e to i n c r e a s e u n t i l its r a t e o f propagation equals the speed of the opposing current. T h e thickness increases with increasing current speed until breaking w a v e s develop. T h e s e are r e s p o n s i b l e for s h e d d i n g d r o p l e t s into the w a t e r

flowing

Ah

U

dx oil

i

pressure

f-H

water depth x turbulent Fig.

7-5:

stress

Approximate hydrostatic arrested by a barrier (no

balance for headwave).

idealized

oil

pool

Liquid Spills on Water - The Problem of Oil Pollution

175

underneath, leading to entrainment loss. T h e process, as depicted b y M i l g r a m ( 1 9 7 7 ) , i s r e p r o d u c e d i n F i g . 7-6. O f p a r t i c u l a r i n t e r e s t i n t h e s t u d y o f o i l b o o m s , w o u l d b e first t o d e t e r m i n e t h e i r c a p a c i t y , t h a t i s h o w m u c h o i l t h e y c a n h o l d for g i v e n conditions and, second, the loss rate o n c e this capacity is e x c e e d e d . F i g . 7-5 a n d t h e h y d r o s t a t i c a n a l y s i s i n d i c a t e d , p e r t a i n t o t h e t w o d i m e n s i o n a l case w h i c h is likely to b e of little interest outside laboratory. Practical oil b o o m s are deployed mostly as a

the

catenary Air

-ZZZZZZZZZZZZZZZZZZZ2 0.5

Water

knot

Barrier

current

Headwave 0.75

knot

current

Headwave

\

Interfacial instability

1.0

knot

current

Droplets off

torn

headwave

1.25

knot

current Fig.

7-6;

O

O

U 0

O ~ °

Entrained

0

*>

O

°

oil droplets

past barrier by

O

o

0

GO

G

swept

current

Headwave formation and oil loss due to entrainment various current velocities (from Milgram, 1977).

for

Chapter 7

176

s h a p e , a n d t h e oil p o o l i s f o u n d t o h a v e a n e a r l y s t r a i g h t l e a d i n g e d g e transverse

as s h o w n in Fig. 7-7. T h e

flow

conditions along the centerline will be nearly two-dimensional

to the direction of

flow,

and

the results from laboratory tests should apply here. This is important as the deepest part o f the oil pool will b e found near the and

centerline

the conditions here will determine the m a x i m u m capacity. A

rather c o m p l e t e r e v i e w o f barrier p e r f o r m a n c e is g i v e n b y F a n n e l o p (1983). This review emphasizes two-dimensional results as very few detailed measurements

for t h r e e - d i m e n s i o n a l

flow

are available, and

most are of poor accuracy. W i t h r e f e r e n c e t o t h e s c h e m a t i c s l i c k c r o s s - s e c t i o n s h o w n i n F i g . 7-

Fig.

7-7:

Headwave

and oil barrier

geometry.

Liquid Spills on Water - The Problem of Oil Pollution

7, w e c a n d i s t i n g u i s h

177

three r e g i o n s : the h e a d w a v e w i t h its

distinct

shape, the n e a r - b o o m r e g i o n w h e r e the s t a g n a n t oil a n d the

flowing

water interact w i t h the solid barrier a n d the large intermediate region w h i c h c o n t a i n s m o s t o f the oil a n d w h e r e hydrostatic e q u i l i b r i u m is a s s u m e d to prevail. It i s p r i m a r i l y t h e responsible boom

headwave and

near-boom

regions which

are

for l o s s e s a n d w h i c h limit the p e r f o r m a n c e . T h e n e a r -

failure

mode

of interest,

i.e.

drainage,

has

already

been

d i s c u s s e d i n S e c t i o n 3.3 o n " S e l e c t i v e w i t h d r a w a l " . The

loss

oil/water

due

to w a v e

interface

of

breaking

the

and

headwave,

droplet is

formation

obviously

a

at

the

stability

phenomenon. Leibovich (1976) and others have carried out the rather complex

analysis

Kelvin-Helmholz critical

speed

required instability

when

to

determine

occurs.

droplets

first

the

critical

T h i s is further appear.

speed

related

when to

the

The problem with

this

analysis is that the results d e p e n d o n slick thickness, b u t a n

answer

has

infinite

been

found

thickness. analysis

o n l y for the

A simple

limiting cases

alternative

for finite t h i c k n e s s ,

to

the

of zero and

complicated

would be

the

use

mathematical

of semi-empirical

criteria from controlled experiments. A g r a w a l a n d Hale (1974)

have

s u g g e s t e d t h e c r i t e r i o n for t h e o n s e t o f d r o p l e t g e n e r a t i o n We

cr

= 28.2

(7-48)

w h e r e the special W e b e r n u m b e r is defined b y

Pw We

cr

=

cr

U c

2

.

r

(7-49)

sj[Pw-Po)9

a

a n d t h e i n d e x (cr) r e f e r s t o d r o p l e t f o r m a t i o n a n d n o t ( a s o f t e n u s e d ) the stability limit. A l t h o u g h t h e h e a d w a v e t h i c k n e s s hj ( F i g . 7 - 7 ) d o e s n o t a p p e a r , w e need

information

on headwave geometry both

slick

cross-section

(and

hence

the

volume

for c a l c u l a t i n g contained)

and

the to

investigate the expected relationship between the headwave thickness and current speed. O n equating the speed of the headwave, seen as the front o f a p r o p a g a t i n g slick, E q . ( 7 - 2 6 ) , a n d the s p e e d o f the current, w e obtain using the usual slick nomenclature

178

Chapter 7

(7-50)

which

can

be rearranged

to give

= k (a constant)

where

the

d e n s i m e t r i c F r o u d e n u m b e r is d e f i n e d b y

a n d A = (p w

p )lp . 0

Milgram and

w

V a n Houton (1978) have carried

out

a series of

d e t a i l e d m e a s u r e m e n t s o f h e a d w a v e g e o m e t r y for v a r i o u s t y p e s o f oil, here roughly classified as l o w viscosity oils ( 1 0 centistoke) a n d highviscosity oils ( 1 0 0 est), ( w h e r e the n u m b e r s o n l y indicate the order of magnitude). F i g . 7-8 s h o w s t h e r e s u l t e x p r e s s e d i n t e r m s o f F r o u d e n u m b e r v s . c u r r e n t s p e e d . A s e x p e c t e d w e find n e a r c o n s t a n t v a l u e s b u t a l s o a t r e n d t o w a r d s l o w e r v a l u e s w i t h i n c r e a s i n g c u r r e n t v e l o c i t y for t h e h i g h - v i s c o s i t y o i l . It i s c h a r a c t e r i s t i c for t h i s f l o w , b o t h h i g h a n d l o w viscosity, that the front is wedge-like rather than r o u n d e d as in the case of the gravity currents discussed g e o m e t r y is s h o w n in Figs.

7-7 a n d

in Chapters

7-9a,b.

4 and

T h e fronts

6. T h e

have

the

character of slender w e d g e s with a w e d g e angle which increases with increasing current velocity. T h e e x c e p t i o n is the Diesel oil w h i c h h a s viscosity not m u c h perhaps

h i g h e r t h a n w a t e r a n d for w h i c h w e r e c o g n i z e

the b e h a v i o u r o f a gravity head, g o v e r n e d b y a balance of

pressure forces and,

not as in the v i s c o u s case, b y pressure

and

v i s c o u s f o r c e s . F o r b o t h t y p e s o f fronts, a d i s t i n c t n e c k r e g i o n a p p e a r s b u t p e r h a p s for d i f f e r e n t r e a s o n s . F o r a g r a v i t y c u r r e n t ( s u c h a s

a

h e a v y gas intruding in air) the flow inside the h e a d is circulatory, in effect a v o r t e x , a n d t h e n e c k i s p r o d u c e d b y i t s i n t e r a c t i o n w i t h t h e main current. For the viscous wedge-like intrusion,

there is little

m o t i o n i n s i d e t h e h e a d , b u t t h e flow g e o m e t r y l e a d s t o s e p a r a t i o n o n t h e w a t e r s i d e a n d w e h a v e i n effect a r e a r w a r d f a c i n g s t e p w i t h i t s a s s o c i a t e d r e c i r c u l a t i n g ( v o r t e x ) flow. T h i s i n t u r n p r o d u c e s t h e n e c k , which

typically

is

one

half

the

headwave

thickness.

These

explanations are tentative b u t plausible, a n d they serve to explain b o t h the differences a n d similarities b e t w e e n the t w o different types of

flow.

Liquid Spills on Water - The Problem of Oil Pollution

• Arzew light Crude, A N 2 Diesel, Light mineral oil,

S

l

2

1.6

179

A = 0.196, A = 0.163, 4 = 0.14,

v = 4.5 c s v = 3.6 c s v=12cs

^

1.4

<•>

J

it

a

1.2 1.0

-

0.8

-

0

0

10

20

low - viscosity

A

h

40

oils

Data/or

\J 9 J

30

50

c

U

(cm/s)

heavy mineral oil, A = 0 . 1 2 , v = 125 c s

1.6 1.4 1.2 h 1.0 h 0.8

0 b)

Fig.

0

•

flagged symbols with surface open symbolsjrom diagrams,

10

high-viscosity

7-8;

Froude Houton,

20

30

^ - S -

tension reducing additive closedfrom photos

40

oil

number vs. velocity. 1978.)

50 c

U

(Data from

(

c

m

/

Milgram

s

)

and

Van

Chapter 7

180

Heavy mineral oil ( c-r with surfactant)

0-

0 ©

0 0 0 ©

0 0 b )

Fig.

10 high - viscosity

7-9;

20

30

oil

Headwave shape vs. Van Houton, 1978.)

40 U

c

velocity.

(Data from

50 (cm/s)

Milgram

and

Liquid Spills on Water - The Problem of Oil Pollution

The minimum thickness

in the n e c k

181

a n d i t s p o s i t i o n xjq a r e

n e e d e d to d e t e r m i n e the oil v o l u m e in the intermediate region. For our purposes the values h

= (1 / 2 ) fy-and x^ = 1.75 Lj suffice. W e c h o o s e

N

to define a t h i c k n e s s

parameter 0 =f

(7-51)

L

f

w h i c h is c o n s i d e r e d k n o w n , for a g i v e n t y p e o f o i l , f r o m F i g . 7 - 9 . T h e volume

of oil per

unit width contained

in

the

headwave can

be

determined from the expression

17 f h

f

h

-

(7 52)

^Je^~^f

A

obtained b y assuming a polygon shape. A

n

is the cross-sectional

area

of the h e a d w a v e in this two-dimensional analysis. In order to find the v o l u m e contained in the intermediate region, we

must

integrate

establish the

the

resulting

hydrostatic

balance

differential equation

prevailing here over

the

and

streamwise

l e n g t h of the slick u s i n g the slick t h i c k n e s s at the n e c k as

boundary

condition. W e will discuss the hydrostatic b a l a n c e w i t h reference to Fig. 7-5. A c o m p l e t e f o r c e b a l a n c e for a n a r r e s t e d s l i c k e l e m e n t i s g i v e n i n V a n Houton's dissertation ( M I T , 1976). A d d i t i o n a l terms arise d u e to the sloping slick b o u n d a r y a n d takes into account the change in direction of the velocity vector (pressure drag) as well as the shear forces. O n assuming the slick to b e slender, the velocity a n d shear forces can b e considered to act in the horizontal direction a n d w e can establish following force balance (Fig. 7-5):

oil On rearranging w e obtain

where terms to order A

2

have been retained.

water

the

Chapter 7

182

Only in near stagnant waters will the w i n d shear b e of primary importance. In o p e n water, the w i n d stress p r o d u c e s a surface current w h i c h exerts stress o n the slick in addition to tidal

components,

t o w i n g s p e e d o r o t h e r c a u s e s o f r e l a t i v e m o t i o n a l o n g t h e oil / w a t e r interface. W i t h o u t m u c h loss in generality, w e can incorporate

the

wind-stress

the

contribution

in

the

water

stress term

to obtain

d i f f e r e n t i a l e q u a t i o n for h (x) i n t h e f o r m

A

g

h

± c = (lTA)2 c U

C

/

M

( ?

-

T h e v a l u e o f cj i s h i g h l y u n c e r t a i n , a v a i l a b l e d a t a differ b y

5 3 )

an

order of magnitude. T h e variation with x deviates considerably from t h a t for t u r b u l e n t (1978) found

flow

(separated

flow)

to a

constant

near

o v e r a flat p l a t e . M i l g r a m a n d V a n H o u t o n

a m a x i m u m value near the head,

a negative value

in the neck region and an asymptotic-like value

in

the

downstream

approach

region. Given

the

u n c e r t a i n t y i n v a l u e a n d f u n c t i o n a l f o r m , it a p p e a r s r e a s o n a b l e to u s e a c o n s t a n t v a l u e o f c j f o r a g i v e n o i l . F r o m a r e v i e w o f all e x i s t i n g data, Fannelop (1983) r e c o m m e n d s

O n a s s u m i n g cjsnd

Cj = 0 . 0 0 4

( l o w - v i s c o s i t y oils)

Cj= 0 . 0 1

( h i g h - v i s c o s i t y oils)

h e a d w a v e g e o m e t r y to be d e p e n d e n t only on the

t y p e o f oil, E q . ( 7 - 5 3 ) c a n b e i n t e g r a t e d to g i v e

h(x) =

^ - ^ ^ ( x - 1 . 7 5 ^ 4

(7-54)

W e h a v e u s e d h e r e t h e initial c o n d i t i o n s x^ = 1.75 Lj, h (x ) = hj/2 0f = hf/Lj is c o n s i d e r e d k n o w n a s n o t e d . N

and

T o obtain the v o l u m e contained per unit width, Eq. (7-54) can b e integrated again from the n e c k to the n e a r - b o o m region. O n e can in fact u s e ( 7 - 5 4 ) all t h e w a y t o t h e b o o m , a s t h e t h i c k n e s s v a r i a t i o n s i n the n e a r - b o o m r e g i o n are relatively unimportant. W h a t is important is t h e d r a i n a g e p r o b l e m w h i c h is a l o c a l p h e n o m e n o n n e a r t h e b o o m . In cases w h e n drainage d o e s not occur, the m a x i m u m v o l u m e of oil is c o n t a i n e d w h e n h e q u a l s the draft d of the barrier in the t w o d i m e n s i o n a l c a s e . W i t h r e f e r e n c e t o F i g . 7-7, t h e m a x i m u m v o l u m e i n

Liquid Spills on Water - The Problem of Oil Pollution

183

the t h r e e - d i m e n s i o n a l c a s e is c o n t a i n e d w h e n b o t h L = D a n d h = d. T h e total v o l u m e is the s u m o f that c o n t a i n e d in the h e a d w a v e V

and

n

that in the m a i n slick V .

F r o m Eq. (7-52) w e have

m

(7-55)

w h e r e W is the w i d t h o f the b o o m (Fig. 7-7). T h e i n t e g r a t i o n o f h o v e r t h e d e p t h a n d w i d t h o f t h e b o o m p o c k e t is tedious. O n e c a n simplify the p r o b l e m b y a s s u m i n g the b o o m s h a p e to be

a

parabola,

rather

than

a

catenary.

With

simplifications a v e r y s i m p l e result is o b t a i n e d

Vr =

5

this

and

other

(Fannelop, 1983)

%dWD

(7-56)

In practical applications outside the laboratory, this v o l u m e will b e m u c h larger than that contained in the headwave. T h e importance of the

latter

lies therein

that

it

mechanism, droplet entrainment, show

h o w the

wave

controls

the

most

important

loss

as already discussed. W e will next

geometry and

the

stability criterion can

be

utilized to correlate existing information on loss rates. Consider the quasi-steady process depicted in the last picture of Fig. 7-6. Oil is lost from the h e a d w a v e d u e to droplet e n t r a i n m e n t the rate q

E

(yet to b e d e t e r m i n e d ) . T h e h e a d w a v e shape

and

at

size

r e m a i n s u n c h a n g e d , w h i c h indicates that oil is fed to the h e a d w a v e from the m a i n slick at the s a m e rate. T h e n o n d i m e n s i o n a l variable q /{h.N E

U ) which

through

c

is

Simpson

ratio

between

the

counterflow

velocity

the n e c k a n d the water velocity, is of potential interest

correlating data. A s h interest

the

and is U . cr

N

Britter

= ( 1 / 2 ) hjwe (1979),

For U

c

in

will use the form suggested b y

i . e . q l(hj

U ).

E

less than U

cr

n o n d i m e n s i o n a l r a t i o i s t h e r e f o r e (U c

A second velocity of

c

no

droplets

occur. A

second

U )/U . cr

c

W e are l e d to attempt a correlation o f form

(7-57)

T o e v a l u a t e t h e f u n c t i o n w e m a k e u s e first o f a l l o f t h e d a t a o b t a i n e d b y the U S C o a s t G u a r d in their o c e a n trials. T h e results o b t a i n e d are

Chapter 7

184

q

°

- 1 0 -

Uh c

f

© calm weather 10

data

•

rough weather

A

Tampa

8

LT = 0.36 cr

( We

r r

test

0

m/sec •

4

/

•

0 Stope:

2

F i g . 7-10:

/

= 28.2 )

6

0

data

/

A

1^

0

1.45- 10"

L

0.1

0.2

Proposed loss Coast Guard.

0.3

0.4

0.5

correlation,

based

0.6

on

data

from

U.S.

s h o w n i n F i g . 7 - 1 0 i n t e r m s o f t h e v a r i a b l e s p r o p o s e d . It i s s e e n t h a t the data c a n b e fitted b y the straight line


U c

c -

U

c r

where C = 1.4510" . 3

I n F i g . 7 - 1 1 a d d i t i o n a l d a t a a r e i n c l u d e d a n d it i s s e e n t h a t

the

c o r r e l a t i o n a p p e a r s t o b e a g o o d fit a l s o for t h e r e s u l t s o b t a i n e d i n t h e l a b o r a t o r y . W h a t i s i m p o r t a n t for t h e v a l i d i t y o f t h i s c o r r e l a t i o n , a n d p e r h a p s surprising, is that the droplets, o n c e entrained, d o not rejoin the slick a l t h o u g h the larger droplets at least are b r o u g h t in contact with the slick d u e to b u o y a n c y . T h e underlying p h e n o m e n o n has b e e n investigated b y M i l g r a m et al. ( 1 9 7 8 ) .

Liquid Spills on Water - The Problem of Oil Pollution

10

Full scale © calm •

9 E * 10

data weather

rough

weather

A Tampa

[m /s] 2

185

test

E> underwater (Miller et

photo al)

O.J

Laboratory

data

in

Graebel-Phelps

c> j I

O.Oi

I

0.2 Fig.

7.5 Oil

7-11:

0.3

I

Haleetal

^

WiJson

0.7

0.8

industries

I

OA

0.5

0.6

Loss data and comparison (For references, see Fannelop,

U

c

with proposed 1983.)

[m/s] correlation.

Oil Spill Drift a n d Ultimate F a t e spilled

in

large

quantities

on

the

ocean

will

eventually

disappear, d u e to a v a r i e t y o f n a t u r a l c a u s e s , or hit a distant coast line. Several billion tons o f oil are transported

in tankers across

the

oceans each year. A g o o d part of the v o y a g e s run parallel to d e n s e l y populated

coastal

regions, in Northern

Europe

as

well as

North

A m e r i c a . T h e well k n o w n accidents w i t h e x t e n s i v e d a m a g e s h a v e all

Chapter 7

186

occurred

in these

waters;

Torrey Canyon, Argo Merchant,

Amoco

Cadiz a n d E x x o n Valdez to m e n t i o n only a f e w n a m e s associated w i t h m a j o r s p i l l s . T h e l o s s r a t e h a s b e e n r e a s o n a b l y c o n s t a n t a t 0 . 1 % for many

years.

The highly publicized major

small fraction of this amount. tropical seas,

the

degradation

spills represent

only

a

F o r oil spilled in the w a r m w a t e r s o f b y the

sun

and

other

"weathering"

a g e n t s o c c u r s o fast t h a t t h e o i l d i s a p p e a r s w i t h l i t t l e v i s i b l e d a m a g e . T h i s is w h y the large losses in the Persian Gulf during recent

wars

h a v e n o t l e d to a n e n v i r o n m e n t a l catastrophy. T h e m i l l i o n t o n s o f oil released in the M e x i c a n Gulf b y the Ixtoc I b l o w o u t , h a d also rather m i l d effects in c o m p a r i s o n w i t h the m u c h smaller A m o c o C a d i z spill w h i c h devastated a large part of the N o r m a n d y coastline. V e r y small spills, of the order of a few tons, numbers

of sea

birds,

far

are k n o w n to have killed large

from shore.

It w i l l for this a n d

other

reasons b e of interest to track a n d forecast the m o v e m e n t of oil spills until their e x p e c t e d d i s a p p e a r a n c e . T h e first s t e p in t h e spreading

and

tracking analysis is the decoupling of the

drifting processes.

o b s e r v i n g or p r e d i c t i n g the

"Tracking" refers in this case

trajectory

of the

centroid of the

to

slick

w h i l e the s p r e a d i n g a n a l y s i s predicts its size. T h e s p r e a d i n g velocity is ( e x c e p t at v e r y early t i m e s ) a n order o f m a g n i t u d e smaller t h a n the c u r r e n t s p e e d r e s p o n s i b l e for t h e drift. T h e c u r r e n t i s u s u a l l y r e s u l t o f s e v e r a l c o m p o n e n t s , w i n d drift, t i d a l c u r r e n t a n d scale o c e a n currents. T h e rapid variations in the tidal

the

larger-

components

m a k e s tracking (even h i n d c a s t i n g ) v e r y difficult n e a r the shore

(an

e x a m p l e i s t h e C h e v r o n spill, d i s c u s s e d b y W a l d m a n e t al., 1 9 7 2 ) . T h e wind-drift c o m p o n e n t can be estimated w h e n the w i n d velocity a n d direction are k n o w n . O n arguing that the turbulent shear stress on both sides of the a i r / w a t e r interface is the same and f r i c t i o n c o e f f i c i e n t s for t h e t w o v e r y h i g h R e y n o l d s n u m b e r

that flows

the also

m u s t b e the same, w e obtain the equation

(7-58)

from w h i c h w e deduce

(7-58a)

F r o m the E k m a n boundary-layer solution w e k n o w that a stress on

Liquid Spills on Water - The Problem of Oil Pollution

187

the o c e a n surface p r o d u c e s m o t i o n in a direction different from the a p p l i e d f o r c e d u e t o t h e E a r t h ' s r o t a t i o n . ( S e e for i n s t a n c e

Prandtl's

Essentials of Fluid Mechanics, 1952.) The importance of this was

effect

first r e c o g n i z e d b y W a r n e r et al. ( 1 9 7 2 ) w h i l e t r a c k i n g the spill

f r o m t h e A r r o w a c c i d e n t ( C h e d a b u c t o B a y , 1 9 7 0 ) . It i s u s u a l t o d a y t o use an empirical correction to the direction of the w i n d . T h e w i n d drift c o m p o n e n t i s t a k e n t o a c t

15 d e g r e e s t o t h e r i g h t o f t h e w i n d

vector at n o r t h e r n latitudes a n d to h a v e a m a g n i t u d e 3.0% o f the w i n d s p e e d . T h e s e v a l u e s i n c l u d e t h e w a v e - i n d u c e d drift c o m p o n e n t , t h e s o c a l l e d S t o k e s ' drift. I n F i g . 7 - 1 2 , t h e "classical" i l l u s t r a t i o n ,

due

to

S v e r d r u p , o f t h e E k m a n l a y e r is r e p r o d u c e d . W h i l e n o r m a l w a v e s h a v e a r a t h e r m o d e r a t e effect o n t h e m o t i o n of drifting slicks, the presence o f b r e a k i n g w a v e s will m a k e the slick disappear altogether. T h e breaking w a v e s beat the surface slick deep into the water m a s s in the form of small droplets. A l t h o u g h the larger

Fig.

7-12;

Wind

induced

current

in the ocean

(Ekman

layer).

Chapter 7

188

d r o p s r e s u r f a c e after a s i n g l e w a v e t o f o r m a n e w slick, a

succession

of b r e a k e r s will m a k e the slick disappear altogether. After a

storm

the slick is u s u a l l y g o n e . T h e r e is a large b o d y o f literature w h i c h deals

with

so-called

"fate

and

effect"

studies

of oil spills.

The

p a r t i c u l a r p r o b l e m o f t h e effect o f b r e a k i n g w a v e s h a s b e e n s t u d i e d i n detail b y N a e s s ( 1 9 8 0 ) .

Special A

Nomenclature constant in p o w e r law, x i

=A t

E

n

A

headwave

c

w a v e v e l o c i t y i n o i l l a y e r , c = yj a 8

n

cross-section

cj

turbulent friction coefficient

D

s i m i l a r i t y v a r i a b l e , D = D (X)

h

thickness o f oil pool

hj

height of headwave

k, k

t

empirical constants

/

e x p o n e n t i n p o w e r l a w for l e a k r a t e

Lj

length of headwave

M

c o n s t a n t i n p o w e r l a w for l e a k r a t e

m

leak rate, time-dependent

q

droplet entrainment rate (loss)

t

"time i n m o t i o n " , t

U

s i m i l a r i t y v a r i a b l e , U = U (X)

E

R

R

U

= (x

LE

source -

x)/u

LE

wind velocity

A

U

relative velocity, water and

U

c r i t i c a l v e l o c i t y for o n s e t o f e n t r a i n m e n t l o s s

UT

drift v e l o c i t y ( m o v i n g s l i c k )

U

w i n d drift c o m p o n e n t ( m o v i n g s l i c k )

V

oil v o l u m e

C

CR

W

air

V

oil v o l u m e contained b y b o o m ( m a x . v a l u e )

We

critical W e b e r n u m b e r (onset drop formation)

m

cr

x, y

slick

XIE

leading-edge position

coordinates

X z

similarity variable, X = x/xi position coordinate (drifting slick)

a 8

r e d u c e d g r a v i t y c o n s t a n t , a =g (p slick thickness

Oj A

s h a p e factor ( h e a d w a v e ) , 6j = hj/Lj d e n s i t y p a r a m e t e r ( s m a l l ) , A - (p - p )l

E

w

w

- p )/p 0

Q

or g Apl p

w

x

p

w

Liquid Spills on Water - The Problem of Oil Pollution

a

net surface

TJJJ

shear stress

N

189

tension

Indices o

oil

w

water h e a d w a v e (front)

Special

Notation

overbar

dimensionless

quantity

REFERENCES A b b o t t , M . B . ( 1 9 6 1 ) O n t h e s p r e a d i n g o f o n e fluid o v e r a n o t h e r . P a r t II. La Houille

Blanche

6, p p 8 2 7 - 4 6 .

A g r a w a l , R . K . a n d H a l e , L . A . ( 1 9 7 4 ) A n e w criterion for

predicting

h e a d w a v e instability of an oil slick retained b y a barrier. Paper

OTC

1983.

Blokker, P.C. (1964) Spreading and evaporation of petroleum o n w a t e r . Proc.

4th International

Harbour

Conference,

products

Antwerp, pp

911-19. Cross, R.H. and National

Hoult, D . P . ( 1 9 7 0 ) C o l l e c t i o n o f oil slicks.

Meeting

on Transportation

Engineering,

Boston,

ASCE, Preprint

1236. F a n n e l o p , T . K . ( 1 9 8 3 ) L o s s r a t e s a n d o p e r a t i o n a l l i m i t s for b o o m s u s e d a s oil b a r r i e r s . Applied

Ocean Research

5, N o 2 , p p 8 0 - 9 2 .

F a n n e l o p , T . K . a n d W a l d m a n , G . D . ( 1 9 7 1 ) T h e d y n a m i c s of oil slicks, o r " c r e e p i n g c r u d e " . AIAA Paper No

71-14.

Fannelop, T . K . a n d W a l d m a n , G . D . (1972) T h e d y n a m i c s of oil slicks. A I A A Journal

10, N o 14, p 5 0 6 .

F a y , J . A . ( 1 9 6 9 ) T h e s p r e a d o f oil o n a c a l m sea. I n Oil on the Sea ( E d D.P. Hoult). Plenum Press. H o u l t , D . P . ( 1 9 6 9 ) C o n t a i n m e n t a n d c o l l e c t i o n d e v i c e s for o i l s l i c k s . I n Oil on the Sea ( E d D . P . H o u l t ) , p p 6 5 - 8 0 . P l e n u m P r e s s . H o u l t , D . P . ( 1 9 7 2 ) O i l s p r e a d i n g o n t h e s e a . Annual Mechanics,

Reviews

of

Fluid

pp 341-68.

Hoult, D.P., Fay, J.A., Milgram, J.H. and

Cross, R . H . (1970) The

Chapter 7

190

s p r e a d i n g a n d c o n t a i n m e n t o f oil s l i c k s . AIAA Paper L a m b , H . ( 1 9 4 5 ) Hydrodynamics,

70-754.

6th Ed, C h VIII, p. 278. Dover.

Lee, R . A . S . (1971) A study of the surface tension controlled regime of oil s p r e a d . M . S . T h e s i s , D e p t M e c h . E n g . , M a s s a c h u s e t t s I n s t i t u t e f

of T e c h n o l o g y . L e i b o v i c h , S. ( 1 9 7 6 ) O i l s l i c k i n s t a b i l i t y a n d t h e e n t r a i n m e n t f a i l u r e o f oil c o n t a i n m e n t b o o m s . J. Fluids Milgram,

J.H.

Technology

Eng. 9 8 , p p 9 8 - 1 0 5 .

(1977) Being prepared

Review

for f u t u r e A r g o

Merchants.

J u l y / A u g u s t , p p 15-27.

Milgram, J.H. and V a n Houton, R. (1978) Mechanics of a layer of

floating

o i l a b o v e a w a t e r c u r r e n t . AIAA

restrained

J.

Hydronautics

12, N o 3, p 9 3 . Naess, A . ( 1 9 8 0 ) T h e m i x i n g o f oil spills into the sea b y

breaking

w a v e s . J.P.T. 3 2 , N o 6. P r a n d t l , L . ( 1 9 5 2 ) Essentials ( N e w G e r m a n E d . Filhrer

of Fluid durch

Dynamics.

Blackie and Son Ltd.

die Stromungslehre.

V i e w e g 1990.)

S i m p s o n , J.E. a n d B r i t t e r , R . E . ( 1 9 7 9 ) T h e d y n a m i c s o f t h e h e a d o f a g r a v i t y c u r r e n t a d v a n c i n g o n a h o r i z o n t a l s u r f a c e . Journal Mechanics

of

Fluid

9 4 , P t 3, p 4 7 7 .

S t o k e r , J.J. ( 1 9 5 7 ) Water Waves,

C h . 10, p 3 0 8 . I n t e r s c i e n c e .

Taylor, G.I. (1950) The formation of a blast wave b y a very intense e x p l o s i o n . I . T h e o r e t i c a l d i s c u s s i o n . Proc. Roy.

Soc. A 2 0 1 , p p 1 5 9 -

74. Van

Houton,

R.

(1976)

Hydrodynamics

of contained

oil

slicks.

Doctoral Thesis, Department of Ocean Engineering, Massachusetts Institute of Technology. Waldman, G.D., Fannelop, T.K. and Johnson, R.A. (1972)

Spreading

a n d t r a n s p o r t o f o i l s l i c k s o n t h e o p e n o c e a n . Offshore

Technology

Conference,

Paper N o 1548.

W a r n e r , J.L., G r a h a m , J . W . a n d D e a n , R . G . ( 1 9 7 2 ) Prediction o f the m o v e m e n t of an oil spill o n the surface Technology

Conference,

Paper N o 1550.

of the water.

Offshore

Liquid Spills on Water - The Problem of Oil Pollution

191

PROBLEMS P r o b l e m 1. B y m e a n s o f t h e s i m i l a r i t y m e t h o d o f F a y , o u t l i n e d i n S e c t i o n 6 . 2 , d e r i v e t h e s p r e a d i n g l a w s for o i l s l i c k s o n w a t e r for, ( a ) the gravity/inertial regime, (b) the g r a v i t y / v i s c o u s regime a n d (c) the surface-tension/viscous Problem

2.

regime.

Determine

the

cross-over times

between

the

flow

r e g i m e s o f P r o b l e m 1, a n d p l o t t h e l e n g t h d i m e n s i o n ( s l i c k d i a m e t e r ) v e r s u s t i m e for a s p i l l o f 1 0 , 0 0 0 t o n s . (Data: a

n e t

= 310"

2

N/m, v

= 10" m / s , p 6

w

2

= 1030 k g / m , p 3

w

Q

k g / m . ) W h y i s t h e k i n e m a t i c v i s c o s i t y for w a t e r v 3

w

more

= 850

relevant

t h a n that o f oil, v ? G

P r o b l e m 3. C o n s i d e r a g r o u n d e d tanker leaking oil at the rate V m / s into a current of velocity U .

D e r i v e t h e s p r e a d i n g l a w s for t h e

3

c

t h r e e flow r e g i m e s o f i n t e r e s t . ( H i n t : R e p l a c e t i m e b y t h e flow

t i m e z/U

equivalent

w h e r e z is the distance d o w n s t r e a m from the point of

c

t h e l e a k . N o t e t h a t c o n t i n u i t y i m p l i e s V~xi&

U 8.) C h e c k y o u r r e s u l t s c

against those listed in Table 7-3. Problem

4.

Consider

an

instantaneous

unstabilized crude, with density p

tons of

= 800 k g / m , spreading on 3

Q

ocean, of higher density p

spill of 1000

the

= 1 0 3 0 k g / m . A b o u t 10% o f t h e m a s s i s 3

w

lost d u e to e v a p o r a t i o n in the first hour, a n additional 2 0 % in next four hours. Determine the

size of the

slick after

5 hrs

the and

c o m p a r e w i t h the case o f "stabilized" c r u d e w h i c h h a s h i g h e r density, say,

Po = 8 5 0

kg/m

3

and

negligible evaporation.

The

flat-slick

a p p r o x i m a t i o n m a y b e u s e d i n t h i s a n a l y s i s w i t h "finite" t i m e s t e p s . P r o b l e m 5. A g r o u n d e d t a n k e r l e a k s o i l o f h i g h v i s c o s i t y a t r a t e 10 m / s i n t o a c u r r e n t U = 0 . 6 m / s . A b o o m o f n e a r 3

c

s h a p e s p a n s the oil p l u m e at a distance

the

parabolic

1500 m d o w n s t r e a m of the

l e a k . D e t e r m i n e t h e oil c o n t a i n e d after 2 h r s . W h a t i s t h e s i z e a n d

the

t h i c k n e s s d i s t r i b u t i o n o f t h e s l i c k c o n t a i n e d b y t h e b o o m o f draft

0.5

m after a s t e a d y s t a t e h a s b e e n r e a c h e d ( a ) w i t h o u t s k i m m i n g a n d ( b ) w i t h s k i m m e r s r e m o v i n g oil at the rate 3 m / s . (Use d a t a available in 3

the text.) P r o b l e m 6. O n e h u n d r e d t o n s o f o i l i s r e l e a s e d

instantaneously

into a current r u n n i n g south at o n e knot a n d w i t h the w i n d from n o r t h w e s t b l o w i n g at 15 k n o t s . D e t e r m i n e t h e s i z e a n d l o c a t i o n o f t h e

192

Chapter 7

s l i c k after 4 h r s . ( U s e t h e f l u i d p r o p e r t i e s s p e c i f i e d i n P r o b l e m 2 . )