A Numerical Model of the Rough Turbulent

8/9/2019 A Numerical Model of the Rough Turbulent

1/14

CHAPTER

65

A

NUMERICAL

MODEL OF THE ROUGH TURBULENT

BOUNDARY

LAYER

IN

COMBINED

WAVE

AND

CURRENT

INTERACTION

HUYNH-THANH

Son

and

TEMPERVILLE

Andre

Institut de

Mecanique

de

Grenoble

B.P. 53X, 38041 Grenoble Cedex, France

ABSTRACT

The

turbulent

boundary-layer flow over

flat

rough beds due

to

wave

or

a

combined wave-current nteraction s

tudied y using implified umerical

second-order

turbulence

model.

The

model

results

re

compared

with many ets

of

experimental

data.

Excellent

predictions

for

ensemble-averaged

velocities

and

favourable redictions

o r

urbulence

uantities re btained.

ariations

f

kinematic

and

dynamic characteristics

of

boundary-layer

flow

with wave,

current

and

ed

oughness

parameters

re

etermined. The

model

s

lso

modified o

simulate

he

scillatory

urbulent

lo w ver ippled eds. The

mean velocity

field nd

he

istribution of

t ime-averaged

urbulence

quantities

re

calculated.

The

validity of he

model

s

verified

hrough omparison with

xperimental

results.

The

performance

and the

limitation

of

the

model

are discussed.

I.

INTRODUCTION

A

knowledge

of th e

boundary

ayer flow n he vicinity of the

ea

bed is

important

fo r

problems

of

coastal

engineering, in

particular

fo r investigations of

coastal erosion, sediment

transport

and

the

transport of pollutants.

Bodies of

water

that

are

subjected

to

currents

and

waves,

according

to

their

characteristics, produce

a

flat, generally rough

bed,

or

a

rippled

bed.

In

order

to

quantify sediment transport, the amplitude and d irection of the velocities

and

shear

stresses

in the

boundary layer

close

to

these

different

shapes

of

bed

must

be

known.

Enquiries

into the

turbulent

boundary layer

generated by a sinusoidal wave are

not

recent.

The

experiments

of

Jonsson

(1963),

Horikawa

and

Watanabe

(1968),

Kamphuis

1975) ,

onsson

nd

arlsen 1976)

re oteworthy.

Recently,

experiments

have

been

performed

using

laser

velocimetry,

e.g.

the experiments of

853


2/14

854

OASTAL ENGINEERING -1990

Sumer

et

al . (1986), Sleath (1987) and Jensen

et

al. (1989).

Theoretically,

numerous

investigations

exist: from

the

analytical models of

Kajiura

(1968),

Brevik

1981),

Myrhaug

(1982), Trowbridge

and Madsen

(1984) to

the

numerical

models

of Bakker

(1974),

Johns

(1975),

Sheng

(1984),

Fredsoe

(1984),

Asano

and

Iwagaki

(1986) ,

Blondeaux

1986) ,

ustesen

1988) , heng

nd

Villaret 1989).

lso

o

e

mentioned

s

he

emi-empirical model

of

Jonsson 1980),

which proposed

universal

distribution

la w

fo r the

velocity

in

the

boundary

layer.

As

far

as

the

boundary

layer

due

to

th e

interaction between

a

current

and

a

wave

is concerned,

few

experiments

are available, among

which only those of

Van

Doom

(1979),

Simon

et

al. (1988) pertain to

the turbulent

and

hydraulically rough case

that

is

of

interest

to

us . After the analytical

model

of

mixing length d ue

to

Bijker (1967),

other an alytical

models are based

on the

time-invariant turbulent viscosity,

as

in

the

case

of

a

wave

:

Lundgren

(1973),

Smith

(1977),

Grant and

Madsen

(1979),

Tanaka

and

Shuto

1984) ,

Myrhaug

1984) , Asano

nd Iwagaki

1984) .

For

numerical

models, hat

of

the mixing

length

due

to

Bakker

and

Van

D oom

1978),

Van

Kerstern

and Bakker (1984) as well as that of

Fredsoe (1984), which assumes a

logarithmic velocity

distribution.

Models

with

more

or

less complicated turbulent

closure

are

also

applied

to

this

problem:

Sheng (1984),

and Davies

et

al .

(1988).

To investigate

the

effect

of

wave and

current

on

the

boundary

layer,

we have

selected he

econd

order

urbulence

model hat

w as

riginally

uggested

by

Lewellen

(1977)

and simplified

by Sheng (1984), Sheng

and Villaret

(1989) fo r

the

one

dimensional

flows.

A

simplified

three

dimensional version

of

th e

model

is

actually developed

fo r

the

case

of

a wave without

and

with

current.

The

numerical

results

are

compared

with

experimental results

in

order to verify

the

validity of

the

model.

In

the

last

section, the model

is

written

in

orthogonal

curvilinear

coordinates

in

order to

investigate oscillatory

turbulent

flow over a

rippled

bed. The

results

obtained

are also

compared

with

the

experimental

results

fo r

the case of symmetric

and

asymmetric

ripples.

II.

BOUNDARY LAYER ABOVE A FLAT BED

II-1.

Equations

f

the

model

The

problem is treated

in

cartesian coordinates (x,y,z) with the z

axis

directed

upwards (Fig. ).

The

flat

horizontal

bottom is

fixed

at z = zo = kN/30, where k^

represents the

equivalent

Nikuradse roughness.

The

system of equations is established with

the

following assumptions: a) the

thickness

of

the

boundary

layer

is

much

smaller

than the

wavelength

of

the wave;

b)

th e

amplitude

of

th e

wave

velocity

Uh is much

smaller

than the

wave

celerity

C.

In

these

conditions,

the

mom entum

equations

for

the

tw o

horizontal components

of velocity

(u,v)

along

x and

y

can be

written:

3u

3P

d T-T.

..

3v

P

d -r-r,

(1 )

=i._+_(.

UW

)

;

(2 )

_ =l_

_(.

V

w)

3t

dx

dz t

dy dz


3/14

ROUGH TURBULENT BOUNDARY

LAYER

855

where th e Reynolds

stresses

-

uV

and

- v'w' can be

modelled

in th e form:

3u

3)

-

uw = v

t

-

3z

—

3v

-vw

=

v

t

dz

where

v

t

represents

th e

turbulent viscosity.

The pressure

gradients

are

expressed as

follows

:

4)

.1

3P

auhx

3P

C

ap

au

2 2 .

.1

3P

C

P 3x

t

P x

y

t

P y

where

(Uhx. Uhy) are

th e

tw o

horizontal

components

of

th e

wave

velocity

and P

c

represents th e pressure due to

the

current.

Turbulent

closure

is

performed

by means of

tw o

equations

fo r

th e

turbulent

kinetic

energy

K

and

fo r

th e

length scale

L

of

th e

turbulence

(Lewellen,

1977):

(5 )

(6 )

3K

3t

3L

3t

=

t

[ff-ff]

L

2

3z 3z

«k

SW.

+

.075

VTK

+ .2

—(v

t

—

3z

3z

0.375 VI

3(VICL)

VK

3z

The

assumption

of

local

equilibrium of th e

turbulence made

by Sheng (1984)

allows v

t

to

be

put

in

the

form:

VKL

7)

v, =

VI-

*c0

wave direction

current

direction

Fig.

1

Outline

of

physical

system a nd reference system ofaxes flat

bed)


4/14

856

OASTAL

ENGINEERING-1990

At

the

bottom

(z = 2 0 )

the

boundary

conditions

in all

cases

are

the

following :

(8 )

= v = 0 ;

dKJdz

= 0 ; L = azo with a 0.67,

where

the

von Karman

constant

is taken

as

k

=

0.4.

The

conditions

at

the

upper

limit

of

the

boundary

layer

depend

on

the

particular

case

studied,

and

will

be

described

later.

The

set

of

equations (1 )

to (8 )

is discretised

using

th e

implicit finite

control

volume method (Patankar,

1980) on a

grid

whose step

size

increases

exponentially

from bottom to

top,

thus giving good resolution near th e

bed

where gradients are

important. The

im e

te p s

constant

over th e

whole

period

of

th e wave.

Each

discretised equation corresponds to a

tridiagonal

matrix

which

can

be solved by

means

of

Thomas's

algorithm

(Roache, 1976).

II-2.

ase

of

the

ave

In

th e

case

of

a unidirectional wave,

th e

above set

of

equat ions

is solved by

taking

v = 0. The

pressure

gradient

is given by:

9

1̂

=

Pax

where

Uh = Uh

sin

c o t

is

th e

wave

velocity

at

the

upper

limit

of

th e

boundary

layer

defined by

Z h

= 5K, R

corresponds

to

the

thickness beyond which

K

is

zero.

The

following

approximation

was

obtained:

0.81

ah.

k

N

For

z

=

z j ,

the

boundary

conditions

are :

(11) =

L

=

0

nd

U

=

U

h

(10) =

0.246

k

N

with h

—

C O

*

Comparison

with

experimental

results The

model

results

were

compared

with

th e

experiment

of

Sumer

et

al. (1986). The

lower boundary

is

at

0

=

kN/30

=

0.0133

cm, and

the

upper boundary is

taken

to

be

at

Z h =

20

cm. The

magnitude

of

th e

wave

velocity

is Uh

= 10 m/s nd he

period

T =

T C / C O

.1

. G o o d

agreement

can

be

seen

in

Figure

2

for

the

velocities, except at

z

=

0.1

cm.

The values

of

the

friction

velocity

u*

=

sign

(-u'w')

V|

-u'w'|

re lightly ower than ound

experimentally

(Fig. 3).

In

Figure 4

the

profiles

of

th e

fluctuating velocities

Vu'

2

and

V

w'

2

are

compared. It

can

be seen

that

there

is agreement

fo r Vu'

2

fo r phases

between

30°

and

120°,

and

fo r

V

w'

2

fo r

the

other

phases.

*

Wave

friction

coefficient:

In

investigations

of

the

wave

boundary

layer,

the

friction

coefficient

fh

introduced by Jonsson

(1963)

is

often

used

-

y

fh

Uh

where

x

is

th e amplitude

of

the

shear

stress

at

th e

bed. The formula fo r fh

that arises


5/14

ROUGH TURBULENT BOUNDARY

LAYER

857

from

th e present model

as follows:

(12)

h

= 0.00278 exp

-0.22

l«

5

I S

For

comparison,

Figure

hows

th e

curves

obtained

from

th e

formulae

of

Kajiura

(1968), Kamphuis

(1975), Jonsson

and

Carsen

(1976).

The

curve given

by

(12)

is

close

to th e

results

of Kamphuis.

Fig.

2

Comparison between th e

calculated

velocity profiles and

those

measured

by

Sumer

et

al 1986)

symbols)

for

the

different phases.

Fig.

3 Comparison

between

the shear

velocity calculated

by the

present

model —

and

that obtained experimentally

by

Sumer et al

1986)

(•)


6/14

858

COASTAL

ENGINEERING-1990

Measurement

30 °

3+ 0

t

•

60°

Present

model

10

*t> 30 *

» /»

1

/2

3

•

C0

V 7 ?

7T- K)

3z

i«

lZ*L

+

K

1 / 2

120°

150°

180°

•P •

£

u

7

*)"

2

(cm/s)

)̂

W

(cm/s)

Fig.

4

omparison

between

th e

fluctuating

velocity

profiles calculated by

th e present

model and and

those

obtained

experimentally b y Sumer et al

1986)

for

th e

different phases.

Present model

Kajiura

Jonsson

Kamphuis

ai>/kN

Fig.

5

Variation

of

the

friction coefficientfy

s

a

function

of

a/ ,

Iks


7/14


8/14

860

COASTAL

ENGINEERING-1990

Figures 7 nd

8

show th e

variation

of f

C

h

and f

c

s

function

of

Sh/kN fo r

different values

of Z C Q / I C N , Uj/Uh.

and

< t > h -

From

these

curves,

it

can

be

noted

that or

fixed

Z c o / k N

and fa, f

c

h

and

f

c

increase for increasing

LyUt,.

The change in f

c

with

< | > h

is

substantial

only

fo r

LVUh

<

.

When

he

current

is

tronger

than

he

wave

(lyOi,

2

.5+2),

th e influence

of

< j > h on f

c

is

not

marked, which means

that

in

this

case

th e

mean

characteristics

of

the

current practically

do

not

change

under

the

action

of

the

wave.

Fig. 7

Variation of th e friction

coefficient

k

as function

ofa

h

k

N

,

W* JV ,

UJUk

nd

j >

h

.

Z c o / k

N

= io o i f c = 0 °

l)U«A4»0 ̂ 2)IMJL-1 3)U«M=1,5 4) UAJk

=

Fig.

8 Variation

of the

friction coefficient as

function

ofahlk^,

ZcolkN

UJUf,

nd

fa .

Parameters

and

notations

as

in

Fig.7.


9/14

ROUGH

TURBULENT

BOUNDARY

LAYER

86 1

III. B O U N D A R Y

LAYER

O N A RIPPLED B ED

III-1.

Formulation

f he model

The

physical

problem

is outlined in

Figure

9 under the action of a wave of

wavelength

Lh,

maximum

velocity

Uj,

and

period T,

two-dimensional

vortex

ripples

are

assumed to

be

present on

th e bed.

Laboratory

and

in

situ

measurements

have

shown

that

Lh »

L^ .

This allows us to

restrict

the

zone

of

the

calculation

rather

than

investigating

the

problem over

th e

whole of

the

wavelength

Lj,, we

shall

only

consider

th e

wavelength

L

t

s hown

n

Figure

9.

Moreover , o implify he

description of

th e

boundary

condit ions

t

th e urface

of

th e

ripple

nd

lso o

eliminate

the

unknown

pressure

gradient due

to

the

bed

form,

it is

convenient

in

this

case

o

ransform

he

artesian

oordinates

x,z)

nto

r thogonal

urvilinear

coordinates

(X,Z),

and

to

use

th e variables

\i/ (stream

function)

and

£ ,

(vorticity)

instead

of

the

velocities

u

and

w.

Potent ia l

f low

Free

surface

Dom ain o f

calculation

Fig. 9

Scheme

of the

physical

system

rippled

bed).

In

general,

th e

coordinate

transformation

is

given by :

N

(16)

=

+

£

n

.

exp

(-

n

k

r

Z).

sin

(n

k

r

X

n

)

n=l

N

Z =

z

-

2^

„ .

exp

(- n

k

r

Z). cos (n k

r

X

n

)

n=l

where

n

nd

8

n

re

he mplitude

nd

phase difference

of

the

th harmonic

describing

the

ripple, and k

r

=

7t/L

r

s

he wave umber ssociated with L,.

The

stream

function

\ |/ and

the

vorticity

£ ,

are

defined by :

(17)

u

d\|r

•=

3u

3w

\ i/

—

w —

dz

x

z

dx

In

coordinates

(X,Z),

the

set

of equations

to

be

solved

is


10/14

86 2

OASTAL

ENGINEERING -1990

(18)

V

2

V

=

(19)

a t

a

x,

z)

(20

» ^M Q

1 > 2

J_L*

+

±

J

t

a x,z)

ax\

x/

az\ az

+

t

J

f a V

2

i

VtK

-IAK

J

2

\a x

2

\az

2

j

The

length

scale L is

assumed

to

vary

as

follows :

(21)

= aZ^Jl

-i

and

th e

turbulent

viscosity

v

t

is

determined

by (7).

In

th e previous set of equations, and K are

terms

pertaining

to

the

partial

2

derivatives

of

\ |/

and

v

t

J

is

the

jacobian

of

the

coordinate

transformation

V

s the

laplacian

operator.

The

following

boundary

conditions

are

applied:

-

At

the

lower

limit of

the

boundary

layer

(Z

=

Zo =

1CN/30)

tyjs

V ̂ „.

_

3K

,

_

2JXII

az ax

Y

az Zi-zo)

2

where

\|/i is

the

stream

function

at height

Zi on the second

node of the

grid

(Roache,

1976).

-

At

the

upper

limit

of

th e

boundary

layer

(Z

=

Zj,)

(23)

=

h

U

h

(t) K ;

-

A t

the

lateral

boundaries

(X = 0

and

X

=

Lr),

we

assume

spatially periodic

conditions fo r r,

\

and K.

The above

se t

of

equations is discretized

using

implicit

finite

difference

schemes

(centred

in

space

and forward in

time).

The

alternating direction implicit

(A .

D. I. )

method is

used

to

solve the

equations fo r

\ and K.

The

Poisson

equation

fo r

y

is

solved

by th e

bloc-cyclic

reduction

method

(Roache, 976)

which

allows

a huge

saving

in

calculation

time

compared

with

the

Gauss-Seidel

iteration

method.

The

spatial grid

contains

M xN

nodes

with step

AX

=

const, and

AZ

varying exponentially

from

he

bottom upwards.

The

im e

te p

s

At

=T/360

.

n

ll

he

est ases,

convergence

is

obtained

after 20

calculation

periods.

111-2.Comparison

ith

he

xperimental esults

f Du oit

nd

Sleath

1981)

The

dimensional

parameters

in

the

test

fo r

comparison are

the

fol lowing:

-

Symmetric

ripple

with

Lr

=

17.2

cm, h

r

=

2.9

cm, d = 0.04 cm

-

Cosinusoidal

wave

U^ =

Uh

cos

c o t , where U h =

4.3 m/s, = J C / C O =

5.37s


11/14

ROUGH

TURBULENT

BOUNDARY

LAYER

863

For

the numerical calculation, a 17x25 node

grid

was

chosen with AX = 1.0625

cm and

AZ

varying

from 0.056

cm

to

0.5

cm.

The

equivalent Nikuradse

roughness

is

I C N =

2.5

d

=

0.1

cm.

The

upper

limit

is

chosen

to

be

equal

to

Zj, =

5

cm.

Fig.

10 Comparison between

the

results ofpresent

model

and

those

of

the

measurements

ofDu

Toit and Sleath 1981)

for

the t ime

variation

of

the horizontal

velocity

u

and of

the horizontal

fluctuating

velocity

V

u

2

.

Measurements

at

height

z =

1.65

cm

above

th e

crest

Pn

Fig.

11 Vertical

variation

of

the

amplitude

of

the velocity

u calculated

(-

• )

and

measured «—»)

by Du Toit and Sleath 1981) above the crest.

At height z =

1.65 cm

above

the

crest

(Fig.

0),

good agreement

can

be

seen

between

the

amplitude

of

the

calculated and measured

horizontal

velocities

u, as well

as

for the fluctuating

horizontal velocity

Vu'

2

.

There

is ,

however,

a

discrepancy

of

25°

between

th e

measured

local peak

(tot

=

140°)

and

that

calculated

c o t

= 165°).


12/14

864 OASTAL

ENGINEERING

-1990

The vertical variation of th e amplitude of th e velocity

u,

designated

by u,

obtained

from

th e

model

coincides

with

that

measured

above

the

crest

(Fig.

1).

Note

that

the

z

axis

is

normalized

by

th e

parameter

P

= y

co/2v

=

7.2

m"

1

and

z\ is

measured

from

the

ripple

crest.

III-3.

Comparison

with

he

xperimental

esults

f

Sato t

l.

(1987)

The

test parameters

are

the

following:

-

The

ripple is

asymmetric

with

Lr =

12

cm,

h

r

=

2

cm,

d

= 0.02

cm.

- T he

potential

flow

is

a

third-order

Stokes

wave

(T

=

2 J C / C O =

4

s):

U

h

= 29,5 (cos

c o t

+ ,258

cos

2cot + ,048

cos

3cot) cm/s)

For

the

modelling,

after

determining

the

amplitudes

a„

and

th e

phase

shifts

0

n

of

th e

simulated

ripple, we choosed a

grid of

13x25 nodes with AX = cm, and AZ

varying

from

0.06

cm to

0.6

cm.

The

time step

is

At

=

0.011 . The upper

limit

is

choosen

at Zh =

6.5

cm .

Figure 12

shows

the

comparison

between th e

results

of

th e

model and

those

of

the

measurement fo r

the

velocity

field

and

the

turbulent

kinetic

energy

K

fo r phase c o t

=

54°.

It can

be

seen

that

the

vortex

obtained

with

the

model

on

the

right hand

leeside

of

the

ripple is

weaker than

that

measured,

and

the

calculated intensity

K

is

smaller in

th e model than found experimentally.

IV. DISCUSSION N D CONCLUSION

We

have examined

the

problem of

the

osc illatory turbulent boundary layer

on

a

rough

ea

bed

using

different

versions

of

a

urbulent

closure

model

with

w o

equations,

one for

the

turbulent

kinetic

energy

K

and

the

other for

the

length scale L.

* erformance

of

th e

model:

For

a

flat

bed,

a

simplified

three

dimensional

model

w as

used to investigate th e hydrodynamic

characteristics

of

th e flow

in

th e

boundary

layer

as

a function

of

the

different wave,

current,

angle

of

interaction

and

bed

parameters.

For

the oblique wave-current interaction,

the

model

requires

further

experimental verifications.

For a rippled

bed,

we

have

used

a

two-dimensional

model

that can

reproduce

the

velocity

and

th e vorticity fields

as

well

as

other

turbulent

quantities. Comparison

with

th e experimental results

shows

that

this

model

is

able

to

predict quite well th e

complex flow

properties over

a rippled bed.

Before applying th e model

to

general

cases,

it

would

be

necessary

to

confirm

the

numerical results

by

conduct ing

further

tests,

particularly fo r

the

Reynolds

stresses

and

the

turbulent

quantities.

*

Limitation

of

the model

: As

fo r

all

models of turbulent closure

(Rodi,

1980),

the present

model

w as originally designed

fo r

permanent flows

in

the

fully

developed

turbulent

regime

t high

Reynolds umbers.

When

he

flow s

oscillatory, he

condition of

local

equilibrium

of the

turbulence,

which

is

valid

fo r a

permanent

flow,


13/14

ROUGH TURBULENT

BOUNDARY

LAYER

865

x cm)

x cm)

Fig. 12

Comparison

etween

he

esults

of

th e

present

model

and

the

measurements

of

Sato et

al.

1987)

at

phase

c a t =54°.

a)

velocity

field;

b)

turbulent

kinetic

energy

field

is

no

longer

completely satisfied,

particularly at

th e

times

when

th e

velocity

of th e

potential flow is small. Consequently

there

is

a

time

variation of th e friction

in

th e

oscillatory

boundary

layer, which induces the

change

of flow regime in th e

course

of

a period. No

such

change

w as

included

in

th e

model.

Thus, to

obtain

more

precise

results,

it

is

necessary to improve th e model not only fo r high Reynolds numbers b ut

also

fo r

moderate

Reynolds

numbers.

In

parallel

with

investigations

into

improvements,

w e

shall apply th e

model to

th e

prediction

of

certain

important

parameters

of

th e

natural

boundary

layer,

together

with

analysis

of sea measurements

in

the framework

of th e

GDR

Manche

project.


14/14

86 6

OASTAL ENGINEERING-1990

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A Numerical Model of the Rough Turbulent

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