Local Scour: Influence of an exponential eddy viscosity distribution on the longitudial flow velocity

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eddy viscosity distribution on the longitudina1 flow velocity Report nr. 14-88

G.J.G.M Hoffmans

Faculty of Givi1 Engineering Hydrau1ic Engineering Delft University of Techno1ogy

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2.1

2. Flow velocity and eddy viscosity distributions (2-D-vertical) 7

Logarithmic flow velocity profile Eddy viscosity (parabolic) profile

7 7 9 10 10 12 13 13 14 Standard concept 2.1.1 2.1.2 2.2 Coles concept 2.2.1 2.2.2

Coles flow velocity profile Eddy viscosity (Coles) profile 2.3 Exponential concept (eddy viscosity)

2.3.1 2.3.2

Exponential form of the eddy viscosity Flow velocity profile (exponential)

3. Eddy viscosity at a free surface (uniform flow) 3.1 General

3.2 Classical theories of Prandtl and Boussinesq 3.3 k-equation

3.4 k-equation and Boussinesq 3.5 ODYSSEE (k-e-model)

17 17 17 17 20 20 4. Assesments

Undisturbed outer flow Disturbed outer flow

22 22 22 26 26 26 29 4.1 4.2 4.3

Open channel flow Pipe flow

Internal boundary layer 4.3.1 4.3.2 5. Conclusions References Appendices 4.3.2.1 Shear-stress distribution 31

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a B C 11 c_,..,. D g i k 1 m L n m q Q Re R* R u,n t u w x y flow depth

width (of flurne)

turbulence constant

local friction coefficient

turbulence constant

constant in turbulence model (k-f)

internal diameter

acceleration of gravity

energy slope

turbulent kinetic energy per unit mass

mixing length (Prandtl) length scale of turbulence

integer, index

pressure fluctuation

stress production of the turbulent energy discharge per unit width

discharge

Reynolds nurnber

u*a/1I ; Reynolds nurnber correction term

time

local time-averaged longitudinal flow velocity longitudinal flow velocity fluctuation

friction or wall- or bed-shear velocity local shear velocity

longitudinal free stream velocity

depth-averaged longitudinal flow velocity

transverse flow velocity fluctuation (y-direction) local time-averaged vertical flow velocity

vertical flow velocity fluctuation longitudinal coordinate transverse coordinate (L) (L) (

-

) (

-

) (

-

) (

-

) (L) 2 (LT ) (-) 2 2 (L T ) (L) (L) (

-

) (

-

) 1 2 (ML T ) 2 3 (L T ) 2 1 (L T ) 3 1 (L T ) (

-

) (

-

) (

-

) (T) 1 (LT ) 1 (LT ) 1 (LT ) (LT ) 1 (LT ) 1 (LT ) 1 (LT ) 1 (LT ) 1 (LT ) (L) (L)

(5)

z a w f IC 1/ 1/ t P Ot o 1/ r t "0 Ó w

n

'i1 w( ) zero-velocity level exponential parameter

rate of energy dissipation per unit mass by turbulence Von Karman's universal constant

kinematic molecular viscosity coefficient

eddy viscosity fluid density

turbulent normal stress

standard deviation (eddy viscosity)

turbulent shear stress

wallor bed-shear stress

(new wall-) boundary layer thickness

Coles's profile parameter

scalar product wake function Subscripts C Coles E Exponential 1/ t max maximum

ref at reference height

s at surface

w at outer edge new wall-boundary layer

(L) (-) 2 3 (LT ) (

-

) 2 1 (L T ) 2 1 (L T ) 3 (ML ) 1 2 (ML T ) (

-

) 1 2 (ML T ) 1 2 (ML T ) (L) (

-

) ( - ) (

-

)

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1. Introducti~n

The general purpose of this research project is to model mathematica11y

the loca1 scour downstream of a structure (2-D). The model has to

simu1ate the deve10pment of the scour as a function of the time,

(figure 1). Basica11y two mode1s are necessary name1y a flow model and a

morphologica1 model. The latter model has to describe the bed- and

suspended load and the eros ion of the bed.

In the present study a function (prescription) for the eddy viscosity

with a variable parameter, which is a parameter in the exponential

function, is discussed. A sensitivity study has been made in order to

fit the parameter mentioned above for several flow types.

In this study the hypothesis of Boussinesq and a 1inear shear stress

distribution are applied.

sz

>

sz

--uniform flow ...

-

relaxation zone

mixing layer

figure 1 Definition sketch

Using this eddy viscosity concept, it follows, that the profile of the

longitudinal flow velocity is a logarithmic profile with a correction

term. The differences between the traditional logarithmic flow velocity profile and the Coles flow velocity profile are small near the wall. A large difference occurs at the surface of a uniform flow or at the

center of a pipe flow. There the gradient of the longitudinal flow

(7)

regardless of the value of the parameter in the exponential function for the eddy viscosity.

In section 4.0 some applications are given for a uniform flow, a pipe

flow and a new wall-boundary layer with a disturbed and an undisturbed

outer flow (no turbulence). In paragraph 4.3.2 a function for the eddy

viscosity is discussed, which describes the eddy viscosity in a new

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2. Flow velocity and eddy viscosity distributions (2-D-vertical)

2.1 Standard concept

2.1.1 Logarithmic flow velocity profile

In the (turbulent) boundary layer two regions may distinguished. A region adjacent to the wall, in which the flow is directly affected by the condition at the wall. This condition is expressed by the wall shear stress. This region is referred to as the 'wa11' or 'inner' region. Beyond this wall region there is an another region, where the flow is only indirectly affected by the wall through its wal1 shear stress. This second region is usually referred to as the 'outer' region.

In the inner region there is a thin layer at the wall where the flow is (essentially) viscous. Outside this thin viscous sublayer inertial effects become more and more important compared to the viscous effects, until at some distance from the wall the flow becomes fully turbulent with dominating inertial effects and negligible direct viscous effects on the main flow. The region between the fully turbulent flow and the viscous sub-layer, where the viscous stresses and Reynolds stresses are of the same magnitude, is called the transition or 'buffer' region.

The momentum equation over the boundary layer thickness can be written kinematically in terms of local shear velocity u*z as (Coleman, 1984):

aü

u2 = v +

*z az (1)

in which:

u*z local shear velocity

v kinematic molecular viscosity coefficient ulocal time-averaged longitudinal flow velocity

K 0.4; Von Karman's universal constant z vertical coordinate

Using an asymptopic approach, the boundary conditions at the bottom (z = 0) and for z = ó are respectively:

(9)

Z = 0: u = 0 and (2)

Z = ó :

w u ü(ó w) and

o

(3)

in which:

ó boundary layer thickness w

u* friction velocity or wall-shear velocity

At the origin of z viscous stresses predominate. Dropping the turbulent

.

aü

stress term

(~z)2{az)2

equation (1) reduces to:

Equation (4) can only be integrated if an expression for u*z is

available. To scale

ü

by the friction velocity u*, assuming that u*z is equal to u*, equation (4) is integrated over z. This gives:

(5)

v

which is, of course, the viscous sublayer equation.

Moving outward in the flow, the turbulent stresses predominate, so eliminating the viscous stress term, equation (1) changes into:

(6)

As with equation (4), equation (6) can be integrated only if an expression for u*z is available. Again to scale

ü

by the friction velocity u*, taking u*z = u*, yields:

1

ln ~

~ Zo

(7)

(10)

The assumption u*z = u* was made in order to perform the integration; in short, it is an asymptotic approximation at Z

«

ó , where the shear

w

stress is approximately equal to the shear stress at the wall. In equation (7), Zo is an intercept at which

ü

calculated from the logarithmic function is zero; this is a boundary condition on the

integral and does not indicate that a real velocity profile goes to zero at a finite heigth above the origin of z.

Generally equation (7) can be applied to flows over both rough and smooth boundaries.

2.1.2

Eddy viscosity (parabolic) profile

Considering a uniform flow in an open channel, the flow is driven by the component of the acceleration of gravity in the longitudinal direction. Balance of forces and stresses supplies a linear shear stress

distribution (no external forces):

T = pga(l - ~)i t,xz a (8) in which: T = shear stress t,xz p fluid density g acceleration of gravity a = flow depth i energy slope

The definition of the bed-shear stress (TO) is given by:

pu2

*

(9)

Conversion of (8) and (9) leads to:

T = pu; (1 - -az)

(11)

Originally Boussinesq assumed that the turbulence stresses act like the

viscous stresses, that is, that they are directly proportional to the

velocity gradient. However, in a boundary layer, for instance, the eddy viscosity is not constant near the wall. The hypothesis of Boussinesq reads (vt is a function of z):

T

t,xz (11)

in which:

vt eddy viscosity

w local time-averaged vertical flow velocity

x longitudinal coordinate

Using a logarithmic flow velocity distribution (7) and applying the

hypothesis of Boussinesq (11), with ~:

«

~~

and the assumption

mentioned above (linear shear stress), the dimensionless eddy viscosity reads:

~ (1 - ~)

a a (12)

which is the parabolic eddy viscosity distribution.

2.2 Coles concept

2.2.1 Coles flow velocity profile

The Coles flow velocity profile, which is also known as the law of the wake, over both geometrically smooth and geometrically rough boundaries

can be written as (Hinze 1975, p. 697):

1

[ln ~ K Ó W - II{2 - w(~ ))] w

(~ »1)

Zo (13)

(12)

in which:

local time-averaged longitudial flow velocity (Coles)

longitudial free stream velocity

Coles's profile parameter 1 + sin(2~ - 1)~ - 2sin2{~ ~

Ó 2 2 Ó

w w

wake function (14)

Z

The wake function w(S ), which is a positive function, is a correction w

to the logarithmic flow velocity profile. Experimenta1 evidence suggests that this law has a universal character.

In order to quantify the longitudinal flow velocity for Z = ó and to

w

indicate the differences between the (pure) logarithmic flow velocity profile and the one according to Coles, the following proceeding has been carried out.

Using the approximation of (2.1.1) and substituting z (13), [üC(zo)= OJ, this gives:

Zo into equation

1 Zo Zo

; [In

s- -

rr{2

- w(s-)}J

w w

(15)

Substitution of (15) into (13), the Co1es flow velocity profile can be written as: z z

1

[In

rr { (

)

K zo+ w

s-W (16) A

where term Arepresents the 'Coles' correction to the logarithmic profile.

The parameter

rr

is a profile parameter and does not depend on z. It is re1ated to the loca1-friction coefficient Cf = 2(U*/U)2. In a boundary

layer

rr

is a weak function of x.

The Co1es parameter can be approximated by integrating equation (16) to z (appendix A) :

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ó . - ln ...1!

+

1

Zo

(17)

Here Ü represents the depth-averaged longitudinal flow velocity.

For a boundary layer with zero pressure-gradient Coles assumed

rr

to be independent of x. He obtained

rr -

0.55 with ~ - 0.4.

For an open channel flow

rr

is found to depend on the Reynoldsnumber R* (-u*a/v) up to a certain value of this number: Below R*

=

500, the Co1es parameter

rr

is nearly zero, then it increases with R* and assumes a constant va1ue of approximate1y 0.2 for R*

>

2.000, (~~ 0.419), (Nezu and Rodi, 1986).

It can be noted, that if the Coles parameter

(rr)

is equal to zero equation (16) reduces to (7), (figure 2).

d

z

~ t i :l

i

" d QJI 0.. 1.1 1.2 x - exponential;

rr

= 0.00 ct - 1.50 w

o -

Co1es;

figure 2 Coles and 'exponential' flow velocity profile

2.2.2 Eddy viscosity (Coles) profile

Applying the Coles flow ve10city profile, the hypothesis of Boussinesq and a linear shear stress distribution the eddy viscosity can be written as:

(14)

{l

-

~ } ~ ~

s

w u*ów = K,

s

_w _[₁ _{+ 11" "6}rrz . ( z

₎

₁

S1n 11""6 w w (18)

If the Coles parameter rr is equal to zero, equation (18) reduces to (12),

[ó

-

al

.

w

2.3 Exponential concept

2.3.1 Exponential form of the eddy viscosity

Since the calculations by the turbulence model ODYSSEE (Hoffmans, 1988a) show an exponential character for the eddy viscosity distribution in a number of local scour holes, in 2.3 an analysis is given of the

longitudinal flow velocity using an exponential function for the eddy viscosity.

exponential: K, "'6

z

exp -ow ó

(

~ )

w w

(19)

in which:

o = 'exponential' parameter

w

Examination of equation (19) gives the following result. For 0 < 0 the

w

eddy viscosity increases progressively, that is, if dv /dz and d2v /dz2

t t

are positive and increase with z. For 0 = 0 a linear function will be

w

obtained. For 0

<

0

<

1 the eddy viscosity increases degressively, that w

is, if dvt/dz is positive and increases with z, while d2vt/dz2 is

smaller than zero. For Ow

>

1 a local maximum occurs for z = ó~ow. By

an increase of 0 not only z (z-coordinate where vt is at maximum)

w max

but also the absoLut;evalue of tit will decrease, (figure 3).

Close to the bottom the eddy viscosity is approximately:

z

«

ó

(15)

Assuming a logarithmic flow velocity profile near the wa11 region, this is in agreement with the theorem of Prandtl.

Prandtl: T t,xz (21) in which: 1 - ~z; mixing 1ength m 1.0 0.' 0.6 (lw> I

• 6w

_o.~ O.l O<Clw<1 Clw< 0

0 O.O~ o.oa O.ll 0.16 O.lO O.l~ 0.21 o.n 0.35 o.~ VI

u.liw

figure 3 Eddy viscosity as a function of Ct

w

2.3.2 Flow velocity profile (exponential)

Using the exponentia1 eddy viscosity function and applying the hypothesis of Boussinesq and a 1inear shear stress distribution,

integration of the horizontal velocity gradient from Zo to z results in a logarithmic flow velocity profile with a correction.

aw

aü

Assuming that 8x

«

8z' conversion of (10) and (11), (a - ów)' leads to:

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(16)

Z) ( ~)

"6

exp aw 6

w w

(23)

Using the Taylor-series to the exponential function integration of (23)

from Zo to z, [ü(z) - üE(z)], results in (appendix B):

ÜE(z) z co _z _Zo

1

_[In ₊ _l: _{Ru n}_{{ (}_6"_}_n _{6"}n_)] ₍₂₄₎ u* K. _Zo n=l ' w w B in which:

local time-averaged longitudinal flow velocity (exponential vt)

n-l a a __J!_ [__J!_ _ 1] n = index (25) n! n R u,n

Term B, equation (24), is a correction-term to the logarithmic flow

velocity profile. The higher order terms converge to a limit value,

which is equal to zero, (appendix C). It appears that after 30 terms

(n = 1,...,30), which is satisfactory to converge the sum of the higher

order terms to zero, (appendix D), R is approximately equal to zero

u,n

for a ::::1.50, (1.50286), (figure 4).

w

Difference between the Coles and the exponential concept (velocities) Using this eddy viscosity concept the agreement with a (pure)

logarithmic flow velocity profile is fair in the near wall region,

however, a relatively large difference occurs in the outer region

(z = 6 ), (figure 2). There the gradient of the longitudinal flow

w

velocity is equal to zero, while applying a (pure) logarithmic or the

Coles profile the gradient has a small value.

Differentiation of (24) to z yields: co

1 [

1

+

1

l: K. Z 6 w n=l R u,n (26)

Substitution of z = 6 into (26) gives:

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aÜE(ó

w) co

1

_1_ _[1 ₊

_L

R n1 - 0

u*

az

1(15w n-1 u,n

co

because

L

R n ...-1, (appendix

E).

n=l u,n (27) (28) number of terms n - 30 11

..

3 2 L R u,n 0

r

-1 -2 -3 -4 -2

.

-o 2 4. ... Ct W

figure 4 L Ru,n as function of the parameter Ctw

In order to write the longitudina1 flow velocity as a velocity defect using an exponentia1 eddy viscosity, the fol1owing pr-ocedu'r'ehas been carried out.

Substitution of

z -

15_w into (24),

üE(ó)

can be written as:

üE(ó

w) 15 co Zo

1

_[in ~

₊

_~ _R _{1

_{- (';5)}

_n}

₁

u* I( Zo n-l u,n w

(29)

Subtracting (24) from (29) leads to:

co

1

[ln z.,

I( 15

w

(18)

3. Eddy viscosity at a free surface (uniform flow)

3.1 General

In this section an overview is given of the different values, which the eddy viscosity can have at a free (plane) surface in a uniform flow starting from different theories. The classical theories of Prandtl and Boussinesq yield a zero eddy viscosity at the surface, while stàrting from the k-equation and the k-e-model, the eddy viscosity does not reduce to zero there.

3.2 Classica1 theories of Prandtl and Boussinesq

Using a logarithmic or the Coles flow velocity profile, the mixing

length must be equal to zero at the surface according to the theorem of

Prandtl (21), since T = 0 (no external

t,xz

[ aw

aÜ]

hypothesis of Boussinesq (11),

ax«

az '

the eddy viscosity reads:

forces). Conversion of the and the theorem of Prandtl

(31)

Because 1 equals zero, the eddy viscosity reduces also to zero.

m

Using the longitudinal flow velocity profile, which follows from the exponential eddy viscosity concept, the shear stress equals zero at the

surface, since the gradient ~~ = 0, however, the eddy viscosity does not

reduce to zero (19).

3.3 k-equation

Considering a steady two-dimensional flow with a constant width and a

plane water surface, the turbulent energy equation reads (Kay and

(19)

Dk Dt (32) turbulent diffusion p -t xxa

au

+ -t zxr

au

+ -t xzr

aw

+ _atzz

aw

p'

a

x

p'

az

p'

ax

p'

az

(33) f = (34) in which:

k turbulent kinetic energy per unit mass presssure fluctuation

stress production of the turbulent energy time

t

longitudinal flow velocity fluctuation

transverse flow velocity fluctuation (in y-direction) vertical flow velocity fluctuation

rate of energy dissipation per unit mass by turbulence turbulent normal stress

scalar product

Since there are no changes in the longitudinal direction (uniform flow), all the gradient types with ~x are equal to zero and because the

vertical flow velocity

(w)

and the fluctuating vertical velocity (w1) reduce to zero near the surface (free water surface), thus the normal

stress

[a

= _pw1W1] reduces also to zero, the production term P

t,zz reads:

p -tr zx

au

p' Bz (35)

In case there are no external forces at the surface, the production by the shear stress equals zero. Then equation (32) reduces to:

(20)

The turbulent diffusion term in (36) and the dissipation are modelled (Rodi, 1980):

a

n1w1 -{~+kwl}

aZ

p (37) (38) in which: turbu1ence constant

1ength sca1e of turbu1ence

The definition of the kinetic energy k reads:

(39)

The kinetic energy does not reduce to zero, since UI and vI do not equa1

zero at the surface. Based on measurements in an open channe1 flow UI

and VI are modelled by (Nezu, 1977):

z 2.30 exp

(-'6 )

w (40) z 1.27 exp(

-'6 )

w (41)

ak

a

2k

Then

az

and

az

2 are a1so not reducing to zero. Since there is kinetic

energy, there is a1so dissipation (38). Consequently the diffusion term (diffusion of the kinetic energy, equation 36) has a certain va1ue at

the surface. In case vt is modelled by (Rodi, 1980):

c

Wk

v (42)

Cv is a turbulence constant, it follows, that if L ~ 0, vt is not

reducing to zero. In case L equals zero the eddy viscosity also equals

zero according (42), however, then the dissipation (38) should tend to

infinity, which is hardly conceivable. It is more 1ikely to assume that

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3.4 k-eguation and Boussinesg

Substitution of (11) into (35) yie1ds [8W« 8Ü]:

8x 8z

p (43)

If the production due to the shear stress is equa1 to zero (no external forces), the eddy viscosity or the vertica1 gradient of the longitudinal flow velocity has to reduce to zero. Using the logarithmic or the Coles flow velocity profile in a uniform flow the eddy viscosity has to be

. 8ü. 1 h f

zero, S1nce 8z 1S not equa to zero, at t e sur ace.

3.5 ODYSSEE (k-f-model)

In the turbu1ence model used in ODYSSEE the eddy viscosity is modelled by [conversion of (38) and (42) with c~ ~ cDcv]:

(44)

where c is a constant in the k-f-mode1. ~

The boundary conditions at the water-surface are modelled (Delft Hydraulics, 1987):

8k 8z

II

8z

o

(45)

It appears that in case the grid distance is re1ative1y large the turbulence parameters k and f are also relative1y large (Hoffmans,

1988a). Because k does not reduce to zero the eddy viscosity has a certain va1ue at the surface.

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4. Assesments

4.1 Open channel flow

Delft Hydraulics (1988) carried out turbulence measurements in a sand flume with a Laser Doppier anemometer. The mean velocity distribution and turbulence characteristics were obtained. The experiments (Tl to T4) were conducted in a sand flume with a flat bed. The width of the flume measured 1.5 mand the discharge per experiment was constant. The most important hydraulic data of the experiment (T2), which were used to fit the parameter ow' were:

3 Q 0.040 m

Is

B 1.5 m 2 q 0.0294 m

Is

a 0.0773 m u 0.436

mis

s _3 i 0.084*10 u a ₃ Re ~ __s_ z_30*10 v u* z_0.057 u s ₆ ₂ V = 10 m

Is

discharge width of flume

discharge (integrated velocity profile) (averaged) flow depth

velocity at surface energy slope

Reynolds number bed-shear velocity

kinematic molecular coeficient (water)

Figure 5 shows the values of the computed and measured velocity defect, [ü - ü(z)]/u*, (van Mierlo and de Ruiter 1988, experiment T2), while

max

figure 6 shows the values of the computed and 'measured' (dimensionless) eddy viscosity [v I(u*ó )] as a function of the relative depth [z/ó ].

t w w

The agreement between the computed (0 = 1.50) and the measured velocity

w

defect respectively eddy viscosity is satisfactory.

4.2 Pipe flow

Measurements of turbulence parameters were made as early as 1954 by Laufer. They were executed in a test tube, which was a straight,

(23)

Mean-velocity dï"stributions and turbulence parameters were measured even

into the viscous sublayer.

:

z

d ~ ! ~ ~

..

.

'! 0 V"_lty .... ct (-)

o

van Mierlo (B.2.1) exponential a = 1.5 w

figure 5 Computed (30) and 'measured' velocity defect (uniform flow)

D D :!:

.

..

!

D .!, D ~

.

..

D D D :I D D.02 Q.04 0.06 o.oe 0.1 0.12 iUdJ'wlec..&b- (-)

o

van Mierlo (B.2.l) exponential a - 1.5 w

(24)

The most important hydrau1ic data of the experiment, which were used to

calibrate the parameter ow' were:

u 30 m/s maximum velocity (center pipe) (air)

max

D 0.247 m interna1 diameter

_6

11 15.1*10 m2/s kinematic m01ecu1ar coefficient (air)

u_max D 5 Re - ~ 5*10 Reyn01ds number 11 u* ~ 0.035 wa11-shear velocity u max

The eddy viscosity is obtained directly from the turbu1ence-shear-stress

distribution:

11 (z) ..

-t

(46)

The eddy viscosity computed in this way is shown in figure 7. Not only

Laufer's data for the mean-ve10city distribution have been used but a1so

0.01 + + 0 0.07 0 0.06 + 0 0 :1: 0.05 ~_; 0.04 ~ -; ~v 0.03 '" 0.02 0.01 0 0 0.2 0.4 0.11 0.1 R.IoIlv. é.pUl (-) 0 Nunner + Laufer exponentia1 a

-

2.0 w

(25)

the data obtained by Nunner (Hinze 1975, p. 730). In Nunner's

experiments the wa11-shear velocity amounted to u*/ü

=

0.045. The max

drawn 1ine represents the exponentia1 distribution of the eddy viscosity

(19) for a - 2.00. w

The 'measured' eddy viscosity first increases 1inear1y, then reaches a

maximum at about.2z/D - 0.3, then decreases slight1y, and attains a

near1y constant va1ue beyond 2z/D - 0.5. The same course of the

distribution curve was obtained by Reichardt (Hinze 1975, p. 730) in a two-dimensional channel flow, although the maximum there occurred at about 2z/D ~ 0.4 to 0.5.

Figure 8 shows the velocity defect distribution; the agreement between computed and measured distributions is very satisfactory indeed in the region 2z/D

>

0.2. 6 ~ ~ .!-.! 4 ~ D ~ 3 } ~ 2 o 0.2 0.4 0.6 0.8 R.loU.. a.pUl (-)

o

Laufer exponentia1 a - 2.0 w

figure 8 Computed (30) and 'measured' velocity defect (pipe flow)

Using a logarithmic flow velocity profile or the Coles profile, a discontinuity in the velocity gradient (8ü/8z) occurs in the center of the pipe, which is in conflict with nature. Due to this the eddy

viscosity reduces to zero for the shear stress has to be zero. Then a correction has to be made for the eddy viscosity or for the longitudinal flow velocity.

(26)

Applying the exponential eddy viscosity concept the agreement between the computed and 'measured' eddy viscosity is much better, while the boundary condition for the velocity (aü/az) is also much better in the

center of the pipe (aü/az = Ol).

4.3 Internal boundary layer

4.3.1 Undisturbed outer flow

The most discussed type of boundary layer is the two-dimensional boundary layer flow along a plane surface (plate) with zero pressure

gradient. In front of the leading edge of the plate the velocity of the

undisturbed flow is uniform (aü/az = 0). In case the flow in the outer

region is frictionless, the eddy viscosity reduces to zero. The retarded

layer along the plate is called the boundary layer. With increasing

distance from the leading edge in the downstream direction the thickness of the retarded layer increases continuously. Frictional forces retard the motion of the fluid in a thin layer near the wall. In that thin layer the velocity of the fluid increases from zero at the wall (no slip) to its full value which corresponds to the frictionless flow

(undisturbed outer flow).

Equation (19) describes the eddy viscosity reasonably in case of a well

chosen value of Q . In the lower half of the new wall-boundary layer the

w

modelled v is somewhat smaller, while in the upper half the modelled

t

one is somewhat larger compared with measurements of Klebanoff and Townsend (Hinze 1975, p. 645), figure 9.

4.3.2 Disturbed outer flow

In case the outer flow is disturbed, for instance caused by a mixing layer upstream, the distribution of the eddy viscosity in the boundary layer or the new wall-boundary layer differs from the distribution mentioned in 4.3.1 (wall-boundary layer with undisturbed outer flow),

(27)

especially in the outer region, which is caused by the relative high

turbulence degree in the outer flow (relaxation zone).

I.D ,_,

,,

0.8

,

'-'-

,

'-0.6 '-\

•

\ 6;, 0.4

'.

0.2

0 O.DI OD2 ODJ 0.D4 0.05 0.06 0.07 O.D'

VI

u.ow

4

0 From Klebanoff's data: Re ::::: 7.5 * 10 u*/U

-

0.037

4

X From Townsend's data: Re ::::: (3 to 4) * 10 u*/U - 0.044 according to Coles (18) ; TI - 0.55

according to equation (19); Q - 2.25

w

figure 9 Eddy viscosity: boundary layer with undisturbed outer flow

Measurements above artificial dunes (van Mierlo and de Ruiter, 1988; experiment T5 and T6) show that the eddy viscosity is affected by relative large length scales (vt :: L) in the upper part of the new wall-boundary layer, while in the lower part it is influenced by the wall, where L is relatively small. The following suggestion has been made to model the eddy viscosity:

z 0<-Ó w z

<

ref Ó w (47) z ref v - v Ó Ó 2 [ t,w Ó t,ref]exp{ _[_w~_...:w~]} u*

w

°v

z

ref

<

Ó Ó

w

W

<

1 (48) in which:

v - eddy viscosity at outer edge of boundary layer t,w

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vt,ref eddy viscosity at reference heigth

zref reference heigth

a = standard deviation (eddy viscosity)

v

For z smaller than the reference heigth (z f) the eddy viscosity is re

influenced by the wall, while for z larger than Z_ref it is affected by the óuter flow.

The flow above a dune is characterised by a deceleration and an acceleration zone. Af ter reattachment in the acceleration zone a boundary layer appears. The modelling of the eddy viscosity is satisfactory for, figure 10:

Q 2.5 to 2.8 w

a 0.4 to 0.5· v

z_re

f/ó

_w

-

0.2 to 0.3

In figure 10 the eddy viscosity as well as the thickness of the wall-boundary layer are made dimensionless

{v I(uó ), z/ó*

_t _w _w

J

.

It should be pointed out that the magnitude of the eddy viscosity v_t,w

"

~ _"

_"

::!: D

!

.~_d

.

D ~ D D :l D D i:P D 0.02 0.060 0.1 0.14 0.'. CL22 0.26

...,.

....

_

(-)

o

van Mierlo according to (47) and (48) Q - 2.60,

z

fiS

-

0.2, a - 0.52 w re w v

(29)

is dependent on the developed mixing layer downstream (Hoffmans, 1988b).

Considering a uniform flow the eddy viscosity is proportional to the bed-shear velocity, however, in a new wall-boundary layer with a disturbed outer flow, the bed-shear velocity begins to grow af ter the reattachment point, while simultaneously the eddy viscosity (at the outer edge) will decrease in the longitudinal direction. From equation

(48), it follows, that the eddy viscosity is a 'weak' function of the Z

b de -s ear ve oc~tyh 1 · [v_t_,_ref - ~u*z_refexp{-Q_w rsef}].

w

4.3.2.1 Shear-stress distribution

Assuming a logarithmic flow velocity profile and applying the hypothesis of Boussinesq (11) and the eddy viscosity concept mentioned in 4.3.2 the

shear stress in the new wall-boundary layer reads:

T _L.g = exp(-Q ~ ) TO W S W Z Z ref T v v v t.ref]ex { S S 2 t.XZ ~ _[~

_-

_[ w w

_]

_} TO _~u*z ~u*z ~u z p a * v Z 0<-S w z_ref

<

Ó w (49) Z Z ref Ó

<

Ó w w < 1 (50)

Figure 11 shows various profiles for the shear stress in a new wall-boundary layer where the outer flow is disturbed. The shear stress is

made dimensionless with the bed- or wall-shear stress. From equation

(49), it follows, that for z smaller than z f the shear stress

re

decreases (becomes smaller than TO)' while above Z f it is strong1y

re

dependent on the value of the eddy viscosity at the outer edge of the

boundary 1ayer (v). In case v is decreasing or in case the bed

-t,w t,w

shear velocity as well as the thickness of the new wall-boundary 1ayer

are increasing the shear stress will decrease at the outer edge.

If ó is at maximum or if the boundary layer reattaches the surface, the

w

eddy viscosity shou1d tend to zero there. Then the shear-stress is not

equal to zero according to (50),however, the value of the shear stress

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In case equilibrium flow conditions are dominating the distribution of

the shear stress is not quite correct, since it is not linear.

0.4 0.8 1.2 1.1 2.4

o

= 0.01 + - 0.1 l:>. - 1

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5. Conclusions

Uniform flow

According to the classical theories of Prandtl and Boussinesq and using a logarithmic flow velocity profile the eddy viscosity equals zero at the surface.

Considering the two-dimensional k-equation and the k-f-model the eddy viscosity does not reduce to zero, because the eddy viscosity is related

to the kinetic energy and the kinetic energy is not equal to zero at the surface.

Using the Coles flow velocity profile with TI= 0.20 (Nezu and Rodi, 1986) the velocities are approximated in a better way especially in the outer region (wake function!) than using a pure logarithmic flow or using the velocity profile which follows from an exponential eddy

viscosity.

Pipe flow

It appears that the agreement between a mathematical exponential function for the eddy viscosity (equation 19) and the corresponding

values based on measurements is satisfactory for a = 2.0. In the center

w

of the pipe the stress production of the turbulent energy is equal to zero, since aü/az ~ 0, however, the eddy viscosity is not, because there the kinetic energy (k), the dissipation (f) and the diffusion of the

kinetic energy are not reducing to zero (Hinze 1975, p. 725 and 737).

Using a logarithmic flow velocity profile or the Coles flow velocity profile the boundary condition in the center of the pipe is not satisfactory, because there the gradient of the longitudinal flow velocity is discontinue. Also the eddy viscosity will not be equal to

zero there. Then a correction has to be made on the eddy viscosity

distribution which follows from the Coles flow velocity profile (linear

shear stress).

Internal boundary layer

The Coles flow velocity profile using TI= 0.55 describes the

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viscosity very well in a new wall-boundary layer with an undisturbed

outer flow. However, the description is poor in case the outer flow is

disturbed. Using the equations (47) and (48) the modelling of the eddy

(33)

References

Coleman, N.L. (1984), Velocity profiles with suspended sediment, J. of Hydraulic Research, Vol. 22, No. 4, p. 263-289.

Delft Hydraulics, Delft (1987), Documentation to the computerprogram ODYSSEE, part 3b: Mathematical and numerical description of the code Ulysse, 2e version.

Hinze, J.O. (1975), Turbulence, Second edition, Mc. Graw HilI Baak Co., New York.

Hoffmans, G.J.C.M. (1988a), Flow simulation by the 2-D turbulence model ODYSSEE, No. 2-88, Delft University of Technology, Dept. of Civil Eng.

Hoffmans, G.J.C.M. (1988b), Damping of turbulence parameters in relaxation zone, No. 10-88, Delft University of Technology,

Dept. of Civil Eng.

Hoffmans, G.J.C.M. (1988c), Flow model with prescribed eddy viscosity, No. 11-88, Delft University of Technology, Dept. of Civil Eng.

Kay, J.M. and R.M. Nedderman (1985), Fluid mechanics and transfer processes, Cambridge University Press.

Mierlo, M.C.L.M. and J.C.C. de Ruiter (1988), Turbulence measurements ab ave dunes, Report Q789, Vol. 1 and 11, Delft Hydraulics.

Nezu, I. (1977), Turbulent structure in open-channel flows (Translation of doctoral dissertation published in Japanese, Kyoto

University, Dept. of Civile Eng.

Nezu, I. and W. Rodi (1986), Open-channel flow measurements with a laser dopp1er anemometer, Journalof hydrau1ic Engineering, Vol. 112, No. 5, p. 335-355.

Rodi, W. (1980), Turbulence models and their application in hydraulics,

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In appendix A an elaboration is given of the integration of equation (16) from Zo to Ów in order to write TIas a function of K, Ü, u*, Ów and Zo (for the meaning of the symbols, see notation).

The Coles velocity profile reads:

(Al) in which: w(~ ) = 1 + sin(2~ w w 2sin2{lrL} . 2 ó ' w wake function

Integration of (Al) from Zo to

s

gives: w Ó

s

w w w

I

üC(z)dz u*

[I

ln - dzZ + TI

I

{W("6)Z W("6)}Zo dZ] K _Zo Zo Zo Zo w w (A3) 1 2 3 Term 1: q (A4) ó_w

_s

Zo

Term 2: [z ln ~

-

z]

I

s

ln _!tI

-

Zo ln -

-

(ó - zo)

Zo w Zo Zo w Zo Ó Zo ó [ln _!tI

-

1 + -] (A5) w Zo ó_w Zo

Assuming (-)

«

1 -+ term 2 may reduce to:

s

_w Ó ó [ln_!tI

-

1] (A5a) w Zo

s

w w

I

{l + Z

I

_Zo sin(2"6 - 11" 2sin2{lr-} dz Term 3: 1)2}dz ₂ ó Zo w Zo w Ó Ó Ó Z W Zo w

{z _!tI cos(2"6 _ l)lr

11

Z 2sin2{lr "6

11

11" 2 2

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Zo 1 Zo ó_w[

1 - --

_ó + -_~ _cos

(2

_s-

- 1

)

₂

~

w w Zo If (--)

«

1 ó w (1 - --) 2sin2{~ --}

s

_w 2

s

_w

term 3 may reduce to:

s

w

(A6) (A6a)

written as:

Substitution of (A4), (A5) and (A6) into (A3) (q - Üó ), TI can be

w

:Q

s

K, in ....::t!. + 1 u* Zo TI= Zo

1

Zo 1)~ 1

_s

+ cos(2s-~ 2 w w Zo Assuming

-- «

_s

1,

n

reads:

n

w

s

w Zo Zo (1 - --) 2sin2l~ --} ó 2 ó w w

s

1n....::t!.+ 1 Zo (A7) (AB)

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In appendix B an e1aboration is given of equation (23), where the

horizontal flow velocity gradient to Z is integrated from Zo to z. Using the Tay1or-series to the exponentia1 function e1aboration of (BI)

resu1ts in (for the meaning of the symbo1s, see notation):

1 aü L{l ~ } exp(a ~

)

u* az ICZ 6_w w 6_w L{1 ~ }{1 _{+ a} ~ ₊

_2'

1 ~ )2 1 ~ )3 1 ~ )4 (B1) (a + 3' (a +

4'

(a +

.

) ICZ 6 w 6 . w 6 . w 6 . w 6 w w w w w

Conversion of (BI) leads to:

+ 12,(a ~)2 + -31(a ~6)3 + -41,(a ~6)4 + ... ) + . w 6 ! w . w w w w + a w (~ )2 +

1

a2 (~)3 1 3 (~ )4 ) 6 2! w 6 + 3!aw 6 + ... W W W L{l 1)~ 1 Z )2 1 1 (2) (~ )3 + (a

-

+ (- a2

-

a )(- + (- a3

-

+ ICZ W 6 2! w w 6 3! w 2! w 6 w w w 1 _{1 (}_{3) (~} ₎₄

_...

) + (- a4

-

+ 4! w 3! w 6 w 1 {1 _{+ (a}

_-

1)1 _{+ (- a}1 2 a ) ~ 1 1 ~2 62 + (- a3 - (2) 63 + IC Z w 6 2! w w 3! w 2! w w w w 1 1 ~3 + (- a4 - (3) ₊

...

} 4! w 3! w 64 w (B2)

Integration of (B2) from Zo to z gives (a is independent of z): w z z - Zo

1

{1n + (a - 1)( ) IC Zo w 6_w Z2 - z~ 1)( 62 ) + w + - z~ 64 ) + ... } w

(39)

Z3 -

zg

1 Ow Z4 -

z~

- 1) ( ) + - 03 (- - 1) ( ) + } 63 4! w 4 6···· w w 1 Ow + - 02(-3! w 3 (B3)

In general farm equatian (B3) can be written as [üE(z) = ü(z)]:

(B4) in which: n-l

°

R

w

[w

_

1] u,n n! n (BS)

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In appendix C an argumentation is given of the convergence of the correction-term B.

The correction-term B reads for n ~ ~:

(Cl) in which: n-l Q Q R ~ [~- 1] u,n n! n (appendix B) (C2) Because Zo Z Ó

s

Ó

s

1 w w estimated by: Zo (_}n]

<

R equa l.onti Cl Ó u,n w can be lim n-l Q Q

.s:

[~-

1] = n! n n~~ lim n Q __ w_ n n! lim n-l Q W (C3) n~~ n -+eo n(n-l)! Since, n n n Q Q Q

~

_<

~ _and _lim ~ _{~ 0} _{(standard limit) and}

n n! n! n!

n-+~

ri-T n-l m

Q Q Q

W w

and lim ~ = 0 (standard limit, n-l)

n(n-l)!

<

(n-l)! m! m

m -+eo

(41)

In appendix D an argumentation is given of the convergence of the ~ of

the higher order terms, which are part of the correction-term B in

equation (24).

The correction-term B reads:

(Dl) in which: n-l a a R ...Y!. [...Y!. - 1] u,n n! n (appendix B) (D2) Because Zo Z Ó :S Ó :S 1 w w estimated by: equation Dl can be

z

n=l n-1 a a ...Y!. [...Y!. _ 1] n! n n n-1 eo _a eo _a :l: ...Y!. :l: ...Y!. n=l n n! n-1 n! n n-1 <Xl _a eo a :l: ...Y!. :l: w n=l n n! n=l n(n-1)! n n-1 eo _a a :l:

1

_[n7 _{(n~l)! ]} n=l n n m eo a eo _l_ a :l:

1

...Y!. :l: ...Y!. [m=n-1] n=1 n n! m=O m+1 m! n aO m eo a eo a :l:

1

...Y!.

[

1

...Y!. + :l: _L _m7] n=l n n! 1 O! m=l m+1

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n=l n-1 n co Ct ....:tI

_[1

1 L _{n+1 ]} 1 n-1 n! n n co _Ct L ....:tI [n + 1 - n] 1 n-1 n! n(n + 1) n co _Ct ....:tI 1 L n! [n(n + 1)] 1 (D3) n-1 Since, n n co Ct co Ct

1 ....:tI ....:tI _[_standard _{limi t}_J

L _{[n(n + 1)]}

<

L - exp(Ct )

n=l n! n=l n! w

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In appendix E an argumentation is given of the following power-series: ~ n=l R n ~ -1 u,n (El) in which: n-l 0 0

R

.s:

[....!tI.

_-

_1] _{(appendix B)} _(E2)

u,n n! n

Substitution of (E2) into (El) gives:

n-l co 0 0 ~

.s.

[....!tI.

_-

_l]n n-l n! n n-l n-l co 0 co 0 ~ ....!tI. ~ w n! 0 (n-l)! n=l w n=l n n-l co 0 co 0 ~

.s:

~ w n=l n! n=l (n-l)! n n co 0 co 0 ~ ....!tI. ~ ....!tI. n=l n! n=O n! n 0° n 0° co 0 co 0

~

.s:

_[07 + ~ _n7] ....!tI. -1 [Q.E.D] (E3)

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