Local Scour: Damping of turbulence parameters in relaxation zone

(1)

TU

Delft

Delft University ofTechnology

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in relaxation zone Report nr. 10-88 G.J.C.M Hoffmans

Faculty o~ Civil Engineering Hydraulic Engineering Delft University of Technology

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l. Introduction ₆

2.

The transport equations ₇

3. Calibration-frame 10

4. Damping of the turbulence General 11 11 11 14 15 19 4.1 4.2 4.3 4.4 4.5 5. Kinetic energy (k) Dissipation (E) Eddy viscosity (vt)

Length scale of turbulence (L)

Conclusions 21

References Appendices Tables Figures

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a local flow depth turbulence constant (L) (

-

) (

-

) (

-

) ( - ) (

-

) (

-

) (

-

) (

-

) (-) (

-

) 2 1 (L T ) 2 2 CL T ) C - ) 2 2 CL T ) (L) CL) (

-

) 1 2 CML T ) 2 3 CL T ) 2 1 CL T ) (

-

)

C1f= turbulence constant (k-f-model)

C2f= turbulence constant (k-f-model)

empirical coefficient (CÀ,f z

ë

À Z cÀ)

turbulence constant (k-f-model) turbulence constant f function fl function; 0(1) scale factor F Fr Froude number g k acceleration of gravity kinetic energy per unit mass

kinetic energy per unit mass; 0(1) scale factor k

K k

s equivalent roughness of Nikuradse

length scale of turbulence roughness parameter

fluid pressure fluctuation

stress production of the turbulent energy discharge per unit width

Reynolds number

local-time-averaged longitudinal flow velocity longitudinal flow velocity fluctuation

or longitudinal flow velocity; 0(1) bed-shear stress velocity

scale factor

ü

1 (LT ) 1 (LT ) I (LT ) 1 (LT ) 1 (LT ) 1 (LT ) (

-

) L L X pI P q Re u 1 (LT ) 1 (LT ) (-)

u

U depth-averaged longitudinal flow velocity

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

vertical flow velocity fluctuation or vertical flow velocity; 0(1) scale factor w 1 (LT ) CL) w x longitudinal distance

(5)

x

longitudinal distance; 0(1)

scale factor x

y transverse (horizontal) distance vertical distance z Zl _{vertical distance; 0(1)} scale factor z

z

t

zero-velocity level time

ak damping exponent kinetic energy

aL growth exponent length scale of turbulence a damping exponent dissipation

€

a damping exponent eddy viscosity

v

p

angle mixing layer

~ growth factor mixing layer

€ energy dissipation per unit mass by turbulence

À averaged damping constant (À ~ À ~~)

€

Von Karman's universal constant

v kinematic molecular coefficient eddy viscosity

density fluid

turbulent normal stress

turbulence constant (k-€-model)

T t 6 m 6 w

turbulent shear stress thickness mixing layer

thickness new wall-boundary layer scalar product O( ) = order of magnitude (

-

) (L) (L) (L) (

-

) (L) (L) (T) (

-

) (

-

) (

-

) (

-

) (

-

) (

-

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

-

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

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Subscripts

o transition from a uniform flow to the deceleration zone ~ transition from the deceleration zone to the relaxation zone

Abbreviations

BS Backward-facing step LS Local Scour hole

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1. Introductión

The general purpose of this research project is to model mathematically the scour-hole downstream of a structure (2-D). The model has to

simulate the development of the scour as a function of time. Basically two models are necessary namely a flow model and a morphological model. The latter has to describe the bed- and suspended load and the erosion of the bed.

In the present study the behaviour of the turbulence in the re1axation zone of alocal scour hole (figure 1) is discussed.

The purpose of this note is to find a function (prescription) for the eddy viscosity in the relaxation zone, which will be used in the flow model DUCT, where the eddy viscosity is prescribed (Hoffmans, 1988b). The function for the eddy viscosity is based on the transport equations for the kinetic energy and its dissipation, in which the diffusion and gain (production) terms are neglected.

Computations by means of the k-e-model for the flow in a number of configurations (Hoffmans, 1988a), were used for calibration.

sz

_

..

p

uniform flow

...

laytr

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2. The transport eguations

Considering a steady two-dimensional flow in alocal scour hole with a

constant width and a plane water surface the turbulent energy equation

reads (Kay and Nedderman, 1985, p. 161):

Ok -Bk -Bk - p

B

~

kul}

B

plWl kwl} u- ₊

_wa;

_-

_e

-ax

{

p +

az

{

p + Ot

Bx

turbulent diffusion p 0 Bü r Bü r Bw 0 Bw -t x_P _, x ₊ -t zx + -t,xz + -t zz Bx p , 8z p 8x p ,

Bz

(1) (2) (3) in which:

k kinetic energy per unit mass presssure fluctuation

stress production of the turbulent energy

time

t

u local-time-averaged longitudinal flow velocity

longitudinal flow velocity fluctuation

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

vertical flow velocity fluctuation longitudinal distance vertical distance w x z v

energy dissipation rate per unit mass by turbulence kinematic molecular coefficient

p density fluid

turbulent normal stress turbulent shear stress scalar product

Oownstream of reattachment a rapid decay of Reynold's normal and shear

(9)

upwards through the reattached shear layer (relaxation zone). At some distance downstream the turbulent shear layer reattaches to the surface

(figure 1).

To analyse the damping behaviour of the turbulence parameters in the relaxation zone the ra te of generation of turbulence by mean-strain

effects (P) and the turbulent diffusion are neglected in the transport

equation for the kinetic energy. In that case the following equation may

be used:

Dk

Dt

-ak -ak

u-

_ax

+ w:-

_az

(4)

Based on experimental data of van Mierlo and de Ruiter (1988) the

vertical velocities in the relaxation zone can be neglected with respect

ak

.

ak

to the horizontal ones. Although

az

1S larger than

a

x'

equation (4) may

reduce to (Appendix A): Dk Dt -ak

u-ax

(5)

The f-equation can be induced from the Navier-Stokes equation, but it is much more complicated than (1) (Hanjalic and Launder, 1976). Hence, the

same type of relation as (1) is assumed in the following:

Df Dt

-af

u- +

w:-a

x

az

(6) turbulent diffusion in which: Cl e 1.44; constant in k-e-model C2f 1.92; constant in k-f-model a 1.3 , constant in k-f-model e v eddy viscosity t

Applying the same assumptions as above, equation (6) reduces to:

-af

(10)

It should be noted that for a uniform or equilibrium flow the stress production (P) is equal to the energy dissipation (E) in the near wall region. Then the convective- and the diffusion terms vanish.

The loss (dissipation) and diffusion terms make increase the damping rate of the turbulence. Conversely it decreases by the gain terms

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3. Calibration-frame

Computations by means of the k-f-model for the flow in a number of situations; three local scours holes and one backward-facing step (Hoffmans, 1988a) are used to calibrate turbulence parameters in the

relaxation zone (figure 1). The flow in the different configurations is

characterised by a large Reynolds number and a small Froude number.

Table 1 shows the most important hydraulic data of the various situations. BS2 LSl LS2 LS3 2 q 0.121 0.100 0.119 0.104 m

Is

ao (x xo) 0.30 0.30 0.30 0.30 m a (x x ) 0.45 0.34 0.43 0.58 m '1 '1 x 3.0 3.0 3.0 3.0 m Tl ₃ Re

Üoaolll

121 100 119 104

_*

10 (-) Fr

ÜoIJ(gao)

0.24 0.19 0.23 0.20 (-) 3 k 0.5 0.5 0.5 0.5

*

10 m s

table 1 Hydraulic data

in which:

a flow depth Fr Froude nurnber

g = acceleration of gravity

k equivalent roughness of Nikuradse s

q discharge per unit width Re Reynolds nurnber

Uo depth-averaged longitudinal flow velocity at x = Xo

Xo transition from a uniform flow (fixed bed) to the deceleration zone (erodible bed)

x = transition from the deceleration zone to the relaxation zone

Tl

BS Backward-facing step LS Local Scour hole

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4. Dampin~ of the turbulence

4.1 General

In chapter 4. an analysis is given of the behaviour of the kinetic energy (k), the dissipation (!), the eddy viscosity (vt) and the length

scale of turbulence (L) as a function of the distance in the flow

direction.

4.2 Kinetic ener~y (k)

Assuming that the longitudinal flow velocity (ü) is constant in the

x-direction, differentiation of (5) to x gives:

ft..¬ ..

a

x

(8)

Substitution of (5) and (8) into (7) yields:

(9)

The general solution of (9) is (Appendix B):

k(x) x - x k

I

n + 'I À o l} k _x > x 'I (10) in which: 1

-l~---c--- -1.087; damping exponent kinetic energy

2!

À = damping constant

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The damping constant À can be derived by a linearization of equation

(10) for x = x . 'I Differentiation of (10)to x results in (x x ):'I

ak

a

x

l

_x

₌

_x

'I or k o_n k (12)

ak/

8x

l

x=x 'I

(13)

k-€-predictions, as a function of x for the various geometries and it also forms the basis to determine ~. On dimensional considerations ~ has the dimension of a length scale. Figure A shows that by an increase of the scour-growth or by an increase of the maximum flow depth (a ) the

'1

damping occurs relatively faster; see the differences between the curves of respectively LSI (a - 0.34 m) and LS3 (a - 0.58 m). In other words

'1 '1

the damping of the kinetic energy increases, if the depth-averaged flow

velocity becomes smaller (qLSl ~ qLS3 ~ 0.1 m2/s, table 1), so it is conceivable that ~ is related to

Ü .

'1

However, more downstream in the new wall-boundary layer (acce1eration zone) the velocities are increasing. It shou1d be pointed out that the damping of the kinetic energy (10) refers to the relaxation zone and not to the new wall-boundary 1ayer.

B_{ecause t e parameters}h k_'1 and 8k_8xlx=x _are h_ard t₀ _quant·f₁ _y, th_e

'1

fol1owing suggestion has been made to ca1culate ~.

x '1 (q Ü a ) '1 '1 (13) in which: c~ = empirical coefficient c~

-

c). 8k k 8xl_x=x '1 c). c). '1 3 3

*

10

*

10

*

_% (J/kgm) (J/kg) (m) (

-

) BS2 1.03 4.7 4.96 2.48 -1.2 LSI 0.36 2.2 6.64 2.51 0.0 LS2 0.85 3.9 4.99 2.38 -5.2 LS3 1.13 4.3 4.14 2.67 6.4 c~ 2.51

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It appears that 'the relative difference of cl is relatively small.

probably the damping constant l is proportional to tbe flow velocity

ratio

(Ü /Ü

o), however, it should be noted tbat in all cases the initial

'7

depth-averaged flow velocity (Üo) was approximately 0.35 mis and the

location (x ) measured 3.0 m (tabie 1). '7

Because cl ranges from 2.48 to 2.67 an averaged damping constant has

been introduced. Substitution of cl = 2.5 into (13) yields the following

damping constants for the several geometries:

lBS2 2.5*(0.3/0.45)*3.0 5.00 m

lLSI 2.5*(0.3/0.34)*3.0 6.62 m

lLS2 2.5*(0.3/0.43)*3.0 5.23 m

lLS3 2.5*(0.3/0.58)*3.0 3.88 m

Figure 2 shows the small differences between the ODYSSEE-predictions and

the kinetic energy according to (10) (tabie A).

1.0 -- k - t - prldiclion. 5.0 --- - occordinll 10(10) 3.0 4.0 2.0

_{11-__}

0.0 '--- ...----'--.._---'- __ '--_-'-_---'- __ .._ ___ lt(m) 0.0 1.0 2.0 3D 5.0 U) 7.0 • .0

figure 2 Damping kinetic energy (BS2)

Generally the kinetic energy computed by the k-€-model is decreasing

somewhat faster than the calculated one using (10). Tbis means that the derived kinetic energy (10) is less dissipating. In its totality the

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4.3 Dissipation (f)

Differentiation of equation (10) to x and substitution of it into (5)

gives: dx) _-_u

- ak

_a

_x

x - x a e ( n + 1) e

n

x

>

x

,.,

(14) in which: a

e -2.087; damping exponent dissipation (15)

App1ying the same procedure to determine À, see 4.1, differentiation of (14) to x gives (x - x

_,.,

):

II

axl_x_=x

,.,

or f ak rz (16)

Tab1e 3 gives an overview of the coefficient c, ,which shows

" , f resemb1ance to cÀ' see (13). (17)

II

_e axl

,.,

x=x

_,.,

3 3

*

10

*

10 (Wjkgm) (W/kg) BS2 0.34 0.76 LS1 0.11 0.36 LS2 0.31 0.71 LS3 0.33 0.66 À f cÀ

-

_cÀ e e c À, f C À,f

*

% (

-

) (m) 4.67 2.34 -5.3 6.83 2.58 4.5 4.78 2.28 -7.7 4.17 2.69 8.9 c 2.47 À, f

tab1e 3 Empirica1 coefficient c,

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It should be noted that there are differences between cÀ,! and cÀ or between À and À, however, they are relatively small

«

10%).

!

Figure B shows the change of the dissipation in the center of the mixing layer as a function of the distance in the flow direction for severa1

geometries (table B). The differences between the calculated dissipation

(14) and the k-E-predictions are relatively smal1 (figure 3). Near the

reattachment point the dissipation (14) is increasing somewhat faster

compared to k-!-calculations. For x

>

4.5 m the computed dissipation

using (14) is less decreasing than the calculated one by the k-E-mode1.

Generally the agreement between the two simulations is fair, provided the dissipation at the relaxation point (E ) has been chosen we11.

TI

4.4 Eddy viscosity

The eddy viscosity can be written by the formu1a, which is known as thc

Kolmogorov-Prandtl expression (Rodi, 1980).

c

Wk

11

(18)

C

11 turbulence constant (uniform flow:

C :::::0.55)

11

L length scale of turbulence

.l(w ) C IlO kg 0.' 11.- t ,rtcli(lion, 0.' 0.' 0.1

..

~ 0.1

-.

0.1 ouo,'ing ,•. (14I 0.0"__-'-_-'-_-'-_---L_--'__ "--_""__-'-__ alm) 0.0 1.0 1.0 l.o '.0 '.0 7.0 •.0 figure 3 Damping dissipation (BS2)

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The dissipation f is modelled by the expression (dimensiona1

considerations):

f = ₍₁₉₎

in which cD is a turbu1ence constant (uniform flow: cD Z 0.16). Substitution of (19) into (18) resu1ts in:

(20)

in which c

IJ 0.09 is a turbu1ence constant of the k-f-mode1.

Substitution of (10) and (14) into (20) gives:

x -

X

!l ₊

₁₎

cr

v x > x

TI (21)

in which:

-0.087; damping exponent eddy viscosity (22)

The figures C and D show the behaviour of the eddy viscosity given by

the k-f-mode1 for the four configurations. Figure C shows the eddy

viscosity in the center of the mixing layer as a function of the flow

direction, whi1e figure D shows the maximum value of the eddy viscosity.

It can be noted that for large va1ues of x the predicted eddy viscosity

(k-f-mode1) at the surface is somewhat larger compared with values in

the middle of the flow, which is not fully correct.

In the center of the mixing layer before the reattachment point the eddy

viscosity can be written by (Kay and Nedderman, 1985):

(23)

in which:

~ = 0.010 to 0.020; empirical constant Ó

(18)

~ 0.175 to 0.225; angle of mixing layer

The eddy viscosity (k-t-model) decreases approximately at x = 3.0 m.

Although there are differences between the configurations for the

location, where the eddy viscosity starts to decrease, they are not

large. In the DUeT-model the (prescribed) eddy viscosity reaches its

maximum, where ó is at maximum, that is where ó is equal to the local

m m

flow depth (relaxation point x ). If the initial flow depth (ao) is

'I

0.3 mand the angle of the mixing layer (~) is 0.20, the calculated x

'I

measures 3.0 m (x = 2ao/~). Then the maximum flow depth in the scour

'I

hole is less than 0.6 m (a ~ 2ao), see table 1. If the maximum scour

'I

depth is larger than 2ao, the relaxation point is calculated by

2(a - ao)/~. 'I Substitution of 11 t,'I 11t,max (25) into (21) yields: (26)

The maximum value of the eddy viscosity (x

for BS2, LS1, LS2, LS3: x ) measures successively 'I 3 2 1/ 0.014*0.403*0.45 2.54 10

mis;

(BS2) t,max ₃ ₂ 11 0.012*0.332*0.34 1.35 10

mis;

(LS1) t,max ₃ ₂ 1/ 0.012*0.396*0.43 2.04 10

mis;

(LS2) t,max ₃ ₂ 1/ 0.013*0.345*0.58 2.60 10

mis;

(LS3) t,max

The agreement between the results of equation (26) compared with the maximum eddy viscosity computed by the k-t-model is quite good, however,

the agreement between the curves mentioned above and the calculated eddy

viscosity (k-t-model) in the mixing layer is poor (figure 4 and table C

for all the configurations). Part of the differences is due to the

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viscosity is small. There the eddy viscosity is small because of the

relative small length scales, despite of the increase of the flow

velocity in the new wall-boundary layer (vt - üL).

In grid turbulence, diffusion and production terms are zero so that c2

e

is the only constant appearing in the equations (1) and (2) yielding

equation (5) and (7) . The constant C2f can therefore be detcrlllillcu

directly from the measured rate of decay of k behind a grid and was

found to lie in the range l.8 to 2.0 (Rodi, 1980). A standard value of

C2f measures 1.92, which is recommended by Launder and Spa1ding (1974).

1.0

2.0.

• _ c _ pr.dlellon. ·'etnl., ",I.lng 101")

k- C_ p,.dlelion. ''''0'''''''''')

•

Geeo,ding 10 '21)

-o.s ~~,-,

_,

...

, ,_, 0.0 ~---~---~---~---~---~---~~---~---~---. 0.0 x (m) 1.0 2.0 3.0 4.0 ,.0 • .0

figure 4 Damping eddy viscosity (BS2)

Since the power-component Q is relatively small, the eddy viscosity

v

will decrease very slowly in the longitudinal direction.

It appears that the damping is very sensitive to the value of the

turbulence constant C2f (table 4).

The slow damping is due to neglecting the diffusion terms, however, the

contribution can not be large.

Downstream of reattachment, where a new wall-boundary layer developes,

the value of the eddy viscosity is relatively small compared to the

value in the layer above (Hoffmans, 1988b).

The boundary layer thickness increases with (Booy, 1986):

dS

~ L - 2

x

(20)

in which:

Ó thickness of the new wa11-boundary layer .

w

IC = Von Karman's universal constant

L = 1n(ó~zo) = 5 to 11; roughness parameter

x Zo = zero-ve10city level C2f ak a a _al e v 2.50 -0.667 -l.667 0.333 0.667 2.00 -l.000 -2.000 0.000 0.500 l.92 -l.087 -2.087 -0.087 0.457 l.80 -l.250 -2.250 -0.250 0.375 l.50 -2.000 -3.000 -l.000 0.000

tab1e 4 Sensitivity of damping exponents to C2f

The roughness parameter L ranges from 5 to 11, so the flow is more or

x

1ess in equilibrium (uniform flow conditions) at a distance of 20 to 50 times the flow depth after the reattachment point, since then the

turbulent shear layer reattaches to the surface

4.5 Length sca1e of turbulence (L)

Conversion of (19) leads to:

L(x) (28)

Substitution of (10) and (14) into (28) gives:

L(x) (x - x ) L { '1 x

>

x'1 (29) in which: C2f aL _C2f - l. 5

(21)

Behind a disturhance in the flow (transition from a fixed bed to an

erodible bed) the length scale of turbulence increases not only in the deceleration zone but also in the relaxation zone. Though the k-f-model

does not compute a length scale of turbulence, L can be determined by

the equations (18) or (28). In the calculations to determine L the

turbulence constants Cv and cD are taken respectively 0.55 and 0.16. The

differences between both manners mentioned above [according to (18) or (28)] to determine L appear to be negligible.

Figure 5 shows the length scale of turbulence for the backward-facing

step (BS2) calculated by equation (29) and using k-f-predictions, while

figure E shows the maximum length scale of turbulence for all the configurations (k-f-predictions). 50 100 .0 k-&-pttdiClionl occo,ding (291

11.,.,....

, ..."....

",....

" ...

"

20 Je(m).

figure 5 Growth length scale (relaxation zone) (BS2)

It can be noted that the predicted length scale of turbulence by the k-f-model is increasing somewhat faster compared with the simulated one (29), however, the agreement is reasonable.

(22)

5. Conclusions

In the relaxation zone the diffusion and the dissipation do have a damping influence on the behaviour of the turbulence parameters (k, E,

vt), while the turbulence increases by the production of the Reynolds stresses. In the derivation of the analytical solutions of the damping functions the production and the diffusion are neglected in the

transport equations for the kinetic energy and its dissipation. The agreement between the results of the perturbated functions is rather good.

For large values of x the kinetic energy as weIl as the dissipation tend to zero, because of the neglect of the production term. The damping of the eddy viscosity passes very slowly. It appears that the damping is very sensitive to the value of the turbulence constant c2 . A slight

E

change in c2 can accelerate the damping of the eddy viscosity;

e

c2 = 1.92 ~ Q -0.087; c2 = 1.80 ~ Q -0.250 or can even

E v E v

decelerate; c2 > 2.00 ~ Q > 0.00.

E v

It should be noted that the modelling of the turbulence parameters

refers to the relaxation zone and not to the new wall-boundary layer. In spite of the relative slow decrease of the eddy viscosity in the

relaxation zone, the overall agreement is fair, because the new wall-boundary layer, where other flow conditions are dominating, reattaches to the surface only at a distance of 20 to 50 times the flow depth after the reattachment point.

It is recommended to do further research in order to verify the

proportion of the damping coefficient ~ with x Ü jÜo' In other words to

"

verify if cÀ is equal to 2.5 (equation 13). A sensitivity analysis (ODYSSEE-computations) has to be made to quantify cÀ for different

values of the initial flow depth (ao) and the discharge (ao ~ 0.3 mand q ~ 0.1 m2js, table 1).

(23)

Referenees

Booy, R. (1986), Turbu1enee, Leeture notes, Delft University of

Teehno1ogy, Dept. of Civi1 Eng. (Duteh)

Hanja1ie, K. and B.E. Launder (1976), Contribution towards a Reyno1 ds-stress e10sure for low-Reyno1ds-number turbu1enee, J. F1uid

Meeh., vol 74, part 4, pp. 593-610.

Hinze, J.O. (1975), Turbu1enee, seeond edition, Me. Graw Hi11 Book Co.,

New York.

Hoffmans, G.J.C.M. (1988a), Flow simulation by the 2-D turbulenee model

ODYSSEE, Delft University of Teehno1ogy, Dept. of Civil Eng.

Hoffmans, G.J.C.M. (1988b), Flow model with preseribed eddy viseosity,

Delft University of Teehno1ogy, Dept. of Civil Eng. Kay, J.M. and R.M. Nedderman (1985), F1uid meehanies and transfer

proeesses, Cambridge University Press.

Launder, B.E. and D.B. Spa1ding (1974), The numerieal eomputation of turbulent flow, Comp. Meth. in App1. Meeh., and Eng. 3, pp. 269.

Mierlo, M.C.L.M. and J.C.C. de Ruiter (1988), Turbu1enee measurements above dunes, Report Q789, Vol land 11, Delft Hydrau1ies.

Rodi,

w

.

(1980), Turbu1enee mode1s and their applieation in hydraulies,

(24)

(25)

(26)

appendix A: Scale analysis of the convective terms in the transport equation for the kinetic energy

appendix B: Elaboration of differential equation (9)

Tables

table A: Damping kinetic energy; k-f-predictionsjequation (10) table B: Damping dissipation k-f-predictionsjequation (14) table C: Damping eddy viscosity; k-f-predictionsjequation (26) table D: Growth length scale k-f-predictionsjequation (29)

Figures (concerns k-f-predictions)

figure A: Kinetic energy as a function of the flow direction figure B: Dissipation as a function of the flow direction figure C: Eddy viscosity as a function of the flow direction

(center mixing layer)

figure D: Eddy viscosity as a function of the flow direct ion (maximum)

figure E: Length scale of turbulence as a function of the flow direction

Abbreviations

BS Backward-facing step LS Local Scour

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Af ter neglecting the stress production of the turbulent energy and the diffusion terms the transport equation for the kinetic energy reads (for the meaning of the symbols, see notation):

-8k -8k

u8x +

wa;

- -f (Al)

The contribution of each single term can be estLmat.ed by using a normalizing method. Each term is represented as the product of a

constant scale factor and a dimensionless variabIe of the order of unity 0(1). The importance of each term is indicated by the relative magnitude of its sca1e factor. A function f(x) is norma1ized as:

f(x) = F· fl(X) (A2)

in which: x = X Xl

X scale factor

F scale factor

The derivative of f(x) is:

df dx

.E

dfl X dxl (A3) in which: dfl _ 0(1) dxl (A4)

The variables of the convective terms (Al) are:

u U Ui

(28)

(29)

(30)

UK akI

- ul-- +

X

ax

l (A5)

Using (van Mierlo and de Ruiter, experiment T5):

u

-

1.0 mjs _2 K - 1.0 10 Jjkg or m2js2

x

-

LOm W - 0.01 mjs

Z

-

0.1 m it fo11ows that: _2 UK 1.0*1.0*10 _2 X 1.0 _{_2} 0(10 ) WK 0.01*1.0*10 _3 Z 0.1 0(10 )

Based on this ana1ysis on1y equation (Al) may reduce to:

-ak

u- - -E

(31)

The differential equation (9) reads:

(BI)

The general solution of (BI) is:

x - X 0: k(x) _ k { v + I} k '1 ). x > x'1 (B2) Verification Differentiation of (B2) to x gives:

ak

ax

x - x __ ...!.LV +

x

0: -1 I} k (B3)

Raise to the square of (B3) gives:

x - x __ ""'-v +

x

20: -2 I} k (B4) Differentiation of (B3) to x gives: x - x __ ~V +

x

0: -2

I} k

(BS)

Substitution of (B2), (B4) and (BS) into (BI) gives:

x - X 0: k ( V + 1) k '1 ). x - X 0: -2 __ .:.LV + 1} k

x

+ x - x __ ..!Jov + ). 20: -2 I} k

_o

(B6) or (B7) or 1 (BB)

(32)

(33)

(34)

k

*

10 k

*

10 k

*

10 k

*

10 (m) (J/kg) (J/kg) (J/kg) (J/kg) 0.00 2.20 2.20 3.90 3.90 0.25 2.10 2.11 3.70 3.71 0.50 2.02 2.03 3.51 3.53 0.75 1.95 1.96 3.34 3.37 1.00 1.88 1.89 3,19 3.22 1.25 1.81 1.82 3.04 3.09 1.50 1.74 1.76 2.90 2.96 1.75 1.67 1.70 2.76 2.85 2.00 1.61 1.65 2.63 2.74

tab1e Al Damning kinetic energy re1axation zone LS1LLS2

LS3 BS2 x - x ODYSSEE ).=3.9 ODYSSEE ).=5.0 TI 3 3 3 3 k

*

10 k

*

10 k

*

10 k

*

10 (m) (J/kg) (J/kg) (J/kg) (J/kg) 0.00 4.26 4.26 4.70 4.70 0.50 3.83 3.74 4.22 4.24 1.00 3.34 3.32 3.81 3.86 1.50 2.90 2.99 3.42 3.53 2.00 2.50 2.72 3.05 3.26 2.50 2.20 2.49 2.75 3.02 3.00 1.95 2.29 2.50 2.82 3.50 1.75 2.12 2.29 2.64 4.00 1.58 1.98 2.11 2.48

(35)

(m) E

*

10 (\ol/kg) f

*

10 (\ol/kg) f

*

10 (\ol/kg) 0.00 0.25 0.50 0.75 l.00 l.25 l.50 0.37 0.34 0.32 0.30 0.28 0.26 0.24 0.71 0.64 0.57 0.51 q.46 0.42 0.39 0.37 0.34 0.32 0.30 0.28 0.26 0.24

tab1e B1 Damping dissipation re1axation zone LS1/LS2

E

*

10 (\ol/kg) 0.71 0.64 0.59 0.54 0.49 0.45 0.42 LS3 BS2 x - x ODYSSEE ).=3.9 ODYSSEE ).=5.0 'I 3 3 3 3 E

*

10 E

*

10 E

*

10 E

*

10

(m) (\ol/kg) (\ol/kg) (\ol/kg) (\ol/kg)

0.00 0.56 0.56 0.67 0.67 0.50 0.51 0.44 0.62 0.55 l.00 0.39 0.35 0.49 0.46 l.50 0.31 0.28 0.40 0.39 2.00 0.23 0.24 0.33 0.33 2.50 0.18 0.20 0.27 0.29

(36)

center mixing maximum center mixing maximum

1ayer 1ayer

3 _3 _3 _3 _3 3

1/ * 10 1/ * 10 1/ _{* 10} 1/ * 10 1/ * 10 1/ _* 10

t 2 t 2 t 2 t 2 t 2 t 2

(m) (m Is) (m Is) (m Is) (m Is) (m Is) (m Is)

0.00 0.88 1. 28 1.35 1.88 1. 95 ·2.04 0.25 0.80 1.28 1. 35 l.88 1. 96 2.03 0.50 0.68 1. 28 1.34 1.8) 1. 97 2.02 0.75 0.60 l.28 l. 34 l.84 1.96 2.02 l.00 0.50 1.27 l.33 l.80 1.96 2.01 1.25 1. 27 1.33 l.75 1.96 2.00 l.50 1.26 l.33 l.68 1.95 2.00 2.50 l.40 1.91 l.97 *LS3* *BS2*

x-x ODYSSEE ODYSSEE ).=3.9 ODYSSEE ODYSSEE ).=5.0

'1

center mixing maximum center mixing maximum

1ayer 1ayer

3 _3 3 _3 _3 _3

1/

*

10 1/ * 10 1/ * 10 1/ * 10 I/t * 10 1/

*

₁₀

t 2 t 2 t 2 t 2 2 t 2

(m) (m Is) (m Is) (m Is) (m Is) (m Is) (m Is)

0.00 2.49 2.56 2.60 2.38 2.44 2.54 0.50 2.59 2.62 2.57 2.40 2.48 2.52 1.00 2.59 2.62 2.55 2.39 2.50 2.50 l.50 2.58 2.60 2.53 2.36 2.50 2.48 2.00 2.54 2.58 2.51 2.32 2.50 2.47 2.50 2.48 2.55 2.49 2.25 2.49 2.45 3.00 2.40 2.52 2.47 2.19 2.47 2.44 3.50 2.30 2.48 2.46 2.12 2.45 2.43 4.00 2.17 2.43 2.45 2.04 2.43 2.41 4.50 2.02 2.38 2.43 1.95 2.39 2.40

(37)

maximum maximum _3 _3 3 3 L

*

10 L

*

10 L

*

10 L

*

10 (m) (m) (m) (m) (m) 0.00 49.6 49.6 56.4 56.4 0.25 50.9 50.4 58.2 57.6 0.50 52.0 51. 3 6,0.0 58.8 0.75 53.0 52.1 61.7 60.0 1.00 53.9 52.9 63.1 61.1 1.25 54.6 53.7 64.9 62.2 1. 50 55.3 54.5 66.0 63.3 2.50 71.0 67.5 LS3 BS2 x - x ODYSSEE ),=3.9 ODYSSEE ).=5.0 '7 maximum maximum 3 _3 _3 _3 L

*

10 L

*

10 L

*

10 L

*

10 (m) (m) (m) (m) (m) 0.00 69.6 69.6 65.0 65.0 0.50 76.0 73.5 69.6 67.9 1.00 81.9 77.3 74.0 70.6 1.50 88.0 80.8 78.0 73.3 2.00 93.4 84.1 81.6 75.8 2.50 98.5 87.3 84.7 78.2 3.00 103 90.3 87.8 80.6 3.50 107 93.3 90.5 82.8 4.00 110 96.1 93.2 85.0 4.50 113 98.8 96.0 87.2