A study concerning the influence of the relative local turbulence on local-scour holes

A study concerning the influence o f the relative turbulence intensity on local-scour holes

Report W - D W W - 9 3 - 2 5 1

G . J . C M . Hoffmans

Road and Hydraulic Engineering Division P.O. Box 5 0 4 4

2 6 0 0 GA Delft, Netherlands february, 1993

A study concerning the

influence of the relative

turbulence intensity on

local-scour holes

Ministry of Transport, Public Works and Watermanagement

Directorate-General of Public Works and Watermanagement

Road and Hydraulic Engineering Division

Voor u ligt een rapport (W-DWW-93-251) dat een modificatie geeft betreffende de ;

turbulentiecoëfficiënt in de Delftse ontgrondingsformule.

In de zestiger jaren is een semi-empirische relatie ontwikkeld om ontgrondingen te

voorspellen ten behoeve van de dimensionering van de Deltawerken in Zeeland. In de

jaren zeventig en tachtig zijn verschillende bureaustudies uitgevoerd om bovengenoemde

turbulentiecoëfficiënt als functie van geometrische parameters zoals lengte van de

bodembescherming en drempelhoogte van de constructie etc. te modelleren. Deze

relaties hebben een zuiver empirisch karakter, waardoor extrapolatie met grote

voorzichtigheid dient te geschieden.

In dit rapport wordt een relatie afgeleid voor de turbulentiecoëfficiënt, welke is

gebaseerd op de transportvergelijkingen van de turbulente kinetische energie en de

dissipatie. De theorie is geverifieerd door gebruik te maken van meer dan 500

gootexperimenten.

\

CONTENTS List of symbols

1 Introduction

2 Semi-empirical scour approach 3 Initiation of movement

4 Turbulence parameters 4.1 General

4.2 Nature o f t h e f l o w

4.3 Turbulence energy and turbulence intensity 4.4 Turbulence coefficient in the scour f o r m u l a 5 Conclusions

Relation between the depth-averaged turbulence energy and the relative turbulence intensity

O p t i m i z a t i o n o f the relative turbulence intensity Optimization of the turbulence coefficient

Verification of the relative turbulence intensitiy (285 experiments)

References Appendix A

Appendix B Appendix C Appendix D

List of symbols b = length of t h e a b u t m e n t [L] B = w i d t h o f the f l o w [L] C| = coefficient [-] q = relaxation coefficient (= 1.2) [-] c = coefficient in k-e-model (= 0.09) [-] CQ = coefficient (= 1.45) [-] C = Chézy coefficient • [L^T^] = coefficient [-] d = particle diameter [L]

D = height of the sill [L] = roughness f u n c t i o n [-]

Fr = Froude number (=U^{ghy-^) [-] g = acceleration o f gravity . [LT'^]

= initial f l o w - d e p t h [L]

k = (kinetic) turbulence energy [L^T'^] k^^ = (kinetic) turbulence energy close t o the bed ( n o n - u n i f o r m f l o w ) [l^T^]

k^ = (kinetic) turbulence energy in the mixing layer [L^T^] k^ = equivalent (or effective) roughness of Nikuradse [L] k^ = (kinetic) turbulence energy close t o the bed ( u n i f o r m f l o w ) [L^T^]

k = (kinetic) turbulence energy in relaxation zone [L^T^] K = coefficient

= coefficient [-]

L = length o f the bed protection [L]

n = number [-]

Q = discharge [ L ' T ^ ] = relative turbulence intensity [-]

R . = hydraulic radius [L] Re = Reynolds number hU^h^/v) [-]

t = time [T]

^^ = characteristic time at w h i c h _ / ^ / 7 Q _ [T] = transport parameter [=iaUQ-U)/UJ [-]

u' - longitudinal t u r b u l e n t velocity c o m p o n e n t [ L T - ]

= bed shear-velocity [ L T - ]

u^^ = critical bed shear-velocity [ L T - ]

= depth-averaged c r i t i c a l j l o w - v e l o c i t y according t o Shields [ L T - ]

= mean flow-velocity = Q/{B [ L T - ]

v' = transverse t u r b u l e n t velocity c o m p o n e n t [ L T - ] w' = vertical t u r b u l e n t velocity c o m p o n e n t [ L T - ]

X = longitudinal coordinate [L] x„ = x-coordinate where the f l o w reattaches the bed [L]

= m a x i m u m scour-depth [L]

= y e = A = K = A = = P .= a = V =

=

Subscripts c • = • / = m R 5 0 = coefficient [-] coefficient (= 0.5) [-]

angle o f t h e mixing layer (= 0.09) _[-]

coefficient (= 0.3 t o 0.4) [-]

error rate [-]

relative density ( = ( P s = p ) / p ) constant o f von Karman (= 0.4)

[-]

relative density ( = ( P s = p ) / p )

constant o f von Karman (= 0.4) [-]

relaxation length ( « y 2 C^/JQ/^S^) material density [ L ] relaxation length ( « y 2 C^/JQ/^S^) material density [ M L - ' ] fluid density mv'] standard deviation [-] kinematic viscosity [ L ^ T - ] coefficient (= 0.3) [-] calculated index measured rough smooth initial

1 Introduction

Scour is t h e lowering o f the sea or river-bed as a result o f non-equilibrium sediment transport conditions and can be divided into several categories (Breusers and Raudkivi, 1991) . Local scour, which may occur at the base of a structure because of the affected f l o w pattern, can severely endanger t h e stability of this structure. M a n y varieties o f local-scour systems downstream f r o m hydraulic structures exist, each w i t h its o w n particular geometry and hence local scour mechanism. Local scour is superimposed on general and constriction scour.

The prediction of local-scour holes t h a t develop downstream f r o m hydraulic structures plays an i m p o r t a n t role in their design. Excessive local scour can progressively undermine the f o u n d a t i o n o f a structure. Because complete protection against scour is t o o expensive generally, the m a x i m u m scour-depth and the upstream slope o f the scour hole have t o be predicted t o minimize t h e risk of failure.

In 1961 a systematical research w i t h respect t o scour holes started at Delft Hydraulics w i t h i n t h e scope o f t h e Dutch Delta works. A f t e r the catastrophic f l o o d disaster in 1953 the Delta plan was made t o protect the Rhine-Meuse-Scheldt delta f o r f u t u r e disasters. Dams w i t h large scale sluices were planned in some estuaries. The severe scour expected necessitated a better understanding of t h e scour process.

To f i n d detailed information a b o u t the physical processes playing a role in scour many experiments were carried o u t , in which various parameters o f t h e f l o w and the scoured material were varied. From the results of experiments in flumes w i t h all difficulties of scale effects and limitations in instrumentation some empirical relations were o b t a i n e d , which describe the erosion process as function of time and place (Prins, 1963 and Breusers, 1 9 6 6 , 1967).

In these empirical relations a not well defined turbulence coefficient was introduced. Up to n o w this coefficient was related t o t h e geometry upstream o f the scour hole, w h i c h relation was based on trial and error. Based on theoretical grounds an analytical relation f o r the depth-averaged turbulence intensity is derived. This relation, w h i c h implies a modification of the turbulence coefficient in the Breusers scour f o r m u l a , is verified using approximately 3 0 0 experiments.

The modified scour f o r m u l a yields results t h a t compare reasonably well t o measured and c o m p u t e d developments of a scour hole in case o f a u n i f o r m f l o w upstream o f the scour hole corresponding w i t h a large protected bed area. The computations were based on the two-dimensional Navier-Stokes and convection-diffusion equations ( H o f f m a n s , 1992) . The present paper aims at extension o f the domain of application o f t h e scour f o r m u l a t o n o n - u n i f o r m f l o w conditions upstream.

2 Semi-empirical scour relations

Generally the scour process is determined by f l o w and sediment characteristics. The sediment transport is mainly dependent on the bed shear-stress and t h e turbulence condition near the bed on t h e one hand and the density of t h e bed material, the

sediment-size distribution and the porosity of the (non-cohesive) material on the other hand. Based o n many clear-water scour observations Breusers ( 1 9 6 6 , 1967) reported t h a t the scour process could be w n t t e n as:

t

where = (1)

or using the invariables g (acceleration o f gravity) and (kinematic viscosity), (i.e., characteristic time at which = Z?,,) can also be given by:

A'

(2)

In the equations^above is the maximum scour-depth, is the initial f l o w - d e p t h , t is the t i m e , = Q/{Bh^ is the mean flow-velocity, Q is the discharge, B is the width of the flow, is the depth-averaged critical flow-velocity (Shields), A ^(p^p)/p is the relative density, is the niaterial density, p is the fluid density, T^ = {aUQ-U^/U^ is a transport parameter, Fr = U^{gh^'°-^ is the Froude number, Re = u j i j v is the Reynolds n u m b e r and K , and a are coefficients (table 1). Originally the characteristic t i m e was expressed in hours (Breusers, 1 9 6 6 ) , so t h a t the dimension o f K measured [hours m^'"^= s'^'] , (3600/< =/C^g^^V^.)

K Ar/10* _{P. ^ 3 ^ 4} a

Hinze (1961) 90 0.94 4.0 1.62 2.05 2.7 0.3 1+3r„

Breusers (1966, 1967) 90 0.94 4.0 1.62 2.05 2.7 0.3 1+3r„

Dietz (1969) 48 9.96 4.0 1.5 1.75 2.5 0.5 1+3r„

van der Meulen and Vinjé (1975) 250 12.9 4.3 1.7 2.0 2.87 0.43

Zanke (1978) 4.0 1.33 2.0 2.67 0.33

de Graauv\/ and Pilarczyk (1981) 330 17.1 4.3 1.7 2.0 2.87 0.43

Jorissen and Vrijling (1989) 330 17.1 4.3 1.7 2.0 2.87 .0.43 1.5+5ro Table 1 Empirical coefficients in scour formula (equations 1 and 2)

The sediment transport (bed load and suspended load) is largely d e t e r m i n e d by the bed turbulence d u r i n g the fluid inrush phase (sweeps) and the t u r b u l e n t outrush phase (ejections). During the t u r b u l e n t outrush phase the bed layer is disrupted locally by bursts o f f l u i d , so t h a t then the sediment particles are in suspension. In the transport parameter , w h i c h can be interpreted as a measure f o r the erosion capacity in the scour hole, the turbulence is represented by the turbulence coefficient a . According

- , siU

bedprobction

i l i a

t o Hinze (1961) t h e turbulence coefficient a is related t o (i.e., relative turbulence intensity) at t h e transition of t h e fixed t o the erodible b e d . Hence t h e local bed turbulence in t h e scour hole is n o t included in a . T h o u g h this is s o m e w h a t controversial, t h e used approach is f o l l o w e d because o f its simplicity. In section 4 . 4 a comprehensive analysis is given regarding turbulence parameters in n o n - u n i f o r m f l o w . Zanke (1978) has given a clear definition of t h e different scour phases. He assumed t h a t the scour process can be divided in f o u r phases. In t h e first phase a scour hole arises, in which t h e m a x i m u m scour-depth increases progressively compared t o t h e corresponding longitudinal distance , i.e., t h e distance f r o m t h e end o f t h e bed protection t o t h e point w h e r e the scour hole is at m a x i m u m . In t h e second phase t h e f o r m s of t h e scour hole are similar. Then t h e ratio yjx^ is more or less constant. In t h e third phase t h e development of t h e scour d e p t h increases deoressively and in t h e last phase an equilibrium is achieved, figure 1 . d

For/thre^dimjgn^ional scour c o r f f i c i e f i t / )<' is depêndent^on .the geometry -(van dér MeuleiT,:ahd Vinje, 1 ^75^.

The definition o f t h e e x p o n e n t y in equation (1) is not obtained u n a m b i g u - " ously. For two-dimensional f l o w t h e coefficient y measures a b o u t 0 . 3 8 , which is based o n an extensive analysis of the bed levels measured at which the m a x i m u m scour-depth is about 0.5/?o . A smaller value of y (=0.32) is more appropriate f o r experiments, where t h e m a x i m u m scour-depth is approximately equal t o t h e initial f l o w - d e p t h . According to Dietz (1969) t h e coefficient y lies in the range of 0.34 t o 0 . 4 0 , which can be considered as a confirmation of t h e Breu-sers' results. W h e n t h e time boundaries of t h e second phase o f t h e scour process are more specified, t h e value of y m i g h t be more unambiguously. To include t h e time evolution of local scour (Driegen et al., 1987).

A f t e r a further extensive evaluation of t h e enormous a m o u n t of data o f b o t h t w o and three-dimensional scour experiments t h e coefficients in t h e scour relations were readjusted (van der M e u l e n and Vinjé, 1 9 7 5 ; de Graauw and Pilarczyk, 1981 and Jorissen and Vrijling, 1989). Besides these Dutch research activities many other investigators (Dietz, 1969 and Zanke, 1978) contributed t o t h e solving o f t h e problem of scour w i t h empirical relations. The research activities o f Dietz and Zanke ( H o f f m a n s , 1992) confirmed t h e considerations of Breusers. A l t h o u g h t h e f o r m u l a e are identical, Dietz proposed different values f o r t h e empirical coefficients (table 1). The differences

phase I

(inili ation)

Figure 1

phase2 phasc3 phase4

(dcvelopmenl) (slabiluadcui) (Mimlibrium)

Development of tlie scour process

between t h e coefficients obtained by calibrating the measurements are small and may be due t o t h e different m e t h o d calculation, where the n u m b e r o f experiments and t h e reach o f t h e hydraulic and material parameters are not insignificant considering t h e ' i m p r o v e d ' constants.

3 Initiation of movement

In 1 9 3 6 , Shields published his criterion f o r beginning o f m o v e m e n t o f u n i f o r m granular material on a flat bed. The experimental data used by Shields was mostly obtained by extrapolating curves o f sediment transport versus applied shear stress t o t h e zero transport condition. Originally t h e data points were plotted by Shields and t h e curve (averaged critical value) was d r a w n by Rouse constituting t h e 'Shields d i a g r a m ' as usually q u o t e d (Neill, 1 9 6 8 ) . Actually Shields drew not a single curve, but a broad belt f o r reasons o f t h e n o n - u n i f o r m distribution of the mixtures and t h e effects of grain and imbrication (i.e., the preferred orientation o f natural sands and gravel particles under certain conditions of transport.

In the sixties Delft Hydraulics studied t h e initiation o f m o v e m e n t o f bed material in detail and distinguished seven qualitative criteria. These introduced criteria all lie in t h e broad belt as given by Shields originally c o n f i r m i n g t h e earlier research activities o f Shields. To determine the critical bed shear-velocity u^^ a criterium was selected, w h i c h was characteristic f o r a f l o w in which the bed particles move permanent at every location. This criterium almost agreed w i t h the averaged critical value d r a w n by Rouse.

The critical bed shear-velocity can be obtained graphically directly f r o m t h e Shields diagram. However, t o avoid more t h a n one possible interpretation u^^ can also be determined using analytical expressions (e.g., van Rijn, 1 9 8 4 ) , w h i c h f i t t h e Shields diagram. W i t h the aid o f Chézy's f o r m u l a and applying t h e Chézy coefficient, as proposed by Thijsse ( 1 9 4 9 ) , the depth-averaged critical f l o w - v e l o c i t y reads:

tj:

= U^^£^ (3)

C

-K

in w h i c h C is the Chézy coefficient, K is the constant o f v o n Karman, R is t h e hydraulic radius and is the equivalent (or effective) roughness o f Nikuradse. Usually t h e equivalent roughness o f a plane bed is related t o t h e largest particles o f the bed dgg, d^, d^ . The influence o f the gradation, the shape o f t h e particles and t h e f l o w conditions are generally disregarded. In the systematical research o n scouring t h e equivalent roughness was taken equal t o the mean particle diameter. However, in t h e literature several values f o r k^ can be f o u n d , table 2; According t o van Rijn (1982) the

12K

3.3v

u _»,c

equivalent roughness o f a plane movable bed varies f r o m about 1 t o 10 times o f the bed material. These values, which are rather large, show t h a t a completely plane bed does n o t exist f o r conditions w i t h active sediment transport. Probably, the relatively large scatter o f t h e equivalent roughness is caused by t h e initial unevenness (initial bed f o r m s ) . Here t h e equivalent roughness is supposed t o be twice as large as the mean particle diameter, w h i c h almost agrees w i t h the assumption made by Engelund-Hansen.

Ackers-Wfiite 1.25^33 Einstein As Engelund-Hansen 2d,^ Hey 33d,, Kamphuis Mali mood van Rijn 3d^

Table 2 Equivalent rougliness (van Rijn, 1982)

4. Turbulence parameters

4.1 General

Usually a sill has the f u n c t i o n of a f o u n d a t i o n f o r a closure d a m in an alluvial river or estuary. In a river the f l o w over a sill is mostly unidirectional. Sills w i t h a broad or a sharp crest and a sill w i t h and w i t h o u t a bed protection can be distinguished. Normally t h e , f l o w above a sill is subcritical, b u t depending on the waterlevel downstream f r o m the sill t h e f l o w can become supercritical. In this study only subcritical f l o w is considered.

deodeiation zooe reallacfaed sbear-layer

4.2 Nature of the flow

In analogy of the distribution of characteristic f l o w patterns in scour holes ( H o f f m a n s , 1992) t h e f o l l o w i n g f l o w zones can be distinguished downstream f r o m a sill: a mixing layer, a recirculation zone, a relaxation zone and a new w a l l - b o u n d a r y layer, figure 2. In t h e deceleration zone, the separated

shear-layer appears t o be much like an ordinary plane mixing layer. A recirculating f l o w develops behind the sill. The underside of the shear layer curves sharply d o w n w a r d s t o the reattachment point. Both in the mixing layer and in the recirculation zone the f l o w is very unsteady and highly t u r b u -lent. Downstream f r o m the point of

leattachinenl point/^

reattachment, the turbulence energy and t h e dissipation rate o f t h e turbulence energy decay rapidly. Simultaneously, a new w a l l - b o u n d a r y layer develops and spreads into t h e relaxation zone (the outer part of t h e reattached shear-layer). Measurements o f T r o u t t et al. (1984) have shown t h a t behind a backward-facing step t h e f l o w in the relaxation zone still has most o f t h e characteristics o f a free shear layer f l o w as much as 50 step heights d o w n s t r e a m f r o m reattachment. This observation demonstrates t h e persistence of the large-scale eddies, w h i c h are developed in the mixing layer.

4.3 Turbulence energy and turbulence intensity

Based on theoretica:! grounds a relation is derived f o r t h e turbulence coefficient a in t h e transport parameter in equation 1 (section 4.4). In this relation a is related t o r^ at the transition o f t h e fixed t o t h e erodible bed. The parameter r^ is t h e d e p t h -averaged relative turbulence intensity o f the longitudinal turbulence velocity c o m p o n e n t . The longitudinal c o m p o n e n t is only considered because this was t h e only c o m p o n e n t measured in general.

Before examining t h e turbulence energy downstream f r o m a sill in more detail some definitions are given. The (kinetic) turbulence energy k and r^ respectively are defined by (Hinze, 1975):

k = y2{lFLF + 77 + wW)

1 'rl

' ° = ^ ^ ^ ^ ^ ^ (6)

in which the terms u' , v' and w' represent the f l u c t u a t i n g flow-velocities in the longitudinal, transverse and vertical direction respectively.

C o m b i n i n g equations (5) and (6) and using measurements o f Nezu (1977) t h e turbulence energy (averaged over the depth) is f o r u n i f o r m - f l o w (Appendix A):

''oi

For n o n - u n i f o r m f l o w , measurements o f van M i e r i o & de Ruiter (1988) have s h o w n t h a t t h e turbulence energy k^ in the centre o f the mixing layer g r o w s rapidly t o a m a x i m u m . Downstream f r o m the point of reattachment t h e turbulence energy in t h e relaxation zone decreases gradually again and becomes small compared t o the turbulence energy k^^ generated by the bed in the developing n e w w a l l - b o u n d a r y layer. The turbulence energy generated in the mixing layer vanishes f o r relatively large values of X , i.e., where the new w a l l - b o u n d a r y layer is well developed. Then t h e turbulence energy tends t o an equilibrium value k^ , w h i c h largely consists of turbulence generated a t the bed.

Earlier studies of Hoffmans (1992) have shown t h a t in a scour hole k^^ can be repre-sented by a combination o f the turbulence energy generated at t h e bed and a certain

part o f t h e turbulence energy f r o m t h e m i x i n g layer, figure 3.

k^^ix) = w,/c^(x) + k^ix) w h e r e k^{x) = (8)

w h e r e ( = 0.3) is a t u r

-bulence coefficient, is t h e bed shear-velocity and

c ( = 0 . 0 9 ) is a coefficient used in k-e-models (Rodi, 1 9 8 0 ) .

T o analyze t h e decay o f t h e turbulence energy in t h e re-laxation zone an analogy w i t h t h e decay o f k and t h e dissipation in grid turbulence can be used (Launder and Spalding, 1 9 7 2 ) . W h e n t h e zone d o w n s t r e a m f r o m t h e p o i n t o f reattachment is c o n -sidered and t h e p r o d u c t i o n and diffusion terms in t h e transport equations o f t h e t u r -bulence energy and t h e dissi-pation are neglected, k

O.0S 0.04 0.03 0.01 aoo A A ^ in mixing layer _ in mixing layer — albed H k|, = u . ' / V c 7 A A ' I ^ in mixing layer _ in mixing layer — albed H k|, = u . ' / V c 7 M A A ^ in mixing layer _ in mixing layer — albed H k|, = u . ' / V c 7 I

/

_s ^ in mixing layer _ in mixing layer — albed H k|, = u . ' / V c 7 I

/

_s

/

V ' V A ' S ti V V t ' I i tl V B B H BE a 3 10 14 16 18 20

Figure 3 Calculated and measured k as function of x above an artificial dune (van Mierio & de Ruiter, 1988)

can be given by ( H o f f m a n s , 1 9 9 2 ) :

i x - x , ^ ^ ₍₉₎

in w h i c h x is t h e longitudinal coordinate, X R ( = 6 D ) is t h e x - c o o r d i n a t e w h e r e t h e f l o w reattaches t h e bed, D is t h e height o f t h e sill, A i^Vi c^h^/fij is a relaxation l e n g t h , yÖ„ ( = 0 . 0 9 ) is t h e angle o f t h e mixing layer, c, ( = 1 . 2 ) is a relaxation coefficient and ( = - 1 . 0 8 ) is a coefficient, w h i c h is directly related t o t h e turbulence coefficients used in k-e-models.

The hypothesis o f self-preservation ( T o w n s e n d , 1976) requires a constant turbulence energy in the m i x i n g layer up t o t h e p o i n t w h e r e t h e boundaries have reached t h e surface and t h e bed. A n appropriate value is ( H o f f m a n s , 1 9 9 2 ) :

(10)

in w h i c h q ( = 0 . 0 4 5 ) is a coefficient and U is t h e d e p t h - a v e r a g e d flovy velocity above t h e sill.

can be determined d o w n s t r e a m f r o m a sill, can be given by:

l . ik{x,z)dz = fi,l<<ix) ^ c,ul{x) /^''^

in w h i c h yff^ (=0.5) is a coefficient (Appendix B).

Actually the bed shear-velocity varies in t h e streamwise direction. In t h e recirculation zone t h e bed shear-velocity is relatively small and even zero in t h e reattachment point. Downstream f r o m t h e point o f reattachment the bed shear-velocity tends rapidly t o t h é equilibrium value corresponding t o u n i f o r m - f l o w conditions f o r w h i c h applies = 1.45. The length o f the bed protection L will f o r safety reason always extend beyond t h e point o f reattachment. C o m b i n i n g equations (7) t o (11) in t h e zone d o w n s t r e a m f r o m t h e p o i n t o f reattachment only shows can be represented by:

^0 =

in w h i c h C is t h e Chézy coefficient related t o the bed protection upstream f r o m t h e scour hole.

M o r e t h a n 2 5 0 experiments (Delft Hydraulics, 1972 and Buchko, 1986) were used t o verify the model e q u a t i o n j f o r the turbulence intensity. In these laboratory experiments the hydraulic conditions iU^< h^, k) as well as t h e geometrical parameters (L, D, B) were varied (appendix D). M o r e o v e r tests were executed w i t h an a b u t m e n t in permanent f l o w introducing three-dimensional scour. In these tests t h e length o f t h e a b u t m e n t measured 6 = 0.16, figure 4 .

Figures 5 t o 8 presents the results o f t h e measured and calculated values o f , where the influence o f b o t h the height o f t h e sill and t h e length o f the b e d protection can be observed.

The standard deviation of f o r three-dimensional experiments

(fig-ures 6 and 8) is s o m e w h a t larger t h a n f o r two-dimensional ones (fig-ures 5 and 7 ) . This can partly be ascribed t o periodical vortices, w h i c h can occur at the boundaries o f t h e m i x i n g layer in the transverse

direction k n o w n as a Karman Vortex-Street (Vinjé, 1969) and partly t o the measuring procedure.

In t h e two-dimensional experiments the velocities in the main direction o f the f l o w w e r e

measured w i t h a propellor-type current meter (Schuyf,

1966) and were taken at about 10 points along the vertical axis in t h e centre o f the f l u m e . In t h e case o f three-dimensional scour the l o n g i t u d i n a l f l o w - v e l o c i t i e s were also measured at several locations along t h e transverse axis. However, t h e data given in figures 6 and 8 are not averaged values over the w i d t h o f t h e f l u m e , but c o n -cerns measurements carried o u t in t h e axis, where the depth-averaged f l o w velocity is at m a x i m u m .

The measurements o f the longitudinal flow-velocity near the bed, w h e r e large gradients occur, were inaccurate. Due t o the relatively large dimensions of the measuring instrument the f l o w pattern was disrupted introducing severe errors. Also the number o f measuring points in the vertical was not enough especially in the mix-ing layer t o calculate f f o m the measurements accu-rately (figure 7 ) . 0.5 0.4 0.3 Ui I I 0.2 0.1 0.0 0.0

/

V I z «. rough D/hf, = 0 1 rough 0<D/h^<(i.35 A rough D/hg>0.35 B smooth TVho-O o smooth 0<DAi(^0.35 V smooth D/hg>0J5 Pk = 0.50 0.1 0.2 0.3 measurements — 0.4 0.5

Figure 5 Relative turbulence intensity (2-D experiments)

0.5 A 0.4 0.3 0.2 0.1 0.0 V A A A , , A V ' * rough D/hQ= 0 X rough 0<D/ho<.0.35 A rough D/hQ>0.35 R smooth D/ho=0 o smooth 0<D/hoi0.35 7 smooth D/ho>OJ5 P* = Ó.50 0.0 0.1 0.2 0.3 measuiEineBts • 0.4 0.5

Figure 6 Réiative turbulence intensity (3-D experiments)

4.4 Turbulence coefficient in the scour formula

The value o f a in the scour f o r m u l a can be obtained using the relation between a and f r o m Jorissen and Vrijling (1989):

a = 1.5 + 5/-„ (13)

This value is based on the use of a local depth-averaged velocity U^^ in t h e scour f o r m u l a . If a three-dimensional f l o w is considered and t h e mean f l o w - v e l o c i t y (figure 4) is used in the scour f o r m u l a , the value o f a has t o be multiplied by A re-examination of more than 2 5 0 experiments (Delft Hydraulics, 1972 and Buchko, 1986) shows t h a t reasonable results are t h e n achieved f o r b o t h t w o and

three-dimensional experiments (table B I , appendix B). • •D/hc=0.20-, — D/h(,= 0.30 calculatioos — n/h(f=O.SO -• 0.17 < D/ho< 0.23 B 0.30 <D/hf,< 0.35 * D/h(i=0.50 h = 0.5

Figure 7 Relative turbulence intensity as function of L/h^ (2-D experiments; rougii)

a = ^.5 + 4.4rJ^

Already Hinze (1961) remarked t h a t n o t only t h e i n fluence o f t h e turbulence i n -tensity b u t also t h e influence o f t h e f l o w - v e l o c i t y profile near t h e bed is significant f o r t h e d e v e l o p m e n t o f t h e scour process. T o include t h e i n f l u -ence o f a s m o o t h bed a simple expression is i n t r o d u c e d , w h i c h is verified applying a b o u t 5 5 0 experiments (Delft Hydraulics, 1 9 7 2 ; Dietz, 1 9 6 9 ; Buchko, 1 9 8 6 and H o f f m a n s , 1 9 9 0 ) : (14)

in w h i c h = C/C^ represents a roughness f u n c t i o n and = 4 0 m ^ / s . For hydrauli-cally-rough conditions, i.e., f o r C < 40m*^/s regarding t h e fixed bed before t h e scour hole. 0.5 0.4 0.3 ro(-) 0.2 0.1 0.0 B B alculatior ï • • • . . 10 • D/ho = 0.30 (rough) -D/hg^ 0.60 (rough) « D/llg = 0.30 (rough) a D/ho = 0.60 (rough) A D/ho = 0.30 (smooth) 0.60 (smooth) P t = 0 . 5 20 30 40 50 T h o u g h t h e influence o f three-dimensional effects are n o t included in t h e model equation f o r (equation 12) satisfactory results are o b t a i n e d , especially f o r t w o -dimensional scour (appendix B).

Conclusions

Figure 8 Relative turbulence intensity as function of L/ho (3-D experiments)

A model equation is given f o r t h e turbulence coefficient (equation 12) in t h e Breusers scour f o r m u l a , w h i c h is based on theoretical grounds and f i t t e d t o results o f velocity measurements. This study shows a w a y t o calculate t h e relative turbulence intensity at t h e transition o f t h e fixed bed t o t h e erodible b e d , w h i c h can be used t o predict t w o -dimensional scour downstream f r o m a sill.

The particular case of scour o f three-dimensional f l o w considered here is f l o w a r o u n d an a b u t m e n t w i t h consequent scour development d o w n s t r e a m . A l t h o u g h some

char-acteristics o f three-dimensional f l o w and additional phenomena such as vortices w i t h a vertical axis in particular have not been taken into account in the model equation f o r t h e turbulence intensity, promising results regarding three-dimensional scour are obtained. For example, in t h e centre of the f l o w where t h e scour d e p t h is a b o u t at m a x i m u m , the influence of t h e Karman Vortex-Street can be neglected. However, prudence has t o be called f o r complex hydraulic structures. Then it is recommended t o carry o u t experiments using a scale model t o find detailed information a b o u t the development o f a scour hole.

Appendix A Relation between the depth-averaged turbulence energy and the relative turbulence intensity

For u n i f o r m f l o w t h e relative turbulence intensity o f the transverse and vertical c o m p o n e n t can be w r i t t e n as (Nezu, 1977):

\ / 7 7 ( i ) = Kv ^^^^

\lw'w'iz) =

s

IIF

l

FW)

in w h i c h y^ (= 0.71) and y^ (= 0.55) are coefficients.

C o m b i n i n g equations ( A l ) and (A2) and the definition o f t h e turbulence energy (equation 5 ) , t h e turbulence energy can be given by:

kiz) = WWiX) ^^^^

in w h i c h q (= Vzd + + j ^ ) = 0.90) is a coefficient.

The variance o f the longitudinal instantaneous flow-velocity reads (Nezu, 1 9 7 7 ) :

u'u'iz) = ^ , exp 2z (A4)

in w h i c h y^{=^.^2) is a coefficient.

Hence the depth-averaged turbulence energy can be given by:

\ '

J - \k{z)dz = u\ ^^^^

in w h i c h = y 2 ( 1 - e-^)c^K^ ^ 1.44 .

A p p l y i n g equation (A4) and the definition o f the relative turbulence intensity (equation 6 ) , can be represented by: .

= ^ (A6)

in w h i c h C3 = (1 - e''')^^ = 1.21 .

y\Kz)dz

= cjr.U,)

''o i

in which = c / c ^ = = 1 . 0 .

Appendix B Optimization of the relative turbulence intensity

Three methods are used t o optimize in equation 12, w h i c h are (table B I ) :

Least Absolute Value Method (LAVM):

r o s ^ / ^ k ) - ' • 0 ^ , 1

^ L A V M ^ k ^ 2 ^ r

Least Square Method (LSM):

1=1 '^0^^

Least Standard Deviation Method (LSDM):

^ L S D M ^ ^ k ^

>

M - 1

Table BI Optimize methods

In table BI and are the calculated and the 'measured' respectively. Actually the 'measured' is n o t a measurement but calculated using measurements, since the turbulence intensity is here defined as the integration of t h e measured local turbulence intensities (equation 6). The error f u n c t i o n e{^J) as discussed above is

K

L A V M 0.357 0.043

LSM 0.386 0.042

primary estimation 0.500 0.040

LSDM 0.596 0.042

minimized applying n (=285) f l u m e experiments (Delft Hydraulics, 1972 and Buchko, 1986) yielding t h e f o l l o w i n g results f o r y?^ , table B2.

From this analysis it can be deduced t h a t is slightly dependent on fi^ . The methods L A V M and LSM reflect the measurements in a better w a y f o r relatively small values o f (appendix C). Conversely the m e t h o d LSDM gives better results f o r

relatively large values of r^, .

Besides this optimization of t h e standard deviation oi^^) is d e t e r m i n e d . Overall, t h e smallest error is obtained f o r = 0.5 (table B2).

Appendix C Optimization of the turbulence coefficient

The n u m b e r o f laboratory experiments used t o f i n d t h e best compromise a m o n g the 'measured' and calculated a in equation (14) using the 'measured' values o f , amounts t o 2 8 5 (Delft Hydraulics, 1 9 7 2 ) , table C l . Several model relations f o r a are introduced, w h i c h are:

a = (1.079 + 5^^8r^)f^ ( C l )

a = 1.483 + 4.392rJ^ (C2)

a = ^.^20f^ + 5.854ro (C3)

The coefficients in the equations above were obtained after minimizing t h e error function using t h e least standard deviation m e t h o d , appendix B.

2D-rough (^2^ = 154) 2D-smootfi in,, =64) 3D-rough ("3^ = 50) 3D-smootii (A73, = 17) (13) 0.392 _(-) 0.462 _(-) (CD 0.329 0.430 0.547 0.856 (C2) 0.353 0.270 0.458 0.862 (C3) 0.313 0.328 0.526 0.721.

Table C l Standard deviation of a based on 'measured' values of r,

It is noted that the'measured' a is based on measured parameters ( Q , h^, B, A , t^, d^^ and determined w i t h equation (1), The kinematic viscosity is supposed t o be equal t o

v = ^0'^m^/s , whereas the depth-averaged critical flow-velocity according t o Shields is

c o m p u t e d as discussed in section 3 (van Rijn, 1 9 8 4 ) .

The results o f the standard deviation o f a are given in table C l .

2D-rough (n2, = 154) 2D-smooth (A7,,=64) 3D-rough

K

= 50) 3D-smooth ( " 3 5 = 17) (13) 0.448 (-) 1.060 (-) ( C l ) 0.307 0.446 0.596 0.568 (C2) 0.395 0.291 0.487 0.735 (C3) 0.326 0.340 0.550 0.574

Table C2 Standard deviation of a based on calculated values of ( ^^^ = 0.5 )

introduced by Jorissen and Vrijling (1989) is o f t h e same order, Generally t h e error rate f o r three-dimensional f l o w is s o m e w h a t larger than f o r t w o - d i m e n s i o n a l f l o w , especially f o r hydraulically-smooth conditions.

Table C2 presents the standard deviation o f a , which is based on c o m p u t e d using equation 12. Hence the f o l l o w i n g remarks can be made.

Considering the model relations C l t o C3 the difference between t h e calculated and measured standard deviation o f are marginal. However, f o r 3 D - s m o o t h cases

P. 2D-rougii ( n , , = 240) 2D-smooth ( " 2 3 = 132) 3D-rougii ( " 3 ^ = 54) 3D-smootli ( " 3 5 = 104) (13) 0.357 0.411 (-) 1.090 (-) (13) 0.500 0.441 (-) 1.060 (-) (13) 0.596 0.468 (-) 1.042 (-) (CD 0.357 0.329 0.563 0.682 0.598 (CD 0.500 0.301 0.572 0.596 0.520 (CD 0.596 0.299 0.578 0.555 0.492 (C2) 0.357 0.374 0.268 0.566 0.645 (C2) 0.500 0.387 0.271 0.487 0.554 (C2) 0.596 0.404 0.275 0.447 0.509 (C3) 0.357 0.306 0.470 0.635 0.633 (C3) 0.500 0.313 0.473 0.550 0.541 (C3) 0.596 0.335 0.475 0.511 0.498

(three-dimensional f l o w and hydraulically-smooth conditions) the error rate becomes smaller w h e n t h e turbulence intensity is calculated, which is probably due t o t h e measuring procedure, as discussed in section 4 . 4 .

Finally t h e influence o f the coefficient (equation 12) on the standard deviation a is determined using approximately 525 experiments (Delft Hydraulics, 1 9 7 2 ; Dietz, 1 9 6 9 ; Buchko, 1 9 8 6 and H o f f m a n s , 1 9 9 0 ) , table C3. From this analysis it can be concluded t h a t equation C2 (or equation 14) gives the best results compared t o t h e o t h e r relations

A p p e n d i x D V e r i f i c a t i o n o f t h e r e l a t i v e t u r b u l e n c e i n t e n s i t y r e p o r t s e r i e s c o n d i t i o n D/h,

c

''om ^0. m648b S8-20 sinooth2D .00 10. 70, , 06 . 05 m648b S8-21 smooth2D . 00 10. . 7 1 . , 08 ,05 m648b S8-22 smooth2D . 00 10. 72 , , 06 . 05 • m648b S8-25 smooth2D .00 10. 7 0 . .08 , 05 m648b S8-24 smooth2D . 00 10. 7 1 . , 08 , 05 m648b S8-64 smooth2D .00 10. 62. , 06 . 06 in648b S8-63 smooth2D .00 10. 63, , 07 . 06 m648b S8-61 smooth2D . 00 10. 65. , 06 , 06 m648b S8-62 smooth2D . 00 10, 66. . 07 . 06 m648b S8-60 smooth2D . 00 10. 67. , 08 ,06 m648b S8-59 smooth2D .00 10. 68, .07 . 06 m648b S8-58 smooth2D . 00 10. 69, , 08 , 05 m648b S8-56 smooth2D .00 10. 71, .08 . 05 m648b S8-31D smooth2D . 00 10. 6 5 . , 06 ,06 m648b S8-31A sinooth2D . 00 10. 66. , 04 .06 in648b S8-33B smooth2D . 00 10. 67. , 06 , 06 m648b S8-49 smooth2D . 00 10. 7 0 . .07 .05 m648b S B - 5 3 sinooth2D , 00 10. 71,, , 06 , 05 m648b S8-52 sinooth2D . 00 10. 7 2 . , 06 . 05 m648b S8-54 smooth2D . 00 10. 7 3 . ,07 , 05 m648b S l l - 3 rough2D . 33 4. 4 1 . , 20 , 22 m648b s l l - 1 rough2D .33 4. 41, , 22 ,22

in648b S11-2A rough2D . 33 4. 41, , 22 , 22

m648b S l l - 2 7 rough2D .33 4. 40, .23 ,22

m648b S l l - 2 9 rough2D .33 4. 40, .23 , 22

m648b S l l - 2 8 rough2D .33 4. 40, . 24 .22

r e p o r t s e r i e s c o n d i t i o n L/h^

c

_''nm

m648b s l l - 1 5 rough2D .33 4. 4 1 . .22 . 22

m648b S 1 3 - 1 1 rough2D .50 4. 38. .32 ,30

m648b S 1 3 - 1 0 rough2D .50 4. 38. . 30 .30

in648b S13-9- rough2D .50 4. 38. .28 .30

ra648b S13-8A rough2D .50 4. 38. .31 .30

m648b S13-7 rough2D .50 4. 38. .31 . 30 m648b S13-6 rough2D .50 4. 38. . 28 .30 m648b S13-3 rough2D . .50 4. 38, .31 .30 in648b S13-2 rough2D .50 4. 38. .29 .30 m648b S13-1 rough2D .50 4. 38. .30 .30 m648b S14-4 rough2D . 00 10. 36. .10 . 10 m648b S14-1 rough2D .00 10. 36. .09 .10 m648b S 1 4 - 2 rough2D .00 10. 36. .09 . 10 m648b S 1 4 - 3 rough2D . 00 10. 36. .09 . 10 m648b S 1 4 - 8 rough2D . 00 10. 36. . 12 .10 m648b S 1 4 - 7 rough2D .00 10. 36. . 12 , 10 m648b S 1 4 - 5 A rough2D . 00 10. 36. . 12 .10 m648b , S15-13 rough2D .00 10. 39. . 09 .10 m648b S15-14 rough2D . 00 10. 39. .10 .10 m648b S 1 5 - 1 5 rough2D . 00 10. 39. . 09 . 10 ni648b S 1 5 - 1 1 rough2D . 00 10. 39. . 09 . 10 m648b S15-12 rough2D . 00 10. 39. .09 .10 m648b S15-9 rough2D . 00 10. 39. . 10 . 10 m648b S15-8 rough2D .00 10. 39. .09 .10 m648b S15-5 rough2D . 00 10. 39. . 09 . 10 m648b S15-7 rough2D .00 10. 39. . 09 . 10 m648b S15-6 rough2D .00 10. 39. . 10 . 10

r e p o r t s e r i e s c o n d i t i o n D/h, Z.//7,

c

m648b S15-4 rough2D . 00 10. 39. . 08 . 10 m648b S15-1 rough2D , 00 10. 39. . 08 .10 iQ648b S15-2 rough2D . 00 10. 39. .09 . 10 m648b S15-3 rough2D . 00 10. 39. . 08 . 10 m648b S15-29 rough2D .00 10. 39. .09 .10' m648b S15-30 rough2D .00 10. 39. .08 . 10 m648b S 1 5 - 3 1 rough2D .00 10. 39. .09 . 10 m648b S15-27 rough2D . 00 10. 39. . 10 . 10 m648b S15-26 rough2D . 00 10. 39. .11 .10 m648b S 1 5 - 2 5 rough2D . 00 10. 39. . 10 . 10 m648b S15-23A rough2D . 00 10. 39. .09 .10 m648b S15-36 rough2D . 00 10. 39. . 09 . 10 m648b S15-32 rough2D .00 10. 39. . 10 . 10 in648b S15-33 rough2D .00 10. 39. . 09 .10 m648b S15-34 rough2D . 00 10. 39. .09 . 10 m648b S16A-13 rough2D . 17 4. 41. . 15 . 18 m648b S16A-14 rough2D . 17 4. 41. . 15 . 17 m648b S16A-15 rough2D .17 4. 41. . 15 .17 m648b S16A-35 rough2D .17 4. 41. . 13 .17 m648b S16A-31 rough2D . 17 4. 41. .13 . 17

in648b S16A-30A rough2D . 17 4. 41. .13 ,17

ID648b S16A-33 rough2D .17 4. 41. .14 . 17

in648b S16A-32 rough2D .17 4. 41. .13 .17

m648b S16A-23 rough2D . 17 4. 41. . 13 ,17

m648b S16A-24 rough2D . 17 4. 41. . 14 , 17

m648b S16A-20 rough2D . 17 4. 41. . 13 ,18

r e p o r t s e r i e s c o n d i t i o n D/h,

c

m648b S16A-21 rough2D .17 4. 4 1 . . .13 .18

m648b S16A-18 rough2D .17 4, 4 1 . .12 ,18

iti648b S16A-17 rough2D . 17 4. 41, , 13 .17

m648b S16A-25 rough2D .17 4. 40. .11 ,18 m648b S16A-26 rough2D . 17 4, 40. .11 ,18 • m648b S16A-27 rough2D . 17 4, 4 1 . .11 .18 m648b S16B-10 rough2D . 17 4, 4 1 . . 12 , 18 m648b S16B-11 rough2D . 17 4, 41, . 13 .17 m648b S16B-12 rough2D .17 4. 41. .13 . 17 m648b S16B-9 rough2D ,17 4. 41, , 12 . 18 m648b S16B-7 rough2D .17 4. 41. , 13 .17 m648b S16B-6 rough2D ,17 4. 41. .12 . 18 in648b S16B-5 rough2D ,17 4. 41. , 13 . 17 m648b S16B-4 rough2D , 17 4. 41. . 13 .17 m648b S16B-3 rough2D , 17 4. 41. , 13 . 18 m648b S16B-2 rough2D .17 4. 41. , 12 . 17 m648b S16B-1 rough2D . 17 4. 41. .13 ,17

in648b S17A-23A rough2D .33 4. 4 1 . ,24 ,22

m648b S17A-13 rough2D .33 4 . 4 1 . ,21 ,22

m648b S17A-12 rough2D .33 4. 4 1 . ,23 ,22

in648b S17A-11 rough2D . 33 4. 41, .22 , 22

m648b S17A-9 rough2D .33 4. 41, .25 , 22 m648b S17A-20 rough2D .33 4. 40, . 19 .22 m648b S17A-16A rough2D .33 4. 40. .20 . 22 m648b S17A-17 rough2D . 33 4. 4 0 . ,20 ,22 m648b S17A-18 rough2D .33 4. 41. ,20 .22 m648b S17B-5 rough2D .33 4. 4 1 . . 24 . 22

r e p o r t s e r i e s c o n d i t i o n D/h, L/h,

c

''nm m648b S17B-6A rough2D .33 4. 41. ,21 .22 m648b S17B-4 rough2D .33 4. 4 1 . , 19 .22 m648b S17B-3 rough2D .33 4. 41, ,16 .22 m648b S17B-2 rough2D .33 4. 41. .20 , 22 m648b S17B-22 rough2D .33 4. 4 0 . ,22 .22 m648b S 1 9 - 1 1 rough2D . 00 12. 41, .07 ,09 m648b S19-10 rough2D . 00 12. , 4 1 . , 08 .09 m648b S19-7A rough2D .00 12. 41, ,07 , 09 m648b S19-9A rough2D .00 12. 4 1 . .07 .09 m648b S19-4 rough2D . 00 12 . 4 1 . . 10 , 09 m648b S19-5 rough2D .00 12. 41, , 10 ,09 m648b S19-6 rough2D . 00 12. 41, , 10 . 09 in648b S19-1 rough2D . 00 12 . 4 1 . ,10 . 09 ra648b S19-2 rough2D . 00 12. 41. .09 ,09 m648b S19-3 rough2D . 00 12 . 4 1 . . 11 , 09 m648b S19-20 rough2D .00 12. 4 3 . .07 , 09 in648b S19-17 r o u g h 2 D . 00 12 . 43 . . 06 , 09 m648b S 1 9 - 2 1 rough2D .00 12. 43, . 06 ,09 m648b S19-18 rough2D .00 12. 43, .06 ,09 m648b S19-22A r o u g h s D . 00 12 . 41, ,07 ,09 m648b S19-23 rough2D .00 12 . 4 1 . ,07 , 09 m648b S19-24 rough2D . 00 12. 41. . 08 .09 m648b S19-25A rough2D . 00 12. 41. ,08 .09 m648b S19-D rough2D . 00 12 . 41, . 08 . 09 m648b S19-C rough2D . 00 12. 41, ,08 , 09 m648b S19-B rpugh2D . 00 12 . 41, ,09 .09 m648b S19-A rough2D . 00 12. 4 1 . .09 .09

r e p o r t s e r i e s c o n d i t i o n D/h, L/h,

c

m648b S 2 0 - 9 r o u g h 2 D .23 5. 41. ,13 . 18 m648b S 2 0 - 1 0 rough2D .23 5. 4 1 . ,16 , 18 m648b S 2 0 - 5 rough2D .23 5. 4 1 . ,15 . 18 m648b S 2 0 - 4 rough2D .23 5. 4 1 . , 16 ,18 m648b S 2 0 - 6 r o u g h 2 D .23 5. 4 1 . .17 .18 in648b S 2 1 - 3 rough2D . 33 5. 4 1 . .23 ,21 m648b S 2 1 - 2 rough2D .33 5. . 4 1 . .21 ,21 m648b S 2 2 - 5 smooth2D . 00 12 . 74. .04 ,05 in648b S 2 2 - 6 smooth2D .00 12. 7 5 . ,04 ,05 m648b S 2 2 - 3 smooth2D .00 12 . 8 0 . . 06 , 05 m648b S 2 2 - 4 smooth2D .00 12 . 82 . .06 , 05 m648b S 2 2 - 1 smooth2D .00 12 . 88. ,05 , 04 m648b S 2 2 - 8 smooth2D . 00 12 . 65. .04 , 06 m648b S 2 2 - 9 sinooth2D .00 12 . 65. ,03 , 06 m648b S 2 2 - 1 0 smooth2D . 00 12. 66. ,04 , 06 m648b S 2 3- 6 smooth2D . 00 2. 63. , 03 . 06 m648b S 2 3- 5 smooth2D . 00 2. 64. , 03 , 06 in548b S 2 3- 4 smooth2D . 00 2'. 65, , 02 . .06 m648b S 2 3- 3 smooth2D .00 2 . 65. .02 .06 m648b S24-4 smoothsD . 00 8. 63 , ,04 . 06 m648b S24-3 smooth2D . 00 8. 64. , 03 . 06 m648b S24-2 smooth2D . 00 8. 65, , 05 .06 m648b S24-1 , smooth2D . 00 8. 65, .05 . 06 m648b S 2 5 - 4 smooth2D . 00 14. 63, .04 . 06 m648b S 2 5 - 3 smooth2D . 00 14. 64, . 05 • 06 m648b S 2 5 - 2 smooth2D . 00 14. 65, ,06 .06 m648b S 2 5 - 1 smooth2D . 00 14. 65. . 05 . 06

r e p o r t s e r i e s c o n d i t i o n D/h, L/h,

c

m648b S 2 5- 5 smooth2D . 00 14. 66. .04 . 06 m648b S 2 6- 3 smooth2D .00 14. 56. .04 , 07 m648b S 2 6- 2 smooth2D . 00 14. • 5 7 . .04 ,07 m648b S 2 6- 1 - smooth2D .00 14. 57. , 05 .07 ffl648b S 2 6- 4 smoothsD .00 1 4 . 5 7 . ,04 _• m648b S27-4 smoothsD . 00 14.• 5 3 . .04 ,07 m648b S27-3 smooth2D . 00 14. 5 3 . , 05 .07 m648b S27-2 smooth2D . 00 14. 53, , 05 . 07 m648b S 2 7- 1 smooth2D . 00 14. 53, .05 ,07 T Q 6 4 8 b S 2 8- 4 smooth2D . 00 14. 49, .05 ,08 m648b S 2 8- 3 smooth2D . 00 14. 49. , 04 .08 m648b S 2 8- 2 smooth2D . 00 14. 49. .05 . 08 m648b S 2 8- 1 smooth2D . 00 14. 4 9 . . 05 . 08 m648b S 2 9- 5 smooth2D .00 14. 44. ,06 .09 m648b S 2 9- 4 smooth2D . 00 14. 44. . 05 .09 m648b S 2 9- 3 smooth2D . 00 14. 44. , 05 , 09 m648b S 2 9 - 2 smooth2D . 00 14. 44. . 06 .09 m648b S 2 9 - 1 smoothSD .00 14. 44. .05 .09 in648b S 3 0 - 5 rough2D .00 14. 39. .06 .10 in648b S 3 0 - 4 r o u g h s D . 00 14. 39. . 06 .10 m648b S 3 0 - 3 rough2D . 00 14. 39. . 06 . 10 m648b S 3 0 - 2 rough2D . 00 14. 39. . 06 , 10 m648b S 3 0 - 1 rough2D . 00 14. 39, , 06 , 10 in648b S 3 1 - 5 rough2D .00 14. 34, , 07 ,11 m648b S 3 1 - 3 rough2D . 00 14. 34, , 07 , 11 m648b S 3 1 - 1 rough2D . 00 14. 34. , 08 • 11 m847b SOGLAD smooth2D .00 10. 7 3 . , 04 .05

r e p o r t s e r i e s c o n d i t i o n D/h, L/h, C _''om

m847b SOMOLO rough2D .00 10. 46. .07 . 08

m847b S0M1L2 smooth2D .00 10. 75. .07 ,05

iQ847b SOMOLO rough2D . 00 10. 46. .08 , 08

m847b S0M1L2 smooth2D . 00 10. 7 3 . .07 , 05

iQ847b S I G I A D . smooth2D .30 10. 71, ,10 ,15

m847b SIMOLO rough2D .30 10. 46, ,13 , 16

m847b S1M1L2 smooth2D .30 10. 7 3 . ,14 , 15

in847b SIMOLO rough2D .30 10. 46, . 14 . 16

m847b S2GLAD smooth2D . 60 10. 7 0 . .20 .27 in847b S2M0L0 rough2D , 60 10. 46, .22 .27 m847c FOVIM r o u g h s D . 00 5- 46, .08 .08 m847c FOVIM r o u g h s D .00 5. 46, .08 .08 m847c FOVIM r o u g h s D . 00 5. 47, .08 .08 I Q 8 4 7 C FOVIM rough3D . 00 5. 47, ,08 ,08 in847c F I V I M rough3D . SO 5. 46, . 12 , 19 m847c F I V I M rough3D . SO 5. 46, , 12 , 19 m847c F l V l M r o u g h s D .30 5. 46, ,12 , 19 m847c F2V1M r o u g h s D .60 5. 4 5 . .48 .35. m847c F2V1M rough3D . 60 5. 4 5 . .48 .35 m847c F2V1M r o u g h s D . 60 5. 46. .48 .35 m847c F0V2M r o u g h s D . 00 10. 46. .14 . 08 in847c F0V2M rough3D . 00 10. 46. ,14 . 08 in847c F0V2M r o u g h s D . 00 10. 47. .14 . 08 m847c F0V2M r o u g h s D .00 10. 47. ,14 .08 m847c F1V2M r o u g h s D .30 10. 46, , 14 . 16 m847c F1V2M r o u g h s D .30 10. 46, , 14 . 16 ni847c F1V2M r o u g h s D .30 10. 46. . 14 .16

r e p o r t s e r i e s c o n d i t i o n D/h, L/h, C _''nm m847c F1V2M rough3D .30 10. 46. • .14 ' .16 m847c F2V2M r o u g h s D . 60 10. 46. .32 . 27 m847c F2V2M r o u g h s D .60 10. 46. .32 .27 m847c F2V2M r o u g h s D . 60 10. 46. . 32 .27 m847c F0V3M r o u g h s D .00 15. 47. .12 . 08 ' m847c F0V3M r o u g h s D . 00 15. 4 7 . .12 . 08 m847c F0V4M r o u g h s D .00 20. 47. . 13 .08 m847c F0V4M rough3D . 00 20. 4 7 . . 13 ,08 in847c F0V4M rough3D . 00 20. 47. .13 .08 m847c F1V4M rough3D . SO 20. 46. .17 , 13 m847c F1V4M r o u g h s D .30 20. 47. .17 .13 m847c F1V4M rough3D .30 20. 47. .17 , 13 m847c F2V4M rough3D . 60 20. 46. .17 ,21 m847c F2V4M r o u g h s D . 60 20. 46. . 17 , 21 m847c F2V4M roughSD . 60 20. 46. . 17 .21 m847c DOVIG smoothSD .00 5. 65. . 08 .06 m847c DOVIG smoothSD . 00 5. 66. . 08 .06 m847c D I V I G smoothSD .30 5. 65. . 18 , 18 m847c D I V I G smoothSD .30 5. 66. . 18 ,18 in847c D2V1G smoothSD .60 5. 65. .19 . 34 in847c D0V2G smoothSD . 00 10. 65. . 10 ,06 m847c D0V2G smoothSD .00 10. 66. . 10 . 06 m847c D0V2G smoothSD . 00 10. 67. . 10 .06 m847c D1V2G smoothSD .30 10. 63 . . 12 . 15 m847c D1V2G smoothSD .30 10. 65. .12 .15 m847c D1V2G smoothSD .30 10. 66. . 12 . 15 m847c D2V2G smoothSD . 60 10. 61. .36 ,27

r e p o r t s e r i e s c o n d i t i o n D/h, L/h,

c

m847c D2V2G smoothsD . 60 10. 63 . .36 .27 m847c D2V2G smoothsD , 60 10. 6 5 . .36 .27 m847c D0V4G smoothSD . 00 20. 74. . 15 . 05 m847c D1V4G smoothSD .30 20. 74. . 12 . 12 m847c D2V4G smoothSD .60 20. 74. .20 .2 0' in847c DOVIR r o u g h s D .00 5. 42. .12 . 09 m847c DOVIR roughSD . 00 5. 42. .12 .09 m847c DOVIR r o u g h s D . 00 5. 4 2 . . 12 .09 m847c D I V I R roughSD . SO 5. 42. .18 .20 m847c D I V I R roughSD .30 5. 4 2 . . 18 . 20 m847c D2V1R roughSD . 60 5. 4 2 . .46 .35 m847c D0V2R roughSD .00 10. 42 . . 16 . 09 m847c D0V2R roughSD . 00 10. 42. . 16 .09 m847c D0V2R roughSD . 00 10. 42 . . 16 .09 m847c D1V2R roughSD .30 10. 42. . 18 . 17 m847c D1V2R roughSD . SO 10. 42. .18 . 17 m847c D1V2R roughSD .30 10. 42. . 18 . 17 m847c D2V2R roughSD . 60 10. 42, .40 .28 m847c D2V2R roughSD . 60 1 0 . 42. .40 .28 m847c D2V2R roughSD . 60 10. 4 2 . .40 . 28 m847c D0V4R roughSD .00 20. 42 . .18 . 09 m847c D1V4R roughSD . 30 20. 4 2 . .18 .14 m847c D2V4R roughSD . 60 20. 4 2 . .28 .21 q239 t o i r o u g h s D . 00 8. 4 3 . . 05 .09 q239 t 0 2 rough2D . 00 8. 43 . .05 . 09 q239 t 0 3 rough2D . 00 8. 43 . . 05 , . 09 q239 t 0 4 rough2D .00 8. • .40. , 06 .09

r e p o r t s e r i e s c o n d i t i o n D/h, L/h, C ''nr q239 t 0 5 rough2D .00 8. 40. . 06 .09 q239 t 0 6 rough2D . 00 8. 40. .06 .09 q239 t 0 7 rovigh2D .34 4. 40. .21 . 22 q239 t 0 8 rough2D .34 4. 40. .21 .22 q239 t 0 9 rough2D . 34 2. 40. .21 .25 q239 t i o rough2D .34 1. 40. .21 .27 q239 t i l rough2D .34 4. .40. .21 .22 q239 t l 2 rough2D . 35 4. 43 . . 16 .22 q239 t l 3 rough2D ,35 4. 43 . .16 .22 q239 t l 4 rough2D .35 4. 4 3 . . 16 .22 q239 t l 5 rough2D .35 4. 43. • 16 .22 q239 t l 6 rough2D .35 4. 4 3 . . 16 .22 q239 t l 7 rough2D .35 4. 43 . . 16 .22 q239 t l 8 rough2D . 35 4. 4 3 . .16 .22 q239 t l 9 rough2D . 00 9. 40. .05 .09