A hydraulic and morphological criterion f o r upstream slopes in local-scour holes
Report W - D W W - 9 3 - 2 5 5
G . J . C M . Hoffmans
Road and Hydraulic Engineering Division P.O. Box 5 0 4 4
2 6 0 0 GA Delft
CONTENTS
1 I n t r o d u c t i o n
2 T i m e phases of t h e scour process 2.1 General 2.2 Initial phase 2.3 Development phase 2.4 Stabilization phase 2.5 Equilibrium phase 3 Transport mechanism 3.1 General
3.2 Bed boundary condition 3.3 Mass-balance equation 3.4 Bed load
3.5 Shear stresses 4 Upstream scour slopes
4.1 General
4.2 Earlier investigations
4.3 Hydraulic and morphological stability criterion 4.4 U n d e r m i n i n g 5 Verification w i t h p r o t o t y p e data 5.1 General 5.2 Hydraulic conditions 5.3 Discussion 6 Conclusions
Appendix A Stability criterion
A p p e n d i x B Effective bed shear-stress Appendix C Undermining parameters
A p p e n d i x D Results of prototype experiments Brouwersdam
References
(
i
1 I n t r o d u c t i o n
Scour is a natural p h e n o m e n o n caused by t h e f l o w o f water in rivers and streams. Scour occurs naturally as part of t h e morphological changes o f rivers and as result of structures m a n - m a d e . Several types of scour can be distinghuished.
Experience has shown t h a t due t o sand or f l o w slides or micro instabilities at t h e end of the bed protection, the scour process can progressively damage t h e bed protection, leading eventually t o t h e failure o f the hydraulic structure f o r w h i c h t h e bed protection was meant. The length of the bed protection depends on the permissible a m o u n t o f scour ( m a x i m u m scour d e p t h and the upstream scour slope) and t h e geotechnical structure of the soil involved (Pilarczyk, 1984).
In t h e scope of the Dutch Delta works, a systematical investigation o f t i m e scale f o r t w o and three-dimensional local scour in loose sediments was conducted by Delft
Hydraulics and t h e Department of Public W o r k s (Rijkswaterstaat). From model experiments on different scale and bed materials, relations were derived in order t o predict the steepness of the upstream scour slope (De Graauw and Pilarczyk, 1981 and De Graauw, 1983).
Since t h e predictability of these relations is poor, especially f o r p r o t o t y p e conditions, a theoretical study concerning upstream scour slopes is carried o u t . In t h e present study a stability criterion is deduced which is based on the mass-balance equation and a stochastical bed-load predictor (Van Rijn, 1985) and fitted using approximately 2 5 0 clear-water scour experiments. In addition a empirical relation f o r u n d e r m i n i n g is discussed.
The stability criterion f o r upstream scour slopes and the model relation f o r u n d e r m i n i n g are verified applying some prototype experiments (Delft Hydraulics, 1979 and De Graauw & Pilarczyk, 1981).
2 T i m e phases of the scour process 2.1 General
To give some insight into the scour process behind hydraulic structures, t h e f l o w
pattern and the sediment transport along the upstream slope o f the scour hole are described f o r several phases in the scour process. Based on experiments at scale model w i t h small Froude numbers (Breusers, 1966 and Dietz, 1969) Zanke (1978) distin-guished f o u r phases in the evolution of a scour hole: an initial phase, a d e v e l o p m e n t phase, a stabilization phase and an equilibrium phase.
2.2 Initial phase
In the initial phase the f l o w in the scour hole is nearly u n i f o r m in the longitudinal direction. This phase of the scour process can be characterized as the phase in w h i c h the erosion capacity is most severe compared t o the erosion capacity in t h e remaining phases o f the scour process.
Observations showed (e.g. Breusers, 1966) t h a t at the beginning o f the scour hole development a certain a m o u n t of bed material near the upstream scour slope goes into suspension. M o s t of these suspended particles are convected w i t h t h e main f l o w and remain in suspension due t o the internal balance b e t w e e n t h e upwards diffusive flux and the convective (downwards) flux, figure 1. Some of these particles wijl deposit and
will g o into suspension again due t o t h e large bursts o f t h e t u r b u l e n t f l o w near t h e bed, .and some particles w i t h a j u m p height smaller t h a n a defined saltation o r reference height are transported as bed load.
i^J^"^ reference height initial phase development pha.se stabilization phase Figure 1 equilibrimn phase S c h e m a t i z a t i o n o f s e d i m e n t t r a n s -p o r t a l o n g u -p s t r e a m scour slo-pe 2.3 Development phase In t h e d e v e l o p m e n t phase t h e f o r m s o f t h e scour hole are similar. A t this time t h e ratio o f the m a x i m u m scour d e p t h and t h e distance f r o m the end o f t h e bed t o t h e point where t h e scour hole is at m a x i m u m is, more or less, constant.
A f t e r t h e transition o f t h e bed protection t o t h e erodible b e d , t h e separated shear layer appears t o be much like an ordinary plane mixing layer. The centre o f t h e mixing layer is at t h e very beginning slightly curved caused by t h e influence o f t h e bed. The curvature increases w i t h t h e distance f r o m t h e end o f t h e bed protection, especially near t h e reat-t a c h m e n reat-t poinreat-t. A recirculareat-tion zone develops w i t h a f l o w direction opposite t o t h e mean f l o w direction, figure 2.
Measurements o f H o f f m a n s (1990) showed t h a t the upper part o f t h e upstream scour slope is in equilibrium, whereas the lower part is still in m o t i o n . In t h e recirculation zone the
suspended load close t o t h e bed is decreased significantly compared t o t h e conditions in t h e initial phase. This can mainly be ascribed t o t h e lowering o f t h e bed f l o w veloci-ties in t i m e , despite t h e increase o f t h e turbulence energy.
T h o u g h bed particles are picked up and convected by t h e f l o w , t h e time-averaged value o f t h e sediment transport in t h e upper part o f t h e upstream scour slope is negligibly small, since t h e contribution o f t h e sediment transport due t o t h e instan-taneous f l o w velocities in t h e main direction equals approximately t h e transport due t o t h e instantaneous f l o w velocities against t h e main direction.
2.4 Stabilization phase
In the stabilization phase t h e develop-m e n t o f t h e develop-m a x i develop-m u develop-m scour depth increases degressively. The erosion ca-pacity in t h e deepest part o f the scour hole is o f no importance compared t o the erosion capacity downstream f r o m the point o f reattachment, so t h a t t h e dimensions o f t h e scour hole increase , more in t h e streamwise direction than in the vertical direction.
new wall-boundnry layer
reattachment point
Figure 2 F l o w regions
The more t h e scour process continues, the more t h e f l o w velocities above t h e lower part of t h e upstream scour slope decrease. In t h e stabilization phase t h e equilibnum situation f o r b o t h the upstream scour slope and the m a x i m u m scour d e p t h is almost achieved.
2.5 Equilibrium phase , , , . t The equilibrium phase can be defined as the phase in w h i c h t h e dimensions o t the scour hole do no longer change significantly.
Generally in this phase o f the scour process the bed particles at the upstream scour slope are only rolling and sliding beneath a saltation height.
3 Transport mechanism
3.1 General . The transport of sediment by a f l o w can be divided into t w o categories: t h e bed-load
(transport) and the suspended-load (transport). Usually bed load is defined as the transport o f particles of bed material which are sliding and rolling immediately above the bed If under given f l o w conditions sediment particles are j u m p i n g above a defined saltation height, then these particles are assumed to be transported as suspended load^ The problem of defining critical f l o w conditions associated w i t h initial instability and entrainment of bed sediment particles is of f u n d a m e n t a l importance t o predict the sediment transport mechanism.
The first k n o w n treatise on initial bed grain instability using t h e concepts o f Prandtl and V o n Karman on boundary f l o w was produced by Shields ( 1 9 3 6 ) , w h o described the
problem using the f o l l o w i n g basic parameters: the fluid density, t h e sediment density, the kinematic viscosity, the mean particle size and the bed shear-stress.
W h e n the f l o w velocity over a bed of non-cohesive material has increased sufficiently, individual grains begin t o move in an intermittent and random f a s h i o n . T h e initial bed instability results f r o m the interaction between t w o statistically distributed random variables. A t first every grain on, the bed surface can be assumed t o be potentially susceptible t o an instantaneous critical.bed shear-stress. The equilibrium o f the grain becomes unstable if the instantaneous bed shear-stress exceeds t h e critical one. Due t o t h e random shape, w e i g h t and placement of the individual grains, these critical shear stresses will have a probability distribution, which defines the initial m o v e m e n t characteristics of the bed material. The other random variables in the process of initial bed instability result f r o m the variations in the action of t h e instantaneous bed shear-stresses generated by the f l o w . The probability t h a t the instantaneous bed shear-stress is larger t h a n a characteristic critical one is a measure for the transport of sediment. 3.2 Bed boundary condition
O n e of the most f u n d a m e n t a l problems of sediment transport is t h e process t h a t controls t h e exchange of sediment particles between the bed load and suspended load layer (boundary condition). A particle leaving the bed starts its trajectory by f o l l o w i n g a saltation. A particle can enter the suspension layer w h e n it is lifted t o a level at which the u p w a r d turbulence-induced forces are comparable t o or higher t h a n the
sub-merged particle w e i g h t , r U J l J
A l t h o u g h in a f l o w w i t h solid transport no sharp distinction can be made f o r bed load 3
and suspended load, a suspension layer and a bed layer can be defined f o r reasons o f simplicity. In t h e upper layer sediment particles are conveyed in suspension due t o the t u r b u l e n t eddy-diffusivity, which is prevailing over t h e vertical m o t i o n . In t h e bed layer the bed load depends on t h e local composition o f t h e sediment and t h e local characteristics o f the f l o w .
3.3 Mass-balance equation
The continuity equation per unit w i d t h f o r t h e total sediment transport s (i.e. the sum o f bed load and suspended load) in the scour hole can be expressed as:
. ^ . ^ = 0 (1) dt dx
in which is t h e bed level, t is time and x is the longitudinal coordinate. In the equilibrium phase of the scour process w h e n t h e bed in t h e recirculation zone is in equilibrium, i.e., w h e n the erosion capacity is nil, the sediment transport equals approximately the upstream sediment supply.
3.4 Bed load
Several expressions, more or less empirical, have been suggested d u r i n g t h e last century t o c o m p u t e bed load as a f u n c t i o n of f l o w characteristics and particle diameter. Here a bed load f o r m u l a introduced by Van Rijn (1985) and modified by H o f f m a n s (1992) is used to predict upstream scour slopes in the equilibrium phase o f t h e scour process.
According t o Van Rijn ( 1 9 8 4 ) , bed load is c o m p u t e d as the p r o d u c t o f t h e saltation height, the particle velocity and the bed-load concentration. The equations of m o t i o n f o r a solitary particle, as given by W h i t e and Schuiz ( 1 9 7 7 ) , are solved numerically t o determine the saltation height and particle velocity. Simple relations f o r t h e saltation height are proposed and calibrated by b o t h a mathematical model and a large number o f f l u m e experiments.
In the equilibrium phase of the scour process the (time-averaged) bed load at the upstream scour slope is negligibly small, since the bed shear-stress is marginal in comparison w i t h t h e critical one. However, due t o sweeps and ejections (Lu and W i l l m a r t h , 1 9 7 3 ) , which occur d u r i n g burst, bed particles are lifted up. Due t o the acceleration of gravity the bed particles are deposited, so these particles are h o p p i n g randomly at the upstream scour slope w i t h o u t m o v i n g significantly w i t h t h e main f l o w . 3.5 Shear stresses
The standard deviation of the instantaneous bed shear-stress t-^ determines, t o g e t h e r w i t h the effective (mean) bed shear-stress , t h e bed load t o a large extent. In addition, the sediment transport is also influenced by the sediment charac-teristics such as the density o f the material, the sediment-size distribution, t h e shape o f the particle and the porosity o f the (non-cohesive) material.
Bed shear-stress
Generally the (mean) bed shear-stress or the bed shear-velocity u decreases enormously f r o m the, more or less, u n i f o r m f l o w at the fixed bed t o t h e erodible bed (figure 3 ) . Downstream f r o m the separation point the absolute value o f t h e bed
stress at t h e upstream slope Is relatively small compared t o t h e bed shear-stress in t h e n e w wall-boundary layer. In t h e recirculation zone t h e bed shear-stress is directed against t h e main f l o w , and in t h e n e w wallboundary layer d o w n -stream f r o m t h e point o f reattachment the bed shear-stress increases gradual-
ly-Characteristic critical bed shear-stresses
Applying a classical approach o f sedi-ment mechanics, i.e. t h e sedisedi-ment par-ticles are only m o v i n g and rolling along the bed surface (no suspension), t h e f o l l o w i n g relations can be given f o r a sediment particle resting on a t w o d i -mensional slope, figure 4 (downslope):
sin(0 - e) sin0 (2) and (upslope): sin(0 B) (m/s) O.lOOr surface V* 0.075 , 0.050 0.025 0.000 -0.025 (Meijer el al, rKXKKKK}
\ upst cam scoi r slope
/ / / / / / / 1 separali / 30 point / /
eattachn ant point
(ni) 0.00 -0.25 -0.50 •-0.75 -1.00 -1.25 10
— lay-out scour bole 1992)
— ^ X ( m )
bed shear velocity
sin0
(3) Figure 3 Betd s h e a r - v e l o c i t y as a f u n c t i o n o f t h e l o n g i t u d i n a l distance
in which r., and are characteristic critical bed shear-jtresses, ( = I . S r ^ is the characteristic critical bed shear-stress f o r u n i f o r m f l o w , (= pu\) is t h e (mean) critical bed shear-stress according t o Shields, p is t h e fluid density, u^^ is t h e critical bed shear-velocity, 0 is t h e angle o f repose and Ö is t h e slope angle, i.e. t h e angle between t h e upstream scour slope and t h e horizontal.
Instantaneous bed shear-stress
The influence o f turbulence on bed load has been investigated by several researchers in the past. As given by Kalinske (1947) and Einstein (1950) t h e instantaneous f l o w velocity varies according t o a Gaussian distribution. The idea o f Kalinske was picked up by Van Rijn (1986) w h o postulated an instantaneous transport parameter, w h i c h is an expression f o r t h e fluctuating mobility o f t h e particles in terms o f t h e stage o f t h e fluctuating m o v e m e n t relative t o t h e critical stage f o r initiation o f m o t i o n .
For reasons o f simplicity Van Rijn assumed t h a t t h e instantaneous bed shear-stress TQ is normally distributed. However, this distribution can be questioned, since measure-ments o f Lu and W i l l m a r t h (1973) show t h a t t h e influence o f sweeps a n d ejections, whose contribution is larger than t h e contribution o f t h e inward a n d o u t w a r d interaction, is n o t included in t h e Gaussian distribution. The p h e n o m e n a sweeps, w h i c h are directed t o t h e bed, and ejections, which are m o v i n g away f r o m t h e b e d , contribute most t o t h e t u r b u l e n t shear stresses. M o r e details concerning sweeps and ejections can be f o u n d in Lu and W i l l m a r t h (1973) and H o f f m a n s ( 1 9 9 2 ) .
4 Upstream scour slopes 4.1 General
The upstream slope in a local-scour hole is defined here as t h e slope between t h e coordinates
X = h^/30 and x = hJ2 . Gen-erally this slope reaches an equi-librium and is less steep t h a n t h e t a n g e n t at t h e transition f r o m t h e fixed t o t h e erodible bed. Based on theoretical grounds a hydraulic and morphological stability criterion is derived f o r predicting t h e steepness of upstream scour slopes. In addi-tion a simple criterion f o r under-mining is f o u n d . These criteria are calibrated using a large n u m -ber o f f l u m e experiments, in which t h e material properties and t h e hydraulic and g e o m e t r i -cal conditions were varied.
iéy-Ws^-^o] measure for bed load due to-I Tol r - : < ; • : ! niea.sure for bed load due to | T,||
Figure 4 Scliematization o f p r o b a b i l i t y d i s t r i b u t i o n o f f l o w a n d material characteristics in n o n - u n i f o r m f l o w 4.2 f a r / / e r investigations
Based o n a t h o r o u g h investigation of scour downstream f r o m an apron Dietz, ( 1 9 6 9 , 1973) reported t h a t the upstream scour slope depends o n a turbulence level, t h e f o r m of the f l o w velocity profile and a dimensionless parameter given by:
6 = _ ( ^ 0 - Ü,)d, (4)
w
0 (initial d e p t h - a v e r a g e d
w (fall velocity o f t h e
which is related t o the f l o w a n d sediment characteristics, U. f l o w velocity), (critical depth-averaged f l o w velocity), bed material) and (sedimentological diameter).
For nearly identical hydraulic structures, Dietz f o u n d a relation between t h e slope angle and 6 . However, this relation is not unambiguous, w h e n t h e relative turbulence intensity is n o t constant b u t varies due t o different geometrical conditions.
As given by Breusers et al. (1977) many parameters can be distinguished w h i c h may influence t h e scouring p h e n o m e n o n . S o m e w h a t arbitrary. Kolkman (1980) and Buchko (1986) combined some of these parameters, resulting in:
cotan e = f 50 . ^^50 ^ 0 f ^1 —.— r (5)
accel-eration of gravity, v is the kinematic viscosity and h, is t h e initial f l o w d e p t h . The idea of this concept was t h a t each parameter could be varied independently f r o m the other ones Kolkman argued t h a t f o r fine sand t h e influence o f t h e kinematic viscosity (term 1) could also be expressed differently by introducing t h e fall velocity, so that:
cotan Ö = f (6)
T h o u g h t h e dimensionless parameters are important, t h e theoretical consideration can be questioned, since the influence of turbulence in t h e recirculation zone is not taken into account. , „ , , ^ i Based on t h e research activities of Dietz, Kolkman and the socalled systematical scour research' data (Delft Hydraulics, 1 9 7 2 , 1 9 7 9 ) , the slope angle was represented by (De Graauw and Pilarczyk, 1981):
w cotan Ö = 5 . 5 — - 1/3 V -2.5 + 0.75 • -2.5 + a - 1.32 (7)
The value o f the turbulence coefficient a can be obtained f r o m previous model investigations or f r o m scale models f o r complex hydraulic constructions.
Several expressions for a have been deduced f r o m t h e tests in the systematical series. These relations, which include t h e influence of the roughness of t h e bed protection and the effects o f b o t h t w o and three-dimensional f l o w , are summanzed in a scour manual (Van der W a l et al., 1991).
Later, after evaluating the enormous a m o u n t of data of the scour expenments, De Graauw (1983) f o u n d t h a t t h e upstream scour slope is a f u n c t i o n of the turbulence coefficient only:
1 (8) cotan Ö = 2.3 +
a - 1.3
4 3 Hydraulic and morphological stability criterion
The stability of the upstream scour slope is the result of the interaction between fluid m o t i o n and material properties. The equilibrium situation of upstream scour slopes f o r non-cohesive material is achieved here by equalization o f bed load due t o t h e instan-taneous bed shear-stresses sloping d o w n w a r d and bed load due t o t h e instaninstan-taneous bed shear-stresses sloping u p w a r d .
Assuming a Gaussian (symmetrical) distribution f o r , and if only clear-water scour is considered, thus no upstream sediment transport is present, the f o l l o w i n g relation can be denved (appendix A):
- 2m^o
0 (9)
where p is an efficiency factor where the influence of the bed roughness Is taken 7
into account (appendix B). Generally the bed o f t h e upstream scour slope is hydrauli-cally s m o o t h f o r w h i c h applies p=^1 .
In order t o include t h e influence o f sweeps and ejections it seems reasonable t o assume t h a t t h e probability distribution o f r^, has long tails f o r extreme values f o r r . The skewness o f the probability distribution f o r could be expressed by (figure°4):
-1 i + f = 0 (10)
Consequently the slope angle can be w r i t t e n as (appendix B):
e = arcsin/f + Vif) w i t h f = ^ 2 . 9 * 1 0 ^ - ^ ( ' ' I )
in which is a measure f o r t h e skewness.
T h o u g h the effective bed shear-stress p7o determines the slope angle, its influence on the steepness o f the upstream scour slope compared t o the influence o f sweeps and ejections (modelled by f ) is relatively small, especially f o r experiments at scale model w i t h small Reynolds numbers (appendix B).
W h e n e exceeds a critical value, i.e. f o r arcsin/f + y2f\>0' w i t h 0' is the angle o f internal friction, micro instabilities could occur, u n J e r m i n i n g the end o f the bed protection. Sand and f l o w slides of the soil under t h e bed protection may even be possible, however, these p h e n o m e n a are strongly dependent on the soil properties, e.g. the contraction and t h e elastic compressibility of loose sand (De G r o o t et al., 1 9 9 2 ) . Since little information was available regarding the material properties o f the scour experiments (e.g. the porosity, the angle of repose, angle o f internal friction were not measured), the aforementioned criterion is not extensively examined.
A simple criterion f o r gradual u n d e r m i n i n g is discussed in section 4 . 4 which i less, based on trial and error.
Closure problem
To optimize the f u n c t i o n f in equation 10, the hydraulic para-meters have t o be k n o w n .
Since the instantaneous bed shear-stress is not unambiguously defined ( H o f f m a n s , 1 9 9 2 ) , it is assumed f o r reasons o f simplicity t h a t the parameter f is closely related t o the relative^turbulence intensity r^ just upstream f r o m the scour hole and the effective bed roughness o f the bed protection.
The material properties are characterised here by the mean bed particle d^^ , t h e particle
.0 ^ D O. B 1
/
t / • 1 -• I t * + 1 * M *. / / / / / + t ** g : + / / / / 2D-rough + 2D-smoolh =4: -3D-fough Q 3D-snnooth 3 4 Meosurements Figure 5 Calculated a n d m e a s u r e d c o t a n 6 8diameter d,, (for which 9 0 % of t h e mixture is smaller t h a n d^ ) and t h e relative density, so the influence of the distnbution of t h e m i x t u r e ' a n d t h e density of t h e bed material (polystyrene, bakelite, sand) is taken into account.
M o r e t h a n 2 5 0 experiments (Delft Hydraulics, 1972 , 1 9 7 9 and Buchko, 1986) were used t o f i n d the best compromise between the measured and calculated angle o f t h e slope, resulting in (figure 5):
f^ = (0.22 - 1 X . K c w i t h ! - ^^^^
in which r is the measured relative turbulence intensity, 6' is the displacement thickness °(: is the Chézy coefficient and = 4 0 m V s . For hydraulically-rough condition's, i.e. f o r C < 40m^Vs regarding t h e fixed bed before t h e scour hole, the roughness function measures fc = '^ • . L. ^ j In these laboratory experiments not only the hydraulic conditions (discharge Q and initial f l o w depth ) were varied but also the geometrical conditions (length o f t h e bed protection L , height of the sill D and the bed roughness). 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.
To determine the predictability of the aforementioned relations f o r upstream scour slopes, t h e discrepancy ratio r (i.e. the ratio between t h e measured and calculated upstream scour slope) is computed using experiments of Delft Hydraulics ( 1 9 7 2 , 1979) and Buchko (1986). The measured upstream scour slopes were obtained directly f r o m the measurements, whereas the calculated ones (equations 7 and 8) were determined
using a 'measured' turbulence coefficient (e.g. Hoffmans, 1993).
This analysis, in which a distinction has been made between t w o and three-dimensional f l o w (table 1), shows t h a t compared to the other empirical relations, upstream scour slopes can be calculated more accurately w i t h the model relation given in this study.
0 . 9 0 < r < 1.11 0 . 8 0 < r < 1.25 0 . 6 7 < r < 1.50 references 2 D ' 3 D 2D . 3 D 2 D 3 D n - 2 0 1 n = 6 5 (201) (.65) ( 2 0 1 ) (65) De G r a a u w a n d Pilarczyk ( 1 9 8 1 ) 1 5 % 6 % 34%. 5 1 % . 5 8 % 6 5 % De G r a a u w ( 1 9 8 3 ) 5 0 % 2 5 % 8 5 % 4 0 % . 9 7 % . 7 8 % H o f f m a n s ( 1 9 9 3 ) 5 9 % 3 7 % 8 5 % 1 7 1 % 9 5 % . . . _ i _ 7 8 % Table 1 Comparison of c o m p u t e d and measured upstream scour slope 4.4 Undermining
The p h e n o m e n o n of undermining can be defined as the erosion w h i c h occurs at t h e end of t h e bed protection. In addition t o the gradual u n d e r m i n i n g due t o t h e composi-tion of t h e bed, a sudden undermining may occur (sand or normal slide), w h e n the slope angle of the bed is larger than the angle of internal friction.
According t o Delft Hydraulics (1979) and Blazejewski ( 1 9 9 1 ) , t h e dangerous u n d e r m i n -ing of t h e vertical edge of the bed protection results f r o m the higher turbulence energy
and the larger erosion capacity o f the f l o w in t h e recirculation zone due t o t h e higher turbulence energy directly upstream f r o m t h e scour hole. Measurements at scale m o d e l w i t h sand as scour material (Delft Hydraulics, 1 9 7 9 and Buchko, 1986) have s h o w n t h a t the end o f t h e bed protection is undermined if a dimensionless parameter a defined as:
O.m
w i t h a = 1.5 + AAr^f^ (13)
exceeds a critical value
For t w o - d i m e n s i o n a l scour, m a x i m u m _ depth-averaged the f l o w velocity whereas_ scour U U o.m f o r equals a b o u t U^, three-dimensional depends strongly on the geometry upstream f r o m the scour hole, f o r example by an obstacle or an a b u t m e n t in the f l o w .
In this study the characteristic u n -d e r m i n i n g z^it^) is -define-d as t h e u n d e r m i n i n g which w o u l d occur w h e n the m a x i m u m scour d e p t h equals approximately the initial f l o w d e p t h . The dimension-less u n d e r m i n i n g z^{t^)/h^ as a function o f is shown in figure 6. T h o u g h a distinction has been made between flexible and fixed beds, the type o f bed protection seems t o be o f secondary i m p o r t -ance compared t o a . . 0.35 0.30 -J'l) 0.25 0.20 0.15 0.10 0.05 flexible bed § fixed bed 4> flexible bed 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 6 U n d e r m i n i n g as a f u n c t i o n o f 5 Verification w i t h p r o t o t y p e data 5.1 General
W i t h i n the scope o f research activities w i t h respect t o scour behind t h e storm surge barrier and c o m p a r t m e n t dams in the Eastern Scheldt, some experiments on p r o t o t y p e scale were carried o u t (De Graauw and Pilarczyk, 1981). For this purpose t h e sluice in the Brouwersdam was chosen which was built t o refresh t h e brackish w a t e r in the Grevelingen lake f o r environmental reasons. The experiments were executed t o study t h e influence of clay layers t o scour and to verify scour relations obtained f r o m scale models.
5.2 Hydraulic conditions
The discharges and f l o w velocities regarding the t w o experiments were almost similar, 10
whereas the soil characteristics were different.
The discharges, the f l o w levels and the bed configuration were measured frequently. Also some f l o w velocity and concentration measurements in t h e centre o f t h e sluice were carried out. During the experiment t h e sea w a t e r was discharged into t h e lake during t h e f l o o d and o u t d u r i n g the ebb. The last had no influence on t h e develop-ment o f t h e scour hole, because of the relatively small f l o w velocities above the scour hole d u r i n g the ebb. The suspended load f r o m t h e sea into t h e lake was also neg¬ ligible
The sluice consisted of a sill 5.4 m height w i t h t w o side constrictions equal t o 2.5 m on the left side and 1.5 m on the right side. The f l o w d e p t h was a b o u t 10 m and t h e length o f t h e bed protection f r o m the toe of t h e sill measured about 5 0 m. The effective roughness of the bed protection is estimated t o be 0.4 m. The other dimensions o f the sluice are presented in figure 7. ^ ^ ^ j t u The soil characteristics w i t h respect t o experiment A were measured beforehand. The diameter of t h e bed material varied w i t h the depth f r o m 0.2 t o 0.3 m m . Some thin clay lenses on different levels were present, especially in the upper soil-layer between 2
Noflh S»o H.H.W, g * V t G r e v e l i n g e n L o k * E L E V A T I O N Si. Figure 7 P r o t o t y p e situation ( B r o u w e r s d a m )
and 4 meters below the original bed. The thickest clay layer of a b o u t 0.2 m was situated at a b o u t 3.5 m below the bed. The other clay lenses were mostly in t h e range of 0.01 t o 0.02 m of about 0.2 m.
The developed scour hole was refilled w i t h loosely-packed material. The bed material regarding experiment B consisted of fine sand w i t h a mean particle diameter o f a b o u t 0.260 m m . The particle diameter for which 9 0 % of the mixture is smaller t h a n d^^
measured 0.29 m m . . As a result of ebb and f l o o d not only the f l o w velocity varies in t i m e b u t also the
sediment transport. To simulate the scour process a characteristic depth-averaged f l o w 11
water surface Z (m) experiment A experiment B S after 140i) O after 265h V af(er 844h A after 140h B after 486li * after 8501i
Figure 8 Bed profiles o f scour holes ( B r o u w e r s d a m ) (111)
velocity is introduced which is d e -fined as t h e depth-averaged f l o w velocity, w h i c h w o u l d occur if t h e sediment transport does n o t vary in time.
5.3 Discussion
Figure 8 shows some measured bed profiles o f t h e . considered pro-t o pro-t y p e experimenpro-ts apro-t differenpro-t moments, whereas t h e gradual u n d e r m i n i n g , including a sand slide, is shown in figure 9. The calculations o f b o t h t h e upstream scour slopes (cotanö ==2.) and the u n d e r m i n i n g (zjh^ ^ 0.25) agree w i t h t h e measurements rea-sonably. The computations are obtained w i t h equations 11 and 13. M o r e details of experimental and c o m p u t a t i o n a l results can be f o u n d in appendix D.
Due t o t h e tidal influence the f l o w velocities vary in time. In experi-m e n t A and B t h e experi-m a x i experi-m u experi-m f l o w velocities averaged over about 140 tides measured by approximation 1.2 m/s ( H o f f m a n s , 1992). W i t h this assumption and using t h e stability criterion (equation 11) it f o l -lows t h a t c o t a n ö = 1 . 8 . Gen-erally t h e angle o f internal friction f o r sand lies in t h e range of 3 0 t o 4 0 degrees and depends on the porosity, t h e particle diameter and the distribution of t h e mixture. M i c r o instabilities can be predicted w h e n t h e f l o w velocities are larger than 1.3 m/s ( 0 ' = 3 0 ° ) or 1.9 m/s f o r 0' = 4 0 ° . D u r i n g t h e ex-periments f l o w velocities were measured varying f r o m 1.5 t o 2.0 m/s. Since micro instabilities and a sand slide after approximately 4 5 0
hours (nett-scour time) were observed, t h e stability criterion seems t o be feasible f o r practical engineering. W h e n the sub-soil consists of clay and sand layers, t h e results obtained f r o m equation 11 must be interpreted carefully, because t h e influence o f t h e cohesion o f t h e sub-soil is n o t taken into account.
p r a .snnd /
/
/ 0 -—i/
J
— ]/
—©=I
1 xperinw xperim sill A ent B 0 100 200 .100 4 0 0 .500 600 700 800 9 0 0 • nett scour-tinic (liours)Figure 9 U n d e r m i n i n g as a f u n c t i o n o f t i m e
6 Conclusions
A hydraulic and morphological stability criterion f o r upstream scour slopes has been denved which is calibrated using a large number of f l u m e experiments and verified f o r t w o prototype experiments. Despite the simplifications made in t h e closure problem, this study shows a w a y t o calculate the steepness of upstream slopes in t h e equilibrium phase of the scour process f o r non-cohesive material. M o r e o v e r , t h e criterion is able t o predict micro instabilities.
j
Appendix A Stability criterion
A hydraulic and morphological stability criterion f o r upstream scour slopes in t h e equilibrium phase of t h e scour process is derived applying a stochastical bed-load predictor (Van Rijn, 1 9 8 6 ) . The stability criterion regards clear-water scour only. The equilibrium situation o f upstream scour slopes f o r non-cohesive material is achieved by
equalization o f bed load due t o t h e instantaneous bed shear-stresses sloping d o w n w a r d and bed load due t o sloping u p w a r d . Thus t h e time-averaged bed load
along t h e slope Is assumed t o be nil.
As given by Van Rijn ( 1 9 8 6 ) , bed load can be related t o a stochastical e n t r a i n m e n t parameter f o r b o t h u n i f o r m and n o n - u n i f o r m f l o w . The contribution, o f E^ caused by sloping d o w n w a r d reads:
(Al)
and the contribution due t o against the main f l o w direction is:
(A,)
In equations A^ and A j , expresses the mobility of bed particles as a f u n c t i o n o f
, P is the probability distribution o f and k 2.0) is a constant.
Assuming P t o be Gaussian distributed, a further elaboration of equations (A^) and (Aj) yields ( H o f f m a n s , 1992): 1 Ye-"'' df (A3) y -t (A4)
where f i = (M7"o - ^)/^^o ^"d f j = (-pro + rj)/^^ are dimensionless parameters,
7-^ , and 7-^ are characteristic.critical is t h e standard deviation o f t h e instantaneous p is an efficiency facto> ( a p p e n d i x ' s ) , 7 - , " , \ and are characteristic.critical bed shear-stresses (section 3) and a^^
bed shear-stress.
W i t h t h e assumption t h a t E^^ E^^ , the f o l l o w i n g stability criterion f o r upstream scour slopes In the equilibrium phase o f the scour process is o b t a i n e d :
(A5) - 2M7o = 0
Appendix B Effective bed shear-stress
In appendix B a discussion is given concerning t h e influence of t h e effective bed shear-stress pro on upstream scour-slopes in t h e equilibrium phase o f t h e scour process. The influence of pro is compared t o the influence of sweeps and ejections.
Assuming a Gaussian distribution f o r the instantaneous bed shear-stress ro t h e f o l l o w i n g relation can be deduced (appendix A ) :
! l l \ l 3 } ^ = 0 (Bi)
in which r , r^ and r, (= a j ) are characteristic critical bed shear-stresses
(section 3.5)! r is the critical bed shear-stress and a, = 1.5 is a constant (De Ruiter, 1 9 8 2 , 1 9 8 3 ) . The efficiency factor p reflects t h e roughness of the bed and can be approximated by ( H o f f m a n s , 1992):
^ , 12/7 In
in which h is the f l o w d e p t h , is the effective bed roughness, w h i c h is by approximation equal t o the mean dune height, k'^ = 3^90 and d^^ is t h e particle diameter f o r which 9 0 % of the mixture is smaller than d^^ .
The influence of the phenomena sweeps and ejections in the probability distribution for ro
could be described by:
r, . r , - 2 p 7 o ^ ^ (B3)
in which f is a measure f o r the skewness of the probability distribution.
Expressing ' and in terms of 6 (slope angle) and 0 (angle o f repose) yields:
sin(0 - 6) sin(0 + 6) 2pro
f =
sin0 sin0 r^ "
^ _ 2 s i n ö c o s 0 _ 2 p r o ^ ^ ^ Q (B^)
sin0 r^ ^
Hence 9 can be given by:
Ö = arcsin(c^f^ + y 2 g ) . w i t h = - j j ^^'^
= 1.0 . Based on the results o f various investigators, there seems t o be dependency o f the 0 - v a l u e on t h e slope angle. Lysne (1969) reported a value o f 0 = 3 8 ° f o r a bed sloping u p w a r d and a value of 0 === 5 0 ° f o r a bed sloping d o w n w a r d . Fernandez-Luque and Van Beek (1976) f o u n d a value o f 0 = 4 7 ° f o r a bed sloping d o w n w a r d .
To proceed w i t h examining the influence o f the effective bed shear-stress on upstream scour slopes some definitions are given. The friction coefficient and t h e critical bed shear-stress r are defined as:
Cf =
V2pu: (Bg)
50 (By)
in which p is the fluid density, is the time-averaged f l o w velocity in the o u t e r f l o w (of course, = U^ and is t h e initial depth-averaged f l o w velocity) 4»^ is the critical mobility-parameter,. A is the relative density, g is the acceleration o f gravity and d^^ is the mean particle diameter.
In t h e equilibrium phase of the scour proces is directed against t h e main f l o w , thus f > 0 . C o m b i n i n g equations Bg, Bg and B^ gives:
f = ^ f ^ 0
o'c^c Age/.
(Bs)
Computations w i t h b o t h the f l o w models DUCT and ODYSSEE have s h o w n t h a t in t h e equilibrium phase o f the scour proces the bed shear-stress at upstream scour slopes is, more or less, constant. For nearly in equilibrium scour holes t h e friction coefficient measures approximately 5.5 * 1 0 " ' (Meijer et al., 1992). However, f o r scour holes where the expansion ratio hjh^ (ratio between the m a x i m u m f l o w d e p t h and t h e initial f l o w depth) is about 1.5 the friction coefficient is significantly lar-ger t h a n 5 . 5 * 1 0 - ^ . Predictions of the f l o w model DUCT as well as laser Doppler measurements show t h a t Cf is related t o the geometry o f the scour hole, table B l .
Vanoni et al. (1967) have noted t h a t the critical mobility-parameter f o r the f u l l y - r o u g h t u r b u l e n t zone, i.e. f o r 4^^ = 0.06 , corresponds t o a l o w but measurable bed load. A t
m 0 slope ^ 0 reference ( m m ) ( m / s ) ( • 1 0 ' ) ( * 1 0 - ' ) 5 0.5 1 V : 2 . 5 H 0 . 7 7 5.5 M e i j e r e t a!., 1 9 9 2 5 0.5 1 V : 2 . 5 H 1.16 5.5 2 6.0 1 V : 2 H 0 . 4 0 31 1 0 H o f f m a n s , 1 9 8 8 3 6.0 1 V : 4 H 0.41 7.5 3.1 1.4 2.5 1 V : 2 H 0 . 5 2 5 9 . 3 6 V a n M i e r l o a n d De 1.3 2.5 1 V : 2 H 0.63 73 32 Ruiter, 1 9 8 8
values of 0.03 and even less, occasional m o v e m e n t o f single grains may occur Assuming t h a t the bed of the upstream scour slope is hydraulicaJly smooth^ j p the parameter f due t o the bed shear-stress is (C, = 5.5 * 10"^ HJ^ = 0.045)
r . 2 / = cU Agd. 50 - 1) (Bo) in which c ( - 2 . 9 * 1 0 ^ ) is a constant. . ' ' . . M o r e t h a n 3 0 0 experiments were used t o calibrate resulting in (section 4 . 3 ) :
= 0.22 + 1:5r^
in which is the relative turbulence intensity at t h e transition of the fixed t o the erodible bed.
Consequently the ratio between f and measures:
2.9 * 10"* Ut (Bii)
f 0.22 + 1.5r^ Lgd,,
Generally the turbulence upstream f r o m the scour hole is dying o u t t o u n i f o r m f l o w conditions, provided the length o f the bed protection is sufficient large, t h a t is approximately 80 t o 100 times the initial f l o w d e p t h .
Figure B l shows combinations of U^ and f o r which applies ^ / ^ ^ = 0 . 1 . The computational results are obtained using sand as bed material A = 1.65 . In addition the relative turbulence intensity at the transition of the fixed t o the erodible bed is assumed to be
ro = 0 . 1 5 _ . For experiments at scale model ( L / o < 1 m / s ) the influence of the bed shear-stress is of no importance compared t o the influence of the large i n s t a n t a n e o u s bed shear-stresses. However, f o r prototype conditions (Brouwersdam) where the sub-soil consists of fine sand and where f l o w velocities larger than 1 m/s occur, prudence has t o be called. Then the influence o f the bed shear-stress can not simply be neglected. 2.5 (m/s) 2.0 1.5 1.0 0.5 fx (fy> OJ fy<o.i san r o = f=0.1 d 0.15
/
400 800 1200 1600 —>-Jso(."an) 2000 Figure B1 C o m b i n a t i o n s o f OQ a n d d^o f o r ^T^^ =i
!
Appendix C Undermining parameters
Tables C l and C2 provide an overview o f t h e most relevant parameters w h i c h can undermine hydraulic constructions. M o r e details concerning t h e hydraulic conditions of the experiments such as the f l o w velocities, the length o f t h e bed protection the roughness of t h e bed etc. can be f o u n d in Delft Hydraulics (1979) and Buchko ( 1 9 8 6 ) . The turbulence coefficient a , the relative turbulence intensity and t h e dimensionless parameter a , are c o m p u t e d according t o f o r m u l a e given in section 4 . The relative undermining z^{Q/h, is obtained by interpolation o f t w o profile measurements o f the P R O V O . In t h e last column of b o t h tables t h e condition of t h e bed protection just upstream f r o m the scour hole is given.
series O ^0 0, c o n d i t i o n t 0 1 1.93 0 . 0 9 1.70 0 f i x e d t 0 2 1.93 0 . 0 9 1.74 0 f i x e d t 0 3 1.93 0 . 0 9 1.62 0 f i x e d t 0 4 1.93 0 . 0 9 1.68 0 f i x e d t 0 5 1.93 0 . 0 9 1.76 0 f i x e d t 0 6 1.93 0 . 0 9 1.80 0 f i x e d t 0 7 2.49 0.22 2 . 2 4 0 . 1 0 1 f i x e d t 0 8 2.49 0.22 2.32 0 . 1 2 8 f i x e d t 0 9 2.61 0.25 2 . 3 6 0 . 1 3 2 f i x e d tio 2 . 7 0 0 . 2 7 2.45 0 . 1 2 6 f i x e d til : 2.49 0.22 2 . 2 4 0 . 1 3 7 f i x e d t 1 2 2.57 0:22 2 . 2 6 Ö.087 f i x e d t 1 3 2.57 0.22 2.41 0 . 1 1 9 f i x e d t 1 4 2.57 0 . 2 2 2.41 0 . 1 3 5 f i x e d t 1 5 2.57 0.22 2.37 0 , 0 8 4 f i x e d t 1 6 2.57 0.22 2 . 1 6 0 . 0 5 6 f i x e d t 1 7 2.57 0.22 2.36 0 . 0 7 7 f i x e d tie 2.57 0.22 2.16 0 . 0 6 8 f i x e d t 1 9 1.93 0 . 0 9 1.61 • 1 1 0 f i x e d Table C l Computational and experimental results (Buchko, 1986)
series a ^0 c o n d i t i o n 1 t 0 . 1 2 . 5 0 0 . 2 3 2 . 2 5 0 . 0 7 1 f i x e d 1 t 2 2 . 5 0 0.23 2.21 0 . 0 7 0 flexible 1 t 1 2 . 5 0 0.23 2 . 2 5 0 . 1 2 0 flexible 1 t 3 2 . 5 0 0 . 2 3 2 . 2 9 0 . 1 1 0 flexible 1 t 1 1 2 . 5 0 0 . 2 3 2.33 0 . 1 7 0 flexible 2 t 0 . 2 2.21 0 . 1 6 1.97 0 . 0 1 8 f i x e d 2 t 4 2.21 0 . 1 6 1.97 0 . 0 1 6 flexible 2 t 7 2.21 0 . 1 6 2.01 0 . 0 6 0 flexible 3 t 5 2.32 0 . 1 9 1.99 0 . 0 8 0 flexible 3 t 6 2 . 3 2 0 . 1 9 2 . 1 0 0 . 0 9 7 flexible 4 t 8 2 . 5 0 0.23 2 . 2 0 . 0 . 1 0 0 flexible 4 t 9 2 . 5 0 0.23 2.81 0 . 3 2 0 flexible 5 t 1 2 2.21 0 . 1 6 1.95 0 . 0 2 6 flexible 5 t 1 3 2.21 0 . 1 6 2 . 0 0 0 . 0 4 3 flexible Table C2 Computational and experimental results (Delft Hydraulics, 1979)
Appendix D Results of p r o t o t y p e experiments Brouwersdam
Both experimental and computational results of some p r o t o t y p e experiments near t h e Brouwersdam are given (tables D l and D2). M o r e details regarding the c o m p u t a t i o n of the characteristic discharge, the critical depth-averaged velocity and the relative turbulence intensity (table D2) can be f o u n d in H o f f m a n s ( 1 9 9 2 , 1993).
e x p e r i m e n t a l parameters e x p e r i m e n t A e x p e r i m e n t B initial f l o w d e p t h (scour hole) ( m ) 1 0 . 6 9.6
h e i g h t sill ( m ) 5.4 5.4
length o f t h e bed p r o t e c t i o n (m) 5 0 5 2 effective roughness o f bed p r o t e c t i o n ( m ) 0.4 0.4
averaged discharge ( m V s ) 2 7 1 2 7 0 m a x i m u m discharge (m^/s) 3 8 0 3 8 0 particle d i a m e t e r ( m m ) c/50 = 0.25 c/90 = 0 . 2 9 d,o = 0 . 2 6 = 0 . 2 9
u p s t r e a m scour slope at e n d of test 1 V : 2 . 2 H 1 V : 1 . 1 H u n d e r m i n i n g just before sand slide ( m ) 2.9 u n d e r m i n i n g at e n d of test (m) 2.3 5.0 c o n d i t i o n sub-soil c l a y / s a n d sand
Table D l Experimental results
c o m p u t a t i o n a l parameters e x p e r i m e n t A e x p e r i m e n t B characteristic discharge ( m V s ) 9 . 8 9 9 . 8 9 characteristic d e p t h - a v e r a g e d velocity ( m / s ) 0.93 1.03 critical d e p t h - a v e r a g e d velocity ( m / s ) 0.41 0.41 relative t u r b u l e n c e intensity (-) 0 . 2 8 0 . 2 9 t u r b u l e n c e c o e f f i c i e n t (-) 2.91 2.92 roughness f u n c t i o n (-) 1.13 1.12 p a r a m e t e r 2.47 2.52 p a r a m e t e r 0 . 0 9 0 . 1 0 p a r a m e t e r 1.03 1.04 u p s t r e a m scour slope 1 V : 2 . 1 H 1 V : 2 . 1 H u n d e r m i n i n g at f = ( m ) 2 . 0 ± 1 . 0 1 . 8 ± 1 . 0
References
Breusers, H.N.C., 1966, C o n f o r m i t y and time scale in t w o - d i m e n s i o n a l local scour, Proceedings Symposium on model and p r o t o t y p e c o n f o r m i t y . Hydraulic Research
Laboratory, Poona, p.1-8. j ,. , . , .
Breusers, H.N.C., G. Nicollet and H.W. Shen, 1 9 7 7 , Local scour around cylindrical piers. Journal o f Hydraulic Research, lAHR, Vol.15 No.3.
Blazejewski R, 1 9 9 1 , Prediction of local scour in cohesionless soil d o w n s t r e a m o f outlet works,' Doctoral thesis. Agriculture University Poznan, Poland.,
Buchko, M.F., 1 9 8 6 , Investigation of local scour in cohesionless sediments by using a tunnel model. Report N o . Q 2 3 9 , Delft Hydraulics, Delft.
Delft Hydraulics, 1972, Systematical investigation o f t w o and three-dimensional local scour. Investigation M 6 4 8 / M 8 6 3 / M 8 4 7 (in d u t c h ) . Delft Hydraulics, Delft. Delft Hydraulics, 1 9 7 9 , Prototype scour hole 'Brouwersdam', Investigation M l 5 3 3 Part
1,11,111 and IV (in dutch). Delft Hydraulics, Delft.
Dietz, J.W., 1 9 6 9 , Kolkbildung in feinen oder leichten Sohlmaterialien bei strömendem ' Abfl'uR, Mitteilungen Heft 155, Universitat Fridericiana Karlsruhe.
Dietz, J.W., 1 9 7 3 , Sicherung der FluBsohle unterhalb von W e h r e n und Sperrwerken, Wasserwirtschaft 63, No.3.
Einstein, H.A., 1 9 5 0 , The bed load function f o r sediment transportation in open channel fiows, Technical Bulletin, N o . 1 0 2 6 , US Department o f Agriculture, W a s h i n g t o n
D . C .
Fernandez-Luque, R. and R. van Beek, 1 9 7 6 , Erosion and transport of bed load sediment. Journal of Hydraulic Research, lAHR, V o l . 1 4 , No.2.
De Graauw, A.F.F., and K.W. Pilarczyk, 1 9 8 1 , M o d e l - p r o t o t y p e c o n f o r m i t y of local scour in non-cohesive sediments beneath o v e r f l o w - d a m , 19th lAHR-congress, New Delhi (also Delft Hydraulics, Publication No.242).
De Graauw, A.F.F., 1983, A review of research on scour, Report S562 (in d u t c h ) , Delft Hydraulics, Delft.
De Groot, M . , Den Adel, H., T.P. Stoutjesdijk and K.J. van W e s t e n b r u g g e , 1 9 9 2 , Risk analysis of f l o w slides, 23rd International Conference on Coastal Engineering, Venice.
Hoffmans, G.J.C.M., 1988, Flow model w i t h prescribed eddy viscosity. Report N o . 1 1 - 8 8 , Faculty of Civil Engineering, Hydraulic and Geotechnical Engineering Division, Delft University of Technology, Delft.
Hoffmans, G . J . C M . , 1990, Concentration and f l o w velocity measurements in a local scour hole. Report N o . 4 - 9 0 , Faculty of Civil Engineering, Hydraulic and Geotechnical Engineering Division, Delft University of Technology, Delft.
Hoffmans, G . J . C M . , 1 9 9 2 , Two-dimensional mathematical modelling of local-scour holes. Doctoral thesis. Faculty of Civil Engineering, Hydraulic and Geotechnical Engineering Division, Delft University of Technology, Delft.
Hoffmans, G . J . C M . , 1993, A study concerning the influence of t h e relative turbulence intensity on local-scour holes. Report W - D W W - 9 3 - 2 5 1 , Ministry o f Transport, Public W o r k s and W a t e r m a n a g e m e n t , Road and Hydraulic Engineering Division, Delft.
Kolkman, P.A., 1 9 8 0 , Parameters which may influence the scour process, unpublished notes. Delft Hydraulics, Delft.
Kalinske, A.A., 1 9 4 7 , M o v e m e n t of sediment as bed load in rivers, Transactions, American Geophysical U n i o n , V o l . 2 8 , No.4.
(
Lysne, D.K., 1 9 6 9 , M o v e m e n t of sand in tunnels. Journal of Hydraulic Division, N 0 . H Y 6 , ASCE
1'Lu SS and W . W . W i l l m a r t h , 1 9 7 3 , Measurements of t h e structure of t h e Reynolds 1 ' stress in a t u r b u l e n t boundary layer. Journal of Fluid Mechanics, V o l . 6 0 , Part 3,
P - 4 8 1 - 5 1 1 . . . . . . J n- •
Meijer, D . C , Hoffmans, G.J.C.M. and A.K. O t t a , 1 9 9 2 , Eddy viscosity modelling in 'nearly equilibrium scour holes, Delft Hydraulics, Delft.
Van Mierlo, M . C . L . M . and J . C C De Ruiter, 1 9 8 8 , Turbulence measurements above dunes. Report Q 7 8 9 , Vol.1 and 2 , Delft Hydraulics, Delft.
Pilarczyk, K.W., 1 9 8 4 , Interaction w a t e r m o t i o n and closing elements. The Closure of Ti'dal Basins, p . 3 8 7 - 4 0 5 , Delft University Press, Delft. \ \ , Van Rijn, L . C , 1 9 8 4 , Sediment Transport, Part I: Bed load transport. Journal o f Hydraulic
Engineering, V o l . 1 1 0 , No.10, ASCE . ' . Van Rijn, L . C , 1 9 8 5 , Mathematical models f o r sediment concentration profiies in steady
f l o w , Euromech 192, M u n i c h / N e u b i b e r g , Germany, (also Delft Hydraulics, Publication No.365).
Van Rijn L C , 1 9 8 6 , Mathematical modelling of suspended sediment in n o n - u n i t o r m flows, Journal of Hydraulic Engineering, V o l . 1 1 2 , No.6, ASCE, (also Delft Hydraulics, Communication No.365).
De Ruiter, J . C C , 1 9 8 2 , The mechanism of sediment transport on bed f o r m s , Euromech 156, Mechanics of Sediment Transport, p . 1 3 7 - 1 4 2 , Istanbul.
De Ruiter, J . C C , 1 9 8 3 , Incipient m o t i o n and pick-up of sediment as f u n c t i o n o f local variables, unpublished notes. Delft Hydraulics, Delft.
Shields A , 1936, A n w e n d u n g der Ahnlichkeitsmechanik und der Turbulenzforschung ' a u f ' d i e Geschiebebewegung, M i t t e i l u n g e n Preussischen Versuchsanstalt f ü r Wasserbau and Schiffbau, Nr.26, Berlin.
Vanoni, V.A. e t a l . , 1 9 6 7 , Closure t o discussion on "Sediment transportation mechanics: Initiation of m o t i o n " , Journal of Hydraulic Division, ASCE.
W h i t e , B.R. and J.C. Schuiz, 1 9 7 7 , M a g n u s e f f e c t in saltation, J o u r n a l of Fluid 'Mechanics, V o l . 8 1 , Part 3, p.497-512.
W a l , M . van der, G. van Driel and H J ; Verheij, 1 9 9 1 , Scour manual. Report Q 6 4 7 , Delft Hydraulics, Delft. .. _ Zanke, U., 1 9 7 8 , Zusammenhange zwischen S t r ö m u n g und Sedimenttransport Teil 2:
' Berechnung des Sedimenttransportes hinter befestigten Sohlenstrecken, Sonderfall zweidimensionaler Kolk, M i t t e i l u n g e n des Franzius-Instituts der TU Hannover, Heft 4 8 ,
I
(
List of symbols C C, coefficient Chézy coefficient friction coefficient d particle diameter L U D height o f sill 3 U sedimentological diameter; d^J^Ag/yr) L-J £ * stochastical entrainment parameter [-J
roughness function t"^ ƒ skewness parameter _ f"]
f dimensionless bed shear-stress; -pVr, [-] ^
g acceleration of gravity f - T ]
/? f l o w d e p t h tL] initial f l o w depth f'-j effective or equivalent bed roughness t U
kl effective bed roughness related t o the grains t U
L length o f bed protection f - ]
n number (of experiments) ["3 P probability distribution ^'^ ^
Q discharge '•!:"''•'
r discrepancy factor t-J
relative turbulence intensity ^'^
s sediment transport per unit w i d t h [L T 3
t time JI| f characteristic time at which the m a x i m u m scour depth equals [TJ
instantaneous transport parameter [ - ] ^
ul bed shear-velocity tLT^]
u*^ critical bed shear-velocity (Shields) [LT'^] (j'-' initial depth-averaged f l o w velocity [LT'^l L/° critical depth-averaged f l o w velocity '•'""'''i-'
. w fall velocity [LT ]
X longitudinal coordinate I^L] z vertical coordinate . z undermining at end of bed protection [ U
a
Y
turbulence coefficient; ^.5 + 4ArJ^ [-] coefficient , _ _ v _
dimensionless parameter; \aU^^ - U^/U^ [-] ^ coefficient ,_ _ >
J dimensionless parameter; {u^ - U^DJw [-]
Ó' displacement thickness ["^
A relative density; (p^ - p)/yo f-^
Ö slope angle [•J p efficiency factor
V kinematic viscosity [L T ]
f dimensionless parameter [-]
List of symbols (continued)
material density (To standard deviation of
instantaneous bed shear-stress
T critical bed shear-stress (Shields)
f[ characteristic critical bed shear-stress; aj^
T\ characteristic critical bed shear-stress; r . s i n U - ö ) / s i n 0 T\ characteristic critical bed shear-stress; -7-^sin(0 + ö)/sin0
To bed shear-stress 0 angle of repose
0' angle of internal friction
HJ^ critical mobility-parameter Subscripts b bed c critical m m a x i m u m or measured 0 initial or reference
/
(
