Stability of stones under non-uniform flow

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STABILITY OF STONES

UNDER NON-UNIFORM

FLOW

20 Oktober 2006

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1. Report number DWW-2006-085 2. ISBN number ISBN-10 90-369-5619-6 ISBN-13 978-90-369-5619-2 3. Title:

Stability of Stones Under Non-Uniform Flow

4. Date of report: 23 October 2006

5. Author: Dr.Ir. Gijs J.C.M. Hoffmans 6. Project name: ICSE-3

7. Name and address of client Rijkswaterstaat

Dienst Weg- en Waterbouwkunde Postbus 5044

2600 GA DELFT

8. Comments on Distribution: Ministerie V&W (Ministry of Transport, Public Works and Water Management

7. Paper

In the Netherlands hydraulic engineering structures are not only designed, but are also managed and maintained (M&M). In the nineteen-nineties in the United States during periods of high water scores of bridges collapsed, because American law forbids the use of bed protection around bridge piers and moreover, no maintenance plans were available. Despite the fact that in Korea there are always strong tides recently during periods of high water, various bridge piers and abutments have been undermined with dramatic consequences. This emphasises the need for good M&M plans for both bed and bank protection. This publication contributes to the International Conference on Scour and Erosion (ICSE-3) in Amsterdam (1-3 November 2006) and contains information on the loads imposed on bed protection by currents and on the amount of damage that may occur in normative situations 8. Key words: 9. Distribution system Obtainable from:

Stability of stones, non-uniform flow, Dienst Weg- en

Waterbouwkunde

turbulence, erosion modelling Attn. Mw. M.A. Schomaker

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1. ABSTRACT

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2. RECENT DEVELOPMENTS

As a superintendent of dikes and hydraulic engineer, Brahms (1767) tested the initiation of motion of particles experimentally. His results with respect to flow velocity and size of the particles are summarized in Figure 1. After a period of about 150 years Shields (1936) introduced his well-known stability diagram for bed particles (Figure 2). Several model relations have been developed to improve the classical approach of Shields to bed stability, and to predict erosion in non-uniform flow regions, e.g. Izbash (1970) and Pilarczyk (1987, 1993). However, no satisfactory physical model relation has been established to provide a definititive answer to the question of bed stability for practical engineering purposes.

Figure 1 Initiation of motion according to Brahms (1767) and Hjulström (1935)

Usually when the bed shear stress (τ0) is close to its critical value, moving

particles can be observed. Shields relation is based on the dimensionless ratio between load and strength that can be derived from considering the equilibrium of the forces on a stone.

(

ρ ρ

)

gd τ strength load s − = = 0 Ψ with (1) 2 * 0 ρu τ =

in which Ψ is the mobility parameter, ρs is density of the particle, ρ is

density of water, g is acceleration of gravity, d (or d50) is the mean

particle diameter and u* is the bed shear velocity. Although the bed

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difficult to observe the nature of particle movement. This difficulty is a consequence of a phenomenon which is random in both time and place. In the Shields diagram, the influence of fluctuating shear stresses on bed particles is not directly specified. In the sixties WL|Delft Hydraulics studied the initiation of movement of bed particles in detail and

distinguished 7 qualitative criteria from “occasional bed particle movement at some locations” to “general transport”. These introduced criteria all lie in the broad belt originally given by Shields, thus confirming the earlier research activities of Shields.

For large particles he found a value of Ψc = 0.06. In such situations general

transport will occur, however in prototype situations this must be prevented. When some damage can be accepted, a value which is half of this (or criterion 1 or 2 according to Delft Hydraulics, 1972) should be taken. When riprap structures are loaded up to design conditions more frequently, regular maintenance is necessary. Sometimes a lower value of Ψc (Ψc < 0.02) could

be more appropriate. However, this depends on the costs, on the one hand, and the risk of failure of the hydraulic structure on the other hand. A cost-benefit analysis must be made to support a decisive answer.

Figure 2 Diagram of Shields (1936) D* =d(∆g/ν2)⅓

∆ is the relative density, ν is the kinematic viscosity

The forces acting on a bed particle under turbulent flow conditions are its self-weight, friction, and hydraulic forces characterized as fluctuating drag and lift forces. Since the submerged weight (Fg) of the bed particle and

the lift force (Fl) are most important for particle stability, they will be

considered in greater detail (Booij, 1998).

max 2 4 1 _π_d _p Fl = ₍₂₎

(

ρ ρ

)

g d π F_g ₌ 1₆ 3 _s (3)

in which pmax is the difference between the positive and negative

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According to Emmerling (1973) in turbulent flow the standard deviation of the fluctuating pressure (σp) on the bed is about three times the bed

shear stress. He also found that the positive and negative pressure peaks could be up to 6σp. With the estimations of pmax/σp ≈ 6 and σp ≈ 3τ0, the

maximum pressure peaks can reach up to pmax ≈ 18τ0. Consequently,

loads from pmax gives:

(

ρ ρ

)

gd τ F F s g l -0 Ψ 1 = with (4) 037 . 0 = Ψ

Papanicolaou (2000), investigated the characteristics of near-bed

turbulence experimentally. Incipient flow conditions prevailed throughout the experiments. From his experiments he concluded that the turbulent normal stress (= ρσw2, see also equation 9 and Figure 18) should be

considered as the most dominant stress responsible for particle entrainment.

Unsteady eddying motions that are in constant motion with respect to each other characterize turbulent flow. At any point in the flow, eddies produce fluctuations in the flow velocity and pressure and exchange momentum and energy. Kalinske (1947) proposed a model of bed load transport that incorporated fluctuations in the bed shear velocity. Grass (1970) hypothesized that the value of instantaneous bed shear stress follows a Gaussian distribution. The overlapping region between the two distributions defines the critical shear stress values that correspond to particles’ initiation of motion (Figure 3).

Figure 3 Probability functions of the load and strength parameters,

σ0 is the standard deviation of the instantaneous shear

stress, σc is the standard deviation of instantaneous

critical shear stress (Grass, 1970)

In the sixties Raudkivi showed that movement of a bed particle is governed by the bed shear stress (steady part) and the turbulence intensity (fluctuating part) near the bed (It). A bed particle could be

moved by τ0 alone (no extra bed turbulence from e.g. a mixing layer) or

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flow velocity equals zero. The two effects are not superimposed linearly, but more probably (Raudkivi, 1992, 1998).

) Ψ ( ₂ 2 0 0 t I U ρ τ f = + ₍₅₎

in which f(Ψ) represents equal conditions for initiation of motion and U0

is the depth-averaged flow velocity. Jongeling et al. (2003) introduced a new mobility parameter (ΨWL):

(

)

gd k α u_b _b WL _∆ Ψ 2 + = (6)

in which ūb is the local mean velocity near the bed, kbis the turbulent

kinetic energy near the bed (see also equation 10) and α is a coefficient. Calculations with the turbulence model CFX showed that for different overflow and underflow conditions α is about 6 and the order of magnitude O(ΨWL) = 10. Note that O(Ψ) is 0.01.

Hofland (2005) studied a single stone under non-uniform flow conditions in detail. He found that besides the bed shear stress (horizontal direction) there were also pressure fluctuations in the flow attributable to the stone movement (vertical direction). These results are consistent with other reported turbulent measurements in flow conditions well above those corresponding to particle incipient conditions. He extended equation (6) by assuming that the mobility parameter is also correlated to a

characteristic length scale.

In the nineties Annandale established a model relation between stream power (load) and erosion index (strength) by analysing published and field data for a wide variety of types of earth material (Annandale, 2000, 2006).

In this study the resultant force on particles is correlated to the turbulent kinetic energy (equation 10), so the steady part in equation (5) is

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3. MODELLING UNIFORM FLOW

Turbulent flow is characterized by unsteady motions that are in constant motion with respect to each other. At any point in the flow, the eddies produce fluctuations in the flow velocity. Nezu (1977) investigated the turbulent structure in open channel flows. Systematic measurements of turbulent open-channel flows over both smooth and rough beds were carried out using hot-film anemometers and hydrogen bubble tracers. For uniform flow Nezu found the following model relations for zu*/ν > 70

(Figure 4): h z e u γ z σ_u( )= _u _* − ₍₇₎ h z e u γ z σ_v( )= _v _* − ₍₈₎ h z e u γ z σ_w( )= _w _* − (9) in which, σu , σvand σware the standard deviations of the fluctuating

velocities in the x, (= longitudinal), y (= transverse) and z (= vertical) directions respectively, h is the flow depth and γu (≈ 1.92), γv (≈ 1.06)

and γw (≈ 1.34) are coefficients.

Figure 4 Distributions of turbulent flow fluctuations in uniform flow

Very close to the bed viscosity dominates. In the viscous sublayer for

zu*/ν < 5 the turbulence is damped out. The bed shear stress is almost

entirely viscous, resulting in a linear velocity profile. In the buffer layer for 5 < zu*/ν < 70 the velocity profile is logarithmic and in the outer

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Combining the relations for predicting fluctuating velocities and the definition of the turbulent kinetic energy, k as a function of z reads (for

zu*/ν > 70):

( )

(

σ z σ z σ z

)

ω u e zh z k _u _v _w _k 2 2 * 2 2 2 2 1 ₍ ₎ ₍ ₎ ) ( = + + = − (10) in which ωk = ½(γu2 + γv2 + γw2) ≈ 3.3. Hence, the depth-averaged

turbulent kinetic energy is:

( )

(

)

(

)

(

)

2 * 0 2 0 2 * 2 1 0 2 1 2 * 0 2 1 _he _ω _u _e _e _c _u h u ω dz z k h k k h _k h ave h z = = = =

∫

_- − _- (11)

with c0 (≈ 1.2) is a coefficient.The depth-averaged relative turbulence

intensity (r0) can be rewritten with the equation of Chézy u*/U0 = √g/C:

C g c U u c U k r ave ₀ 0 * 0 0 0 = = = (12)

where C is the Chézy coefficient. For hydraulically smooth conditions, that is for C = 75 m½_{/s, r}

0 ≈ 0.05 and for hydraulically rough conditions,

i.e. C = 25 m½_{/s, r}

0 ≈ 0.15. Rewriting equations (1) and (12) yields:

(

)

gd U r ∆ 7 . 0 = Ψ 2 0 0 (13)

The reference level δ where the load (r0U0) acts on the bed is at about

40% of the flow depth:

Flow velocity: δU = 0.37h (equation 14)

( )

e z h h κ u dz z z κ u dz z u h U q h h 0 * 0 0 * 0 0 = = ln = ln =

∫

( )

<=> = <=> = 0 * 0 * 0 ln ln_z δ κ u e z h κ u δ u U U U _z δ h δ e z h U U ₀_.₃₇ 0 0 = <=> = Turbulence intensity: δr = 0.41h

( )

h ω c h δ ω c h δ ω c e e ω c e u ω c δ k r k r k r k k k U U u r U h r δ h r δ h r δ 41 . 0 ln ln 2 2 0 2 1 2 0 2 0 0 2 * 1 0 1 0 2 2 2 0 0 * 0 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = <=> ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = <=> = <=> = <=> = <=> = − − − where z0 (=ks/33) is the zero velocity point, ks is the effective roughness,

u is the local flow velocity in the streamwise direction, q is the discharge and κ (= 0.4) is the constant of von Kármán

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3.4)τ0 and thus confirming the earlier results of Emmerling. Grass (1970)

found a model relation between the standard deviation of the

instantaneous bed shear and the bed shear stress, σ0/τ0 = constant, see

also Figure 3. Following Emmerling, Hinze and Grass, the measure of damage could be modelled by only fluctuating parts. Consequently, equation (13) is valid for all types of uniform flow. The mobility parameter in equation (13) represents the transport or measure of damage caused by a combination of a steady part and a fluctuating part. The advantage of equation (13) with respect to all other stability

predictors is that the designer can easily estimate the strength of the top layer of the bed protection provided that he knows both the load (r0U0)

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4. VALIDATING UNIFORM FLOW

In the seventies Aguirre Pe conducted laboratory experiments to

determine the influence of the bed shear stress on the sediment transport rate for different materials near the critical threshold for motion. These experiments for large relative roughness lie in the range of 0.2 < h/d < 30 (Appendix A to E).

In this study these experiments are used to validate equation (13). The depth-averaged relative turbulence intensity is calculated by applying equation (12), in which the Chézy coefficient is determined by C =

U0/√(Ri) in which R is the hydraulic radius and i is the energy slope.

About 90% of the experiments lie in the range of 0.02 < Ψ < 0.05 (Figure 5), which confirms the earlier results of Aguirre (Aguirre Pe, 1993). However, there is one difference. The vertical and horizontal axes give information on load and strength respectively. The load is

determined by U0 and r0 both depth-averaged values and the strength by

∆, g and d.

Figure 5 Initiation of motion, experimental data from (Aguirre Pe, 2006)

Figure 6 shows the influence of these parameters on the movement of bed particles, using the experimental data from both Aguirre Pe (2006) and Maynord (1988) (Appendix A to F). The broad belt represents the initiation of motion. If r0 increases and U0 decreases following the trend,

equal conditions for transport are obtained. In the 5 experimental series the mean particle diameter is about 5 cm and the relative density is approximately 1.65. Usually, for uniform flow the depth-averaged turbulence intensity varies from 0.05 to 0.15, however, Aguirre Pe carried out experiments in which the ratio of flow depth and particle diameter is approximately 1, resulting high turbulence intensities (0.2 <

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Figure 6 Initiation of motion, r0 as function of U0 for equal strength

conditions, see also Figure 7, (d ≈ 0.05 m and ∆ ≈ 1.65)

Figure 7 Initiation of motion, r0 as function of U0 for equal strength

conditions, see also Figure 6 (d ≈ 0.05 m and ∆ ≈ 1.65) however, in his experiments h0/d > 10 yielding turbulence intensities 0.10

< r0 < 0.15.

Figure 7 again shows the influence of load on the movement of bed particles. The broad belt of data represents the measure of damage and is split up into 5 criteria, the mobility parameter Ψ varies from 0.01 to 0.06. If r0 or U0 increase, the damage at the bed protection will also

increase.

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Figure 8 Stability on top of broad-crested weir (Pilarczyk,1987) In order to validate equation (13) the following assumptions have been made for both the nominal particle diameter (dn) and the depth-averaged

flow velocity at the crest (sub critical flow):

d ρ M V d_n 3 _s 0.84 50 3 50 = = = (16)

(

H hb

)

g µ U₀ = 2 ₀ (17) where V50 and M50 are the mean volume and mean mass of the particle

respectively, µ is the discharge coefficient, H0 and hbrepresent the

upstream and downstream flow depth above the crest and Ū0 is the

depth-averaged flow velocity above the crest. Rewriting equations(13), (16) and (17) gives:

( )

2 0 0 ₊₀_.₈₆ Ψ ∆ = ∆ d µr h d H n b n (18)

In the experiments Ψ is approximately equal to 0.035. For sub critical flow or large flow depths, the depth averaged turbulence intensity r0 is

about 0.12 for hydraulic rough flow (0.10 < r0 < 0.15). With µ = 1

equation (18) reads: 2 ∆ ∆ 0 ₌ ₊ n b n d h d H (19)

The transition between sub and super critical flow will occur if hbequals

0.67H0 giving ∆dn > 0.15H0. If hbdecreases, the bed roughness

increases. Thus the load at the crest is larger for super critical flow than for sub critical flow. Measurements show that H0/(∆dn) varies from ½ to

3 (or 0.33H0 < ∆dn < 2 H0) for negative values of hb. In such situations

the depth-averaged relative turbulence intensity lies in the range of 0.15 < r0 < 0.4, which can easily be verified applying (18) and assuming Ψ =

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5. MODELLING OF NON-UNIFORM FLOW

The centre of the mixing layer has a very definite curvature, increasing with the distance from the top of the sill (Figure 9). In the reattachment point in particular the curvature is strong. In the relaxation zone the turbulence decreases in the downstream direction e.g. Hoffmans (1988). After the point of reattachment a new wall-boundary layer which spreads in the relaxation zone develops. The flow is more or less in equilibrium (uniform flow) if this boundary layer thickness equals the local flow depth. The point of reattachment is largely determined by the magnitude of the initial slope of the sill and the ratio of the height of the sill and the flow depth.

Figure 9 Flow pattern downstream of a backward facing step Non-uniform flow measurements of Van Mierlo and De Ruiter (1988) showed that the turbulence energy (km) in the centre of the mixing layer

(with horizontal axis) grows rapidly to a maximum and vanishes where the new wall-boundary layer is well developed. The turbulence energy (k0) then approaches an equilibrium value, which consists largely of

turbulence generated at the bed (Figures 10 and 11).

Downstream of the point of reattachment, the turbulence energy (kη) in

the relaxation zone decreases gradually and becomes small compared to the bed turbulence energy (k0) for uniform flow conditions. Earlier

studies of Hoffmans (1992) have shown that in a scour hole the

turbulence energy (kb) near the bed can be represented by a combination

of a certain part of the turbulence energy (kη) and the turbulence energy

(k0) caused by the bed roughness.

) ( ) ( ) (x ωk x k₀ x k_b = _η + with ( ) ( ) (20) 2 * 0 x ω u x k = _k

in which ω coefficient. In the recirculation zone ω is about 0.33.To analyse the decay of the turbulence here, an analogy with the decay of the turbulence energy and the dissipation in grid turbulence can be used (Launder and Spalding, 1972).

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turbulence energy and the dissipation are neglected, kηcan be given by (Booij, 1989): 1 ) ( k α R m η _λ x x k x k _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − = (21)

where xR (≈ 6D) is the x-coordinate where the flow reattaches the bed, D

is the height of the sill, λ (≈ ½ cλh/βm) is a relaxation length, cλ (≈ 1.2) is a

relaxation coefficient, βm (≈ 0.09) is the angle of the mixing layer and αk

(≈ -1.08) is a coefficient that is directly related to the turbulence coefficients used in k-ε-models.

Figure 10 k as function of x and z above a dune (T5); k(x,z) = ¾(σu2(x,z) + σw2(x,z)), σu and σw are measured values

(Van Mierlo and De Ruiter, 1988)

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The hypothesis of self-preservation (Townsend, 1976) requires constant turbulence energy in the mixing layer up to the point where the

boundaries have reached the surface and the bed. An appropriate value is: 2 0 U C k_m = _k (22) in which Ck (≈ 0.045) is a coefficient. For a backward facing step the

value of Ck is approximately equal to 0.045, whereas for a sill with slopes

gradually Ck is smaller than 0.045. In fact the value of Ck depends on the

configuration of the sill. However, this dependency is not examined here. In analogy to (11), the turbulence energy averaged over the depth, from which r0 can be determined downstream of a sill, can be given by:

x u c x k β z z x k h η h k ( ) ( ) d ) , ( 1 2 * 2 0 0 + =

∫

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where βk (≈ 0.5) is a coefficient. If the geometry of the tests consists of a

horizontal bed where the flow is sub-critical above a sill, the following relation for r0 can be deduced for L > 6D:

2 08 . 1 2 0 ₆_.₆₇ 1 1.4 6 1 0225 . 0 C g h D L h D r ⎟ + ⎠ ⎞ ⎜ ⎝ ⎛ − ₊ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − − (24) For reasons of safety the length L of the bed protection will always extend beyond the point of reattachment. More than 250 experiments were used to calibrate and verify (24). In these laboratory experiments both the hydraulic conditions and the geometrical parameters were varied (Hoffmans, 1993). In the re-circulation the contribution of the steady part (equation 5) is negligible with respect to the fluctuating part, see also Figures 12 and 13, so equation (24) simplifies to:

2 0 0.0225 1 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = h D r ₍₂₅₎

The turbulence intensity increases when the height of the sill increases in relation to the tail water depth. Equation (25) yields reasonable results if the ratio between D to h is smaller than 0.7 (r0 < 0.5).

According to Emmerling (1973), particles are unstable if pmax is about 6kb.

However, for non-uniform flow no information about this lift criterion is available from literature. Therefore this criterion has been analysed in greater detail.

Though kb is at maximum in the re-circulation zone, experiments show

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small flow gradients close to the bed. Particles must first be lifted, before they can be transported in the downstream direction. Thus lifting

particles alone is not enough for transportation.

In the region 5 < x/D < 20, kb decreases owing to the acceleration of the

flow. Although the bed shear stress reaches its maximum value at the top of the dune, the flow is still not in equilibrium at x/D = 20 (Figures 12 and 13), since r0 > 0.1. The flow conditions were hydraulically smooth,

since the bed of the flume was plastered with sand (d = 0.0016 m). Applying the Chézy coefficient:

k h κ g C s 12 ln = (26)

For hydraulically smooth conditions (ks = 2d), r0 equals about 0.07 for

uniform flow conditions, see also Table 1.

Figure 12 Turbulence parameters as function of x/D, test T5,

r0(x) = kave(x)/U0(x), k0(x) = ωk·u*2(x); u* is determined

from the flow velocity profile; see also Figures 10 and 11 (Van Mierlo and De Ruiter, 1988);

Figures 12 and 13 show that the turbulent kinetic energy kb as function

of the downstream direction is strongly related to the depth-averaged relative turbulence intensity: kb ≈ (r0U0)2. For uniform flow kb is about

2(r0U0)2 at z = 70ν/u*. If the bed shear velocity varies from 0.001 to 0.01

m/s and using z = 70ν/u* the vertical coordinate lies in the range of 10 to

100 mm. Van Mierlo and De Ruiter (1998) measured flow characteristics at a mean interval of 1 cm. Very close to the bed measurements were carried out at 2 to 5 mm (Figures 10 and 11). If the chosen reference level of kb is not that level proposed by Van Mierlo and De Ruiter, but at

about 10 mm in the outer region, kb increases with a factor 2 resulting kb

≈ 2(r0U0)2. If both kb ≈ 2(r0U0)2 and pmax = 6kb are valid pmax = 12(r0U0)2 ,

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Downstream of the reattachment point r0 gradually decreases. At x/D =

10, the bed shear stress equals about half the maximum value (τ0 ≈

0.5τ0,max) and r0 equals approximately r0,max (r0 ≈ 0.8r0,max) (see also Table

1), in which r0,max is predicted by equation (25). Note that the differences

between the calculated and measured r0,max for both T5 and T6 are

marginal. At x/D = 20, τ0 reaches its maximum and r0 ≈ 0.5r0,max. At the

top of the dune as well at x/D = 10 the load, that is the lift force

(expressed by r0) and the drag force (expressed by τ0), is large enough to

transport particles.

Figure 13 Turbulence parameters as function of x/D, (T6) see also Figure 12

Since measurements of Xingkui and Fontijn (1992) illustrate that the heaviest attack occurs at 10 < x/D < 20, equation (13) is valid not only for predicting the stability of bed protection in uniform flow, but also for flow downstream of a backward facing step. It is recommended to validate the assumption of pmax ≈ 6kb in non-uniform flow as well as to

precise the reference level for kb .

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6. VALIDATION FOR NON-UNIFORM FLOW

Within the framework of the Dutch Delta Works, experience was

obtained in predicting the dimensions of rock fill closure dams under tidal conditions. At Delft Hydraulics (1982) several experiments were

performed on broad-crested weirs over which the discharge was constant in time. Variable parameters were the height of the sill D, the tail water depth h, the discharge, the form of the sill, the density of the bed particles and the size of the particle diameter (Appendix G and H). The objective here is to verify equation (13) for non-uniform flow conditions. Since no turbulence parameters were measured, r0 is calculated using

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Figure 14 shows the load parameters U0 and r0 for different stages of

transport. Both only non-uniform flow (Delft Hydraulics, 1982), and uniform flow (Aguirre Pe, 2006, Brown and Chu, 1968) are considered. Because of the huge scatter, the experiments are re-plotted in such a way that information is given about the mobility parameter. Figure 15 shows regions of equal transport in which no difference can be distinguished between experiments representing uniform and non-uniform flow. Raudkivi (1992) stressed this phenomenon qualitatively, equation (5). Though the experiments do not confirm his hypothesis in a quantitative way, the results are promising.

Figure 14 r0 as function of U0 for critical conditions of movement

UF ( Uniform Flow), experimental data series 3A, 4A and 11 from Aguirre Pe (2006), BFS (Backward Facing Step). Experimental data from Delft Hydraulics 1982, see also Figure 15

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between 4.6 and 11.8 mm) on a flat bed, and at various turbulence levels (0.3 m/s < U0 < 1.2 m/s; 0.15 m < h < 0.35 m), see also Appendix

I.

Figure 15 r0 as function of U0 for critical conditions of movement,

7. ABUTMENTS AND BRIDGE PIERS

The flow pattern at an abutment and pier is characterised by a horseshoe vortex at the base and wake vortices downstream of the structure. Not only do turbulence intensities play a role on the stability of particles, but also acceleration and deceleration effects. Ariens (1993) investigated the stone stability downstream of a vertical wall abutment. Figure 19 shows iso-lines of characteristic flow velocities:

(

)

∫

+ = h u kar _h u σ dz U 0 3 1 (27)

Figure 19 Characteristic mean velocity downstream of a vertical abutment. The heaviest attack occurs in the region where Ukar is about 82 cm/s.

The heaviest attack occurs where Ukar is at maximum (= 0.82 m/s).

According to Ariens the corresponding coordinates are U0,m = 0.68 m/s

and the ratio between r0,m and r0 is 1.36 (r0 is the depth-averaged relative

turbulence intensity upstream of the abutment). The flow velocity in the constriction is 0.54 m/s. Consequently the correction for turbulence and velocity is Kt = 1.36 and Kv = 1.26 respectively. Equation (13) can be

rewritten as:

(

)

gd U r m m ∆ 7 . 0 Ψ 2 , 0 , 0 = with m m t v constriction U K K r U r₀_, ₀_, = ₀ (28)

According to Ariens Kv varies from 1.1 for streamlined abutments to 1.4

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8. PRACTICAL APPLICATION

Various methods are available for sizing riprap. Most of the model relations from the 18th_{centure have the form (Brahms, 1767):}

d u_b_,_c =4.5−5.0

(29) in which the dimensions of d and ūb,c (is critical near bed velocity) are

(m) and (m/s) respectively. Equation (29) can be rewritten in a dimensionless form as:

7 . 0 77 . 0 62 . 0 ) ∆ 2 /( 2 , gd = − ≈ u_b_c (30)

Following Izbash (1935; Izbash and Khaldre, 1970) the stability relation for embedded and exposed stones on a sill reads:

φ c

b gd K

u2, /(2∆ )= (31) For exposed stones Kφ is 0.7 and for embedded stones Kφ is 1.4.

Equation (31) can be used directly provided the near bed velocity is known. Following Pilarczyk (1993) the reference level is at 1 m above the bed in the prototype situation. Using equation (13) the reference level lies in the range of 0.37h to 0.41h. Combining these two stability relations and assuming ūb,c = 0.6U0, Kφ can be rewritten as: Kφ =

0.26Ψc/(r0)2, see also table 3.

Table 3 Relation between Kφ (equation 29) and r0 (equation 13) Kφ (Ψc = 0.03) Kφ (Ψc = 0.04) r0 Remarks 3.09 4.11 0.05 low turbulence 1.40 1.40 embedded 0.77 1.03 0.10 normal turbulence 0.70 0.70 exposed 0.34 0.46 0.15 high turbulence Although Izbash (1935) did not specify the turbulence near the bed in a qualitative way, from a theoretical point of view r0 increases if the bed

becomes rougher, thus from embedded to exposed stones.

Pilarczyk (1993) presented an overview of practical design considerations for stones against current attack for various civil engineering

applications. He combined all these aspects to one general form:

g U K K K K d_n _t _v _s 2 ∆ 1 02 Ψ − = (32) in which Kψ (= 0.035φc/Ψc) is a mobility factor, Kt is a turbulence factor,

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θ α K_s 2₂ sin sin 1− = ₍₃₃₎

in which α is a slope angle and θ is the angle of repose of the riprap stones. The following values of φc are recommended:

φc = 0.50 to 0.75 for continuous protection;

φc = 0.75 can be treated as a common reference for stones;

φc = 1.0 to1.5 for exposed edges and/or transitions.

The turbulence factor Kt varies from 1 to 3, see also table 4. According to

Pilarczyk the factor Kv includes the effects of the flow velocity profile.

However, Kv also takes into account the influence of bed turbulence. For a

logarithmic velocity profile under uniform flow r0 = 1.2√g/C = 0.15√Kv

with Kv = 2/log2(1+12h/ks). For a not fully developed velocity profile

Pilarczyk found Kv = (1+h/dn)-0.2, see also table 5. Pilarczyk models the

effects of turbulence by both Kt and Kv in which Kt represents the extra

turbulence of for example vortices of horizontal and vertical axises and Kv

characterizes the turbulence generated near the bed.

Table 4 Relation between Kt(equation 32) and r0 (non-uniform flow) Kt r0 remarks

0.05 - 0.10 uniform flow (hydraulic smooth) 1.0 - 1.5 0.10 - 0.15 uniform flow (hydraulic rough), rivers

1.5 - 2.0 0.15 - 0.20 increased turbulence as below stilling basins, outer bends in rivers

2.0 - 3.0 0.20 - 0.30 high turbulence as below hydraulic jumps, local disturbances, sharp outer bends 3.0 – 5.0 0.30 - 0.50 very high turbulence, i.e. inside hydraulic

jumps, sharp river bends, steep channels with limited flow depth, impact of flow jets, standard screw races by propellors

10 1 more or less maximum turbulence as can be found in specific action of screw races by ship propellers (i.e. screw race near the bed at harbour keys), ski-jump at high dams Table 5 Relation between Kv(equation 32) and r0 (uniform flow)

h/ks Kv = 2/log2_(1+12h/k s) r0 h/dn Kv = (1+h/dn )-0.2 r0 1 1.61 0.19 0.33 0.94 0.14 10 0.46 0.10 3.33 0.74 0.13 100 0.21 0.07 33.3 0.49 0.11

For practical applications Hoffmans model relation can be written as:

(

)

s c n _g _K U r d 1 Ψ 7 . 0 84 . 0 ∆ 2 0 0 ⋅ = ₍₃₄₎

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9. EXAMPLES OF PRACTICAL APPLICATION

Example1: Calculate stable stone size of riprap for protection in sharp river bend.

- Current: U0 = 2.5 /s (maximum depth-averaged flow velocity);

- High turbulence: r0 = 0.15;

- Bank slope: cotan α = 2; (or α = 26.60_);

- Sediment density: 2600 kg/m3_{, thus}_{∆ = 1.6.}

72 . 0 0 . 40 sin 5 . 26 sin 1 sin sin 1 2 2 2 2 = − = − = o o s θ α K Izbash

with r0 = 0.15 => Kφ = 0.40, see table 3:

(

)

₀_.₂₁ 72 . 0 1 81 . 9 6 . 1 2 40 . 0 5 . 2 6 . 0 84 . 0 1 ∆ 2 84 . 0 2 2 = ⋅ ⋅ ⋅ ⋅ = = s φ b n _K _g _K u d m Pilarczyk

with r0 = 0.15 => Kt = 1.5, see table 4;

assuming that: Kv = 0.34;

for edges: φc = 1.5, thus

21 . 0 6 . 1 81 . 9 2 5 . 2 72 . 0 34 . 0 5 . 1 035 . 0 035 . 0 5 . 1 ∆ 2 2 2 0 1 Ψ _⋅ _⋅ = ⋅ ⋅ = = − g U K K K K d_n _t _v _s m Hoffmans (r0 = 0.15)

(

)

(

)

₀_.₂₁ 72 . 0 1 6 . 1 035 . 0 81 . 9 5 . 2 15 . 0 7 . 0 84 . 0 1 ∆ Ψ 7 . 0 84 . 0 2 2 0 0 ₌ ⋅ ⋅ ⋅ ⋅ = ⋅ = s c n _g _K U r d m

Conclusion: The methods of Izbash, Pilarczyk and Hoffmans yield the same result. In the next examples the influence of the flow depth will be taken into account.

Example 2: Calculate stable stone size of riprap for protection at the crest of a sill.

- Flow depth upstream of the sill: 8 m; - Flow depth downstream of the sill: 7.8 m; - Sill height: 5 m;

- Sediment density: 2600 kg/m3 _{; thus}_{∆ = 1.6.}

Current (sub critical flow):

(

)

1.0 2 9.81

(

3 2.8

)

2.0 2 ₀

0 = µ g H hb = ⋅ − =

U - m/s

(32)

for nearly uniform flow: φc = 0.75, thus 040 . 0 6 . 1 81 . 9 2 0 . 2 42 . 0 0 . 1 035 . 0 035 . 0 75 . 0 ∆ 2 2 2 0 Ψ = ⋅ ⋅ ⋅ _⋅ _⋅ = = g U K K K d_n _t _v m Hoffmans

(

)

(

)

₀_.₀₂₅ 6 . 1 035 . 0 81 . 9 0 . 2 077 . 0 7 . 0 84 . 0 ∆ Ψ 7 . 0 84 . 0 2 2 0 0 ₌ ⋅ ⋅ ⋅ ⋅ = ⋅ = c n _g U r d m Verification of first estimation of roughness:

48 025 . 0 3 9 . 2 12 ln 4 . 0 81 . 9 3 12 ln = ⋅ ⋅ = = d h κ g C n m 1/2_{/s (OKAY)}

Conclusion: Equations (19) and (32) predict a larger stone size than Hoffmans method. Equation (19) includes a default value for the depth-averaged relative turbulence (r0 = 0.12), thus it can not correct the stone

size for r0 is 0.077. The model relation of Pilarczyk is only valid for r0 >

0.1.

Example 3: Calculate stable stone size of riprap for protection at the crest of a sill.

- Flow depth upstream of the sill: 8 m; - Flow depth downstream of the sill: 7 m; - Sill height: 5 m;

Current: (super critical flow):

(

)

1.0 2 9.81

(

3 2

)

4.43

2 ₀

0 = µ g H hb = ⋅ − =

U - m/s

First estimation of roughness bed at the crest of sill is C = 24 m1/2_{/s, thus} r0 is: 15 . 0 024 . 0 24 81 . 9 4 . 1 4 . 1 ₂ ₂ 0 = = = = C g r Equation (19): 31 . 0 6 . 1 1 3 ∆ 12 0 2 1 − ₌ − ₌ = b n h H d m Pilarczyk

assuming that: dn = 0.46 m => Kv = (1 + 3/0.46)-0.2 = 0.67;

for nearly uniform flow: φc = 0.75, thus

46 . 0 6 . 1 81 . 9 2 4 . 4 67 . 0 5 . 1 035 . 0 035 . 0 75 . 0 ∆ 2 2 2 0 Ψ = ⋅ ⋅ ⋅ _⋅ _⋅ = = g U K K K d_n _t _v m Hoffmans

(

)

(

)

₀_.₄₇ 6 . 1 035 . 0 81 . 9 4 . 4 15 . 0 7 . 0 84 . 0 ∆ Ψ 7 . 0 84 . 0 2 2 0 0 ₌ ⋅ ⋅ ⋅ ⋅ = ⋅ = c n _g U r d m Verification of first estimation of roughness:

24 47 . 0 3 5 . 2 12 ln 4 . 0 81 . 9 3 12 ln = ⋅ ⋅ = = d h κ g C n m 1/2_{/s (OKAY)}

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default value of r0 (= 0.12). The predictors of Pilarczyk and Hoffmans

give nearly the same result.

Example 4: Calculate stable stone size of riprap for protection downstream of a sill.

- Current downstream of sill: U0 = 2.5 /s (maximum depth-averaged

flow velocity)

- Flow depth downstream of sill: 8 m; - Sill height: 4 m;

- Length of fixed bed protection (concrete) downstream of sill: 25 m; - Sediment density: 2600 kg/m3 _{; thus}_{∆ = 1.6.}

First estimation of roughness bed is C = 30 m1/2_/s

27 . 0 0153 . 0 98 . 0 0576 . 0 30 81 . 9 4 . 1 1 8 67 . 6 4 6 25 8 4 1 0225 . 0 4 . 1 1 67 . 6 6 1 0225 . 0 2 08 . 1 2 2 08 . 1 2 0 = + ⋅ = = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ₊ ⋅ ⋅ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = = + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ₊ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − − − − C g h D L h D r Pilarczyk

assuming that: dn = 0.5 m => Kv = (1 + 4/0.5)-0.2 = 0.64;

for edges φc = 1.5, thus

(

)

(

)

₀_.₄₉ 6 . 1 035 . 0 81 . 9 5 . 2 27 . 0 7 . 0 84 . 0 ∆ Ψ 7 . 0 84 . 0 2 2 0 0 ₌ ⋅ ⋅ ⋅ ⋅ = ⋅ = c n _g U r d m The predictors of Pilarczyk and Hoffmans yield the same result. Example 5: Calculate stable stone size of riprap for protection in an irrigation channel.

- Mean flow depth (above bed): 0.035 m; - Current: 0.5 m/s

First estimation of roughness bed is C = 10 m1/2_/s

37 . 0 1373 . 0 10 81 . 9 4 . 1 4 . 1 ₂ ₂ 0 = = = = C g r Pilarczyk with r 0= 0.37 => Kt = 3.7;

according to Pilarczyk: Kv = 1 and φc = 1.25 for h/dn < 5, thus

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10 037 . 0 3 035 . 0 12 ln 4 . 0 81 . 9 3 12 ln = ⋅ ⋅ = = d h κ g C n m1/2_{/s (OKAY)}

Again the method of Pilarczyk and Hoffmans predict similar stone size, however, the coefficients of Pilarczyk are not unambiguous. Based on experimental data of Aguirre Pe the particle diameter should be approximately dn = 0.042 m (d = dn /0.84 = 0.05 m), see also figures 6

and 7.

Example 6: Calculate stable stone size of riprap for protection downstream of a strong hydraulic jump.

- Flow depth: 2 m;

- Sediment density: 2600 kg/m3 _{; thus}_{∆ = 1.6;}

- Ratio of flow depth and conjugate depth: h/hc = 4, thus: Fr1 = 3.16

and r0 = 0.19, see table 2.

Conjugate depth: hc = 2/4 = 0.5 m; Current: Ujet = Fr1.√(ghc) = 3.16√(9.81.0.5) = 7.0 m/s; Current: U0 = 7.0 · 0.5/2 = 1.75 m/s. Pilarczyk with r0 = 0.19 => Kt = 1.9; assuming that: dn = 2 m => Kv = (1 + 2/0.15)-0.2 = 0.59;

for edges φc = 1.5, thus

(

)

(

)

₀_.₁₂ 6 . 1 035 . 0 81 . 9 75 . 1 19 . 0 7 . 0 84 . 0 ∆ Ψ 7 . 0 84 . 0 2 2 0 0 ₌ ⋅ ⋅ ⋅ ⋅ = ⋅ = c n _g U r d m

Usually for large flow depths the bed turbulence Kv can be neglegted

with respect to the extra turbulence Kt. However for relative small flow

depths (say h/dn <2,) equation (32) may overestimate the dimensions of

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10. CONCLUSIONS

The previous analysis has shown that to make an accurate determination of the size of bed protection in non-uniform flow it is necessary to take explicit account of the influence of turbulence. Downstream of hydraulic structures the flow velocity profile differs from both the standard

logarithmic profile and also from the profiles of the turbulent kinetic energy. A wide variety of velocity and turbulence profiles may occur (Figure 9).

The depth-averaged method discussed in the present paper has to be considered as a useful tool for designers. The influence of turbulence is given by the parameter r0 and represents the hydraulic characteristics on

particles in the vertical direction. For sills an analytical expression can be used to estimate the depth-averaged relative turbulence intensity (equation 24). The depth-averaged flow velocity represents the magnitude of the drag force in the horizontal direction.

For uniform flow conditions equation (13) confirmed earlier

investigations and supports the hypothesis of Raudkivi (equation 5) for both uniform and non-uniform flow. Though the simplified model assumptions for a backward facing step still present a challenge, the validation of equation (13) using the experiments of Delft Hydraulics (1982) are promising. It is recommended to validate the assumption of

pmax ≈ 6kb in non-uniform flow and to precise the reference level for the

turbulent kinetic energy kb.

For both weak hydraulic jumps (Escarameia and May, 1992) and strong hydraulic jumps (Russian experimental research carried out by Kumin) equation (13) yields remarkably good results. For other hydraulic structures such as abutments and bridge piers the first results of Ariens, 1993) are encouraging.

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11. REFERENCES

Aguirre Pe, J. and R. Fuentes, 1993, Stability and weak motion of riprap at a channel bed, Proceedings River, Coastal and Shoreline Protection, John Wiley Sons, New York.

Aguirre Pe, J. 2006, Personal Communications.

Annandale, G.W., Smith, S.P and Tamara Butler, 2000, Scour of rock and other earth materials at bridge pier foundations, Proceedings ISSMGE, TC33, Melbourne.

Annandale, G.W., 2006, Scour technology, Mechanics and engineering practice, McGraw Hill, New York.

Ariens, E.E., 1993, Interaction between local scour and stone stability of a top layer (in Dutch), Master thesis, Technical University Delft, Delft.

Booij, R., 1989, Depth-averaged k-epsilon model in Odyssee, Report No. 1-89. Technical University of Delft, Delft.

Booij, R., 1998, Erosion under a geometrical open filter, Report 2-98 (in Dutch), Technical University of Delft, Delft.

Brahms, A., 1767, Anfangsgründe der Deich und Wasserbaukunst, Herausgegeben vom Marschenrat bei Schuster in Leer, (in German) Aurich (Provincial Library Fryslan).

Brown, B. J. and Y.H. Chu, 1968, Boundary effects of uniform size relative roughness, Journal of Hydraulic Research, Vol. 14, No. 2, p.115-126. Delft Hydraulics, 1972, Systematical investigation of two- and three dimensional local scour, Investigation

M648/M863 (in Dutch), Delft Hydraulics, Delft.

Delft Hydraulics, 1982, Stability of bed protection downstream of hydraulic structures, Investigation M1834/S543, part I and II (in Dutch), Delft Hydraulics, Delft.

Emmerling, A.,1973, The instantaneous structure of the wall pressure under a turbulent boundary layer flow (German), Max-Plank- Institut für Strömungsforschung, Bericht 9.

Escarameia, M. and R.W.P. May, 1992, Channel protection; turbulence downstream of structures, Report SR 313, HR Wallingford. Grass, A.J., 1970, Initial instability of fine bed sand, Hydraulic Division,

Proceedings of the ASCE, Vol.96, No.HY3, p.619-632. Hinze, J.O., 1975, Turbulence, Second edition, McGraw-Hill Book

Company, New York.

Hjulström, F., 1935, The morphological activity of rivers as illustrated by River Fyris, Bulletin Geological Institute Uppsala, Vol. 25, Ch III.

Hoffmans, G.J.C.M., 1988, Damping of turbulence parameters in relaxation zone, Report No.10-88, Technical University of Delft, Delft.

Hoffmans, G.J.C.M., 1992, Two-dimensional mathematical modeling of local scour holes, Doctoral thesis, Technical University of Delft, Delft.

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Hofland, B., 2005, Rock & Roll, Turbulence-induced damage to granular bed protections, Doctoral thesis, Technical University of Delft, Delft.

Izbash, S.V. and Kh.Yu. Khaldre, 1970, Hydraulics of river channel closure, Buttersworth, London.

Jongeling, T,H,G., Blom, A. Jagers, H.R.A, Stolker, C. and H.J. Verheij, 2003, Design method granular protections, Technical Report Q2933/Q3018 (in Dutch), Delft Hydraulics, Delft.

Kalinske, A.A., 1947, Movement of sediment as bed load in rivers, Transactions, American Geophysical Union, Vol.28, No.4. Launder, B. E. and D.P. Spalding, 1972, Mathematical models of

turbulence, Academic Press, London.

Maynord, S.T., 1988, Stable riprap size for open channel flows, Technical Report HL- 88-4, U.S. Army Engineering Waterways Experiment Station, Vicksburg, Mississippi.

Mierlo, van M.C.L.M., and J.C.C. de Ruiter, 1988, Turbulence measurements above dunes, Report Q789, Vol. 1 and 2, Delft Hydraulics, Delft.

Nezu, I., 1977, Turbulent structure in open-channel flows (translation of doctoral thesis published in Japanese), Department of Civil Engineering, Kyoto University, Kyoto.

Papanicolau, A.N., 2000, The Role of Near-Bed Turbulence in the Inception of Particle Motion, International Journal of Fluid Dynamics (2000), Vol. 4, Article 2.

Pilarczyk, K.W., 1987, Interaction water motion and closing elements, Proceedings: The closure of tidal basins, Closing of Estuaries, Tidal inlets and dike breaches, Delft University Press, Delft, The Netherlands.

Pilarczyk, K.W., 1993, Simplified Unification of stability formulae for revetments under current and wave attack, Proceedings: River, Coastal and Shoreline Protection, John Wiley Sons, New York. Raudkivi, A.V., 1992, Personal Communications.

Raudkivi, A.V., 1998, Loose Boundary Hydraulics, Balkema, Rotterdam. Shields, A., 1936, Anwendung der Aehnlichkeitsmechanik und der

Turbulenzforschung auf die Geschiebebewegung, Mitteilungen der Preusischen Versuchsanstalt für Wasser-bau und Schiffbau, (in German), Heft 26, Berlin NW 87.

Townsend, A.A.,1976, The structure of turbulent shear flow, Cambridge University Press

Xingkui, W. and H.L. Fontijn, 1993, Experimental study of the

hydrodynamic forces on a bed element in an open channel with a backward facing step, Journal of Fluid and Structures, Vol 7, p.299-318.

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12. SYMBOLS

B width of flume c0 (≈ 1.2) coefficient cλ (≈ 1.2) coefficient C Chézy coefficient Ck (≈ 0.045) coefficient

d (or d50) mean particle diameter dn nominal particle diameter

D height of sill

D* sedimentological diameter Fg submerged weight of bed particle

Fl lift force on particle

g acceleration of gravity

h flow depth or tail water depth

hb downstream flow depth above crest

hc conjugate depth (flow depth downstream of hydraulic jump)

H0 upstream flow depth above crest i energy slope

It turbulence intensity near the bed

k turbulent kinetic energy

kave depth-averaged turbulent kinetic energy

kb turbulent kinetic energy near the bed

km turbulent kinetic energy in the mixing layer

kmax maximum turbulent kinetic energy near the bed ks effective roughness

kη turbulent kinetic energy in the relaxation zone

Khj coefficient

Ks slope factor

Kt correction coefficient for turbulence

Kv correction coefficient for velocity

M50 mass of mean particle L length of bed protection

pmax difference between positive and negative pressure peaks

q discharge

r0 depth-averaged relative turbulence intensity R hydraulic radius

u local longitudinal flow velocity

u* bed shear velocity

uA characteristic flow velocity near stone

ūb local mean velocity near the bed

ūb,c critical velocity near the bed

U0 depth-averaged flow velocity (uniform flow) Ukar characteristic flow velocity

Ū0 depth-averaged flow velocity at crest V50 volume of mean particle

x longitudinal coordinate

xR (≈ 6D) is the x-coordinate where flow reattaches

y transverse coordinate

z vertical coordinate

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Symbols (continued)

α slope angle or (≈ 6.0) coefficient αk (≈ -1.08) coefficient used in k-ε models

βk (≈ 0.5), coefficient

βm (≈ 0.09), angle of mixing layer

γu (≈ 1.92), coefficient γv (≈ 1.06), coefficient γw (≈ 1.34), coefficient δ reference level ∆ relative density κ constant of von Kármán λ relaxation length µ discharge coefficient ν kinematic viscosity

θ angle of repose of riprap stones

ρ density of water

ρs density of particle

σc standard deviation of the instantaneous critical bed shear stress

σp standard deviation of the fluctuating pressure

σu standard deviation of the fluctuating velocity in x direction

σv standard deviation of the fluctuating velocity in y direction

σw standard deviation of the fluctuating velocity in z direction

σ0 standard deviation of the instantaneous bed shear stress τ0 bed shear stress

Ψ mobility parameter or damage parameter or Shields parameter Ψc critical mobility parameter

ΨWL mobility parameter according to Delft Hydraulics

ω (≈ ⅓), coefficient

ωk (≈ 3.3), coefficient

Abbreviations

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Appendix A

Experimental data of Aguirre Pe (2006), series 3A, 3B and 3C

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Appendix B

Experimental data of Aguirre Pe (2006), series 4A, 4B and 4C

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Appendix C

Experimental data of Aguirre Pe (2006), series 8

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Appendix D

Experimental data of Aguirre Pe (2006), series 11 and series 12

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Appendix E

Experimental data of Aguirre Pe (2006), series 15 and series 16

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Appendix F

Experimental data of Maynord (1988)

Test i (-) U0 (m/s) h (m) d (m) ∆ (-) B (m) A26 0.0085 1.103 0.251 0.048 1.65 2.44 A27 0.0138 1.323 0.209 0.048 1.65 2.44 A28 0.0197 1.558 0.192 0.048 1.65 2.44 A29 0.0076 1.457 0.371 0.048 1.65 2.44 A30 0.0109 1.597 0.338 0.048 1.65 2.44 B34 0.0119 1.387 0.208 0.051 1.65 2.44 B35 0.0186 1.606 0.182 0.051 1.65 2.44 B36 0.0100 1.533 0.380 0.051 1.65 2.44 B37 0.0138 1.868 0.311 0.051 1.65 2.44 B38 0.0152 1.939 0.301 0.051 1.65 2.44 B39 0.0180 2.045 0.285 0.051 1.65 2.44 B40 0.0078 1.929 0.452 0.051 1.65 2.44 B41 0.0094 2.076 0.434 0.051 1.65 2.44

Experimental data of Brown and Chu (1968)

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Appendix G

Experimental data of Delft Hydraulics 1982 (criterion 1-2)

(47)

Appendix H

Experimental data of Delft Hydraulics 1982 (criterion 5-6)

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Stability of stones under non-uniform flow