Settling-velocity based criteria for incipient sediment motion

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1 Settling-velocity based criteria for incipient sediment motion Nian-Sheng Cheng School of Civil and Environmental Engineering Nanyang Technological University (NTU), Singapore 2008

2 Settling velocity is used for many applications - Critical condition for incipient sediment motion (Cheng and Chiew 1999; Egiazaroff 1965; Komar and Clemens 1986; Le Roux 1998; Yang 1973) - Sediment transport rate (Bagnold 1973) and sediment transport capacity (see Qian and Wan 1999) - Clarification of bedforms (Liu 1957)

3 Incipient sediment motion Rubey (1933) noticed that u bc is approximately of the same order of w. This approximation was later adopted by Egiazaroff (1965). Other studies include Yang (1973); Komar and Clemens (1986); Le Roux (1998).

4 Bed-load function Einstein (1950) used w for characterizing the time scale of sediment entrainment Bedform Liu (1957) used w in identifying the formation of ripples Sediment transport capacity w was involved in many empirical formulas (see Qian and Wan 1999)

5 An intuitive view (Komar and Clemens, 1986) A connection between threshold condition and w is unlikely or at most spurious Reason 1: threshold occurs when the grain is essentially at rest Reason 2: settling becomes important only when the particle is suspended

6 w is a good representation of drag Fluid flowing around a bed particle induces a drag This drag is comparable to the resistance exerted by fluid as the particle settles

7 Objective of this study To simplify the theoretical result from the probabilistic studies by Cheng and Chiew (1998; 1999) To propose a formula that relates critical shear velocity (u *c ) to settling velocity (w) To compare with other results available in the literature

8 Cheng and Chiew (1999) 10 Initial motion Initial suspension 1 τ Suspended load 0.1 Bed load Stationary bed Re *

9 From Cheng & Chiew (1999), the following function is derived, τ * c = ( ) α[re Re exp( 0.093Re )] + 5 * c * c 1.32 Re 2 * c * c / 2 α = ( 2.5π ln 4 p 4 p ) 0 c c α = for incipient sediment motion at p c = 10-7 α = 5.07 for threshold of suspension at p c = 0.01

10 Dimensionless parameters: τ* = u* 2 /(ΔgD) dimensionless shear stress (Shields number) Re* = u*d/ν Δ = ρ s /ρ-1 shear Reynolds number effective specific gravity

11 10 1 Shields curve (Vanoni, 1975) Yalin and Karahan (1979) Chien and Wan (1983) P = 10^(-7) τ Re* Critical condition for incipient sediment motion (Cheng and Chiew 1999)

12 Consider two extreme conditions For fully-rough boundaries, Re *c is very large and Eq. (1) reduces to τ *c = For fully-smooth boundaries, Re *c is very small and Eq. (1) is simplified to τ *c = Re *c -0.3

13 If using the settling velocity, we get u c = w * D* for D * < 2 u* c = 0.21 for D w * > 20

14 10 u * c w 1 u c = w * D* u c 8.6 = D * * w 0.16 u* c = 0.21 w D *

15 Critical condition defined as a w-based function 1/ n 1/ n u * c 8.6 1/ n = w + D * (n = 0.05) or u w * c 1/ n 41 = D * n

16 Settling velocity is computed by Re w = ( D * 5) (Cheng 1997) Re w = wd/ν Appliable for naturally worn sediment grains It reduces to Re w = 0.042D *3 for very small D * Re w = 1.15D * 1.5 for very large D *

17 Optimization analysis indicates that with n = 0.05, the simplified eq agrees with the original τ *c Re *c within ±3.3% for Re *c =

18 τ *c 10 1 Lower limit, Qian & Wan (1999) Upper limit, Qian & Wan (1999) Data used by Shields (1936) Yalin & Da Silva (2001) Whitehouse et al (2000) This study Re *c

19 Comparisons with Other w-based Formulas

20 (1) Egiazaroff s (1965) assumption Critical near-bed velocity u bc = Settling velocity w It appears very crude and has been considered not correct, e.g. by Bagnold (1973), Komar and Clemens (1986). Theoretical analysis of the threshold condition for the entrainment of non-uniform sediment, performed based on the assumption, fits well some experimental observations

21 ) ( ln ) ( ) ( = + s k k s e y y u Computation of u b u + = u/u *, y + = u * y/ν, k s + = u * k s /ν Generalized law-of-the-wall function

22 25 u + = y + u u + = 2.5lny This study (m = -3) Spalding (1961) oooo van Driest (1956) y + 3 Law of the wall for smooth boundary

23 25 u k s + = y + 3 Law of the wall for smooth and rough boundaries

24 The generalized law-of-the-wall function is applicable for viscous sublayer buffer layer logarithmic layer and also for fully-smooth bed fully-rough bed transitional bed.

25 Computing steps Given D*, first compute u *c /w and Re *c Then find out u bc /u *c Finally work out relation of u bc /w with D * Cases considered y b /D = 0.25, 0.30,, 0.75 k s /D = 1, 1.25, 1.5,, 4.0 The results obtained for over 100 cases were then averaged to yield an average relation between u bc /w and D *

26 u bc w a = 1, b = 0.25 a = 2.5, b = 0.25 a = 4, b = 0.25 a = 1, b = 0.75 a = 2.5, b = 0.75 a = 4, b = 0.75 average D *

27 Observations u bc /w varies slightly for D * > 2, but increases significantly for smaller D *. The average curve is very close to the relation computed with y b /D = 0.5 and k s /D = 2.5

28 Implications The relation u b = w is very reasonable for natural sediment in the size of sand or larger particles. The size range of sand and gravel is D = mm, and if ν = 10-6 m 2 /s and ρ s /ρ = 2.65, then D * = (i.e. D* > 2)

29 Implications The relation (u bc = w) is helpful in the formulation of hydrodynamic forces, lift and drag, for bed particles at the threshold of motion. u bc depends on the near-bed flow structure that is generally complex By engaging w rather than u bc, some simplifications are possible

30 Appreciation of Bagnold s bedload argument Bedload transport rate is proportional to the velocity difference, u b -w Because u b and w have the different directions, the use of the velocity difference in such a way is not easy to understand With w u bc, the Bagnold s argument can be appreciated in terms of (u b u bc ) or equivalently (u * u *c ).

31 (2) Yang s (1973) formula U c w = log( u 2.05 * 2.5 D / ν) for u * ν D 70 for 1.2 < u * ν D < 70 U c is the critical depth-averaged velocity

32 U c w This study (h/d=10) (h/d=20) (h/d=200) Yang (1973) Data used by Yang (1973) Re *c 110 3

33 (3) Le Roux s (1998) formula τ * c = ln w* ln w* for for for w * < < w * w * > < w * is dimensionless settling velocity defined as w 3 * = w Δgν = Rew/ / D *

34 τ *c This study Le Roux (1998) Yalin & Da Silva (2001) Whitehouse et al (2000) w *

35 Conclusions u *c for incipient sediment motion, when scaled with w, is solely related to the dimensionless sediment diameter. w can be used to represent the effective u bc, which is experienced by a typical bed sediment particle under the threshold condition, but only for large sediment sizes such as sand and gravel.

36 Conclusions Yang s (1973) formula is suitable for flows with small relative flow depth Le Roux s (1998) formula may underestimate the critical condition for very fine sediments The generalized wall-layer function predicts the near-bed velocity distribution throughout the viscous sublayer, buffer layer and logarithmic layer for fully-smooth, fully-rough and transitional boundary conditions.

Cheng, N. S. (2006). Influence of shear stress fluctuation on bed particle instability. Physics of Fluids. 18 (9): Art. No

Cheng, N. S. (2006). Influence of shear stress fluctuation on bed particle instability. Physics of Fluids. 18 (9): Art. No Cheng, N. S. (006). Influence of shear stress fluctuation on bed particle instability. Physics of Fluids. 8 (9): Art. No. 09660. Influence of shear stress fluctuation on bed particle mobility Nian-Sheng