• Why is bed shear stress important? • Provides an index of fluid force per unit
area on the stream bed, which has been related to sediment mobilization and transport in many theoretical and empirical treatments of sediment transport
Calculation of Bed Shear Stress
• Various methods based on – Reach-averaged relations– Theoretical assumptions about structure of turbulence
– Direct measurements of turbulence
Calculation of Bed Shear Stress
Reach-Averaged MethodMean Bed Shear Stress - force per unit area exerted by a “block” of water on the channel boundary as it moves downstreamF = γWDXsinθ (N) [MLT-2)
τb = γRS (N m-2)(downstream oriented component of the weight of the block)
Reach-Averaged Method
Advantages –Serves as an index of the total resistance by ALL frictional influences on the flow (particle-, bedform-, bar-, and planform-scale effects)
Relatively easy to measure
Disadvantages –Does not provide information on spatial variation in resistance at sub-reach scale
Is not necessarily a good index of the competence of the stream to move sediment
“Law of the Wall” Method
• Based on the assumption that the velocity profile in the lower portion (15-20%) of an open channel flow has a logarithmic structure:
“Law of the Wall” Method
⎟⎟⎠
⎞⎜⎜⎝
⎛κ
=0
* lnzzuu
u = mean velocity (in vertical), u* = shear velocity , κ = von Karman’s constant, z = distance above bed, z0= roughness height (height above bed where velocity goes to zero)
bgRS( / )= τ ρ
“Law of the Wall” Method
00**
0**
0
*
lnln ,
where
lnlnlnln
zmzubum
bzmzuzuzzuu
−=−==
+=−=⎟⎟⎠
⎞⎜⎜⎝
⎛=
κκ
κκκ
“Law of the Wall” Method
Measure mean velocities (u) at various heights above bed in lower 15-20% of the flow
Regress the values of u against the logarithms of z to get estimates of m and bCalculate values of shear velocity, bed shear stress, and roughness height
m/b*b* e,u,mu −=ρ=τκ= 02 z
“Law of the Wall” MethodAdvantages
Provides local measure of shear stress
Can be used to map spatial patterns of shear stress and roughness height at subreach scale
Standard error of estimate of regression can provide an estimate of error in u*
Disadvantages
Flow must conform with logarithmic velocity profile
Errors in measurement of u and z can influence results (least precise of “law of wall” methods)
Variants on “Law of the Wall”
finer is material of % for which size particle d
30/
ln*
p
adzuu
p
p
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
κ=
Advantage
requires only a single near-bed velocity reading in lower 20% of flow for estimate of u*
Disadvantage
requires information on the grain-size distribution of bed material
Applies to gravel-bed rivers only and assumes that empirical relation z0 = (adp/30) applies to all such rivers
a = 3, p = 84 Whiting and Dietrich, 1990
a = 2.85, p = 90 Wilcock et al. 1996
Variants on “Law of the Wall”
logarithms natural of base depthflow
velocityaveraged-depth
)30/(
ln*
===
⎟⎟⎠
⎞⎜⎜⎝
⎛
κ=
ehU
adehuUp
Advantage
Has less variability than other “law of the wall” methods
Disadvantage
requires measurement of velocity profile to determine mean [could perhaps be used with a single measure of U (6/10th depth)]
Evaluation of “Law of the Wall” Precision (Wilcock, 1996)
Lowest precision – slope of velocity profile
Highest precision –depth-averaged velocity
Says nothing about accuracy of the various methods
Direct Measurement: Near-bed Reynolds
Shear Stress
nfluctuatio velocity verticalbed-near nfluctuatio velocity downstream bed-near
==
ρ−=τ'b
'b
'b
'bb
wu
wu
( ) ( )( ) ( ) ( )
2 22 2
2
2 2 22 2 2
3
b d bxz bxy b b b b
b d bxz bxy byz b b b b b b
b
can also look at resultants in 2D and 3D
u w u v
u w u v v w
v near-bed lateral velocity fluctuation
' ' ' '
' ' ' ' ' '
'
τ = τ + τ = −ρ +
τ = τ + τ + τ = −ρ + +
=
Direct Measurement: Near-bed Reynolds Shear Stress
Advantage
Direct measurement of turbulent shear stress near the bed
Disadvantage
How close to the bed do you need to be? (seems to depend on roughness characteristics and purpose of measurement)
Many measurement devices cannot measure velocity fluctuations accurately close to the bed
Need 2-D measurements of turbulent fluctuations
Rough Boundary Layers
Outer LayerLogarithmic LayerRoughness Layer
Form-Induced Sublayer
Interfacial Sublayer(pressure in these two
regions may deviate from hydrostatic and form drag components of the total stress emerge)
Subsurface Layer Boundary Layer Structure over Rough Beds (Nikora et al. 2001)
Shear stress Profile method
Recall τb = γDSFor any level in the flow
i.e. shear stress varies linearly with height above the bed (assumes hydrostatic conditions)
Project profile of shear stresses measured over depth to the bed
1b
g D z Sor
z D
( )
( / )
τ = ρ −
τ = τ −
Shear Stress Profile Method
Advantages
No need to estimate roughness height
Based on shear stress measurement over fluid column
DisadvantagesRequires 2D velocity measurements
Viscous effects or near-bed effects disrupt linear profile of shear stresses near the bed
Turbulent Kinetic Energy Method
19.0
)(5.0
1
1
2'2'2'
≈ρ=τ
++==
CkC
wvukTKE
b
9.02
2'2
=ρ=τ
CwCb
Alternative Formulation
Turbulent Kinetic Energy Method
Advantages
No need to estimate roughness height
Single near-bed reading of 3-D velocities
DisadvantagesHow close to bed
3-D velocity measurements
Values of C1 and C2 not derived from streams or rivers (oceans)
Shear Stress PartioningBed forms and large roughness elements produce form drag (resistance) that differs from the drag required to mobilize grains on the bed of the river
Connecting shear stress to sediment movement requires isolating the portion of the shear stress associated with grain drag
12
20
1 12
Dsf b
sf
C H Hz
ln( )
−⎧ ⎫⎡ ⎤⎪ ⎪τ = τ + −⎨ ⎬⎢ ⎥λκ ⎣ ⎦⎪ ⎪⎩ ⎭
τsf = bed shear stress due to skin frictionτb = total bed shear stressCD = drag coefficient for bedforms (0.21 separated
flows; 0.84 unseparated flows)λ= bedform wavelengthH = bedform height
(from Wiberg and Smith 1989)
Is Turbulent Shear Stress the Right Index?
Some recent studies have questioned whether looking at turbulence stresses is the right approach for understanding sediment transport
Instead look at actual sediment mobilization and transport and relate it to metrics other than shear stress (readings for today)
instantaneous longitudinal velocity and its duration above a threshold level for transport
u = <u> +u’
Thorne et al., 1989
High bedload transport rates occur during sweeps (quad IV) and outward interactions (quad I) when instantaneous U is high and sustained
ReferencesBauer, B. O., D. J. Sherman, and J. F. Wolcott (1992), Sources of uncertainty in shear stress and roughness length estimates derived from velocity profiles, The Professional Geographer, 44, 453-464.
Bergeron, N. E., and A. D. Abrahams (1992), Estimating shear velocity and roughness length from velocity profiles, Water Resources Research, 28, 2155-2158.
Biron, P. M., S. N. Lane, A. G. Roy, K. F. Bradbrook, and K. S. Richards (1998), Sensitivity of bed shear stress estimated from vertical velocity profiles: the problem of sampling resolution, Earth Surface Processes and Landforms, 23, 133-139.
Biron, P. M., C. Robson, M. F. Lapointe, and S. J. Gaskin (2004), Comparing different methods of bed shear stress estimates in simple and complex flow fields, Earth Surface Processes and Landforms, 29, 1403-1415.
Kim, S.-C., C. T. Friedrichs, J. P.-Y. Maa, and L. D. Wright (2000), Estimating bottom stresses in tidal boundary layer from acoustic doppler velocimeter data, Journal of Hydraulic Engineering, 126, 399-406.
Nikora, V. and Goring, D. 2000. Flow turbulence over fixed and weakly mobile gravel beds. Journal of Hydraulic Engineering, 126, 679-690.
Nikora, V., Goring, D., McEwan, I, and Griffiths, G. 2001. Spatially averaged open channel flow over rough bed. Journal of Hydraulic Engineering, 127,123-133.
Thorne, P.D., Williams, J.J. and Heathershaw, A.D. (1989) In situ acoustic measurements of marine gravel threshold and transport. Sedimentology, 36, 61-74.
Wiberg, P.L. and Smith, J.D. 1989. Model for calculating bed load transport of sediment. Journal of Hydraulic Engineering, 115, 101-123.
Wilcock, P. R. (1996), Estimating local bed shear stress from velocity observations, Water Resources Research, 32, 3361-3366.
