Carl Friedrichs, Virginia Institute of Marine Science;

Self-Burial of Objects on Sandy Beds by Scour: A Synthesis of Observations
Carl Friedrichs, Virginia Institute of Marine Science; Sarah Rennie & Alan Brandt, Johns Hopkins University Applied Physics Laboratory 1. Introduction: Data sets for equilibrium self-burial and relation to classic work 2. Equilibrium self-burial of objects under currents 3. Equilibrium self-burial of objects under waves or waves plus currents (Rennie & Brandt 2014)

Data Sources for this Synthesis

Relation to Classic Work
e /D , S /D = 0.1 (KC) 0.5 L Sumer et al. (2001) D Cylinders (pipe segments) with waves only N = 25 observations (for e/D) dsed = 0.18 mm Umax = 0.13 to 0.50 m/s D = 2 to 10 cm L/D = 6 to 30 T = 1.5 to 3.5 sec robj = 1.3 to 6.0 g/cm3 q = to 0.35 (2001) EMBEDMENT KC = Umax T/D

Relation to Classic Work B / D = 0.1 (KC) 0.5
// Relation to Classic Work B / D = 0.1 (KC) 0.5 (Sumer et al. best-fit) 1.0 L D B/D Cylinders with waves only & D ≥ 3 cm N = 234 observations dsed = 0.18 to 0.30 mm Umax = 0.13 to 0.80 m/s D = 4 to 25 cm L/D = 2 to 12 T = 1.9 to 12 sec robj = 1.3 to 7.8 g/cm3 q = to 0.54 0.1 // 3 10 100 ∞ KC = Umax T/D

// Relation to Classic Work B / D = 0.1 (KC) 0.5 1.0 a Wave orbital Cylinders in field under waves + current (all others from lab experiments) B/D Cylinders under waves with a < 60o (all others a ≥ 60o) 0.1 Cylinders with D < 2.6 cm for waves only & Cylinders with D < 2.6 cm for currents only (all others have D ≥ 3.0 cm) Cylinders & spheres with waves only // Cylinders with waves + currents 3 10 100 ∞ Cylinders & spheres with currents only KC = Umax T/D

// Relation to Classic Work B / D = 0.1 (KC) 0.5 Cylinders in field under waves + current (all others from lab experiments) 1.0 Cylinders under waves with a < 60o (all others a ≥ 60o) Cylinders with D < 2.6 cm for waves only & Cylinders with D < 2.6 cm for currents only (all others have D ≥ 3.0 cm) B/D Cylinders & spheres with waves only 0.1 Cylinders with waves + currents Cylinders & spheres with currents only Cones with waves only Cones with waves + currents // Tapered cylinders with waves only 3 10 100 ∞ Tapered cylinders with currents only KC = Umax T/D

Object Burial Under Currents Only
Cylinders with D < 2.6 cm for currents only 1.0 N = 91 observations dsed = 0.22 to 0.42 mm Umax = 0.13 to 0.60 m/s D = 1.27 or 2.54 cm (0.5 or 1 inch) L/D = 4 to 18 robj = 1.8 to 10.6 g/cm3 B/D Cylinders with D > 8 cm for currents only N = 24 observations dsed = 0.18 to 0.42 mm Umax = 0.20 to 0.59 m/s D = 9 to 11 cm L/D = 2 to 20 robj = 1.2 to 6.0 g/cm3 0.1 0.01 0.1 q = b / [(sed – w) g d50]

Cylinders with D < 2.6 cm for currents only 1.0 N = 91 observations dsed = 0.22 to 0.42 mm Umax = 0.13 to 0.60 m/s D = 1.27 or 2.54 cm (0.5 or 1 inch) L/D = 4 to 18 robj = 1.8 to 10.6 g/cm3 B/D Cylinders with D > 8 cm for currents only N = 24 observations dsed = 0.18 to 0.42 mm Umax = 0.20 to 0.59 m/s D = 9 to 11 cm L/D = 2 to 20 robj = 1.2 to 6.0 g/cm3 0.1 qcr = 0.040 Clear water scour Live bed scour 0.01 0.1 q = b / [(sed – w) g d50]

B / D = a q b = fq a = 3.4, b = 0.54 Cylinders with D < 2.6 cm for currents only 1.0 a = 2800, b = 2.6 N = 91 observations dsed = 0.22 to 0.42 mm Umax = 0.13 to 0.60 m/s D = 1.27 or 2.54 cm (0.5 or 1 inch) L/D = 4 to 18 robj = 1.8 to 10.6 g/cm3 a = 830, b = 2.4 B/D a = 1.3, b = 0.36 Cylinders with D > 8 cm for currents only N = 24 observations dsed = 0.18 to 0.42 mm Umax = 0.20 to 0.59 m/s D = 9 to 11 cm L/D = 2 to 20 robj = 1.2 to 6.0 g/cm3 0.1 qcr = 0.040 Clear water scour Live bed scour 0.01 0.1 q = b / [(sed – w) g d50]

B / D = a q b = fq a = 3.4, b = 0.54 Cylinders with D < 2.6 cm for currents only 1.0 a = 2800, b = 2.6 Cylinders with D > 8 cm for currents only Spheres for currents only N = 6 observations dsed = 0.19 mm Umax = 0.37 to 0.40 m/s D = 6 to 7 cm robj = 2.0 to 4.0 g/cm3 a = 830, b = 2.4 B/D a = 1.3, b = 0.36 a = 1.1, b = 0.36 0.1 Tapered (bullet-shaped) cylinders for currents only qcr = 0.040 N = 5 observations dsed = 0.42 mm Umax = 0.30 to 0.58 m/s D = 8 cm robj = 2.4 g/cm3 Clear water scour Live bed scour 0.01 0.1 q = b / [(sed – w) g d50]

B / D = a q b = fq a = 3.4, b = 0.54 Cylinders with D < 2.6 cm for currents only 1.0 a = 2800, b = 2.6 Cylinders with D > 8 cm for currents only Spheres for currents only Tapered (bullet-shaped) cylinders for currents only a = 830, b = 2.4 B/D a = 1.3, b = 0.36 a = 1.1, b = 0.36 Next, divide B/D by a q b = fq and plot on y-axis. (B / D) fq-1 0.1 qcr = 0.040 Clear water scour Live bed scour 0.01 0.1 q = b / [(sed – w) g d50]

(B / D) fq-1 robj /rw Object Burial Under Currents Only:
1.4 Object Burial Under Currents Only: Effect of Object Density (B / D) fq-1 = 0.9 (robj /rw ) 0.1 = fro 1.2 Cylinders with D > 8 cm for currents only N = 24 observations dsed = 0.18 to 0.42 mm Umax = 0.20 to 0.59 m/s D = 9 to 11 cm L/D = 2 to 20 robj = 1.2 to 6.0 g/cm3 (B / D) fq-1 1.0 0.8 R2 = 0.18, p < 0.05 1.0 3.0 10.0 robj /rw

(B/D)predicted = fro fq
1.4 Object Burial Under Currents Only: Effect of Object Density (B / D) fq-1 = 0.9 (robj /rw ) 0.1 = fro 1.2 Cylinders with D > 8 cm for currents only N = 24 observations dsed = 0.18 to 0.42 mm Umax = 0.20 to 0.59 m/s D = 9 to 11 cm L/D = 2 to 20 robj = 1.2 to 6.0 g/cm3 (B / D) fq-1 1.0 The final best-fit equation as a function of q and is then robj /rw 0.8 (B/D)predicted = fro fq R2 = 0.18, p < 0.05 = 0.9 (robj /r) 0.1 a q b 1.0 3.0 10.0 robj /rw

1.0 The final best-fit equation as a function of q and is then robj /rw B/D (B/D)predicted = fro fq 1:1 = 0.9 (robj /rw) 0.1 a q b 0.1 where a and b are a function of object type and clear-water vs. live-bed scour, and density correction is applied to cylinders with D > 8 cm. R2 = 0.85 0.01 0.01 0.1 1.0 (B/D)predicted = fro fq

B/D Burial w/ Waves or Waves + Currents KC = Umax T/D 1.0 0.1 2 10 100
Cylinders in field under waves + current (all others from lab experiments) 1.0 Cylinders under waves with a < 60o (all others a ≥ 60o) Cylinders with D < 2.6 cm for waves only & (all others have D ≥ 3.0 cm) Cylinders with waves only B/D Spheres with waves only Cylinders with waves + currents 0.1 Cones with waves only Cones with waves + currents Tapered cylinders with waves only 2 10 100 KC = Umax T/D

B / D = a (KC) b = fKC B/D Burial w/ Waves or Waves + Currents
Cylinders in field under waves + current (all others from lab experiments) 1.0 a = 0.10, b = 0.51, R2 = 0.62 Cylinders under waves with a < 60o (all others a ≥ 60o) Cylinders with D < 2.6 cm for waves only & (all others have D ≥ 3.0 cm) B/D Cylinders with waves only Spheres with waves only Cylinders with waves + currents 0.1 Cones with waves only Cones with waves + currents a = , b = 0.78, R2 = 0.63 Tapered cylinders with waves only 2 10 100 KC = Umax T/D (Catano-Lopera et al. 2011)

B / D = a (KC) b = fKC B/D Burial w/ Waves or Waves + Currents
Cylinders in field under waves + current (all others from lab experiments) 1.0 a = 0.10, b = 0.51, R2 = 0.62 Cylinders under waves with a < 60o (all others a ≥ 60o) Cylinders with D < 2.6 cm for waves only & (all others have D ≥ 3.0 cm) B/D Cylinders with waves only Spheres with waves only Cylinders with waves + currents 0.1 Cones with waves only Cones with waves + currents a = , b = 0.78, R2 = 0.63 Tapered cylinders with waves only Next, divide B/D by a (KC) b = fKC and plot (B / D) fKC-1 on y-axis. 2 10 100 KC = Umax T/D

(B/D)predicted = fUc|| fKC
Burial w/ Waves or Waves + Currents: Effect of Parallel Currents R2 = 0.07, p < 0.001 1.0 Cylinders w/ waves + currents Cones with waves + currents (B / D) fKC-1 = e-1.1 (Uc|| /Umax) = fUc|| N = 189 observations, dsed = 0.25 mm Umax = 0.27 to 0.77 m/s D = 5 to 25 cm, L/D = 2 to 4 robj = 2.0 to 7.8 g/cm3 (B / D) fKC-1 The best-fit equation is then: (B/D)predicted = fUc|| fKC = e-1.1 (Uc|| /Umax) a KC b 0.1 0.0 0.2 0.4 0.6 Uc|| /Umax

(B/D)predicted = fUc|| fKC
Burial w/ Waves or Waves + Currents: Effect of Parallel Currents R2 = 0.07, p < 0.001 1.0 Cylinders w/ waves + currents Cones with waves + currents (B / D) fKC-1 = e-1.1 (Uc|| /Umax) = fUc|| N = 189 observations, dsed = 0.25 mm Umax = 0.27 to 0.77 m/s D = 5 to 25 cm, L/D = 2 to 4 robj = 2.0 to 7.8 g/cm3 (B / D) fKC-1 The best-fit equation is then: (B/D)predicted = fUc|| fKC = e-1.1 (Uc|| /Umax) a KC b 0.1 Next, divide (B / D) fKC-1 by fUc|| & plot on y-axis. (B / D) fKC-1fUc|| -1 0.0 0.2 0.4 0.6 Uc|| /Umax

(B/D)predicted = fa fUc|| fKC
Burial w/ Waves or Waves + Currents: Effect of Wave Angle R2 = 0.48, p < 10-7 1.0 Cylinders under waves with a < 60o N = 49 observations, dsed = 0.25 to 0.30 mm Umax = 0.19 to 0.77 m/s, a = 0 to 52 deg D = 4 to 25 cm, L/D = 2 to 4 robj = 2.0 to 7.8 g/cm3 (B / D) fKC-1 fUc||-1 The best-fit equation is then: (B/D)predicted = fa fUc|| fKC = e-3.4 (cos a – 0.6) e-1.1 (Uc|| /Umax) a KC b (B / D) fKC-1fUc||-1 = e-3.4 (cos a – 0.6) = fa 0.1 a Wave orbital 0.0 0.1 0.2 0.3 0.4 cos a – 0.6

(B/D)predicted = fa fUc|| fKC
Burial w/ Waves or Waves + Currents: Effect of Wave Angle R2 = 0.48, p < 10-7 1.0 Cylinders under waves with a < 60o N = 49 observations, dsed = 0.25 to 0.30 mm Umax = 0.19 to 0.77 m/s, a = 0 to 52 deg D = 4 to 25 cm, L/D = 2 to 4 robj = 2.0 to 7.8 g/cm3 (B / D) fKC-1 fUc||-1 The best-fit equation is then: (B/D)predicted = fa fUc|| fKC = e-3.4 (cos a – 0.6) e-1.1 (Uc|| /Umax) a KC b (B / D) fKC-1fUc||-1 = e-3.4 (cos a – 0.6) = fa 0.1 Next, divide (B / D) fKC-1fUc||-1 by fa & plot (B / D) fKC-1fUc||-1fa-1 on y-axis. 0.0 0.1 0.2 0.3 0.4 cos a – 0.6

Burial w/ Waves or Waves + Currents:
Effect of q Cylinders in field under waves + current (all others from lab experiments) Cylinders under waves with a < 60o (all others a ≥ 60o) 1.0 Cylinders with D < 2.6 cm for waves only & (all others have D ≥ 3.0 cm) (B / D) fKC-1 fUc||-1 fa-1 Cylinders with waves only Cylinders with waves + currents Spheres with waves only Tapered cylinders with waves only Bed fluidization effects Cones with waves only 0.1 Clear-water scour Cones with waves + currents Live-bed scour 0.03 0.1 0.3 1.0 q = b / [(sed – w) g d50]

(B / D) fKC-1 fUc||-1 fa-1 Burial w/ Waves or Waves + Currents:
Effect of q (B / D) fKC-1 fUc||-1 fa-1 = c q d = fqW c = 3.6, d = 0.33 Cylinders in field under waves + current (all others from lab experiments) tapered Cylinders under waves with a < 60o (all others a ≥ 60o) 1.0 Cylinders with D < 2.6 cm for waves only & (all others have D ≥ 3.0 cm) (B / D) fKC-1 fUc||-1 fa-1 R2 = 0.29, p < 10-25 Cylinders with waves only Cylinders with waves + currents cylinders Spheres with waves only c = 1.8, d = 0.33 Tapered cylinders with waves only c = 1.4, d = 0.33 Bed fluidization effects spheres Cones with waves only c = 1, d = 0 0.1 Clear-water scour Cones with waves + currents Live-bed scour 0.03 0.1 0.3 1.0 q = b / [(sed – w) g d50]

(B/D)predicted = fa fUc|| fKC fqW (B / D) fKC-1 fUc||-1 fa-1
Burial w/ Waves or Waves + Currents: Effect of q (B / D) fKC-1 fUc||-1 fa-1 = c q d = fqW c = 3.6, d = 0.33 tapered The final best-fit equation is then: 1.0 (B/D)predicted = fa fUc|| fKC fqW (B / D) fKC-1 fUc||-1 fa-1 = e-3.4 (cos a – 0.6) e-1.1 (Uc|| /Umax) a KC b c q d cylinders c = 1.8, d = 0.33 c = 1.4, d = 0.33 Bed fluidization effects spheres 0.1 Clear-water scour Live-bed scour 0.03 0.1 0.3 1.0 q = b / [(sed – w) g d50]

(B/D)predicted = fa fUc|| fKC fqW B/D
Burial w/ Waves or Waves + Currents: 1.0 The final best-fit equation as a function of KC, Uc|| , a and q under waves or waves + currents is then 1:1 (B/D)predicted = fa fUc|| fKC fqW B/D 0.1 = e-3.4 (cos a – 0.6) e-1.1 (Uc|| /Umax) a KC b c q d where a, b, c and d are a function of object type. R2 = 0.85 0.01 0.01 0.1 1.0 (B/D)predicted = fKC fUc|| fa fqW

Carl Friedrichs, Sarah Rennie & Alan Brandt
Self-Burial of Objects on Sandy Beds by Scour: A Synthesis of Observations Carl Friedrichs, Sarah Rennie & Alan Brandt Conclusions 1. The main factor which increases B/D under steady currents is increasing q, with different power laws for clear-water and live-bed scour. 2. Under steady currents, greater B/D is observed as function of q for small cylinders (D < 3 cm) and for tapered cylinders, and smaller B/D is observed for spheres. 3. The main factor which increases B/D under waves or waves + currents conditions is increasing KC, with distinct power laws for cones vs. other shapes, such that cones bury less. 4. Under waves, B/D decreases as the strength of Uc|| relative to Umax increases, B/D decreases as the wave angle a decreases, and B/D increases as q increases.

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