Particulate Gravity Currents with Resuspension · 2016-05-06 · Particulate Gravity Currents with...

Types Experiments DNS KSB Debris Debris Flows Conclusions

Particulate Gravity Currents with

Resuspension

Jim McElwaine

Professor of GeohazardsDepartment ofEarth Sciences

Durham University


Acknowledgments

Betty Sovilla Barbara Turnbull

Kouichi Nishimura Christophe Ancey

Dieter Issler Takahiro Ogura

Eckart Meiburg Shane Byrne

Nathalie Vriend Jan-Thomas Fischer

Liz Bowman


Plan of Talk

• Transition Experiments

• Direct Numerical Simulations

• Dynamic Models


Powder Avalanche on K2 Pierre Beghin


Head of Powder Snow Avalanche Cemagref


Slab Avalanche Fracture Line


Test Chute in Davos film


Schematic of Avalanche, Turbidity Current, Pyroclastic Flow

Base

Powder cloud

Nose

Head

Body

Transition regionDense flow

ρ1

ρ2

gθ

x

z

umax

uh

uf


Aim

Understand formation of Suspension Currents

Use steep slopes to give a low Richardson number for large

density difference

Transition to suspension

Limited entrainment

Steady flows

Understand air interaction

Comparison of field observations with experiments


Similarity Criteria

Experiment Materials Rep Ri St∆ρ

ρa

Re

Powder snow avalanches Snow-air 3000 1 0.02 10 109

Ancey (2004) sawdust 50 1.7 0.006 0.05 104

water

Bozhinskiy (1998) aluminium 0.1 20 0.03 1 103

air

Beghin (1981) Brine-water/ – 5 – 0.02 104

Beghin (1983) Sand-waterBeghin (1983) suspension

Nishimura (1998) Ping-pong balls 2 × 104 2 10 50 107

McElwaine (2001) air

Hampton (1972) Kaolinite and – < 0.5 – 0.1 –water slurry

Hermann (1987) Polystyrene powder 1.5 0.1 10−4 0.002 104

water

Hopfinger (1977) Brine – 2 – 0.01 103

Tochon-Danguy (1974) water

Present Study Snow-air 150 1 10 10 105

Present Study Polystyrene-air 150 2 1 5 104


Similarity Summary

Exact similarity of Ri and ∆ρ/ρa

Re and Rep not matched but qualitatively similar

St is different ⇒ sedimentation is important

Slope angles are different.

Appears unimportant at high Re


Experiments

Air pressuresensor

��

2m

3m

Sieve

Tube

Funnel

front view

Funnel

Sieve

Tube

��

Angles 45° to 90°

Hoist

side view


Side View 8 Litre Avalanches

31.5o slope

58.5o slope

91.0o slope

100 ml side 100 ml front8000 ml side 8000 ml front


Non-dimensional Height — h = hV 1/3

45 50 55 60 65 70 75 80 85 900

0.1

0.2

0.3

0.4

0.5

0.6

slope angle (o)

no

nd

ime

nsio

na

l fin

al head h

eig

ht 0.5 l

1.0 l2.0 l5.0 l


Polystyrene balls


Polystyrene balls on 70◦ surface

g


Ping-Pong Avalanches


Non-Dimensional Velocity — u = u

V16 g

12

50 60 70 80 900

2

4

6

θ (o)

~ u 0

0.5 l

1.0 l

2.0 l

5.0 l

50 60 70 80 900

2

4

6

θ (o)

~ u 1


Front Velocities at the K-Point

100

102

104

106

2

3

4

5

10

15

Number of balls (log scale)

Ve

locity (

ms−

1)

(lo

g s

ca

le)

v =1.8n1/6

experiment


Pressure Theory Comparison

0 0.05 0.1 0.15 0.2 0.25

−2

−1

0

1

2

3

Time (s)

Pre

ssure

(pa)

data dynamic fit hydrostatic fit

Air pressure in a small powder cloud


DNS simulations and Ping-Pong experiments

−0.4 −0.2 0 0.2−140

−120

−100

−80

−60

−40

−20

0

20

40

60

Time (s)

Overp

ressure

(P

a)

experimental data p = 1/2ρ

a v

2 (R/r)

3 [2−(R/r)

3]

v=10.2m/s, R=1.0m, r=R−vt

Air pressure data from aping-pong ball avalanches

and comparison with theory

0 200 400−0.01

−0.005

0

0.005

0.01

0.015

0.02

TimeP

ressure

Air pressure data from a di-

rect numerical simulation ofa gravity current


Front Instability, 5 l polystyrene, 51 ◦ slope


Conclusions

Transition to suspension can be achieved in the laboratory

Can deduce avalanche length, height, speed, and

front angle from pressure data

Good agreement between theory, experiments and field

observations

Pressure measurements can distinguish suspended from

dense flows

Coherent internal velocities can be twice front velocity

Take care estimating forces !


Direct Numerical Simulations

• Meiburg Code

2d spectral with compact finite differences

• Diablo from John taylor

3d spectral with low order finite differences

• Simulation region 8×1

• Release area 2×0.5

• Slope angles 0–90◦

• Boussinesq and non-Boussinesq

Test hypothesis:

stagnation point is lowest point as Re→ ∞


Time evolution, Re=32,000, Slope=10◦film front

0

4

8

Positio

n

0 5 10 15 20

−50

0

50

Angle

Time

0

0.1

0.2

Heig

ht

nose

lowest

stagnation


Time evolution, Re=32,000, Slope=60◦film front

0

4

8

Positio

n

0 5 10 15 20

−50

0

50

Angle

Time

0

0.1

0.2

Heig

ht

nose

lowest

stagnation


Re Comparison at slope 0◦

2k 00deg, s=30, t= 5.80 2knb 00deg, s=30, t= 5.80

4k 00deg, s=30, t= 5.80 4knb 00deg, s=30, t= 5.80

8k 00deg, s=30, t= 5.80 8knb 00deg, s=30, t= 5.80

16k 00deg, s=30, t= 5.80 16knb 00deg, s=30, t= 5.80

32k 00deg, s=30, t= 5.80 32knb 00deg, s=30, t= 5.80

64k 00deg, s=30, t= 5.80



2k 20deg, s=30, t= 5.80 2knb 20deg, s=30, t= 5.80

4k 20deg, s=30, t= 5.80 4knb 20deg, s=30, t= 5.80

8k 20deg, s=30, t= 5.80 8knb 20deg, s=30, t= 5.80

16k 20deg, s=30, t= 5.80 16knb 20deg, s=30, t= 5.80

32k 20deg, s=30, t= 5.80 32knb 20deg, s=30, t= 5.80



2k 40deg, s=30, t= 5.80 2knb 40deg, s=30, t= 5.80

4k 40deg, s=30, t= 5.80 4knb 40deg, s=30, t= 5.80

8k 40deg, s=30, t= 5.80 8knb 40deg, s=30, t= 5.80

16k 40deg, s=30, t= 5.80 16knb 40deg, s=30, t= 5.80

32k 40deg, s=30, t= 5.80 32knb 40deg, s=30, t= 5.80


Stagnation Point Angle

0 20 40 60 80−5

0

5

10

15

20

25

Slope angle

Sta

gnation A

ngle

Devia

tion

2k

4k

8k

16k

32k

64k


Front Speed — High angles evaporate

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

Slope angle

Fro

nt S

peed

2k

4k

8k

16k

32k

64k


Hindered Sedimentation, with Particle Pressure

∂φ

∂t+∇ · q = 0,

where particle flux

q = uφ+ usφ(1 − αφ)− D∇φ

1 − βφ

Sedimenting Boundary condition n · ∇(n · q) = 0 or ∇φ = 0

Resuspending boundary condition n · q = 0

Use mixed compact finite differences and finite volume

schemes to exactly conserve mass.


Sedimentation vs Resuspension

2 3 4 5

0 0.5 1 1.5 2

0 1 2 30

0.5z

(b)x

φ

FSOF

HSZF

x

t = 5 t = 15


2D with no-slip and hindered sedimentation, Re = 2 000


2D with slip and hindered sedimentation, Re = 2 000


3D with no-slip and hindered sedimentation, Re = 4 000


3D with slip and hindered sedimentation, Re = 4 000


Comparison

2D Re=2 000 no-slip

2D Re=4 000 no-slip

2D Re=2 000 slip

2D Re=4 000 slip

3D Re=2 000 no-slip

3D Re=4 000 no-slip

3D Re=2 000 slip

3D Re=4 000 slip


Kulikovskiy–Sveshnikova–Beghin (KSB)Three conservation equations

volumedV

dt= qs + qa

buoyancydB

dt= (ρs − ρa)qs

momentumd

dt

{

[

B + (1 + χ)Vρa

]

u

}

= Bg sin θ

ρs snow density ρa air density χ added mass

g gravity θ slope angle

qs snow flux qa air flux


Geometric Closure

s

efu δt

ρ

ρa

h

yh

ρ

xu

s

θ

L

k = h/L aspect ratio

V = πhL volume

χ = k added mass

uf = u + dLdt front velocity


Flux Closures

qs = uf he − βu√

V snow entrainment/detrainment

qa = (αu − us)√

V air entrainment/detrainment

α(Ri) =

{

e−λRi

2Ri ≤ 1

e−λ

RiRi > 1

Ri =ρ− ρa

ρa

gh cos θ

u2

he erodible snow depth β mass loss coefficient

us sedimentation velocity


Comparison with Velocity Data

0 200 400 600 800 1000 1200 1400 1600 18000

20

40

60

80

100

x (m)

uf (

m s

−1)

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

s (m)

uf (

m s

−1)

Avalanche no. 200 Avalanche no. 509

0 200 400 600 800 1000 1200 14000

10

20

30

40

50

60

70

x (m)

uf (

m s

−1)

Avalanche no. 628


Comparison with Ping-Pong Avalanches

0

5

10

15

2021,000 balls

Velo

city (

m/s

)

0 50 100 1500

5

10

15

2042,000 balls

Velo

city (

m/s

)

Flow distance (m)

84,000 balls

168,000 balls

0 50 100 150

320,000 balls

Flow distance (m)

0

5

10

15

2010,500 balls

Velo

city (

m/s

)measurements with curvature without curvature


Illgraben Situation


Illgraben Bridge


Height and Stress Data

22:44:24 22:46:28 22:50:00 22:55:000

0.5

1

1.5

2

2.5

h [m

]

height

22:44:24 22:46:28 22:50:00 22:55:000

10

20

30

N,P

[K

N/m

2]

normal stress

fluid pore pressure

Ratio of shear force to normal force


Motivation – Zeroth order shallow water

∂u

∂t= g sin θ − µg cos θ

Chezy µ(Fr,h/d , θ) ∝ Fr2

Viscous µ(Fr,h/d , θ) ∝ Fr

µ should also depend on solids concentration at the bottom

Fr = u√gh cos θ


Two phase debris flow model

After much algebra . . .

vertical equation of mass conservation is

∂tα = ∂y [Vα(1 − α) + D∂yα] = ∂yD [Peα(1 − α) + ∂yα]

α density of solids relative to max

V sedimentation velocity

D diffusion

Pe = V/D Peclet number

This is a diffusion equation with hindered settling and the same

as Gray’s segregation theory


Reduction to slow manifold

Define the vertical centre of mass for the solids fraction

hp(t) =

∫ H

0

yα(t , y) dy .

Then use previous result and more algebra to get

dhp(t)

dt=

h∗

p − hp(t)

T

[

1 + ǫ(h∗

p − hp(t) + · · ·]

.

For resuspension dominated regime h∗

p = H2

and mixture

becomes well mixed

For sedimenation dominated regime h∗

p =Hp

2 where Hp is total

height of particles at maximum packing fraction

Excess pore pressure is

p

ρf gy=

c1(2hp − Hp) + c2(hp − h∗

p)

H − Hp


Complete Set of Depth Averaged Equations

∂tM + ∂x (MU) = 0

∂t (cM) + ∂x [cMU (1 + a1m1)] = 0

∂t (UM) + ∂x

[

MU2(a2 + a3m21) + gyhM

]

= Mgx − µMgy

∂tm1 + ∂x [m1U + (a4 + a5m21)U] =

m∗

1 − m1

Tm[1 + · · · ]

m1 = 2hp/H − 1 dimensionless deviation from mixed

M total mass hold up

c relative concentration of particles

h centre of mass


Conclusions

Resuspension very important in many geophysical flows

DNS expensive but can reproduce two layer structure

Integral point mass models work well for some cases

Depth integrated equations with vertical resdistribution a

good compromise

Easy to construct empirical models

Also works for n component mixtures and segregation


Publications

• N.A. Konopliv and S.G. Llewellyn-Smith and J.N.McElwaine and E. Meiburg. Modeling gravity cur-rents without an energy closure. J. Fluid Mech.,789:806–829, 2016.• B. Sovilla, J.N. McElwaine, and M.Y. Louge. Thestructure of powder snow avalanches. ComptesRendus Physique, 16(1):97–104, 2015.• B. Turnbull, E.T. Bowman, and J.N. McElwaine.Debris flows: Experiments and modelling. ComptesRendus Physique, 16(1):86–96, 2015.• M. Ash, P.V. Brennan, C.J. Keylock, N.M. Vriend,J.N. McElwaine, and B. Sovilla. Two-dimensionalradar imaging of flowing avalanches. Cold RegionsScience and Technology, 102:41–51, 2014.• A.J. Hafiz, J.N. McElwaine, and C.P. Caulfield.The instantaneous froude number and depth of un-steady gravity currents. J. Hydraulic Res., 51:432–445, 2013.• J. Kowalski and J.N. McElwaine. Shallow two-component gravity-driven flows with vertical varia-tion. J. Fluid Mech., 714:434–462, 2013.• N. M. Vriend, J.N. McElwaine, B. Sovilla, C. J.Keylock, M. Ash, and P. V. Brennan. High resolutionradar measurements of snow avalanches. Geo. Res.Let., 40(4):727–731, 2013.

• B. Turnbull and J.N. McElwaine, 2008. Exper-iments on the non-Boussineq Flow of Self-IgnitingSuspension Currents on a Steep Open Slope., J.Geophys. Res., 113(F01003).• B. Turnbull, J.N. McElwaine and C. Ancey,2007. The Kulikovskiy–Sveshnikova–Beghin Modelof Powder Snow Avalanches: Development and Ap-plication, J. Geophys. Res., 112(F01004).• Turnbull, B., and J.N. McElwaine, 2007. A Com-parison of Powder Snow Avalanches at Vallée de laSionne with Plume Theories, J. Glaciol., 53(30)• J.N. McElwaine, and B. Turnbull, 2006. PlumeTheories Versus Compact Models for Powder SnowAvalanches, Sixth International Symposium on Strat-ified Flows, Perth, December 11-14,• J.N. McElwaine, 2005. Rotational flow in grav-ity current heads, Phil. Trans. R. Soc. Lond., 363,1603–1623, 10.1098/rsta.2005.1597.• J.N. McElwaine and B. Turnbull, 2005. Air Pres-sure Data from the Vallée de la Sionne Avalanchesof 2004, J. Geophys. Res., 110(F03010).• J.N. McElwaine and K. Nishimura, 2001. Particu-late Gravity Currents, Blackwell Science, chap. Ping-pong Ball Avalanche Experiments, no. 31 in SpecialPublication of the International Association of Sedi-mentologists, 135–148.

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