university of south carolina erosion rate formulation and modeling of turbidity current ao yi and...

UNIVERSITY OF SOUTH CAROLINAUNIVERSITY OF SOUTH CAROLINA

Erosion rate formulation and Erosion rate formulation and modeling of turbidity currentmodeling of turbidity current

Ao Yi and

Jasim Imran

Department of Civil and Environmental Engineering


Recent Progress in numerical modeling of density currents considering the vertical

structure• Stacey & Bowen (1988): Temporal evolution of vertical structure of density and turbidity current

using zero Equation turbulence model. Convective and diffusive terms in the flow direction neglected.

• Meiburg et al. : (2000) Used DNS to solve the Navier Stokes equations for density current at low Reynolds number (with Boussinesq assumption)

• Imran, Kassem, & Khan AND Kassem & Imran (2005): Used the commercial flow solver FLUENT to simulate density current in confined and unconfined straight and sinuous channel. The model used RANS Equations and k-e turbulence closure

• Felix (2001): Solved the RANS equations for lock exchange particulate currents at large scale using 2 equation Mellor-Yamada level 2.5 model

• Choi and García (2002): Simulated continuous dilute saline density current on a ramp by solving 2-D steady state boundary layer equation using the 2 equation k-epsilon turbulence model

• Huang, Imran & Pirmez (2005): Solved 2 equation k-epsilon model for turbidity current for well-sorted sediment. Considered bed level change by adjusting the bed boundary and the grids during the computation. Successfully reproduced Garcia’s (1990) experiment including the flow structure and bed level changes.


For conservative density currents, numerical modeling has certainly reached very high level of

sophistication.

Where are we in terms of modeling turbidity currents and the morphological changes at the

field scale ?

Does not matter how much detail of the current one can churn out from the most sophisticated

solution of the Navier-Stoke Equations, there is no escape from van Rijn or Akiyama-Fukushima, or Garcia-Parker or Smith-McLean or some other ‘empirical relationship’ that defines sediment

entrainment from the bed!


Suppose we make numerical model runs

keeping the boundary conditions and computational grid same but choose different entrainment relationships in different runs

Can we expect to see the same result?


We need to pay serious attention to

• the modeling of turbulent kinetic energy (TKE)

&• the sediment entrainment from the bed


Peak velocity region poses a strong barrier to

mixing

Barrier to diffusion

If there is a sediment pick up, can The particles diffuse above the ‘fish-trap’?

t (m/s)2

Dep

th,

z(m

)

0 1E-05 2E-05 3E-050

0.02

0.04

0.06

0.08

0.1

0.12

0.14k-eps ( 2 eqn model)

k (m/s)20.0E+00 1.0E-04 2.0E-04 3.0E-040

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

k-eps ( 2 eqn model)


How turbulence closure affects the result?


Consider three different closure models

k – l model ( 1 eqn model) Turbulence kinetic energy (k) is calculated using transport equation where turbulence length scale (l) is calculated using an empirical formula. q2– l model ( 1 eqn model) Turbulence velocity scale (q) is calculated using transport equation where turbulence length scale (l) is calculated using an empirical formula.

k – ε model ( 2 eqn model) Both turbulence kinetic energy (k) and its dissipation rate (ε) are calculated using transport equations.


u/U

z/h

0 0.5 1 1.5 20

0.5

1

1.5

2

DAPER6 (Garcia 1993)

k-l ( 1 eqn model)

q2-l ( 1 eqn model)


Supercritical flow condition :vertical section is taken at x = 3.0 meter from inletafter 2400 sec

u/U

z/h

0 0.5 1 1.5 20

0.5

1

1.5

2


k-l ( 1 eqn model)

q2-l ( 1 eqn model)


Subcritical flow condition :vertical section is taken at x = 8.0 meter from inletafter 2400 sec

c/C

z/h

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3


k-l ( 1 eqn model)

q2-l ( 1 eqn model)


Subcritical flow condition :vertical section is taken at x = 8.0 meter from inletafter 2400 sec

c/C

z/h

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3


k-l (1 eqn model) : with bridge

k-l ( 1 eqn model)

q2-l ( 1 eqn model)


Supercritical flow condition :vertical section is taken at x = 3.0 meter from inletafter 2400 sec


Let us revisit the Parker et al. (1986) work


Four equation model

Uex

hU

t

hw

2*

22

2

1uRghCS

x

ChRg

x

hU

t

hU

)( 0CrEvx

hUC

t

hCss

)(2

1

2

12

1

0

032

*

CrERghvRghCUehCRgv

heUUux

hKU

t

hK

ssws

w


Three erosion rate relationships

1. Akiyama and Fukushima (1985) –E-I

0

1103

3.01012

Z

ZZE c

s

c

mc

m

ZZ

ZZZ

ZZ

sp v

uRZ *5.0


2. Garcia and Parker (1993)-E-II

57

57

1033.41

103.1

Z

ZEs

2*1

ps

Rv

uZ

36.2for 23.1

586.0

36.2for 6.0

0.1

2

1

2

1

p

p

R

R


3. Smith and McLean (1977) –E-III

11

1

65.0

*

*

0

*

*

0

c

s

c

s

sE

0024.00

;

ss RgD

u**


Four Equation model under steady-state condition

Ri

RirU

v

Uu

RieRiS

dx

dhes

w

1

121

)21

2( 02

2*

1

es

hU

v

dx

d

h

U

Ri

RirU

v

U

uRieRiS

dx

dUes

w

1

12

1)

2

11( 02

2*

h

URir

U

v

U

vRi

U

h

U

Ke

U

uRie

dx

dK

ess

ww

2

0

30

22

2*

]12

1

)1(2

1[


What is ignition condition?

For known values of current thickness, sediment size, and bed slope, there is a combination of velocity, turbulent kinetic energy, and concentration at which the flow will ignite or become erosional.


Set the left side of these equations equal to zero and find solution for U,and K

Pick the lowest positive value

1

es

hU

v

dx

d

h

U

Ri

RirU

v

U

uRieRiS

dx

dUes

w

1

12

1)

2

11( 02

2*

h

URir

U

v

U

vRi

U

h

U

Ke

U

uRie

dx

dK

ess

ww

2

0

30

22

2*

]12

1

)1(2

1[


Ignition Condition


103

104

105

106

0.2

1

2

UI (

m/s)

h0/ D

s

E-I I I / 0.1E-I I I / 0.05

E-I I I / 0.025

E-I / 0.025

E-I / 0.05

E-I / 0.1

E-I I / 0.025

E-I I / 0.1

E-I I / 0.05

Ds = 0.1 mm

E-II-I

E-III-I

E-III-II


103

104

105

106

0.1

0.5

1

UI (

m/s)

h0/ D

s

E-I I I / 0.1

E-I I I / 0.05

E-I I I / 0.025

E-I / 0.025E-I / 0.05

E-I / 0.1

E-I I / 0.025

E-I I / 0.1E-I I / 0.05

Ds = 0.03 mm


UI vs. h0/Ds for Ds = 0.03 mm, 0.06 mm, and 0.1 mm. S and Cf* are respectiv

ely set equal to 0.05 and 0.004

103

104

105

106

0.1

0.5

1

1.5

h0/ D

s

UI

E-I / 0.1mm E-I I / 0.1mm

E-I I I / 0.1mm

E-I / 0.06mm

E-I I / 0.06mm

E-I I I / 0.06mm

E-I I / 0.03mm

E-I / 0.03mm E-I I I / 0.03mm


Phase diagram computed for the case: Ds = 0.1 mm, h0 = 5 m, Cf* = 0.004 an

d S = 0.1. A-U0 = 0.6 m/s, 0 = 2.0×10-4 m 2 /s, and K0 = 8.0×10-3 m 2/s 2

0 0.5 1 1.5 20

0.5

1

1.5

2

0 0.5 1 1.5 20

0.5

1

1.5

2

/ I

U/U I

Subcritical

E-Combine/0.1 mm

Igniting field

subsidingfieldCL

CL

AGL-E-I

AGL-E-II

AGL-E-III

A-III

A-II

A-I


Phase diagram computed for the case: Ds = 0.03 mm, h0 = 1 m, Cf* = 0.004

and S = 0.05. C-U0 = 0.2 m/s, 0 = 2.0×10-5 m 2/s, and K0 = 1.0×10-3 m 2/s 2

0 0.5 1 1.5 20

0.5

1

1.5

2

U/U I

/ I

Subcritical

E-Combine/0.03 mm

Igniting field

Subsidingfield

CL

CL

AGL-E-III

AGL-E-II

AGL-E-I

C-I

C-II

C-III


Cross field for case Ds = 0.1 mm, h0 = 5 m, Cf* = 0.004, and S = 0.1

10-6

10-5

10-4

10-3

10-2

0.5

1

U(m

/s)

(m2/ s)

Subcritical

AGL-I

AGL-E-II

AGL-E-III

Ds = 0.1mm

Subsiding field

Igniting field

A

Cross field

B

A-E-I

B-E-I B-E-I I

B-E-I I I

A-E-I I I

A-E-I I


Cross field for case Ds = 0.03 mm, h0 = 1 m, Cf* = 0.004, and S = 0.05 10

-610

-510

-410

-3

0.2

0.4

0.6

0.8

U(m

/s)

(m2/ s)

Subcritical

AGL-I

AGL-E-II

AGL-E-III

Ds = 0.03mm

Subsidingfield

Igniting field

C

Cross fieldD

D -E-I

C-E-ID -E-I I

D -E-I I I

C-E-I IC-E-I I I


0.3

0.4

0.5

0.6

0.7

U (

m/s

)

0 100 200 300 400 50010

-4

10-3

10-2

x (m)

(m2 /

s)

5

10

15

h (

m)

E-I

E-II

E-III

A


0.1

0.2

0.3

0.4

U (

m/s

)

0 100 200 300 400 50010

-6

10-5

10-4

10-3

x (m)

(m2 /

s)

2

4

6

8

10

h (

m/s

)

E-I

E-II

E-III

C


Conclusions

• 1. The ignition values obtained with different models can vary widely.

• 2. A turbidity current predicted to be subsiding by one entrainment relation could turn out to be igniting when a different entrainment model is used.

• 3. Important implication in the numerical modeling of turbidity current

• 4. Further research on the topic of sediment entrainment is crucial


ACKNOWLEDGMENT

Funding from the National Science Foundation

(OCE-0134167) is gratefully acknowledged.

university of south carolina erosion rate formulation and modeling of turbidity current ao yi and...

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