wTo-Phase Continuum Models for Geophysical Particle-Fluid ... · Guilherme H. FIOROT geo o16 March...

Two-Phase Continuum Models for Geophysical

Particle-Fluid Flows

Time-dependent measurements for incipient

bed load discharge on shallow open channel

�ows

Guilherme H. FIOROT

supervisors :Pascal DUPONT

Geraldo de F. MACIEL

LGCGM - INSA de Rennes & PPGEM - UNESP de Ilha Solteira

March 14-18, 2016

Guilherme H. FIOROT geo�o16 March 14-18, 2016 1 / 35

Introduction

General framework

τb

ws

τc

z

bedload

suspension

wash load

u(z)ρ, μ ∆ρ∗ = ρs

ρ − 1

q0 ; u0 ; h0 ;→ τb ; u∗ ;

u∗ > u∗c :

Transport !

Shields number

Sh = u2∗/(∆ρ∗gD) ;

Transport

→ q∗ = A(Sh− Shc)B .

(Shields, 1936; Paphitis, 2001; Beheshti and Ataie-Ashtiani, 2008)(Meyer-Peter and Müller, 1948; Wong and Parker, 2006)


Introduction

Dimensionless analysis

Reynolds : Re = 4u0h0ν

�ow dynamics ;

Froude : Fr =u0√

gh0 cos θ�ow hydraulic regimen ;

Shields : Sh =u2∗

∆ρ∗gDparticles initiation of motion ;

Particle Reynolds : Rep =u0D

νparticle-�ow interaction ;

Stokes : St =Tp

Tfparticles-�ow interaction ;

Particle Froude : Frp =u0√gD

�ow capacity of transport ;

Rouse : ws

κu∗;ws

u∗;u∗ws

dominant mode of sediment transport ;

(suspension or movability)


Introduction

Non-stationary transport - Turbulence

Figure 1 � Fluctuations of the instantaneous velocity around the mean value.Particle entrainment occurs sporadically. (Ancey et al., 2008).

u∗ ∝ u0

u∗ < u∗c u∗ > u∗c


Introduction

Non-stationary transport - Grain-size distribution

Figure 2 � Solid discharges of materials with di�erent granulometry (Frey et al.,2003).

Heterogeneous Homogeneous


Introduction

Non-stationary transport - Flow instabilities

Free surface instabilities & Bedforms

Figure 3 � Evolution of roll waves andsediment transport (Davies, 1990). Figure 4 � Comparison of stability

lines for roll waves and bedforms(Colombini and Stocchino, 2005).


Introduction

Main objective

Non-stationary sediment transport

Experimentally study time-dependent e�ects on sediment transport forruno� �ows.

Close to threshold cases :

Shields number

Sh < 2.5Shc ;

(Recking et al., 2009)

Turbulence e�ects :


Experimental Project

Experimental project

Figure 5 � Sketch of experimental setup.

I shallow water �ow h0/l � 1 ;

I PIV method for local �ow dynamics measurements ;

I contact needle for global measurements ;

I combined shadowgraph and PTV methods for local sediment transportmeasurements ;

I measurement of total weight of particle as value of reference.Guilherme H. FIOROT geo�o16 March 14-18, 2016 8 / 35


Test sand

Figure 6 � Histogram for particles used in experiments and MEV images.

I non-cohesive ;

I regular shape ;

I 0.15 < D < 0.25 mm ;

I Shc = 0.062 ;

I favorable to bedload (given the exp.limitations, ws ≈ 2cm/s) ;



Dimensionless parameters

Dimensionless Range Interpretation

Re 3000 < Re < 4000 Turbulent (close to transition) ;

Fr 1.1 < Fr < 1.5 Torrential �ows

Sh 0.068 < Sh < 0.165 particles initiation of motion ;

Rouse : 2.4 <ws

u∗< 3.6 bed-load transport ;

Shc = 0.062 (Paphitis, 2001)


Methodology

Methodology debrief

For �ow dynamics

I determination of friction velocityu∗ based on theoretical turbulentpro�le ; assumptionu∗(t) = F{< u > (y0, t)} ;correction of results based onglobal values (contact needle) ;

I Obtaining of time-dependentfriction velocity u∗(t).

For sediment transport

I Image acquisition fromfast-recording camera ; imagesprocessing on Matlab ; PTValgorithm ; correction of particlessize/velocities ; time dependentdischarge computation ;comparison to global values frommean weighted discharge ;

I Obtaining of time-variabletransport rate q(t).

Statistically...

q(t)?= f {u∗(t)}


Methodology Hydraulic properties

Friction velocity obtaining

100

101

102

0

5

10

15

20

25

y+

u+

u+ = κ�1 y+ +5

u+ = y+

gb1

gb2

gb3

gb4

gb5

gb6

gb7

gb8

In wall coordinates :u+ = u(y)/u∗andy+ = u∗y/ν

For viscous sublayer :u+ = y+ ;For log region :u+ = κ−1 log y+ + 5 ;

Figure 7 � Turbulent characteristics of average pro�le of mean �ow velocity. Linesindicate theoretical values : solid line is u+ = κ−1 log y+ + 5 for log region ;dashed line is u+ = y+ for viscous layer.



Turbulent intensities - u+RMS

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

y/h0

u+ R

MS

gb1

gb8

gb1

gb2

gb3

gb4

gb5

gb6

gb7

gb8

(Antonia and Krogstad, 2001)u+RMS

=

A exp(− y+

Re∗

) [1− exp

(− y+

B

)]+

Cy+ exp(− y+

B

)A = 2, B = 8, and C = 0.34 ;Re∗ = u∗PIVh0/ν is the Reynoldsnumber using friction velocity asreference.

Figure 8 � Turbulent intensities for runs gb1 to gb8. Lines indicate computedvalues following empirical results from Antonia and Krogstad (2001). Longitudinalturbulent intensities u+

RMS; dark line represent run computed value for gb1, and

gray line, for gb8.Guilherme H. FIOROT geo�o16 March 14-18, 2016 13 / 35


Friction velocity u∗(t)

Bottom shear stress :τb ∼ ρu2∗ ∼ ρgh0 sin θ

For our experiments :

I u∗ ∼ 0.01 ms−1 ;→ τb ∼ (103)(10−2)2 ∼ 0.1 Pa ;

I h0 ∼ 0.01 m ;→ τb ∼ (103)(101)(10−2)(10−2) ∼ 1 Pa ;

The precision required tomeasure �uctuationswould be ∼ 0.01 Pa

! ! !

(Detert et al., 2010; Amiret al., 2014)

So, an indirect measure is pursued...



Friction velocity correlation

τb(t) ?

I u2∗ ∝ τb ∝ u20

I u(y0) ∝ u0 → u∗ ∝ u(y0)

Dimensionally and statistically wecan assume that :

I u2∗(t) ∝ τb(t) ∝ u2(y0, t)

I so there is a function F that :u∗(t) = F{u(y0, t)}

(Ould Ahmedou et al., 2007)

I the hypothesis : F is also valid forinstantaneous variables, so that :u∗(t) = F{u(y0, t)}

0 0.2 0.4 0.60

1

2

3

4

5

6

7

Mean particle size(to scale)

u (y) [m/s]

y[m

m]

gb1

gb2

gb3

gb4

gb5

gb6

gb7

gb8

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

u (y0)

u*

PIV

Linear fit

95% pred bounds

Figure 9 � Correlation between 〈u〉(y0)and u∗PIV. First-order polynomialapproximation and 95% con�denceboundaries.

y0 = 0.69 mm



Friction velocity signal

Figure 10 � Signals of u∗(t) and Sh(t) for a close to threshold experiment run.The dotted dark line represent Shc .


Methodology Instantaneous discharge measurement


Top view of the trap.Side view, camera perspective.

(red arrow)

(Ho et al., 2014)




Top view of the trap.Side view, camera perspective.

(red arrow)(Ho et al., 2014)



Particles identi�cation methodOriginal image

Linear histogram normalization

Binary Image

I Based on maximum andminimum images from series,each image gray level isadjusted ;

I Canny �lter is applied, usingOtsu's threshold method, toidentify particle edges ;

I Morphological operations(closing, �lling, opening,watershed) are performed.

(Frey et al., 2003)



Particle Tracking Velocimetry

t

ΔyROI

ΔxROI

pi

k-1

pi

k

Δri

k

k

x

yα

Particle mass :mk

i = πρs6Dk

i3

Particle displacement :∆~rki

Particle velocity :

~vki =

∆~rki∆t

Total mass at time tk :

M(tk) =

Np∑i

mki ;

Time dependent discharge :

q(tk) =M(tk)

∆tkesc,

Time scale of particles permanence :

∆tkesc =∆yROIvkyi

;

Mean : vkyi = 1

Np

∑Np

i vkyi ;

Weighted-mean : vkyi = 1

M(tk )

∑Np

i mki v

kyi .


Results

Instantaneous discharge measurement

0 0.5 1 1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

qsWg

qs P

TV

qsPTV

= 2.35qsWg

Figure 11 � Correlation betweensediment discharge computation basedon images and weighted mass.

Measure u∗ Sh Fr qlW qlPTV

gb1-2 0.014 0.068 1.16 0.065 0.064gb1-3 0.014 0.075 1.20 0.179 0.386gb1-4 0.015 0.079 1.10 0.204 0.336gb2-1 0.021 0.158 1.55 1.187 2.967gb2-2 0.021 0.159 1.54 0.896 2.107

Top view of the trap.


Results

Sh = 0.079,Fr = 1.10

0 50 100 1500

0.5

1

1.5

Time (s)

qs

(gm

1s

1)

Initiation stage

First wave of sedimentsSteady solid transport

Disturbed transport


Results

Sh = 0.079,Fr = 1.10


Results

Sh = 0.159,Fr = 1.54

0 20 40 60 80 100 120 1400

2

4

6

8

Time (s)

qs

(gm

1s

1)

Initiation stage

First wave of sediments

Steady solid transport

Disturbed transport


Results

Sh = 0.159,Fr = 1.54


Results

Comparison to classical formulas

0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

3.5

4

Sh

qs

(gm

1s

1)

Meyer-Peter & Muller (1948)

Wong & Parker (2006)

Steady transport

Bedforms

Meyer-Peter & Muller (1948) :

qs = 8(Sh− Shc)3

2

Wong & Parker (2006) :

qs = 4.93(Sh− Shc)8

5


Results

PSD

103

102

101

100

101

107

106

105

104

103

102

101

100

Frequency (Hz)

PS

D

Steady transport

u*(t)

5

3


Further analysis...

Discussions

I qsRMS and u∗RMS ;

I characteristic time scales ;

I additional e�ect : pulsating �ows ;


Further analysis...

Thank you !

g�[email protected]

gh�[email protected]


Références

M. Amir, V. I. Nikora, and M. T. Stewart. Pressure forces on sediment particles in turbulentopen-channel �ow : a laboratory study. Journal of Fluid Mechanics, 757 :458�497, 2014.ISSN 0022-1120. doi : 10.1017/jfm.2014.498. URLhttp://www.journals.cambridge.org/abstract_S0022112014004984.

C. Ancey, a. C. Davison, T. Böhm, M. Jodeau, and P. Frey. Entrainment and motion of coarseparticles in a shallow water stream down a steep slope. Journal of Fluid Mechanics, 595(August), 2008. ISSN 0022-1120. doi : 10.1017/S0022112007008774.

R. A. Antonia and P.-r. Krogstad. Turbulence structure in boundary layers over di�erent typesof surface roughness. Fluid Dynamics Research, 28(2) :139, 2001. URLhttp://stacks.iop.org/1873-7005/28/i=2/a=A06.

A. A. Beheshti and B. Ataie-Ashtiani. Analysis of threshold and incipient conditions forsediment movement. Coastal Engineering, 55(5) :423�430, may 2008. doi :10.1016/j.coastaleng.2008.01.003. URLhttp://linkinghub.elsevier.com/retrieve/pii/S0378383908000021.

M. Colombini and A. Stocchino. Coupling or decoupling bed and �ow dynamics : Fast and slowsediment waves at high froude numbers. Physics of Fluids, 17 :036602(1�19), 2005.

T. R. H. Davies. Debris-�ow surges - Experimental Simulation. Journal of Hydrology, 29(1) :18�46, 1990.

M. Detert, V. Nikora, and G. Jirka. Synoptic velocity and pressure �elds at the water-sedimentinterface of streambeds. Journal of Fluid Mechanics, 660 :55�86, 2010. ISSN 0022-1120.doi : 10.1017/S0022112010002545.

P. Frey, C. Ducottet, and J. Jay. Fluctuations of bed load solid discharge and grain sizedistribution on steep slopes with image analysis. Experiments in Fluids, 35(6) :589�597,2003. ISSN 07234864. doi : 10.1007/s00348-003-0707-9.Guilherme H. FIOROT geo�o16 March 14-18, 2016 28 / 35

http://www.journals.cambridge.org/abstract_S0022112014004984

http://stacks.iop.org/1873-7005/28/i=2/a=A06

http://linkinghub.elsevier.com/retrieve/pii/S0378383908000021

Further analysis...

T. D. Ho, a. Valance, P. Dupont, and a. Ould El Moctar. Aeolian sand transport : Length andheight distributions of saltation trajectories. Aeolian Research, 12 :65�74, 2014. ISSN18759637. doi : 10.1016/j.aeolia.2013.11.004.

E. Meyer-Peter and R. Müller. Formulas for bed-load transport. In Proceedings of the 2nd

Meeting of the International Association for Hydraulic Structures Research, pages 39�64,Stockholm, 1948.

D. Ould Ahmedou, a. Ould Mahfoudh, P. Dupont, a. Ould El Moctar, a. Valance, and K. R.Rasmussen. Barchan dune mobility in Mauritania related to dune and interdune sand �uxes.Journal of Geophysical Research : Earth Surface, 112(2) :1�18, 2007. ISSN 21699011. doi :10.1029/2006JF000500.

D. Paphitis. Sediment movement under unidirectional �ows : an assessment of empiricalthreshold curves. Coastal Engineering, 43(3-4) :227�245, Aug. 2001.

A. Recking, P. Frey, A. Paquier, and P. Belleudy. An experimental investigation of mechanismsinvolved in bed load sheet production and migration. Journal of Geophysical Research, 114(F3) :F03010, Aug. 2009. ISSN 0148-0227.

A. Shields. Application of Similarity principles and Turbulence Research to bed-load movement.Mitteilungen der Preussischen Versuchsanstalt für Wasserbau und Schi�bau, 1936. Originalversion in german (Anwendung der Aehnlichkeitsmechanik und der Turbulenz Forschung aufdie Geschebebewegung).

M. Wong and G. Parker. The bedload transport relation of meyer-peter and müller overpredictsby a factor of two. Journal of Hydraulics Engineering, 132 :1159�1168, 2006.Guilherme H. FIOROT geo�o16 March 14-18, 2016 29 / 35

Further analysis...

Mean �ow velocity and turbulent intensities

0 0.2 0.4 0.60

1

2

3

4

5

6

7

u (y) [m/s]

y[m

m]

gb1

gb2

gb3

gb4

gb5

gb6

gb7

gb8

0 0.05 0.1 0.150

1

2

3

4

5

6

7

uRMS

(y) [m/s]

y[m

m]

Figure 12 � Results for average pro�les of mean �ow velocity 〈u〉(y) and standarddeviation 〈uRMS〉(y) for runs gb1 to gb8.


Further analysis...

Turbulent intensities - v+RMS

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y/h0

u+ R

MS

Antonia and

Krogstad (2001)gb1

gb2

gb3

gb4

gb5

gb6

gb7

gb8 (Antonia and Krogstad, 2001)

v+RMS

= 1.14 exp(

−0.76yh0

)

Figure 13 � Turbulent intensities for runs gb1 to gb8. Lines indicate computedvalues following empirical results from Antonia and Krogstad (2001). Verticalturbulent intensities v+

RMS; dark line represent computed values.


Further analysis...

Friction velocity correction

Figure 14 � Correlation betweenfriction velocity computedthrough both methods u∗CN andu∗PIV. Solid line represent linearrelation between both methodsfor friction velocity calculation.Dashed line represents equalityu∗PIV = u∗CN.

u∗

u∗ = u∗CN = 1.08u∗PIV − 0.006with a correlation coe�cientR2 = 0.96.


Further analysis...

Grain-size distribution

Figure 15 � Probability distribution function from captured images in comparisonto calibrated grain-size distribution.

For low Shields number

→ Correction of particles diameter.


Further analysis...

∆yROI in�uence

0 5 10 15 200

0.2

0.4

0.6

0.8

1

∆yROI

/(ws∆t)

qs

∆yROI

/(ws∆t) 3.5

~10%

∆yROI

/(ws∆t) 15

~1%

35 35.5 36 36.5 37 37.5 38 38.5 39 39.5 400

1

2

3

4

5

6x 10

6

time scale (s)

soli

d d

isch

arg

e (k

g/s

)

∆yROI

/(ws∆t) ≈ 3.5

∆yROI

/(ws∆t) ≈ 15

Figure 16 � In�uence of ∆yROIon qs(t).

Filtering of data, reducingdiscretization noise.

→ Imposes time window of signal acquisition : ∆tesc.In order to be comparable to PIV signal : ∆yROI = 6.5mm

(∆tesc = 0.13− 0.14s)


Further analysis...

Experimental run

1. arrange the experiment ;

2. partially block the outlet ;

3. set �ow discharge and slope ;

4. start data acquisition (PIV andPTV) ;

5. release the outlet ;

6. after 1'30�, data acquisitionstops ;

7. block outlet ;


Further analysis...

PSD

103

102

101

100

101

107

106

105

104

103

102

101

100

Frequency (Hz)

PS

D

Steady transport

Non steady transport (bedforms)

u*(t)

5

3


wTo-Phase Continuum Models for Geophysical Particle-Fluid ... · Guilherme H. FIOROT geo o16 March...

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