Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Hydrodynami s of Lakes and e osystems

Jair REYES

Institut Jean le Rond d'Alembert - UPMC

Do toral Advisor: Mauri e ROSSI

November 12, 2014

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Introdu tion

The role of hydrodynami s on the dynami s of algae populations in

lakes.

What are the determining fa tors in the explosive growth of

ertain algae ?

This thesis is divided into two parts

Analysis of the �uid dynami s in lakes (Presented here).

Coupling biology models with hydrodynami s models

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Content

1.- Comparison Hydrostati and Nonhydrostati model

2.- Sediment resuspension using GERRIS ode.

2.a The model and its validations

2.b Example of sediment resuspension due to urrents

2. Example of sediment resuspension due to Internal Solitary

Waves breaking

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Equations and Boundary Conditions

Hydrodynami s of Lakes

Physi al Models

Hydrostati model (FORTRAN Code):

Model widely used in marine s ien e

Verti al momentum equation repla ed by the hydrostati

approximation.

Verti al velo ity al ulated from the ontinuity equation.

Primitive equations solved by using sigma oordinates

Is the hydrostati model orre t near boundaries?

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Equations and Boundary Conditions

Hydrodynami s of Lakes

Physi al Models

Nonhydrostati model (GERRIS Code):

Navier-Stokes equations

E�e tive vis osity

Surfa e: Rigid-Lid Approximation

We have ompared the hydrostati and nonhydrostati model.

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Equations and Boundary Conditions

Hydrodynami s of Lakes

Di�erent simpli�ed topographies: �at and paraboli basins.

We are interested in

Stru ture of urrents

Shear on the bottom

Surfa e:

(

µ∂u∂z,w

)

= (τs

, 0)

W

a

l

l

s

:

S

l

i

p

o

n

d

i

t

i

o

n

Bottom: No-slip ondition

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Di�erent ases

Di�erents Reynolds numbers: Re = ρ[U][L]µ

= ρτs

H

2

µ2.

H [m℄ µ [

kg

ms

℄ U

wind

[

km

h

℄ Re Re

max

10 1

3.2 100 24.0

6.4 400 88.7

10.3 1 000 198.5

17.8 3 000 493.6

32.5 10 000 1213.3

With Re

max

as a Reynolds number based in the highest

horizontal velo ity u

max

.

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Unsteady �ow

-1-0.8-0.6-0.4-0.2

0

-4 -3 -2 -1 0 1 2 3 4

y

x

Velocity field Re=10 000, t = 1240

-1

0

1

2

3Vorticity

Video Re = 10000

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Steady velo ity �eld

Steady velo ity �eld orresponding to paraboli lake of Re = 10

3

-1-0.8-0.6-0.4-0.2

0

-4 -3 -2 -1 0 1 2 3 4

y

x

0 0.04 0.08 0.12 0.16 0.2

Velocity Norm

t = 100

-1-0.8-0.6-0.4-0.2

0

-4 -3 -2 -1 0 1 2 3 4

y

x

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Velocity Norm

NH-M

H-M

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Steady velo ity �eld

Steady velo ity �eld orresponding to re tangular lake of

Re = 4 · 102

-1-0.8-0.6-0.4-0.2

0

-4 -3 -2 -1 0 1 2 3 4

y

x

0 0.04 0.08 0.12 0.16 0.2 0.24 0.28

Velocity Norm

-1-0.8-0.6-0.4-0.2

0

-4 -3 -2 -1 0 1 2 3 4

y

x

0 0.05 0.1 0.15 0.2 0.25

Velocity Norm

-1

-0.9

-0.8

-0.7

-0.6

-0.5

3.8 3.85 3.9 3.95 4y

x

0

0.005

0.01

0.015

0.02

0.025

NH-M

H-M

Zoom H-M

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Upwelling and downwelling regions

NH model H model

P

R

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

-4 -3 -2 -1 0 1 2 3 4

UD

W(x

)

x

Re = 102

4·102

103

3·103

104

Lx = 8

-1-0.8-0.6-0.4-0.2

0 0.2 0.4

-4 -3 -2 -1 0 1 2 3 4

UD

W(x

)

x

Re = 102

4·102

103

3·103

104

Lx = 8

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6

-4 -3 -2 -1 0 1 2 3 4

UD

W(x

)

x

Re = 102

4·102

103

-1-0.8-0.6-0.4-0.2

0 0.2 0.4

-4 -3 -2 -1 0 1 2 3 4

UD

W(x

)x

Re = 102

4·102

103

Upwelling and Downwelling Fun tion

w̄(x) = 1

top−bottom

∫

z=topz=bottom w(x , z) dz .

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

Modeling Sediment

Dimensionless sediment equation

φt

+ u ·∇φ =1

S Re

∆φ, (1)

with S = νDφ

Dimensionless boundary onditions:

Surfa e and walls:

∂φ∂n

= 0

Bottom: Suspension and deposition depending on shear.

∂φ∂n

= −[Eb

(τb

) + D

b

(τb

)φ].

where τb

is the shear stress on the bottom

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

Modeling Sediment

Bottom boundary ondition

Erosion of Sediment:

E

b

=

0 |τb

| < τ e

e

b

(

|τb

|τ e

− 1

)

|τb

| ≥ τ e

,

Deposition of Sediment:

D

b

=

0 |τb

| > τ d

d

b

n · ez

(

1− ‖τb

‖τ d

)

|τb

| ≤ τ d

,

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

Validation: Two dimensional In lined Couette Flow

Di�erents angles θ = 0,10, 20

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

2D Couette Flow

Sediment omputation

Sediment mass in the total volume

Φ =∫

φ dV = τs

t.

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

Φ

t

Analytical solution Numerical solution, Run 1

23

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

2D Couette Flow

θ = 20, Re�ne=2

6

θ = 20, Re�ne=2

8

PDF of relative error in shear omputation:

η =τs

analiti

−τs

numeri

τs

analiti

0

50

100

150

200

250

300

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02

P.D

.F.

Run 123

Refine = 26

0

50

100

150

200

250

300

350

400

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02

P.D

.F.

η

Run 123

Refine = 28

0

0.2

0.4

0.6

0.8

1

-0.04 -0.02 0 0.02

C.F

.D.

η 0

0.2

0.4

0.6

0.8

1

-0.04 -0.02 0 0.02

C.F

.D.

η

0

0.2

0.4

0.6

0.8

1

-0.04 -0.02 0 0.02

C.F

.D.

η

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

3D Poiseuille Flow

θ ≈ 0, Re�ne=2

8

θ ≈ 40, Re�ne=2

8

0

5

10

15

20

25

30

-0.1 -0.05 0 0.05 0.1

P.D

.F.

η

Run 123

0 0.2 0.4 0.6 0.8

1

-0.05 0 0.05

C.D

.F.

η

Errors lo ated at the MPI boundaries

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Validation of sediment al ulation

Erosion of sediment in the lake

Erosion of sediment in the lake

Video Re = 10

5

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

ISW Breaking

Internal solitary wave (ISW) modeling

1

Momentum equation with Boussinesq approximation

2

Adve tion-di�usion equation for temperature

Boundary onditions

Surfa e:

(

v , ∂u∂x, ∂ρ∂y

)

= (0, 0, 0);

Right wall:

(

u, ∂v∂y

, ∂ρ∂x

)

= (0, 0, 0);

Bottom:

(

u, v ,∂ρ∂n

)

= (0, 0, 0).

G. Ri kard,J. Callaghan, S. Popinet, OMJ (2009).

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

ISW Breaking

ISW Breaking

Suspension of sediment du to ISW breaking and run-up

Video suspension

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25

Φ=∫

φdV

t

Sediment concentration evolutionRun 1 Run 2 Run 3

Sediment

Φ =∫

φdV

Jair REYES Presentation

Introdu tion

Physi al models

Comparison between H and N-H models

Modeling Sediment

Internal solitary wave

Con lusions

Con lusion

Comparison of the Hydrostati and Nonhydrostati Models

Sediment resuspension du to urrents in the lake.

Sediment resuspension du to ISW breaking and run-up.

Jair REYES Presentation