Laboratoire de Sondages Electromagnétiques de l’Environnement Laboratoire de Sondages Electromagnétiques de l’Environnement TerrestreTerrestre

(Université de Toulon et du Var)(Université de Toulon et du Var)

Philippe Fraunié Sabeur BERRABAA

Jose Manuel Redondoet al

Stratified turbulent flows in Stratified turbulent flows in Ocean and Atmosphere : Ocean and Atmosphere :

Processes, observations and Processes, observations and CFDCFD

ObservationsObservations

Basic processesBasic processes

KH instabilityKH instability

Kelvin-Helmholtz instability : Richter (1969)

Holmboe instabilityHolmboe instability

Ri > ¼Ri > ¼

Su > 2 SbSu > 2 Sb

Possibility of Holmboe instabilityPossibility of Holmboe instability

Holmboe instabilityHolmboe instability

Richardson numberRichardson number

Global Richardson numberGlobal Richardson number

Turbulence scalesTurbulence scales

Measurements in Measurements in AtmosphereAtmosphere

Profiles of Profiles of temperature temperature mesured by mesured by baloons : weakly baloons : weakly and srongly and srongly stratified layers stratified layers (Dalaudier (Dalaudier et al., et al., 1994) 1994)

Measurements in OceansMeasurements in Oceans Temperature profiles in Temperature profiles in

Malta sea : Contribution Malta sea : Contribution of K.-H. instabilities to of K.-H. instabilities to mixed layers (Woods, mixed layers (Woods, 1969) 1969)

Korotayev et Korotayev et Panteleyev (1977), Panteleyev (1977), Indian and Pacific Indian and Pacific oceans, Alford et Pinkel oceans, Alford et Pinkel (2000) California(2000) California

Measurements in OceanMeasurements in Ocean

Temperature Temperature profiles in Japan sea profiles in Japan sea : Contribution of : Contribution of internal waves to internal waves to mixed layers mixed layers (Navrotsky, 1999) (Navrotsky, 1999)

Laboratory Experiments : Laboratory Experiments : the layering effectthe layering effect

Generation of Generation of turbulence (grids) in a turbulence (grids) in a stratified flow at reststratified flow at rest

Interaction betweenInteraction betweenturbulence and turbulence and stratificationstratification

Computational Fluid Computational Fluid DynamicsDynamics

Focused on Kelvin-Focused on Kelvin-Helmholtz instability Helmholtz instability (Palmer (Palmer et al.,et al., 1996) 1996)

Only few numerical Only few numerical experiments experiments concerning internal concerning internal waves (Koudella et waves (Koudella et Staquet, 1996 ; Staquet, 1996 ; Bouruet-Aubertot Bouruet-Aubertot et et al., 2001)al., 2001)

Navier-Stokes solverNavier-Stokes solver

Based on JETLES DNS Code (Versico, Based on JETLES DNS Code (Versico, Orlandi) adapted to stratified flows :Orlandi) adapted to stratified flows :

cartésian coodinatescartésian coodinates sreamwise non périodic bc (Ox)sreamwise non périodic bc (Ox) transport equations for salinity and transport equations for salinity and

temperature)temperature) LES LES Smagorinsky subgrid modelSmagorinsky subgrid model

LES equationsLES equations

Continuity equation : Continuity equation :

Momentum equations :Momentum equations :

0j

jx

v

ijt

j

z

i

i

i

d

j

ji T

xdzzyx

xg

xP

xv

vtv

2'',,11

00

3

0

i

Transport of scalar fieldsTransport of scalar fields

Temperature and Salinity :Temperature and Salinity :

State Equation : State Equation :

j

TtT

jj

j xT

xxT

vtT

; 1,, 00 SSzyx S

j

t

jj

j xS

xxS

vtS

00 1,, TTzyx T

LES numerical codeLES numerical code

Continuity equation : Continuity equation :

Momentum equations :Momentum equations :

0j

jx

v

ijt

j

z

i

i

i

d

j

ji T

xdzzyx

xg

xP

xv

vtv

2'',,11

00

3

0

i

Turbulence closureTurbulence closure

Smagorinsky model :Smagorinsky model :

1coù et

2.0

2 : avec

),,,(

tt

3

s

2/1

2

SSc

zyx

C

TTT

TCtzyx

t

t

ijij

st

DiscretizationDiscretization

Time marching :Time marching : three steps three steps Runge-Kutta Runge-Kutta scheme,scheme, third order third order accurate accurate

Spacial discretization :Spacial discretization : second second order centered finite differencesorder centered finite differences

AlgorithmAlgorithm

Computational domain Computational domain

Taille du domaine:Taille du domaine:2 < Lx < 4 m ; Ly = 0.1 m ; 0.1 < Lz < 0.2 m

Maillage :Maillage :x = 3.9 mm ; y mm ; z = 1 mm

Taille de la barre :Taille de la barre : cmdb 2ou 1

Boundary conditions Boundary conditions En surface et au fond :En surface et au fond :

A la frontière droite :A la frontière droite :

A la frontière gauche :A la frontière gauche :

0),(),(

; 0

surfaceenfondau

surfacefond

zvu

zvu

ww

0

surfaceaufondau zS

zS

0 ; 0),,(

droitefrontière

droitefrontière xS

wvu

0),,,(),,,(

xSwvu

tSwvu

U a

aUgauchefrontièreu si 0 gauchefrontièreu

0 gauchefrontièreu0 si

avec

Homogeneous flow :Homogeneous flow :Von Karman streetsVon Karman streets

1cmscm 4.4440Re dU bb ; / ;

Champs d’iso-vitesses horizontales, d’iso-vitesses verticales et d’iso-vorticités d’axe (Oy)

3D structures low 3D structures low Reynlods numberReynlods number

- en rouge et bleu, les - en rouge et bleu, les surfacessurfaces

Surfaces d’iso-vorticité :Surfaces d’iso-vorticité :

1dU bby )//(

60)//( 1dU bbx

- en vert et noir, les surfaces- en vert et noir, les surfaces

440Re

cm ; cm/s b 1 d 4.4Ub

3D structures larger 3D structures larger Reynolds numberReynolds number

Surfaces d’iso-vorticité :Surfaces d’iso-vorticité :cm2 ; cm/s2 dU bb 4.4

- en rouge et bleu, les - en rouge et bleu, les surfacessurfaces 1dU bby )//(

205)//( 1dU bbx

- en vert et noir, les surfaces- en vert et noir, les surfaces

4880Re

2D du computational 2D du computational domaindomain

Turbulence collapse  (1) Turbulence collapse  (1)

Champs d’iso-vorticité d’axe (Oy)Champs d’iso-vorticité d’axe (Oy)cm 2 ; cm/s 4.4 ; 880Re dU bb

Turbulence collapse  (2)Turbulence collapse  (2)

Transformée de Fourier de l’évolution temporelle des composantes de vitesseTransformée de Fourier de l’évolution temporelle des composantes de vitesse dans le sillage proche : - Diminution du nombre de Strouhal dans le sillage proche : - Diminution du nombre de Strouhal avec l’augmentation de la stratificationavec l’augmentation de la stratification

Turbulence collapse (3) :  Turbulence collapse (3) :  physical process  physical process 

Temporal evolution of the near wake width Temporal evolution of the near wake width for Richardson numbers less than 1/4 : for Richardson numbers less than 1/4 : the wake grows following a tthe wake grows following a t1/3 1/3 law as for law as for

homogeneous flow homogeneous flow coolapse occurs when the wake width is coolapse occurs when the wake width is

maximummaximum the wake widh decreases up to an constant the wake widh decreases up to an constant

value value

Physical collapse (4) Physical collapse (4)

oooooo RiRi00 = 0.03 = 0.03 ; ; oooooo RiRi00 = 0.039 = 0.039

L’épaisseur du sillage proche atteint une valeur L’épaisseur du sillage proche atteint une valeur

maximale pour maximale pour NNBVBVt t 2 2 Ri Ri00 < 1/9 < 1/9

D ’après Lin D ’après Lin et al. (1992)et al. (1992)

Physical collapse (5)Physical collapse (5)

NNBVBVtt (maximum wake (maximum wake width) depends on width) depends on RiRi00 (Xu (Xu et al.,et al., 1995) : 1995) :

RiRi00 < 1/9 < 1/9 : : NNBVBVt t varies in the range varies in the range 1.5 1.5 - - 2.52.5

1/9 < Ri1/9 < Ri00 < 1/4 < 1/4 : : NNBVBVt t varies between varies between 33 and and 55

RiRi00 > 1/4 : > 1/4 : the wake the wake width is constantwidth is constant

Physical collapse (6) :Physical collapse (6) :

La taille de la zone perturbée dans le cas La taille de la zone perturbée dans le cas

n’évolue pas contrairement au casn’évolue pas contrairement au cas

40 Ri0.1Ri 0

Gravity internal Gravity internal wave :wave :

weak initial weak initial stratification (1)stratification (1)

Iso-density fields for Iso-density fields for différent Richardson différent Richardson numbers :numbers :

Ondulation occurs at Ondulation occurs at the starting pointthe starting point

)et ( 0 0.25 0.1 0.015,Ri

19.5t ; cm 2d

; cm/s 4.4U ; 880Re

ab

b

Gravity internal wave :Gravity internal wave :weak initial stratification (2)weak initial stratification (2)

Profiles of local Profiles of local Richardson number :Richardson number :

Waves occur for Waves occur for RiRi > 1 : > 1 : stratification dominates stratification dominates turbulenceturbulence

2)zu

max(

zgRi

ρ

ρ

137x/d ; 19.5t ; cm 2d

; cm/s 4.4U ; 880Re

bab

b

Gravity internal wave :Gravity internal wave :strong initial stratificationstrong initial stratification (1) (1)

39t ; cm 2d ; cm/s 4.4U ; 1Ri ; 880Re abb0

Gravity internal wave :Gravity internal wave :strong initial stratificationstrong initial stratification (2) (2)

Iso-density and d’iso-vorticity - transverse Iso-density and d’iso-vorticity - transverse axis (Oy)axis (Oy) ondulatory motion imposed by internal ondulatory motion imposed by internal

waveswaves Remember Lee waves (Atkinson) : Remember Lee waves (Atkinson) :

BV

b

NU 2

Ri 0 = 1 et N BV = 2.2 (s -1) th = 6.28 d b

nu 6.5 d b

Ri 0 = 4 et N BV = 4.4 (s -1) th d b

nu 3.5 d b

Mixing Processes in the near Mixing Processes in the near wake : weak initial stratification wake : weak initial stratification

(1)(1)

Iso-vorticity - transverse Iso-vorticity - transverse axis (Oy) in the near axis (Oy) in the near wakewake

Shear instabilityShear instability

overturningoverturning

cm 2d ; cm/s 4.4U

; 0.1Ri ; 880Re

bb

0

Mixing Processes in the near Mixing Processes in the near wake : weak initial stratification wake : weak initial stratification

(2) (2)

OverturningOverturning : : time evolution time evolution of two density of two density surfacessurfaces

Roll up Roll up

cm 2d ; cm/s 4.4U

; 0.1Ri ; 880Re

bb

0

Mixing Processes in the near Mixing Processes in the near wake : weak initial stratification wake : weak initial stratification

(3)(3)

Unstable Unstable situationsituation

OverturningOverturningLocal convective Local convective

instability instability

Mixing Processes in the near wake : Mixing Processes in the near wake : strong initial stratification (1)strong initial stratification (1)

Time evolution Time evolution of two density of two density surfacessurfaces

Breaking Breaking internal wavesinternal waves

cm 2d ; cm/s 4.4U ; 4Ri ; 880Re bb0

Mixing Processes in the far wake Mixing Processes in the far wake : weak initial stratification: weak initial stratification

Iso-density field in the far wakeIso-density field in the far wake Mushroom type structures collapse due to Mushroom type structures collapse due to

stratificationstratification

cm 2d ; cm/s 4.4U ; 0.1Ri ; 880Re bb0

Sillage lointain

Mixing Processes in the far Mixing Processes in the far wake : strong initial stratification wake : strong initial stratification

(1)(1)

Iso-density field in the far wakeIso-density field in the far wake Mixed fluid inside the elliptic zonesMixed fluid inside the elliptic zones

Sillage lointain

cm 2d ; cm/s 4.4U ; 4Ri ; 880Re bb0

Mixing Processes in the far wake Mixing Processes in the far wake : strong initial stratification (2): strong initial stratification (2)

Iso-density Iso-density fields at fields at different timesdifferent times

interaction interaction betyween betyween shifted internal shifted internal waves : waves :

BreakingBreaking

Layering effect : computational Layering effect : computational domaindomain

Succession de passages d’une ou de plusieurs barresSuccession de passages d’une ou de plusieurs barres

« sheets & layers »« sheets & layers »

Density profiles for Density profiles for weak and strong weak and strong initial stratification initial stratification

Layering effect Layering effect weakly depends on weakly depends on initial stratificationinitial stratification

1760Re 880Re

Strongly stratified layersStrongly stratified layers

??

Stratified layers of another typeStratified layers of another type

cm 4D ; cm/s 4.4U ; 1Ri ; 880Re bb0

Unstable Unstable stratificationstratification

Convergence Convergence of density of density isolinesisolines

Successive wakesSuccessive wakes

Density profiles Density profiles and gradients and gradients after each after each cylinder towcylinder tow

Sratification Sratification increases after increases after each towingeach towing

cm 2d ; cm/s 2U ; 1Ri ; 400Re bb0

Successive wakesSuccessive wakes

Time evolution of Time evolution of the density the density gradientgradient

The maximum The maximum value increasesvalue increases

Damped Damped oscillations oscillations

cm 2d ; cm/s 2U ; 1Ri ; 400Re bb0

Infinitesimal perturbation Infinitesimal perturbation (1)(1)

Champ de densité après trois passages de la perturbationChamp de densité après trois passages de la perturbation

Successive infinitesimal Successive infinitesimal perturbation (2)perturbation (2)

Density profiles Density profiles and gradients after and gradients after 4 tows4 tows

Growth of the Growth of the perturbation after perturbation after each towingeach towing

cm 122x ; cm/s 2U b

Time evolution of the density Time evolution of the density and velocity gradientsand velocity gradients

Oscillation is Oscillation is dampeddamped

The stratification is The stratification is evolving following evolving following three stepsthree steps

The layering The layering increase is due to increase is due to the initial state the initial state before new before new perturbationperturbation

Vertical cylinder: Vertical cylinder: computational domaincomputational domain

Laboratory experimentsLaboratory experiments

Density profileDensity profile Towed vertical Towed vertical

cylindercylinder

Vertical cylinderVertical cylinder zig-zag zig-zag

instabilityinstability Layering effectLayering effect

Conclusion Conclusion

Caractéristics of stratified flows : Caractéristics of stratified flows : turbulence collapseturbulence collapse internal waves occuringinternal waves occuring

Mixing processes :Mixing processes : overturning collapseoverturning collapse breaking internal wavesbreaking internal waves

Layering effect :Layering effect : sheets & layers sheets & layers reorganizing layersreorganizing layers

PerspectivesPerspectives CFD improvements :CFD improvements : boundary conditions (open problem)boundary conditions (open problem)

long time computation : statistics and long time computation : statistics and budgetsbudgets

subgrid models (Babiano et al)subgrid models (Babiano et al)

Energy spectrumEnergy spectrum

Ri0 NBV (s-1) (m2.s-3) (mm)0.015 0.27 7.8 10-6 0.59

0.1 0.69 1.3 10-4 0.29

0.25 1.1 5.3 10-4 0.2

1 2.2 4.2 10-3 0.12

4 4.4 3.4 10-2 0.07

4/132

2/1

30 10 2

BVNl

Velocity components and Velocity components and gradientsgradients

440Re 4880Re

Processus de mélange dans le Processus de mélange dans le sillage proche : zones sillage proche : zones

mélangéesmélangées

Evolution Evolution temporelle d’un temporelle d’un profil vertical de profil vertical de densité dans les densité dans les cas de faible et de cas de faible et de forte stratificationforte stratification

cm 2d ; cm/s 4.4U ; 880Re bb