Rotating D’Arcy Convection

Manel Naspreda MartíDepartament de Física Fonamental

Universitat de Barcelona

Group of Statistical Physics

Revisor: Michael C. Cross,

Caltech University (California) USA

H.Philibert Gaspard Darcy

(1803-1858)

Outline:

• Motivation

• Historical Introduction

• Basic equations

• Rotating Rayleigh-Bénard convection

• D’Arcy convection and rotating D’Arcy convection

• Conclusions

Motivation:

• In rotating Rayleigh-Bénard convection there are predictions of chaos (wall state) in terms of the angular velocity and with small Prandtl number that it doesn’t work because of the mean flow. On the other hand, in D’Arcy convection there is not mean flow!

• Petrol!!• Analytical motivation!!!

Rc

KL instability

stripes

wall state

1202no pattern

Historical Introduction:

• Kuppers and Lorz (1969) predicted that there is some a critical value of the dimensionless angular velocity, c, when the stripes become unstable (KL instability), rotating an angle c60º.

• Busse and Heikes (1980) proposed a dynamical model that was introducing tree amplitudes.

• Yuhai Tu and M. Cross (1992) introduced in the last model the spatial dependences and predicted spatiotemporal chaos and within the amplitude equation one can see that the scaling in time and in space are:

• Actually, the experiments are not in a very well agreement with those predictions!!

12/1 ;

Basic equations

• Navier-Stokes equation:

extfvpvvtv

2·

• Continuum equation, incompressible equation:

0· v

•Density profile:

pT

1

,1)(

z

dT

z midcond

where is the fluid’s coefficient of thermal expansion at a constant pressure:

and T is the temperature measured from the temperature from the mid-depth: (T1–T2) / 2.

From the Navier-Stokes equation ( ), it is found that the pressure field is:

T

0v

.2

)( 2 conszdT

zgzp midcond

zd

TTTzTcond

21

2)(

•Temperature profile ( ):0v

•Balance momentum equation:

extfvpvvtv

2· zvzTTgvPvv

tv

D ˆ·2ˆ· 2

•Temperature evolution equation(Fourier Law):

TTvtT 2·

gzpP

Rotating Rayleigh-Bénard Convection:

d

T1

T2

D

0·

·

ˆ·2ˆ/·2

211

v

Rwv

zvzvvvv

t

t

where it has been considered the thermal diffusion time d2/, the pressure in units of d/, the temperature in units of /gd3( is the kinematic viscosity, g is the gravitational acceleration and is the effective thermal diffusivity).

•The adimentional constants are:

2

3

:

d :velocity angular alAdimension

:number Prandtl

TgdRnumber Rayleigh

D

zcondcond vw PP TT

D’Arcy Convection

• Hypotesis:• Isotropic porous medium

• Average porous size is sufficiently large

• The boundaries conditions are the same as Rayleigh-Bénard convection, unrealisitc free slip boundary conditions.

• It’s Rayleigh-Bénard convection, but the fluid is flowing in a porous media.

• Momentum balance equation (D’Arcy approximation):

vKzTTgP

ˆ0 •K is called the permeability of the porous medium, it is a positive constant and depends on the viscosity of the fluid and structure of the pores.

•Acceleration term is neglected compared to the damping term.

•No mean flow!!

0·ˆ vz

The other equations do not change.

Then if the time is measured in units of d2/, the temperature in units of K/gd, and the pressure in terms of K, one can find:

0·

·

ˆ02

v

Rwv

z v

t

KTdg

R :number Rayleigh porous

2

22

·

0

Rwwv

w

zt

Linear Analysis

• Antsatz:

0 2 4 6 8-50

-40

-30

-20

-10

0

10

20

q

q

R

R

R

xqitee

z

zwwq

·

)(

)(

• Free slip b.c.:

)cos()(

)(z

w

z

zw

• Critical Rayleigh number: ccc q R

q

qqR ;4)( 2

2

222

Amplitude equation

• Define an expansion parameter:c

c

RRR

• Expand every quantity in terms of this parameter:

)(

)(

)(

2/310

2/1

2/310

2/1

2/310

2/1

Ouuu

Owww

O

• In a multi-scale analysis:

..)(),,(

..)(),,(

..)(),,(

000

000

000

cczueTYXAu

cczweTYXAw

cczeTYXA

xiq

xiq

xiq

c

c

c

wherer the slow variables scales as: X=1/2x; Y=1/4y; T= t

0

·222

22 zt

z

z wu

w

R

• From the matricial equation:

+ some algebra...

0000

222

0000 2AAgA

iAA YXT

where the differents parameters are: 0=(2)-1; 0=-1; g0=1.

Rotating D’Arcy convection

• Coriolis force contribution in balance momentum:

zvvKzTTgP ˆ·2ˆ0

• Z-component of the vorticity:

zw

vz 2·ˆ

• The dynamics are described by:

2

2222

·

410

Rwwv

ww

zt

z

Linear Analysis

• Antsatz + f.s.b.c:

)cos(· zeeww xqitq

• Critical Rayleigh number:

4/12222 41411412 cc q R

2

22222 41)(

q

qqqRc

0 2 40

500

1000

1500

2000

2500

3000

Rc

Amplitude equation

• Define the perturbative parameter:)(

)(

c

c

RRR

• The way is the same as the D’Arcy convection, in sense of the perturbative theory and the seperation the magnitudes in seperate scales.

• The dynamics is described by:

0

·

41 2222

22 zt

z

z wu

w

R

Some algebra, and...

0000

222

0000 21

AAgAiq

AA Yc

XT

where the differents parameters are:

2220

22

0

2

2

cc

c

c

c

c

qR

qg

R

R

q

Conclusions:

• Not enough time to work in this field.

• In this framework, it has shown that there is an invariant in the sign of the angular velocity!!

Future works:

• Introduce the time dependence and the non-linear term in the momentum balance equation, to find the possible efect of the sign of the angular velocity.

• Some computer simulations, to check the analytical framework.

• Introduction of more than one amplitude in the seperate scale spaciotemporal dependence, tending to BH.

References

• Li Ning and Robert E. Ecke, PRE, 47, 5 (1993).• Yuhai Tu and M. C. Cross, PRL, 69, 17 (1992).• Yuchou Hu, Robert E. Ecke and Guenter Ahlers,

PRL, 74, 25 (1995).• E. Y. Kuo and M. C. Cross, PRL, 47, 4 (1993).• M. C. Cross, M. Louie and D. Meiron

(unpublished).