# Presentation: 002688

## Phenomenology of Rayleigh-Taylor Turbulence

- _Misha Chertkov_   (Theory Division, Los Alamos)
- Igor Kolokolov (Landau Institute,  Moscow)
- Vladimir Lebedev (Landau Institute,  Moscow)

- MC                Phys.Rev.Lett **91,** 115001 (2003)
- MC,IK,VL     Phys.Rev.E **71**, 055301 (2005)

_*Boussinesq appr. (At<<1,miscible)  *_
_*Immiscible => effects of surface tension*_

_*U of Warwick: Sep 15, 2005*_

## Condition:  

- The developed (mixing) regime of
-  Rayleigh-Taylor instability (turbulence)

_*Question:*_
- Explain/understand spatio-temporal hierarchy of scales in velocity and density (temperature) fluctuations inside the mixing zone.

_***_

## Menu

- _Cascade picture_ (phenomenology)

_*Menu*_

- _Navier-Stokes__ Turbulence_   (Kolmogorov,Obukhov’41)
- _Passive Scalar Turbulence_     (Obukhov’48, Corrsin’51)

- _Rayleigh-Taylor turbulence phenomenology_

- _3d_

- _2d_

- _Low Atwood number_
- _incompressible  Boussinesq_

- _2d_
- vs
- _3d_

- Immiscible, steady turbulence
- Richtmyer-Meshkov + decay turb.
- Chemical reactions

-  Anisotropy
-  Intermittency
-  Mixing

- vs     Rayleigh-Benard

_*Plans:*_

- _Immiscible, incompressible_
-  [_effects of surface tension]_ *

## Navier-Stokes Turbulence  (steady 3d)

- _Menu_

- kinetic energy flux
-  scale independent !!!
-   time independent !!!

- typical velocity fluctuation on scale  “r”

- Kolmogorov ’41
- Obukhov ‘41

## Passive scalar turbulence    (steady)

- _Menu_

-       scalar flux
-  scale independent !!!
-   time independent !!!

- typical temperature fluctuations on scale  “r”

- Obukhov ’48
- Corrsin ‘51

## Low Atwood number, Boussinesq approximation

- e.g. Landau-Lifshitz
- ``Hydrodynamics”

_*Free convection*_* **(one fluid)*

_*Navier-Stokes*_

- Oberbeck 1879
- Lord Rayleigh  1883
- J. Boussinesq 1903

- _unstable_

- G.I. Taylor ‘1950
- Chandrasekhar ‘1961
-   ...

- _Menu_

- Rayleigh-Taylor  vs Rayleigh-Benard
-  (different initial/boundary conditions)

## Rayleigh-Taylor turbulence. Boussinesq.   3d

- L(t)  ~ turbulent (mixing) zone width
-            also energy-containing scale

- Adiabatic picture:
-        decreases with r

- _Menu_

- Sharp-Wheeler ’61 +
- Review:  Sharp ‘84

## Rayleigh-Taylor turbulence. Boussinesq.  

-                   3d   smallish scales

- velocity is smooth
- passive scalar adveciton is
- ``Batchelor”

- _Menu_

_*viscous scale*_

_*dissipative scale*_

## Rayleigh-Taylor turbulence. Boussinesq.   2d

- L(t)  ~ turbulent (mixing) zone width
-            also energy-containing scale

- _Menu_

- _so far the same as in 3d_

- _Two false attempts_

## Rayleigh-Taylor turbulence. Boussinesq.  2d  active scalar regime

-  Bolgiano ‘59-Obukhov ‘59
- (Rayleigh-Bernard turb scenario)

- _Menu_

*consistent with RB numerics*
*     **Celani,Matsumoto,*
*    **Mazzino,Vergassola ‘02*

## New numerical confirmations of the Bolgiano-Obukhov spectra

- A. Mazzino, A. Celani, L. Vozella  - Castel Gandolfo Sep 1-4, 2005

- _Menu_ *

## RTT. Immiscible case (effects of surface tension) d=3. 

_*Early stage*_

_*Late stage*_

- surface tension
- coefficient

_*capillary scale*_

_*viscous scale*_

_*wave turbulence range*_  *
_*density fluctuations*_*  **

- _Menu_

## Surface/capillary wave dynamics

_*Zakharov, Filonenko ’66*_
_*Zakharov,Pushkarev ’96,’00*_
- At=1  (water-air)

- Potential, incompresible flow

-  surface elevation
-  velocity potential

- 3-wave interaction

- _Menu_

## Surface/capillary wave turbulence

_*Zakharov and coll. ’66--*_

- Kolmogorov-Zakharov
- flux solution

*asymptotically weak*

- _Menu_

*locality is ok*

## Cascades at the early stages of immiscible RTT

- _Menu_

*Injection scale, L =*
*mixing zone width*

*K41 *

*K41 (bulk)*

*KZ (interfaces)*

*Capillary scale, l*

*wave dissipation*
*scale,  *

*viscous scale,  *

*Surface turb. is decoupled from*
*fluctuations in the bulk*
*Bulk (K41) turb. is not affected *
*by fluctuations on the surface(s)*
*[capilary fluct. of the scale r decay *
*distance r away from an interface]*

- _[Immis](slide12.xml)__[.](slide12.xml)_

## Density fluctuations. Immiscible. 3d.

*Obukhov-Corrsin*

- _Menu_

_*Immis.*_

_*passive scalar*_

## RTT. Immiscible. 2d

*Injection scale, L =*
*mixing zone width*

*BO *

*BO (bulk)*

*KZ (1d !!)*

*Capillary scale, l*

*wave dissipation*
*scale,  *

*viscous scale,  *

*Contact lines turb. is decoupled *
*from fluctuations in the bulk*
*Bulk (BO) turb. is not affected *
*by fluctuations on the line(s)*
*[capilary fluct. of the scale r decay *
*distance r away from an interface]*

- _Menu_

_*Immis.*_

## 2003 Dirac Medal

- On the occasion of the birthday of P.A.M. Dirac
- the Dirac Medal Selection Committee takes pleasure in announcing that
- the 2003 Dirac Medal and Prize
- will be awarded to:
**Robert H. Kraichnan**
- (Santa Fe, New Mexico)
- and
**Vladimir E. Zakharov**
- (Landau Institute for Theoretical Physics)
*The 2003 Dirac Medal and Prize is awarded to Robert H. Kraichnan and Vladimir E. Zakharov for their distinct contributions to the theory of turbulence, particularly the exact results and the prediction of inverse cascades, and for identifying classes of turbulence problems for which in-depth understanding has been achieved.*
* *
*Kraichnan’s most profound contribution has been his pioneering work on field-theoretic approaches to turbulence and other non-equilibrium systems; one of his profound physical ideas is that of ****the inverse cascade for two-dimensional turbulence****.  Zakharov’s achievements have consisted of putting the theory of wave turbulence on a firm mathematical ground by finding turbulence spectra as exact solutions and solving the stability problem, and in introducing the notion of ****inverse**** and dual cascades in wave turbulence.*
* *
* *
* *
* *
- 8 August 2003

- _Menu_

*2d Boussinesq *

*SurfaceWaves **

## Miscible case

- _Menu_

- compatibility
-   condition