Loading in 5 sec....

Statistical Fluctuations of Two -d imensional TurbulencePowerPoint Presentation

Statistical Fluctuations of Two -d imensional Turbulence

Download Presentation

Statistical Fluctuations of Two -d imensional Turbulence

Loading in 2 Seconds...

- 99 Views
- Uploaded on
- Presentation posted in: General

Statistical Fluctuations of Two -d imensional Turbulence

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Statistical Fluctuations of Two-dimensional Turbulence

Mike Rivera and Yonggun Jun

Department of Physics & Astronomy

University of Pittsburgh

Table of Contents

- Introduction
- Experimental Setup
- Experimental Results
- • Average Behavior
- • Fluctuations
- Comparison with 3D Results
- Conclusion

Soft-Condensed Matter Physics Group

What is Turbulence?

- Turbulence: irregularly fluctuating and unpredictable motion which is made up of a number of small eddies that travel in the fluid.
- Eddy: volume where the fluid move coherently.

Leonardo da Vinci

Soft-Condensed Matter Physics Group

Evolution to Turbulence

At low Reynolds numbers, the flow past the rod is regular.

Re=UL/n

U: typical velocity

L: typical length

n: viscosity

As Reynolds number increases, the size of traveling vortices also increases.

Re>50

Finally, the flow becomes irregular.

Soft-Condensed Matter Physics Group

15 oA

Freely Suspended Film is 2DL

*Non-equilibrium Films: 1<h<100 m

h/L ~ 10-4 - 10-3

Soft-Condensed Matter Physics Group

Flows in Earth Atmosphere is 2D

Soft-Condensed Matter Physics Group

vy

7 cm

Soft-Condensed Matter Physics Group

Forced 2D Turbulence- Applied voltage : f = 1 Hz.
- Taylor microscale Reynolds number
- Rel= 110, 137, 180 and 212
- - Energy injection scale linj=0.3cm,
- outer scale lo~2cm

Soft-Condensed Matter Physics Group

Experimental Setup

CCD Camera

Nd-YAG Laser

Magnet array

Soap film frame

Soft-Condensed Matter Physics Group

Transitions to Turbulence

Soft-Condensed Matter Physics Group

Particle Image Velocimetry

Dt=2 ms

Soft-Condensed Matter Physics Group

Soft-Condensed Matter Physics Group

Typical Velocity Field

Soft-Condensed Matter Physics Group

Soft-Condensed Matter Physics Group

Stability of the Flow

Soft-Condensed Matter Physics Group

Fluctuations increases with Re

Soft-Condensed Matter Physics Group

Navier-Stokes Equation

: incompressible condition

v : velocity of fluid

p : reduced pressure

n : the viscosity

a : drag coefficient between the soap film and the air

f : reduced external force

Reynolds Number Re

Soft-Condensed Matter Physics Group

Injection length linj

Energy flux e

Dissipative length ldis

………………………………….….

Energy Cascade in 3D TurbulenceSoft-Condensed Matter Physics Group

2D

Energy Spectrum in 2D and 3DE(k)

E(k)

Ev~k-5/3

E~k-5/3

k-3

k3

k

kd

kd

ki

ki

Soft-Condensed Matter Physics Group

Physics of 2D Turbulence

Vorticity Equation

Since no vortex stretching in 2D ( ),

, w is a conserved quantity when n=0.

Soft-Condensed Matter Physics Group

l

Consequence of Enstrophy Conservationk1

k0

k2

E0=E1+E2

k02E0=k12E1+k22E2

k0=k1+k2

Let k2=k0+k0/2 and k1=k0-k0/2

Soft-Condensed Matter Physics Group

Longitudinal Velocity Differences

Urms (cm/s)

10

8.0

5.5

4.0

3.0

1.9

Soft-Condensed Matter Physics Group

2nd Order Structure Function

Soft-Condensed Matter Physics Group

Topological Structures

Soft-Condensed Matter Physics Group

Vorticity and Stain-rate Fields

Enstrophy Fields, w2

Squared strain-rate Fields, s2

Soft-Condensed Matter Physics Group

Pressure Fields

Soft-Condensed Matter Physics Group

- In 3D turbulence, intermittency stems from the non-uniform distribution of the energy dissipation rate by vortex stretching.

(a) velocity fluctuations from a jet and (b) velocity fluctuationsafter high-pass filtering which shows intermittent bursts (Gagne 1980).

Soft-Condensed Matter Physics Group

Soft-Condensed Matter Physics Group

Intermittency

- From velocity time series and assuming homogeneity/isotropy of flows, e can be calculated.
- In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of the velocity difference are close to Gaussian.

The turbulent plasma in the solar corona

E. Buchlin et.al A&A 436, 355-362 (2005)

Soft-Condensed Matter Physics Group

The PDFs of dvland Sp(l)

Soft-Condensed Matter Physics Group

The Scaling Exponents

Red: Our data;

Blue: 2D turbulence by Paret and Tabeling (Phys. of Fluids, 1998)

Green: 3D turbulence by Anselmet et. al. (J. of Fluid Mech. 1984)

Soft-Condensed Matter Physics Group

Log-Normal Model

In 1962, Kolmogorov suggested log-normal model.

Soft-Condensed Matter Physics Group

The PDFs of el

The el has broad tails, but log(el) is normally distributed.

Soft-Condensed Matter Physics Group

Cross-correlation Function between dvl and el

The velocity difference dvl is

correlated with the local

energy dissipation rate. But

such a dependence decreases

as l increases.

Soft-Condensed Matter Physics Group

The Scaling Exponent zp/ z3

- Red diamonds are calculated by velocity difference vlp
- ~ zp
- blue circles are obtained by local energy dissipation elp
- ~ p/3+tp
- Solid line indicates the slope 1/3 by the classical Kolmogorov theory.
- The dash line indicates the fit based on lognormal model, m~0.11

Soft-Condensed Matter Physics Group

Conclusions

- We demonstrated that it is possible to conduct fluid flow and turbulence studies in freely suspended soap films that behave two dimensionally.
- The conventional wisdom suggests that turbulence in 2D and 3D are very different. Our experiment shows that this difference exists only for the mean quantities such as the average energy transfer rate. As far as fluctuations are concerned, they are very similar.
- Intermittency exists and can be accounted for by non-uniform distribution of saddle points similar to 3D turbulence.

Soft-Condensed Matter Physics Group

Acknowledgement

- Mike Rivera
- Yonggun Jun
- Brian Martin
- Jie Zhang
- Pedram Roushan

- Walter Goldburg
- Hamid Kelley
- Maarten Rutgus
- Andrew Belmonte

This work has been supported by NASA and NSF

Soft-Condensed Matter Physics Group