The Abdus Salam International

Center for Theoretical Physics

ICTP operates under a tripartite agreement between two United Nations Agencies—UNESCO and IAEA—and the Government of Italy.

ICTP VISITORS STATISTICS, 1970-2006IC

TP P

ublic I

nfo

rmation O

ffic

e,

Febru

ary

2007

DC: 50% (v), 70% (p-m)

aaaa

v=visitors p-m=person-months

area visitors p-m area visitors p-mNorth America AfricaLatin America Middle East and South AsiaWestern Europe South East Asia and the PacificEastern Europe Far East

9030 51309080 13870

34080 1875015050 16250

10450 16810

13350 204101580 19908000 14960

4000-6000 per year

6441 in 2004

permanent

scientists

temporary scientific staff: post-docs,

staff associates, long-term visitors

short-term visitors:

associates, guest scientists

participation in conferences,

schools and workshops;

diploma students

research and training

coordination

mostly research

do own or collaborative research, or

receive training through participation

receive training through

schools, conferences, and

exchange of information

30

100

420

+ about 140 general staff

Programs at ICTP

Diploma (ICTP),

Masters (with Italian partners)

Ph.D. (both Italian University system and IAEA SANDWICH).

Post-doctoral appointments

Scientific Associates

Training and research in Italian Laboratories Fellowships

Schools; Colleges; Workshops; Conferences

Electronic access to scientific literature

Support for regional conferences, often with other donors

UNESCO high school physics teacher -training workshops

Turbulence in classical and quantum fluids

Introduction

I. Real and ideal fluids in motion

II. Turbulence

III. Helium as a working fluid for generating high Re

IV. Classical turbulence experiments at high Re using helium

V. Quantum turbulence experiments in helium

VI. Development of instrumentation and large scale cryogenic experiments

OUTLINE

Fluids and fluid instabilities, including turbulence, appear in a wide

range of natural contexts as well as engineering systems.

“Perhaps the fundamental equation that describes the swirling

nebulae and the condensing, revolving, and exploding stars is

just a simple equation for the hydrodynamic behavior of nearly

pure hydrogen gas”. -- Richard Feynman

Consideration of fluid instabilities (Rayleigh Taylor instability) was needed explain

the first detailed observations of a supernova explosion: 1987a.

From Science and Technology Review (2000)

Simulation of Rayleigh Taylor instability when a

dense fluid is initially placed on top of a lighter

one. Here the dense fluid falls in spikes and the

ligher fluid rises in ―bubbles‖ to take its place.

Hydrodynamic (continuum) approximation

Lhydro>> mfp

Fluids can be considered as continuous fields if the characteristic

macroscopic motions are much larger than molecular motions, i.e. if

In this case, we can use continuum mechanics and express the

equations of motion in terms of partial differential equations (which will

make the problem nearly intractable but better than the alternative!)

There needs to be a sufficient number of atoms so that a density can be

assigned and further that relative fluctuations in density will be negligibly

small (they decrease as N-1/2, where N is the number of atoms in a volume

with the characteristic dimension of the length scale we are considering).

This criterion is not always satisfied in flows, e.g., a number of large scale

(turbulent) astrophysical flows.

VS

dvdt

dsdu

Basic Fluid Equations

The equations of mass, momentum and energy conservation are

written down in a coordinate system that is fixed in space (―Eulerian‖

description).

Considering a fluid of density moving at a velocity u: the mass

flux through an element of surface area ds of a volume V of fluid

must be equal to the rate of mass loss in the volume:

0)u(t

Mass conservation then takes the form:

Assuming constant density (note: we will assume this even in the

case where there is thermal expansion due to heating) this reduces to

the divergenceless condition:

0u

For the conservation of momentum we write down the equations in

terms of a force per unit volume f.

Like the magnetic field B in electrodynamics the fluid velocity has zero

divergence (there is a close analogy to the equations of electrodynamics).

fonaccelerati

extvisc ffpf

Before writing down the viscous forces we consider the acceleration term.

Note: the acceleration is not simply t

v

since a fluid particle can also change its momentum by flowing to a place where

the velocity is different.

In fact, the rate of change of any quantity, say ―G‖, is given by

z

Gu

y

Gu

x

Gu

t

G

Dt

DGzyx

Gut

G

where the operator

utDt

D is the substantial or convective derivative

Viscous forces

If you apply a shearing force to a fluid it will move— the shear forces are

described by the viscosity. Consider a layer of fluid between two plates, one

stationary and one moving at a slow speed v0

d

v0

A

F where F is the force required to keep the upper plate

moving and η is the dynamic or shear viscosity.Shear stress:

A more practical viscometer

due to Mallock and Couette

For an incompressible fluid a stress

tensor is given by

ijijij sp 2

i

j

j

iij

x

u

x

us

2

1

ext

iij

jj

ij

ii fxx

uu

t

u

Dt

Du

The momentum equation for incompressible fluids is then

Fluids are like graduate students….constantly under stress

ext

i

jj

i

ij

ij

i fxx

u

x

p

x

uu

t

u 11 2

Applying the continuity equation, the simplified NS equations are then:

Which also can be written in the more familiar form :

extpt

F1

u1

uuu 2

Flow past a circular cylinder. (a) Re = 26. (b) Re = 2000.

rr

'L

vv

'U

pp

U2'

ttU

L'

Re = UL/

v1

vvv 2pt

vRe

1vv

v 2pt

Consider simple pressure-driven flows:

(Navier-Stokes)

Reynolds

If we consider two simple flows that are geometrically similar, then they are also

dynamically similar if the corresponding Re is the same for both, regardless of the

specific velocities, lengths and fluid viscosities involved.

Matching such parameters between laboratory testing of a model and the actual

full-scale object is the principle upon which aerodynamic model-testing is based.

We will touch on these applications later.

Above left: wake behind a flat plate in the laboratory inclined 45 degrees to the direction

of the flow (left to right). Above right: A foundered ship in the sea inclined 45 degrees to

the direction of the current.

Dynamical similarity

0)( ut

pt

1uu

u

where we have assumed that the external forces can be written as the gradient

of a potential per unit mass Φ (like gravity).

Ideal fluids

Neglecting viscous forces we have the Euler equations for incompressible

potential flow:

The simplified equations can be simplified further—we can consider ―ideal‖

fluids, having no viscosity. Away from solid boundaries this can actually be a

good approximation to many flows and is especially useful for flow around

airfoils.

This rotation of the fluid is called the vorticity .

We define a new vector field Ω as the curl of u

u

A

dAd ulu

BY Stokes theorem we can relate the sum of vorticity over a given area to a

circulation around any reducible loop in the fluid bounding that area:

Taking the curl of both sides (why we wanted to consider that the

external force was conservative) we can eliminate the pressure and

obtain an equation for just the velocity:

0uΩΩ

t

If Ω =0 everywhere at any time t , then its time derivative also vanishes, so that Ω

is still zero at t+Δt. That is, the flow is permanently irrotational, so that if it starts

with zero rotation it will always have zero rotation. This can be understood by

considering that in the invisid fluid there are only normal pressure stresses (the

diagonal elements of σij and these cannot induce any rotation.

We can now rewrite the equations of motion , taking advantage of vector

identities to obtain

pt

1

2

1 2uuΩ

u

Taking the curl of both sides (why we wanted to consider that the

external force was conservative) we can eliminate the pressure and

obtain an equation for just the velocity:

0uΩΩ

t

If Ω =0 everywhere at any time t , then its time derivative also vanishes, so that Ω

is still zero at t+Δt. That is, the flow is permanently irrotational, so that if it starts

with zero rotation it will always have zero rotation. This can be understood by

considering that in the invisid fluid there are only normal pressure stresses (the

diagonal elements of σij and these cannot induce any rotation.

We can now rewrite the equations of motion , taking advantage of vector

identities to obtain

pt

1

2

1 2uuΩ

u

Bernoulli’s theorem

Consider steady flows—flows for which the velocity at

any one place never changes, i.e. for

0u

t

Taking the dot product of u into the equation of motion we obtain

02

1u 2u

p

This says that along streamlines (lines which are tangent to the fluid

velocity) we have

constup 2

2

1Bernoulli’s theorem

For steady irrotational flow Bernoulli’s theorem applies everywhere in the fluid (not

just along streamlines). It is simply a statement of the conservation of energy.

The relation between pressure and

velocity can be demonstrated with a

Venturi tube (right). In this case the fluid

has viscosity and there is some loss so

that the pressure is reduced in the

downstream section from what it was

before the constriction.

An invisid, irrotational fluid cannot cause

drag but can exert a lift force. Note that the

circulation in (b) is allowed because the

circuit cannot be reduced to a point. The

linear combination of the two possible

solutions for the flow past a cylinder is also

a solution and is shown in (c) where a

higher velocity on the top means a lower

pressure and hence a vertical lift force.

Lift of an airfoil

Vortex lines

Analogous to streamlines, vortex lines can be drawn which have

everywhere the direction of the vorticity Ω and whose density is

proportional to the magnitude of Ω.

The divergence of Ω is zero so lines of Ω does not start of stop in

the fluid unless they start and stop in closed loops (Helmholtz’ first

theorem). Otherwise they must begin and end on fluid boundaries

(the fluid would need viscosity in order to dissipate the vorticity

within the fluid).

Vortex lines move with the fluid.

We can imagine that we draw a

cylinder around an amount of fluid

which is parallel to the vortex lines.

The same bit of fluid will later be

someplace else and the cylinder

enclosing it may change shape (but

not volume since the fluid is

incompressible). So if the cylinder

crass-sectional area decreases the

density of vortex lines will increase

(because the magnitude of Ω is the

same).This is a statement of

conservation of angular momentum for

the fluid. Kelvin Vortex Theorem

0lu dDt

D

The circulation about any fluid line contour remains

constant in time as we move along with the fluid

The Potential vortex

The potential vortex is an example of irrotational

flow. Consider a flow in cylindrical coordinates in

which the velocity is entirely azimuthal and given by

01

zur

Cr

rr

However, at r=0 there is a singularity. In reality, the vortex cannot have infinite velocity

at its center. In real fluids with potential flow we might have zero vorticity

everywhere except on the central core can rotate as a rigid body. This qualitatively

describes tornadoes and perhaps hurricanes to a good approximation (the eye of the

hurricane --the core of the vortex-- is in fact relatively calm).

In a potential vortex the line integral of the velocity gives a constant value for the

circulation of 2πC independent of r.

0r

r

CV where C is a constant

This flow which describes, e.g. whirlpools, is irrotational; i.e. curl of the velocity is zero

Vortex line in an

inviscid Euler fluid

rv

2

For superfluids, as we will see later on, the core is approximately 1 Angstrom!

Reconnections between vortices qualitatively distinguish superfluid flows from

flows of classical fluids for which this process dissipates energy through viscosity

(superfluids have been called ―reconnecting Euler fluids‖)

Too thick and too vicous…

How to make a vortex yourself that closes on itself

Real Fluids

While ideal fluids are useful to understand flow away from boundaries,

we will need to consider actual viscous fluids if we are to understand

turbulence.

In certain circumstances viscosity can be neglected, for instance when

the Reynolds number is large –then the effects of viscosity are negligible

compared to inertia. However, the effects of viscosity become important

near boundaries where viscous boundary layers act as a buffer between

the flow and the no-slip condition that must occur at the solid-fluid

interface.

That the velocity must go to zero at a boundary is not obvious but us

easily demonstrated by examining the fine layer of dust on a fan blade

(the larger dust particles stick out of the boundary layer and are blown

off).

where kf / CP is the thermal diffusivity of the fluid with thermal conductivity kf

and specific heat CP.

We need one more equation for the energy: this is the advection diffusion equation

for conservation of energy and in the Boussinesq approximation [A. Oberbeck,

Annalen der Physik und Chemie 7, 271 (1879)]. Is given by

TTt

T 2u

Adding temperature to the equations: convection problems

To describe gravitational buoyancy in thermally driven flows, we need to specify

the external; body force term in the NS equations. There is only one component

of this force in the vertical direction given by Fext = g T where g is the

acceleration of gravity, is the isobaric coefficient of thermal expansion and T

is the temperature difference across a layer of fluid in the direction of gravity.

3

2

B

v THgRa

2H

D

H

Fluid

T+ T

T

D

HViscous diffusion time scale:

Thermal diffusion time scale:

Buoyancy time scale:

v

Pr

CONTROL PARAMETERS FOR CONVECTION

Rayleigh-Benard Convection

21

ff

BTg

H

V

H/

2

v

H

For ―small‖ temperature differences (Ra<1708) there is only pure conduction

For large enough temperature differences (Ra=1708 or

greater) there is a bifurcation to a convective state

RBC near onset: linear temperature gradients, up/down symmetry, steady patterns

Convective patterns in Nature

Solar granulation

atmosphere

Imprint in dry lake bed

Waffle House coffee

Some of the examples of convective flows in nature

Fluid Earth

The action takes place outside of the

core region which is relatively calm. The

hurricane acts as a heat engine

converting latent heat of evaporation of

warm ocean water into kinetic energy of

the vortex. Buoyancy drives the motion.

Hurricanes

The giant red

spot is a storm

which has

lasted at least

400 years

Over land (like Gainesville) hurricanes

lose their source of energy and quickly

dissipate (although usually not quick

enough!)

―Simple fluids are easier to drink than understand.‖

--A.C. Newell and V. E. Zakharov, in Turbulence: A

Tentative Dictionary ( Plenum Press, NY 1994)

G. Ihas and J. Niemela, in Tiny’s Bar,Santa Fe, NM 2003

―Simple fluids are easier to understand if you drink.‖