An Axiomatic Approach toTurbulence:

Future Problems

Motivation

We start with an experimental reportsummarized by the three plots in Fig.1 .

We take off from this experimentalobservation to suggest a proposition,which we now state:

The failure of the continuum theoryin describing turbulence is signaled bya singularity in the form: accelerationof an infinitesimal volume at a pointis –infinity.

0 0.1 0.2 0.3 0.4 0.5 0.6-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4 3b: Air dV/dt

Time (s )

d

V

/

d

t

0 0.1 0.2 0.3 0.4 0.5 0.6-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

41b: N2 dV/dt

Time (s )

d

V

/

d

t

0 0.1 0.2 0.3 0.4 0.5 0.6-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2b: CO2 dV/dt

Time (s )

d

V

/

d

t

dV/dt( a

rb. u

nits

)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (s)

0.5

0

!0.5

!1.0

!1.5

!2.0

!2.5

1.0

0 .0

!1 .0

!2 .0

!3 .0

!4 .0

!5 .0

!6 .0

!7 .0

0.5

0

!0.5

!1.0

!1.5

!2.0

!2.5

!3.0

(a)

(b)

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4 3b: Air dV/dt

Time (s )

d

V

/

d

t

0 0.1 0.2 0.3 0.4 0.5 0.6-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

41b: N2 dV/dt

Time (s )

d

V

/

d

t

0 0.1 0.2 0.3 0.4 0.5 0.6-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

2b: CO2 dV/dt

Time (s )

d

V

/

d

t

dV/dt( a

rb. u

nits

)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Time (s)

0.5

0

!0.5

!1.0

!1.5

!2.0

!2.5

1.0

0 .0

!1 .0

!2 .0

!3 .0

!4 .0

!5 .0

!6 .0

!7 .0

0.5

0

!0.5

!1.0

!1.5

!2.0

!2.5

!3.0

(a)

(b)

(c)

Fig.1 – Occurrence of transition spikes in efflux experiments in (a) argon, (b) carbon dioxide and (c) air.

Taking the failure of continuumtheory as a given, we nowattempt to re-define turbulencein our own way.

Proposition I

• A gas is turbulent when the steady-state solutions of the relevant transportequation produce a multi-valuedvelocity field. Each steady-statesolution will be realized as a snapshotof the velocity field. Other snapshotscan be produced by other allowedsteady states. In time, the velocity fieldchanges as transitions occur from oneallowed steady state solution toanother.

Fig. 2 – An example: velocity field inside a toroidal model from the exact solutions for the steady-state post-Navier Stokes equation proposed by Getreur, Albano and Muriel [10]. The top and bottom panels differ in the probability of kicks σ from the “quantum paddle” Adopted from Ref. [10], courtesy of Elsevier.

Proposition II

• A gas consisting of ground statemolecules is laminar. A similar gasconsisting of excited molecules isturbulent. By virtue of the Boltzmanndistribution, a gas will consist oflaminar and turbulent flow. Thisproposition is consistent with a loreorally transmitted by Russianresearchers from Lev Landau, whosuggested that there is alwaysturbulence in a real gas, it is a matter ofdegree.

We will go back to Proposition I later. In the meantime Proposition II is encapsulated by a simple diagram below:

Laminar Turbulent

This single Proposition II hasresulted in the followingexperimental results:

(1) All critical Reynolds experiments arespecies-dependent. For example:

0 2 4 6 8

P/Pc

2600

2800

3000

3200

Re

N2

CO2

SF6

H2O & D

2O

1000 2000 3000 4000

Re

0.08

0.07

0.06

0.05

0.04

0.03

0.02

Friction

fac

tor

H2O

D2O

Re(H2O) = 3020

Re(D2O) = 3480

First conclusion from thishypothesis:

Scale Invariance is dead!Scale invariance isusually in the first chapterof most hydrodynamicstextbooks.

For example:

New Laws andPhenomena discovered

using the MolecularTheory of Turbulence

(1) Modification of textbook Law of PartialPressuresLaw of Partial Pressures

In a mixture of gases, each gas contributes itsown partial pressure. The total pressure is the

sum of all partial pressuresLaw of Superposition of Laminar-Turbulent

TransitionsIn a mixture of gases, each gas contributes itsown laminar-transition independent of the

other gases (Battat, Dadap, Hinkle, Muriel, submitted for publication)

(2) New Laws of Critical Pressures

(a) (Muriel, Physica A)

(b)

(Muriel, Physica A)

3/1

2

1

3/1

2

1

2

1

!!"

#$$%

&=!!

"

#$$%

&=

P

P

v

v

c

c

'

'

3/4

1

2

3/1

2

1

2

1

!!"

#$$%

&!!"

#$$%

&=

m

m

v

v

c

c

'

'

(3) New Scaling Law on Turbulent Efflux

Novapashin, Muriel (unpublished)

NEW SCALING LAW FOR TURBULENT FLOW

n

ooP

P

f

f!"

#$%

&=

New Scaling Law for

Turbulent Flow

n

• Argon 0.602

• Helium 0.690

• Carbon Dioxide 0.569

• Nitrogen 0.587

• Oxygen 0.595

New Principle Enunciated:The Principle of Limiting Excitation• In a physical phenomenon that threatens a

runaway situation, the system reacts to reducethe runaway situation by awakening its internaldegrees of freedom. (Muriel, Physica D)

• This is analogous to Le Chatelier’s Principleand Lenz Law

(4) Possible Detection of Stealth Plane

Back to Proposition I:A gas is turbulent when the steady-statesolutions of the relevant transport equationproduces a multi-valued velocity field.Each steady-state solution will be realizedas a snapshot of the velocity field. Othersnapshots can be produced by otherallowed steady states. In time, the velocityfield changes as transitions occur from oneallowed steady state solution to another.

Examples of transport equationsfrom which turbulence may

arise:Navier-Stokes equation

( ) fpuuut

u=!+!•+"#=

$

$%

Navier-Stokes Equation: Solve! $1million Clay Institute, Massachusetts

Examples of transport equationsfrom which turbulence may

arise:Burgers Equation a la Imperio-Esguerra

Examples of transport equationsfrom which turbulence may

arise:Solon-Esguerra Equations

Examples of transport equationsfrom which turbulence may

arise:GAM transport equation (Getreur, Albano,

Muriel)

Analytic demonstration of turbulence a la Muriel

mu

x

u

t

u !="

#

#+

#

# $$

2

2

1

Examples of transport equationsfrom which turbulence may

arise:Jirkovsky-Muriel transport equation

Analytic and numerical demonstration of turbulent velocity profile in pipes

mPUUU

t

Uijjiijj

i !"#"=+#+

#

# $

%$

1

( ) ( ) ( )

( ) ( ) !!"

#$$%

&'

(()

*

++,

-

!!"

#$$%

&'+

+(()

*

++,

-

!!"

#$$%

&+'!!

"

#$$%

&''=

.

/

.

/

ktzk

kzk

k

g

zkk

zkkt

k

gtzU

expsinh2

tanhcosh

sinh2

tanhcoshexp1,

This is the first result that reproduces real data.

Others:• Fokker-Planck equation• Boltzmann equation• Langevin equation

Next:• Solis-Esguerra transport equation

Experimental

In addition to Oldenburg, Trieste andNovosibisrk, there is now a Hinkleapparatus in the Chemical EngineeringDepartment of the College of Engineering

Possibly to do the following:

• Verify old results• Control the onset of turbulence by sound• Find new scaling laws of turbulence• Detect far infra red radiation from

turbulent CO2.• Possible numerical work: confront

analytic results with numerical results atthe Department of Computer Science inthe College of Engineering