Flows Infected by Disturbance…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Turbulent Flows

The Infection

This change, however, does not occur suddenly.

The instability triggers a transition process, which is characterized by its

intermittently laminar-turbulentnature.

Instability may drastically change the

flow pattern from laminar to turbulent

An intermittently laminar-turbulent flow

Intermittency Factor

On Set of Turbulence

6

Definition

• A Fluid motion in which velocity, pressure, and other flow quantities fluctuate irregularly in time and space.

• “Turbulent Fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be observed.” - Hinze

• “Turbulence is due to the formation of point or line vortice on which some component of the velocity becomes infinite.:”

-Jean Leray

Time

What is turbulence?

• Unsteady, aperiodic motion in which all three velocity components fluctuate, mixing matter, momentum, and energy.

First Methods on Analyzing Turbulent Flow

Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation

- Likewise

stands for any scalar: u, v, w, T, p, where:

)(x,y,z,τuU(x,y,z)u(x,y,z,τ(

d

1

Time averaged Scalar

Averaging Navier Stokes equations

ρ ρ ρ

uUu

p Pp

v'Vv

w'Ww

Substitute into Steady incompressible Navier Stokes equations

Continuity equation:0

z

w

y

v

x

u

Instantaneous velocity

Averagevelocity

fluctuationaround averagevelocity

time

0

z

w')(W

y

v')(V

x

u')(U

0

z

w'

y

v'

x

u'

z

W

y

U

x

U

Averaging of Continuity Equations

0

z

w'

y

v'

x

u'

z

W

y

V

x

U

d

10

1

d

d

1

0

z

W

y

V

x

U 0

z

W

y

V

x

U

'

0'

2121221121 '')')('(

0'

Time Averaging Operations

)(

)( )( )( '2

'12121

Average x-momentum Equation

uμ z

u

y

u

x

uμ

2

2

2

2

2

2

vu )v(u

=0 continuity

2

2

2

2

2

2

z

u

y

u

x

uμ

x

p

z

uw

y

uv

x

uu

τ

uρ

uμ x

pvu

τ

uρ

Write x-momentum equations in a short format:

Short format of momentum equation in x direction:

)v(u

Averaging of x-momentum Equation

uμ x

pvu

τ

uρ

uμ x

pvu

τ

uρ

uUuU

τ

uUρρ

'

uU

uμ x

pvu

τ

uρ

vVuUvu

vuVuvUVV

vuVuvUVU

vuVU

kwujvuiuuvu ˆˆˆ

z

wu

y

vu

x

uu

z

wu

y

vu

x

uu

z

wu

y

vu

x

uuVU

uμ x

pvu

τ

uρ

uUu

uU

U