CFD 7 – 1 David Apsley

7. TURBULENCE SPRING 2012

7.1 What is turbulence?

7.2 Momentum transfer in laminar and turbulent flow

7.3 Turbulence notation

7.4 Effect of turbulence on the mean flow

7.5 Turbulence generation and transport

7.6 Important shear flows

Summary

Examples

PART (a) – THE NATURE OF TURBULENCE

7.1 What is Turbulence?

Instantaneous Mean

• A “random”, 3-d, time-dependent eddying motion with many scales, superposed on an

often drastically simpler mean flow.

• A solution of the Navier-Stokes equations.

• The natural state at high Reynolds numbers.

• An efficient transporter and mixer ... of momentum, energy, constituents.

• A major source of energy loss.

• A significant influence on drag and boundary-layer separation.

• “The last great unsolved problem of classical physics.” (variously attributed to

Sommerfeld, Einstein and Feynman)

CFD 7 – 2 David Apsley

7.2 Momentum Transfer in Laminar and Turbulent Flow

In laminar flow adjacent layers of fluid slide past each other

without mixing. Transfer of momentum occurs between

layers moving at different speeds because of viscous stresses.

In turbulent flow adjacent layers continually mix. A net

transfer of momentum occurs because of the mixing of fluid

elements from layers with different mean velocity. This

mixing is a far more effective means of transferring

momentum than viscous stresses. Consequently, the mean-

velocity profile tends to be more uniform in turbulent flow.

7.3 Turbulence Notation

The instantaneous value of any flow variable can be decomposed into mean + fluctuation.

is decomposed into

mean + fluctuation

Mean and fluctuating parts are denoted by either:

• an overbar and prime: uuu

′

+

=

or

• upper case and lower case: uU

+

The first is useful in deriving theoretical results but becomes cumbersome in general use. The

notation being used is, hopefully, obvious from the context.

By definition, the average fluctuation is zero:

0=

′

u

In experimental work and in steady flow the “mean” is usually a time mean, whilst in

theoretical work it is the probabilistic (or “ensemble”) mean. The process of taking the mean

of a turbulent quantity or a product of turbulent quantities is called Reynolds averaging.

The normal averaging rules for products apply:

222 uuu ′

+= (variance)

vuvuuv ′′

+= (covariance)

Thus, in turbulent flow the “mean of a product” is not equal to the “product of the means” but

includes an (often significant) contribution from the net effect of turbulent fluctuations.

v

u

laminar

turbulent

CFD 7 – 3 David Apsley

7.4 Effect of Turbulence on the Mean Flow

Engineers are usually only interested in the mean flow. However, turbulence must still be

considered because, although the averages of individual fluctuations (e.g. u

′

or v

′

) are zero,

the average of a product (e.g. vu

′

′

) is not and may lead to a significant net flux.

Consider mass and momentum fluxes in the y direction across surface A. For simplicity,

assume constant density.

7.4.1 Continuity

Mass flux: vA

Average mass flux: Av

The only change is that the instantaneous velocity is replaced by the mean velocity.

The mean velocity satisfies the same continuity equation as the instantaneous velocity.

7.4.2 Momentum

x-momentum flux: AuvuvA )(

)(

=

Average x-momentum flux: AvuvuuvA )()( ′′

+=

The average momentum flux has the same form as the instantaneous momentum flux …

except for additional fluxes Avu ′′ due to the net effect of turbulent fluctuations. These

additional terms arise because of the averaging of a product of fluctuating quantities.

A net rate of transport of momentum Avu ′′ from lower to upper side of an interface ...

• is equivalent to a net rate of transport of momentum Avu ′′

− from upper to lower;

• has the same dynamic effect (i.e. same rate of transfer of momentum) as a stress (i.e.

force per unit area) vu ′′

−.

This apparent stress is called a Reynolds stress. In a fully-turbulent flow it is usually much

larger than the viscous stress.

Other Reynolds stresses ( uu ′′

−, vv ′′

−, etc.) emerge when considering the average flux of

the different momentum components in different directions.

The mean velocity satisfies the same momentum equation as the instantaneous velocity,

except for additional apparent stresses: the Reynolds stresses jiuu ′′

−

vA

ρ

u

vA

ρ

CFD 7 – 4 David Apsley

In a simple shear flow the total stress is

{

321

stress

turbulent

stress

viscous

vu

y

u′′

−

∂

∂

= (1)

In fully-turbulent flow turbulent stress is usually substantially bigger than viscous stress.

can be interpreted as either:

• the apparent force (per unit area) exerted by the upper fluid on the lower,

or

• the rate of transport of momentum (per unit area) from upper fluid to lower.

The dynamic effect – a transfer of momentum – is the same.

The nature of the turbulent stress can be illustrated by considering the motion of particles

whose fluctuating velocities allow them to cross an interface.

If particle A migrates upward (v′ > 0) then it tends to retain its original

momentum, which is now lower than its surrounds (u′ < 0).

If particle B migrates downward (v′ < 0) it tends to retain its original

momentum which is now higher than its surrounds (u′ > 0).

In both cases, vu

′

′

−

is positive and, on average, tends to reduce the

momentum in the upper fluid or increase the momentum in the lower

fluid. Hence there is a net transfer of momentum from upper to lower

fluid, equivalent to the effect of an additional mean stress.

Velocity Fluctuations

Normal stresses: 222 ,, wvu ′′′

Shear stresses: vuuwwv ′′′′′′ ,,

(In slightly careless, but extremely common, usage both vu ′′

− and vu ′′ are referred to as

“stresses”.)

Most turbulent flows are anisotropic; i.e. 222 ,, wvu ′′′ are different.

Turbulent kinetic energy: )( 222

2

1wvuk ′

+

′

+

′

=

Turbulence intensity: U

k

U

u

itymean veloc

ctuationsquare fluroot-mean- rms 3

2

=

′

==

y

U

v'

B

A

v

u

y

U

τ

τ

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