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

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