Turbulence
Lecture 21
Nonlinear Dynamics
Strong nonlinearity is a key feature of turbulence.
1. Unstable, chaotic behavior.
2. Strongly vortical (vortex stretching)
30’s & 40’s – Taylor’s work on homogeneous turbulence
Kolmogorov.
60’s & 70’s – Kraichnan and followers addressed nonlinear stochastic processes.
Deductive theory hasn’t been developed.
1. Learned much about nonlinear dynamics.
2. Theories are based on ad hoc assumptions.
Simplify the problem and consider homogeneous turbulence.

Contains essential physics of the nonlinear dynamics.

Simple as possible.
Homogeneous in all spatial directions.
If it is then:
a. Energy exchange with mean flow.
b. Diffusion terms are not important.
Lose some large structures pathological (?) flow.
(Could be too much of a simplification.)
Ideas at least apply to smallscale in high Re no. flows.
Can generate a homogeneous flow in a lab.
1
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View Full DocumentScreen; mesh size, small compared to D.
/
U
⇒
Possibly with constant mean shear to provide energy to perturbations downstream.
– Not homogeneous
1. Taylor introduced the concepts in the 30’s and developed the experiment to study it.
2. Also can study homogeneous turbulence on the computer.
Solve the N.S. equations
directly at a high enough Re. No.
Re
1,000
λ
∼
realizable based on Taylor microscale.
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 Spring '09
 MEI
 Dynamics, Fluid Dynamics, Chaos Theory, Kinetic Energy, homogeneous turbulence

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