1. Introduction to Symmetry Analysis
Chapter 13 -Similarity Rules for Turbulent
Shear Flows
Brian Cantwell
Department of Aeronautics and Astronautics
Stanford University

2. Reynolds number
invariance

5. Turbulent kinetic energy spectrum

6. The Reynolds averaged Navier-Stokes equations
Reynolds stresses

7. Integral length and velocity scales
Dissipation scales with production

8. Derivation of the turbulent kinetic energy equation
Momentum and continuity equations
Kinetic energy equation - project the momentum equation onto the velocity vector

9. Decompose the flow into a mean and fluctuating part.
Project the mean momentum equation onto the mean velocity vector
Subtract

10. Dissipation of TKE equals production of TKE
The turbulent kinetic energy (TKE) transport equation is
Consider homogeneous shear flow
All gradients of correlations are zero
Dissipation of TKE equals production of TKE

11. Invariant group of the Euler equations

12. One parameter flows

13. Temporal similarity rules
Reduced equations

14. Spatial similarity rules - jets
Spatial similarity rules - wakes

15. Reynolds number scaling
The idea of an eddy viscosity

17. Scaling of a turbulent vortex ring

18. Scaling of a turbulent vortex ring

19. Integral of the motion - the hydrodynamic impulse

20. Streamlines
and
Particle Paths

21. Recall that the incompressible Navier-Stokes equations are invariant under a group of arbitrary translations in space.

22. Instantaneous flow field
in the wake of a
circular cylinder as
seen by two observers.

23. Reduced equations
Particle paths
Frames of reference

24. Particle paths in similarity coordinates do not depend on the observer

25. Fine scale motions responsible for kinetic energy dissipation.
Recall
Introduce the Taylor microscale

26. Introduce the Kolmogorov microscales
Fine scale velocity gradients
Large scale velocity gradients

27. Turbulent kinetic energy spectrum

28. Scaling the inertial subrange. Assume
The wavenumber of an eddy is essentially the inverse of its length scale.

29. The TKE as a function of wavenumber should scale as
This is the scaling of TKE first postulated by Kolmogoriv in 1941
and seems to agree with measurements in high Reynolds number flows.

33. Derivation of the turbulent kinetic energy equation

34. Turbulent kinetic energy equation
Momentum and continuity equations
Kinetic energy equation - project the momentum equation onto the velocity vector

35. Viscous and pressure terms
Kinetic energy equation

36. Insert fluctuations and average

37. Project the mean momentum equation onto the mean velocity vector

40. Dissipation of TKE equals production of TKE
The turbulent kinetic energy (TKE) transport equation is
Consider homogeneous shear flow
All gradients of correlations are zero
Dissipation of TKE equals production of TKE