1. Reynolds-Averaged Navier-Stokes Equations -- RANS
4 equations; 7 unknowns
Similar situation as when we went from Cauchy’s eq to N-S eq

2. Aj = eddy viscosity
[m2/s]
Turbulence Closure

3. Turbulent Kinetic Energy (TKE)
Total flow = Mean plus turbulent parts =
Same for a scalar:
Start with momentum equation (balance) for total flow:
and subtract momentum equation for mean flow:
yields the momentum equation for turbulent flow:
An equation to describe TKE is obtained by:
multiplying the momentum equation for turbulent flow times the turbulent flow itself (scalar product)
and then do ensemble averages

4. Turbulent Kinetic Energy (TKE) Equation
Multiplying turbulent flow momentum equation times ui and dropping the primes (all lower case letters are turbulent or fluctuating variables)
Total changes of TKE
Transport of TKE
Shear
Production
Buoyancy
Production
Viscous
Dissipation
fluctuating strain rate
Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport

5. interaction of Reynolds stresses with mean shear;
represents gain of TKE
represents gain or loss of TKE, depending on covariance
of density and w fluctuations
represents loss of TKE

6. In many ocean applications, the TKE balance is approximated as:

7. Turbulence Production and Cascade
Injective range -- large scales where forcing injects the energy
Inertial range -- where the time required for energy transfer is shorter than the dissipative time and the energy is thus conserved and transported to smaller scales.
Dissipative range -- where the energy dissipation overcomes the transfer and the cascade is stopped.
Inertial range
“Big whorls have little whorls
That feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity.” (Lewis F Richardson, 1920)

9. - distance from boundary
The largest scales of turbulent motion (energy containing scales) are set by geometry:
- depth of channel
- distance from boundary
The rate of energy transfer to smaller scales can be estimated from scaling:
u velocity of the eddies containing energy
l is the length scale of those eddies
u2 kinetic energy of eddies
l / u turnover time
u2 / (l / u ) rate of energy transfer = u3 / l ~
At any intermediate scale l,
But at the smallest scales LK,
Kolmogorov length scale
Typically,
so that

11. Turbulence Cascade has a well defined structure – Kolmogorov’s K-5/3 law
Spectral power
Time (secs) S
Frequency (Hertz)
Spectral Amp (e.g. m2/Hz)
S = sin(2 π t /30)
T = 30 s

12. S = sin(2 π t /30) + sin(2 π t /12) S
Time (secs)
Spectral Amp (e.g. m2/Hz)
Frequency (Hertz)

13. S = sin(2 π t /30) + sin(2 π t /12) + sin(2 π t /43)
Time (secs)
Spectral Amp (e.g. m2/Hz)
Frequency (Hertz)

14. S (m3 s-2)
Wave number K (m-1)

15. P P & Kolmogorov’s K-5/3 law
small in inertial range -- vortex stretching
(Monismith’s Lectures)
equilibrium range
inertial
dissipating range

16. Data from Ichetucknee River
Hour from 00:00 on
-5/3

17. Kolmogorov’s K-5/3 law -- one of the most important results of turbulence theory
(Monismith’s Lectures)