Physics of turbulence at small scales

Turbulence is a property of the flow not the fluid.

1. Can only be described statistically.2. Dissipates energy and mixes properties in rates much higher than laminar flows.3. Intermittent in time and space.4. Threshold phenomena (Re>Rec)

Governing equations:

Navier-Stokes:

Continuity:

Presence of viscosity requires supply of energy to maintain a given turbulence.

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Reynolds decomposition:

velocity:

Scalars:

<> Denotes an averaging operator (usually time average).

0' , ,' iiiiii uUuuUu

0' , ,' pPppPp

Energy cascade:

Inertial terms cascade the energy from large scale eddies which are inviscid to smaller scales and those cascade energy to smaller eddies until the smallest eddies (Kolmogorov scale) are dissipated by viscosity.

The ratio of the inertial term to the viscous term is the ReLU/.

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ij x

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Kolmogorov’s theory:Assume large (‘integral-’) scale eddies at which energy is fed into the system-L, with characteristic velocity UL.

The cascade hypothesis assumes that these eddies becomes unstable at TL~L/ UL and dump their energy into smaller eddies (say of size L/2) and so on.

The energy transferred (per unit mass, per unit time) is ~UL2/TL=UL

3/L.

In equilibrium, for all eddies in the cascade ~U2/T=U

3/.

The 2 length-scales bounding the cascade are:Integral length scale: Lu’3/Kolmogorov (dissipation) scale: Tdisipation=Tcascade

Characteristic scales (Jimenez, 1997):

Range Length Velocity shear Decay t

Largest L u’ /u’2 u’2/

inertial L (L) (L2) (L2/)

Kolmogorov

Dissipative L L L2/Kinetic energy is associated with the large scale eddies.Shear and viscous dissipation with the smallest scales.Time scale of small eddies is shorter allowing them to adjust fast to the slowly changing larger scale fluctuations.

Transfer from 2-D to 3-D:

Large scale eddies are 2-D, while small scale turbulent eddies are isotropic (3-D). This change can be accomplished by the interaction of shear with vortices:

Stretching means faster rotation and enhanced viscous dissipation.

The TKE dissipation rate ():

2

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5.72

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For isotropic, homogeneous turbulence it can be shown (Batchelor, 1953) that:

This is the principle of measurement of using shear probes.

Turbulent diffusivity and :

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22

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The mixing of scalars occur at higher rate then molecular diffusion and is related to the . For density, mixing goes against gravity and:

~0.25.

Turbulence at small scales

Turbulent cascade of energy from scales where inertia dominates to scales where viscosity dominates.

Rapid adjustment at small scales to changes in energy input

Can (should) be quantified using ADCP’s and/or CTD casts.