6.7 Energy cascade

The process of large eddies becoming smaller is introduced in Sec. 6.6 . The process involves a transfer of kinetic energy from larger to smaller eddies, known as a the energy cascade.

PICT\relax \special {t4ht=

The variation in kinetic energy with eddy size is often illustrated by the energy spectrum graph above. The horizontal axis is wave number eqn,10 which represents the number of eddies per unit length, i.e. eqn where eqn = eddy size.

The cascade starts with the largest eddies of size eqn with the highest level of turbulent kinetic energy per unit mass (TKE). In the range, eqn, TKE = eqn.

PICT\relax \special {t4ht=

Large eddies are usually anisotropic due to the way turbulence is generated. For example, flow past a cylinder causes shedding of vortices whose shape are similar to the cylinder, i.e. longer in the direction of the cylinder axis.

Once moving through open space, eddies quickly stretch, bend, rotate and break, so that quite soon they become “blobs” of vorticity. The turn-over time, initially associated with shedding, decreases with eddy-size — by a factor of eqn at the smallest scales, according to Eq. (6.9 ). Any dominant frequencies, e.g. those due to shedding, are thereby quickly lost.

Kolmogorov’s hypothesis was that the turbulence then becomes isotropic and that eqn is then only a function of and eqn when eqn is large. From this hypothesis emerges his five-thirds law for the inertial subrange eqn

E(k) =
where eqn for eqn. The exponents of eqn and eqn are chosen to match the dimensions of eqn which are eqn, and noting eqn.

Experimental data generally supports Eq. (6.10 ) — remarkably, given its simplicity.

10The symbol eqn, is commonly used in science to denote wave number; it should not be confused with its later use to denote turbulent kinetic energy.

Notes on CFD: General Principles - 6.7 Energy cascade