Homogeneous-Isotropic Turbulence

• The term homogeneous turbulence implies that the velocity fluctuations in the system are random.
  
• The average turbulent characteristics are independent of the position of the fluid, i.e., invariant of axis translation.
  
• In addition to its homogeneous nature, if the velocity fluctuations are independent of axis of reference, i.e., invariant to axis rotation and reflection, then the turbulence can be termed as homogeneous and ISOTROPIC. By definition ISOTROPIC turbulence is always homogeneous.

Consider the root mean square velocity fluctuations

In homogeneous turbulence the rms values of the velocity components can all be different, but each value must be constant over the entire turbulent field.

In isotropic turbulence fluctuations are independent of the direction of reference and

or