| Abstract: | It has long been observed that after a short initial transient period of time the decay of the velocity .uctuations  of high Reynolds number homogeneous isotropic turbulence follows closely the algebraic law  ∼ t–n. From experiments and DNS data it is noticed that the numerical value for the exponent is n ≈ 1.2. In the last decade values close to the latter one have been adopted to calibrate turbulence models such as the k‐ε‐model. It will be shown that if such an algebraic decay exists it corresponds to the invariant (similarity) solution of the equations of fluid dynamics under the group of scaling. Most important it is shown that if such solutions exist the decay exponents are fixed due to certain invariants. For the Navier‐Stokes equation the invariant is a constant Reynolds number and it follows n = 1. For the Euler equation the value is prescribed to  owing to the conservation of energy with finite initial energy. It is interesting to note that the experimental and DNS data indicate towards a decay induced by the Euler equations rather than the one conformal to the Navier‐Stokes equations. |
