A refinement of the previous hypothesis of turbulent energy spectrum

Abstract

To refine the previous hypothesis of turbulent energy spectrum (Nan-niti, 1970), further theoretical consideration and a simple experiment using electric fans was carried out. From the hypothesis ofq=δε/δk andεq=const., the following relations are derived for the spectrumF, and the autocorrelation functionR:F(k)∼k _{−(5−2α)/3} andR(t)∼1−(t/τ)_2(1−α)/3, whereq is the local energy supply,ε the energy dissipation rate, andk the wavenumber,t the time,τ the lifetime of the largest eddy. The following further relations are also obtained:ε∼l _−α,v∼l _β,v∼τ _γ andγ=(1−α)/(2+α)=β/(1−β), wherel is the length scale of the phenomenon,v turbulent velocity,α=1/2,β=1/6, andγ=1/5. The experimental results support these relations.