A Suggested Modification of Millionshchikov’s Zero-Fourth Cumulant Hypothesis

Authors: Christos Mamaloukas, Amitabha Chanda, Himandri Pai Mazumdar

Abstract: Over a few decades researchers, working in the field of turbulence, are examining the zero-fourth cumulant model, proposed by Millionshchikov. Numbers of modifications have been suggested. In some cases the problem of negative energy at the initial stage is observed and further modification is suggested. Some have tried to point out that the value of fourth cumulant may not remain zero all through the hierarchy of eddies vis-à-vis over all length scales and suggested different modification. In the present paper we have tried to suggest a new model. Its viability has also been examined to some extent. In the final period decay of isotropic turbulence, the smallest eddies of the largest wave number only are active. Viscous dissipation of turbulent energy in the form of thermal energy takes place in this final period. The largest wave number is defined as $$k_{∞}=\left(\frac{V^3}{2}\right)^{-\frac{1}{4}}$$. In the final period decay, neither inertia force nor pressure gradient takes part in the decay process. But these two terms, in fact, are the sources of third order moment. Keeping this in mind we have tried to develop a model for fourth cumulant, expressed as $$\widetilde{K_{4}}= \overline{\widetilde{u}^4}-\overline{\left( \widetilde{u}^2 \right)^2}$$ in wave number space. In the present case we would use the symbol $$\widetilde{u}$$ only, since in isotropy $$\overline{\widetilde{u_{1}}^4}=\overline{\widetilde{u_{2}}^4}=\overline{\widetilde{u_{3}}^4}$$ and $$\overline{\widetilde{u_{1}}^2}=\overline{\widetilde{u_{2}}^2}=\overline{\widetilde{u_{3}}^2}$$. With targets given above, we present a model of $$\widetilde{K_{4}}$$.

Pages: 56-59

DOI: 10.46300/91015.2022.16.11

International Journal of Systems Applications, Engineering & Development, E-ISSN: 2074-1308, Volume 16, 2022, Art. #11