Numerical study on comparison of Navier-Stokes and Burgers equations

Abstract

We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematic viscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, “On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid,” Izv. Akad. Nauk SSSR Ser. Mat. 21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl. 24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term \documentclass[12pt]{minimal}\begin{document}$-{\bm u}\cdot \nabla p$\end{document}−u·∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with \documentclass[12pt]{minimal}\begin{document}$\frac{1}{2}|{\bm u}|^2$\end{document}12|u|2 and with \documentclass[12pt]{minimal}\begin{document}$\frac{1}{2}|{\bm \omega }|^2$\end{document}12|ω|2 are also found to be symmetric, fluctuating at the same times as the probability density function of \documentclass[12pt]{minimal}\begin{document}$-{\bm u}\cdot \nabla p$\end{document}−u·∇p. A power-law relationship is found in the mathematical bound for the enstrophy growth \documentclass[12pt]{minimal}\begin{document}$\dfrac{dQ}{dt} + 2 \nu P \propto \left(Q^a P^b\right)^\alpha ,$\end{document}dQdt+2νP∝QaPbα, where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis.