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Title: Numerical study on the regularity of the Navier-Stokes equations
Author: Dowker, Mark
ISNI:       0000 0004 2723 2389
Awarding Body: University of Sheffield
Current Institution: University of Sheffield
Date of Award: 2012
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This thesis is mainly focused on the regularity problem for the three-dimensional Navier-Stokes equations. \\\\ The three-dimensional freely decaying Navier-Stokes and Burgers equations are compared via direct numerical simulations, starting from identical {\it incompressible} initial conditions, with the same kinematic viscosity. From previous work by Kiselev and Ladyzenskaya (1957), the Burgers equations are known to be globally regular thanks to a maximum principle. In this comparison, the Burgers equations are split via Helmholtz decomposition with consequence that the potential part dominates over the solenoidal part. The nonlocal term $-{\bm u}\cdot\nabla p$ invalidates the maximum principle in the Navier-Stokes equations. Its probability distribution function and joint probability distribution functions with both energy and enstrophy are essentially symmetric with random fluctuations, which are temporally correlated in all three cases. We then evaluate nonlinearity depletion quantitatively in the enstrophy growth bound via the exponent $\alpha$ in the power-law $\frac{dQ}{dt}+2\nu P\propto(Q aP b) {\alpha}$, where $Q$ is enstrophy, $P$ is palinstrophy and $a$ and $b$ are determined by calculus inequalities. \\\\ Caffarelli-Kohn-Nirenberg theory defines a local Reynolds number over parabolic cylinder $Q_r$ as $\delta(r)=1/(\nu r)\int_{Q_r}|\nabla {\bm u}| 2\,d{\bm x}\,dt$. From this we determine a cross-over scale $r_* \propto L\left( \frac{ \overline{\|\nabla \bm{u} \| 2_{L 2}} } {\| \nabla \bm{u} \| 2_{L \infty}} \right) {1/3}$, corresponding to the change in scaling behavior of $\delta(r)$. Following the assumption that $E(k)\propto k {-q}$ $(1
Supervisor: Ohkitani, Koji Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available