# Numerical Study on the Regularity of the Navier-Stokes Equations

Dowker, Mark (2012) Numerical Study on the Regularity of the Navier-Stokes Equations. PhD thesis, University of Sheffield.

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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 $E(k)\propto k^{-q}$ $(1<q<3)$, it is shown that $r_*\propto \nu^a$ where $a=\frac{4}{3(3-q)}-1$. Direct numerical simulations of isotropic turbulence with $R_{\lambda}\approx 100$ and random initial data result in the scaling $\delta(r)\propto r^4$, which extends {\it throughout the inertial range}. This follows from the smallness of the intermittency parameter $a\approx 0.26$. From this value, the $\beta$-model predicts a dissipation correlation exponent $\mu=\frac{4a}{1+a}\approx 0.8$ which is much larger than the experimental observations of $0.2-0.4$. This suggests that the $\beta$-model is valid qualitatively but not quantitatively. The scale $r_*$ gives a practical method for estimating intermittency. \\\\ By studying the steadily propagating shock wave solutions of the one-dimensional Burgers equation with passive scalar, we determine a relationship between the dissipation rate $\epsilon_\theta$ of passive scalar and Prandtl number $P_r$ as $\epsilon_\theta\propto1/\sqrt{P_r}$ for large $P_r$. The profile of the passive scalar manifests as a sum of $\tanh^{2n+1}x$ for suitably scaled $x$ when $\nu\to 0$, implying that we must distinguish between different orders of the Heaviside function $H$ and $H^n$. If we do not account for this, we obtain the incorrect relationship $\epsilon_\theta\propto1/P_r$. The correct evaluation of this dissipation anomaly therefore requires Colombeau's theory for multiplication of distributions.