One of the most prominent open problems in mathematical physics is determining whether solutions to the incompressible three-dimensional (3D) Navier-Stokes system, corresponding to arbitrarily large smooth initial data, remain regular for arbitrarily long times. A promising approach to this problem relies on the fact that both the smoothness of classical solutions and the uniqueness of weak solutions in 3D flows are ultimately controlled by the growth properties of the $H^1$ seminorm of the velocity field U, also known as the enstrophy. In this context, the sharpness of analytic estimates for the instantaneous rate of growth of the $H^2$ seminorm of U in two-dimensional (2D) flows, also known as palinstrophy, and for the instantaneous rate of growth of enstrophy in 3D flows, is assessed by numerically solving suitable constrained optimization problems. It is found that the instantaneous estimates for both 2D and 3D flows are saturated by highly localized vortex structures. Moreover, finite-time estimates for the total growth of palinstrophy in 2D and enstrophy in 3D are obtained from the corresponding instantaneous estimates and, by using the (instantaneously) optimal vortex structures as initial conditions in the Navier-Stokes system and numerically computing their time evolution, the finite-time estimates are found to be uniformly sharp for 2D flows, and sharp over increasingly short time intervals for 3D flows. Although computational in essence, these results indicate a possible route for finding an extreme initial condition for the Navier-Stokes system that could lead to the formation of a singularity in finite time. / Thesis / Doctor of Philosophy (PhD)

One of the key open problems in the field of theoretical fluid mechanics concerns the possibility of the singularity formation in solutions of the 3D Navier-Stokes system in finite time. This phenomenon is associated with the behaviour of the enstrophy, which is an L2 norm of the vorticity and must become unbounded if such a singularity occurs. Although there is no blow-up in the 2D Navier-Stokes equation, we would like to investigate how much enstrophy can a planar incompressible flow in a bounded domain produce given certain initial enstrophy. We address this issue by formulating an optimization problem in which the time derivative of the enstrophy serves as the objective functional and solve it using tools of the optimization theory and calculus of variations. We propose an efficient computational approach which is based on the iterative steepest-ascent procedure. In addition, we introduce an easy-to-implement method of computing the gradient of the objective functional. Finally, we present computational results addressing the key question of this project and provide numerical evidence that the maximum enstrophy growth exhibits the scaling dE/dt ~ C*E*E for C>0 and very small E. All computations are performed using the Chebyshev spectral method. / Thesis / Master of Science (MSc) / For many decades, scientists have been investigating fundamental aspects of the Navier-Stokes equation, a central mathematical model arising in fluid mechanics. Although the equation is widely used by engineers to describe numerous flow phenomena, it is still an open question whether the Navier-Stokes system always admits physically meaningful solutions. To address this issue, we want to explore its mathematical aspects deeper by analyzing the behaviour of the enstrophy, which is a quantity associated with the vorticity of the flow and a convenient measure of the regularity of the solution. In this study, we consider a planar and incompressible flow bounded by solid walls. Using basic tools of mathematical analysis and optimization theory, we propose a computational method enabling us to find out how much enstrophy can such a flow produce instantaneously. We present numerical evidence that this instantaneous growth of enstrophy has a well-defined asymptotic behavior, which is consistent with physical assumptions.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20154号 / 理博第4239号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 上田 哲生, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Motivated by the results concerning the regularity of solutions to the fractional Navier-Stokes system and questions about the influence of noise on the formation of singularities in hydrodynamic models, we have explored these two problems in the context of the fractional 1D Burgers equation. First, we performed highly accurate numerical computations to characterize the dependence of the blow-up time on the the fractional dissipation exponent in the supercritical regime. The problem was solved numerically using a pseudospectral method where integration in time was performed using a hybrid method combining the Crank-Nicolson and a three-step Runge-Kutta techniques. A highlight of this approach is automated resolution refinement. The blow-up time was estimated based on the time evolution of the enstrophy (H1 seminorm) and the width of the analyticity strip. The consistency of the obtained blow-up times was verified in the limiting cases. In the second part of the thesis we considered the fractional Burgers equation in the presence of suitably colored additive noise. This problem was solved using a stochastic Runge-Kutta method where the stochastic effects were approximated using a Monte-Carlo method. Statistic analysis of ensembles of stochastic solutions obtained for different noise magnitudes indicates that as the noise amplitude increases the distribution of blow-up times becomes non-Gaussian. In particular, while for increasing noise levels the mean blow-up time is reduced as compared to the deterministic case, solutions with increased existence time also become more likely. / Thesis / Master of Science (MSc)

Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasigeostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of two-dimensional and quasigeostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascade in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasigeostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the unexpecteddependency on an infrared Reynolds number in the spectral scaling of theenergy spectrum in the inverse energy cascade, and the validation of Charney’spredictions on the dynamics of quasigeostrophic turbulence. It has also beenshown that the scaling of the energy spectrum in the enstrophy cascade isinsensitive to intermittency in higher order statistics, but that corrections mightapply to the ”universal” Batchelor-Kraichnan constant.

<p>Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasigeostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of two-dimensional and quasigeostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascade in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasigeostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the unexpecteddependency on an infrared Reynolds number in the spectral scaling of theenergy spectrum in the inverse energy cascade, and the validation of Charney’spredictions on the dynamics of quasigeostrophic turbulence. It has also beenshown that the scaling of the energy spectrum in the enstrophy cascade isinsensitive to intermittency in higher order statistics, but that corrections mightapply to the ”universal” Batchelor-Kraichnan constant.</p>

Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasi-geostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of twodimensionaland quasi-geostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascades in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasi-geostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the sensitivity toan infrared Reynolds number in the spectral scaling of the energy spectrumin the inverse energy cascade range, and the validation of Charney’s predictionson the dynamics of quasi-geostrophic turbulence. It has also been shownthat the scaling of the energy spectrum in the enstrophy cascade is insensitiveto intermittency in higher order statistics, but that corrections apply to the”universal” Batchelor-Kraichnan constant, as a consequence of large-scale dissipationanomalies following a classical remark by Landau (Landau &amp; Lifshitz1987). Another finding is that the inverse energy cascade is maintained bynonlocal triad interactions, which is in contradiction with the classical localityassumption. / QC 20101029