Singularities and Turbulence in Hydrodynamical Models
Three hydrodynamical models are presented in direct comparison, the Navier-Stokes equation, the Burgers' equation and a new and intermediate model, the Euler-Burgers' equation. The ultimate ambition is to shed light on the Navier- Stokes/Euler equation regarding the problem of turbulence and the question of possible finite time singularities. By proving existence and regularity of its solutions, the Euler-Burgers' equation is shown to be a prototype of a hydrodynamical model that circumvents the problems posed by the Navier-Stokes equation. With the help of numerical simulations it demonstrated, that this equation nevertheless exhibits turbulent statistics and intermittency that is successfully described by a generalized She-Leveque model.
Summary
In this thesis, three hydrodynamical models are presented in direct comparison, the Navier-Stokes equation, the Burgers' equation and a new and intermediate model, the Euler-Burgers' equation. The ultimate ambition is to shed light on the Navier- Stokes/Euler equation regarding the problem of turbulence and the question of possible finite time singularities. Since these problems are unsolved despite enduring and intensive research, the introduction of the Euler-Burgers' equation can be considered as a new and different approach to the topic. While being entirely theoretical in nature and without any application in reality, the Euler-Burgers' equation is shown to be a prototype of a hydrodynamical model that circumvents the problems posed by the Navier-Stokes equation, without turning out to be overly simple in its structure and dynamical behavior.
All three models are compared analytically and numerically under the aspects of general characteristics, turbulence and possible singularities. The close connection of the structure of Navier-Stokes and Euler-Burgers' equation is pointed out for several aspects. Both are similar in terms of the procedure of energy dissipation, since the identical viscous term of both hydrodynamical models is solely responsible for the annihilation of kinetic energy. The nonlinearity is shown to leave the total energy untouched and rather change the scale the energy is distributed on, finally leading to the process that was introduced as the Richardson cascade. Furthermore, both equations feature a nonlinearity local in position space, accompanied by a non-local projection operation. While the projection for the Navier-Stokes equation is just a circumscription for the physical pressure and concurrently causes mathematical and numerical inconvenience due to its instantaneous nature, the same operation for Euler-Burgers' equation is mathematically treatable. It is shown that global solutions of the Euler-Burgers equation exist for all times when sufficiently smooth external forcing and initial conditions. It is, furthermore, deduced that the solutions are smooth for all times and are unique. The Euler-Burgers' equation is, therefore, proved to be a full-fledged predictive hydrodynamical model that behaves well for all times.
For all further analysis and comparison of the model equations, numerical simulations were used. Two different implementations are added to the already existent framework racoon II. The first allows for accurate modeling of the maximum vorticity, maximizing the resolution by means of adaptive mesh refinement. While automatically ensuring solenoidality of the velocity field, the vorticity formulation, furthermore, provides reliable data for the most important finite time blow-up criterion, at the expense of execution speed. The second implementation focuses on the direct comparison of the three models. No adaptive refinement is employed but all model equations are implemented at once, allowing for an authentic comparison with improved execution speed. Both numerical formulations demand the solution of a Poisson-type equation that was obtained by implementing a Multigrid algorithm. It is capable of solving the Poisson equation parallel on an adaptive grid at arbitrary precision with adjustable error smoothing and interpolation methods. The simulations were mostly run on the local Linux Cluster, the obtained data was utilized to study general properties of the flows, to generate statistics and to analyze specific scenarios for possible finite time blow-ups.
No strict theory of turbulence is known today for the Navier-Stokes equation. During the last century, starting with the famous results of Kolmogorov, more and more refined phenomenological descriptions of turbulence based thereupon were developed. The most promising of these theories, regarding the successful description of numerical and experimental data, is the model of She and Leveque. This model was modified to the generalized She-Leveque model to be applied to all three considered hydrodynamical models. For Burgers' equation, the energy spectrum and structure function were measured and the numerical data fits well on the theory. For the Euler-Burgers' equation, two-dimensional folded sheets were identified as the most dissipative structures. Inserting these into the generalized She-Leveque model, predictions can be made for energy spectra and structure functions of turbulent Euler-Burgers' flows. The numerical data agrees very well with these predictions, showing that Euler-Burgers' turbulence is closely connected to Navier-Stokes turbulence.
Regarding the question of finite time singularities of the Euler equation, the mathematical criteria presented hint to monitor the peak vorticity of the flow. Exactly this was carried out in an adaptive simulation to probe the Kida-Pelz initial conditions for a finite time singularity at a very high resolution, compared to simulations run so far. Though no definite conclusion could be drawn whether a finite time singularity is possible, the numerical data allows statements regarding the necessary grid resolution. Larger simulations will be able to finally settle the issue definitely.