Fluid dynamics on logarithmic lattices

Abstract

Open problems in fluid dynamics, such as the existence of finite-time singularities (blowup), explanation of intermittency in developed turbulence, etc., are related to multi-scale structure and symmetries of underlying equations of motion. Significantly simplified equations of motion, called toy-models, are traditionally employed in the analysis of such complex systems. In such models, equations are modified preserving just a part of the structure believed to be important. Here we propose a different approach for constructing simplified models, in which instead of simplifying equations one introduces a simplified configuration space: velocity fields are defined on multi-dimensional logarithmic lattices with proper algebraic operations and calculus. Then, the equations of motion retain their exact original form and, therefore, naturally maintain most scaling properties, symmetries and invariants of the original systems. Classification of such models reveals a fascinating relation with renowned mathematical constants such as the golden mean and the plastic number. Using both rigorous and numerical analysis, we describe various properties of solutions in these models, from the basic concepts of existence and uniqueness to the blowup development and turbulent dynamics. In particular, we observe strong robustness of the chaotic blowup scenario in the three-dimensional incompressible Euler equations, as well as the Fourier mode statistics of developed turbulence resembling the full three-dimensional Navier-Stokes system.

LogLatt: A computational library for the calculus and flows on logarithmic lattices

“…From(L.1) to (L.3), both the density of nodes and the number of triads per each node increase, thus providing finer resolution. In effect, we have shown that these are all possible lattices important for applications -consult [4] for the precise statement and its proof. We say that the lattice is nondegenerate if every two nodes interact though a finite sequence of triads.…”

“…The library LogLatt has been already applied to several important problems in Fluid Dynamics: blowup and shock solutions in the one-dimensional Burgers equation [1], the chaotic blowup scenario in the three-dimensional incompressible Euler equations [3,4], and turbulence in the three-dimensional incompressible Navier-Stokes equations [4]. A possible extension to isentropic compressible flow [4] was also considered. Here we show how to implement the library applicabilities on two classical equations, in order to validate the library and attest its efficiency.…”

“…As an alternative method, we proposed to simulate the governing equations on multi-dimensional lattices of logarithmically distributed nodes in Fourier space [3]. A special calculus was designed for this domain [4], so that the equations of motion preserve their exact form and retain most of the properties of the original systems, like the symmetry groups and conserved quantities. Their strong reduction in degrees of freedom allows the simplified models to be easily simulated on a computer within a surprisingly large spatial range.…”

“…Their strong reduction in degrees of freedom allows the simplified models to be easily simulated on a computer within a surprisingly large spatial range. This general technique was successfully applied to important problems in Fluid Dynamics, such as the blowup and shock solutions in the Burgers equation [1], the chaotic blowup in ideal flow [3], and the Navier-Stokes turbulence [4], and it is ready-to-use on any differential equation with quadratic nonlinearity. However, some special operations on logarithmic lattices may result in nontrivial computational implementations.…”

“…This paper is divided as follows. In Section 2 we briefly review the calculus on logarithmic lattices developed in [4]. We describe the library, its implementation and computational cost in Section 3 and apply it to some problems of fluid flow in Section 4.…”