Numerical Experiments on the Interaction Between the Large- and Small-Scale Motions of the Navier-Stokes Equations

• Published in: Multiscale modeling & simulation Vol. 1; no. 1; pp. 119 - 149
• Main Authors: Henshaw, William D., Kreiss, Heinz-Otto, Yström, Jacob
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2003
• Subjects: Energy; Laboratories; Navier-Stokes equations; Numerical analysis; Weather forecasting
• ISSN: 1540-3459; 1540-3467
• DOI: 10.1137/S1540345902406240

We consider solutions to the unforced incompressible Navier-Stokes equations in a $2\pi$-periodic box. We split the solution into two parts representing the large-scale and small-scale motions. We define the large scale as the sum of the first kc Fourier modes in each direction and the small scale as the sum of the remaining modes. We attempt to reconstruct the small scale by incorporating the large-scale solution as known forcing into the equations governing the evolution of the small scale. We want to find the smallest value of kc for which the time evolution of the large scale sets up the dissipative structures so that the small scale is determined to a significant degree. Existing theory based on energy estimates gives a pessimistic estimate for kc that is inversely proportional to the smallest length scale of the flow. At this value of kc the energy in the small scale is exponentially small. In contrast, numerical calculations indicate that kc can often be chosen remarkably small. We attempt to explain why the time evolution of a relatively few number of large-scale modes can be used to reconstruct the small-scale modes in many situations. We also show that similar behavior is found in solutions to Burgers' equation.