Self-similarity in incompressible Navier-Stokes equations

Ali Ercan; M Levent Kavvas

doi:10.1063/1.4938762

Self-similarity in incompressible Navier-Stokes equations

Chaos. 2015 Dec;25(12):123126. doi: 10.1063/1.4938762.

Authors

Ali Ercan¹, M Levent Kavvas²

Affiliations

¹ J. Amorocho Hydraulics Laboratory, Department of Civil and Environmental Engineering, University of California, Davis, California 95616, USA.
² Hydrologic Research Laboratory and J. Amorocho Hydraulics Laboratory, Department of Civil and Environmental Engineering, University of California, Davis, California 95616, USA.

PMID: 26723165
DOI: 10.1063/1.4938762

Abstract

The self-similarity conditions of the 3-dimensional (3D) incompressible Navier-Stokes equations are obtained by utilizing one-parameter Lie group of point scaling transformations. It is found that the scaling exponents of length dimensions in i = 1, 2, 3 coordinates in 3-dimensions are not arbitrary but equal for the self-similarity of 3D incompressible Navier-Stokes equations. It is also shown that the self-similarity in this particular flow process can be achieved in different time and space scales when the viscosity of the fluid is also scaled in addition to other flow variables. In other words, the self-similarity of Navier-Stokes equations is achievable under different fluid environments in the same or different gravity conditions. Self-similarity criteria due to initial and boundary conditions are also presented. Utilizing the proposed self-similarity conditions of the 3D hydrodynamic flow process, the value of a flow variable at a specified time and space can be scaled to a corresponding value in a self-similar domain at the corresponding time and space.