Unsteady Separation and Navier-Stokes Solutions at High Reynolds Numbers

Abstract

We compute the numerical solutions for Navier-Stokes and Prandtl’s equations in the case of a uniform bidimensional flow past an impulsively started disk. The numerical approx- imation is based on a spectral methods imple- mented in a Grid environment. We investigate the relationship between the phenomena of unsteady separation of the flow and the exponential decay of the Fourier spectrum of the solutions. We show that Prandtl’s solution develops a separation singularity in a finite time. Navier-Stokes solutions are computed over a range of Reynolds numbers from 3000 to 50000. We show that the appearance of large gradients of the pressure in the stream- wise direction, reveals that the viscous-inviscid interaction between the boundary layer flow and the outer flow starts before the singularity time for Prandtl’s equation. We observe a relationship between the formation of large gradients of pres- sure and the various stage of unsteady separation with the loss of exponential decay of the spectrum of the solutions.