Analysis of complex singularities in high-Reynolds-number Navier–Stokes solutions

Abstract

Numerical solutions of the laminar Prandtl boundary-layer and NavierStokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions that can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the complex singularity tracking method to Prandtl and NavierStokes solutions and analyze the previous interactions from a different perspective.

Enhanced effects from tiny flexible in-wall blips and shear flow

“…The response produces only a small disturbance from λ over the quasi-steady temporal scale (i) of §3 until a critical value λ = λ c is encountered at which stage (ii) weakly nonlinear amplification can occur ( §4), leading to strongly nonlinear evolution over the faster time scale (iii) of (2.4a-2.4d), (2.6a-2.6b). This is followed by finite-time blowup as in Smith (1988), Peridier, Smith & Walker (1991 which provokes the even faster evolution (iv) described by Davies, Bowles & Smith (2003) (see also Cassel & Conlisk (2014), Gargano, Sammartino, Sciacca & Cassel (2014)) with further restructuring and deep transition towards turbulence taking place. The time scales (i)-(iv) etc for response a are slow, fast, faster, faster, etc.…”

Unsteady separation in vortex-induced boundary layers

“…In particular, although 'real' singularities presumably do not occur in the Navier-Stokes solutions, singularities can be tracked in the complex plane. Extending the earlier work of Cowley [52], Rocca et al [53] and Gargano et al [54][55][56] have tracked complex singularities in non-interactive boundary-layer and NavierStokes solutions involving unsteady separation. In the most recent of these investigations, the evolving wall shear stress distribution is analysed for complex singularities.…”

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

“…It is well known how Prandtl solutions develop a singularity (see [17] and references therein), and one could therefore expect that the boundary layer solution that we have constructed to show similar blow-up phenomena, and to do this also if initialized with analytic data. In the classical high Reynolds number Navier-Stokes theory the boundary layer singularities (or, to be more precise, the appearance of complex singularities that are precursor of the blow-up [18]) signal the interaction stage-when the pressure profile at the boundary is modified by the vorticity generated in the boundary layer; this interaction ultimately leads to separation of the boundary layer and to the break-up of asymptotic structures like (1.1). It would be interesting to explore if the phenomenology typical of the high Reynolds number Navier-Stokes flows is also shown by the primitive equations solutions.…”

“…[7,[15][16][17][18]28,41]), we use the following version of the Abstract CauchyKowalevski Theorem (ACK) (cf. [1,35,47] and references therein).…”