High-Reynolds-number asymptotics of steady separated flows

Overview

There are problems that are easy to pose but difficult to solve. Among them is the problem of high-Reynolds-number asymptotics of steady incompressible flow past a bluff body.

 [Steady Navier-Stokes equations]

Steady Navier-Stokes equations involve only one parameter, Re, and there are only two asymptotic limits: Re tending to zero and Re tending to infinity. The limit of Re tending to zero was found by George Gabriel Stokes in 1851. For streamlined bodies (like aerofoils) the limit of Re tending to infinity was found by Ludwig Prandtl in 1904. For bluff bodies (like a cylinder) the limit of Re tending to infinity was found only in 1988.

 [Streamlines of a separated flow past a cylinder, with the eddy much bigger than the cylinder]

Fig.1. Steady flow past a circular cylinder at Reradius=300 calculated numerically by B.Fornberg in 1985.

In contrast to the Stokes theory and the boundary layer theory, bluff body high-Re steady solutions never correspond to real flows quantitatively, since real separated flows at high Re are always turbulent. This is illustrated by the long and wide wake in Figure 1. In averaged turbulent flow the eddy is narrower and shorter. Nevertheless, the steady asymptotics of separated flows always arose great interest not only because of its beauty and difficulty but also because its solution throws light on general properties of the Navier-Stokes equations. Moreover, in spite of the differences, real and steady high-Re flows have enough in common for the steady high-Re asymptotics to be used (with caution) for qualitative analysis of real turbulent flows.

 [Sketch of the distinguished limits of the high-Re steady separated flow]

Fig.2. Asymptotic structure of steady separated flow.

The theory proves that in steady separated flows the eddy length and width are both proportional to Re, while the drag coefficient tends to zero as 1/Re. The asymptotic flow structure consists of many distinguished limits. Following the arrows in Figure 2 one can go step-by-step from the Sadovsky flow on the Re x Re eddy scale to the Kirchhoff flow on the 1 x 1 body scale.

More information:

  • Chernyshenko, S.I. 1988. The asymptotic form of the stationary separated circumfluence of a body at high Reynolds numbers. J. Appl. Math. Mech. Vol:52, Issue 6, 746–753.
  • Chernyshenko, S.I. 1998. Asymptotic theory of global separation, Applied Mechanics Reviews, Vol:51, 523-536.
  • Chernyshenko, S.I. and Castro, I.P. 1993. High-Reynolds-number asymptotics of the steady flow through a row of bluff bodies. J. Fluid Mech., Vol:257, 421-449.
  • Chernyshenko, S.I.1995. Asymptotics of steady axisymmetric flow of incompressible fluid past a bluff body at high Reynolds number. Fluid Dynamics. Vol:30, Issue 1, 28-34.
  • Chernyshenko, S.I. and Castro, I.P. 1996. High-Reynolds-number weakly stratified flow past an obstacle. J. Fluid Mech., Vol:317, 155-178.
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