The topology of skin friction and surface vorticity fields in wall-bounded flows

Abstract

In previous studies, the three invariants (P, Q and R) of the velocity gradient tensor have been widely used to investigate turbulent flow structures. For incompressible flows, the first invariant P is zero and the topology of turbulent flow structures can be investigated in terms of the second and third invariants, Q and R, respectively. However, all these three invariants are zero at a no-slip wall and can no longer be used to identify and study structures at the surface in any wall-bounded flow. An alternative scheme is presented here for the classification of critical points at a no-slip wall; the skin friction vector field at the wall is given by the wall normal gradients of the streamwise and spanwise velocity components; at a critical point, these gradients are simultaneously zero. The flow close to critical points in the surface skin friction field can be described by a no-slip Taylor series expansion and the topology of the critical point in the skin friction field is defined by the three invariants (, and ) of the ‘no-slip tensor’. Like the invariants of the velocity gradient tensor, the no-slip tensor invariants can be easily computed and these invariants provide a methodology for studying the structure of turbulence at the surface of a no-slip wall in any wall-bounded flow.

Keywords:

• turbulence
• wall-bounded flows
• surface skin friction fields
• surface vorticity fields
• local solutions of Navier–Stokes equations

Acknowledgements

The authors would like to acknowledge the use of data from computations of wall bounded flows, especially the data from torroja.dmt.upm.es (described in [9]). This work is funded by ARC Grant DP1093585.

Notes

¹The Poincaré–Hopf index theorem states that the topology of a smooth vector field on a surface is given by ∑(Nodes)−∑(Saddles)=2−2g, where g is the genus of the surface.