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RE-LAMINARIZATION AND RE-TRANSITION OF A TURBULENT BOUNDARY LAYER FROM A VORTICITY POINT OF VIEW

Garry L. Brown
Department of Mechanical and Aerospace Engineering Princeton University Princeton, NJ 08544

Saurabh S. Patwardhan
Department of Aerospace Engineering Indian Institute of Science Bangalore, Karnataka, India

O.N. Ramesh
Department of Aerospace Engineering Indian Institute of Science Bangalore, Karnataka, India

Sinopsis

The presentation at TSFP 9 by Brown, Lee and Moser (2015) on the near wall transfer of viscous stress to Reynolds stress, based on a vorticity transport perspective, was motivated by the connection between the transport of vorticity and the Reynolds stress, i.e.

d/dy(−u'v') = v'ω'z−w'ω'y,

(for plane, parallel turbulent flow) to which Taylor (1915) first drew attention. That work has now been expanded to include Couette flow and is being submitted for publication (BLM (2017)). We consider here the role played by these vorticity transport terms in an understanding of the mechanics of a highly accelerating, re-laminarizing, turbulent boundary layer. An initial, zero pressure gradient, turbulent boundary layer is accelerated by a very favorable pressure gradient, which provides an increase by a factor of three in free stream velocity. It is followed by a relaxation towards zero pressure gradient at this much higher free stream velocity. A recent experimental and numerical investigation (Patwardhan (2015), Patwardhan and Ramesh (2017)) shows a dramatic fall in local skin friction coefficient and, particularly, a local fall in the actual wall friction due to this acceleration. Since an acceleration and fall in static pressure is a source of spanwise vorticity at the wall this decrease in wall vorticity demands a mechanistic explanation. A 're-laminarization' of the near wall flow, as the flow accelerates, is found, which is then followed by a re-transition to turbulence in the subsequent, approximately zero pressure gradient, flow.
This relaminarizing flow attracted early experimental and theoretical attention by Sreenivasan (1972), and Narasimha and Sreenivasan (1973, 1979)) who provided a mechanistic description, largely based on momentum considerations. They developed a two layer model with the idea of a 'laminar sub boundary layer' near the wall and a 'rapid distortion' model for the outer flow, but it was not possible at that time to measure the components of vorticity near the wall. DNS computions now offer detailed and complementary insights. In particular these computations provide results for the vorticity field and for the vorticity transport terms in the above equation.
As pointed out in (BLM) (2015), in a channel flow the two vorticity transport terms are equal at the location ym where the Reynolds stress is a maximum. Nearer the wall, w'ω'y dominates and in the outer flow, (y >>ym), v'ω'z has the larger magnitude. Importantly, v'ω'z, acts to transport the mean spanwise vorticity in the same direction as laminar diffusion, whereas near the wall w'ω'y acts to transport the spanwise vorticity against the mean vorticity gradient! (Both effects cancel at y=ym and the vorticity is transported there only by the viscous diffusion, as discussed by BLM (2015, 2017). These two vorticity transport terms have now been calculated from the Direct Numerical Simulation database of the re-laminarizing turbulent boundary layer flow (initial Rτ = 461) of Patwardhan (2015).