Nonlinear instability of developing streamwise vortices with applications to boundary layer heat transfer intensification through an extended Reynolds analogy

Abstract

The intent of the present contribution is to explain theoretically the experimentally measured surface heat transfer rates on a slightly concave surface with a thin boundary layer in an otherwise laminar flow. As the flow develops downstream, the measured heat transfer rate deviates from the local laminar value and eventually exceeds the local turbulent value in a non-trivial manner even in the absence of turbulence. While the theory for steady strong nonlinear development of streamwise vortices can bridge the heat transfer from laminar to the local turbulent value, further intensification is attributable to the transport effects of instability of the basic steady streamwise vortex system. The problem of heat transport by steady and fluctuating nonlinear secondary instability is formulated. An extended Reynolds analogy for Prandtl number unity, PrZ1, is developed, showing the similarity between streamwise velocity and the temperature. The role played by the fluctuation-induced heat flux is similar to momentum flux by the Reynolds shear stress. Inferences from the momentum problem indicate that the intensified heat flux developing well beyond the local turbulent value is attributed to the transport effects of the nonlinear secondary instability, which leads to the formation of 'coherent structures' of the flow. The basic underlying pinions of the non-linear hydrodynamic stability problem are the analyses of J. T. Stuart, which uncovered physical mechanisms of nonlinearities that are crucial to the present developing boundary layers supporting streamwise vortices and their efficient scalar transporting mechanisms.

Modification of mean wake flow behind a very slender axisymmetric body of revolution by imposed nonlinear unstable helical modes

“…This is entirely in the spirit of Stuart [19] where the instability characteristics are given by the linear theory, whereas the amplitude functions are solved nonlinearly. This has also found to be the case in other nonlinear stability problems [20] where the amplification is far from weak. The details of the linear theory, including applications [2,3], appear in the electronic supplementary material, appendix B.…”

“…Stuart [9] and Meksyn & Stuart [21] advanced the idea that a dominant mode suffices to represent the nonlinear disturbances which enter into the Reynolds stress for mean flow modification. This idea is found, in many instances, to hold for developing flows and in situations far from any weak nonlinear assumptions [20,22]. For the present, the most amplified helical n = ±1 modes according to the linear theory are assumed to represent the developing nonlinear modes (including consequential generation of other modes through nonlinear interactions), causing the modification of mean flow characteristics.…”

Three-dimensional secondary instability of a spatially developing von Kármán vortex street in a far wake

“…Cimbala et al 1988) for understanding the mechanisms discussed in this paper. The general discussions of the nonlinear interaction between two-dimensional primary flow and the three-dimensional disturbance in § §2 and 6 could be the starting point of leapfrog nonlinear parabolic computations, as has been performed in other secondary instability problems in shear flows (Girgis & Liu 2002, 2006Liu 2008). …”

Heat Transfer Analysis in a Flow Over Concave Wall With Primary and Secondary Instabilities

“…Studies to explain theoretically the rate of heat transfer in a boundary layer flow over a slightly concave surface were conducted by Liu 12 . In his study it was observed that one can greatly enhance the heat transfer, paying the price of almost one to one in drag.…”