An extended Reynolds analogy for excited wavy instabilities of developing streamwise vortices with applications to scalar mixing intensification

Abstract

Studies are presented to elucidate the role of steady streamwise vortex structures, initiated upstream from weak Görtler vortices in the absence of explicit vortex generators, and their excited nonlinear wavy instabilities in the intensification of scalar mixing in a spatially developing mixing region. While steady streamwise vortex flow gives rise to significant mixing enhancement, the excited nonlinear wavy instabilities, which in turn modify the basic three-dimensional streamwise vortices, give rise to further mixing intensification which is quantitatively assessed by a mixedness parameter. Possibility of similarity between the dimensionless streamwise momentum and scalar transport problems leading to an extended Reynolds analogy is sought. This similarity is shown earlier to hold for the steady streamwise vortex flow in the absence of nonlinear wavy instabilities (Liu & Sabry 1991 Proc. R. Soc. A 432, 1-12). In this paper, the momentum conservation equations for the nonlinear wavy or secondary instabilities together with the advected fluctuation scalar problems are examined in detail. The presence of the streamwise fluctuation pressure gradient, which prevents the similarity, is estimated in terms of the fluctuation dynamical pressure and its relative importance to advective transport. It is found from scaling that the fluctuating streamwise pressure gradient, though not completely negligible, is sufficiently unimportant so as to render similarity between fluctuation streamwise velocity and fluctuation temperature and concentration a distinct possibility. The scalar fluctuations are then inferable from the fluctuation streamwise velocity and that the Reynolds stresses of the nonlinear fluctuations and the scalar fluxes are also similar. The nonlinear instabilitymodified mean streamwise momentum and the modified mean heat and mass transport problems are also similar, thus providing a complete 'Reynolds analogy', rendering possible the interpretation of the scalar mixedness for a gaseous medium for which the Prandtl and Schmidt numbers are near unity. It is found that the nonlinearity of the wavy instability, which induces scalar fluxes modifying the mean scalar transport, further intensifies scalar mixedness over a significant streamwise region which is well above that achieved by the steady, unmodified streamwise vortices alone for the numerical example corresponding to the most amplified wavy-sinuous mode.

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

“…Through the effects of eddy heat and mass flux from secondary instability, significant enhancement of momentum and scalar mixing has been found possible in free mixing layers (Girgis & Liu 2002;Liu 2007). It is anticipated that much more complex fluid properties and geometry would have to be taken into consideration of intensifying mixing in practical combustors and in jet exhausts in a well-controlled manner for minimizing pollutants of combustion and of aerodynamic sound.…”

“…This reduces to arguing for similarity between (2.8) and (2.11), leading to u 0 Zq 0 . The author (Liu 2005(Liu , 2007 has shown that the pressure gradient term in (2.8), which is the impediment to similarity, is of the order of the magnitude of u 0 /1 relative to the inertia terms. In this case, (2.8) is similar to (2.11), and, for PrZ1 and similar initial and boundary conditions, the Reynolds analogy follows:…”

“…In the present application, Re/cmZ3425/cm, l z /d 0 Z 13.85. A detailed discussion of the parameters is given in Liu (2007). In order to explain the overshoot beyond the local turbulent value in the absence of turbulence, it is necessary to appeal to the effect of the 'eddy heat flux' owing to the transport effects of the secondary instability.…”

Some fundamental considerations of streamwise vortices found useful in mixing enhancement and jet noise suppression

“…9 The passive scalar iso-lines are similar to the velocity structures for Prandtl and Schmidt numbers near unity in the mixing region supporting steady streamwise vortices and their nonlinear secondary instabilities. 22

Figure 1.The flow is from left to right. Local cross sectional plane “streamlines” and iso-u contour “mushrooms”.

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“…10 Iso-u contour “mushrooms” are also that of iso-scalars contours (dimensionless temperature and concentration for a binary mixture) for Prandtl and Schmidt numbers unity. 11 …”