Role of coherent structures in multiple self-similar states of turbulent planar wakes

Abstract

We study the nature of archetypal, incompressible, planar splitter-plate wakes, specifically the effects of the exit boundary layer state on multiple approximate self-similarity. Temporally developing direct numerical simulations, at a Reynolds number of 1500 based on the volume-flux defect, are performed to investigate three distinct wake evolution scenarios: Kelvin-Helmholtz transition, bypass transition in an asymmetric wake, and an initially fully turbulent wake. The differences in the evolution and far-wake statistics are analysed in detail. The individual approximately self-similar states exhibit a relative variation of up to 48 % in the spread rate, in second-order statistics, and in peak values of the energy budget terms. The multiplicity of self-similar states is tied to the non-universality of the large-scale coherent structures. These structures maintain the memory of the initial conditions. In the far wake, two distinct spanwise-coherent motions are identified: (i) staggered, segregated spanwise rollers on either side of the centreplane, dominant in wakes transitioning via anti-symmetric instability modes; and, (ii) larger spanwise rollers spanning across the centreplane, emerging in the absence of a near-wake characteristic length scale. The latter structure is characterized by strong spanwise coherence, cross-wake velocity correlations and a larger entrainment rate caused by deep pockets of irrotational fluid within the folds of the turbulent/non-turbulent interface. The mid-sized structures, primarily vortical rods, are generic for all initial conditions and are inclined at ∼±33 • to the downstream, shallower than the preferential ±45 • inclination of the vorticity vector. The spread rate is driven by the inner-wake dynamics, more specifically the advective flux of spanwise vorticity across the centreplane, which depends on the large-scale coherent structures.

Compressibility effects on the structural evolution of transitional high-speed planar wakes

“…The cross-wake location of the normalized TKE peak increases with increasing Ma (see figure 11); a similarly increasing monotonic trend in the peak location is observed in lower-Re simulations. As rollers develop, the typical double peaks of the TKE profile is lost and the maximum turbulence intensity is located about the centreplane, as seen in figure 12; the double peak is soon regained after transition, see Hickey et al (2013) for similar profiles for the incompressible wake. The centreline peak of the TKE profile during transition is the result of the induced cross-wake velocity (caused by the coherent spanwise rollers), see the time evolution of ρvv in figure 9.…”

“…All wakes show a well-defined peak in vv and uv during the nonlinear stage of the transition; the peak is attributable to the roll-up process during transition. As for the incompressible cases, the peaks in the streamwise and spanwise normal stresses are rather blunt, and slowly decay to a far-field plateau which is approximately half of the peak value (Hickey et al 2013). One of the principal quantitative differences in the turbulence statistics is found in the peak values reached during transition.…”

“…The numerical code was extensively verified against the analytical solution of a viscous shock, Taylor-Green vortex and decaying compressible isotropic turbulence. Validation with experimental wake data in the incompressible limit was performed in Hickey et al (2013).…”

“…Admittedly, the choice of a Gaussian, instead of a double Blasius profile, may inhibit the development of near-wake instability modes originating from a double boundary layer profile (Papageorgiou & Smith 1989;Papageorgiou 1990). In Hickey et al (2013), the evolution of a Gaussian and double Blasius wake profile were compared in the incompressible limit; other than a delayed transition, the main characteristic structures and statistics remain surprisingly similar. Based on these conclusions, we deemed that an initial Gaussian profile allows the greatest generality for the study of the temporally evolving wake.…”

“…Even though the wake tends toward a fully incompressible state as the velocity defect becomes null, it does not necessarily mean that the self-similar states of the incompressible and high-speed wakes are identical. Since free-shear flows maintain a 'memory' of their origin, it can be inferred that a differing far-wake structural organization and statistics found by Bonnet et al (1991) and Hickey, Hussain & Wu (2013) may be related to the near-wake and/or transitional evolution.…”