Simulations of natural transition in viscoelastic channel flow

Abstract

Orderly, or natural, transition to turbulence in dilute polymeric channel flow is studied using direct numerical simulations of a FENE-P fluid. Three Weissenberg numbers are simulated and contrasted to a reference Newtonian configuration. The computations start from infinitesimally small Tollmien–Schlichting (TS) waves and track the development of the instability from the early linear stages through nonlinear amplification, secondary instability and full breakdown to turbulence. At the lowest elasticity, the primary TS wave is more unstable than the Newtonian counterpart, and its secondary instability involves the generation of $\unicode[STIX]{x1D6EC}$-structures which are narrower in the span. These subsequently lead to the formation of hairpin packets and ultimately breakdown to turbulence. Despite the destabilizing influence of weak elasticity, and the resulting early transition to turbulence, the final state is a drag-reduced turbulent flow. At the intermediate elasticity, the growth rate of the primary TS wave matches the Newtonian value. However, unlike the Newtonian instability mode which reaches a saturated equilibrium condition, the instability in the polymeric flow reaches a periodic state where its energy undergoes cyclical amplification and decay. The spanwise size of the secondary instability in this case is commensurate with the Newtonian $\unicode[STIX]{x1D6EC}$-structures, and the extent of drag reduction in the final turbulent state is enhanced relative to the lower elasticity condition. At the highest elasticity, the exponential growth rate of the TS wave is weaker than the Newtonian flow and, as a result, the early linear stage is prolonged. In addition, the magnitude of the saturated TS wave is appreciably lower than the other conditions. The secondary instability is also much wider in the span, with weaker ejection and without hairpin packets. Instead, streamwise-elongated streaks are formed and break down to turbulence via secondary instability. The final state is a high-drag-reduction flow, which approaches the Virk asymptote.

Perturbative expansions of the conformation tensor in viscoelastic flows

“…cases with modal growth rates below, at, and above the Newtonian value. Lee & Zaki (2017) examined the evolution of the modes using the (spatially averaged) perturbation kinetic energy and found, for all cases, that it initially grew at a rate consistent with predictions from linear theory. In all but the cases with Wi = 4.50, the kinetic energy eventually saturated; our focus is on conditions where an equilibrium solution is possible, and hence Wi = 4.50 is not discussed further.…”

“…The saturation energy for the Newtonian case and W 1.83 coincided, while W 6.67 saturated at a much lower energy. Superficially, the saturation appeared to be similar, but Lee & Zaki (2017) found that the behaviour of the flow before and after saturation was distinctly different in W 1.83 compared to W 6.67.…”

“…That study motivates the present focus on TS waves and, for this reason, its main relevant findings are outlined. Lee & Zaki (2017) performed linear stability analyses of channel flow of FENE-P fluid, with β = 0.90 and L max = 100. They reported the growth rates of the instability waves in (Re, Wi )-space, and identified the neutral curve which exhibited non-monotonic behaviour: The critical Re initially decreased and subsequently increased with increasing Wi .…”

The mean conformation tensor in viscoelastic turbulence

“…The algorithmic details are the same, as are all the flow parameters in table 1 except for Wi and the resulting Re τ . The initial conditions for the two cases Wi ∈ {1.83, 4.50} were obtained from DNS of natural transition to turbulence via Tollmien-Schlichting waves reported by Lee & Zaki (2017), and were extracted from the fully turbulent regime. In the case with Wi = 8.00, the initial condition was obtained by parameter continuation from Wi = 6.67.…”

“…In addition, the resulting mean conformation tensor appears directly in the mean momentum equation for some common constitutive models. As such, the arithmetic mean conformation tensor has often been adopted in the analysis and modelling of viscoelastic turbulence (Housiadas & Beris 2003;Masoudian et al 2013;Lee & Zaki 2017).…”