DYNAMICS OF ELASTO-INERTIAL TURBULENCE IN FLOWS WITH POLYMER ADDITIVES

Abstract

The dynamics of elasto-inertial turbulence is investigated numerically from the perspective of the coupling between polymer dynamics and flow structures. In particular, direct numerical simulations of channel flow with Reynolds numbers ranging from 1000 to 6000 are used to study the formation and dynamics of elastic instabilities and their effects on the flow. Based on the splitting of the pressure into inertial and polymeric contributions, it is shown that the trains of cylindrical structures around sheets of high polymer extension that are characteristics to elasto-inertial turbulence are mostly driven by polymeric contributions. INTRODUCTION Polymer additives are known for producing upward of 80% drag reduction in turbulent wall-bounded flows through strong alteration and reduction of turbulent activity (White & Mungal, 2008). The changes in flow dynamics induced by polymers do not lead to flow relaminarization but, at most, to a universal asymptotic state called maximum drag reduction (MDR, Virk et al. 1970). At the same time, polymer additives have also been shown to promote transition to turbulence (Hoyt, 1977), or even lead to a chaotic flow at very low Reynolds number as in elastic turbulence (Groisman & Steinberg, 2000). These seemingly contradicting effects of polymer additives can be explained by the interaction between elastic instabilities and the flow’s inertia characterizing elastoinertial turbulence, hereafter referred to as EIT (Samanta et al., 2012; Dubief et al., 2013). EIT is a state of smallscale turbulence that exists by either creating its own extensional flow patterns or by exploiting extensional flow topologies. EIT provides answers to phenomena that current understanding of MDR cannot, such as the absence of log-law in finite-Reynolds numbers MDR flows (White et al., 2012), and the phenomenon of early turbulence. Moreover, it supports De Gennes (1990)’s picture that drag reduction derives from two-way energy transfers between turbulent kinetic energy of the flow and elastic energy of polymers at small scales, resulting in an overall modification of the turbulence energy cascade at high Reynolds numbers. As shown by the viscoelastic pipe experiment of Samanta et al. (2012), an elastic instability can occur at a Reynolds number smaller than the transition in Newtonian pipe flow if the polymer concentration and Weissenberg number are sufficiently large. Moreover, it was observed that the measured friction factor then follows the characteristic MDR friction law. These findings were also confirmed by direct numerical simulations as shown in Figure 1 (Dubief et al., 2013). The analysis of these simulations showed that thin sheets of locally high polymer stretch, tilted upwards and elongated in the flow direction, create trains of spanwise cylindrical structures of alternating sign, as shown in Figure 2. This feature of EIT disappears when the flow is too turbulent or the polymer solution not elastic enough, which led to the hypothesis that EIT is an asymptotic state that should occur when the elasticity of the solution can efficiently control and contain the growth of turbulence. Dubief et al. (2013) suggested that the formation of sheets of polymer stretch results from the unstable nature of the nonlinear advection of low-diffusivity polymers. These sheets, hosting a significant increase in extensional viscosity, create a strong local anisotropy, with a formation of local low-speed jet-like flow. The response of the flow is through pressure, whose role is to redistribute energy across components of momentum, resulting in the formation of waves, or trains of alternating rotational and straining motions. Once triggered, EIT is self-sustained since the elas-