## Final Report Summary - NNJETS (Stability of Non-Newtonian jets and implications for the onset of turbulence)

The secondary instability of streaks and transition to turbulence in viscoelastic Cou- ette flow are studied using direct numerical simulations (DNS). Viscoelasticity is mod- elled using the FENE-P constitutive equations, and both the polymer concentration and Weissenberg number are varied in order to assess their effect on transition at moderate Reynolds number, Re = 400.

The base streaks are obtained from nonlinear simulations of the Couette flow response to a streamwise vortex, and can be classified as quasi-Newtonian streaks according to the terminology introduced by Page & Zaki (2014). We choose the initial amplitude of the vortex in order to have a constant maximum amplitude of the streaks during their temporal evolution. Nonetheless the mean energy of the streaks decreases at higher Weissenberg. The growth of streaks in both Newtonian and non-Newtonian flows is due mainly to the longitudinal vorticity. However torque exerted by the polymers along the longitudinal direction is opposing the vorticity (Page & Zaki 2015) lowering the streak growth at large Weissenberg.

At every streak amplitude of interest, harmonic forcing is introduced to trigger the secondary instability and breakdown to turbulence. We notice that (i) at low Weissenberg number the critical amplitude of this forcing decreases while (ii) at large Weissenberg number the critical amplitude of this forcing increases instead. However the degree of stabilisation due to a strong elasticity is depending on the initial value of the streak am- plitude. For low streak amplitudes the critical amplitude increases much more significantly than for large streak amplitudes. This is explained by the two different mechanisms for triggering transition to turbulence which are active at low and large amplitudes (Cossu et al., 2011). In particular for low amplitudes the transition is triggered by a two steps mechanism as explained by Waleffe (1997). The streaks are initially distorted by the sinuous perturbation but after a short period, they reach a maximum in the energy and return to a nearly stable state. However, they ultimately reach a higher amplitude and break down to turbulence. The normal vorticity plays a fundamental role in this second growth and so in the breakdown. At large Weissenberg the polymer torque along the normal direction is opposing the vorticity with hindering the transition.

Finally the increase of the polymer concentration show a clear decrease in the critical amplitude of the forcing for every analysed streak amplitude.

The base streaks are obtained from nonlinear simulations of the Couette flow response to a streamwise vortex, and can be classified as quasi-Newtonian streaks according to the terminology introduced by Page & Zaki (2014). We choose the initial amplitude of the vortex in order to have a constant maximum amplitude of the streaks during their temporal evolution. Nonetheless the mean energy of the streaks decreases at higher Weissenberg. The growth of streaks in both Newtonian and non-Newtonian flows is due mainly to the longitudinal vorticity. However torque exerted by the polymers along the longitudinal direction is opposing the vorticity (Page & Zaki 2015) lowering the streak growth at large Weissenberg.

At every streak amplitude of interest, harmonic forcing is introduced to trigger the secondary instability and breakdown to turbulence. We notice that (i) at low Weissenberg number the critical amplitude of this forcing decreases while (ii) at large Weissenberg number the critical amplitude of this forcing increases instead. However the degree of stabilisation due to a strong elasticity is depending on the initial value of the streak am- plitude. For low streak amplitudes the critical amplitude increases much more significantly than for large streak amplitudes. This is explained by the two different mechanisms for triggering transition to turbulence which are active at low and large amplitudes (Cossu et al., 2011). In particular for low amplitudes the transition is triggered by a two steps mechanism as explained by Waleffe (1997). The streaks are initially distorted by the sinuous perturbation but after a short period, they reach a maximum in the energy and return to a nearly stable state. However, they ultimately reach a higher amplitude and break down to turbulence. The normal vorticity plays a fundamental role in this second growth and so in the breakdown. At large Weissenberg the polymer torque along the normal direction is opposing the vorticity with hindering the transition.

Finally the increase of the polymer concentration show a clear decrease in the critical amplitude of the forcing for every analysed streak amplitude.