Upper edge of chaos and the energetics of transition in pipe flow

In the past two decades, our understanding of the transition to turbulence in shear flows with linearly stable laminar solutions has greatly improved. Regarding the susceptibility of the laminar flow, two concepts have been particularly useful: the edge states and the minimal seeds. In this nonlinear picture of the transition, the basin boundary of turbulence is set by the edge state's stable manifold and this manifold comes closest in energy to the laminar equilibrium at the minimal seed. We begin this paper by presenting numerical experiments in which three-dimensional perturbations are too energetic to trigger turbulence in pipe flow but they do lead to turbulence when their amplitude is reduced. We show that this seemingly counterintuitive observation is in fact consistent with the fully nonlinear description of the transition mediated by the edge state. In order to understand the physical mechanisms behind this process, we measure the turbulent kinetic energy production and dissipation rates as a function of the radial coordinate. Our main observation is that the transition to turbulence relies on the energy amplification away from the wall, as opposed to the turbulence itself, whose energy is predominantly produced near the wall. This observation is further supported by the similar analyses on the minimal seeds and the edge states. Furthermore, we show that the time evolution of production-over-dissipation curves provides a clear distinction between the different initial amplification stages of the transition to turbulence from the minimal seed.

Figure 1

(a) Time series of perturbation kinetic energy for five initial conditions from $\text{Re}=10000$. Annotated snapshots are the flow structures for one of the simulations at times $t=0.0,25,50$. Visualized here are the isosurfaces of streamwise velocity at $50\%$ of its maxima and minima (red and blue) as well as the streamwise vorticity isosurfaces at $25\%$ of its maxima and minima (green and purple) at the respective instances. (b) Time evolution of perturbation kinetic energy for initial conditions obtained from rescaling turbulent initial conditions at $\text{Re}=10000$ such that trajectories neither become turbulent nor laminarize for longer and longer times. Inset: Zoom-in to the region $t\in [0,16]$ and $k\in [0.255,0.3]$. (c) A “cartoon” of the state space where initial conditions that relaminarize are separated from those that develop into turbulence by the stable manifold of the “edge state.”

Figure 2

(a) $\mathcal{P}$, (b) $\mathcal{P}/\mathcal{D}$, (c) $U_z$, as a function of radial position for initial conditions $a\left(0\right)=c{a}_0$, where $a_0$ is a turbulent state at $\text{Re}=10000$ and $c\in [0.3,1.2]$. Different colors correspond to different $c$ values shown in the legend of (b). Blue and orange dashed curves correspond to the critical initial conditions, respectively, on the lower and the upper edge of chaos with corresponding coefficients $c=0.531856455058$ and $c=0.972761377146$. The arrows in the inset of (a) are there to emphasize that the production in the bulk region decreases for $c>0.9$. (d) $\mathcal{P}$ as a function of $c$ at $r=0.5R$ and $r=0.9R$, normalized by the respective maxima of the curves. Dots correspond to the data points and they are connected with line segments in order to guide the eye.

Figure 3

(a) $[\mathcal{P}/\mathcal{D}]\left(r\right)$ for traveling waves $S_1$ (red/solid) and $S_{1N}$ (blue/dashed) on the laminar-turbulent boundary. (b) Evolution of $[\mathcal{P}/\mathcal{D}]\left(r\right)$ as a trajectory proceeds towards turbulence after spending a long time on the “edge.” Colors indicate the direction of time (initial: blue; final: red); each curve is separated by $\mathrm{\Delta }t=5$ in time. (c) Time series of perturbation kinetic energy with the same color coding for reference.

Figure 4

(a) Time evolution of $[\mathcal{P}/\mathcal{D}]\left(r\right)$ for the minimal seed at $\text{Re}=3000$. Colors indicate the direction of time (initial: blue; final: red); each curve is separated by $\mathrm{\Delta }t=2$ in time. (b) Time series of perturbation kinetic energy with the same color coding for reference. The gray line corresponds to the energy time series for the largest tested value of $E_0$ below which the transition never occurred.

Figure 5

(a) Time evolution of $[\mathcal{P}/\mathcal{D}]\left(r\right)$ for the minimal seed at $\text{Re}=3000$. Colors indicate the direction of time (initial: blue; final: red); each curve is separated by $\mathrm{\Delta }t=0.1$ in time. (b) Time series of perturbation kinetic energy with the same color coding for reference. (c) Time evolution of the peak ${(\mathcal{P}/\mathcal{D})}_{\text{max}}$ (red) and of its radial position $r_{\text{max}}$ (blue) for the minimal seed.

Figure 6

(a) Time evolution of $[\mathcal{P}/\mathcal{D}]\left(r\right)$ for the minimal seed at $\text{Re}=3000$ with flattened base profile [refer to (3)]. Colors indicate the direction of time (initial: blue; final: red); each curve is separated by $\mathrm{\Delta }t=0.1$ in time. (b) Time series of perturbation kinetic energy with the same color coding for reference. (c) Time evolution of the peak ${(\mathcal{P}/\mathcal{D})}_{\text{max}}$ (red) and of its radial position $r_{\text{max}}$ (blue) for the minimal seed with flattened base profile.