First coherent structure in elasto-inertial turbulence

Two-dimensional channel flow simulations of FENE-P (finitely extensible nonlinear elastic-Peterlin) fluid in the elasto-inertial turbulence (EIT) regime reveal distinct regimes ranging from chaos to a steady traveling wave which takes the form of an arrowhead structure. This coherent structure provides insights into the polymer/flow interactions driving EIT. A set of controlled numerical experiments and the study of transfer between elastic and turbulent kinetic energies highlight the role of small- and large-scale dynamics in the self-sustaining cycle of chaos in EIT flows.

Figure 1

Characteristic snapshots of the four distinct regimes: CR, chaotic regime (a, b), CAR chaotic arrowhead regime (c, d), IAR, intermittent arrowhead regime (e, f), and SAR, steady arrowhead regime (g, h), identified in the $L_x=2\pi$-computational domain. The left column shows contours of first normal stress difference on a log-scale color map, the right column provides the corresponding pressure contours on a linear scale. In panels (a) and (b), the green dots identify the location of low- and high-pressure regions in the lower half of the channel.

Figure 2

Temporal signals of drag increase $DI$ for examples of the four regimes highlighted in Fig. 1. Colors correspond to simulations introduced in Fig. 1.

Figure 3

(a) Drag increase for different $L$ and Wi, $\beta =0.9$, $\text{Re}=1000$ and a domain length of $L_x=2\pi$. $•$, $L=50$; $\blacksquare$, $L=100$; ▲, $L=200$; ▼, $L=500$. Symbols are color coded by states as defined in Fig. 1, and gray defines the laminar regime. (b) Flow regimes in the Wi-$L$ phase space. ${\text{Re}}_b=1000$ and $\beta =0.9$.

Figure 4

(a) Profiles of mean streamwise velocity as a function of the distance from the wall $\xi =h-z$. (b) Profiles of mean trace of the configuration tensor normalized by the polymer maximum extensibility as a function of $\xi$. Colors correspond to simulations introduced in Fig. 1.

Figure 5

Profiles of the mean Kolmogorov scale [Eq. (14)] as a function of the distance from the wall $\xi =h-z$. Colors correspond to simulations introduced in Fig. 1.

Figure 6

Profiles of the mean energy transfer term ${\mathrm{\Pi }}_e$ (a) and (b) the mean fluctuating energy transfer term ${\overline{{\mathrm{\Pi }}^{\prime }}}_e$ (solid lines) and dissipation rate of TKE $\overline{\varepsilon }$ (dashed lines) as a function of the distance from the wall $\xi =h-z$. Colors correspond to simulations introduced in Fig. 1.

Figure 7

Streamwise cospectra of the fluctuations of the energy transfer term defined in Eq. (15) as a function of the distance from the wall $\xi$. The streamwise wave number is normalized by the minimum mean Kolmogorov length scale from Fig. 3. (a) CR, (b) CAR, (c) IAR, (d) SAR. Each graph corresponds to simulations introduced in Fig. 1.

Figure 8

Snapshots of $N_1$ fields in a domain twice the length of the domain used in simulations used for Fig. 1. For all simulations, the Reynolds number is 1000.

Figure 9

Snapshots of $N_1$ fields of steady arrowhead regime at two different Reynolds numbers.

Figure 10

Snapshots of $N_1$ fields illustrating the effect of the parameter $\beta$. Panels (a) and (b) share the same $L=100$ and $\text{Wi}=100$ as SAR simulations identified for $\beta =0.9$ [see Fig. 3]. $\beta =0.5$ sustains the SAR (a), whereas $\beta =0.97$ destabilizes the flow to the CR. (c) The SAR for $\beta =0.97$ at $L=500$ and $\text{Wi}=100$.

Figure 11

(a) Superimposition of streamlines computed from the fluctuating velocity field and contours of $N_1$ for $\text{Re}=1000$, $L=200$, $\text{Wi}=100$, and $\beta =0.9$ [Fig. 1]. (b) Superimposition of the same streamlines and contours of the transfer of energy fluctuations ${\mathrm{\Pi }}_e^{\prime }$.

Figure 12

Power spectral density plots of times series sampled at a fixed location: (a) wall pressure, (b) centerline streamwise velocity. Colors correspond to simulations introduced in Fig. 1.

Figure 13

Samples of the spectral analysis used in the resolution study for ${\mathrm{Re}}_b=1000,L=50,\text{Wi}=100,\beta =0.9$. (a) Effect of grid resolution on the power density spectrum of pressure fluctuations at the wall. (b) Effect of the mesh size at the wall on the power density spectrum of pressure fluctuations at the wall. (c) Influence of the Schmidt number on the power density spectrum of turbulent kinetic energy at the centerline.

Figure 14

(a) Imaginary (negative curves) and real (positive curve) components of the modified wave numbers for the upwind compact scheme [19] (solid lines) and the staggered second-order scheme (dashed) for the spatial first derivative used for the advection term in the transport equations for the conformation tensor and the momentum, respectively. The imaginary and real components quantify the numerical dispersion and dissipation, respectively, of each scheme as a function of the wave number. (b) Power spectra of first normal stress difference at the wall of two SAR simulations ($L=100,200$, $\text{Wi}=100$, $\beta =0.9$). (c) Power spectra of turbulent kinetic energy for the same simulations.