Effect of Weissenberg number on polymer-laden turbulence

Direct numerical simulation of polymer-laden turbulent flow is performed for the investigation of the two-way interaction. Lagrangian dynamic simulation is adopted for a finitely extensible nonlinear elastic (FENE-2) dumbbell model to observe polymer dynamics in turbulent flow. The impact of the polymers on fluid momentum is described using the elastic force between two beads. The elasticity of the polymer is characterized using the Weissenberg number, ${\text{We}}^{*}=\frac{{\tau }_p}{{\tau }_k}$, where ${\tau }_p$ and ${\tau }_k$ denote the elasticity timescale and the Kolmogorov timescale, respectively. We observe that for ${\text{We}}^{*}\le 1.0$ most of the polymers are in a coiled state and the hydrodynamic properties of fluid remain unchanged. The coil-stretch transition is observed at approximately ${\text{We}}^{*}=3.0$. The highly stretched polymers in turn contribute to significant turbulence modification for large values of the Weissenberg number. The effect of the Weissenberg number on various turbulent statistics is computed and analyzed. The highly stretched polymers tend to rotate around the vortical structures. The feedback force effectively suppresses the turbulence structures for large values of the elasticity parameter. The effect of ${\text{We}}^{*}$ on the alignment between the end-to-end distance vector, vorticity vector, and the eigenvector of the rate of strain tensor is presented. A better insight into the effect of polymers on turbulence is obtained through the direct force modeling in stationary turbulence although the effect itself has been known for a long time.

Figure 1

Numerical test with different $N_r/N_c$: (a) time-averaged magnitude of the end-to-end distance vector; (b) PDF of magnitude of end-to-end distance vector; and (c) time-averaged mean dissipation, kinetic energy, and rms of vorticity.

Figure 2

Polymer statistics of the end-to-end distance magnitude (a) mean and (b) rms; here the solid line with symbol represents the one-way coupling case results.

Figure 3

Effect of ${\text{We}}^{*},\text{We}$ on variations in (a) $Q$ and (b) $\sigma$ for both one-way and two-way coupling cases.

Figure 4

Probability density function of the end-to-end distance vector magnitude for different values of the elasticity parameter.

Figure 5

Conditional PDF of cosine angle between end-to-end distance vector and eigenvectors of rate of strain tensor for (a) ${\text{We}}^{*}=5.0$ and (b) ${\text{We}}^{*}=20$. Owing to symmetry, only half of the PDF range is shown. Open symbols are for one-way coupling and closed symbols are for the two-way coupling case.

Figure 6

Conditional PDF of cosine angle between the vorticity vector and eigenvectors of rate of strain tensor at polymer position for (a) ${\text{We}}^{*}=5.0$ and (b) ${\text{We}}^{*}=20$. Owing to symmetry, only half of the PDF range is shown. Open symbols are for one-way coupling and closed symbols are for two-way coupling case.

Figure 7

Effect of the Weissenberg number on (a) $E_p$ and (b) $q$, $\varepsilon$, and ${\psi }_p$.

Figure 8

Effect of polymers on the probability density function of local fluid dissipation for different values of We.

Figure 9

PDF of the cosine angle between ${\omega }_i$ and (a) ${\omega }_j{s}_{ij}$ and (b) ${\nu }_f{\partial }^2{\omega }_i/\partial x_j\partial x_j$ for various We.

Figure 10

Comparison of the isosurfaces of $|{\omega }_i|$ in flow (a) without polymers and [(b), (c), (d)] with polymers at $\text{We}=2.97,4.64,8.08,$. The level is chosen by $|{\omega }_i|/{\sigma }_{\omega }=3.5$. where ${\sigma }_{\omega }$ is the standard deviation of $|{\omega }_i|$.

Figure 11

(a) Distribution of end-to-end distance vector along with fluid vorticity component in an $x\text{−}z$ plane for $\text{We}=2.97$ and (b) PDF of the cosine angle between the end-to-end distance and vorticity vector, open symbol corresponds to the one-way case, while closed represents two-way coupling case. Inset shows an enlarged view for $0.9\le cos{\theta }_3\le 1.0$.

Figure 12

Influence of the Weissenberg number on (a) average enstrophy production, dissipation, the polymer contribution, and the sum of all three and (b) PDF of $\mathrm{\Omega }$ with inset in linear vertical scale.

Figure 13

(a) Three dimensional kinetic energy (b) dissipation spectra for different values of We.

Figure 14

Polymers influence on (a) energy transfer function (b) and two-way interaction energy rate due to polymers.