Polymer concentration and properties of elastic turbulence in a von Karman swirling flow

We report detailed experimental studies of statistical, scaling, and spectral properties of elastic turbulence (ET) in a von Karman swirling flow between rotating and stationary disks of polymer solutions in a wide, from dilute to semidilute entangled, range of polymer concentrations $\varphi$. The main message of the investigation is that the variation of $\varphi$ just weakly modifies statistical, scaling, and spectral properties of ET in a swirling flow. The qualitative difference between dilute and semidilute unentangled versus semidilute entangled polymer solutions is found in the dependence of the critical Weissenberg number ${\text{Wi}}_c$ of the elastic instability threshold on $\varphi$. The control parameter of the problem, the Weissenberg number $\text{Wi,}$ is defined as the ratio of the nonlinear elastic stress to dissipation via linear stress relaxation and quantifies the degree of polymer stretching. The power-law scaling of the friction coefficient on $\text{Wi}/{\text{Wi}}_c$ characterizes the ET regime with the exponent independent of $\varphi$. The torque $\mathrm{\Gamma }$ and pressure $p$ power spectra show power-law decays with well-defined exponents, which has values independent of $\text{Wi}$ and $\varphi$ separately at $100\le \varphi \le 900$ ppm and $1600\le \varphi \le 2300$ ppm ranges. Another unexpected observation is the presence of two types of the boundary layers, horizontal and vertical, distinguished by their role in the energy pumping and dissipation, which has width dependence on $\text{Wi}$ and $\varphi$ differs drastically. In the case of the vertical boundary layer near the driving disk, $w_v^v$ is independent of $\text{Wi}/{\text{Wi}}_c$ and linearly decreases with $\varphi /\varphi *$, while in the case of the horizontal boundary layer $w_v^h$ its width is independent of $\varphi /\varphi *$, linearly decreases with $\text{Wi}/{\text{Wi}}_c$, and is about five times smaller than $w_v^v$. Moreover, these $\text{Wi}$ and $\varphi$ dependencies of the vertical and horizontal boundary layer widths are found in accordance with the inverse turbulent intensity calculated inside the boundary layers $V_{\theta }^h/V_{\theta }^{h\mathrm{rms}}$ and $V_{\theta }^v/V_{\theta }^{v\mathrm{rms}}$, respectively. Specifically, the dependence of $V_{\theta }^v/V_{\theta }^{v\mathrm{rms}}$ in the vertical boundary layer on $\text{Wi}$ and $\varphi$ agrees with a recent theoretical prediction [S. Belan, A. Chernych, and V. Lebedev, Boundary layer of elastic turbulence (unpublished)].

Figure 1

Experimental setup.

Figure 2

[(a), (b)] PDFs of $\delta \mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{rms}}$ at different $\text{Wi}$ for polymer solutions at $\varphi =100$ ppm and $\varphi =3000$ ppm of PAAm, respectively. [(c), (d)] PDFs of $\delta p/p_{\mathrm{rms}}$ at different $\text{Wi}$ for polymer solutions at $\varphi =100$ ppm and $\varphi =3000$ ppm of PAAm, respectively.

Figure 3

Average torque $\langle \mathrm{\Gamma }\rangle$ vs $\text{Wi}$ for several values of $\varphi$. The arrows indicate the instability onset for four values of $\varphi$.

Figure 4

Dependence of the onset of the elastic instability ${\text{Wi}}_c$ on $\varphi$. The line is the fit to the part of the data of a lower polymer concentration solutions ${\text{Wi}}_c\sim {\varphi }^{\delta }$ and $\delta \simeq 0.37\pm 0.02$.

Figure 5

(a) Average torque $\overline{\mathrm{\Gamma }}$ versus ${\text{Wi/Wi}}_c$. A solid line is the fit to the data $\langle \mathrm{\Gamma }\rangle \sim {\left({\text{Wi/Wi}}_c\right)}^{1.5}$. (b) Friction coefficient $C_f$ as a function of ${\text{Wi/Wi}}_c$ for all polymer concentrations $\varphi$. A solid lid line is the fit to the data at $\varphi =300$ ppm $C_f\sim {\left({\text{Wi/Wi}}_c\right)}^{-0.35}$. The power-law fits to the rest of the data for $C_f$ at ${\text{Wi/Wi}}_c>1$ and all $\varphi$ except 700 ppm give about the same exponent.

Figure 6

(a) ${\mathrm{\Gamma }}_{\mathrm{rms}}/\langle \mathrm{\Gamma }\rangle$ as a function of ${\text{Wi/Wi}}_c$ and (b) $p_{\mathrm{rms}}/\langle p\rangle$ as a function of ${\text{Wi/Wi}}_c$ for all polymer concentrations $\varphi$. A solid line is the fit $p_{\mathrm{rms}}/\langle p\rangle \sim 2{\left({\text{Wi/Wi}}_c\right)}^3$ at $\varphi \le 700$ ppm.

Figure 7

The average azimuthal velocity $\langle V_{\theta }\rangle$ radial profiles close to the side wall for various $\text{Wi}$ and four values of $\varphi$: (a) 100, (b) 300, (c) 900, and (d) 3000 ppm.

Figure 8

The radial profiles of the rms azimuthal velocity gradients ${(d{V}_{\theta }/dr)}_{\mathrm{rms}}$ close to the side wall for various $\text{Wi}$ and four values of $\varphi$: (a) 100, (b) 300, (c) 900, and (d) 3000 ppm.

Figure 9

The normalized boundary layer width by the disk radius $R_c$ near side walls of the average azimuthal velocity in the horizontal plane $w_v^h/R_c$ as a function of (a) ${\text{Wi/Wi}}_c$ for different $\varphi$ and (b) $\varphi /\varphi *$ at the largest $\text{Wi}$. A solid line in panel (a) is the linear fit $w_v^h\sim -0.0018{\text{Wi/Wi}}_c$. The normalized boundary layer width of the rms azimuthal velocity gradient fluctuations in the horizontal plane $w_{dv/dr}^h/R_c$ as a function of (c) ${\text{Wi/Wi}}_c$ for different $\varphi$ and (d) $\varphi /\varphi *$ at the largest $\text{Wi}$. A solid line in panel (c) is the linear fit $w_{dv}^h\sim -0.0017{\text{Wi/Wi}}_c$.

Figure 10

The ratio of the average azimuthal velocity to the rms azimuthal velocity fluctuations in the horizontal plane $\langle V_{\theta }^h\rangle /V_{\theta }^{h\mathrm{rms}}$ as a function (a) of $\text{Wi}$ for $\varphi =500$ ppm; (b) of $\varphi$ at the largest $\text{Wi}$. A solid line is the fit $\langle V_{\theta }^h\rangle /V_{\theta }^{h\mathrm{rms}}=7.25-(0.038\pm 0.001)\text{Wi}$. The ratio of the average azimuthal velocity to the rms azimuthal velocity fluctuations in the vertical plane $\langle V_{\theta }^v\rangle /V_{\theta }^{v\mathrm{rms}}$ as a function (c) of $\text{Wi}$ for $\varphi =300$ ppm and six values of $z/d$ close to the cylinder wall; (d) of $\varphi$ at $\text{Wi}=90$ and $z/d=0.98$. A solid line is the fit $\langle V_{\theta }^v\rangle /V_{\theta }^{v\mathrm{rms}}=14.4+(0.005\pm 0.001)\varphi$.

Figure 11

The rescaled azimuthal velocity profiles in the vertical plane for different $\text{Wi}$ and $\varphi$: (a) 100, (b) 300, (c) 900, and (d) 3000 ppm.

Figure 12

The normalized vertical boundary layer width near the upper disk of (a) the azimuthal velocity $w_v^v/d$ versus ${\text{Wi/Wi}}_c$ for different $\varphi$; (b) $w_v^v/d$ vs $\varphi /\varphi *$ at the largest $\text{Wi}$. A solid line is the linear fit $w_v^v/d=0.17-(0.016\pm 0.001)\varphi /\varphi *$.

Figure 13

${\mathrm{Wi}}_{\mathrm{loc}}^h={\omega }_z^{\mathrm{rms}}\lambda$ in the horizontal plane as a function of ${\text{Wi/Wi}}_c$ for different $\varphi$ and averaged: (a) at $r=1$ cm and $\varphi \le 900$ ppm; (b) at four values of $r$ and $\varphi =3000$. ${\text{Wi}}_{\mathrm{loc}}^v$ based on the measurements in the vertical plane and averaged over a whole profile as a function of ${\text{Wi/Wi}}_c$ for (c) $\varphi \le 900$ ppm; (d) $\varphi \ge 1600$ ppm.

Figure 14

Power spectra of (a) torque $\mathrm{\Gamma }$ and $p$ fluctuations as a function of the normalized frequency $f/f_{\mathrm{vor}}$ for $\varphi =100$ ppm and various $\text{Wi}$; (b) torque $\mathrm{\Gamma }$ and $p$ fluctuations as a function of the normalized frequency $f/f_{\mathrm{vor}}$ for $\varphi$ in the semidilute entangled regime at maximum $\text{Wi}$. Solid lines are the power-law fits to the pressure and torque frequency spectra give the following value for the exponents: $E_{pp}\sim {(f/f_{\mathrm{vor}})}^{-2.2}$ and $E_{\mathrm{\Gamma }\mathrm{\Gamma }}\sim {(f/f_{\mathrm{vor}})}^{-4.6}$. The value of the main vortex frequency $f_{\mathrm{vor}}$ is obtained from the peaks in $\mathrm{\Gamma }$ and $p$ spectra vs $\text{Wi}$.

Figure 15

Correlation time of pressure fluctuations as a function of ${\text{Wi/Wi}}_c$ for all values of $\varphi$. Inset: The normalized correlation time ${\tau }_{\mathrm{corr}}/\lambda$ vs ${\text{Wi/Wi}}_c$.

Figure 16

Structure functions of the pressure increments $S_n\left(t\right)$ up to $n=6$ order vs the $\delta t$ for (a) $\varphi =500$ ppm and $\text{Wi}=90$. Solid lines are fits to the data in the scaling range to define the exponents ${\zeta }_n$. For $S_2\sim {\left(\delta t\right)}^{{\zeta }_2}$ the power-law fit gives the exponent ${\zeta }_2=\approx 1.9\pm 0.1$. (b) $\varphi =2300$ ppm and $\text{Wi}=225$.

Figure 17

Normalized scaling exponent ${\zeta }_n/{\zeta }_2$ obtained from the plots similar to those in Fig. 16 vs $n$ for various $\varphi$ and the highest $\text{Wi}$. The dashed line corresponds to a linear function ${\zeta }_n/{\zeta }_2=n/2$.