Oscillating grid generating turbulence near gas-liquid interfaces in shear-thinning dilute polymer solutions

Understanding the behavior of liquid phase turbulence near gas-liquid interfaces is of great interest in many fundamental, environmental, or industrial applications. For example, near-surface liquid side turbulence is known to enhance the mass transfers between the two phases. Descriptions of this behavior for air-water systems exist in the literature, but the case of turbulence in a shear-thinning liquid phase below a flat gas-liquid interface has never been considered to the best of our knowledge. This paper consists in an experimental characterization of low Reynolds number, oscillating grid generated, near-surface turbulence in shear-thinning dilute polymer solutions, in the surface-influenced and in the viscous sublayers. The energy transfer mechanism, known in the water case, is evidenced in dilute polymer solutions. A horizontal damping mechanism, similar to the one introduced by surfactants, is evidenced. The evolution of the viscous sublayer depth can be explained by both viscous and shear-thinning effects, and it appears that a critical polymer concentration may exist within the dilute regime.

Figure 1

Sketch of the SPIV setup for liquid phase velocity measurements.

Figure 2

Example of velocity fluctuation snapshot fields $u^{\prime }$ for water in 2D2C [x-z plane (a)] and 2D3C (b). The colorbar represents the out-of-plane velocity component $u_y^{\prime }$. Only one of every four vectors is shown in both the $x$ and $z$ dimensions for the sake of readability.

Figure 3

(a) Profiles of normal and tangential turbulent rms velocities $u_n^{\prime }$ (full lines) and $u_t^{\prime }$ (dashed lines) along z/h, scaled by a grid reference velocity $fS$. Typical uncertainties on velocity values are shown for the clean water run near the interface with black error bars. $h$ is the distance between the average grid position and the surface. (b) Profiles of vertical/normal integral length scale $L^n$ (full lines) and horizontal/tangential integral length scale $L^t$ (dashed lines) along $z/h$, scaled by the grid mesh parameter $M$. Markers in (a) and (b) are plotted on one of every five data points for the sake of readability. For 100 ppm, length scales become much larger than the ROI, which prevents the integral length-scale computation. Horizontal thin gray lines are plotted at each viscous sublayer depths reported, and colored ticks on the central bar indicate which working fluid they correspond to. (c) Evolution of $L^t/L^n$ with $C_{\mathrm{XG}}/C_D$ at $\mathrm{z}={\delta }_v$.

Figure 4

(a) Profiles of normal and tangential turbulent rms velocities $u_n^{\prime }$ (full lines) and $u_t^{\prime }$ (dashed lines) normalized by the undisturbed velocity scale (based on horizontal velocity fluctuations) $u^{\prime }$, along the depth $z$ itself normalized by the undisturbed integral scale ${Ł}_{\infty }$. The dashed-doted vertical gray line shows is plotted at $u_n^{\prime }=u^{\prime }$ or $u_t^{\prime }=u^{\prime }$. (b) Evolution of the measured kinetic energy $k_m$ at $z={\delta }_v$ divided by the predicted, undisturbed, kinetic energy $k_u$, as a function of polymer concentration, the dashed-doted line is plotted at $k_u=k_m$. Markers in (a) and (b) are plotted on one of every five data points for the sake of readability. (c) Profiles of normal an tangential integral length scales $L^n$ and $L^t$ normalized by $L_{\infty }$ plotted versus $z{Ł}_{\mathrm{infty}}$. The color code is the same as in Fig. 3.

Figure 5

Evolution of the measured viscous sublayer depth ${\delta }_v$ (log scale, black dots) with scaled polymer concentration compared to predictions based on undisturbed scales and constant scale viscosities ${\mu }_0$ (right-pointing triangles), ${\mu }_m=\left({\mu }_0+{\mu }_{\infty }\right)/2$ (down-pointing triangles), and ${\mu }_{\infty }$ (squares).

Figure 6

Modeling of terms A (a), B (b), and C (c) in terms of grid-based Deborah number. A represents the ratio of undisturbed velocity scale in polymer over that for water. B is the ratio of undisturbed integral scales, and C the ratio between zero shear rate (maximum) viscosity and viscosity of water. Scaling laws are obtained by least-square fitting and shown on each associated plot. Shaded areas represent scaling uncertainty areas of, respectively, $\pm 5\%$ for the darker and $\pm 15\%$ for the lighter.

Figure 7

Modeling of the viscous sublayer depth using Eq. (16). Black dots are measured points. The error bars on measured ${\delta }_v/{\delta }_v^w$ are deduced by adding relative uncertainty on ${\delta }_v$ for each polymer case to that for water. The shaded region corresponds to $\pm 5\%$ uncertainty area on the model.