Near-wall lubricating layer in drag-reduced flows of rigid polymers

The current theories on the mechanism for polymer drag reduction (DR) are generally applicable for long-chain flexible polymers that form viscoelastic solutions. Rigid polymer solutions that generate DR seemingly lack prevalent viscoelastic characteristics. They do however demonstrate higher viscosities and a noticeable shear-thinning trend, approximated by generalized Newtonian models. The following experimental investigation scrutinizes the flow statistics of an aqueous xanthan gum solution in a turbulent channel flow, with friction Reynolds numbers ${\text{Re}}_{\tau }$ between 170 and 700. The amount of DR varies insignificantly between 27% and 33%. The velocity field is measured using planar particle image velocimetry and the steady shear rheology is measured using a torsional rheometer. The results are used to characterize the flow statistics of the polymer drag-reduced flows at different ${\text{Re}}_{\tau }$ and with negligible changes in DR, a parametric study only previously considered by numerical simulations. Changes to the mean velocity and Reynolds stress profiles with increasing ${\text{Re}}_{\tau }$ are similar to the modifications observed in Newtonian turbulence. Specifically, the inner-normalized mean velocity profiles overlap for different ${\text{Re}}_{\tau }$ and the Reynolds stresses monotonically grow in magnitude with increasing ${\text{Re}}_{\tau }$. Profiles of mean viscosity with respect to the wall-normal position demonstrate a thin layer that consists of a low-viscosity fluid in the immediate vicinity of the wall. Fluid outside this thin layer has a significantly higher viscosity. We surmise that the demarcation in the mean shear viscosity between the inner lubricating layer and the outer layer cultivates fluid slippage in the buffer layer and an upward shift in the logarithmic layer, a hypothesis akin to DR using wall lubrication and superhydrophobic surfaces.

Figure 1

Isometric cross section of the glass test section.

Figure 2

(a) Steady shear viscosity measurements of 170 ppm XG solution and water. (b) Skin-friction coefficient as a function of Reynolds number for water and the 170 ppm XG solution.

Figure 3

Isometric three-dimensional model of the planar PIV setup relative to the glass test section and channel section.

Figure 4

Inner-normalized distributions of (a) mean streamwise velocity and (b) the indicator function, for Newtonian flows.

Figure 5

Inner-normalized profiles of (a) streamwise Reynolds stress and (b) wall-normal and Reynolds shear stresses, for Newtonian flows.

Figure 6

Inner-normalized distributions of (a) mean streamwise velocity and (b) the indicator function, for flows with 170 ppm XG solution.

Figure 7

Pseudomean viscosity normalized by wall viscosity as a function of inner-normalized wall location.

Figure 8

Instantaneous contour of (a) streamwise velocity fluctuations and (b) viscosity for XG at ${\text{Re}}_{\tau }=170$.

Figure 9

Probability density function of fluctuating viscosity taken at $x$ of $0.1h$ and $y$ of (a) $0.07h$ and (b) $0.42h$.

Figure 10

Wall-normal profiles of (a) mean viscosity and (b) the root mean square of the fluctuating viscosity. Error bars are shown at $y/h$ of 0.07 and 0.42.

Figure 11

Two-point correlation of viscosity fluctuations along the streamwise direction at wall-normal locations of (a) $y/h=0.07$ and (b) $y/h=0.42$.

Figure 12

Two-point correlation of viscosity fluctuations along the wall-normal direction with a reference wall-normal location of (a) $y_0=0.07h$ and (b) $y_0=0.42$.

Figure 13

Inner-normalized profiles of (a) streamwise Reynolds stress and (b) wall-normal and Reynolds shear stresses, for flows with 170 ppm XG solution.

Figure 14

Inner-normalized mean stress balance of XG at three of the seven ${\text{Re}}_{\tau }$ conditions. The lines correspond to $\cdots ,1-y^{+}/{\text{Re}}_{\tau }$;$$——$$, ${\tau }^{+}$; $•••,{\langle uv\rangle }^{+}$;$––, {\tau }_v^{+}$; and $–·, {\tau }_v^{\prime +}$. Error bars are shown at $y/h$ of 0.07 and 0.42.

Figure 15

Shear rheology of the 170 ppm XG solution measured for the PP geometry with different $h_{\text{PP}}$.

Figure 16

Shear rheology of the 170 ppm XG solution measured for the PP geometry with and without Tween 20 and an $h_{\text{PP}}$ of 0.2 mm.

Figure 17

Statistical convergence of (a) $\langle U\rangle$, (b) $\langle u^2\rangle$, (c) $\langle v^2\rangle$, (d) $\langle uv\rangle$, (e) $\langle \dot{\gamma }\rangle$, and (f) $\langle \mu \rangle$, for the flow of XG at its smallest and largest ${\text{Re}}_{\tau }$ cases of 170 and 700 and at a $y^{+}$ of 100.