Reynolds number effects in pipe flow turbulence of generalized Newtonian fluids

The turbulent pipe flow of inelastic shear-thinning fluids has many practical applications; however, there is a deficit in understanding of how shear-thinning rheology modifies turbulence structure in the near-wall boundary layer (affecting shear stress and pressure drop) and in the core (affecting mixing). While previous direct numerical simulation studies have examined the effect of shear-thinning rheology at low Reynolds number (${\text{Re}}_{\tau ,\text{max}}=323$), the way in which these effects vary with ${\text{Re}}_{\tau }$ was unknown. In particular, from earlier work it was unclear if inner-scaled mean axial velocity profiles for Newtonian and shear-thinning fluids could collapse to a common curve with increasing Reynolds number. Via direct numerical simulations of Newtonian and one shear-thinning rheology for friction Reynolds number ${\text{Re}}_{\tau }=323\text{–}750$ (${\text{Re}}_G=10000\text{–}28000$), the present study investigates how increasing Reynolds number modifies turbulent pipe flow of a power-law fluid with particular focus on the boundary layer profiles. The results show that the inner-scaled mean axial velocity profiles for Newtonian and shear-thinning fluids cannot collapse to a common curve with increasing Reynolds number, which is consistent with predictions from the Dodge-Metzner correlation [Dodge and Metzner, Turbulent flow of non-Newtonian systems, AIChE J. 5, 189 (1959)]. In inner-scaled coordinates, mean viscosity profiles are shown to become independent of Reynolds number except close to the pipe center. The contribution of viscosity fluctuations in the mean shear budget and in the mean flow and turbulence kinetic energy budget remains small at all $\text{Re}$. Both increasing Reynolds number and shear thinning influence the turbulence kinetic energy budget near the wall; however, the region where shear thinning is important is much wider than the region where increasing Reynolds number influences the results. The persistence of shear-thinning effects on turbulence modification in pipe flow requires consideration in the development of suitable turbulence models for such fluids. The current results suggest that the effect of shear-thinning rheology in turbulence models can be captured via a Reynolds-number-independent mean viscosity model in the inner region.

Figure 1

Detail of spectral element meshes used to discretize the pipe cross section. The mesh in (a) has 300 spectral elements with 11th-order element interpolation functions and was used for ${\text{Re}}_{\tau }=323$. The mesh in (b) has 1188 spectral elements and was used for ${\text{Re}}_{\tau }=500$ with an eighth-order interpolation function; for ${\text{Re}}_{\tau }=750$, tenth-order interpolation functions were used. In each panel, spectral element boundaries are shown on the left and collocation points on the right.

Figure 2

Viscosity rheograms plotted for Newtonian and PL fluids on (a) lin-lin and (b) log-log axes.

Figure 3

Two-point correlation coefficient of axial velocity fluctuations plotted as a function of separation distance $\mathrm{\Delta }z/D$ at (a) $r/D=0.35$ and (b) $r/D=0.48$.

Figure 4

Inner-scaled statistical profiles from DNS of Newtonian fluid at ${\text{Re}}_{\tau }=323$ (solid lines), compared to experimental results of den Toonder and Nieuwstadt [22] (open circles, ${\text{Re}}_{\tau }=314$) and DNS results of El Khoury et al. [20] (closed circles, ${\text{Re}}_{\tau }=360$): (a) mean axial velocity, (b) rms of axial and radial velocity fluctuations, (c) Reynolds shear stress and azimuthal velocity fluctuations, and (d) turbulent kinetic energy budget.

Figure 5

Inner-scaled statistical profiles from DNS of Newtonian fluid at ${\text{Re}}_{\tau }=500$ (solid line), compared to DNS results of El Khoury et al. [20] (closed circles, ${\text{Re}}_{\tau }=550$) and Chin [23] (open squares, ${\text{Re}}_{\tau }=500$): (a) mean axial velocity, (b) rms of axial and radial velocity fluctuations, (c) Reynolds shear stress and azimuthal velocity fluctuations, and (d) turbulent kinetic energy budget.

Figure 6

Contours of inner-scaled instantaneous axial velocity $u_z^{+}$ plotted in (a)–(c) inner coordinates on a developed cylindrical surface $z^{+}-r{\theta }^{+}$ at $y^{+}=10$ and (d)–(f) outer coordinates on a cross section for a Newtonian fluid at (from top to bottom) ${\text{Re}}_{\tau }=323$, 500, and 750. For (a)–(c), the flow is from left to right and the region is 7000 wall units long and 1600 wall units wide. The contour levels vary from blue to red with the values 8–20.

Figure 7

Contours of inner-scaled instantaneous axial velocity $u_z^{+}$ plotted in (a)–(c) inner coordinates on a developed cylindrical surface $z^{+}-r{\theta }^{+}$ at $y^{+}=10$ and (d)–(f) outer coordinates on a cross section for PL fluid at (from top to bottom) ${\text{Re}}_{\tau }=323$, 500, and 750. For (a)–(c), the flow is from left to right and the region is 7000 wall units long and 1600 wall units wide. The contours levels vary from blue to red with the values 8–20.

Figure 8

(a) Streamwise and (b) azimuthal integral length scales of axial velocity fluctuations plotted as a function of $y^{+}$.

Figure 9

Profiles of the inner-scaled (a) mean axial velocity and (b) mean axial velocity gradient plotted for Newtonian (black lines) and PL fluids (orange lines). Blue lines in (a) show the law of the wall with the slope parameter determined in Fig. 10 (a) for ${\text{Re}}_{\tau }=750$.

Figure 10

The (a) log-law and (b) power-law indicator functions for Newtonian and PL fluids. Vertical lines show the location where the labeled values are read.

Figure 11

Profiles of the velocity defect plotted in outer coordinates for Newtonian and PL fluids at different ${\text{Re}}_{\tau }$.

Figure 12

Profiles of the normalized mean viscosity plotted for PL fluid at different ${\text{Re}}_{\tau }$. For the line legend see Fig. 9.

Figure 13

Comparison of the friction factor obtained via DNS (circles) and those of the Dodge-Metzner correlation (solid lines) and the Anbarlooei et al. correlation (dashed lines) for Newtonian (black lines) and PL fluids (orange lines).

Figure 14

Ratio of friction factors for Newtonian and PL fluids obtained from the correlation (13) plotted against ${\text{Re}}_{\tau }$ with the inset showing a closer look in the ${\text{Re}}_{\tau }$ range considered in this study.

Figure 15

Profiles of the (a) mean viscous stress ${\tau }^{v^{+}}$, (b) Reynolds shear stress ${\tau }^{R^{+}}$, and (c) turbulent viscous stress ${\tau }^{f{v}^{+}}$ plotted for Newtonian and PL fluids at different ${\text{Re}}_{\tau }$.

Figure 16

Profiles of turbulence intensities plotted as a function of (a)–(c) $y^{+}$ and (d)–(f) $y/R$ for Newtonian and PL fluids at different ${\text{Re}}_{\tau }$.

Figure 17

Profiles of the rms viscosity fluctuations normalized by (a) the nominal wall viscosity and (b) the local mean viscosity plotted for PL fluid at different ${\text{Re}}_{\tau }$.

Figure 18

Profiles of the terms which appear only in the mean flow kinetic energy budget (A1) plotted for Newtonian (black lines) and PL fluids (orange lines).

Figure 19

Profiles of the turbulent kinetic energy budget terms [see Eq. (A2)] plotted for Newtonian (black lines) and PL fluids (orange lines) at different ${\text{Re}}_{\tau }$.

Figure 20

Profiles of the sum of the Newtonian and non-Newtonian transport and the dissipation terms plotted for the Newtonian (black lines) and the shear-thinning fluid (orange lines).