Turbulent channel flows over porous rib-roughed walls

Abstract

To describe permeable roughness effects on turbulence, particle image velocimetry measurements of turbulent channel flows over porous roughness are carried out. We consider fully developed turbulent channel flows over porous bottom walls with transverse square porous-ribs whose heights are 10% of the channel height. The considered ratios of the rib spacing w to the rib height k are $w/k=$1, 3, 7, 9 and 19. The porous ribs are also made of the same material as that for the bottom wall. Three kinds of metallic foam materials are applied as the porous media. The mean pore diameter of the most permeable medium is 77% of the rib height. Its permeability is approximately five times as large as that of the least permeable one, while the porosities of all three media are 0.95. Solid impermeable rib-roughness cases are also measured for comparison. The measured flows are in the range of the bulk Reynolds numbers of 5000–20000. At $w/k=1$, the levels of turbulence quantities and the drag coefficient become larger as the permeability increases, while such a trend becomes unclear at $w/k=3$ and a reversed trend is seen at $w/k\ge 7$. At a higher permeability, however, it is found that the sensitivity to the rib spacing on the profiles of the turbulence quantities and the drag coefficient is weakened. Even at a lower permeability, turbulence tends to be less sensitive to the rib spacing at $w/k>3$. The equivalent sand grain roughness height hence follows these trends. It is found that there is an almost linear correlation between the zero-plane displacement and the hydraulic roughness scale of the logarithmic law of the mean velocity over the roughness. It is also shown that the effective displacement, which is introduced for isolating the permeability effects, and the von Kármán constant have reasonable correlations with the permeability and pore Reynolds numbers.

Graphical abstract

Appendices

Appendix A: Comparison of the present results with the data in the literature

Figure 19 shows the close agreement between the present data with the DNS data of Burattini et al. (2008) for the impermeable case at $w/k=3$. The ratio of channel height to the rib height in Burattini et al. (2008) was $H/k=21$ which is twice as large as the present geometry. The shown agreement suggests that the present measurements are reliable and the present channel height is large enough to capture the roughness sublayer.

Fig. 19

Comparison between the present and DNS data of the r.m.s. velocities for the impermeable case at $w/k=3$. The DNS data of Burattini et al. (2008) are their case D2 at $k^+=63$ while the present data are at $k^+=43.5$

Fig. 20

Comparison between the present and the previous r.m.s. velocities of the permeable cases at $w/k=9$, Re$_b\simeq 15000$: a low permeability cases, b high permeability cases. The present data are for cases #30m and #13m while the previous data (Okazaki et al. 2020) are for cases #30 and #13. See a for the legend

Figure 20 shows comparison between the present and previous (Okazaki et al. 2020) data of the permeable cases at $w/k=9$. The porosity of Okazaki et al. (2020) was at $\varphi \simeq 0.8$, while the present cases are at $\varphi =0.95$. The permeabilities of cases #30 and #13 were 0.004 mm$^2$ and 0.033 mm$^2$, while those of cases #30m and #13m are 0.007 mm$^2$ and 0.033 mm$^2$. Hence, porous parameters of the two sets of data are similar and the measured profiles show close agreement.

Appendix B: Fitting method for logarithmic velocity profiles

Fig. 21

Example of the logarithmic layer indicated by $(y+d)^+\mathrm{d}U^+/\mathrm{d}y^+$. In the range of 550 $\le (y+d)^+ \le$ 920, a flat plateau at 1/0.23 is seen with $d^+=310$ and hence 310 and 0.23 are the best fitted values for $d^+$ and $\kappa$, respectively

Differentiating both sides of equation (4) by $y^+$ gives an equation for the velocity gradient in the logarithmic region as

$$\begin{aligned} \frac{\mathrm{d}U^+}{\mathrm{d}y^+}=\frac{1}{\kappa (y+d)^+}. \end{aligned}$$

(7)

When the friction velocity is known, the extent of the logarithmic layer can be determined from plots of $(y+d)^+ \mathrm{d}U^+/\mathrm{d}y^+$ as a function of $y^+$ for several values of $d^+$. Since $(y+d)^+ \mathrm{d}U^+/\mathrm{d}y^+$ should be a constant equal to $1/\kappa$ inside the logarithmic layer, a value of $d^+$ giving a flat plateau in the profile must be the best fitted value. In Fig. 21, among the candidates $d^+=310$ gives a flat plateau at 1/0.23 in the range of 550 $\le (y+d)^+ \le$ 920, and hence the best fitted value is $d^+=310$ with $\kappa =0.23$. Then, with these $d^+$ and $\kappa$, equation (4) is best fitted to the velocity profile to obtain the equivalent roughness height $h^+$.