longitudinal mapping 1
5.4 Comparison with the bed-load experiments
Comparisons of the two granular stress relations with the two smooth-wall experimental data are shown in Figure 5.12 and 5.13. For the granular pressure pS in a and granular shear stress τS in b, the data agree reasonably well with the relations Eq.(5.8) and (5.9).
Some of the scatter is likely due to particle layering, observed earlier to be more pro-nounced at high concentrations, the scatters are shown in different degree between SWSf and SWRF. The kinetic theory relations, however, overestimate the data at low concentra-tions (for the shear stress) and conversely underestimate the data at high concentraconcentra-tions (for both the pressure and shear stress). This suggests that streaming and binary col-lisions, the two mechanisms assumed by kinetic theory, do not fully describe immersed granular behavior in the dilute upper zone, where particles saltate, or in the dense basal zone where contact chains may connect multiple grains. For both relationships, agree-ment is slightly improved by taking added mass effects into consideration (solid lines), as compared with curves calculated without added mass effects (dashed lines).
Figure 5.12 Granular stresses data for selected runs in SWSF (distinguished by color) compared with the constitutive relations: a dimensionless granular pressure ˆpS compared with kinetic theory relation Eq. (5.8) with (solid line) and without (dashed line) added mass effect; b dimensionless granular shear stress ˆτS likewise compared with kinetic theory relation.
Figure 5.13 Granular stresses data for all runs in SWRF (distinguished by color) com-pared with the constitutive relations: a dimensionless granular pressure ˆpS compared with kinetic theory relation Eq. (5.8) with (solid line) and without (dashed line) added mass effect; b dimensionless granular shear stress ˆτS likewise compared with kinetic theory relation.
Next, we compare measurements of the Reynolds stress τL with the mixing length re-lation Berzi and Fraccarollo (2015). In Figure 5.14a and 5.15a, we examine a possible concentration-dependence by plotting the dimensionless ratio ˆ`m = `m/D against the concentration cS. In contrast with the relation proposed by Berzi and Fraccarollo (2015) (dashed line), the present data are found to vary little with concentration, and to be well approximated by the constant ratio `m/D ≈ 0.2 (continuous line). This falls within the range suggested by Wiberg and Smith (1991), indicating that wake effects dominate the vertical momentum balance within the transport layer.
With confidence on the empirical drag law after validation test, for simplicity, we approx-imate the Reynolds number Re in these drag formulas by the terminal Reynolds number Ret= ωD/ν. The mean longitudinal drag force per unit volume experienced in turbulent bed-load can therefore be written
fD = cS
VSFDx= ˆfD(cS)ρLD−1vR(uL− uS) , (5.15) where VS = πD3/6 is the volume of one grain, ˆfD(cS) = 3/4CDcS(1 − cS)2−β is a dimen-sionless function retaining the concentration dependence, and vR(uL− uS) represents the influence of the relative velocity between the liquid and solid phases. To calculate this influence, we assume that granular velocity fluctuations are isotropic and uncorrelated, and neglect liquid velocity fluctuations. Taylor expansion to second order with respect to the mean (without fluctuations) then yields the approximation
vR(uL− uS) ≈
hence the granular temperature T increases the mean drag force. The resulting relation-ship is compared with the experimental data on Figure 5.14 and 5.15b. Although some scatter is apparent, the data deduced from all experiments agree well with the expected relationship (solid line). Without accounting for the granular temperature in (5.16), the drag force would be significantly underestimated. For comparison, we also plot in Figure 5.14 and 5.15b the drag forces (triangles) determined by Teng (2003) from fluidization cell experiments carried out with the same solid and liquid materials, with T set to zero in this case. Provided that granular temperature is taken into account, we find that drag in turbulent bed-load satisfies the same empirical function as fluidization cell flows.
Figure 5.14 Liquid stress and drag force data for selected runs in SWSF (distinguished by color) compared with the constitutive relations: a dimensionless turbulent mixing length
`ˆm compared with the constant approximation ˆ`m ≈ 0.2 (solid line), and the relation proposed by Berzi and Fraccarollo (2015) (dashed line);b dimensionless drag force ˆfD compared with relation (5.15) (solid line) and fluidization cell data (triangles).
Figure 5.15 Liquid stress and drag force data for all runs in SWRF (distinguished by color) compared with the constitutive relations: a dimensionless turbulent mixing length
`ˆm compared with the constant approximation ˆ`m ≈ 0.2 (solid line), and the relation proposed by Berzi and Fraccarollo (2015) (dashed line);b dimensionless drag force ˆfD compared with relation (5.15) (solid line) and fluidization cell data (triangles).
5.5 Conclusion
In this Chapter, we used the maps of smooth wall experiments to obtained the depth profiles, we found that the convective acceleration appeared in experiments and can’t be neglect. Take the convective acceleration term into account, we determined each terms in the two dimensional, steady state momentum equations of two phases, deduced the depth profiles of granular pressure, solid shear stress and liquid shear stress. By com-paring the granular stresses with the kinetic theory, it shows in good agreement which is confirmed that the granular pressure which is created by grain collisions could support the submerged weight of the solid grains in turbulent flow without the effect of turbulence or lift force exerted by liquid. For the solid shear stress it could be also described by kinetic theory.
For the liquid stress, the Reynold stress were dominant and it could be scaled by grain diameter, suggesting that the wake effect dominate in the transport layer. For drag force, we adpoted the empirical law derived by Di Felice (1994) and used the refractive-index-matched materials to verify the law, from the fludization cell experiments and the seepage tests we confirmed the applicability of this relation, and we found out that to apply this relation to the turbulent bed-load flow, the effect of granular temperature must be taken into consideration, the velocity agitations in solid phase increase the drag between liquid and solid.
As the comparisons presented in this Chapter, we confirmed that for steady state
tur-bulent bed-load transport, and with Reynolds number range 5000-12000, the stresses of the turbulent bed-load flow could be described by the constitutive relations of the kinetic theory. We also found that the drag force relation derived from fixed pack and fluidiza-tion cell could be applied to turbulent bed-load if the contribufluidiza-tion from solid velocity fluctuation was considered.