The drag force of a particle in steady, unidirectional, two-phase flow can be simplified as the multiplication of a function of the relative velocities and a function of the other parameters.
FD = f (ρL, d, cS, ωT, Rep)g(uR, uS) = 1
2ρLASCD|uR| (¯uL− uS) (5.17) where AS is the cross-sectional area of a particle, CD is the drag coefficient determined by the solid fraction, the liquid viscosity, the terminal velocity and the shape of the particle,
|uR| is the relative speed which is defined as |uR| =p(¯uL− uS)2+ (¯vL− vS)2+ ( ¯wL− wS)2, we use the Lagrangian solid velocities here to include the effect of solid velocity fluctua-tion, and ¯uL− uS is the relative velocity in stream-wise direction.
Supposing uS, vS, wS be the normal random variables which are mutually independent, then the solid velocities are expressed as uS = N [¯uS, σu], vS = N [¯vS, σv], wS = N [ ¯wS, σw], where the ¯uS, ¯vS, ¯wS are the means and σu, σv, σw are the standard deviations. ¯vL , ¯wL and ¯vS, ¯wS equal to zero because of the unidirectional flow, let the function g(uS, vS, wS) =
|uR| (¯uL− uS), we can estimate the expectation of g(uS, vS, wS) by the means and the variances of the solid velocities and average liquid velocities.
E [g(uS, vS, wS)] ≈ (¯uL− ¯uS)2+1 then the expectation of g(uS, vS, wS) can be expressed as the function of mean liquid velocities, mean solid velocities and granular temperature.
E [g(uS, vS, wS)] ≈ (¯uL− ¯uS)2+ 2T (5.20)
Chapter 6 Conclusion
To increasing our understanding for turbulent bed-load transport, we used the experimen-tal method to study the phenomena under idealized condition: steady state and nearly uniform turbulent flow with the identical solid spheres transported inside. To make it more idealized, we use the refractive index-matched solid and liquid combination to in-vestigate the internal flow structure. Three different channel boundaries were applied to the experiments to see how the flow responds to these changes. Although it is more complex to deal with the para-cymene and PMMA grains than water-opaque grain combi-nation, there is no problem for RIM materials to do the successful steady state turbulent bed-load experiments. The uniform flow condition was failed to achieve for the reason of the limited length of our channel. The measurements during experiment yields reasonable results, for measurements of slope and outlet discharge, original methods were modified to obtain the better results. The characteristic tests also gives us the basic properties of solid and liquid which can be use for numerical and theoretical modelling.
We also introduce the experimental set-up and procedures for refractive-index -matched experiments, the steps and cautions were explained and mentioned. Although it is more complex to deal with the para-cymene and PMMA grains than water-opaque grain com-bination, now it seems plausible for RIM materials to do something more complex. The uniform flow condition was failed to achieve for the reason of the limited length of our channel. The steady re-circulation of solid grains was achieved by jet entrainment mech-anism. it was verified again that the temperature of the mixture is crucial to the index matching condition, in our case is about 17◦C. The uniform flow condition was achieved by changing channel slope, since the variation of the slope is little (1◦–2◦), checking the uniform condition by ruler may not as accurate as we expected. We measured the channel slope, flow depth and the outlet discharge for solid and liquid during the experiments, these data were used to verify or check the data obtained from imaging analysis. A series of tests were also performed to obtained the basic properties of both grains and liquid, the data were not only used for the turbulent bed-load experiment itself, but also be used for the numerical modelling or theory verification.
In the aspect of imaging analysis, we proposed a new approach to measure the inter-nal flow of the refractive-index-matched solid-liquid mixture, based on the combination of transverse and longitudinal laser scanning. We applied the two dimensional particle tracking velocimetry to capture the motion of solid and liquid by transverse scans, ac-quired two dimensional velocity over a three dimensional volume, different masks were
developed and applied to reduce the wrong data captured. By identifying grain crossing events on the longitudinal scan, it is possible to deduce the 3 dimensional velocities and spatial/temporal distributions over scanned volume, obtaining the accurate solid fraction distribution, both algorithm is automatic. As shown in results, over 104 data of solid and 105 data of liquid were obtained for one run. with large number of data, we used phase-averaging method to calculate the averaged maps of velocities and solid fraction with sub-grain resolution, The volumetric discharge integrated from the maps were compared with measurement at outlet and in good agreement. However the under-estimated veloc-ity were found around the vertical cylinders in RWRF, this remind us that the refractive index could be different for the objects made of the same material, but be manufactored by different process.
From these flow maps, we investigate the relationship of bed-load transport rate with variables, the solid and total discharge maps were used to define different domains of transport. integration the different physical quantities over those domains to deduce these map to the factors called outcome variables for three channel boundaries. We show that the outcome variables, CS, US and ABwere scaled with bed-load transport rate QS in the same manner that followed Eq. (4.16). The dimensional analysis was applied to these outcome variables and compared with two dimensioless parameters: Mobility parameter Θ and Shields number ˆˆ τ , for all three channel boundaries, the outcome variables showed its relationship with both Mobility parameter and Shields number having the power of 0.5, and for the dimensionless discharge ˆQS, the power was about 1.5.
To clarify the principle contributions to the momentum balance of each phase, nearly two dimensional flow experiments were selected 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 dimen-sional, steady state momentum equations of two phases, deduced the depth profiles of granular pressure, solid shear stress and liquid shear stress. By comparing 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 adopted 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 contribution from the granular velocity fluctuations must be taken into consideration, this effect increased the drag force between liquid and solid. As the comparisons resented in last Chapter, we confirmed that the stresses of the turbulent bed-load flow could be described by the constitutive relations of the kinetic theory, and we could decompose the contributions of each phase using continuum two phase flow theory.
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