Dimensional analysis

To further characterize the relations between bed-load transport and physical variables within the flow, we selected the related variables and see how they respond to the dimen-sionless parameters which represent the kinematic and dynamic conditions for the flow.

first we chose the variables obtained in Section 4.2, C_S, A_B and U_S, and the physical variables related to the dimensions:

O = F (I, S, ρS, ρL, g, D, B, P ), where I = UC, UT, for kinematic similarity. (4.17)

For the RHS of the relation are the input and dimensional variables, S is the channel slope, g is the gravitational acceleration, D is the particle diameter, B is the channel width and P is the perimeter, these variables represent the physical dimensions of the experiments, for variables, ρ_S and ρ_L and g, could be reduced to g⁰ = ^ρS_ρ^−ρL

L g to include the buoyancy and gravity effects. For LHS of Eq. (4.17), O represents the outcome variables which were composed with

O : (Q_S, C_S, A_B, U_S) (4.18) we selected these variables to see the responds of the bed-load transport rate to the input parameter and we tell which components dominate this variation. We make the outcome variables dimensionless by g⁰, D and B

Qˆ_S = Q_S

where ˆQ_S is the dimensionless parameter called Einstein number, and C_S is dimension-less itself. I in RHS is the input variable, and we chose Mobility parameter Θ as the dimensionless parameter which is defined by

Θ =ˆ U²

g⁰D (4.22)

where U is the amplitude of the velocity outside the boundary layer adopted by Asano (1995) to compare the relation with sediment transport rate in oscillatory sheet flow, the relation between Mobility parameter and the dimensionless bed-load transport rate was also suggested by Asano (1995)

Qˆ_S ∝ ˆΘ^3/2 (4.23)

therefore from Eq. (4.23) we could test the log-law of the solid discharge Eq. (4.16) and plotted with data. For each outcome variables it scales with Mobility parameter in this way:

CS, ˆUS, ˆAB ∝ ˆΘ^1/2 (4.24) In our cases we selected the velocity in the clear liquid domain U_C and the total transport velocity U_T as the velocity scale of the Mobility parameter Eq. (4.25) and Eq. (4.26),

and compared with the normalized outcome variables, the results were plotted in Figure 4.17 and Figure 4.19 on linear scale, in Figure 4.18 and Figure 4.20 on log-log scale.

Θˆ_C = U_C²

g⁰D (4.25)

Θˆ_T = U_T²

g⁰D (4.26)

Figure 4.17 Relations with the clear liquid domain Mobility parameters ˆΘC of the outcome variables on linear scale: a C_S; b ˆU_S; c ˆA_B; d ˆQ_S; symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.18 Relations with the clear liquid domain Mobility parameters ˆΘ_C of the outcome variables on log-log scale: a CS; b ˆUS; c ˆAB; d ˆQS; symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.19 Relations with the clear liquid domain Mobility parameters ˆΘ_T of the outcome variables in linear scale: a C_S; b ˆU_S; c ˆA_B; d ˆQ_S; Symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.20 Relations with the clear liquid domain Mobility parameters ˆΘ_T of the outcome variables in log-log scale: a, C_S; b, ˆU_S; c, ˆA_B; d, ˆQ_S; symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

The range of Mobility parameters ˆΘC are from 0.25 to 15, for all case, the data collapse for all four outcome variables, and for all four outcome variables, the SWRF data shows the lowest responds to the Mobility parameter, the lines plotted in Figures represent the relations of Eq. (4.23) to (4.24), the exponents we assumed are in agreement with data.

For Mobility parameters ˆΘ_T,data shows less correlated, in Figure 4.20cd the slopes of the data seem steeper than we assumed, while in Figure 4.20a the slope of data seems more gentle.

For the dynamic similarity, we chose the classical Shields number to illustrated the bed-load transport variables with shear stress, Shields number can be expressed in the following equation

ˆ

τ_B = H_FS

g⁰D (4.27)

where H = A_F/B represents the flow depth, to represent the 3 dimensional flow doamin for the RWRF case, we also chose the Shield number expressed by the perimeter

ˆ

τ_P = A_F P

S

g⁰D (4.28)

where P is the perimeter which can be obtained from the flow domain, the outcome variables ˆA_B and ˆQ_S need to be modified by replacing B with P .

Aˆ_BP = AB

P D (4.29)

Qˆ_SP = Q_S

Ppg⁰D³ (4.30)

and we assumed the same relationship for dimensionless bed-load transport rate and Shields number:

Qˆ_S ∝ ˆτ^3/2 (4.31)

the relationship is in the same form as the empirical law of Meyer-Peter and M¨uller.

(1948), and was compared with the experiments and verified by Capart and Fraccarollo (2011). The outcome variables scale with ˆτ as below, the relations with Shields number ˆ

τB and ˆτP of the outcome variables were plotted in Figure 4.21 and Figure 4.23 on linear scale, in Figure 4.22 and Figure 4.24 on log-log scale.

C_S ∝ ˆτ^1/2 (4.32)

Uˆ_S ∝ ˆτ^1/2 (4.33)

Aˆ_B ∝ ˆτ^1/2 (4.34)

Figure 4.21 Relations with Shields number ˆτ_B of the outcome variables on linear scale: a, C_S; b, ˆU_S; c, ˆA_B; d, ˆQ_S; Symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; solid lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.22 Relations with Shields number ˆτB of the outcome variables on log-log scale: a C_S; b ˆU_S; c ˆA_B; d ˆQ_S; symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.23 Relations with Shields number ˆτ_P of the outcome variables on linear scale: a C_S; b ˆU_S; c ˆA_B; d ˆQ_S; symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

Figure 4.24 Relations with Shields number ˆτP of the outcome variables on log-log scale:

a, C_S; b, ˆU_S; c, ˆA_BP; d, ˆQ_SP; Symbol definitions: red for SWSF, green  for SWRF, and blue 4 for RWRF; lines: from a to c fitting results for β = 1/2, d fitting result for β = 3/2.

The data show better correlation with the Shields numbers than with Mobility numbers.

In Figure 4.21a and 4.23a it shows that C_S of SWRF case and RWRF case collapse to-gether, and for the SWSF case the value are larger. In Figure 4.21 b and 4.23 b the value of ˆU_S collapse for all three cases, indicating that the three types of channel boundary we adopted would not alter the relation between ˆU_S and ˆQ_S or ˆQ_SP. For the transport area Aˆ_B or ˆA_BP, the RWRF data show the lowest value, it represents the influence of boundary roughness. As the combination of the C_S, ˆU_S and ˆA_B results, SWRF data and RWRF data collapse in the relation of dimensionless bed-load transport rate and Shields number, and for SWSF case the data deviate from the other cases and show higher bed-load trans-port rate, from a to c, we can say the primary variable that dominates the deviation is C_S.

4.4 Conclusion

In this Chapter, we investigate the relationship of bed-load transport rate by the flow maps, we used volumetric solid and total discharge to cut different domains of transport, and integrated the different physical quantities over those domains to deduce these map to the factors called outcome variables for three channel boundaries. It shows that the

outcome variables, C_S, U_S and A_B were scaled with bed-load transport rate Q_S 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 their relationship with both Mobility parameter and Shields number having the exponent of 0.5, and for the dimensionless discharge ˆQ_S, the exponent is about 1.5.

Chapter 5

Internal stresses and drag in turbulent bed-load ¹

In this Chapter, we considered the nearly two-dimensional experiments SWSF and SWRF for further analysis. The objective is to establish the momentum balance equations for solid and liquid phase by the measured data. First we used the phase-averaging maps ob-tained in Chapter 3 to calculate the vertical profiles for both phases by depth integration, then recovered each terms in the two dimensional momentum equations of steady state, but non-uniform flow, These derivations would be introduced in Section 5.1. In Section 5.2 we tested the stresses relations and drag force relation using the deduces stresses profiles, which are the granular pressure PS, liquid shear stress τL, the solid shear stress τ_S and the drag force f_D. For shear stresses and granular pressure, we normalized these data and compared with the predictions of kinetic theory (Jenkins and Hanes, 1998). For drag force, we adopt the empirical relation proposed by Di Felice (1994), to verify the empirical law with the materials we used, a series of seepage test were performed, and we modified and verified this empirical law to be applied to the drag force in turbulent bed-load.