Numerical Study of Forcing Mechanisms in Oscillatory Sheet Flows

 

KEFENG ZHANG and PING DONG

 

Department of Civil Engineering, The University of Dundee, Dundee DD1 4HN, UK

Tel: 0044 1382 344349, Fax: 0044 1382 344816, p.dong@dundee.ac.uk

 

 

ABSTRACT

In this paper, a detailed analysis of the stresses in the oscillatory sheet flow layer is carried out using a complete two-phase flow model. The stresses considered include the intergranular stress and the turbulent stress, both of which are believed to have significant influences on the momentum transfer in the sheet layer. Also, the various forcing terms exerted on both fluid and sediment phases are analysed. It was found that the stresses play different roles in different regions for momentum transfer and the sediment motions. In the region with high concentration the pressure gradient is balanced by the interaction force for fluid phase and the force caused by intergranular action is balanced by the interaction force and the pressure gradient for sediment phase. In the region where the concentration varies significantly the forces caused by turbulence and intergranular action are identified the major sources for the movements of fluid and sediment.

 

Keywords: two-phase flow, sheet flow, momentum transfer, turbulent stress, intergranular stress

 

INTRODUCTION

Among all the transport modes of coastal sediments the sheet flow is the most complicated and least understood because of the large sediment concentrations involved. Most mathematical models that have been developed for providing an estimate of the sediment transport rate have based on over-simplified assumptions treating the fluid/particle motions as a single-phase flow. Despite that such models are generally capable of predicting concentration profiles, the calculation of the sediment transport rate has to rely on inaccurate flow kinematics. Most importantly, these models are unreliable for evaluating the detail momentum transfer processes in the sheet flow layer because of the incomplete description of fluid and sediment kinematics. The situation has now changed with the recent development in modelling techniques based on two-phase flow approach (Asano, 1990; Li and Sawamoto, 1995; Dong and Zhang, 1998). Models of this kind can address the interaction between fluid and sediment in a fundamental way, and therefore are capable of revealing the moving mechanism of the sediment in this high concentration region. However, in all previous published studies the main focus was on the prediction of primary flow parameters such as flow kinematics and concentration, little information was available on the detailed forcing structures without which it is difficult to explain adequately the momentum transfer mechanism within and between the phases.

 

In this study, an investigation of the forcing terms for the sediment motions on a flat bed under oscillatory flow conditions is presented. The numerical model used is the complete linearised two-phase flow model developed by Dong and Zhang (1998). All major forces in the sheet flow layer such as the inter-phase forces, the forces caused by intergranular action and the turbulence and the pressure gradient exerted on fluid and sediment are examined in details.

 

THE MODEL

Based on the continuum assumption and considering uniform cohesionless sediments, the continuity and momentum equations for both sediment phase and fluid phase under two-dimensional circumstances can be written as follows (Dong and Zhang, 1998).

 

(1)

(2)

(3)

(4)

(5)

(6)

 

where x is the axis taken in the horizontal direction and z is the axis taken in the vertical direction, u, w, u_s, and w_s represent the velocities of fluid and sediment in the horizontal and vertical directions, respectively, t is time, r and r_s are the densities of fluid and sediment, C is the volumetric concentration of sediment, p is the full pressure, g is the gravitational acceleration, T_xz and T_zz are the turbulent stresses, T_sxz and T_szz are the intergranular stresses, f_x and f_z are the interaction forces per unit volume between fluid and sediment, and the prime is the fluctuation quantity.

 

Due to the space limitation the expressions for calculating the interaction force (f_x, f_z), the intergranular stress (T_sxz, T_szz) and the turbulent stress (T_xz, T_zz) are not listed here. The detail formulations and the finite difference method employed for solving the coupled equations, Eqs. (1) - (6), can be found in Dong and Zhang (1998).

 

RESULTS AND ANALYSIS

The results to be presented are based on the experiments (Test 2) by Horikawa et al (1982). The adopted parameters for the calculations are as follows: sediment diameter d=0.02 cm, density r_s=2.66 g/cm³, the main fluid velocity u=Usin(wt) with U=127.0 cm/s and w=1.75 1/s. In the following the model validation results are shown first to given an indication of the confidence level of model predictions. Then, the forcing terms exerted on both fluid and sediment predicted by the model will be examined.

 

SEDIMENT VELOCITY AND CONCENTRATION

The calculated and measured sediment velocity distributions and the concentration profiles at different flow phases are plotted in Fig. 1. It can be seen that the agreements between the calculated and measured values are fairly good for both the velocity and the concentration. Although, these results are by no means sufficient to prove the validity of the formulation of each individual component of the model, it at least indicates that the orders of magnitudes of all the major forcing terms are correctly represented, which allows a meaningful numerical investigation of these forcing terms to be carried out.

 

Fig. 1: Sediment velocity distributions and concentration profiles

 

INTERGRANULAR STRESS AND TURBULENT STRESS

Figures 2 and 3 show the phase variations of the intergranular shear stress and the turbulent stress. Clearly, both the intergranular stress and the turbulent stress vary periodically. For the intergranular stress (Fig. 2) the significant variation occurs in the range of z/d<1.25. The magnitude of the intergranular stress increases with decreasing vertical coordinate. However, the variation of the turbulent stress (Fig. 3) occurs in the range of -2.5<z/d<50, and the magnitude of the turbulent stress increases with the vertical coordinate below the elevation of z/d=1.25 and decreases with increasing the vertical coordinate upwards.

 

As the intergranular stress is of comparable magnitude as the turbulent stress near z/d=0.0, it is of interest to examine their relation in details. The phase variations of both the intergranular stress and turbulent stress in the region -2.5<z/d<5.0 are shown in Fig. 4. It reveals that in this range the intergranular stress and the turbulent stress have the same order of magnitude. Above the elevation of z/d=1.25 the magnitude of the turbulent stress is much larger than that of the intergranular stress, while the situation is reversed when z/d<-2.5.

 

Fig. 2: Phase variation of shear stress

 

Fig. 3: Phase variation of turbulent stress

 

Fig. 4: Comparison between intergranular stress and turbulent stress

 

From the above, it can be concluded that in the vicinity of the bed where the concentration is high the intergranular stress is mainly responsible for the momentum transfer and in the upper part of the sheet flow layer where the concentration is relatively lower the turbulent stress plays a greater part in the momentum transfer. However, in the very small transitional region with a height of few particle diameters near z/d=0.0, both the intergranular stress and the turbulent stress contribute to the momentum transfer.

 

FORCING TERMS EXERTED ON FLUID AND SEDIMENT

Although the information presented above concerning the vertical structures of the shear stress and turbulent stress can reveal the relative importance of these stresses, a complete understanding of the momentum transfer process would require further study of the all forcing terms that have influences on the motions of the sediment and fluid. Figure 5 shows the forces exerted on both phases, in which, and are the forces exerted on fluid by the turbulence and on the sediment by the intergranular action, are the interaction force between sediment and fluid, and and are the pressure gradients exerted on fluid and sediment, respectively.

 

 

Fig. 5: Phase variations of forces

 

From Fig. 5 it is clear that for fluid phase the pressure gradient makes the greatest contribution to the fluid movement in the region with lower sediment concentration. In the region with high concentration the turbulent force is virtually absent. The force caused by the interaction between fluid and sediment is balanced by the pressure gradient and the rate of change of the flow. It is in the region near the vertical co-ordinate z/d=0.0 where the sediment concentration varies sharply that the situation is more complicated. In this region at the phase close to wt=0p, all forces have considerable effects on the movement of fluid. However, at any other phases only the forces that are caused by turbulence and the interaction are significant.

 

For the sediment phase it can be found that in the region with very high concentration the force caused the intergranular action is balanced by the interaction force and the pressure gradient. In the region near z/d=0.0 the influence of the pressure gradient is insignificant. Both the interaction force and the force caused by intergranular action dominate the movement of the sediment.

 

CONCLUSIONS

Two-phase flow model is a useful tool for understanding the underlining physics in the sheet flow regime and for making accurate predictions for sediment motions.

 

Except for the region in the vicinity of the immovable bed with the high sediment concentration where the intergranular shear stress is mainly responsible to the momentum transfer, the turbulent stress seem to dominate the momentum transfer.

 

In the region with high concentration, the pressure gradient in the fluid phase is balanced by the interaction force while for sediment phase the balance is between the interaction force and the pressure gradient as well as intergranular force. In the region where the concentration varies significantly the forces caused by turbulence and intergranular action are responsible for the movements of both fluid and sediment.

 

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial support from Leverhulme Trust (Grant No. F/143/L) for carrying out this work.

 

REFERENCES

Asano, T. (1990), Two-phase flow model on oscillatory sheet flow, Proceedings of 22nd International Conference on Coastal Engineering, ASCE, pp. 2372-2384.

Dong, P. and Zhang, K. (1998), Two-phase flow modelling of sediment motions in oscillatory sheet flows, Under review for publication.

Horikawa, K., Watanabe, A. and Katori, S. (1982), Sediment transport under sheet flow condition, Proceedings of 18th International Conference on Coastal Engineering, ASCE, pp. 1335-1352.

Li, L and Sawamoto, M. (1995), Multi-phase model on sediment transport in sheet-flow regime under oscillatory flow, Coastal Engineering in Japan, Vol. 38, No. 2, pp. 157-178.