SIMULATION OF SEDIMENT DEPOSITION IN A CAVITY WITH FREE SURFACE*

ZHANG Xiao-feng

State Key Laboratory of Water Resource and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: Zhangxfwuhee@263.net

CUI Zhan-feng

Yangtze River Scientific Research Institute, Wuhan 430010, China

LU Xin-hua

State Key Laboratory of Water Resource and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

SIMULATION OF SEDIMENT DEPOSITION IN A CAVITY WITH FREE SURFACE*

ZHANG Xiao-feng

State Key Laboratory of Water Resource and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: Zhangxfwuhee@263.net

CUI Zhan-feng

Yangtze River Scientific Research Institute, Wuhan 430010, China

LU Xin-hua

State Key Laboratory of Water Resource and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

(Received May 24, 2009, Revised July 27, 2010)

Studies of the flow and sediment movement in a cavity with free surface were mostly limited to physical modeling experiments. In this study, the sediment movement is characterized in detail using a 3-D turbulent numerical model. To close the Reynolds equations, the standard k-ε model is employed. The VOF method is adopted to capture the time varying free surface and the porosity method is introduced to deal with the irregular boundary and the varying bed deformation. The computation results agree well with the experimental data in major aspects such as the vertical distribution of the sediment concentration and the deposition topography in the cavity. The comparisons show that this model can well predict the flow structure and the sediment movement and also the river bed deformation in a cavity.

3-D turbulent flow, cavity flow, free surface, VOF, porosity, sediment transport

1. Introduction

The simulation of the lid-driven flow inside an enclosed cavity is a very hot issue in computational fluid dynamics. Related studies can be classified into four categories: (1) the evaluation of numerical methods and the verification of computer codes[1-3], (2) steady solutions of a driven cavity at moderate or high Reynolds numbers[3-4], (3) the bifurcation of the flow in a cavity from a steady pattern to an unsteady pattern[5-7], (4) the natural convection flow in an enclosed cavity[8-11]. Nearly all these studies are restricted to the vertical 2-D steady flow in andepth-averaged flow and sediment mathematical model for cavity flows with free surface. It was found that the flow circulation has an asymmetric and non-closed structure, and the characteristics of the flow circulation determine the concentration of the sediment and the deposition in the cavity. His study involves many limitations because of the very complicated 3-D nature of the structures of flow and sediment. The 2-D model can not be used to capture the circulation in the vertical direction, which would change the spatial distribution of the sediment concentration and the deposition in the cavity, from the free surface to the bottom, as can be seen later in the simulation results in this article.

Besides the few studies of numerical simulation, valuable results were obtained through physical models. By using a generalized model test, Xu[13]made a thorough experimental observation of the velocity field of the circulating flow in a cavity, carried out a preliminary theoretical analysis of its hydrodynamic features, and proposed an empirical estimation formula for the critical grain size of the circulating flow sediment deposition. Liu[14,15]conducted an experimental study and a theoretical analysis in detail of several aspects such as the sediment mixture near the opening of a cavity, the flow structure of the circulating flow, the sediment startup and movement in the circulating flow region, and sediment-carrying features of the circulating flow.

In this study, a 3-D fluid-sediment mathematical model is built to study the fluid-sediment movement in a cavity with free surface. The standard k?ω turbulence model is employed to close the Reynolds equations, and the VOF method is used to accurately capture the time varying free surface, and the porosity method is adopted to treat the river bed deformation.

Fig.1(a) Schematic diagram of the lid-driven flow in an enclosed cavity

Fig.1(b) Schematic diagram of the cavity flow with free surface

2. Governing equations and numerical methods

2.1Governing equations

The Reynolds equations of flow are as follows

is the turbulent kinetic energy production, the widelyaccepted values of coefficients are chosen in this article:

The governing equations for the sediment movement and the river bed deformation are:

The boundaries in this simulation include the inlet and outflow of the mainstream, the free surface, the wall boundaries, and the sediment concentration near the bottom. At the inlet, the velocity, the turbulent kinetic energy, the diffusion rate and the sediment concentration distribution are given, and at the outflow, a zero gradient boundary condition is imposed. At the walls and the riverbed, the wall function is used, as for u( v, w):

where u?=is the friction velocity,z0refers to the elevation where the velocity is zero, z0= ks/30,ksdepends on the riverbed: for a smooth bed without sand wave, ksis taken as d50, otherwise, the value ofksdepends on the height and length of the sand wave.

To deal with irregular boundaries and river bed deformations, the porosity method[17]is used. On the free surface, the VOF model is used to determine the time varying surface level. The method used in Ref.[18] is used here to obtain the sediment concentration near the bottom.

3. Simulation of sediment movement and deposition in the cavity

Xia[19]conducted a series of experiments to study the fluid-sediment in a cavity with free surface. In the experiments, the main channel is 16 m long and 1.5 m wide with a longitudinal slope of 1/2 000. The cavity is 1m long in the X-direction and 5 m long in the Y-direction, which is perpendicular with the main channel. The sediment particles have a median diameter d50=0.0165 mm and d90=0.0546 mm, with density of 2 650 kg/m3. The gradation is shown in Fig.2. Two cases are chosen for this simulation with parameters as shown in Table 1.

Fig.2 Sediment gradation

Table 1 Parameters in experiments

The simulated region covers the whole area with a dimension of 16 m×6.5 m×0.3 m, and a 160×65×30 mesh is used for the study.

3.1 Flow structure and spatial distribution of concentration

Figure 3(a) shows the flow field and the spatial distribution of the concentration of sediment on X-Y plane near the surface and z =0.03m. The mainstream flows into the cavity near the downstream of the opening of the cavity due to the deflecting flowat the upstream corner. It can be clearly seen that there are two big vortexes with one near the opening of the cavity and the other at the end of the cavity formed by the solid wall. These two vortexes entrain a great amount of sediment into the cavity from the mainstream.

Fig.3(a) Flow field and spatial distribution of concentration on X-Y plane near the surface

Figure 3(b) shows the distribution of the turbulent viscositytμ on X-Y plane near the surface. There is a high turbulent viscosity region at the downstream of the opening of the cavity, which means a heavy turbulent mixing of sediment in this area. With respect to the flow field and the turbulent mixing, a high concentration range can be seen in the downstream range near the opening in Figs.3(a) and 4(a).

Fig.3(b) Turbulent viscositytμ on X-Y plane near the surface

Figure 4 shows the flow field and the spatial distribution of the concentration of sediment, and the distribution of turbulent viscosity on X-Y plane at z =0.03m. The basic feature is similar with that near the surface but with two main differences: (1) the concentration of sediment is higher than that near the surface, as is consistent with the well known law of the sediment concentration distribution: “upper portion is diluted and lower portion is concentrated”, (2) the turbulent viscosity is very small compared with that near the surface, which indicates a heavy turbulent mixing of sediment mostly at the upper part of the flow.

Fig.4(a) Flow field and spatial distribution of concentration on X-Y plane atz=0.03m

Fig.4(b) Turbulent viscosity μton X-Y plane at z =0.03m

The vertical sediment concentrations are compared with those measured in the selected 9 locations, of which I-1, II-1, III-1 are in the opening of the cavity, I-2, II-2, III-2 are in the middle part and I-3, II-3, III-3 are in the end part. The details are shown in Table 2 and Fig.5.

Table 2 Locations

Fig.5 Representative locations and the simulated area

Fig.6 Sediment concentration vs. relative depth at locations I-1, II-1, III-1 in Case 1

Fig.7 Sediment concentration vs. relative depth at locations I-2, II-2, III-2 in Case 1

Figures 6-8 show a comparison between the computed vertical sediment concentration distribution and the experimental data at the selected 9 locations. As can be seen in these figures, the simulation results generally agree well with the experimental data: (1) the vertical distribution of the sediment concentration shows a trend of increase from the top to the bottom and it is more even in the opening area than in the middle and tail areas, (2) the average sediment concentration is the highest in the opening area andthe lowest in the tail area, and is lower near the upstream side wall (I) than near the downstream side wall (III). This is related to the complicated 3-D flow field and the turbulent mixing between the cavity flow and the mainstream. Also several minor discrepancies are observed between the simulated and measured results. The computed sediment concentration is slightly smaller than the measured value, especially near the upstream side wall (I). On the whole, the simulated distribution of the sediment concentration is more even than the measured one, which may be due to the presence of the density current flow in the experiment. The simulation of such density current flow is not very accurate in our current model.

Fig.8 Sediment concentration vs. relative depth at locations I-3, II-3, III-3 in Case 1

3.2 Deposit characteristics in the cavity

Figures 9 and 10 show the deposit topographic map after 4 h’s running in Case 1 and Case 2, respectively. It is shown that the sediment deposition has an irregular cone shape, which is resulted from the circulation and the suspended sediment movement. The maximum sediment deposition usually occurs along the downstream side wall. Heavy sediment deposition occurs in the upstream side wall region near the opening area, and the downstream side wall region with exception of the opening area due to a sharp decrease of flow velocity near the interface. Comparing Case 1 and Case 2, the sediment concentration increases 3.6 times from 2 kg/m3to 7.2 kg/m3, but the sediment deposition increases 5 times accordingly, indicating a strong influence of sediment concentration in the mainstream on the sediment deposition in the cavity.

Fig.9 Contour of deposit bathymetry in Case 1 (m)

Fig.10 Contour of deposit bathymetry in Case 2 (m)

Fig.11 Deposit height vs. Y-coordinate at Section I in Case 1

Figures 11-13 show the deposit height vs. Y-coordinate in 4 h’s running in Case 1 at Sections I, II and III, respectively. As shown in Figs.11-13, the simulated morphology of sediment deposition in cross-sections agrees well with the measured one.

Fig.12 Deposit height vs. Y-coordinate at Section II in Case 1

Fig.13 Deposit height vs. Y-coordinate at Section III in Case 1

Figure 14 shows the deposition per unit area in the cavity vs. time in Case 1. Because the carrying capacity of the sediment in the cavity is pretty small, one sees a sustained deposition. Unlike the sediment transport in sediment laden flows in natural rivers, the increase of the sediment deposition in the cavity is nearly linear during all running time.

Fig.14 Deposit per unit area in the cavity vs. time in Case 1

4.Conclusions

In this study a 3-D turbulence model is built to simulate the fluid-sediment movement in a cavity with free surface. To capture the time varying free surface, the VOF method is used. The porosity method is employed to treat the irregular boundary and the varying bed deformation. The following conclusions are drawn:

(1) The mainstream flows into the cavity near the downstream of the opening of the cavity due to the deflecting flow at the upstream corner. A heavy mixing of sediment is found near the downstream of the opening. With respect to the flow field and the turbulent mixing, there is a high concentration range in the downstream range near the opening.

(2) The vertical distribution of sediment concentration in the cavity sees a general trend of increase from the top to the bottom, which is most prominent in the middle and tail areas of the cavity as compared to the relatively even distribution in the opening area. The average concentration of sediment in the cavity is the highest in the opening area and the lowest in the tail area, and higher in the downstream side wall than in the upper side wall.

(3) The sediment deposition in the cavity is an accumulative process and the sediment concentration of the mainstream is the main factor determining the deposition in the cavity.

Acknowledgement

This worked was supported by the Ph. D. Independent Research Fund of Wuhan University (Grant No. 20102060101000064).

[1] BARRAGY E., CAREY G. F. Stream function-vorticity driven cavity solutions using p finite elements[J]. Computers and Fluids, 1997, 26(5): 453-468.

[2] CHEN Ying, LU Chuan-jing. A homogenousequilibrium-model based numerical code for cavitation flows and evaluation by computation cases[J]. Journal of Hydrodynamics, 2008, 20(2): 186-194.

[3] ERTURK E., CORKE T. C. and GOKCOL C. Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers[J]. International Journal for Numerical Methods in Fluids, 2005, 48(7): 747-774.

[4] ERTURK E., GOKCOL C. Fourth order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers[J]. International Journal for Numerical Methods in Fluids, 2006, 50(4): 421-436.

[5] FORTIN A., JARDAK M. and GERVAIS J. J. et al. Localization of Hopf bifurcations in fluid flow problems[J]. International Journal for Numerical Methods in Fluids, 1997, 24(11): 1185-1210.

[6] SAHIN M., OWENS R. G. A novel fully-implicit finite volume method applied to the lid-driven cavity problem. Parts I and II[J]. International Journal for Numerical Methods in Fluids, 2003, 42(1): 79-88.

[7] TIESINGA G., WUBS F. W. and VELDMAN A. E. P. Bifurcation analysis of incompressible flow in a driven cavity by the Newton–Picard method[J]. Journal of Computational and Applied Mathematics, 2002, 140: 751-772.

[8] D’ORAZIO A., CORCIONE M. and CELATA G. P. Application to natural convection enclosed flows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary condition[J]. International Journal of Thermal Sciences, 2004, 43(6): 575-586.

[9] AYDIN O., UNAL A. and AYHAN T. Numericalsolutions for buoyancy-driven flow in a 2-D square enclosure heated from one side and cooled from above[C]. Proceedings of the Advances in Computational Heat Transfer Symposium. New York: Begell House, 1997, 387-394.

[10] GANZAROLLI M. M., MILANEZ L. F. Natural convection in rectangular enclosures heated from below and symmetrically cooled from the sides[J]. Inter. J. Heat Mass Transfer, 1995, 38(6): 1063-1073.

[11] AYDIN O., UNAL A. and AYHAN T. Natural convection in rectangular enclosures heated from one side and cooled from the ceiling[J]. Inter. J. Heat Mass Transfer, 1999, 42(13): 2345-2355.

[12] DONG Yao-hua. Calculation and analysis on flow and sediment transport in cavity recirculation[J]. Journal of Sediment Research, 1999, (2): 34-39(in Chinese).

[13] XU Jian-yi. Experiment research of sediment in cecum[D]. Master Thesis, Wuhan: Wuhan University, 1984(in Chinese).

[14] LIU Qing-quan. A study of sediment transport from the equations of turbulent two-phase flows -A discussion on diffusion theory and diffusion[J]. Journal of Hydraulic Engineering, 1995, (4): 68-76(in Chinese).

[15] LIU Qing-quan. A study on sediment diffusion in turbulent mixing region at the mouth of cecum reach[J]. Journal of Sediment Research, 1995, (2): 11-21(in Chinese).

[16] TAO Wen-quan. Numerical heat transfer[M]. Xi’an: Xi’an Jiangtong University Press, 2005(in Chinese).

[17] CUI Zhan-feng, ZHANG Xiao-feng. Flow and sediment simulation around spur dike with free surface using 3-D turbulent model[J]. Journal of Hydrodynamics, Ser. B, 2006, 18(3): 237-244.

[18] XIA Yun-feng. Research on application of 3D hydrodynamics, sediment transport model with non-staggered curvilinear grid for tidal rivers[D]. Ph. D. Thesis, Nanjing: Hohai University, 2003(in Chinese).

[19] XIA Ren-jie. Research of muddy water invasion at the mouth of cecum reaches[D]. Master Thesis, Nanjing: Hohai University, 1982(in Chinese).

10.1016/S1001-6058(09)60096-2

* Project supported by the National Natural Science Foundation of China (Grant No. 51079104).

Biography: ZHANG Xiao-feng (1962-), Male, Ph. D., Professor

LU Xin-hua, E-mail: xhlu@yahoo.cn