Numerical simulation of sediment erosion by submerged jets using an ...

SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

Dcember 2010 Vol.53 No.12: 3324–3330 doi: 10.1007/s11431-010-4165-3

Numerical simulation of sediment erosion by submerged jets using an Eulerian model QIAN ZhongDong1*, HU XiaoQing1, HUAI WenXin1 & XUE WanYun1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Received December 14, 2009; accepted September 20, 2010

The erosion of loose beds by submerged circular impinging vertical turbulent jets is simulated using an Eulerian two-phase model which implements Euler-Euler coupled governing equations for fluid and solid phases, and a modified k-ε turbulence closure for the fluid phase. Both flow-particle and particle-particle interactions are considered in this model. The predictions of eroded bed profiles agree well with previous laboratory measurements and self-designed experiments. Analysis of the simulated results reveals that the velocity field of the jet water varies with various scouring intensities, that the scour depth and shape are mainly influenced by the driving force of the water when the density, diameter and porosity of the sand are the same, and that the porosity is an important contributor to sediment erosion. In this study, the scour depth, the height of dune and the velocity of the pore water increase with increasing porosity. sediment erosion, jet flow, Eulerian model, numerical simulation Citation:

1

Qian Z D, Hu X Q, Huai W X, et al. Numerical simulation of sediment erosion by submerged jets using an Eulerian model. Sci China Tech Sci, 2010, 53: 3324−3330, doi: 10.1007/s11431-010-4165-3

Introduction

Localized scour around hydraulic structures is a severe problem in civil engineering. Local flow acceleration, secondary flow, jet flow and vortex flow are considered to be important contributors to the localized scour. The general approach to obtaining the features of a scour hole is a hydraulic modeling test, due to the complex nature of the flows and their interaction with the sediment bed [1–7]. Experiments on the features of scour holes under the actions of jet flows were conducted by Rajaratnam et al. and their conclusions can be found in refs. [1–5]. With the development of computational fluid dynamics, the computational research on this problem has been conducted effectively. Sun et al. [8] simulated the flow field in a scour hole around a bridge pier using the VOF method for the free sur*Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2010

face, and the scour hole profile was determined based on experimental data. Deng et al. [9, 10] provided a moving mesh method based on the balance of pressure at the bottom of the scour hole to capture the water-sediment interface behavior. Roulund et al. [11] simulated the scour process allowing for automatic mesh deformation based on the moving bed theory. In the studies of Deng and Roulund, the water flow above the water-sediment interface was simulated and the scour process was determined using an empirical approach. The flow-particle and particle-particle interactions below the interface were neglected. In this paper, an Eulerian two-phase model is introduced to simulate the scour process in a loose bed by submerged jets. The jet flow above the water-sediment interface, the flow field of the pore water, and the movement of the sediment are all simulated with this model and the profile of the scour hole is determined automatically based on the balance of the interactive forces. The model is verified by experimental data and the scour mechanism is investigated. tech.scichina.com

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QIAN ZhongDong, et al.

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2 Mathematical description of Eulerian model The mathematical model used in this simulation is an Eulerian two-phase model. It assumes that the sediment-laden flow consists of sediment s and water f phases, which are separate yet form interpenetrating continua. The laws for the conservation of mass and momentum are satisfied by each phase individually. Coupling is achieved through pressure and interphasal exchange coefficients. 2.1

Governing equations

The continuity equations for both the water and sediment phases take the form ∂ (α t ρt ) + ∇ ⋅ (α t ρt vt ) = 0, ∂t

(1)

where t = s, f refer to the sediment and water, respectively, and α s , α f are the volumetric fractions of sediment and water, respectively. The momentum equations for the water and sediment phases, respectively are ∂ α f ρ f v f + ∇ ⋅ α f ρ f v f v f = −α f ∇P ∂t

(

)

(

)

(

)

(2)

+∇ ⋅τ f + α f ρ f g + K sf v s − v f , ∂ (α s ρ s vs ) + ∇ ⋅ (α s ρ s vs vs ) = −α s ∇P ∂t

(

)

−∇Ps + ∇ ⋅τ s + α s ρ s g + K fs v f − v s ,

(3)

where P is the pressure shared by the two phases and τ f and τ s are stress tensors for the water phase and sediment phases, respectively, represented by 2 ⎞ ⎛ τ s = α s μ s ∇v s + ∇v + α s ⎜ λs − μ s ⎟ ∇ ⋅ v s I , 3 ⎠ ⎝

(

t s

)

where λs is the bulk viscosity of the sediment, and μs is the shear viscosity of sediment, represented by μ s = μ s ,col +

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and the rate of dissipation are ∂ αf ρf kf +∇⋅ ∂t μt , f ⎛ = ∇ ⋅ ⎜α f ∇k f σk ⎝ + α f ρ f Πk f ,

(

)

∂ αf ρfε f +∇⋅ ∂t μt , f ⎛ = ∇ ⋅ ⎜α f ∇ε f σε ⎝

(

)

+α f

εf kf

(C

1ε

(α

f

ρfU f kf )

⎞ ⎟ + α f Gk , f − α f ρ f ε f ⎠

(α

(4) f

ρfUfε f )

⎞ ⎟ ⎠

)

Gk , f − C2ε ρ f ε f + α f ρ f Πε f .

To predict the turbulence in the solid phase, Tchen’s [13] theory on the dispersion of discrete particles in homogeneous and steady turbulent flows is used. Dispersion coefficients, correlation functions, and the turbulent kinetic energy of the solid phase are represented in terms of the characteristics of continuous turbulent motion in the fluid phase based on two time scales. The first time scale is relevant to the inertial effect acting on the particle, and is represented by ⎛ ρs + CV ⎜ ρf ⎝

τ F , sf = α s ρ f K sf−1 ⎜

⎞ ⎟. ⎟ ⎠

The second time scale is the characteristic time for correlated turbulent motion or the eddy-particle interaction time, which is written as

τ t , sf = τ t , f

1 1 + Cβ ξ 2

.

The ratio between the two time scales is written as

η sf =

τ t , sf . τ F , sf

Thus, the turbulent kinetic energy for the solid phase is

μ s ,kin + μ s ,fr , where μ s ,kin is the kinetic viscosity, μ s ,fr is

⎛ b 2 + η sf ks = k f ⎜ ⎜ 1 + η sf ⎝

the frictional viscosity, and μ s ,col is the collisional viscos-

and the eddy viscosity for the solid phase is specified by

ity [12]. 2.2

Turbulence equations

A modified k-ε model is adopted to enclose the equations for the water phase. Additional terms that consider the interfacial turbulent momentum transfer are included in the standard k-ε model. The transport equations for the turbulent kinetic energy

(5)

⎞ ⎟ ⎟ ⎠

1 ⎛2 Ds = Dt , sf + ⎜ k s − b k sf 3 ⎝3

(6)

⎞ ⎟τ F , sf . ⎠

(7)

3 Physical model and computational considerations In this section, the Eulerian two-phase model is used to

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simulate an axisymmetric submerged jet scour and a submerged plane jet scour. The numerical results are compared with experimental results. 3.1


is ρ =2699 kg/m3, the median particle size of the sand is D=2.42 mm, and the porosity is n=25%. A 2D non-uniform structured grid system with 25000 elements is used for the computational domain.

Axisymmetric submerged jet scour

This experiment was originally conducted by Aderibigbe et al. in 1996 [3] and the experimental setup is shown in Figure 1. The impinging jet nozzle was located in the center of the upper side of the box and submerged just below the water level. The impinging distance h was varied by keeping the nozzle position fixed and altering the thickness of the sand bed. Based on the axisymmetry of the experimental setup, a 2D computation is conducted, with the computational domain shown in Figure 2. The width and height of the domain are 0.6 m, based on the size of the box. Two computational series’ are performed at d=4 mm and d=8 mm, with h=116 mm and h=500 mm, respectively. For each series, the jet velocity at the nozzle is U0=2.74 m/s, the density of the water is ρ0=1000 kg/m3, the density of the sand

3.2

Submerged plane jet scour

The experiment setup for the submerged plane jet scour is shown in Figure 3. A glass flume with a length of 910 mm, a width of 300 mm and a height of 500 mm is used. The thickness of the sand bed is 90 mm and depth of water is 240 mm. Two experimental series’ were performed with the jet velocities at the nozzle of U0=1.78 m/s and U0=3.1 m/s, and impinging distances h=144 mm and h=117 mm, respectively. For each series, the width of the jet at the nozzle is d=2.5 mm, the density of water is ρ0=1000 kg/m3, the density of sand is ρ =2650 kg/m3, the median size of the sand is D=2 mm, and the porosity is n=37%. A 2D computation was conducted to match the experiment. The computational domain is shown in Figure 4. A non-uniform structured 2D grid system with 35000 elements was used. In both cases, the SIMPLE algorithm was used for discretizing the equations, a second-order upwind scheme was used for the convective term, and a central difference method was used for the diffusion term. A velocity inlet boundary condition was used for the exit at the nozzle, and a symmetry boundary condition was used for the water

Figure 1 Aderibigbe’s experimental setup [3].

Figure 3

Figure 2 Sketch of Aderibigbe’s experimental setup.

Experimental setup for plane jet scour.

Figure 4

Sketch of experimental setup.


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surface where zero normal velocity and zero normal gradients for all variables are satisfied.

4 4.1

Results and analysis Verification of the mathematical model

In the experiments by Aderibigbe and Rajaratnam, the jet velocity at the nozzle U0, the impinging distance h, the jet diameter of the nozzle d, the median particle size of the sand D, and the density were thought to be the main contributors to the jet scour. An erosion parameter was also U 0 ( d h) in their research. The erodefined by Ec = gDΔρ ρ0 sion parameter is typically used to identify features of the scour hole. In this research, the porosity n was also found to be an important contributor to the jet scour. 4.1.1 Axisymmetric submerged jet scour Two computational profiles of the scour hole are shown in Figure 5. Figure 5(a) is the weakly deflected jet scour and Figure 5(b) is the strongly deflected jet scour based on the erosion parameter definition [2]. A sketch defining the parameters of the scour profile is shown in Figure 6, where r is the radius of the scour hole, ε is the scour depth and εm is the maximum scour depth. The computational and experimental non-dimensional scour profiles are shown in Figure 7. It is shown in Figure 7 that the computational profile where Ec =0.22 agrees well with the experimental profile, but when Ec =0.47 there are some differences between the computational and experimental profiles. The computational profile shown in Figure 7 is dynamic while the experimental profile is static. The scour process when Ec =0.47 is strongly deflected jet scour, so there are many sand particles moving in the water and the dynamic scour profile is slightly different from the static profile.


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4.1.2 Submerged plane jet scour The computational and experimental non-dimensional scour profiles for the plane jet are shown in Figure 8. It can be seen that the computational profiles for Ec =0.18 and Ec = 0.38 agree with the experimental profiles. 4.2

Analysis of flow field

Figures 9 and 10 show the flow fields above the watersediment interface at equilibrium for Ec =0.22 and Ec =0.47, respectively. As the jet flows out of the nozzle, the momentum is diffused. At the bottom of the scour hole, the jet divides into two parts and flows in the opposite directions along the water-sediment interface. When both parts flow as far as the crest of the dune, the velocity reduces. The jet reduces to two symmetric flow recirculations and the locations of the vortex are different for different Ec values. For Ec =0.22, the center of the vortexes are located nearly 0.5 h below the water surface. For Ec =0.47, the two vortexes are smaller and center of the vortexes are near the crest of the dune in the vertical direction. 4.3

Movement of sediment

The theory of rock and soil mechanics states that the area immediately below the water-sediment interface is composed of sand and pore water. However, sands located at the interface are driven by both jet water and pore water. In this paper, the forces from the jet water and pore water are referred to as the driving force of water to simplify the statements below. The movement of the sand is determined by the driving force, gravity and the frictional force. The scour profile is determined by the movement of the sand. The velocity contour and velocity vector of the sand and the pore water for Ec =0.22 are shown in Figure 11. It can be seen that sand at the interface moves in the same direction as the water with the maximum velocity. Thus, it can be concluded that the driving force is dominant as compared

Figure 5 Profiles and sketches of scour hole (U0/d/D/h/Ec).

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Figure 6 Sketch of parameter definitions for scour profile.

with gravity and the frictional force. The sand in the lower part moves downward while pore water flows upward, indicating that the driving force and frictional force are not enough to balance the effect of gravity on the sand. The velocity field contains two vortexes: the sand and the pore water. The sand vortex is just below the crest of the dune

Figure 7 Non-dimensional scour profiles for axisymmetric jet.

Figure 9


while the pore water vortex is closer to the centerline of the jet. The velocity of the pore water below the crest of the dune is almost zero and sand in this position moves downward. Thus, it can be concluded that, here, the driving force is too small to balance gravity and the frictional force. The velocity contours and velocity vectors of the sand and the pore water for Ec =0.47 are shown in Figure 12. There are two vortexes in the sand and pore water velocity fields. In most of the area, the sand moves in the same direction as the pore water and the velocity of the sand is slightly smaller than the pore water velocity. In areas where the velocity of the pore water is low, such as the bottom of the scour hole and the center of the vortex, the velocity of the sand is also small. Therefore, it can be concluded that the driving force is dominant and gravity and the frictional force are negligible for strongly deflected jet scour. From the comparison of Figures 11 and 12, it can be seen that given the same density, particle size and porosity of the sand, the driving force of water is the most important factor in the movement of sand and the scour profile is determined

Figure 8 Non-dimensional scour profiles for plane jet.

(a) Velocity contour of jet flow (Ec=0.22); (b) velocity field of jet flow (Ec =0.22).


Figure 10

Figure 11

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(a) Velocity contour of jet flow (Ec =0.47); (b) velocity field of jet flow (Ec =0.47).

Velocity contours for (a) sand; (b) pore water; Ec=0.22; velocity fields for (c) sand; (d) pore water; Ec =0.22.

accordingly. The scour hole for weakly deflected jet scour is shallow and flat, and for strongly deflected jet scour the scour hole is deep and steep. In the experiment on plane jet scour, the permeability of sand was also found to be important in the scour process. The particle size, grading and porosity of the sand are directly related to the permeability of the sand. In previous research on jet scour, the size and grading of the sand were usually considered. In this paper, the effect of the porosity of the sand, n, on the scour process was also studied. Figure 13 shows the non-dimensional scour profile for Ec

=0.22 with different sand porosities, n. The scour hole for n=0.37 has a larger depth, slope, and height of the dune than that for n=0.25. With increasing porosity, the drag force of the pore water on the sand reduces and the velocity of the pore water increases. The driving force then increases and the scour process changes.

5

Conclusions 1) The Eulerian two-phase model, in which sand and

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Figure 12

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Velocity contours for (a) sand, (b) pore water; Ec =0.47. Velocity vectors for (c) sand, (d) pore water; Ec =0.47. This work was supported by the National Basic Research Program of China (“973” Program) (Grant No. 2007CB714106) and the National Natural Science Foundation of China (Grant Nos. 51079106, 10972163). The authors thank HAN Yaqiong for her help in the plane jet scour experiment. 1 2

3

4 Figure 13

Non-dimensional scour profiles with different porosities. 5

water are treated as interpenetrating continua, and both flowparticle and particle-particle interactions are considered, was used to simulate the scour process, and computational results agreed well with the experimental results. 2) The movement of sand is controlled by the driving force of the water, gravity and the frictional force. With the same density, particle size and porosity of the sand, the driving force is the most important factor for changing the scour process. 3) The movement of sand changes the scour profile. The scour hole for a weakly deflected jet scour is shallow and flat and for a strongly deflected jet scour, the scour hole is deep and steep. 4) The porosity of sand is also important in the scour process. As the porosity increases, the velocity of the pore water also increases and so do the depth and slope of the scour hole and the height of the dune.

6 7

8

9

10 11

12

13

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