Numerical Simulation of Shock Wave Bubble ...

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Keywords: Ship ballast water, Shock wave, Micro bubble, Two-phase flow, Volume of Fluid method, Ghost ... 1 Conceptual diagram of shock-bubble treatment.
Proceedings of the 3rd International Symposium of Maritime Sciences Nov. 10-14, 2014, Kobe, Japan

Numerical Simulation of Shock Wave Bubble Interaction for Ballast Water Treatment Nao TANIGUCHI*, Ryuki FURUHATA**, Akira SOU*, Akihisa ABE* *Graduate School of Maritime Sciences., Kobe University, **Dept. of Maritime Management., Kobe University, 5-1-1, FukaeMinamimachi, Higashinada-ku, Kobe 658-0022, Japan

Keywords:

Ship ballast water, Shock wave, Micro bubble, Two-phase flow, Volume of Fluid method, Ghost Fluid Method

1. INTRODUCTION Transportation of marine species in ship ballast water may damage marine ecosystem. To protect marine environment, International Maritime Organization adopted Ballast Water Management Convention in 2004[1]. When the convention becomes effective, all ships must equip with a ballast water management system which can sterilize planktons and viruses in the ballast water. The most popular ballast water treatment system uses chemical processing. However, chemicals can cause environmental destruction if medication outflow occurs. In addition, chemical processing needs post-processing before returning ballast water into the sea. Abe et al. [2] proposed a new treatment system using the interaction between microbubbles and shock waves as illustrated in Fig. 1. A shock wave is introduced to induce motion and stronger shock waves. Marine bacteria’s cell wall is damaged by the shock waves. This reaction enables to sterilize planktons and viruses ecologically and effectivity. Since, the interaction between microbubbles and shock waves occurs at a small space and a short time, it is difficult to make clear the phenomena and to optimize the system only through the experiments. Shock wave

Bubble

Shock wave

Shock wave

Expansion a

Contraction

Bubble .

Bubble Marine bacteria Shock wave front

(a) Shock wave

(b) Contraction

(c) Expansion and shock wave

Fig. 1 Conceptual diagram of shock-bubble treatment system There are some numerical methods to capture and track the gas-liquid interface of a bubble, such as the Level Set (LS) method[3], the front tracking method[4], and high-order advection schemes to track a color-function[5]. However, these methods cannot accurately calculate three-dimensional long-time phenomena. The Volume of Fluid (VOF) method is a volume tracking schemes, which can track a sharp interface for a long time with mass conservation. We proposed the Non-uniform Subcell Scheme (NSS)[6], which is an advanced VOF method. To

capture and convect shock waves, the Total Variation Diminishing (TVD) scheme[7] is often used. As a method to accurately treat the jump condition at a gas-liquid interface, the Ghost Fluid Method (GFM)[8-9] has been proposed. In this study, we developed a new numerical model on compressible gas-liquid two-phase flows based on NSS, TVD and GFM, which may enable us the numerical optimization of the ballast water treatment system. The validities of the several functions of the proposed method were verified by comparing the numerical results and experimental results of various shock tube problems. And then, we calculate the interaction between a shock wave and an air bubble in water. 2. NUMERICAL MODEL 2.1 Governing equation The following local-instantaneous basic equations for compressible inviscid two-phase flow are solved. ∂U ∂t

+

∂F(U) ∂x

+

𝜕𝐺(𝑈) 𝜕𝑦

=0

(1)

where t is the time, x and y are the positions in the horizontal and vertical axes. The U, F, and G are given by 𝜌𝑢 𝜌𝑣 𝜌 2 𝜌𝑢𝑣 𝜌𝑢 + 𝑝 𝜌𝑢 U = [ ] , 𝐹 = [ 𝜌𝑢𝑣 ] , 𝐺 = [ 𝜌𝑣 2 + 𝑝 ] 𝜌𝑣 (𝐸 + 𝑝)𝑢 (𝐸 + 𝑝)𝑣 𝐸

(2)

where is density, u and v are velocities in x and y directions, respectively. p is pressure and E is total energy per unit volume. To close the above equation, the stiffened equation of state[10] is used. p = (γ − 1)ρe − γΠ

(3)

Here  is the specific heat ratio, is the constant describing fluid characteristic and e is internal energy per unit volume. For air, helium and water,  and are given as follows. Air : γ = 1.4 ,Π = 0 Helium : γ = 1.667 , Π = 0 Water : γ = 4.4 , Π = 6 × 108

For space discretization of Eq. (1) we use TVD scheme, and we use operator splitting, for time marching. 2.2 Non-uniform Subcell Scheme (NSS) In order to capture a gas-liquid interface, we use one of a high order VOF[11] scheme, Non-uniform Subcell Scheme (NSS). NSS is one of the Piecewise Linear Interface Calculation (PLIC) schemes, in which the interface gradient in a cell is taken into account. In VOF method, gas void fraction G is defined in each computational cell. If G equals to 1, the cell is full of gas. If G equals to 0, the cell is filled with water. When G takes a value between 0 and 1, there is an interface in the cell. Conceptual diagram of NSS is illustrated in Fig. 2. First, a calculation cell with an interface is divided into the convected and non-convected regions with subcells in each region. Second, the gradient of the interface is calculated by G in the surrounding cells. Third, the position of the interface is determined using an iterative method. Finally, the volume illustrated by deep blue is transported to the next cell. NSS keeps the interface sharply with a good mass conservation.

0.0

u

Liquid

∇ αG

G=0.9 1.0

0.9

0.05

1.0

0.9

0.03

Gas

Helium

Density Velocity Pressure Specific-Heat Ratio

= 1.0

Shock Wave

u = 0.0 p = 1.0 = 1.4

=0.125

u =0.0 p =0.1 =1.667

0.5

0

1.0

Fig. 4 Air-helium shock tube problem

Exact solution Hu and Khoo Present study

Velocity

Liquid

Gas 0.3

Interface Air

uΔt

Computational Domain

G=0.7

3. VERIFICATION 3.1 Shock tube problem in air-helium system To verify the function of calculating a compressible two-phase flow, we simulated an air-helium shock tube problem shown in Fig. 4[12]. As shown in Fig. 5, numerical results of velocity and density around a shock wave in an air-helium two-phase flow agree well with the exact solution and those by Hu and Khoo[12].

Transported

X

Interface

Sub-Cell

(a) Velocity

Fig. 2 Non-uniform Subcell Scheme (NSS)

Ghost Fluid 2

Void fraction

Fluid 2

Exact solution Hu and Khoo Present study

Density

2.3 Ghost Fluid Method (GFM) To treat the jump condition at a gas-liquid interface, we use Ghost Fluid Method (GFM) proposed by Fedkiw et al. [8-9] As shown in Fig. 3, when fluid 1 is calculated, ghost fluid 1 is set across the interface. GFM allows a calculation with an interface as if in a single medium domain.

Expansion wave Shock wave Contact

Fluid 2

discontinuity

Fluid 1

x Ghost Fluid 1

interface Real Fluid 2 Real Fluid 2

V: Velocity P: Pressure S: Entropy

P,V

P, V

S Real Fluid 1 Real Fluid 1 Ghost Fluid 1 Ghost Fluid 1

i-1

i

i+1

i+2

Fig. 3 Ghost Fluid Method (GFM)

X

(b) Density Fig. 5 Result of air-helium shock tube problem 3.2 Shock tube problem in water-air system To verify the function of calculating a gas-liquid two-phase flow, we simulated a water-air shock tube problem shown in Fig. 6[13]. As shown in Fig. 7, the proposed method gives a good prediction for density, velocity and pressure around a shock wave in a water-air two-phase flow agree well with the exact solution and those by Nourgaliev[13].

Interface

Water Density Velocity Pressure Specific-Heat Ratio Constant number

= 1000

u = 0.0 p = 1.0×109 = 4.4 Π = 6.0×108

Air

=50

Shock Wave

u p  Π

=0.0 =1.0×105 =1.4 = 0

0.7

0

1.0

Fig. 6 Water-air shock tube problem

the 2D computational domain is divided into 650 x 180 cells. Upper and lower boundaries are the slip walls and the left and right boundaries are continuous. Time step size t is 1.0 x 10-3. The nondimensionalized initial conditions are[15]: Pre-shocked air: Post-shocked air:

=1, u=0, v=0, p=1 =1.3764, u=0.394, v=0, p=1.5698

Density [kg/m3]

Helium Bubble:

Expansion wave

Contact discontinuity

100mm

Exact

25mm

90mm

Nourgaliev

Shock wave

High pressure

(a) Density

Helium bubble

High density room

y X

Velocity [m/s]

25mm

Shock wave

Present study

Low pressure Low density room

325mm

Fig. 8 Computational domain for helium bubble-air simulation

t = 32s

Exact Nourgaliev Present study

(b) Velocity

Pressure [Pa]

=0.1819, u=0, v=0, p=1

t = 52s

P 1.5698

t = 62s

0

t = 72s

Exact

Nourgaliev

(a) Experimental shadowgraph(14)

(b)Present method

Fig. 9 Results on a helium bubble in air Present study

(c) Pressure Fig. 7 Result of water-air shock tube problem 3.3 Helium bubble and air shock interaction To verify the validity of the 2D calculation function with a bubble, we simulated an interaction between a cylindrical helium bubble and a shock wave in the air. It has been investigated experimentally by Hass and Sturtevant[14]. Fig. 8 shows the schematic of the problem. 325mm x 90mm of

Fig. 9 (a) shows Hass and Sturtevant’s shadowgraph images, and Fig. 11 (b) the pressure contours by the present method. A part of shock wave transmitted through the right side of the bubble and a refraction wave transmitted through the left side bubble. A pressure wave is reflected bubble at t = 62 s. It was confirmed that the proposed method can predict 2D interaction of a shock wave and an interface.

4. SHOCK WAVE AND AIR BUBBLE IN WATER Finally, we simulated the interaction of shock wave and air bubble in water. Fig. 10 shows the schematic of this problem. Computational domain of 15mm x 12mm is divided into 250 x 200 cells. Upper and lower boundaries are the slip walls and the left and right boundaries are continuous. Time step size t is 2.0 x 10-8 s. The initial conditions are the same as that by Takahira and Yuasa[16]: Pre-shocked water: ρ=1000 kg/m3, u=0 m/s, v=0 m/s, p=105 Pa Post-shocked water: ρ=1324 kg/m3, u=682 m/s, v=0 m/s, p=1.9×109 Pa Air Bubble: ρ=1 kg/m3, u=0 m/s, v=0 m/s, p=105 Pa

P (Pa) t = 0.00 s

t = 0.64 s

t = 1.20 s

t = 1.68 s

t = 2.24 s

t = 2.88 s

t = 3.20 s

t = 3.44 s

t = 3.68 s

1.0×105

(a) Pressure

1.8mm 1.2mm

ρ (kg/m3)

Shock wave 12mm

High pressure High density water

X

t = 0.00 s

t = 0.64 s

t = 1.20 s

t = 1.68 s

t = 2.24 s

t = 2.88 s

t = 3.20 s

t = 3.44 s

t = 3.68 s

1323.65

3mm

Air bubble

y

1.9×109

Low pressure Low density water

1000.00

15mm

(b) Density

Fig. 10 Computational domain for air bubble in water Fig. 11 (a), (b) and (c) show pressure, density and velocity distributions respectively. The left side of the air bubble is dented by the incident shock wave. At t = 3.20 s, a liquid jet is formed and impacts the rear of the air bubble with high speed of around 2890 m/s. Then, the bubble is broken up by the liquid jet and the transmitted shock inside the air bubble interact with the rear of the air bubble and transmits into the right side of the water. The highest pressure of around 6 GPa occurs through this interaction which is about three times as high as the initial pressure. The shrinking behavior of the air bubble by the shock wave is predicted well. Some studies using Level Set (LS) method have claimed that mass conservation is not guaranteed in LS simulation. On the other hand, NSS has good mass conservation, potential to be extended to three-dimensional simulations and enables to calculate long time phenomena.

u (m/s) t = 0.00 s

t = 0.64 s

t = 1.20 s

t = 1.68 s

t = 2.24 s

t = 2.88 s

t = 3.20 s

t = 3.44 s

t = 3.68 s

3000

-500

(c) Velocity Fig. 11 Result of shock wave and air bubble in water 5. CONCLUSION In this study, we proposed a new numerical method on compressible gas-liquid two-phase flows to simulate the interaction between shock waves and microbubbles in the ballast water. The method is constructed using a TVD scheme, Ghost Fluid Method and a high-order Volume of Fluid Method. We verified the validities of the several functions of the proposed method by comparing the experiment and numerical results of various shock tube problems. Finally, we simulated the interaction between a shock wave and an air bubble in water. The result

confirmed the applicability of the proposed numerical model to the microbubble-shock wave interaction in ballast water treatment equipment. REFERENCES 1. International Maritime Organization. “Ballast Water Management”, (online), available from http://www.imo.org/OurWork/Environment/BallastWa terManagement/Pages/Default.aspx, (accessed 2014-9-21 ). 2. Abe, A., Nishio, S., Sou, A. and Mimura, H., “Application of Shock Wave Technique to Sterilization Treatment of Ship Ballast Water – Approach to Establishment of Eco-friendly Technique”, Journal of The Japan Institute of Marine Engineering, Series 488, Vol.46, No.4, (2011), pp. 72-77. 3. Osher, S. and J, Sethian. A., “Fronts Propagating with Curvature Dependent Speed Algorithm Based on Hamilton-Jacobi Formation”, Journal of Computational Physics, Vol. 79, (1988), pp. 12-49. 4. Terashima, H., “A front-tracking method with projected interface conditions for compressible multi-fluid flows (in Japanese)”, Proceedings of The 18th Symposium (ILASS-Japan), (2009), pp. 95-101. 5. Johnse, E. and Colonius, T., ”Numerical simulations of non-spherical bubble collapse”, The Journal of Fluid Mechanics, Vol. 629, (2009), pp. 231-262. 6. Hayashi, K., Sou. A. and Tomiyama, A., “A volume tracking method based on non-uniform subcells and continuum surface force model usig a local level set function”, Computational Fluid Dynamics Journal, Vol. 15, Iss.2, (2006), pp. 95-101. 7. Yee, H. C., Warming, R. F. and Harten, A., “Implicit Total Diminishing (TVD) Schemes for Steady-State Calculations”, Journal of Computational Physics, 57, (1985), pp. 327-360. 8. Fedkiw, R., Liu, X. D. and Osher, S., “A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method)”, Journal of Computational Physics, Vol. 152, (1999), pp. 457-492. 9. Fedkiw, R., “Coupling an Eulerian Fluid Calculation to a Lagrangian Solid Calculation with the Ghost Fluid Method”, Journal of Computational Physics, (2002), Vol. 17. 10. Saurel, R. and Abgrall, R., “A Simple Method for Compressible Multifluid Flows“, Journal on Scientific Computing, Vol. 21, No. 3, (1999), pp. 1115-1145. 11. Hu, X.Y. and Khoo, B.C., “An Interface Interaction Method for Compressible Multifluids“, Journal of Computational Physics, Vol. 198, (2004), pp. 35-64. 12. Hirt, C. W. and Nichols, B. D., “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries “, Journal of Computational Physics, Vol. 39, No. 1, (1981), pp. 201-225. 13. Nourgaliev, R., Dinh, N. and Theofanous, T., “Direct Numerical Simulation of Compressible Multiphase Flows: Interaction of Shock Waves with Dispersed Multimaterial Media“, 5th International Conference on Multiphase Flow, (2004), No. 494. 14. Haas, J. -F. and Sturtevant, B., “Interaction of Weak Shock Waves with Cylindrical and Spherical Gas

Inhomogeneities“, Journal of Fluid Mechanics, Vol. 181, (1987), pp. 41-76. 15. Takahira, H. and Yuasa, S., “Investigations of Numerical Methods for Compressible Two-Phase Flows with the Ghost Fluid Method (1st Report, Influence of the Numerical Diffusion on Interface Capturing) (in Japanese)”, The Japan Society of Mechanical engineers, Vol. 72, No. 723, (2006), pp. 46-54. 16. Takahira, H. and Yuasa, S., “Investigations of Numerical Methods for Compressible Two-Phase Flows with the Ghost Fluid Method (2nd Report, Improvement for Gas-Liquid Two-Phase Flows) (in Japanese)”, The Japan Society of Mechanical engineers, Vol. 72, No. 723, (2006), pp. 55-63. ACKOWLEDGMENT A part of this study was supported by the advanced research grants for the research project entitled in ‘the creation of an international marine transportation system integrating tree-principles of transportation’ offered by the Ministry of Education, Culture, Sports, Science and Technology of Japan, and by a Grant-in Aid for Scientific Research (B) (No.22360367) of the Japan Society for the Promotion Science (JSPS). NOMENCLATURE E : Total energy per unit volume e : Internal energy per unit volume P : Pressure S : Entropy t : Time u : Velocity in x direction v : Velocity in y direction x : Horizontal axis y : Vertical axis G : Void fraction for gas  : Specific heat ratio Constant describing fluid characteristic  : Density