Application of the pseudo-potential lattice Boltzmann model for simulating interaction of moving solids with liquids

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Application of the pseudo-potential lattice Boltzmann model for simulating interaction of moving solids with liquids Soroush Fallah Kharmiani , Hojjat Khozeymeh Nezhad , Hamid Niazmand PII: DOI: Reference:

S0045-7930(19)30362-7 https://doi.org/10.1016/j.compfluid.2019.104404 CAF 104404

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Computers and Fluids

Received date: Revised date: Accepted date:

17 July 2019 25 September 2019 2 December 2019

Please cite this article as: Soroush Fallah Kharmiani , Hojjat Khozeymeh Nezhad , Hamid Niazmand , Application of the pseudo-potential lattice Boltzmann model for simulating interaction of moving solids with liquids, Computers and Fluids (2019), doi: https://doi.org/10.1016/j.compfluid.2019.104404

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Highlights    

Development of the pseudo-potential multiphase model for simulating water entry and exit problems. Formation and propagation of pressure wave is captured and discussed. The slamming coefficient agrees well with experiments. Effects of liquid viscosity, surface tension, impact velocity, and gravity are investigated.

Application of the pseudo-potential lattice Boltzmann model for simulating interaction of moving solids with liquids Soroush Fallah Kharmiani, Hojjat Khozeymeh Nezhad, *Hamid Niazmand Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Emails : [email protected], *[email protected]

Abstract Previous lattice Boltzmann model (LBM) studies on solid entry/exit problems are limited to the free-surface LB model in which effects of the surface tension and gas phase are neglected, the surface wettability cannot be adjusted, boundary conditions need to be applied on the interface, and the interface has to be tracked during time. In addition, conventional macroscopic models for simulating the phenomenon such as Volume Of Fluid (VOF) and Constrained Interpolation Profile (CIP) have the same difficulties with the interface, besides the higher computational cost and time. Therefore, for the first time in this paper, a robust pseudo-potential based multi-phase LB model is coupled with moving boundary LB schemes for simulating liquid entry/exit of solids with the circular cylinder as a selected case study without losing generality. The current model has none of the free-surface LBM limitations and is also superior over the conventional models by automatic interface capturing and lower computational cost and time. Furthermore, the integrated model is capable of simulating the phenomenon at relatively high We and Re numbers and density ratios as high as the water/air one. Formation and propagation of the pressure wave in the case of liquid entry are shown and discussed. Cavity and subsequent pinch-off and jets formations for a hydrophobic surface are also captured and quantified. Effects of the We, Re, Fr, and impact velocity on the pinchoff time and depth, and velocity of subsequent jets are investigated, plotted, and discussed in details. Results show that the pinch-off time and depth are independent of the surface tension and liquid viscosity, but are increased linearly with the impact velocity. Furthermore, the

velocity magnitude of both downward and upward jets after the pinch-off is increased with Re and We numbers and is decreased with Fr number. Keywords LBM, pseudo-potential, multi-phase, moving solid, surface tension

1. Introduction Fluid-solid interactions (FSI) are important and practical in various engineering applications, especially in the ocean and naval areas. The extension of FSI applications is wide, including water entry/exit [1], structure responses to waves [2], wave-induced flow around a cylinder [3], vortex-induced vibrations [4], landslide generated waves [5], and interaction between the fluid and hydraulic gates [6]. Among these applications, the water entry/exit has attracted the attention of many researchers due to practical and widespread applications such as hydrodynamic loading on ships, ship slamming [7], see-landing of aerial vehicles [8], and projectiles impact upon entering water [9]. In this regard, numerous experimental, analytical and numerical studies have been conducted to gain further insight into the physics of such complicated and practical problems. Many researchers investigating the physics of water entry/exit have been utilizing solid spheres. Worthington and Cole [10] in 1897 investigated air cavity formation and different splashing behaviors during the vertical entry of solid spheres into the water using the singlespark photography. After this pioneering research, other experiments were initiated and extended, such as those of Gilbarg and Anderson [11], May [12], and Abelson [13]. Surprisingly, the water entry topic has been continuing to be an open and active research area since the initial years of research performed more than a century ago. In 2007, Duez et al. [14] reported that the cavity is formed above a critical velocity which is dependent on the contact angle, liquid viscosity, and surface tension discovered from experiments on free fall impact of spheres. In 2009, Aristoff and Bush [15] observed and categorized different cavity

shapes in the free fall of hydrophobic spheres by changing the impact velocity, in the low Bond number limit, and made a theoretical analysis as well. A year later, Aristoff et al. [16] studied effects of the sphere density as well. Recently, Speirs et al. [17] experimentally studied effects of various contact angles on the cavity shape and further analyzed the critical velocity. Evaluation of the generated impact forces during spheres impact was performed by Watanabe [18] as an early experimental work in 1934. Subsequent experimental studies such as May and Woodhull [19] and Moghisi and Squire [20] followed this initial research. For more details and information on water entry characteristics and further related experimental researches, interested readers are referred to the review paper of Truscott et al. [21]. Although experimental works are very valuable, the feasibility of full-scale experiments is not usually possible because of the long-time physics to be captured, and limited apparatus. On the other hand, performing scaled-model experiments is also limited due to the lack of full analogy and limitation of the experimental equipment [22]. Besides the experimental studies, considerable analytical ones have been carried out for evaluating the water entry problems. Von Karman [23] in 1929 presented the first formula for calculating the maximum pressure on floats during landing. Wagner [24] developed the first mathematical model for the water impact based on the velocity potential and verified the Von Karman formula for the impact force. He also determined the wetted width of the wedge, by taking into account the free surface elevation. Cointe and Armand [25] and Cointe [26] developed another formula for calculating the impact force, using matched asymptotic expansions. For more information on analytical models and developments, one can refer to the review paper of Xu and Duan [27]. Analytical solutions, however, are approximate solutions, simplifying and ignoring some physics and are also limited to simple geometries [28].

Due to the limitations of experimental and analytical studies, numerical models have developed to further reveal the real physics and effective parameters of the FSI problems. Generally, numerical methods for solving these problems are classified into two categories: approximate and direct numerical simulations (DNS) [29]. Approximate methods are established based on the potential and Stokes flows, and point-particle assumptions which simplify the real physics by neglecting some important effects including wakes, viscosity, and separation and stagnation points [30]. Esmaeeli and Tryggvason [31] and Hu [32] presented a good review of the approximate methods. DNS methods solve the full flow governing equations, i.e., the Navier-Stokes equations usually using finite difference, volume and element discretization methods. VOF, Smoothed Particle Hydrodynamics (SPH) and CIP are the typical and widely used conventional DNS numerical methods for simulating FSI problems reported in the literature. Some good works using VOF are those of Zhang et al. [33] who simulated wedge entry in two dimensions, Mirzaii and Passandideh-Fard [29], investigating sphere entry in the axisymmetric coordinates, and Iranmanesh and Passandideh-Fard [34], simulating horizontal circular cylinder entry in three dimensions. In terms of the CIP method, Zhu et al. [35] simulated twodimensional water entry and exit of a circular cylinder, Yang and Qiu [28,36] investigated entry of 2D and 3D bodies, and Wen and Qiu [37] made a parallel 3D water entry simulation. Finally, using SPH Gong et al. [38] investigated cylinder entry, Shao et al. [39] simulated vertical and oblique water entries in two dimensions, and Sun et al. [40] simulated 2 and 3D cylinder entries. The conventional DNS methods, however, have some limitations such as the common problem of the relatively high computational cost and time. The SPH method is not capable of simulating different contact angles or surface wettability up to present [40], which is an important limitation because surface wettability plays an important role and changes the impact dynamics considerably. Furthermore, effects of fluid viscosity and surface tension are

neglected in the FSI simulations using SPH. In terms of VOF and CIP, the interface needs to be captured and reconstructed mathematically, adding the complexity and computational time and cost considerably. The LBM is an alternative robust DNS solver for simulating fluid flow and heat transfer. The method has some advantages over the conventional ones, including the simplicity, computational efficiency, and the kinetic nature. There are very few papers investigating FSI problems with moving boundaries using the LBM. Zarghami et al. [41] simulated wedge entry with a constant velocity at a very low impact Mach number of 0.02 using the freesurface LBM. Very recently, using the same free-surface LB model, Hao et al. [42] studied the water exit of a sphere with constant velocity. The few previous LBM studies on the phenomenon used the free-surface LB model in which only the liquid phase is considered, neglecting effects of the gas phase. Furthermore, the interface must be tracked and proper boundary conditions need to be applied to it, which are common problems for the conventional methods like VOF and CIP as well. Another limitation of the free-surface LB model is neglecting the surface tension force. However, the LBM is confirmed to be powerful for simulating multi-phase flows and interfacial dynamics due to its kinetic nature. Therefore, in this paper, a robust pseudo-potential two-phase LB model reported in the literature is coupled with moving boundary LB schemes for the first time for the DNS simulation of liquid entry and exit of a cylinder, as a case study without losing generality for the solid shape, with the capability of simulating the phenomenon at relatively high We and Re numbers and high density ratios around that of water/air. The current LB model takes advantage of automatic interface capturing, relatively low computational cost and time, and considering the gas phase and surface tension effects, simultaneously. 2. Numerical Model

The two-phase LB model used in this paper is based on the pseudo-potential one proposed by Li and Luo [43], and is almost the same as the authors previous works [44,45] in which the model is given in details. Therefore, main description of the model is presented in the following material and then implemented moving boundary LB schemes in this paper are presented. 2.1. Two-phase pseudo-potential based model As mentioned, only the main description of the two-phase model is presented in this section. The multi-relaxation-time (MRT) LB model is given by [44]: ̂

̂

∑̂

(̂

̂ )

̂ ) ̂

∑(

(1)

where fα is the particle distribution function in the α direction, and superscript *, eq, and ^ respectively show the post-collision, equilibrium, and momentum space values. I is the identity matrix and ̂ is the diagonal relaxation matrix in the momentum space. The source terms ̂ and C are the force term and surface tension adjustment term arrays, respectively. The terms of Eq. (1) are given in continue. ̂

̂

| |

̂

| |

(2)

(

∑

)

∑

(3) (4) (5)

where

is the fluid-fluid interaction force as follows: [∑

|

|

]

(6)

√

where

(7)

⁄

is the lattice speed and

.

The Carnahan-Starling equation of state (EOS) is used in this paper:

(8)

where

is the temperature. The fluid-solid interaction force is given by:

[∑

|

|

]

where S equals one for solid nodes and zero elsewhere and

(9)

is varied to achieve different

wettability. The gravity acceleration is simply given by: (10)

The force term in the momentum space is as follows:

(

| |

)

(

⁄ | |

)

⁄

̂

(11)

(

)

[

where

]

is an input parameter that adjusts the mechanical stability condition to achieve the

thermodynamic consistency, it is equal to

(

throughout this paper [46,47].

)

(

) (12)

( [

) ]

Qxx, Qxy , and Qyy are obtained using the following equation [43]:

[∑

|

| [

]

]

(13)

where κ value varies the surface tension strength, G = -1 [43,47] and

w 1  1/ 3 , w  2   1/12

are

the weights in the D2Q9 lattice model. The streaming step is performed in the velocity space as follows: ̂

(14)

2.2. moving boundary schemes The no-slip wall boundary condition is applied by using the half-way bounce-back scheme proposed by Ladd [48] as follows: ̅(

)

(

(15)

)

where ̅ is the opposite direction of i which is unknown, fluid boundary node,

is the post-collision value, xf the

the wall velocity, and cs the speed of sound (

√ . This

discrete representation of the solid curved shape becomes more accurate as the solid radius becomes larger [48]. Furthermore, as the solid boundary moves, some solid nodes become fluid ones which need to be quantified properly. According to Lallemand et al. [49] the unknown distributions of these fresh nodes can be taken as the equilibrium ones with the wall velocity and a local density, which is adopted in this paper.

3. Model Validation In this section, cylinder exit and entry respectively at

√| |

and Fr = -0.39 are

simulated and compared with corresponding numerical results of Lin [50]. where constant velocity of the cylinder,

its radius and

is the

the gravity acceleration. The

computational domain is 200*300 with symmetric boundary condition applied on the left

side, bounce-back at right and bottom sides, and free slip at the top side. Applying these boundary conditions are easy and can be found in every LBM book such as that of A.A. Mohamad [51] and Kruger et al. [52]. The relaxation times are set to s1 = s4 = s6 =1, s2 = s3 = 0.51 and s5 = s7 = 1.1 throughout the paper [45]. The parameters in the model are selected as all in LB units, leading to Fr = 0.39. The liquid and gas kinematic viscosities are respectively set to

and

, leading to

gas to liquid kinematic viscosity ratio of 5. The density ratio approximately equals 760, corresponding to θ = 0.5θc. The parameter ε = Rs/d equals 0.8, where d is the initial distance of the cylinder center from the interface. Figure 1 and Fig. 2 compare results of the current LBM simulation with those of Lin [50] for the cylinder entry and exit, respectively. As observed, there is a good agreement between the two results in both cases. The nondimensional time T, is defined as

and the axis values are made dimensionless by

dividing them by d. Notice that the images are cropped from top, left and right sides in both Lin [50] and the current simulations to have a better vision, the original domains are much larger.

𝒚 𝒅 𝒙 𝒅

𝒚 𝒅

𝒙 𝒅 Fig. 1 Cylinder entry at Fr = -0.39; (up) Lin [50], (down) current LBM.

Fig. 2 Cylinder exit at Fr = 0.39; (up) Lin [50], (down) current LBM.

4. Results and Discussion Figure 3 shows dynamics of the cylinder forced entry at

√| |

tension,

, where

is the solid diameter,

the surface

the static contact angle, and H the initial height of the liquid column. The

computational domain is 240*1000 with the same boundary conditions as those mentioned in the validation section. As observed, sometime after the impact secondary droplets are

separated from the rising sheet due to the relatively high We number of the impact. Furthermore, as

H

t* = 0.13

t* = 0.53

t* = 1.07

t* = 1.47

t* = 2

t* = 2.93

initial interface

h pinch-off point

t* = 3.6

t* = 4.27

t* = 4.93

Fig.3 Dynamics of a hydrophobic cylinder impact on a liquid film at Re = 1620, We = 1250, Fr = 4, H/Ds = 7.1, νg/ νl = 5, ρl/ρg = 760; (Left) pressure contour, (Right) velocity field.

the cylinder moves thorough the liquid, the displaced mass tends to fill the cavity behind the cylinder and a vortex is formed as a result (see velocity vectors at t* = tVs/Ds = 2.93). Finally, the pinch-off occurs and a bubble is formed behind the cylinder. As reported in the literature such as [16,29], two liquid jets moving in the opposite directions are formed immediately after the pinch-off, which is captured in the current simulation indicated with dashed circles at t* = = 4.39. In addition, as observed from the pressure contours, a pressure wave is formed immediately after the impact (t* = 0.13) due to the liquid compression, which propagates into the liquid with relatively a high speed (t* = 0.53). The high-pressure region observed on the side wall at t* = 1.07, indicated by the dashed white curve, is due to the pressure wave impact on the wall. This region is extended inward due to the wave reflection. The pressure wave finally impinges the bottom surface and reflects, making high pressure regions nearby (t* = 2). The reflected wave from the bottom impacts the high-pressure region ahead of the cylinder and the pressure becomes maximum there (t* = 2.93). The pressure wave then gradually gets weak and dissipated in the liquid. Figurse 4 shows variations of the slamming coefficient during time, for the impact condition shown in Fig. 3, compared with an experimental correlation reported in the literature [53]. The slamming coefficient is defined as follows [53]:

(16)

where Fs is the slam force, which is computed from the pressure field through multiplying a computed average pressure by the projected wetted surface, which indeed all are transient variables. The experimental correlation is extracted from a curve fit on experimental data obtained from experiments on impacts of fouled cylinders on water [53]. According to [53], the measured impact velocity is almost constant up to t* = 1. As observed from the figure, there 5 Current LBM

4.5

Experiment 4 3.5

Cs

3 2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t* Fig. 4 Comparison of slamming coefficient between the current LBM and an experimental correlation [53].

is a good agreement between two results and the maximum slamming coefficient, which is of great important happening at the time of impact, is properly predicted by the current LBM model. Figure 5 shows the non-dimensional time ( ) and depth (

) of the pinch-off

versus the We number, which is varied by changing the surface tension strength, the other non-dimensional parameters are therefore kept constant. As observed, the pinch-off time and depth are independent of the We number. Effect of the Re number, varied by changing the liquid viscosity, on

and

is also shown in Fig. 6, indicating that the pinch-off time and

depth are independent of the Re number as well. In addition, the constant time and depth of the pinch-off

Fig. 5 Effect of We number on the pinch-off time and depth at Re = 1620 and Fr = 4, νg/ νl = 5, ρl/ρg = 760.

Fig. 6 Effect of Re number on the pinch-off time and depth at We = 1250 and Fr = 4, νg/ νl = 5, ρl/ρg = 760.

are the same in Fig. 5 and Fig. 6, indicating that the surface tension and the liquid viscosity almost have no effect on the pinch-off time, depth, and also the cavity volume at the moment of pinch-off, for the forced impact considered in this paper. Effect of the Fr number, changed by the gravity acceleration, on

and

is plotted in Fig.

7, indicating that the former is increased with Fr with a decaying slope, but the latter almost remains constant because as shown in Fig. 8, the cavity volume at the time of pinch-off is

Fig. 7 Effect of Fr number on the pinch-off time and depth at Re = 1620, We = 1250, νg/ νl = 5, ρl/ρg = 760.

Fig. 8 Pressure distribution and interface position at the pinch-off time for different Fr numbers; Re = 1620, We = 1250, νg/ νl = 5, ρl/ρg = 760.

increased with Fr as well such that the pinch-off location,

, almost remains constant. Effect

of the impact velocity, expressed in terms of the Mach number for generality, on the pinchoff is shown in Fig. 9, indicating that both parameters are increased almost linearly with the impact velocity. Therefore, according to Figs. 5-9, it can be concluded that the pinch-off time is changed only with the Fr number, and impact velocity, the former with a parabolic and the

latter with a linear trend. The pinch-off depth is affected considerably only with the impact velocity with a linear trend.

Fig. 9 Effect of impact velocity (Ma number) on the pinch-off time and depth; νg/ νl = 5, ρl/ρg = 760.

In continue effect of the same parameters, We, Re, Fr, Mas, on the upward and downward jet velocities, Vj*u = Vju / Vs and Vj*d, formed immediately after the pinch-off are shown and discussed. Figure 10 and Fig. 11 respectively show effect of We and Re numbers indicating that Vj*u and absolute value of Vj*d are increased with We and Re numbers with decaying slopes. Because the surface tension and liquid viscosity which are placed in dominator of the We and Re numbers respectively are resisting forces and therefore, decreasing their value increases the liquid jets velocities, but not linearly.

Fig. 10 Effect of We number on the upward and downward jet velocities; Re = 1620, Fr = 4, νg/ νl = 5, ρl/ρg = 760.

Fig. 11 Effect of Re number on the upward and downward jet velocities; We = 1250, Fr = 4, νg/ νl = 5, ρl/ρg = 760.

Effect of the Fr number on the upward and downward jet velocities, shown in Fig. 12, is observed to be on the contrary of the We and Re numbers, i.e., the magnitude of the jet velocities is decreased with Fr, although with similar decaying slope trends. Because the jet velocities are mainly affected by the velocity of the two liquid flows moving across each other to fill the cavity formed behind the cylinder. The velocity movement of these flows is

affected by the gravity, the higher the gravity effect, the higher their velocity. Impact of these two flows creates the liquid jets and determines their velocity magnitudes as a result. Therefore, at higher Fr numbers that the gravity effect is lower, the velocity of the filling flows is decreased leading to a weaker impact inertia and consequently formation of liquid jets with lower velocity magnitudes.

Fig. 12 Effect of Fr number on the upward and downward jet velocities; Re = 1620, We = 1250, νg/ νl = 5, ρl/ρg = 760.

Effect of the cylinder impact velocity on the jet velocities is plotted in Fig. 13 which is more complicated. Increasing the impact velocity increases all We, Re, and Fr numbers. According to Figs. 10 and 11, increasing We and Re numbers increases Vj*u, while it is decreased with Fr number according to Fig. 12. Figure 13 implies that the increase in Vj*u with We and Re numbers overcomes the decrease in Vj*u with Fr number. Therefore, Vj*u is increased with the solid impact velocity with a decaying slope. Similarly, magnitude of Vj*d is increased with We and Re numbers, and decreased with Fr number as well. However, the consequent result is reduction in the magnitude of Vj*d with the impact velocity, indicating that effect of the Fr number is dominant on Vj*d and overcomes the opposite effects of We and Re numbers. Furthermore, on the contrary of the previous decaying slope trends of the jet velocities, Vj*d magnitude is decreased with a rising slope with the impact velocity.

Fig. 13 Effect of impact velocity on the upward and downward jet velocities; νg/ νl = 5, ρl/ρg = 760

5. Conclusions A pseudo-potential based MRT LB model was implemented in this paper and coupled with the moving boundary LB schemes for the first time to study the forced liquid entry/exit of solids with the circular cylinder as a case study. The current model removes the limitations and difficulties of the free-surface LB model and the conventional methods by automatic interface capturing and considering the surface tension force and the gas phase. The model is also capable of simulating the problem at relatively high We and Re numbers and highdensity ratios as high as water/air density ratio. Results of the liquid entry of a hydrophobic cylinder are summarized as follows: 

After the impact of solid on liquid, a pressure wave is formed and then propagates in the liquid with a higher velocity than the solid velocity.



The wave hits the walls and reflects afterward, and its dynamics govern the main part of the instantaneous pressure distribution.



The slamming coefficient can be properly predicted by the model.



The pinch-off time and depth are independent of the surface tension and liquid viscosity but are increased linearly with the impact velocity.



The upward and downward jet velocities are increased with We and Re numbers and are decreased with Fr number.



The upward and downward jet velocity magnitude is respectively increased and decreased with the impact velocity.

Author declaration 1. Conflict of Interest No conflict of interest exists. We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

2. Funding No funding was received for this work.

3. Intellectual Property We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property.

4. Authorship We confirm that the manuscript has been read and approved by all named authors. We confirm that the order of authors listed in the manuscript has been approved by all named authors.

5. Contact with the Editorial Office The Corresponding Author declared on the title page of the manuscript is: Prof. Hamid Niazmand This author submitted this manuscript using his account in EVISE. We understand that this Corresponding Author is the sole contact for the Editorial process (including EVISE and direct communications with the office). He is responsible for

communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that the email address shown below is accessible by the Corresponding Author, is the address to which Corresponding Author’s EVISE account is linked, and has been configured to accept email from the editorial office of Computers and fluids: [email protected]

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