vapor for multi-phase flows

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Contents lists available atScienceDirect

Physica A

journal homepage:www.elsevier.com/locate/physa

Numerical investigation of the pseudopotential lattice Boltzmann modeling of liquid–vapor for multi-phase flows

Maedeh Nemati

^a

, Ali Reza Shateri Najaf Abady

^b

, Davood Toghraie

^a

, Arash Karimipour

^c,

*

aDepartment of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran

bMechanical Engineering Department, Engineering Faculty of Shahrekord University, Shahrekord, Iran

cDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

h i g h l i g h t s

• Numerical investigation of pseudo-potential lattice Boltzmann method.

• Liquid–vapor simulation for multi-phase flows in LBM.

• Effects of cubic equation of state on the density ratio and velocity.

a r t i c l e i n f o

Article history:

Received 16 November 2016 Received in revised form 14 May 2017 Available online 5 August 2017

Keywords:

Lattice Boltzmann method Pseudopotential model Equation of state

Single-component multi-phase

a b s t r a c t

The incorporation of different equations of state into single-component multiphase lattice Boltzmann model is considered in this paper. The original pseudopotential model is first detailed, and several cubic equations of state, the Redlich–Kwong, Redlich–Kwong–Soave, and Peng–Robinson are then incorporated into the lattice Boltzmann model. A comparison of the numerical simulation achievements on the basis of density ratios and spurious currents is used for presentation of the details of phase separation in these non-ideal single-component systems. The paper demonstrates that the scheme for the inter-particle interaction force term as well as the force term incorporation method matters to achieve more accurate and stable results. The velocity shifting method is demonstrated as the force term incorporation method, among many, with accuracy and stability results. Kupershtokh scheme also makes it possible to achieve large density ratio (up to 10⁴) and to reproduce the coexistence curve with high accuracy. Significant reduction of the spurious currents at vapor–liquid interface is another observation. High-density ratio and spurious current reduction resulted from the Redlich–Kwong–Soave and Peng–Robinson EOSs, in higher accordance with the Maxwell construction results.

1. Introduction 1.1. Literature review

Multi-phase fluid flows can be found in a variety of systems including natural, scientific, and engineering ones. Scientific and industrial applications are frequently faced with the problem of simulating multi-phase flows, considered for decades

*

Corresponding author.

E-mail addresses:[email protected](M. Nemati),[email protected](A.R. Shateri Najaf Abady),[email protected](D. Toghraie), [email protected],[email protected],[email protected](A. Karimipour).

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66 M. Nemati et al. / Physica A 489 (2018) 65–77

as a challenging task. In the late 1980s, a new mesoscopic method known as the lattice Boltzmann method (LBM) was proposed. Based on the discrete kinetic theory, it has proven a powerful tool for numerical simulation and investigation, covering a wide range of complex flows: porous flows, thermal flows, reactive transport, turbulence flows, and multi phase flows [1]. As a mesoscopic method, the LBM intermediates between the microscopic molecular dynamics and macroscopic fluid dynamics. The greatest ability of LBM is solving conservation equations like Navier–Stokes in the bulk flow. Never the less, the mesoscopic nature of the method makes it feasible to include the microscopic dynamics at fluid–fluid and fluid–solid interfaces [2]. There are a number of ways in which a system can be referred to as a multi-phase one. It may consist of the same component with different phases, like a liquid water and water vapor system. There may be multiple components with large density ratios between them, like a liquid water and air system. Alternatively, multiple components may exist with a small density contrast, like a liquid water and oil system [3]. During the past two decades, the LBM community has experienced the development of several multiphase models. Gunstensen et al. [4] proposed the first one. Called Rothman–Keller (RK) LB model, it separates and models the interaction at multiphase interfaces using color gradients based on Rothman and Keller’s [5] lattice gas model. The second model, the pseudo potential model, was proposed by Shan and Chen [6–9]. In the above model, known as the S–C model, the phase transition is explained through introduction of inter-particle interactions which are shown by a pseudo-potential function. Phase separation is an automatic process, and does not require the interface to be tracked. Furthermore, any equation of state can easily be incorporated in this model, which is considered an advantage, as many industrial applications where real gases are involved crucially require that. Since the pseudo potential LB model is simple and computationally efficient, it has appealed to many users; a large number of them, however, have criticized its lack of thermodynamic consistency and large spurious currents [10]. In the free energy model, proposed by Swift et al. [11,12], phase effects are directly introduced into the collision process. For this purpose, a generalized equilibrium distribution function including a non-ideal pressure tensor term is considered. Both the pseudo-potential model [6–9] and the free energy model [11,12] are major models with sound physical bases.

More researchers have suggested the various single component multi-phase S–C models [6–9]. In their investigation, Yuan and Schaefer [13] incorporated different formulations of the pseudo-potential. According to the results, they obtained the maximum density ratio and other useful pseudo-potential model which are dependent upon the specific equation. For the state equation, the stable solutions can be obtained which is explained in terms of the large spurious currents. On the other hand, it is possible to obtain stable solutions for high density ratios of about 1000 applying Peng–Robinson (P–R) and Carnahan–Starling (C–S) states. In a new scheme proposed by Kupershtokh et al. [14], the density ratio can be obtained in a stationary case as high as 10⁷for the van der Waals equation of state as well as the modified Kaplun–Meshalkin (mKM) EOS, and 10⁹for Carnahan–Starling. There is considerable decrease in spurious velocities. Importantly, the body forces are incorporated into the lattice Boltzmann model. In order to incorporate electric forces in LBM, they proposed exact difference method (EDM) [15]. According to Kim et al. [16], in the case of large density ratios between two phases, considerable divergence can be generated in the velocity field by the incompressible multi-phase LB method. Since the derivative operator used in the method does not satisfy the differentiation-by-parts condition, the above error occurs in the incompressibility condition. As a solution to this problem, the equilibrium function was expanded using not the momentum but the velocity through formulation of a new proposal.

Incorporation of a number of equations of state, including Redlich–Kwong, Redlich–Kwong–Soave, and Peng–Robinson, on Kupershtokh [14] force has been investigated in the present paper. Shan and Chen’s initially proposed method of velocity shifting [6] was used for incorporation of forces in LBM, and the force incorporation method was used in Yuan and Schaefer model [13]. The SC single component multi-phase model is first briefly reviewed here. The results of the simulation are then presented for a comparison to be made between different EOSs. Further comparison is made between the coexistence plots from simulations and theoretical and experimental results for a number of EOSs. The final section of the paper discusses the applicability of the LBE method and perspectives in multi-phase flow simulation with a range of EOS.

1.2. Numerical procedure

A flow with two separated phases is considered which have various densities. The lower density and higher density are defined asρminandρmax. It is well known that the flow field can be simulated by Navier–Stokes equations as follows [17,18]:

∂ⁿ

∂^t + ⃗∇.( nu⃗)

=0 (1)

∂( n_u⃗)

∂^t + ⃗∇.( nu⃗u⃗)

= − ⃗∇.^P+µ∇⃗²+ ⃗F_b (2)

∂ (ϕ)

∂^t + ⃗∇.( ϕ⃗u)

=θM∇²µϕ. ⁽³⁾

µϕshows the chemical potential,θMillustrates the mobility and P represents the pressure. Body force is presented by_F⃗

b; while n andϕare introduced by the following equation [18]:

n=ρA+ρB

2 ϕ= ρA−ρB

2 . ⁽⁴⁾

The density of fluid A and density of fluid B are shown byρAandρB. They also might be the same asρminandρmaxwhich will depend on initial conditions.

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2. Model description

As mentioned, the scheme for the inter-particle interaction force term as well as the force term incorporation method matters to achieve more accuracy. Also, the kind of equation of states for describing non-ideal fluids has a great importance role. So we used Kupershtokh inter-particle interaction force [15] and velocity shifting method [6] to obtain the density ratio in a stationary case and the cubic equations of state such as Redlich–Kwong, Redlich–Kwong–Soave, and Peng–Robinson EOS for describing non-ideal fluids. In this section, LBM will first be briefly introduced [6,13,19], and several schemes available for inter-particle interaction force terms are reviewed.

2.1. The lattice Boltzmann method

Eq.(5)illustrates the equation of particle distribution functions by using the BGK approximation [20]:

f_k(^x+C_k1^t,^t+₁t)−f_k(^x,^t)=

(f_k^eq(^x,^t)−f_k(^x,^t))

τ +₁f_k (5)

wheref_k(x,^t) is the particle distribution function with velocityC_kat positionxand timet,τis the relaxation time,1^fkis the body force andf_k^eq(^x,^t)is the equilibrium distribution function which is given by:

f_k^eq=ωkρ [

1+^C^k.^u

C_s² +(^Ck.^u)² 2C_s² − ^u

2

2C_s² ]

. ⁽⁶⁾

Whereωkis the weighting coefficient, andC_sis the lattice sound speed. D2Q9 and D3Q19, respectively, are selected for 2D and 3D simulations. AsFig. 1demonstrates, the discussion here centers around two-dimensional space with the D2Q9 lattice structure. The weighting coefficientsωkfor D2Q9 lattice are given byωk=4/^{9 for k}=0;ωk=1/^{9 for k}=1−4, andωk=1/^{36 for k}=5−8. Furthermore, the discrete velocity vectors are given as in Eq.(7):

C_k=

{(⁰,⁰) ^k=0 (±1,⁰)^c, (⁰,±1)^c ^k=1,²,³,⁴

(±1,±1)^c ^k=5,⁶,⁷,⁸ ⁽⁷⁾

where c=δx/δtis the lattice speed, withδxas the lattice spacing andδtas the time spacing, both usually set to 1. Based on Eq.(8), the relaxation time specifies the kinematic viscosity:

v=C²_s(τ−¹

2)δt. ⁽⁸⁾

The following equations are used for obtaining fluid density and velocity:

ρ=

N

∑

k=₀

f_k (9)

ρ^u=

N

∑

k=₀

C_kf_k. ⁽¹⁰⁾

2.2. Inter-particle interaction forces

Phase transition is explained by the inter-particle interaction force in the pseudo-potential model. The inter-particle interaction forces will be reviewed below. Eq.(11)provides the inter-particle interaction in S–C [6] and Yuan and Schaefer approaches [13]:

F_int(^x)= −C₀9(^x)^g∇₉(^x) ⁽¹¹⁾

where9(^x)shows the effective mass, g is the inter-particle interaction power and the value of C₀is specified by the lattice structure which is C₀=6.0 at D2Q9 and D3Q19. The fluid state equation that corresponds to Eq.(11)follows:

P=ρ^Cs²+^g

2C₀9(^P) . ⁽¹²⁾

This can be rewritten as:

9(ρ)=

√ 2(

P−ρ^C²_s)

C₀g . ⁽¹³⁾

If pressure P is solved from a given equation of state, and P is substituted into Eq.(9), the effective mass will obtain.

Eq. (11)can thus be used for easy incorporation of an arbitrary formulation of state into the inter-particle interaction

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Fig. 1.Lattice structure and velocity vector of the D2Q9 model [21].

Fig. 2.Schematic of supercritical and subcritical behaviors and Maxwell construction [25]. (Vlrepresents the saturated liquid specific volume,Vgdenotes thats of saturated of vapor andPsrepresents the saturated pressure corresponding to to temperature T).

scheme. Different inter-particle interaction force terms would then be obtained from different equations of state. Eq.(11)is discretized for numerical implementation as Eq.(14):

F_int(^x)= −₉(^x)∑

x′

G( x,^x^′)

9( x^′) (

x^′−x)

(14) where G(x,^x^′) is Green function satisfying G(x,^x^′⁾ = G(x^′,x), which reflects the strength of the interaction between neighboring fluid particles. A negative G(x,^x^′) value denotes attractive force. That divides the fluid into two phases with different densities, including the liquid and vapor. Better results with higher isotropy can be obtained by the form below [22,23] for G(x,^x^′). Hence droplet shape reveals more smooth:

G( x,^x^′)

=

⎧

⎨

⎩

g₁ ⏐

⏐x−x^′⏐

⏐=1

g₂ ⏐

⏐x−x^′⏐

⏐=

√ 2 0 otherwise.

(15) In Zhang and Chen’s [24] model, suggesting a straightforward approach of incorporating an arbitrary state formulation, force term is presented as below:

F_int(^x)= −∇U= −∇( P−ρ^C²s

). ⁽¹⁶⁾

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(a) Density contours of two circular liquid droplets embedded in a vapor phase in a gravity-free free periodic field at P–R P EOS,_T^T

c = 0.^{7 and} iteration=35 at 3 different time steps.

(b) Velocity contours of two circular liquid droplets embedded in a vapor phase in a gravity-free free periodic field at P–R P EOS, _T^T

c = 0.^{7 and} iteration=35 at 3 different time steps.

Fig. 3.

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Fig. 4.Relaxation of liquid droplets with some oscillations, a:6250 iteration. b:6500 iteration. c:6750 iteration. d:7300 iteration. e:8450 iteration.

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Fig. 5.Coexistence plots from simulations with lattice Boltzmann method for P–R, R–K–S and R–K EOS. a: P–R EOS, b:R–K–S s EOS, c:R–K EOS.

Where U=P−ρ^C²_s. Direct substitution of P from the equation of state to Eq.(16)can lead to incorporation of an arbitrary equation of state. Discretization of Eq.(16)for numerical implementation obtains:

F_int(^x)= −¹ C₀

∑

x′

G( x,^x^′)

g U( x^′) (

x^′−x)

. ⁽¹⁷⁾

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Fig. 6.Maximum amount of spurious current versus density ratio at various EOS; (a) comparison between the R–K–S and R–K EOS; (b) comparison between the R–K–S and P–R–S EOS.

The body forces were incorporated in LBM by Kupershtokh et al. [14], who proposed exact difference method (EDM) to incorporate electric forces. Now force is written as Eq.(18):

F_int(^x)= ¹ α^h

[ A

N

∑

k=₁

G_k

g 9²(^x+C_k)^Ck+(¹−2A)9(^x)

N

∑

k=₁

G_k

g9(^x+C_k)^Ck

]

. ⁽¹⁸⁾

The model that is used determines coefficients G_kfor diagonal directions and numerical coefficientα. The latter is equal toα=1,α=3/^{2, and}α=3, respectively, in models D1Q3, D2Q9, and D3Q19. As for coefficients G_kfor diagonal directions, G_k = G/4 in model D2Q9 and G_k =G/2 in model D3Q19, where weighting factor A depends on the specific equation of state. The simulation results almost coincide with the theory in the cases of vdW and C–S EOSs with A=152. Moreover, h is the lattice spacing, often taken in LBE simulations as unities together with time step. A new scheme was proposed by Gong and Cheng [25] for the inter-particle interaction force term, which is written as follows:

F_int(^x)= −β^C0g∇₉(^x)−(¹−β)^C0g∇₉²(^x)

2 . ⁽¹⁹⁾

Whereβ is the weighting factor determined by the specific equation of state. Weighting factor β is specified for vdW, P–R, and S–C EOSs in Eq.(19):

F_int(^x)= −β9(^x)∑

x′

G( x,^x^′)

9( x^′) (

x^′−x)

−¹−β 2

∑

x′

G( x,^x^′)

9²( x^′) (

x^′−x)

. ⁽²⁰⁾

The ‘‘effective mass’’ is involved in Eq.(20)both in the neighboring lattices and in the local one.

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Fig. 7. Comparison of coexistence curves obtained from simulations with theoretical values obtained from Maxwell construction for: (a) R–K, (b) R–K–S, (c) P–R EOS.

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3. Proposed model performance

It should be noted that lattice unit represents all the variables in the above LBM simulations. They are related to real physical properties using the principles of reduced properties. That is, the reduced properties are the same in lattice unit and in the real unit. SinceρR = _ρ^ρ

C. X obtains the reduced density, as stated before,^ρ^Lu

ρ^Luc

= ^ρ^real

ρ^realc

can be obtained, which in turn givesρ^real= ^ρ^Lu^ρ^real^c

ρ^Luc

, where the lattice unit is represented by the superscript Lu, and the real unit by real. Similarly, the equations P^real= ^P^Lu^P^real^c

P^Luc

and T^real=^T^Lu^T^real^c

T^Luc

hold.

The following three conditions in the critical point (atρR=1 and T_R=1) should be met (dP_R

dρR

)

TC

=0 (d²P_R

dρ²_R )

TC

=0P_R=1 (21)

The Peng–Robinson (P–R) equation of state obtains from:

P= ρ^RT

1−bρ− ^aρ²α(^T, ω)

1+2bρ−b²ρ² ⁽²²⁾

where α(^T, ω) = [

1+(

0.³⁷⁴⁶⁴+1.⁵⁴²²⁶ω−0.²⁶⁹⁹²⁶ω²)( 1−

√ T)]2

, ω is the acentric factor specified by the substance under consideration, and the values of a and b are:

a=0.⁴⁵⁷²⁴^R²^T^cr²

P_cr b=0.⁰⁷⁷⁸^RT^cr

P_cr (23)

The critical properties are thus given,ρc =2.^54186,^Tc =0.^{07292 and}^Pcis 0.05957. In the present paper,ω, the acentric factor of water, is set to 0.344. The following obtains the Redlich–Kwong (R–K) equation of state:

P= ρ^RT

1−bρ− ^aρ²

√

T(¹+bρ) ⁽²⁴⁾

where

a=0.⁴²⁷⁴⁸^R²^T^cr²

P_cr b=0.⁰⁸⁶⁶⁴^RT^cr

P_cr . ⁽²⁵⁾

The critical properties are thus given,ρc =2.^78009,^Tc=0.^{19613 and}^Pcis 0.^17843.

The Redlich–Kwong–Soave (R–K–S) equation of state obtains from:

P= ρ^RT

1−bρ−^aρ²α(^T, ω)

1+bρ ⁽²⁶⁾

whereα(^T, ω)= [

1+(

0.⁴⁸⁰+1.⁵⁷⁴ω−0.¹⁷⁶ω²)( 1−

√ T

)]2

, and the values ofaandbare:

a=0.⁴²⁷⁴⁸^R²^T^cr²

P_cr b=0.⁰⁸⁶⁶⁴^RT^cr

P_cr . ⁽²⁷⁾

The critical properties are thus given,ρc =2.^78009,^Tc=0.^{08686 and}^Pcis equal to 0.^07902.

R–K, RKS, and P–R are presented as the cubic EOSs in the above EOS. In the R–K EOS, two parameters are used:T_candP_c; the RKS and P–R three-parameter EOSs include an additional parameterω, the acentric factor, presenting more flexibility in modeling various fluids. The settingsa=2/^49,^b=2/^{21, and}^R=1 hold in our simulations for all of these cubic EOSs.

If pressure P from the three EOSs is now solved and substituted into Eq.(13), the corresponding ‘‘effective mass’’ obtains.

Shifting the velocity in the equilibrium distribution can incorporate the interaction force into the model. Hence, Eq.(28) replaces the velocity in Eq.(6)[13]:

u^eq=u+τ^F

ρ ⁽²⁸⁾

where F is the body force, like the above-mentioned F_int(^x). Averaging the moments before and after the collision obtains the whole fluid velocity U as

ρ^U=ρ^u+δ^t

2F_int. ⁽²⁹⁾

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3.1. Maxwell construction of p–v curves for non-ideal state formulation

Sketch inFig. 2displays the p–v curve of a pure substance at specified temperature for any of non-ideal state formulation.

In case the substance is supercritical, that is, T is larger thanT_c, the critical temperature, the p–v curve at fixed temperature is monotonic, and every pressure value corresponds to only one specific volume or density. It is impossible in these conditions to distinguish between distinct vapor and liquid phases. If T<^Tc, that is, the substance temperature is subcritical, however, the EOS curve is not monotonic, which allows different densities or specific volumes to coexist at a unique pressure. Thus, the liquid and vapor phases are separated from one another. Maxwell construction [19] can be used for specification of the density or specific volume of saturated vapor and that of saturated liquid in the given temperature of T. In the construction, a horizontal line 1−3−2 substitutes the unphysical oscillatory part 1−4−3−5−2 . Eq.(30)expresses the basic idea in Maxwell construction:

∫ Vg Vl

PdV=P_s( V_g−V_l)

. ⁽³⁰⁾

Clearly, solving Eq.(17)iteratively can give liquid density and vapor density at a given temperature provided that the EOS is specified.

4. Simulation results and discussion 4.1. Incorporating equations of state

Now, we investigate the influences of the equation of state in the pseudo potential LB modeling of liquid–vapor flows.

Periodic boundary condition is considered along all directions as the infinite field. To have more accuracy in a suitable time, a 160×160 lattice nodes was selected for the next calculations applying multi-phase LBM with Kupershtokh [14] scheme for inter-particle interaction force terms and velocity shifting [6] method. At the steady state, two liquid droplets with a radius=20 lattice (with densityρlsurrounded by vapor (with densityρg) were generated in the central space. The density contours of two circular liquid droplets embedded in a vapor phase in a gravity-free periodic field at P–R EOS,_T^T

c =0.^{7 and} three different time steps are shown inFig. 3(a). Moreover For P–R EOS, at _T^T

c = 0.9163 and for R–K EOS, at _T^T

c = 0.⁸⁰⁸

and 0.8545, the two liquid droplets remained separated at equilibrium and their velocity fields for P–R EOS, _T^T

c =0.^{7 for} three different time steps were shown inFig. 3(b). Simulation results ofρlandρg are compared versus those of Maxwell construction in Section3.1.

After a number of oscillations, the drops should relax to an equilibrium circular shape. The oscillations of two liquid drops at different times are displayed inFig. 4. The two droplets gradually approach each other, and a collision occurs between them as a result of the spurious current produced by pseudo potential model. As the density ratio increases, so does the spurious current amplitude. This not only leads to numerical instability (or divergence in cases of strong spurious currents) but also limits the maximum density ratios that can be achieved. Furthermore, the spurious currents cannot be easily distinguished from the real flow velocities, which impeaches the accuracy of the simulated flow field [13]. There is a unit circular droplet shape at steady state. The spurious current amplitude can be controlled through selection of a proper inter particle interaction force.

4.2. Comparison of various types of state equation

Fig. 5illustrates the coexistence plots from the different EOS using Kupershtokh inter-particle interaction force [14]

and velocity shifting method [6]. The Kupershtokh inter-particle interaction force [14] with A = 0.01 for R–K EOS and A = −0.0456 for P–R and R–K–S EOS shows the most accurate achievements at less temperatures, and the simulation results virtually coincide with the theory for _T^T

C ≥0.45. The plots of the RKS and R–K EOS are close and the same acentric subject is used for R–K–S and P–R EOS which can apply a higher temperature domain than RKS EOS due to P–R EOS has a great smaller|u^s|_maxunder the same density ratio. It should be noted thatρr=1 at T_r=1 is at the critical state in the coexistence curves. Hence,ρr<^{1 and}ρr>1 denote the saturated vapor branch and the saturated liquid branch, respectively. There is also considerable improvement in stability. For these cubic EOSs, the maximum density ratio larger than 10⁴can be achieved.

The non-zero vortex-like fluid velocity in proximity to phase interface is accompanied by the spurious current. Also known as artificial current or parasitic flow, this is a problem frequently encountered in many multi-phase flow models.

Static droplet (bubble) simulation in a gravity-free fully periodic domain, as displayed inFig. 3, is the method often used for investigation of the spurious current. The droplet (bubble) and the surrounding gas (liquid) are at rest from the physical aspect, and the pressure difference inside and outside the droplet (bubble) should be balanced by the surface tension, according to the Laplace law. The balance is not precise, however, as the discretization in calculating the corresponding gradient is limited. This accounts for the artificial vortex-like velocity at the droplet (bubble) interface displayed inFig. 6.

Fig. 6gives the|u^s|_maxchanges with density ratio at various cubic EOS.ωis chosen as 0.^{344 at}^R−K−Sand P–R EOS which is the water acentric factor. The|u^s|_maxof the P–R EOS is even smaller than R–K–S and R–K EOS. The P–R EOS can provide a density ratio as much as 10⁴, with|u^s|_max =0.43785 while density ratio equals to 40188.41202 . The|u^s|_maxfor R–K EOS

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and R–K–S EOS are higher at lower density ratios. It is observed that whenT_r <⁰.65, spurious currents raise drastically in the Kupershtokh inter-particle interaction force [14] while the raise in spurious currents at smallT_r in suggested model is relatively smooth and little.

A comparison made between the coexistence plots of numerical simulations and the theoretical ones from Maxwell construction for non-ideal gases, as inFig. 7, for the purpose of evaluating these EOSs also helps.Fig. 7 demonstrates that relatively satisfactory results are obtained by all three equations of state. However, as temperatures get closes to the critical temperature (i.e. for smaller T_r > ⁰.75), simulation results of R–K EOS start to deviate a little from those of Maxwell construction. Kupershtokh inter-particle interaction force [14] provides the most accurate achievements at low temperatures of P–R and R–K–S state equations. As can be observed inFigs. 7(b) and (c), the simulated coexistence plots fit well with the theoretical amounts for R–K–S and P–R EOS. At A= −0.0456 for P–R andR−K−SEOS presents accurate achievements at low temperatures and the simulation results virtually coincide with the theory for T_r≥0.45. The necessary value of A depends only on the EOS used.

5. Concluding remarks

The present paper has investigated the force term and method of incorporating the force term individually in the single- component multi-phase LBM. According to the achieved results, they both play important roles in applying the numerical simulation of multi-phase flows. Effects of the cubic equation of state on the density ratio and spurious current in the pseudo potential LB modeling of liquid–vapor flows have then been investigated. More appropriate ability of LBM for the simulation of multi-phase flow domain was developed at present article which usually was ignored in the previous works concerned this approach [26–36]. A summary of the obtained results is presented below:

Using Kupershtokh inter-particle interaction force and velocity shifting method allows one to obtain the density ratio in a stationary case as high as 10⁴for the Redlich–Kwong, Redlich–Kwong–Soave, and Peng–Robinson EOS. The largest deviations of the density ratio at the coexistence curve from theoretical values obtained from Maxwell construction for R–K–S and P–R and R–K EOS are very little, while those for the P–R and R–K–S EOS give more suitable findings. Hence it might be said that a more suitable performance of LBE is achieved while the chosen EOS becomes more realistic. Spurious velocities are also substantially reduced. The|u^s|_maxof the P–R EOS is even smaller than R–K–S and R–K EOS. The P–R EOS can provide a density ratio as much as 10⁴, with|u^s|_max = 0.43785 when density ratio= 40188.^{41202. u}^s|_maxof R–K EOS and R–K–S EOS are higher at lower values of density ratios.

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