Consistent lattice Boltzmann equations for phase transitions

Summary This paper presents a framework for incorporating arbitrary implicit multistep schemes into the lattice Boltzmann method. While the temporal discretization of the lattice Boltzmann equation is usually derived using a second‐order trapezoidal rule, it appears natural to augment the time discretization by using multistep methods. The effect of incorporating multistep methods into the lattice Boltzmann method is studied in terms of accuracy and stability. Numerical tests for the third‐order accurate Adams‐Moulton method and the second‐order backward differentiation formula show that the temporal order of the method can be increased when the stability properties of multistep methods are considered in accordance with the second Dahlquist barrier.

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“…When the coefficients of the respective column of Table are implemented into Equation , the two‐step AM3 method is written as

f^{[+ 1]} = f^{[0]} + \frac{13 δ_{t}}{12} {(I - \frac{5 δ_{t}}{12} R)}^{- 1} Ω (f^{[0]}) - \frac{δ_{t}}{12} {(I - \frac{5 δ_{t}}{12} R)}^{- 1} Ω (f^{[- 1]}) .

For the BGK collision model, and when taking Equation into account, this AM3‐LBM simplifies to

f^{false[+ 1 false]} = f^{false[0 false]} - \frac{1}{τ_{0}} ((), f^{false[0 false]} - f^{eq, false[0 false]}) + \frac{1}{τ_{- 1}} ((), f^{false[- 1 false]} - f^{eq, false[- 1 false]})

with the relaxation times

τ_{0} = \frac{12}{13} \frac{λ}{δ_{t}} + \frac{5}{13}

and

τ_{- 1} = 12 \frac{λ}{δ_{t}} + 5

, as in the work of Siebert et al…”

Section: Methodsmentioning

confidence: 99%

Multistep lattice Boltzmann methods: Theory and applications

Wilde

Krämer

Küllmer

et al. 2019

Numerical Methods in Fluids

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“…The Maxwell equal-area construction can be derived from the above pressure tensor [251]. It is obvious that Eq.…”

Section: The Mechanical Stability Conditionmentioning

confidence: 99%

Lattice Boltzmann methods for multiphase flow and phase-change heat transfer

Luo

Kang

et al. 2016

Progress in Energy and Combustion Science

805

373

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Over the past few decades, tremendous progress has been made in the development of particle-based discrete simulation methods versus the conventional continuum-based methods. In particular, the lattice Boltzmann (LB) method has evolved from a theoretical novelty to a ubiquitous, versatile and powerful computational methodology for both fundamental research and engineering applications. It is a kinetic-based mesoscopic approach that bridges the microscales and macroscales, which offers distinctive advantages in simulation fidelity and computational efficiency. Applications of the LB method are now found in a wide range of disciplines including physics, chemistry, materials, biomedicine and various branches of engineering. The present work provides a comprehensive review of the LB method for thermofluids and energy applications, focusing on multiphase flows, thermal flows and thermal multiphase flows with phase change. The review first covers the theoretical framework of the LB method, revealing certain inconsistencies and defects as well as common features of multiphase and thermal LB models. Recent developments in improving the thermodynamic and hydrodynamic consistency, reducing spurious currents, enhancing the numerical stability, etc., are highlighted. These efforts have put the LB method on a firmer theoretical foundation with enhanced LB models that can achieve larger liquid-gas density ratio, higher Reynolds number and flexible surface tension. Examples of applications are provided in fuel cells and batteries, droplet collision, boiling heat transfer and evaporation, and energy storage. Finally, further developments and future prospect of the LB method are outlined for thermofluids and energy applications.3

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“…As an important remark, note that even though we are not considering the high-order approximations in the advective term of the kinetic equation, the errors due to the possible thermodynamic inconsistencies of the model as studied by Wagner in 2006 [11] and Siebert et al in 2014 [12] become more representative on systems with phase-segregation processes, but less influential when the system starts from two separated phases, as in our case. Even more, the inclusion of an index function actually generates a sharp interface, and our simulations show a uniform pressure at both sides of the interface, solving the two main drawbacks for thermodynamic inconsistency pointed out by Wagner; a thorough analysis of the correct retrieving of the coexistence curves, as pointed out by Siebert et al, is let for future work.…”

Section: Model Descriptionmentioning

confidence: 84%

“…Now, from Eq. (9) and the definition of ψ , it is possible to write p = ψ + ρ R T (11) or, equivalently ψ = p − ρ R T (12) then, the potential ψ actually corresponds to the deviation of the pressure compared to the value it would take if the system were an ideal gas (p ideal = ρ R T ). Eq.…”

Section: Model Descriptionmentioning

confidence: 99%

Study of hydrodynamic instabilities with a multiphase lattice Boltzmann model

Velasco

Muñoz

2015

Comptes Rendus. Mécanique

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Rayleigh-Taylor and Kelvin-Helmholtz hydrodynamic instabilities are frequent in many natural and industrial processes, but their numerical simulation is not an easy challenge. This work simulates both instabilities by using a lattice Boltzmann model on multiphase fluids at a liquid-vapour interface, instead of multicomponent systems like the oil-water one. The model, proposed by He, Chen and Zhang (1999) [1] was modified to increase the precision by computing the pressure gradients with a higher order, as proposed by McCracken and Abraham (2005) [2]. The resulting model correctly simulates both instabilities by using almost the same parameter set. It also reproduces the relation γ ∝ √ A between the growing rate γ of the Rayleigh-Taylor instability and the relative density difference between the fluids (known as the Atwood number A), but including also deviations observed in experiments at low density differences. The results show that the implemented model is a useful tool for the study of hydrodynamic instabilities, drawing a sharp interface and exhibiting numerical stability for moderately high Reynolds numbers.

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Consistent lattice Boltzmann equations for phase transitions

Cited by 26 publications

References 40 publications

Multistep lattice Boltzmann methods: Theory and applications

Multistep lattice Boltzmann methods: Theory and applications

Lattice Boltzmann methods for multiphase flow and phase-change heat transfer

Study of hydrodynamic instabilities with a multiphase lattice Boltzmann model

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