Free-Energy-Based Discrete Unified Gas Kinetic Scheme for van der Waals Fluid

Abstract

The multiphase model based on free-energy theory has been experiencing long-term prosperity for its solid foundation and succinct implementation. To identify the main hindrance to developing a free-energy-based discrete unified gas-kinetic scheme (DUGKS), we introduced the classical lattice Boltzmann free-energy model into the DUGKS implemented with different flux reconstruction schemes. It is found that the force imbalance amplified by the reconstruction errors prevents the direct application of the free-energy model to the DUGKS. By coupling the well-balanced free-energy model with the DUGKS, the influences of the amplified force imbalance are entirely removed. Comparative results demonstrated a consistent performance of the well-balanced DUGKS despite the reconstruction schemes utilized. The capability of the DUGKS coupled with the well-balanced free-energy model was quantitatively validated by the coexisting density curves and Laplace’s law. In the quiescent droplet test, the magnitude of spurious currents is reduced to a machine accuracy of 10−15. Aside from the excellent performance of the well-balanced DUGKS in predicting steady-state multiphase flows, the spinodal decomposition test and the droplet coalescence test revealed its stability problems in dealing with transient flows. Further improvements are required on this point.

Movable and Focus-Tunable Lens Based on Electrically Controllable Liquid: A Lattice Boltzmann Study

“…In this paper, the chemical potential lattice Boltzmann method for multiphase flow is employed [ 41 , 42 , 43 , 44 , 45 , 46 ]. In van der Waals (VDW) fluids, the free energy generalization containing the gradient squared approximation can be expressed as [ 38 , 47 , 48 , 49 ] where the first term on the right-hand side of the equation is the free energy density at a temperature of T , and the second term is the contribution of the density gradient to the free energy in a non-uniform system, is the surface tension coefficient, and is the density. The calculation of the chemical potential can be based on the density and the free energy density: The free energy function determines the diagonal term of the pressure tensor: where the general expression for the fluid state equation is The well-known Peng–Robinson (PR) EOS is superior in expressing the density of the liquid phase [ 50 ]: and its chemical potential is where R is the gas constant, a is the attraction parameter, b is the volume correction parameter, and the temperature function is .…”

Unlocking the Key to Accelerating Convergence in the Discrete Velocity Method for Flows in the Near Continuous/Continuous Flow Regimes

“…To investigate the key factors that contribute to convergence acceleration in DVM calculations, this section introduces and compares three versions of DVM. In these schemes, similar to the DUGKS [27][28][29], the local discrete characteristic solution of the Boltzmann-BGK equation is employed to calculate the numerical flux, addressing the limitation associated with mesh size. This local solution is obtained by integrating Equation (1) from t n = 0 to t n + ∆t p /2 along the characteristic line and approximating the collision term using the trapezoidal rule.…”

“…These methods aim to surmount the limitations tied to mesh size and time step size in the conventional DVM by utilizing the multiscale local solution of the Boltzmann-BGK equation for calculating numerical flux. Notable examples encompass the unified gas kinetic scheme (UGKS) [24][25][26] and the discrete unified gas kinetic scheme (DUGKS) [27][28][29]. The UGKS employs a local integral solution of the Boltzmann-BGK equation in calculating the numerical flux, while the DUGKS adopts a local discrete characteristic solution.…”

Discrete Boltzmann multi-scale modelling of non-equilibrium multiphase flows

“…2019; Yang et al. 2022 b ), collapsing cavitation (Chen, Zhong & Yuan 2011; Falcucci et al. 2013; Kähler et al.…”

“…The interparticle interactions are the underlying engine behind the complex THNE features of multiphase flows. The aforementioned models and their revised versions have been applied successfully to the study of fundamental phenomena and mechanisms of multiphase flows in science and engineering, ranging from droplet evaporation (Ledesma-Aguilar, Vella & Yeomans 2014; Safari, Rahimian & Krafczyk 2014;Zarghami & Van den Akker 2017;Qin et al 2019;Fei et al 2022) to droplet deformation, breakup, splashing and coalescence (Wagner, Wilson & Cates 2003;Wang et al 2015a;Wang, Shu & Yang 2015b;Chen & Deng 2017;Wen et al 2017Wen et al , 2020Liu et al 2018;Liang et al 2019;Yang et al 2022b), collapsing cavitation (Chen, Zhong & Yuan 2011;Falcucci et al 2013;Kähler et al 2015;Sofonea et al 2018;Yang et al 2020Yang et al , 2022a, acoustics levitation (Zang 2020), nucleate boiling Fei et al 2020), ferrofluid and electro-hydrodynamic flows (Falcucci et al 2009;Hu, Li & Niu 2018;Liu, Chai & Shi 2019), hydrodynamic instability (Zhang et al 2001;Fakhari & Lee 2013;Liang et al 2014Liang et al , 2016aLiang, Shi & Chai 2016b;Yang, Zhong & Zhuo 2019;Tavares et al 2021), dendritic growth (Rasin et al 2005;Rojas, Takaki & Ohno 2015;Sun et al 2016a,b), heat and mass transfer in porous media (Chen et al 2015;Chai et al 2016…”