Non-ideal Equation Of State Research Articles

Interfacial vorticity dynamics for Navier–Stokes–Korteweg system: General theory and application to two-dimensional near-wall cavitation bubble

A theory of interfacial vorticity dynamics is developed for the single-component liquid–vapor flows described by the Navier–Stokes–Korteweg equations. The non-ideal equation of state and surface capillarity are collectively considered via the chemical potential gradient force. Explicit formulas of the interfacial enstrophy and chemical potential fluxes are derived to elucidate their generation mechanisms from the interface, which are expressed by using the fundamental surface physical quantities. Then, the transport equation of the Lamb dilatation (namely, the Lamb vector divergence) is rigorously derived, generalizing the previous results of the Lamb-vector dynamics for single-phase incompressible viscous flows. The derived results are evaluated in a near-wall cavitation bubble combined with the van der Waals equation of state. The roles of surface shear stress and vorticity, surface chemical potential gradient, surface acceleration, surface stretching-shrinking motion and dilatational waves are emphasized in generating the interfacial enstrophy and chemical potential fluxes, featured by the strong vorticity concentration, interface deformation and inward-propagating microjet due to the longitudinal pressure difference. These distinct dynamical features observed near the bubble interface are well captured by the spatial distribution of the Lamb dilatation. Analyzing the dynamics of the Lamb dilatation based on the transport equation demonstrates the significance of the high-curvature rotating surface region and its vicinity in view of their dominance to the unsteady diffusion effect of the Lamb dilatation. The present exposition could provide new physical insights into liquid–vapor flows from the perspective of interfacial vorticity dynamics.

Stability of perpendicular magnetohydrodynamic shocks in materials with ideal and nonideal equationsof state.

Magnetized target fusion approach to inertial confinement fusion involves the formation of strong shocks that travel along a magnetized plasma. Shocks, which play a dominant role in thermalizing the upstream kinetic energy generated in the implosion stage, are seldom free from perturbations, and they wrinkle in response to upstream or downstream disturbances. In Z-pinch experiments, significant plasma instability mitigation was observed with pre-embedded axial magnetic fields. To isolate effects, in this work we theoretically study the impact of perpendicular magnetic fields on the planar shock dynamics for different equationsof state. For fast magnetosonic shocks in ideal gases, it was found that the magnetic field amplifies the intensity of the perturbations when γ>2 or it weakens them when γ<2. Weak shocks have been found to be stable regardless of the magnetic plasma intensity and gas compressibility; however, for sufficiently strong shocks the magnetic fields can promote a neutral stability/SAE at the shock if the adiabatic index is higher than 1+sqrt[2]. Results have been validated with numerical simulations performed with the FLASH code.

Towards a consistent lattice Boltzmann model for two-phase fluids

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

Real gas effect on steady planar detonation and uncertainty quantification

The steady planar detonation in hydrogen/air mixture was simulated at elevated pressure. The real gas effect was incorporated by considering the Peng-Robinson equation of state (EoS), nonideal thermodynamic functions and reaction kinetic laws. The nonideal EoS and corresponding thermodynamic functions increase the Chapman-Jouguet (CJ) velocity but decrease the post-shock temperature compared to the ideal gas model, which induces a change of the induction time and distance. The effect of the nonideal reaction rate law depends on the amplitude of compressibility factor and fugacity coefficient and tends to shorten both the induction time and induction distance. These effects counterbalance each other in the complete real gas model which lead to induction time and distance close to the ideal model results. The total heat release was also found to be reduced for the complete model, but the key reactions and their relative importance remain the same as when using the ideal gas model. The nonideal portion of heat release is minor compared with its ideal portion as temperature increases and pressure drops in the reaction zone. The uncertainty of the real gas model, originating from the uncertainty of the parameters in the EoS, was quantified using a Monte Carlo sampling approach. The uncertainty increases linearly with initial pressure and is mainly determined by the species with higher uncertainty factor, larger mole fraction weighted molecular attraction and covolume parameters. Compared to the uncertainty caused by the chemical reaction model, real gas model uncertainty is negligible at low pressure, but becomes of the same order of magnitude at elevated pressure.

Pseudo-potential lattice Boltzmann model for large density and viscosity ratio fluids

The flow of two immiscible fluid components with large density and viscosity ratios is considered. Using the pseudo-potential lattice Boltzmann model with a non-ideal equation of state and a mixture viscosity formulation, we can develop a model that successfully passes a series of benchmark tests, most importantly co-current layered flows, with no discontinuities in the velocity profiles at interfaces. The model is then applied to applications of bubbles rising in denser fluids and shows good quantitative agreement with previous numerical and experimental results. The model is also applied to the flow of two immiscible fluids in a porous medium and shows characteristics that are physically consistent with established experimental observations.

COUPLING MOLECULAR DYNAMICS AND PSEUDOPOTENTIAL LATTICE BOLTZMANN METHOD WITH NONIDEAL EQUATION OF STATE FOR MICROSCOPIC FLUID FLOWS

COUPLING MOLECULAR DYNAMICS AND PSEUDOPOTENTIAL LATTICE BOLTZMANN METHOD WITH NONIDEAL EQUATION OF STATE FOR MICROSCOPIC FLUID FLOWS

Multi-particle collision dynamics with a non-ideal equation of state. II. Collective dynamics of elongated squirmer rods.

Simulations of flow fields around microscopic objects typically require methods that both solve the Navier-Stokes equations and also include thermal fluctuations. One such method popular in the field of soft-matter physics is the particle-based simulation method of multi-particle collision dynamics (MPCD). However, in contrast to the typically incompressible real fluid, the fluid of the traditional MPCD methods obeys the ideal-gas equation of state. This can be problematic because most fluid properties strongly depend on the fluid density. In a recent article, we proposed an extended MPCD algorithm and derived its non-ideal equation of state and an expression for the viscosity. In the present work, we demonstrate its accuracy and efficiency for the simulations of the flow fields of single squirmers and of the collective dynamics of squirmer rods. We use two exemplary squirmer-rod systems for which we compare the outcome of the extended MPCD method to the well-established MPCD version with an Andersen thermostat. First, we explicitly demonstrate the reduced compressibility of the MPCD fluid in a cluster of squirmer rods. Second, for shorter rods, we show the interesting result that in simulations with the extended MPCD method, dynamic swarms are more pronounced and have a higher polar order. Finally, we present a thorough study of the state diagram of squirmer rods moving in the center plane of a Hele-Shaw geometry. From a small to large aspect ratio and density, we observe a disordered state, dynamic swarms, a single swarm, and a jammed cluster, which we characterize accordingly.

Theory of supercritical flames with real-fluid equations of state

We report a theoretical analysis of the propagation and structure of the planar adiabatic/radiative flame in supercritical fluids, employing the Van der Waals and the Redlich-Kwong equations of state to describe the fluid properties. It is demonstrated that the effective molecular volume and attractive force terms in these non-ideal equations of state show opposite effects on the density modification, especially in the upstream unburned region. Furthermore, the non-ideal effect is found to strongly couple with radiative emission and reabsorption, whose respective weakening and strengthening effect can substantially affect the state of radiation-induced quenching. The theory is useful for the description of high-pressure flames as both the non-ideal state and radiation become more pronounced under such conditions.

Simulation and flow physics of a shocked and reshocked high-energy-density mixing layer

Abstract

How planets grow by pebble accretion

During their formation, planets form large, hot atmospheres due to the ongoing accretion of solids. It has been customary to assume that all solids end up at the center, constituting a “core” of refractory materials, whereas the envelope remains metal-free. However, recent work, as well as observations by the Juno mission, indicate that the distinction may not be so clear cut. Indeed, small silicate, pebble-sized particles will sublimate in the atmosphere when they hit the sublimation temperature (T ~ 2000 K). In this paper we extend previous analytical work to compute the properties of planets within such a pebble accretion scenario. We conduct 1D numerical calculations of the atmosphere of an accreting planet, solving the stellar structure equations, augmented by a nonideal equation of state that describes a hydrogen and helium-silicate vapor mixture. Calculations terminate at the point where the total mass in metal is equal to that of the H+He gas, which we numerically confirm as the onset of runaway gas accretion. When pebbles sublimate before reaching the core, insufficient (accretion) energy is available to mix dense, vapor-rich lower layers with the higher layers of lower metallicity. A gradual structure in which Z decreases with radius is therefore a natural outcome of planet formation by pebble accretion. We highlight, furthermore, that (small) pebbles can act as the dominant source of opacity, preventing rapid cooling and presenting a channel for (mini-)Neptunes to survive in gas-rich disks. Nevertheless, once pebble accretion subsides, the atmosphere rapidly clears followed by runaway gas accretion. We consider atmospheric recycling to be the most probable mechanism to have stalled the growth of the envelopes of these planets.

Multi-particle collision dynamics with a non-ideal equation of state. I.

The method of multi-particle collision dynamics (MPCD) and its different implementations are commonly used in the field of soft matter physics to simulate fluid flow at the micron scale. Typically, the coarse-grained fluid particles are described by the equation of state of an ideal gas, and the fluid is rather compressible. This is in contrast to conventional fluids, which are incompressible for velocities much below the speed of sound, and can cause inhomogeneities in density. We propose an algorithm for MPCD with a modified collision rule that results in a non-ideal equation of state and a significantly decreased compressibility. It allows simulations at less computational costs compared to conventional MPCD algorithms. We derive analytic expressions for the equation of state and the corresponding compressibility as well as shear viscosity. They show overall very good agreement with simulations, where we determine the pressure by simulating a quiet bulk fluid and the shear viscosity by simulating a linear shear flow and a Poiseuille flow.

Kinetic modeling of multiphase flow based on simplified Enskog equation

A new kinetic model for multiphase flow was presented under the framework of the discrete Boltzmann method (DBM). Significantly different from the previous DBM, a bottom-up approach was adopted in this model. The effects of molecular size and repulsion potential were described by the Enskog collision model; the attraction potential was obtained through the mean-field approximation method. The molecular interactions, which result in the non-ideal equation of state and surface tension, were directly introduced as an external force term. Several typical benchmark problems, including Couette flow, two-phase coexistence curve, the Laplace law, phase separation, and the collision of two droplets, were simulated to verify the model. Especially, for two types of droplet collisions, the strengths of two non-equilibrium effects, $\bar{D}_2^*$ and $\bar{D}_3^*$, defined through the second and third order non-conserved kinetic moments of $(f - f ^{eq})$, are comparatively investigated, where $f$ ($f^{eq}$) is the (equilibrium) distribution function. It is interesting to find that during the collision process, $\bar{D}_2^*$ is always significantly larger than $\bar{D}_3^*$, $\bar{D}_2^*$ can be used to identify the different stages of the collision process and to distinguish different types of collisions. The modeling method can be directly extended to a higher-order model for the case where the non-equilibrium effect is strong, and the linear constitutive law of viscous stress is no longer valid.

Mesoscopic simulation of three-dimensional pool boiling based on a phase-change cascaded lattice Boltzmann method

In this paper, a three-dimensional (3D) cascaded lattice Boltzmann method (CLBM) is implemented to simulate the liquid–vapor phase-change process. The multiphase flow field is solved by incorporating the pseudopotential multiphase model into an improved CLBM, the temperature field is solved by the finite difference method, and the two fields are coupled via a non-ideal equation of state. Through numerical simulations of several canonical problems, it is verified that the proposed phase-change CLBM is applicable for both the isothermal multiphase flow and the liquid–vapor phase-change process. Using the developed method, a complete 3D pool boiling process with up to hundreds of spontaneously generated bubbles is simulated, faithfully reproducing the nucleate boiling, transition boiling, and film boiling regimes. It is shown that the critical heat flux predicted by the 3D simulations agrees better with the established theories and correlation equations than that obtained by two-dimensional simulations. Furthermore, it is found that with the increase in the wall superheats, the bubble footprint area distribution changes from an exponential distribution to a power-law distribution, in agreement with experimental observations. In addition, insights into the instantaneous and time-averaged characteristics of the first two largest bubble footprints are obtained.

Large Fluctuations in Nonideal Coarse-Grained Systems.

Using the recently introduced molecular dynamics lattice gas approach, we test fluctuations of coarse-grained quantities. We show that as soon as the system can no longer be considered an ideal gas fluctuations fail to diminish upon coarse graining as is usually expected. These results suggest that current approaches to simulating fluctuating hydrodynamics may have to be augmented to achieve quantitative results for systems with a nonideal equation of state. The molecular dynamics lattice gas method gives a guidance to the exact nature of the fluctuation in such systems.

Verification Studies for the Noh Problem Using Nonideal Equations of State and Finite Strength Shocks

The Noh verification test problem is extended beyond the commonly studied ideal gamma-law gas to more realistic equations of state (EOSs) including the stiff gas, the Noble-Abel gas, and the Carnahan–Starling EOS for hard-sphere fluids. Self-similarity methods are used to solve the Euler compressible flow equations, which, in combination with the Rankine–Hugoniot jump conditions, provide a tractable general solution. This solution can be applied to fluids with EOSs that meet criterion such as it being a convex function and having a corresponding bulk modulus. For the planar case, the solution can be applied to shocks of arbitrary strength, but for the cylindrical and spherical geometries, it is required that the analysis be restricted to strong shocks. The exact solutions are used to perform a variety of quantitative code verification studies of the Los Alamos National Laboratory Lagrangian hydrocode free Lagrangian (FLAG).

Hydrodynamics of binary-fluid mixtures —An augmented Multiparticle Collison Dynamics approach

The Multiparticle Collision Dynamics technique (MPC) for hydrodynamics simulations is generalized to binary-fluid mixtures and multiphase flows, by coupling the particle-based fluid dynamics to a Ginzburg-Landau free-energy functional for phase-separating binary fluids. To describe fluids with a non-ideal equation of state, an additional density-dependent term is introduced. The new approach is verified by applying it to thermodynamics near the critical demixing point, and interface fluctuations of droplets. The interfacial tension obtained from the analysis of the capillary-wave spectrum agrees well with the results based on the Laplace-Young equation. Phase separation dynamics follows the Lifshitz-Slyozov law.

Effect of Nonlinear Screening on a Complex Plasma Phase State

Applicability limit of the well-known phase diagram of dusty plasmas in $\kappa $ – $\Gamma $ plane ( $\kappa $ is structural parameter and $\Gamma $ is parameter of Coulomb nonideality) is under discussion. Existence of extensive domains with violation of plasma thermodynamic stability conditions (i.e., with negative isothermal compressibility) was also claimed if one uses well-known nonideal equations of state by (Hamaguchi S. et al. , Phys. Rev. E, 1997) and (Khrapak S. et al. , Phys. Rev. E, 2014). This paper is devoted to analysis of a range of applicability for basic assumption in Hamaguchi’s phase diagram, i.e., linearized (Debye) screening of macroions by microions, which leads to the Yukawa form for effective interactions between macroions. Parameters of nonlinear screening for macroions were calculated within differential Poisson–Boltzmann equation. Two effects were revealed as a result of such calculations: 1) decomposition of all microions onto two subclasses, free and bound ones, and 2) significant reduction of effective charge $\text{Z}^\ast $ of initial bare macroion Z under nonlinear screening by small high-density envelope of bound ions. This effect leads to a renormalization of initial $\Gamma $ and $\kappa $ into $\Gamma ^\ast $ and $\kappa ^\ast $ ( $\Gamma ^\ast , $\kappa ^\ast ). The main physical assumption is phase states of complex plasma under nonlinear screening, which are still the same as on the initial phase diagram, but in $\kappa ^\ast $ – $\Gamma ^\ast $ plane instead of $\kappa $ – $\Gamma $ one. Corresponding calculated shifts of phase states are discussed and illustrated.

Droplet deformation and heat transfer in isotropic turbulence

The heat and mass transfer of deformable droplets in turbulent flows is crucial to a wide range of applications, such as cloud dynamics and internal combustion engines. This study investigates a single droplet undergoing phase change in isotropic turbulence using numerical simulations with a hybrid lattice Boltzmann scheme. Phase separation is controlled by a non-ideal equation of state and density contrast is taken into consideration. Droplet deformation is caused by pressure and shear stress at the droplet interface. The statistics of thermodynamic variables are quantified and averaged over both the liquid and vapour phases. The occurrence of evaporation and condensation is correlated to temperature fluctuations, surface tension variation and turbulence intensity. The temporal spectra of droplet deformations are analysed and related to the droplet surface area. Different modes of oscillation are clearly identified from the deformation power spectrum for low Taylor Reynolds number $Re_{\unicode[STIX]{x1D706}}$, whereas nonlinearities are produced with the increase of $Re_{\unicode[STIX]{x1D706}}$, as intermediate frequencies are seen to overlap. As an outcome, a continuous spectrum is observed, which shows a decrease in the power spectrum that scales as ${\sim}f^{-3}$. Correlations between the droplet Weber number, deformation parameter, fluctuations of the droplet volume and thermodynamic variables are also developed.

A sharp interface method for coupling multiphase flow, heat transfer and multicomponent mass transfer with interphase diffusion

Mixing of partially miscible fluids plays an important role in many physical and chemical processes. The modeling complexities lie in the tight coupling of the multiphase flow, heat transfer and multicomponent mass transfer, as well as diffusions across the phase interface. We present a sharp interface method for modeling such process. The non-ideal equation of state is used to compute the fluid properties such as density, fugacity and enthalpy, and to predict phase equilibrium composition. The phase interface location is tracked using the phase propagation velocity. A third-order one-sided finite difference scheme using a variable grid size according to the interface location is utilized to discretize the partial derivatives immediately next to the interface, while a second-order central scheme is used for the bulk of fluids. An optimization method, the Nelder–Mead method, is applied to search for (1) the phase compositions on both sides of the interface, and (2) the phase propagation velocity based on the coupling of the multicomponent phase equilibrium and the species' balance across the interface. The temperature at the interface is determined by the energy balance. Numerical results are used to demonstrate the convergence of our method and show its capability to simulate the mixing of multicomponent partially miscible fluids.

Thermodynamics of a real fluid near the critical point in numerical simulations of isotropic turbulence

We investigate the behavior of a fluid near the critical point by using numerical simulations of weakly compressible three-dimensional isotropic turbulence. Much has been done for a turbulent flow with an ideal gas. The primary focus of this work is to analyze fluctuations of thermodynamic variables (pressure, density, and temperature) when a non-ideal Equation Of State (EOS) is considered. In order to do so, a hybrid lattice Boltzmann scheme is applied to solve the momentum and energy equations. Previously unreported phenomena are revealed as the temperature approaches the critical point. Fluctuations in pressure, density, and temperature increase, followed by changes in their respective probability density functions. Due to the non-linearity of the EOS, it is seen that variances of density and temperature and their respective covariance are equally important close to the critical point. Unlike the ideal EOS case, significant differences in the thermodynamic properties are also observed when the Reynolds number is increased. We also address issues related to the spectral behavior and scaling of density, pressure, temperature, and kinetic energy.