Nonconserved Variables Research Articles

A computational strategy for nonlinear coupled constitutive relations of rarefied nonequilibrium flows on unstructured grids

The Navier-Stokes-Fourier (NSF) equations are no longer valid in high-Knudsen and high-Mach flows, where the local thermodynamically equilibrium assumption fails, while the nonlinear coupled constitutive relations (NCCR) have been proven to be an efficient approach for rarefied non-equilibrium flows under the structured finite-volume-method computational fluid dynamic (FVM-CFD) solver. In this paper, by adding the time derivatives of non-conserved variables to the NCCR model, a computational strategy for NCCR model under the unstructured FVM-CFD framework is first proposed for further engineering applications. An upwind flux-splitting scheme with the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit time-marching scheme is employed. The linear stability of one-dimensional modified NCCR model in time and space is analyzed subsequently, which indicates the modified NCCR model is unconditionally stable. The accuracy and computational performance of the proposed NCCR solution are assessed by a feat of the numerical tests for hypersonic flow over a blunt cylinder without a wake, supersonic flow past a circular cylinder with a wake, and hypersonic flow around a planetary probe. Computational results show that the newly-developed NCCR solution can get solutions in better agreement with the direct simulation Monte Carlo (DSMC) than NSF results. Moreover, the newly-developed NCCR solution has a better property of convergence than the undecomposed NCCR algorithm under the unstructured FVM-CFD framework. These superior advantages of the present computational strategy are expected to make the NCCR method a promising engineering tool for modeling rarefied non-equilibrium flows.

Multiphase-field modeling of disproportionation performance of ZrCo alloy during hydrogen absorption

This work attempts to develop a multiphase field model to study ZrCoHy, ZrH2, and ZrCo2 phase formation paths in the ZrCoH0.5 matrix. Interestingly, one conserved phase-field variable (c) and four non-conserved phase-field variables (ϕi) are utilized to construct free energy function, where c represents the hydrogen concentration and ϕi stands for the ZrCoHx, ZrCoHy, ZrH2, and ZrCo2 phase. Some valuable material parameters, such as the Gibbs formation enthalpy, are also taken into account. Importantly, the system energy fluctuation is taken as the driving force of phase evolution, and then, governing equations are solved by self-programming complied with the Fortran language. The results demonstrate that disproportionation products are most likely to be derived from the initial ZrCoHy. The presence of ZrH2 and ZrCo2 promotes the nucleation incubation of disproportionation products. Besides, ZrCoHy is more prone to generate ZrH2 and ZrCo2 in comparison with ZrCoHx, which is attributed to the fact that the crystal structure of ZrCoHy is closer to that of ZrCo2. Because ZrCoHy could be regarded as the transition state when ZrCoHx is transformed into ZrCo2, ZrCoHy is easy to be produced in advance during the ZrCoHx disproportionation process, instead of ZrH2 and ZrCo2 directly generated by the ZrCoHx disproportionation.

Dynamics of Nanoparticle Self-Assembly by Liquid Crystal Sorting in Two Dimensions

We study nonlinear dynamical equations for coupled conserved and non-conserved fields describing nanoparticle concentration and liquid crystal order parameter, respectively, and solve them numerically over bidimensional domains. These equations model the rapid segregation of nanoparticles away from nematic domains, which has been observed experimentally in a suspension of gold nanoparticles in 5CB below the isotropic-nematic transition temperature. We contrast the different behaviors obtained when the LC order parameter is treated as a scalar or a tensor, as well as the different rates of evolution observed with each of these. We find, after an instantaneous quench lowering the temperature below the transition one, an initial linear regime where the ordering of the nematic phase proceeds exponentially with time. Only after a lag period the nanoparticle material couples effectively to the LC order parameter and segregates to regions that are less orientationally ordered (extended domain walls for a scalar order parameter, but point disclinations for a tensor one). The lag period is followed by the onset of nonlinear dynamics and saturation of the order parameter. The choice of a scalar or tensor LC order parameter does not change this sequence but results in a clear overshooting of the nonlinear saturation level for the tensor order parameter case. These results are found to be insensitive to weak anchoring due to coupling of gradients of the conserved and non-conserved variables, for the nanoparticle concentrations and anchoring parameters studied. Our modeling approach can be extended in a straightforward manner to cases where the cooling rate is finite and to other systems where a locally conserved concentration is coupled to a orientation field, such as active Langmuir monolayers, and possibly to other examples of nonlinear dynamics in ecological or excitable media problems.

Strong shock as a stringent test for Onsager-Burnett equations.

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)2470-004510.1103/PhysRevE.96.013106] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations

Generalized hydrodynamic equations were originally proposed to describe the rarefied non-equilibrium flows beyond the Navier–Stokes–Fourier (NSF) equations by constructing a non-equilibrium canonical distribution function for the mesoscopic Boltzmann equation. Subsequently, nonlinear coupled constitutive relations (NCCR) were developed under the adiabatic assumption, the Eu’s closure, and Myong’s simplification. NCCR+ was also proposed to include the omitted terms in Myong’s simplification. The goal of this paper is to assess the improvements in the accuracy due to NCCR+ and the influence of the bulk viscosity in one-dimensional steady shock wave structure for monatomic and diatomic gases. In order to solve NCCR+ equations, a coupled solution process based on the time-independent method for non-conserved variables is employed, which is different from the previous uncoupled solution process used for the NCCR equations. Shock structures in argon and nitrogen are calculated up to Mach 50, where the shock profile, inverse shock thickness, asymmetry parameter, and temperature–density separation distance are validated by DSMC and available experimental measurements. The results show that NCCR+ could not provide much improvement in accuracy compared to NCCR but adds to the computational cost, suggesting that Myong’s simplification used in NCCR is satisfactory. Both NCCR and NCCR+ perform better than NSF in computing the one-dimensional shock wave structure at high Mach numbers. It is also shown that the bulk viscosity has significant influence on the accuracy of prediction of the shock wave structure in a diatomic gas using both NSF and NCCR. Stokes’ hypothesis in conventional NSF is valid only for flows in a monatomic gas and for very low Mach number flows in a diatomic gas. Additionally, it is also found that the viscosity exponent s = 0.81 in variable-hard-sphere model provides a good fit with the experimental data for shock wave in argon using NCCR.

A Unified Framework for Modeling Continuum and Rarefied Gas Flows

The momentum and heat transport in rarefied gas flows is known to deviate from the classical laws of Navier and Fourier in Navier-Stokes-Fourier (NSF) equations. A more sophisticated Nonlinear Coupled Constitutive Model (NCCM) has been derived from the Boltzmann equation to describe gaseous and thermal transport both in continuum and rarefied gas flows. We first develop a unified numerical framework for modeling continuum and rarefied flows based on the NCCM model both in two and three dimensions. Special treatment is given to the complex highly nonlinear transport equations for non-conserved variables that arise from the high degree of thermal nonequilibrium. For verification and validation, we apply the present scheme to a stiff problem of hypersonic gas flows around a 2D cylinder, a 3D sphere, and the Apollo configuration both in continuum and rarefied situations. The results show that the present unified framework yields solutions that are in better agreement with the benchmark and experimental data than are the NSF results in all studied cases of rarefied problems. Good agreement is observed between the present study and the NSF results for continuum cases. The results show that this study provides a unified framework for modeling continuum and rarefied gas flows.

Force-driven compressible plane Poiseuille flow by Onsager-Burnett equations

The purpose of this work is to evaluate the recently derived Onsager-Burnett (OBurnett) equations [N. Singh, R. S. Jadhav, and A. Agrawal, “Derivation of stable Burnett equations for rarefied gas flows,” Phys. Rev. E 96, 013106 (2017)] for force-driven compressible plane Poiseuille flow. This classical internal flow problem depicts several non-equilibrium phenomena, for instance, non-constant pressure profile in the transverse direction and tangential heat flux, which are not captured by the classical Navier-Stokes-Fourier equations. The results of OBurnett equations for conserved and non-conserved variables are validated against the existing direct simulation Monte Carlo (DSMC) and molecular dynamics (MD) simulation results. These results suggest that the OBurnett equations are able to predict most of the variables well with respect to DSMC and MD simulation results. We find that the OBurnett equations predict a strictly monotonic pressure profile, in contrast to the bimodal profile predicted by the DSMC results and the conventional Burnett equations, but in agreement with the molecular dynamics simulation results. The equations also recover the non-zero tangential heat flux but fail to capture the peculiar temperature dip at the center, owing to its second order accuracy. These results suggest that the evaluated equations are accurate in predicting the non-equilibrium phenomena observed in the rarefied gas flows for the case considered.

Formation path of δ hydrides in zirconium by multiphase field modeling

A multiphase field model is developed to study the effects of metastable ζ and γ hydrides on the nucleation and growth of the stable δ hydrides in α zirconium matrix. The model incorporates all the possible phases using the Gibbs free energies of formation for each phase and their available material properties. The multiphase field model is constructed by utilizing one conserved phase-field variable to represent the concentration of hydrogen, and six non-conserved phase-field variables to represent the α phase, ζ phase, three orientation variants of γ phase, and δ phase. The evolution equations are coupled with the mechanical equilibrium equations and solved using the Multiphysics Object Oriented Simulation Environment (MOOSE). Nucleation of hydrides is controlled using classic nucleation theory, inserting nuclei randomly with a probability dependent on the competition between the hydride's volume free energy and the interface's area free energy to form critical sized nuclei. The comparison between the results of the multiphase model and a two-phase model (without metastable phases) indicate that the intermediate phases are influential in the initial formation and evolution of δ phase hydrides. Random seed simulations, both in the basal plane and the (101¯0) plane, also indicate that the intermediate metastable phases play a key role in the shape evolution of δ hydrides. Results suggest that quantitative phase field models of δ hydride growth need to include intermediate phases in order to accurately predict the morphology of hydrides.

A phase-field model to study the effects of temperature change on shape evolution of γ-hydrides in zirconium

A temperature-dependent phase-field model is developed to study the effects of temperature change on shape evolution of γ-hydrides in an α-zirconium matrix. To construct the temperature-dependent free energy functional of the phase-field model, Gibbs free energies of formation from previous experiments are employed, and one conserved and three non-conserved phase-field variables are used for hydrogen concentration and hydride orientations, respectively. The mixed order evolution equations of phase-field variables coupled with mechanical equilibrium equations are solved in a finite element framework. Results from isothermal simulations of seeded and random nucleation in single crystal α-zirconium matrix show that the thickness of non-equilibrium hydrides varies with temperature during evolution, and the hydrides are more rod-like (thinner) at higher temperatures and thicker at lower temperatures. Quench simulations with random nucleation indicate that the majority of precipitation occurs at early stages of quenching, but the size and shape of hydrides change as the temperature decreases. Simulations from random nucleation of hydrides in a polycrystalline α-zirconium matrix show a higher concentration of precipitates along high angle grain boundaries.

A quantitative and thermodynamically consistent phase-field interpolation function for multi-phase systems

The aimed properties of the interpolation functions used in quantitative phase-field models for two-phase systems do not extend to multi-phase systems. Therefore, a new type of interpolation functions is introduced that has a zero slope at the equilibrium values of the non-conserved field variables representing the different phases and allows for a thermodynamically consistent interpolation of the free energies. The interpolation functions are applicable for multi-phase-field and multi-order-parameter representations and can be combined with existing quantitative approaches for alloys. A model for polycrystalline, multi-component and multi-phase systems is formulated using the new interpolation functions that accounts in a straightforward way for composition-dependent expressions of the bulk Gibbs energies and diffusion mobilities, and interfacial free energies and mobilities. The numerical accuracy of the approach is analyzed for coarsening and diffusion-controlled parabolic growth in Cu–Sn systems as a function of R/ℓ, with R grain size and ℓ diffuse interface width.

Volume transport and generalized hydrodynamic equations for monatomic fluids

In this paper, the effects of volume transport on the generalized hydrodynamic equations for a pure simple fluid are examined from the standpoint of statistical mechanics and, in particular, kinetic theory of fluids. First, we derive the generalized hydrodynamic equations, namely, the constitutive equations for the stress tensor and heat flux for a single-component monatomic fluid, from the generalized Boltzmann equation in the presence of volume transport. Then their linear steady-state solutions are derived and examined with regard to the effects of volume transport on them. The generalized hydrodynamic equations and linear constitutive relations obtained for nonconserved variables make it possible to assess Brenner's proposition [Physica A 349, 11 (2005); Physica A 349, 60 (2005)] for volume transport and attendant mass and volume velocities as well as the effects of volume transport on the Newtonian law of viscosity, compression/dilatation (bulk viscosity) phenomena, and Fourier's law of heat conduction. On the basis of study made, it is concluded that the notion of volume transport is sufficiently significant to retain in irreversible thermodynamics of fluids and fluid mechanics.

Phase-Field Models for Microstructure Evolution

▪ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

Kinetic model of thermotransport in semiconductors

Abstract In this work we develop a semiclassical kinetic model to construct hydrodynamic equations to study the nonequilibrium behavior of an electron gas in a semiconductor. To describe the system we assume that the Boltzmann transport equation is valid and we solve it in terms of the first 13 moments, which become the basic fields to describe thermotransport in this system. The closure of the transport equations is achieved through an expansion around the nonequilibrium Fermi–Dirac distribution function. The nonconserved variables satisfy relaxation equations and their characteristic times are expressed in kinetic terms. The electric current density and the heat flux can be written as generalized constitutive equations, allowing us to obtain thermotransport coefficients as functions of the frequency, where the relaxation times play a very important role.

Decomposition of entropy and the choice of nonequilibrium variables for relativistic classical and quantum gases

In this paper, beginning from the fundamental principles of general-relativistic kinetic theory of monatomic ideal gases – both classical and quantum, we investigate an expression for the Boltzmann entropy s (per unit volume), with a view to a deeper understanding of the generalized Gibbs relation and other postulates of extended (irreversible) thermodynamics. The main upshot of our analysis is that if M is any finite set of the moments of the distribution function f (by definition, M consists of conserved and nonconserved variables), then there exists a phase-space function z such that s can exactly be written as a sum of two structurally different parts: the first part is a function of M and thus represents the entropy of extended (irreversible) thermodynamics, while the second part gives the functional contribution to s independent of M and vanishing at z=0; this part depends only on z. Originally, the method we develop here resulted from a detailed consideration of the case where M consists of conserved variables (internal energy, density, etc.). This paper extends Banach's previous work (Physica A 275 (2000) 405) in that M contains also nonconserved variables such as, e.g., number flux vector, heat flux, and stress deviator. Our general ideas are illustrated for three important systems: classical ultrarelativistic gases and quantum photon and neutrino gases.

Wave Equations of Elastic Type-II Superconductors

ABSTRACTIt is shown that when the thermodynamic fluxes are included as independent thermodynamic state variables of the generalized entropy density, the constitutive equations for the conserved state variables and the evolution equations for the nonconserved state variables of elastic type-II superconductors can be derived systematically. In particular, the transport equations generalizing the Fourier law for heat transport and the time-dependent Ginzburg-Landau equation for the relaxation of superelectron are proposed in this paper.

Computer simulation of stress-oriented nucleation and growth of θ′ precipitates inAl–Cu alloys

Many structural transformations result in several orientation variants whose volume fractions and distributions can be controlled by applied stresses during nucleation, growth or coarsening. Depending on the type of stress and the coupling between the applied stress and the lattice misfit strain, the precipitate variants may be aligned parallel or perpendicular to the stress axis. This paper reports our studies on the effect of applied stresses on nucleation and growth of coherent θ′ precipitates in Al–Cu alloys using computer simulations based on a diffuse-interface phase-field kinetic model. In this model, the orientational differences among precipitate variants are distinguished by non-conserved structural field variables, whereas the compositional difference between the precipitate and matrix is described by a conserved field variable. The temporal evolution of the spatially dependent field variables is determined by numerically solving the time-dependent Ginzburg–Landau (TDGL) equations for the structural variables and the Cahn–Hilliard diffusion equation for composition. Random noises were introduced in both the composition and the structural order parameter fields to simulate the nucleation of θ′ precipitates. It is demonstrated that although an applied stress affects the microstructural development of a two-phase alloy during both the nucleation and growth stages, it is most effective to apply stresses during the initial nucleation stage for producing anisotropic precipitate alignment.

Morphological evolution of coherent multi-variant Ti 11Ni 14 precipitates in Ti-Ni alloys under an applied stress—a computer simulation study

Coherent precipitation of multi-variant Ti 11Ni 14 precipitates in TiNi alloys was investigated by employing a continuum field kinetic model. The structural difference between the precipitate phase and the matrix as well as the orientational differences between precipitate variants are distinguished by nonconserved structural field variables, whereas the compositional difference between the precipitate and matrix is described by a conserved field variable. The temporal evolution of the spatially dependent field variables is determined by numerically solving the time-dependent Ginzburg–Landau (TDGL) equations for the structural variables and the Cahn–Hilliard diffusion equation for the composition. In particular, the interaction between precipitates, and the growth morphology of Ti 11Ni 14 precipitates under strain-constraints were studied, without a priori assumptions on the precipitate shape and distribution. The predicted morphology and distribution of Ti 11Ni 14 variants were compared with experimental observations. Excellent agreement between the simulation and experimental observations was found.

Diffusion-controlled grain growth in two-phase solids

Microstructural evolution and the kinetics of grain growth in volume-conserved two-phase solids were investigated using two-dimensional (2-D) computer simulations based on a diffuse-interface field model. In this model, a two-phase microstructure is described by non-conserved field variables which represent crystallographic orientations of grains in each phase and by a conserved composition field variable which distinguishes the compositional difference between the two phases. The temporal and spatial evolution of these field variables were obtained through a numerical solution to the time-dependent Ginzburg-Landau (TDGL) equations. The effect of the ratios of grain boundary energies to interfacial energy on the microstructure features was systematically studied. It was found that grain growth in a volume-conserved two-phase solid is controlled by long-range diffusion and follows the power growth law, R m − R m o = kt with m = 3 in the scaling regime for all cases studied, including the microstructures containing only quadrijunctions. The effects of volume fractions and initial microstructures are discussed.

Local formulation of the first law and the second law of thermodynamics for a nonequilibrium system

In this paper we construct a local formulation of the first law and the second law of thermodynamics for a nonequilibrium system such that their global forms are equivalent to Kelvin's principle and Clausius's principle. We also study the relationship between the entropy balance equation and the calotropy balance equation obtained by Eu [Phys. Rev. E 51, 768 (1995)] by examining the model equations of the nonconserved variables proposed by Jou, Casse-Vazquez, and Lebon [Rep. Prog. Phys. 51, 1105 (1988); Extended Irreversibie Thermodynamics (Springer, Heidelberg, 1993)] in extended irreversible thermodynamics.

Note on the nonequilibrium partition function and generalized potentials

In this paper, we present a nonperturbative method of calculating the nonequilibrium partition function appearing in the nonequilibrium ensemble method for dilute gases. A set of first-order partial differential equations is derived for the nonequilibrium partition function. The set can be put into an exact differential form and its integral gives the nonequilibrium partition function with which various macroscopic variables can be expressed. In particular, on solving the aforementioned partial differential equations for the derivatives, the nonequilibrium constitutive relations are directly obtained for the generalized potentials of the substance which are conjugate to the nonconserved variables such as stresses, heat fluxes, diffusion fluxes, and so on. The results for a couple of special cases are presented without proof.