Ellipsoidal Statistical Bhatnagar-Gross-Krook Research Articles

Evaluation of stochastic particle Bhatnagar–Gross–Krook methods with a focus on velocity distribution function

For precise application of Bhatnagar–Gross–Krook (BGK) methods, assessing its accuracy in non-equilibrium flows is necessary. Generally, this assessment relies on macroscopic parameters, which are moments of the velocity distribution function (VDF). However, in non-equilibrium flows, the significance of each moment diminishes as the VDF deviates from the Maxwellian VDF. This study investigates the VDF in non-equilibrium flows. Two Prandtl-corrected BGK methods, the ellipsoidal statistical BGK and Shakhov BGK (SBGK), are compared with the direct simulation Monte Carlo method. To observe the VDF while excluding the effects of convection, the homogeneous relaxation of the initial non-equilibrium state is analyzed. The VDF in Couette flow and normal shock waves, where collision and convection coexist, is then examined. When comparing the accuracy of the BGK methods using higher-order moments, inconsistencies are observed. However, when comparing the VDFs, the SBGK method reproduces the non-equilibrium VDF more accurately. The results demonstrate the importance of the VDF in the evaluation of non-equilibrium flows.

Extension of the Shakhov Bhatnagar–Gross–Krook model for nonequilibrium gas flows

Bhatnagar–Gross–Krook (BGK) models are widely used to study rarefied gas dynamics. However, as simplified versions of the Boltzmann collision model, their performances are uncertain and need to be carefully investigated in highly nonequilibrium flows. In this study, several common BGK models, such as the ellipsoidal statistical BGK (ES-BGK) and Shakhov BGK (S-BGK) models, are theoretically analyzed using their moment equations. Then, numerical comparisons are performed between the Boltzmann collision model and BGK models based on various benchmarks, such as Fourier flow, Couette flow, and shock wave. The prediction performance of the ES-BGK model is better than that of the S-BGK model in Fourier flow, while prediction performance of the S-BGK model is better than that of the ES-BGK model in Couette flow and shock wave. However, with increasing Knudsen number or Mach number, the results of both ES-BGK and S-BGK deviate from the Boltzmann solutions. These phenomena are attributed to the incorrect governing equations of high-order moments of BGK models. To improve the performance of the current BGK models, the S-BGK model is extended by adding more high-order moments into the target distribution function of the original one. Our analytical and numerical results demonstrate that the extended S-BGK (S-BGK+) model provides the same relaxation coefficients as the Boltzmann collision model for the production terms of high-order moment equations. Compared with the other BGK models, the proposed S-BGK+ model exhibits better performance for various flow regimes.

A unified stochastic particle method based on the Bhatnagar-Gross-Krook model for polyatomic gases and its combination with DSMC

Simulating hypersonic flow around a space vehicle is challenging because of the multiscale and nonequilibrium nature inherent in these flows. To effectively deal with such flows, a hybrid scheme combining the stochastic particle Bhatnagar-Gross-Krook (BGK) method with direct simulation Monte Carlo (DSMC) was developed recently, but only for monatomic gases (Fei et al. (2021) [29]). In this paper, the particle-particle hybrid method is extended to polyatomic gas flows. In the near continuum regime, employing the Ellipsoidal–Statistical BGK model proposed by Dauvois et al. (2021) [22] with discrete levels of vibrational energy, the stochastic particle BGK method for polyatomic gases is first established following the idea of the unified stochastic particle BGK (USP-BGK) scheme. It is proven to be of second-order accuracy in the fluid limit. After that, the USP-BGK scheme with rotational and vibrational energies is combined with DSMC to construct a hybrid scheme. The present hybrid scheme for polyatomic gases is validated with numerical tests of homogeneous relaxation, 1D shock structure and 2D hypersonic flows past a wedge and a cylinder. Compared to the other stochastic particle methods, the proposed hybrid scheme can achieve higher accuracy at a much lower computational cost. Therefore, it is a more efficient tool to study multiscale hypersonic flows.

Isogeometric schemes in rarefied gas dynamics context

Geometrical complexity is typical of missile and reentry systems, small-scale devices with intrinsic milli-to-micro scale features, or in general topology-rich systems. Additionally, at high altitudes or at microscales, when the flow becomes rarefied, the conventional continuum/macroscopic equations produce unsatisfactory results. A microscopic/kinetic description becomes necessary. In this work, we introduce isogeometric (IG) kinetic schemes (KS) for the full Boltzmann equation and related kinetic models (Bhatnagar–Gross–Krook (BGK), Ellipsoidal statistical BGK, and Shakov). These schemes are specifically aimed for modeling flows in topologically complex “multi-patch” geometries such as those constructed using computer-aided geometric design tools; high order accurate; unconditionally time-stable (as per our tests) following generalized-α method; non-linearly stable following streamlined upwind Petrov–Galerkin method applied at every patch level with patch-to-patch coupling enforced either strongly by constraining the interface using single degree of freedom, or weakly using numerical fluxes as in discontinuous Galerkin (useful because the underlying system is first order in space); amenable to nearly-linear parallel efficiency owing to the six-dimensional nature of Boltzmann equation, wherein the convection and collision can be, respectively, performed simultaneously over velocity and physical spaces. In particular, since the surface representation is exact in IG, when surface properties are to be computed, which is often the case in aerodynamic applications, these schemes perform well. A series of verification tests for 1D/2D/3D-3V initial–boundary value and boundary-value flow problems, involving momentum, energy, and diffusive transport, including problems posed on non-conforming domains/patches, is performed to illustrate the stability and accuracy of the proposed method.

A particle-based ellipsoidal statistical Bhatnagar–Gross–Krook solver with variable weights for the simulation of large density gradients in micro- and nano-nozzles

This paper demonstrates the efficiency of a modified particle based Ellipsoidal Statistical Bhatnagar–Gross–Krook (ESBGK) solver to simulate micro-nozzles. For this, the common particle ESBGK algorithm is adapted to handle variable particle weights including the creation of additional particles in regions with low statistical samples and merging of particles in dense regions. After the description of the methods and their implementation, the simulation results of a micro-nozzle geometry using the Direct Simulation Monte Carlo, the common particle ESBGK, and the proposed modified ESBGK method are compared concerning accuracy and efficiency. All three methods show good agreement; however, the modified ESBGK method has the highest efficiency, saving a factor of around 500 of computational time to produce a comparable statistical sample size in the rarefied expansion region.

A Benchmark Study of Kinetic Models for Shock Waves

It is well-known that the Boltzmann collision term can be simplified by Bhatnagar–Gross–Krook (BGK) and Fokker–Planck (FP) models. These two kinetic models satisfy the conservation laws; however, their predictions of the Prandtl number are inconsistent with that derived from the Boltzmann equation. A variety of modifications, such as Shakhov BGK (S-BGK), Ellipsoidal Statistical BGK (ES-BGK), Cubic Fokker–Planck (Cubic-FP), and Ellipsoidal Statistical Fokker–Planck (ES-FP), models have been proposed and they can provide correct Prandtl number. However, in the transition regime, the validity of simplified kinetic models is still controversial. In this work, the performance of four kinetic models is investigated through theoretical analysis and numerical comparison of the high-order moments beyond shear stress and heat flux. The simulations of the shock waves show that the results of these modified kinetic models deviate from the Boltzmann equation as Mach number increases. On the whole, the S-BGK model, which gives the most approximated relaxation coefficients for the production terms of high-order moments, provides the best predictions in the transition regime. For the ES-FP and Cubic-FP models, besides the incorrect relaxation coefficients, their deviations at large Mach numbers are further caused by inconstant Prandtl number and overestimated nonlinear terms, respectively.

A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions with second-order accuracy. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contexts

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.

Evaluation of particle-based continuum methods for a coupling with the direct simulation Monte Carlo method based on a nozzle expansion

This paper investigates three different particle-based continuum methods, the ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) and Fokker-Planck (ESFP) methods and the Low Diffusion (LD) method, for a coupling with the direct simulation Monte Carlo (DSMC) method. After a short description of the methods and their implementation, including the coupling concept for the LD-DSMC, simulation results of a nozzle expansion are compared with available experimental measurements and a DSMC simulation. Excellent agreement between ESBGK, ESFP, and DSMC can be observed in the throat of the nozzle, while the LD method fails to predict the correct velocity, temperature, and density profile. Further downstream, only the DSMC and the coupled ESBGK/ESFP-DSMC simulations are able to reproduce the measured rotational temperature profiles. A performance comparison shows the possible computational savings of a coupled ESBGK/ESFP-DSMC simulation, where a speedup of four orders of magnitude can be observed compared to a regular DSMC simulation.

Extending the particle ellipsoidal statistical Bhatnagar-Gross-Krook method to diatomic molecules including quantized vibrational energies

The particle-based ellipsoidal statistical Bhatnagar-Gross-Krook (ESBGK) model is extended to diatomic molecules and compared with the Direct Simulation Monte Carlo (DSMC) method. For this, an efficient method is developed that optionally allows the handling of quantized vibrational energies. The proposed method is verified with a gas in an adiabatic box relaxing from a non-equilibrium state to an equilibrium. It is shown that the analytical Landau-Teller expression as well as DSMC results agrees very well with the new method. Furthermore, the method is compared with DSMC results and experimental measurements of a hypersonic flow around a 70° blunted cone. It is shown that the ESBGK compares very well with the DSMC results while saving up to a factor of ≈35.8 computational time for this low Knudsen number case.

Particle-based fluid dynamics: Comparison of different Bhatnagar-Gross-Krook models and the direct simulation Monte Carlo method for hypersonic flows

The Bhatnagar-Gross-Krook (BGK) model and its extensions (ellipsoidal statistical BGK, Shakhov BGK, and unified BGK) are used in particle-based fluid dynamics and compared with the Direct Simulation Monte Carlo (DSMC) method. To this end, different methods are investigated that allow efficient sampling of the Shakhov and the unified target distribution functions. As a consequence, particle simulations based on the Shakhov BGK and unified BGK models are possible in an efficient way. Furthermore, different energy conservation schemes are tested for the BGK models. It is shown that the choice of the energy conservation scheme has a major impact on the quality of the results for each model. The models are verified with a Couette flow problem at different Knudsen numbers and wall velocities. Furthermore, the models are compared with the DSMC results of a hypersonic flow around a 70° blunted cone. It is shown that the unified BGK model is able to reproduce rarefied gas phenomena. Furthermore, it is shown that the difference in the reproduction of the shock structure is not significant between the ellipsoidal statistical BGK and Shakhov BGK models for the flow around a 70° blunted cone and significantly depends on the energy conservation method. The choice of the energy conservation method is especially crucial for the Shakhov model whereas the ellipsoidal statistical BGK model is much more robust concerning the energy conservation scheme. Additionally, a computational time study is performed to show the efficiency of BGK-based simulations for low Knudsen number flows compared with DSMC.

A coupled ordinates method for solution acceleration of rarefied gas dynamics simulations

Non-equilibrium rarefied flows are frequently encountered in a wide range of applications, including atmospheric re-entry vehicles, vacuum technology, and microscale devices. Rarefied flows at the microscale can be effectively modeled using the ellipsoidal statistical Bhatnagar–Gross–Krook (ESBGK) form of the Boltzmann kinetic equation. Numerical solutions of these equations are often based on the finite volume method (FVM) in physical space and the discrete ordinates method in velocity space. However, existing solvers use a sequential solution procedure wherein the velocity distribution functions are implicitly coupled in physical space, but are solved sequentially in velocity space. This leads to explicit coupling of the distribution function values in velocity space and slows down convergence in systems with low Knudsen numbers. Furthermore, this also makes it difficult to solve multiscale problems or problems in which there is a large range of Knudsen numbers. In this paper, we extend the coupled ordinates method (COMET), previously developed to study participating radiative heat transfer, to solve the ESBGK equations. In this method, at each cell in the physical domain, distribution function values for all velocity ordinates are solved simultaneously. This coupled solution is used as a relaxation sweep in a geometric multigrid method in the spatial domain. Enhancements to COMET to account for the non-linearity of the ESBGK equations, as well as the coupled implementation of boundary conditions, are presented. The methodology works well with arbitrary convex polyhedral meshes, and is shown to give significantly faster solutions than the conventional sequential solution procedure. Acceleration factors of 5–9 are obtained for low to moderate Knudsen numbers on single processor platforms.

A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments

We propose a Bhatnagar–Gross–Krook (BGK) kinetic model in which the collision frequency is a linear combination of polynomials in the velocity variable. The coefficients of the linear combination are determined so as to enforce proper relaxation rates for a selected group of moments. The relaxation rates are obtained by a direct numerical evaluation of the full Boltzmann collision operator. The model is conservative by construction. Simulations of the problem of spatially homogeneous relaxation of hard spheres gas show improvement in accuracy of controlled moments as compared to solutions obtained by the classical BGK, ellipsoidal-statistical BGK and the Shakhov models in cases of strong deviations from continuum.

Experimental measurements and numerical simulations of the Knudsen force on a non-uniformly heated beam

Rarefied thermally-induced flows generate Knudsen thermal force on structures immersed in a gas and are significant for vacuum microbalances as well as for microsystem sensors and actuators. We study experimentally and numerically the influence of transverse thermal gradients on the Knudsen thermal force. A Peltier device provided the thermal gradient in a beam located in close proximity of a large reaction plate, whose displacement has been sensed using a torsional balance. The pressure in the vacuum chamber varied between 1 and 100 Pa giving a range of Knudsen numbers from the slip to nearly free molecular regimes, based on the gap between the beam and the movable plate. Switching polarity of the Peltier heater produces reversal of the Knudsen thermal force, from repulsive to attractive. The numerical solution of Ellipsoidal Statistical Bhatnagar–Gross–Krook (ESBGK) kinetic model of the Boltzmann equation agrees well with experimental data. The obtained non-dimensional Knudsen force coefficient, dependent on the Knudsen number, temperature ratio and the gap height to beam width aspect ratio, quantifies the magnitude of the force for a wide range of generic conditions and could be used for design and analysis of sensors and actuators exploiting this rarefied flow effect.

Multi-scale study of condensation in water jets using ellipsoidal-statistical Bhatnagar-Gross-Krook and molecular dynamics modeling.

Homogeneous water condensation and ice formation in supersonic expansions to vacuum for stagnation pressures from 12 to 1000 mbar are studied using the particle-based Ellipsoidal-Statistical Bhatnagar-Gross-Krook (ES-BGK) method. We find that when condensation starts to occur, at a stagnation pressure of 96 mbar, the increase in the degree of condensation causes an increase in the rotational temperature due to the latent heat of vaporization. The simulated rotational temperature profiles along the plume expansion agree well with measurements confirming the kinetic homogeneous condensation models and the method of simulation. Comparisons of the simulated gas and cluster number densities, cluster size for different stagnation pressures along the plume centerline were made and it is found that the cluster size increase linearly with respect to stagnation pressure, consistent with classical nucleation theory. The sensitivity of our results to cluster nucleation model and latent heat values based on bulk water, specific cluster size, or bulk ice are examined. In particular, the ES-BGK simulations are found to be too coarse-grained to provide information on the phase or structure of the clusters formed. For this reason, molecular dynamics simulations of water condensation in a one-dimensional free expansion to simulate the conditions in the core of a plume are performed. We find that the internal structure of the clusters formed depends on the stagnation temperature. A larger cluster of average size 21 was tracked down the expansion, and a calculation of its average internal temperature as well as a comparison of its radial distribution functions (RDFs) with values measured for solid amorphous ice clusters lead us to conclude that this cluster is in a solid-like rather than liquid form. In another molecular-dynamics simulation at a much lower stagnation temperature, a larger cluster of size 324 and internal temperature 200 K was extracted from an expansion plume and equilibrated to determine its RDF and self-diffusion coefficient. The value of the latter shows that this cluster is formed in a supercooled liquid state rather than in an amorphous solid state.

Near-Contact Gas Damping and Dynamic Response of High-g MEMS Accelerometer Beams

This paper introduces and experimentally validates a new model for near-contact gas damping of microbeams. The model is formulated based on numerical simulations of rarefied gas dynamics using the Boltzmann Ellipsoidal Statistical Bhatnagar-Gross-Krook (ES-BGK) equation. The result is compared with existing models by simulating the motion of beams under high-g acceleration. To experimentally validate the damping models, single crystal silicon MEMS g-switches with cantilever microbeams of various lengths were utilized. The experimental measurements of beam dynamics under peak accelerations of approximately 50,000 g and acceleration ramp rates from 600 to 3,000 g/μs are compared with simulations. Additionally, the damping coefficients are extracted from existing vibrational mode data, and the resulting values are compared to the various models. The new near-contact model was found to predict contact and release times within a root-mean-square deviation from experiment below 9 and 7 for contact and release events, respectively. The damping values for the vibrational modes away from contact were predicted within 33% error, showing a more consistent predictive capability than provided by earlier models.

Assessment of the ellipsoidal-statistical Bhatnagar–Gross–Krook model for force-driven Poiseuille flows

We investigate the accuracy of the ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model for planar force-driven Poiseuille flows. Our numerical simulations are conducted using the deterministic discrete velocity method, for Knudsen numbers (Kn) ranging from 0.05 to 10. While we provide numerically accurate data, our aim is to assess the accuracy of the ES-BGK model for these flows. By comparing with data from the direct simulation Monte Carlo (DSMC) method and the Boltzmann equation, the ES-BGK model is found to be able to predict accurate velocity and temperature profiles in the slip flow regime (0.01<Kn⩽0.1), for both low-speed and high-speed flows. In the transition flow regime (0.1<Kn⩽10), however, the model does not quantitatively capture the viscous heating effect.

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

AbstractA thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small-Mach-number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately $0. 5$. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

Linear theory of sound waves with evaporation and condensation

An asymptotic analysis of a boundary-value problem of the Boltzmann equation for small Knudsen number is carried out for the case when an unsteady flow of polyatomic vapour induces reciprocal evaporation and condensation at the interface between the vapour and its liquid phase. The polyatomic version of the Boltzmann equation of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) model is used and the asymptotic expansions for small Knudsen numbers are applied on the assumptions that the Mach number is sufficiently small compared with the Knudsen number and the characteristic length scale divided by the characteristic time scale is comparable with the speed of sound in a reference state, as in the case of sound waves. In the leading order of approximation, we derive a set of the linearized Euler equations for the entire flow field and a set of the boundary-layer equations near the boundaries (the vapour–liquid interface and simple solid boundary). The boundary conditions for the Euler and boundary-layer equations are obtained at the same time when the solutions of the Knudsen layers on the boundaries are determined. The slip coefficients in the boundary conditions are evaluated for water vapour. A simple example of the standing sound wave in water vapour bounded by a liquid water film and an oscillating piston is demonstrated and the effect of evaporation and condensation on the sound wave is discussed.

Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls

The ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model is investigated for steady gas-phase transport of heat, tangential momentum, and mass between parallel walls (i.e., Fourier, Couette, and Fickian flows). This investigation extends the original study of Cercignani and Tironi, who first applied the ES-BGK model to heat transport (i.e., Fourier flow) shortly after this model was proposed by Holway. The ES-BGK model is implemented in a molecular-gas-dynamics code so that results from this model can be compared directly to results from the full Boltzmann collision term, as computed by the same code with the direct simulation Monte Carlo (DSMC) algorithm of Bird. A gas of monatomic molecules is considered. These molecules collide in a pairwise fashion according to either the Maxwell or the hard-sphere interaction and reflect from the walls according to the Cercignani–Lampis–Lord model with unity accommodation coefficients. Simulations are performed at pressures from near-free-molecular to near-continuum. Unlike the BGK model, the ES-BGK model produces heat-flux and shear-stress values that both agree closely with the DSMC values at all pressures. However, for both interactions, the ES-BGK model produces molecular-velocity-distribution functions that are qualitatively similar to those determined for the Maxwell interaction from Chapman–Enskog theory for small wall temperature differences and moment-hierarchy theory for large wall temperature differences. Moreover, the ES-BGK model does not produce accurate values of the mass self-diffusion coefficient for either interaction. Nevertheless, given its reasonable accuracy for heat and tangential-momentum transport, its sound theoretical foundation (it obeys the H-theorem), and its available extension to polyatomic molecules, the ES-BGK model may be a useful method for simulating certain classes of single-species noncontinuum gas flows, as Cercignani suggested.