Ellipsoidal Statistical Model Research Articles

Navier–Stokes Equations and Bulk Viscosity for a Polyatomic Gas with Temperature-Dependent Specific Heats

A system of Navier–Stokes-type equations with two temperatures is derived, for a polyatomic gas with temperature-dependent specific heats (thermally perfect gas), from the ellipsoidal statistical (ES) model of the Boltzmann equation extended to such a gas. Subsequently, the system is applied to the problem of shock-wave structure for a gas with large bulk viscosity (or, equivalently, with slow relaxation of the internal modes), and the numerical results are compared with those based on the ordinary Navier–Stokes equations. It is shown that the latter equations fail to describe the double-layer structure of shock profiles for a gas with large bulk viscosity.

Study of Shock Structures Using the Unified Gas-Kinetic Wave-Particle Method with Various BGK Models

Recently, the unified gas-kinetic wave-particle (UGKWP) method was proposed to describe rarefied flows efficiently in all Knudsen numbers. However, the initial UGKWP method with the Bhatnagar–Gross–Krook (BGK) model has its own limitations due to the unit Prandtl number. To further improve its performance in non-equilibrium flows, the UGKWP method is extended for the ellipsoidal statistical model (ESBGK) and Shakhov model (SBGK) in this paper. The numerical results of shock structures of the above BGK-type models show that the SBGK model fits the reference data best except the asymmetry parameter, and the ESBGK model comes the second. Meanwhile, in order to overcome the deficiency of negative probability density function (PDF) in SBGK, the UGKWP method with positive-preserving Shakhov PDF is also constructed. It weakens the early rising of temperature phenomenon that existed in the SBGK model, and improves the performance in predicting the stress and heat flux evidently at high Mach numbers.

Erratum to: On the entropic property of the ellipsoidal statistical model with the Prandtl number below 2/3

Erratum to: On the entropic property of the ellipsoidal statistical model with the Prandtl number below 2/3

Boundary conditions for two-temperature Navier-Stokes equations for a polyatomic gas

We previously derived the two-temperature Navier-Stokes system (a set of compressible Navier-Stokes equations with the translational and internal temperatures) for a polyatomic gas with slow relaxation of the internal modes from the ellipsoidal statistical model of the Boltzmann equation. In this paper, the appropriate boundary conditions for the two-temperature Navier-Stokes system are derived. The results are presented in a form that is applicable to practical applications immediately.

An ellipsoidal statistical model for a monoatomic and a polyatomic gas mixture

An ellipsoidal statistical model for a monoatomic and a polyatomic gas mixture

Two-temperature Navier-Stokes equations for a polyatomic gas derived from kinetic theory.

A polyatomic gas with slow relaxation of the internal modes is considered, and the Navier-Stokes equations with two temperatures, the translational and internal temperatures, are derived for such a gas on the basis of the ellipsoidal-statistical (ES) model of the Boltzmann equation for a polyatomic gas, proposed by Andries etal. [Eur. J. Mech. B, Fluids 19, 813 (2000)10.1016/S0997-7546(00)01103-1], by the Chapman-Enskog procedure. Then, the derived equations are applied to numerically investigate the structure of a plane shock wave in CO_{2} gas, which is known to have slowly relaxing internal modes. The results show good agreement with those obtained by the direct numerical analysis of the ES model for moderately strong shock waves. In particular, the results perfectly reproduce the double-layer structure of the shock profiles consisting of a thin front layer with rapid change and a thick rear layer with slow relaxation of the internal modes.

On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3

Entropic property of the Ellipsoidal Statistical model with the Prandtl number Pr below 2/3 is discussed. Although 2/3 is the lower bound of Pr for the H theorem to hold unconditionally, it is shown that the theorem still holds even for $\mathrm{Pr}<2/3$, provided that anisotropy of stress tensor satisfies a certain criterion. The practical tolerance of that criterion is assessed numerically by the strong normal shock wave and the Couette flow problems. A couple of moving plate tests are also conducted.

Kinetic Methods for Solving Unsteady Problems with Jet Flows

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

A Kinetic Model for a Polyatomic Gas with Temperature-Dependent Specific Heats and Its Application to Shock-Wave Structure

The ellipsoidal statistical (ES) model of the Boltzmann equation for a polyatomic gas with constant specific heats (calorically perfect gas), proposed by Andries et al. (Eur J Mech B Fluids 19:813, 2000), is extended to a polyatomic gas with temperature-dependent specific heats (thermally perfect gas). Then, the new model equation is applied to investigate the structure of a plane shock wave with special interest in $$\hbox {CO}_2$$ gas, which is known to have a very large bulk viscosity, and in the case of relatively strong shock waves. A numerical analysis, as well as an asymptotic analysis for large bulk viscosity, is performed in parallel to the previous paper by two of the present authors (Kosuge and Aoki, in: Phys Rev Fluids 3:023401, 2018), where the structure of a shock wave in $$\hbox {CO}_2$$ gas was investigated using the ES model for a polyatomic gas with constant specific heats. From the numerical and analytical results, the effect of temperature-dependent specific heats on the structure of a shock wave is clarified.

Derivation and numerical comparison of Shakhov and Ellipsoidal Statistical kinetic models for a monoatomic gas mixture

Gas mixtures are important for many practical applications. Extending kinetic model equations of the Bhatnagar–Gross–Krook (BGK) type from a single-species gas to a multi-species gas mixture presents a number of significant challenges. First, obtaining the correct species diffusions, viscous stresses as well as heat conduction in the continuum limit requires a careful design of the collision terms in the kinetic equations. Secondly, the derived model collision terms need to guarantee positivity of the macroscopic fields. In the present work, two new kinetic models are introduced and compared: an approach based on the Shakhov kinetic model and an approach involving an anisotropic Gaussian equilibrium function. The two new models are capable of modelling a binary mixture of monoatomic gases, with updated definitions for the relaxation parameters and target species velocities and temperature. Both methods account for separate species-mean velocity such that the species diffusion and velocity drift are accurately represented. The key contribution of the models is the exact recovery of the Fick, Newton and Fourier laws in the continuum limit, while preserving positive temperature fields and crucial properties of the Boltzmann equation. The profile of a normal shock wave is inspected under various flow conditions to numerically validate the two models. The results show improvement upon comparison with a model, which has two correct transport coefficients, and demonstrate the ability to reliably model inert gas mixtures.

A rarefied gas flow around a rotating sphere: diverging profiles of gradients of macroscopic quantities

The steady behaviour of a rarefied gas around a rotating sphere is studied numerically on the basis of the linearised ellipsoidal statistical model of the Boltzmann equation, also known as the ES model, and the Maxwell diffuse–specular boundary condition. It is demonstrated numerically that the normal derivative of the circumferential component of the flow velocity and that of the heat flux diverge on the boundary with a rate $s^{-1/2}$ , where $s$ is the normal distance from the boundary. Further, it is demonstrated that the diverging term is proportional to the magnitude of the jump discontinuity of the velocity distribution function on the boundary, which originates from the mismatch of the incoming and outgoing data on the boundary. The moment of force exerted on the sphere is also obtained for a wide range of the Knudsen number and for various values of the accommodation coefficient.

Kinetic Methods for Solving Non-stationary Jet Flow Problems

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

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.

Slip boundary conditions for the compressible Navier-Stokes equations for a polyatomic gas

The slip boundary conditions for the compressible Navier-Stokes equations for a polyatomic gas are derived from kinetic theory using the ellipsoidal statistical model of the Boltzmann equation. The slip coefficients for some typical polyatomic gases are presented explicitly.

Shock wave structure in rarefied polyatomic gases with large relaxation time for the dynamic pressure

The shock wave structure in rarefied polyatomic gases is analyzed based on extended thermodynamics (ET). In particular, the case with large relaxation time for the dynamic pressure, which corresponds to large bulk viscosity, is considered by adopting the simplest version of extended thermodynamics with only 6 independent fields (ET6); the mass density, the velocity, the temperature and the dynamic pressure. Recently, the validity of the theoretical predictions by ET was confirmed by the numerical analysis based on the kinetic theory in [S Kosuge and K Aoki: Phys. Rev. Fluids, Vol. 3, 023401 (2018)]. It was shown that numerical results using the polyatomic version of ellipsoidal statistical model agree with the theoretical predictions by ET for small or moderately large Mach numbers. In the present paper, first, we compare the theoretical predictions by ET6 with the ones by kinetic theory for large Mach number under the same assumptions, that is, the gas is polytropic and the bulk viscosity is proportional to the temperature. Second, the shock wave structure for large Mach number in a non-polytropic gas is analyzed with the particular interest in the effect of the temperature dependence of specific heat and the bulk viscosity on the shock wave structure. Through the analysis of the case of a rarefied carbon dioxide (CO2) gas, it is shown that these temperature dependences play important roles in the precise analysis of the structure for strong shock waves.

Discrete ellipsoidal statistical BGK model and Burnett equations

To simulate non-equilibrium compressible flows, a new discrete Boltzmann model, discrete Ellipsoidal Statistical(ES)-BGK model, is proposed. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in Burnett level, two kinds of discrete velocity model are introduced; the relations between non-equilibrium quantities and the viscous stress and heat flux in Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, based on the Navier-Stokes, the Burnett equations, etc.

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases

We propose an extension of the Fokker–Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier–Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar–Gross–Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman–Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.

A Fokker–Planck Model of the Boltzmann Equation with Correct Prandtl Number

We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.

Derivation of model kinetic equation

A new form of the model collision operator is derived. One-component and many-component systems are considered. The collision operator proposed takes properly into account the relaxation of the first 13 hydrodynamic moments. A technique for construction of the model collision integral based on a known expression for the linearized model operator is proposed. It is shown that, within our model, the model collision operator does not contain the complicated exponential, common for ellipsoidal statistical models. Boltzmann's H-theorem is proved for our model.

Motion of an array of plates in a rarefied gas caused by radiometric force.

In a rarefied gas in an infinitely long channel between two parallel plates, an array of infinitely many plates, arranged longitudinally with uniform interval, is placed along the channel. The array is assumed to be freely movable along the channel. If one side of each plate is heated, the radiometric force acts on it, and the array starts moving toward the cold sides of the plates. The final steady motion of the array, as well as the corresponding behavior of the gas, is investigated numerically on the basis of kinetic theory using the ellipsoidal statistical model of the Boltzmann equation. As the solution method, a finite-difference method, with a method of characteristics incorporated, that is able to capture the discontinuity in the velocity distribution function is employed. As the result, the local flow field near the edges of the plates and the terminal velocity of the array are obtained accurately for relatively small Knudsen numbers.