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18 pages, 982 KiB

Open AccessArticle

Analysis of Carleman Linearization of Lattice Boltzmann

by Wael Itani and Sauro Succi

Fluids 2022, 7(1), 24; https://doi.org/10.3390/fluids7010024 - 5 Jan 2022

Cited by 18 | Viewed by 3076

We explore the Carleman linearization of the collision term of the lattice Boltzmann formulation, as a first step towards formulating a quantum lattice Boltzmann algorithm. Specifically, we deal with the case of a single, incompressible fluid with the Bhatnagar Gross and Krook equilibrium [...] Read more.

We explore the Carleman linearization of the collision term of the lattice Boltzmann formulation, as a first step towards formulating a quantum lattice Boltzmann algorithm. Specifically, we deal with the case of a single, incompressible fluid with the Bhatnagar Gross and Krook equilibrium function. Under this assumption, the error in the velocities is proportional to the square of the Mach number. Then, we showcase the Carleman linearization technique for the system under study. We compute an upper bound to the number of variables as a function of the order of the Carleman linearization. We study both collision and streaming steps of the lattice Boltzmann formulation under Carleman linearization. We analytically show why linearizing the collision step sacrifices the exactness of streaming in lattice Boltzmann, while also contributing to the blow up in the number of Carleman variables in the classical algorithm. The error arising from Carleman linearization has been shown analytically and numerically to improve exponentially with the Carleman linearization order. This bodes well for the development of a corresponding quantum computing algorithm based on the lattice Boltzmann equation. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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21 pages, 454 KiB

Open AccessArticle

Extraction of Tangential Momentum and Normal Energy Accommodation Coefficients by Comparing Variational Solutions of the Boltzmann Equation with Experiments on Thermal Creep Gas Flow in Microchannels

by Tommaso Missoni, Hiroki Yamaguchi, Irina Graur and Silvia Lorenzani

Fluids 2021, 6(12), 445; https://doi.org/10.3390/fluids6120445 - 9 Dec 2021

Cited by 4 | Viewed by 3096

Abstract

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients,

σ_{1, T}

and

σ_{2, T}

, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation [...] Read more.

In the present paper, we provide an analytical expression for the first- and second-order thermal slip coefficients,

σ_{1, T}

and

σ_{2, T}

, by means of a variational technique that applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator for hard-sphere molecules. The Cercignani-Lampis scattering kernel of the gas-surface interaction has been considered in order to take into account the influence of the accommodation coefficients (

α_{t}

,

α_{n}

) on the slip parameters. Comparing our theoretical results with recent experimental data on the mass flow rate and the slip coefficient for five noble gases (helium, neon, argon, krypton, and xenon), we found out that there is a continuous set of values for the pair (

α_{t}

,

α_{n}

) which leads to the same thermal slip parameters. To uniquely determine the accommodation coefficients, we took into account a further series of measurements carried out with the same experimental apparatus, where the thermal molecular pressure exponent

γ

has been also evaluated. Therefore, the new method proposed in the present work for extracting the accommodation coefficients relies on two steps. First of all, since

γ

mainly depends on

α_{t}

, we fix the tangential momentum accommodation coefficient in such a way as to obtain a fair agreement between theoretical and experimental results. Then, among the multiple pairs of variational solutions for (

α_{t}

,

α_{n}

), giving the same thermal slip coefficients (chosen to closely approximate the measurements), we select the unique pair with the previously determined value of

α_{t}

. The analysis carried out in the present work confirms that both accommodation coefficients increase by increasing the molecular weight of the considered gases, as already highlighted in the literature. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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15 pages, 1214 KiB

Open AccessArticle

Shock Structures Using the OBurnett Equations in Combination with the Holian Conjecture

by Ravi Sudam Jadhav and Amit Agrawal

Fluids 2021, 6(12), 427; https://doi.org/10.3390/fluids6120427 - 26 Nov 2021

Cited by 5 | Viewed by 2227

Abstract

In the present work, we study the normal shock wave flow problem using a combination of the OBurnett equations and the Holian conjecture. The numerical results of the OBurnett equations for normal shocks established several fundamental aspects of the equations such as the [...] Read more.

In the present work, we study the normal shock wave flow problem using a combination of the OBurnett equations and the Holian conjecture. The numerical results of the OBurnett equations for normal shocks established several fundamental aspects of the equations such as the thermodynamic consistency of the equations, and the existence of the heteroclinic trajectory and smooth shock structures at all Mach numbers. The shock profiles for the hydrodynamic field variables were found to be in quantitative agreement with the direct simulation Monte Carlo (DSMC) results in the upstream region, whereas further improvement was desirable in the downstream region of the shock. For the discrepancy in the downstream region, we conjecture that the viscosity–temperature relation (

μ \propto T^{φ}

) needs to be modified in order to achieve increased dissipation and thereby achieve better agreement with the benchmark results in the downstream region. In this respect, we examine the Holian conjecture (HC), wherein transport coefficients (absolute viscosity and thermal conductivity) are evaluated using the temperature in the direction of shock propagation rather than the average temperature. The results of the modified theory (OBurnett + HC) are compared against the benchmark results and we find that the modified theory improves upon the OBurnett results, especially in the case of the heat flux shock profile. We find that the accuracy gain is marginal at lower Mach numbers, while the shock profiles are described better using the modified theory for the case of strong shocks. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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17 pages, 334 KiB

Open AccessArticle

A Review on BGK Models for Gas Mixtures of Mono and Polyatomic Molecules

by Marlies Pirner

Fluids 2021, 6(11), 393; https://doi.org/10.3390/fluids6110393 - 1 Nov 2021

Cited by 8 | Viewed by 2494

Abstract

We consider the Bathnagar–Gross–Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal [...] Read more.

We consider the Bathnagar–Gross–Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

23 pages, 504 KiB

Open AccessArticle

Molecular Extended Thermodynamics of Rarefied Polyatomic Gases with a New Hierarchy of Moments

by Takashi Arima and Tommaso Ruggeri

Fluids 2021, 6(2), 62; https://doi.org/10.3390/fluids6020062 - 1 Feb 2021

Cited by 2 | Viewed by 2091

Abstract

The aim of this paper is to construct the molecular extended thermodynamics for classical rarefied polyatomic gases with a new hierarchy, which is absent in the previous procedures of moment equations. The new hierarchy is deduced recently from the classical limit of the [...] Read more.

The aim of this paper is to construct the molecular extended thermodynamics for classical rarefied polyatomic gases with a new hierarchy, which is absent in the previous procedures of moment equations. The new hierarchy is deduced recently from the classical limit of the relativistic theory of moments associated with the Boltzmann–Chernikov equation. The field equations for 15 moments of the distribution function, in which the internal degrees of freedom of a molecule are taken into account, are closed with the maximum entropy principle. It is shown that the theory contains, as a principal subsystem, the previously polyatomic 14 fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges, instead of to the Grad 13-moment system, to the Kremer 14-moment system. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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22 pages, 2890 KiB

Open AccessArticle

A Note on the Steady Navier–Stokes Equations Derived from an ES–BGK Model for a Polyatomic Gas

by Kazuo Aoki, Marzia Bisi, Maria Groppi and Shingo Kosuge

Fluids 2021, 6(1), 32; https://doi.org/10.3390/fluids6010032 - 8 Jan 2021

Cited by 4 | Viewed by 2051

Abstract

The two-temperature Navier–Stokes equations derived from an ellipsoidal Bhatnagar-Gross-Krook (ES-BGK) model for a polyatomic gas (Phys. Rev. E102, 023104 (2020)) are considered in regimes where bulk viscosity is much greater than the shear viscosity. Possible existence of a shock-wave solution [...] Read more.

The two-temperature Navier–Stokes equations derived from an ellipsoidal Bhatnagar-Gross-Krook (ES-BGK) model for a polyatomic gas (Phys. Rev. E102, 023104 (2020)) are considered in regimes where bulk viscosity is much greater than the shear viscosity. Possible existence of a shock-wave solution for the steady version of these hydrodynamic equations is investigated resorting to the qualitative theory of dynamical systems. Stability properties of upstream and downstream equilibria are discussed for varying parameters. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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17 pages, 549 KiB

Open AccessArticle

Low-Speed DSMC Simulations of Hotwire Anemometers at High-Altitude Conditions

by Christopher A. Roseman and Brian M. Argrow

Fluids 2021, 6(1), 20; https://doi.org/10.3390/fluids6010020 - 2 Jan 2021

Cited by 4 | Viewed by 2622

Abstract

Numerical simulations of hotwire anemometers in low-speed, high-altitude conditions have been carried out using the direct simulation Monte Carlo (DSMC) method. Hotwire instruments are commonly used for in-situ turbulence measurements because of their ability to obtain high spatial and temporal resolution data. Fast [...] Read more.

Numerical simulations of hotwire anemometers in low-speed, high-altitude conditions have been carried out using the direct simulation Monte Carlo (DSMC) method. Hotwire instruments are commonly used for in-situ turbulence measurements because of their ability to obtain high spatial and temporal resolution data. Fast time responses are achieved by the wires having small diameters (1–5

μ

m). Hotwire instruments are currently being used to make in-situ measurements of high-altitude turbulence (20–40 km). At these altitudes, hotwires experience Knudsen number values that lie in the transition-regime between slip-flow and free-molecular flow. This article expands the current knowledge of hotwire anemometers by investigating their behavior in the transition-regime. Challenges involved with simulating hotwires at high Knudsen number and low Reynolds number conditions are discussed. The ability of the DSMC method to simulate hotwires from the free-molecular to slip-flow regimes is demonstrated. Dependence of heat transfer on surface accommodation coefficient is explored and discussed. Simulation results of Nusselt number dependence on Reynolds number show good agreement with experimental data. Magnitude discrepancies are attributed to differences between simulation and experimental conditions, while discrepancies in trend are attributed to finite simulation domain size. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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18 pages, 994 KiB

Open AccessArticle

The Half-Range Moment Method in Harmonically Oscillating Rarefied Gas Flows

by Giorgos Tatsios, Alexandros Tsimpoukis and Dimitris Valougeorgis

Fluids 2021, 6(1), 17; https://doi.org/10.3390/fluids6010017 - 1 Jan 2021

Cited by 3 | Viewed by 1911

Abstract

The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative [...] Read more.

The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative ones for harmonically oscillating half-space and finite-medium flow setups respectively, are solved. The moment equations are derived from the linearized time-dependent BGK kinetic equation, operating accordingly over the positive and negative halves of the molecular velocity space. Moreover, the boundary conditions of the “positive” and “negative” moment equations are accordingly constructed from the half-range moments of the boundary conditions of the outgoing distribution function, assuming purely diffuse reflection. The oscillatory Stokes flow is characterized by the oscillation parameter, while the oscillatory Couette flow by the oscillation and rarefaction parameters. HRMM results for the amplitude and phase of the velocity and shear stress in a wide range of the flow parameters are presented and compared with corresponding results, obtained by the discrete velocity method (DVM). In the oscillatory Stokes flow the so-called penetration depth is also computed. When the oscillation frequency is lower than the collision frequency excellent agreement is observed, while when it is about the same or larger some differences are present. Overall, it is demonstrated that the HRMM can be applied to linear oscillatory rarefied gas flows, providing accurate results in a very wide range of the involved flow parameters. Since the computational effort is negligible, it is worthwhile to consider the efficient implementation of the HRMM to stationary and transient multidimensional rarefied gas flows. Full article

(This article belongs to the Special Issue Rarefied Gas Dynamics)

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