Force-driven Poiseuille Flow Research Articles

A theoretical framework of information preservation method and its application to low-speed nonequilibrium gas flows

Simulations of nonequilibrium gas flows have garnered significant interest in modern engineering problems involving rarefied gas flow characteristics. Despite the popularity of the direct simulation Monte Carlo (DSMC) method in simulating such flows, its use in low-speed flows is limited by statistical noises. The information preservation (IP) method is a promising alternative known for its low noise properties. In this study, a new theoretical framework for the IP method based on kinetic theory is introduced to offer complete understanding for the transport properties of the preserved information. Specifically, we introduce a velocity-information joint distribution function (VIJDF) and derive its governing equation as well as the corresponding macroscopic transport equations. To ensure the accuracy of the IP method, the total stress/heat flux in IP, including information stress/heat flux generated during movement and collision steps and compensation stress/heat flux imposed in the compensation step, is matched to the molecular stress/heat flux in DSMC. To this end, a nonequilibrium model for the VIJDF is proposed to evaluate the compensation stress/heat flux. The parameters in the collision model of IP are theoretically determined by equating the transport coefficients associated with the preserved information to the coefficients of viscosity and thermal conductivity in DSMC. Numerical simulations for a variety of nonequilibrium gas flows, including low-speed Couette flow, Fourier flow, high-speed Couette flow, external force-driven Poiseuille flow, lid-driven cavity flow, and thermal creep flow, demonstrate that the IP method can achieve similar accuracy as the DSMC method with a much smaller sampling size.

On application of the regularized lattice Boltzmann method for isothermal flows with non-vanishing Knudsen numbers

Despite the advances in microscale modelling capabilities, physically observed non-zero Knudsen number (Kn) effects still elude the lattice Boltzmann method (LBM). This article details various Regularized LBM (RLBM) strategies used to model microscale flows and presents a thorough analysis on their applicability for the force-driven Poiseuille flow problem. In particular, we compare the formulations of eight schemes: Projected RLBM (P-RLBM), standard and Galilean Invariant Filtered Collision RLBM, Kn-dependant P-RLBM, Recursively Regularized LBM (R-RLBM), Hybrid R-RLBM, Central-Moment RLBM, and Onsager-Regularized LBM. We demonstrate, at different Kn with discrete Maxwell’s diffuse kinetic wall boundary conditions, the numerical equivalence of the considered RLBM schemes for hard sphere molecules on the D2Q9 lattice. Thereafter, we include a Kn-dependant wall-modified mean-free-path and extend the characterization onto higher-order D2Q17, D2Q21 and Q2Q25-Zero-One-Three, Q2Q49-Zero-One-Two-Three lattices. The study indicates that RLBM schemes are Navier-Stokes solvers which fail to entirely capture all the non-equilibrium phenomenon of slip/transition regime flows.

Third-order accurate 13-moment equations for non-continuum transport phenomenon

The derivation of analytical equations of non-continuum macroscopic transport phenomena is underpinned by approximate descriptions of the particle distribution function and is required due to the inability of the Navier–Stokes equations to describe flows at high Knudsen number (Kn ∼ 1). In this paper, we present a compact representation of the second-order correction to the Maxwellian distribution function and 13-moment transport equations that contain fewer terms compared to available moment-based representations. The intrinsic inviscid and isentropic assumptions of the second-order accurate distribution function are then relaxed to present a third-order accurate representation of the distribution function, using which corresponding third-order accurate moment transport equations are derived. Validation studies performed for Grad’s second problem and the force-driven plane Poiseuille flow problem at non-zero Knudsen numbers for Maxwell molecules highlight an improvement over results obtained by using the Navier–Stokes equations and Grad’s 13-moment (G13) equations. To establish the ability of the proposed equations to accurately capture the bulk behavior of the fluid, the results of Grad’s second problem have been validated against the analytical solution of the Boltzmann equation. For the planar Poiseuille flow problem, validations against the direct simulation Monte Carlo method data reveal that, in contrast to G13 equations, the proposed equations are capable of accurately capturing the Knudsen boundary layer.

OBurnett Equations: Thermodynamically Consistent Continuum Theory Beyond the Navier–Stokes Regime

Abstract Rarefied gas flows are highly nonequilibrium flows whose flow physics cannot be discerned accurately within the framework of the Navier–Stokes equations. The Burnett equations and the Grad moment equations, which form a super-set of the Navier–Stokes equations, have been proposed in the literature to model such flows but not much success has been achieved because of some inherent limitations of these equations. In this review article, we mainly focus on the recently proposed Onsager-Burnett equations (Singh et al., 2017, “Derivation of stable Burnett equations for rarefied gas flows,” Phys. Rev. E 96, p. 013106) for rarefied gas flows, and the progress achieved so far by solving these equations for some benchmark flow problems. Like Burnett and Grad equations, the OBurnett equations form a super-set of the Navier–Stokes equations and belong to the class of higher order continuum transport equations. However, there are two fundamental aspects where the significance of the OBurnett equations is clearly visible. First, the OBurnett equations are unconditionally stable as well as thermodynamically consistent unlike the conventional Burnett and Grad moment equations. Second, the OBurnett constitutive relations for the stress tensor and the heat flux vector do not have any higher order derivatives of velocity, pressure, or temperature. This is quite significant since now the equations need the same number of boundary conditions as that of the Navier–Stokes equations. As such, the OBurnett equations form a complete theory, which cannot be said for the conventional Burnett equations. These two important aspects help to set the OBurnett equations apart from the rest of the higher order continuum theories. The results of the OBurnett equations are compiled for two benchmark rarefied flow problems: force-driven compressible Poiseuille flow and the normal shock wave flow problem. For force-driven compressible Poiseuille flow, the OBurnett equations successfully capture the nonequilibrium effects such as nonuniform pressure profile and presence of normal stresses and tangential heat flux in the flow. The accurate description of highly nonequilibrium internal structure of normal shocks has always been the stringent test for the higher order continuum theories. The results of the OBurnett equations for normal shocks show that there is no theoretical upper Mach number limit for the equations. Further, the equations predict smooth shock structures at all Mach numbers, existence of heteroclinic trajectory, positive entropy generation throughout the shock, and significant improvement over the results of the Navier–Stokes equations. Finally, the recently proposed Grad's second problem, which has the potential to become a benchmark problem, is discussed. The solution of Grad's second problem for different interaction potentials (Maxwell and hard-sphere molecules) within the Burnett hydrodynamics is also presented at length and some important remarks are made in this context.

Improved gas-kinetic unified algorithm for high rarefied to continuum flows by computable modeling of the Boltzmann equation

In this paper, we present an improved gas-kinetic unified algorithm (IGKUA) for high rarefied transition to continuum flows by computable modeling of Boltzmann equation. Compared with the original algorithm, the new method utilizes less needed discrete velocity ordinate points to obtain accurate results and removes the dependency of flow regime on computational time step, which can speed up the convergence in continuum flows. One of the novel strategies adopted in IGKUA is to develop a type of gas-kinetic quadrature rule that can exactly preserve conservation constraint of the model by adjusting the integral weights, increasing efficiency, and reducing nonphysical sources. Another key innovation is to introduce the analytical solutions of colliding-relaxation equation by considering the evolutions of associated macroscopic flow variables first, leading to no limit on the permissible time step. Numerical explicit and implicit schemes for unsteady flows are constructed to solve the physical convective equation, and Fourier spectral method is applied for the molecular-velocity convective movement equation analytically when the flows are under external-force fields. The IGKUA is tested using some numerical examples, including the shock-tube problems, Rayleigh flow, Couette flow, lid-driven cavity, external force-driven Poiseuille flow, and hypersonic flow past an infinite flat plate. Simulation results are in high resolution of the flow fields and match well with the results of the analytical, direct simulation Monte Carlo, Navier–Stokes solvers, and other reference methods. In addition, the new algorithm is better than the original one in the aspects of computational amount and time, which are more obvious when simulating the continuum flows.

Relative entropy based breakdown criteria for hybrid discrete velocity Bhatnagar–Gross–Krook and lattice Boltzmann method

In the present paper, novel breakdown criteria for the assessment of the validity of the continuum-fluid approximation are proposed. The approach is based on relative entropy (Kullback–Leibler divergence). The considered variants of the Kullback–Leibler divergence assess the contribution of non-hydrodynamic components of the gas velocity distribution function to the overall entropy. In addition, the chi-square statistic is introduced as a breakdown variable. As benchmark problems, the plane Couette and force driven Poiseuille flows are considered for various Knudsen and Mach numbers. The modeling is performed with the difference scheme for the kinetic Bhatnagar–Gross–Krook equation and the hybrid approach: the Bhatnagar–Gross–Krook equation is applied in non-equilibrium regions, and the lattice Boltzmann method is applied as the continuum-fluid method. The simulation results show that the proposed criteria can efficiently separate equilibrium and non-equilibrium domains.

Dense gas flow simulations in ultra-tight confinement

Modeling dense gas flows inside channels with sections comparable to the diameter of gas molecules is essential in porous medium applications, such as in non-conventional shale reservoir management and nanofluidic separation membranes. In this paper, we perform the first verification study of the Enskog equation by using particle simulation methods based on the same hard-sphere collisions dynamics. Our in-house Event-Driven Molecular Dynamics (EDMD) code and a pseudo-hard-sphere Molecular Dynamics (PHS-MD) solver are used to study force-driven Poiseuille flows in the limit of high gas densities and high confinements. Our results showed (a) very good agreement between EDMD, PHS-MD, and Enskog solutions across density, velocity, and temperature profiles for all the simulation conditions and (b) numerical evidence that deviations exist in the normalized mass flow rate vs Knudsen number curve compared to the standard curve without confinement. While we observe slight deviations in the Enskog density and velocity profiles from the MD when the reduced density is greater than 0.2, this limit is well above practical engineering applications, such as in shale gas. The key advantages of promoting the Enskog equation for upscaling flows in porous media lie in its ability to capture the non-equilibrium physics of tightly confined fluids while being computationally more efficient than fundamental simulation approaches, such as molecular dynamics and derivative solvers.

Analysis of the Moment Method and the Discrete Velocity Method in Modeling Non-Equilibrium Rarefied Gas Flows: A Comparative Study

In the present study, the performance of the moment method, in terms of accuracy and computational efficiency, was evaluated at both the macro- and microscopic levels. Three different types of non-equilibrium gas flows, including the force-driven Poiseuille flow, lid-driven and thermally induced cavity flows, were simulated in the slip and transition regimes. Choosing the flow fields obtained from the Boltzmann model equation as the benchmark, the accuracy and validation of Navier–Stokes–Fourier (NSF), regularized 13 (R13) and regularized 26 (R26) equations were explored at the macroscopic level. Meanwhile, we reconstructed the velocity distribution functions (VDFs) using the Hermite polynomials with different-order of molecular velocity moments, and compared them with the Boltzmann solutions at the microscopic level. Moreover, we developed a kinetic criterion to indirectly assess the errors of the reconstructed VDFs. The results have shown that the R13 and R26 moment methods can be faithfully used for non-equilibrium rarefied gas flows in the slip and transition regimes. However, as indicated from the thermally induced case, all of the reconstructed VDFs are still very close to the equilibrium state, and none of them can reproduce the accurate VDF profile when the Knudsen number is above 0.5.

Lattice Boltzmann simulations of liquid flows in microchannel with an improved slip boundary condition

Flow characteristics in channels are critical in chemical engineering. Slip velocity has a great influence on the microchannel flows. The lattice Boltzmann simulations of some typical channel flows are conducted with an improved slip boundary condition. Through theoretical analysis, the defects of the existing slip boundary conditions are clarified. To solve these problems, the slip boundary condition of Kuo et al. is ameliorated by considering the tangential momentum change generated by the external force. Besides, the Navier’s slip model is adopted to specify the improved slip boundary condition. With this improved scheme, we study five cases, namely the Couette flow, the force driven Poiseuille flow, the time-dependent force driven Womersley flow, the porous plate flow and the channel flow with a surface-mounted block. Our method is accurate and reliable enough even with a coarse grid. The mass leakage at the wall is avoided and the numerical error generated by the external force is eliminated.

A comparative study of discrete velocity methods for low-speed rarefied gas flows

In the study of rarefied gas dynamics, the discrete velocity method (DVM) has been widely employed to solve the gas kinetic equations. Although various versions of DVM have been developed, their performance, in terms of modeling accuracy and computational efficiency, is yet to be comprehensively studied in all the flow regimes. Here, the traditional third-order time-implicit Godunov DVM (GDVM) and the recently developed discrete unified gas-kinetic scheme (DUGKS) are analysed in finding steady-state solutions of the low-speed force-driven Poiseuille and lid-driven cavity flows. With the molecular collision and free streaming being treated simultaneously, the DUGKS preserves the second-order accuracy in the spatial and temporal discretizations in all flow regimes. Towards the hydrodynamic flow regime, not only is the DUGKS faster than the GDVM when using the same spatial mesh, but also requires less spatial resolution than that of the GDVM to achieve the same numerical accuracy. From the slip to free molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the complicated flux evaluation and the restrictive time step which is smaller than the maximum effective time step of the GDVM. Therefore, the DUGKS is preferable for problems involving different flow regimes, particularly when the hydrodynamic flow regime is dominant. For highly rarefied gas flows, if the steady-state solution is mainly concerned, the implicit GDVM, which can boost the convergence significantly, is a better choice.

Analysis of non-physical slip velocity in lattice Boltzmann simulations using the bounce-back scheme

In this work, we investigate the non-physical velocity-slip at a solid wall associated with lattice Boltzmann simulations using the bounce-back boundary scheme. By comparing the analytical solutions for the force-driven Poiseuille flow problem of two lattice models involving four and nine discrete velocities respectively, we found that the bounce-back scheme fails to define the behavior of certain discrete velocities and gives room for other factors to generate the commonly observed non-physical velocity-slip. To eliminate this deficiency, it is necessary to supplement the missing definition. For a lid-driven cavity flow, this is shown to be very simple although further study is necessary for more complex geometries.

A new curved boundary treatment in lattice Boltzmann method for micro gas flow in the slip regime

A new curved boundary treatment in lattice Boltzmann method is developed for micro gas flow in the slip regime. The proposed treatment is a combination of the nonequilibrium extrapolation scheme for curved boundary with no-slip velocity condition and the counter-extrapolation method for the velocity and its normal gradient on the curved boundary. Taking into consideration the effect of the offset between the physical boundary and the closest grid line, the new treatment is proved to be more accurate than the traditional half-way diffusive bounce-back (DBB) scheme. The present treatment is also more applicable than the modified DBB scheme because the specific gas-wall interaction parameters need to be determined to ensure the validation of the modified DBB scheme.The proposed boundary treatment is implemented to simulate the benchmark problems, which include a Poiseuille flow in the aligned/inclined micro-channel, a flow past a microcylinder and a microcylindrical Couette flow. The results and conclusions are summarized as follows.1) The force-driven Poiseuille flow in an aligned microchannel is simulated separately with different values of wall-grid offset qδx (q=0.25, 0.5, 0.75, 1.0). With the consideration of the wall-grid offset, the numerical results with the new boundary treatment show good agreement with the analytical solutions. However, the results obtained by using the half-way DBB scheme only accord well with the analytical solutions under the condition of a fixed wall-grid offset (q=0.5).2) To demonstrate the capability of the present treatment in dealing with gas flow in a more complex geometry, the force-driven Poiseuille flow in a micro-channel is investigated separately with different inclined angles. The present numerical results fit well with the analytical solutions. However, the discrepancy between the results obtained with the half-way DBB scheme and the analytical solutions can be clearly observed near the inclined boundaries.3) The gas flow past a microcylinder is simulated to prove that the present treatment can deal with the curved boundary. The slip velocity profile along the micro cylinder periphery obtained with the present treatment accords well with the available data in the published literature. However, the results obtained with the half-way DBB scheme show lower values than the results from the published work.4) In the simulations of the microcylindrical Couette flow between two coaxial rotating cylinders for different Knudsen numbers the results obtained by using the present treatment show excellent agreement with the analytical solutions. However, the results obtained with the half-way DBB scheme and the modified DBB scheme deviate obviously from the analytical solutions near the inner and outer cylindrical walls, respectively.In summary, the new boundary treatment proposed in this work is capable of dealing with the complex gas-solid boundary in the slip regime. The new treatment has a higher accuracy than the half-way DBB scheme and shows a better applicability than the modified DBB scheme.

Application of mixed quadrature lattice Boltzmann models for the simulation of Poiseuille flow at non-negligible values of the Knudsen number

We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian quadratures, where the quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range quadratures outperform the full-range Gauss–Hermite quadratures for the direction perpendicular to the diffuse-reflecting plates, while the quadrature on the periodic direction along the flow is the full-range Gauss–Hermite quadrature. Our results are validated against numerical results available in the literature.

Non-equilibrium dynamics of dense gas under tight confinement

The force-driven Poiseuille flow of dense gases between two parallel plates is investigated through the numerical solution of the generalized Enskog equation for two-dimensional hard discs. We focus on the competing effects of the mean free path ${\it\lambda}$, the channel width $L$ and the disc diameter ${\it\sigma}$. For elastic collisions between hard discs, the normalized mass flow rate in the hydrodynamic limit increases with $L/{\it\sigma}$ for a fixed Knudsen number (defined as $Kn={\it\lambda}/L$), but is always smaller than that predicted by the Boltzmann equation. Also, for a fixed $L/{\it\sigma}$, the mass flow rate in the hydrodynamic flow regime is not a monotonically decreasing function of $Kn$ but has a maximum when the solid fraction is approximately 0.3. Under ultra-tight confinement, the famous Knudsen minimum disappears, and the mass flow rate increases with $Kn$, and is larger than that predicted by the Boltzmann equation in the free-molecular flow regime; for a fixed $Kn$, the smaller $L/{\it\sigma}$ is, the larger the mass flow rate. In the transitional flow regime, however, the variation of the mass flow rate with $L/{\it\sigma}$ is not monotonic for a fixed $Kn$: the minimum mass flow rate occurs at $L/{\it\sigma}\approx 2{-}3$. For inelastic collisions, the energy dissipation between the hard discs always enhances the mass flow rate. Anomalous slip velocity is also found, which decreases with increasing Knudsen number. The mechanism for these exotic behaviours is analysed.

Fast spectral solution of the generalized Enskog equation for dense gases

We propose a fast spectral method for solving the generalized Enskog equation for dense gases. For elastic collisions, the method solves the Enskog collision operator with a computational cost of O(Md−1Ndlog⁡N), where d is the dimension of the velocity space, and Md−1 and Nd are the number of solid angle and velocity space discretizations, respectively. For inelastic collisions, the cost is N times higher. The accuracy of this fast spectral method is assessed by comparing our numerical results with analytical solutions of the spatially-homogeneous relaxation of heated granular gases. We also compare our results for force-driven Poiseuille flow and Fourier flow with those from molecular dynamics and Monte Carlo simulations. Although it is phenomenological, the generalized Enskog equation is capable of capturing the flow dynamics of dense granular gases, and the fast spectral method is accurate and efficient. As example applications, Fourier and Couette flows of a dense granular gas are investigated. In addition to the temperature profile, both the density and the high-energy tails in the velocity distribution functions are found to be strongly influenced by the restitution coefficient.

Numerical Simulation of Rarefied Gas Flows with Specified Heat Flux Boundary Conditions

AbstractWe investigate unidirectional rarefied flows confined between two infinite parallel plates with specified heat flux boundary conditions. Both Couette and force-driven Poiseuille flows are considered. The flow behaviors are analyzed numerically by solving the Shakhov model of the Boltzmann equation. We find that a zero-heat-flux wall can significantly influence the flow behavior, including the velocity slip and temperature jump at the wall, especially for high-speed flows. The predicted bimodal-like temperature profile for force-driven flows cannot even be qualitatively captured by the Navier-Stokes-Fourier equations.

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.

Analytic Solution for Microscale Poiseuille Flow Based on Super-Burnett Equations

The previous studies show that the transverse distribution of pressure and temperature in microscale Poiseuille flow cannot be predicted by Navier-Stokes equation with the slip boundary condition. In this paper, we analyzed the planar microchannel force-driven Poiseuille flow by high order continuum model. The super-Burnett constitutive relations were used and the nonlinear ordinary differential Equations of higher-orders were obtained by the hypothesis of parallel flow. With a perturbations theory, we linearized the equations and obtained the analytic solutions. The results show that the solutions can capture the temperature dip which is the same as the DSMC result. However, we also find that the temperature profile near the wall does always not match with the DSMC result. Especially, the difference in the qualitative exists when the Knudsen number is large enough. The non-equilibrium effect near the wall such as Knudsen layer can not be described entirely by continuous model even with high order constitutive relations and this confines the extension of the continuous model such as super-Burnett one.

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.

Multiple Temperature Gas Dynamic Equations for Non-equilibrium Flows

In an early approach, a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)] to simulate non-equilibrium flows was proposed. In this paper, instead of using three temperatures in the x, y and z-directions, we define the translational temperature as a second-order symmetric tensor. Under the new framework, the differences between the temperature tensor and the pressure tensor will be explicitly pointed out. Based on a multiple stage BGK-type collision model and the Chapman-Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The zeroth-order expansion gives 10 moment closure equations similar to that of Levermore [C.D. Levermore, J. Stat. Phys 83, pp.1021 (1996)]. The derived gas dynamic equations can be considered as a regularization of the 10 moments equations in the first-order expansion. The new gas dynamic equations have the same structure as the Navier-Stokes equations, but the stress-strain relationship in the Navier-Stokes equations is replaced by an algebraic equation with temperature differences. At the same time, the heat flux, which is absent in Levermore’s 10 moment closure, is recovered. As a result, both the viscous and the heat conduction terms are unified under a single anisotropic temperature concept. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier-Stokes equations. Our gas dynamic equations are natural extensions of the Navier-Stokes equations to the near continuum flow regime and can be used for microflow computations. Two examples, the force-driven Poiseuille flow and the Couette flow in the transition flow regime, are used to validate the model. Both analytical and numerical results are presented. Theoretically, the Boltzmann equation can be also applied to the current multiple stage gas evolution model to derive generalized macroscopic governing equations in the near continuum flow regime. Instead of using Maxwellian as an expansion point in the Chapman-Enskog method, the multiple temperature Gaussian can be used as an expansion point as well.