Wide Range Of Knudsen Numbers Research Articles

Accuracy and performance evaluation of low density internal and external flow predictions using CFD and DSMC

The Direct Simulation Monte Carlo (DSMC) method was widely used to simulate low density gas flows with large Knudsen numbers. However, DSMC encounters limitations in the regime of lower Knudsen numbers (Kn<0.05). In such cases, approaches from classical computational fluid dynamics (CFD) relying on the continuum assumption are preferred, offering accurate solutions at acceptable computational costs. In experiments aimed at imaging aerosolized nanoparticles in vacuo a wide range of Knudsen numbers occur, which motivated the present study on the analysis of the advantages and drawbacks of DSMC and CFD simulations of rarefied flows in terms of accuracy and computational effort. Furthermore, the potential of hybrid methods is evaluated. For this purpose, DSMC and CFD simulations of the flow inside a convergent–divergent nozzle (internal expanding flow) and the flow around a conical body (external shock generating flow) were carried out. CFD simulations utilize the software OpenFOAM and the DSMC solution is obtained using the software SPARTA. The results of these simulation techniques are evaluated by comparing them with experimental data (1), evaluating the time-to-solution (2) and the energy consumption (3), and assessing the feasibility of hybrid CFD-DSMC approaches (4).

Dimensional homogeneity constrained gene expression programming for discovering governing equations

Data-driven discovery of governing equations is of great significance for helping us understand intrinsic mechanisms and build physical models. Recently, numerous highly innovative algorithms have emerged, aimed at inversely discovering the underlying governing equations from data, such as sparse regression-based methods and symbolic regression-based methods. Along this direction, a novel dimensional homogeneity constrained gene expression programming (DHC-GEP) method is proposed in this work. The DHC-GEP simultaneously discovers the forms and coefficients of functions using basic mathematical operators and physical variables, without requiring preassumed candidate functions. The constraint of dimensional homogeneity is capable of filtering out the overfitting equations effectively. The key advantages of DHC-GEP compared with the original gene expression programming, including being more robust to hyperparameters, the noise level and the size of datasets, are demonstrated on two benchmark studies. Furthermore, DHC-GEP is employed to discover the unknown constitutive relations of two representative non-equilibrium flows. Galilean invariance and the second law of thermodynamics are imposed as constraints to enhance the reliability of the discovered constitutive relations. Comparisons, both quantitative and qualitative, indicate that the derived constitutive relations are more accurate than the conventional Burnett equations in a wide range of Knudsen numbers and Mach numbers, and are also applicable to the cases beyond the parameter space of the training data.

Evaluation of a New Droplet Growth Model for Small Droplets in Condensing Steam Flows

As the condensation phenomenon occurs in the low-pressure stages of steam turbines, an accurate modelling of the condensing flows is very crucial and has a significant impact on the development of highly efficient steam turbines. In order to accurately simulate condensing steam flows, it is essential to choose the right condensation model. Further research to enhance condensation models is of special importance because the outcomes of numerical studies of condensation models in recent years have not been entirely compatible with the experiments and there are still uncertainties in this area. Therefore, the main aim of this paper is to evaluate a proposed droplet growth model for modelling condensation phenomenon in condensing steam flows. The new model is derived to profit from the advantages of models based on the continuum approach for large droplets and those based on the kinetic theorem for small droplets, which results in the model being robust for a wide range of Knudsen numbers. The model is implemented into a commercial CFD tool, ANSYS Fluent 2022 R1, using UDFs. The results of the CFD simulations are validated against experimental data for linear cascades within the rotor and stator blade geometries of low-pressure steam turbine stages. The findings clearly demonstrate the superiority of the new model in capturing droplet growth, particularly for very small droplets immediately following nucleation. In contrast, widely used alternative droplet growth models tend to either underpredict or overpredict the droplet growth rate. This research significantly contributes to the ongoing efforts to enhance condensation modeling, providing a more accurate tool for optimizing the design and operation of low-pressure steam turbines, ultimately leading to a higher energy efficiency and a reduced environmental impact.

Thermophoresis and uniform flow in rarefied polyatomic gases: The role of constitutive relations and boundary conditions

Recently, Rana and Barve [“A second-order constitutive theory for polyatomic gases: Theory and applications,” J. Fluid Mech. 958, A23 (2023)] developed a second-order coupled constitutive relations (CCR) for polyatomic gases that include quadratic nonlinearities in the entropy flux and apply the second law. However, in that work, the boundary conditions were heuristically obtained to match the drag coefficient on a sphere and may not be accurate in situations where thermal transpiration and thermal stress are significant factors, as indicated by their asymptotic analysis. This article presents a systematic approach for deriving thermodynamically admissible boundary conditions for the CCR model. We also propose a set of higher-order boundary conditions based on an asymptotic analysis of the solutions for drag on flow past a sphere and thermophoretic drag. The goal of deriving these boundary conditions is to improve the accuracy of the CCR model when applied to external flows, such as slow flow past particles and thermophoretic motion of a spherical particle and doublet. The results of the study demonstrate that the combination of the newly derived boundary conditions in conjunction with the CCR equations shows excellent agreement with both theoretical predictions and experimental data over a wide range of Knudsen numbers. The study suggests that the approach presented in this article can be used to improve the accuracy of the CCR model in a variety of external flow applications.

Conductive heat transfer through nanoconfined argon gas: From continuum to free-molecular regime

Heat transfer across nanogaps between solid surfaces has received an increased attention driven by both recent experimental investigations and technological applications as well. Most of the studies concerned, however, vacuum conditions and relatively few attention has been paid to the situation where a gas occupied the gap. This latter situation is difficult to handle theoretically because a continuum description of heat transfer across a nanoconfined gas medium completely breaks down. This breakdown maybe related not only to the partially ballistic nature of heat transfer between the surfaces but also to the existence of an adsorbed gas layer which affects energy transfer across the gap. In this work, to better understand the interplay of these effects, we present a series of molecular dynamics simulations of argon gas confined between either metallic or silicon walls held at different temperatures. The gas density is tuned so that the gas experiences a wide range of Knudsen numbers ranging from the continuum to the free molecular regime. It is shown that the Debye temperature of the solid controls the amount of adsorbed gas close to the walls, which in turn controls heat conduction in the middle of the channel. While the nanochannel walls significantly impact the density and temperature distributions of the rarefied gas, the pressure and the heat flux across the gas domain converge toward a plateau as the gas becomes denser. Finally, we also derive new analytical formulas to calculate the gas pressure, induced heat flux, and effective thermal conductivity of the nanoconfined gas. These formulae, which take into account the effects of an adsorbed gas layer, are shown to hold over a broad range of Knudsen numbers in the range 0.05–20. This framework allows one to predict nanoscale heat transfer across a gas confined by a variety of solid walls.

A second-order slip/jump boundary condition modified by nonlinear Rayleigh–Onsager dissipation factor

A newly heuristic form of second-order slip/jump boundary conditions (BCs) for the Navier–Stokes–Fourier (NSF) equations is proposed from the viewpoint of generalized hydrodynamic equations (GHE) to extend the capability of the NSF equations for moderately rarefied gas flows. The nonlinear Rayleigh–Onsager dissipation function appearing in the GHE, which contains useful information about the nonequilibrium flow fields of interest, is introduced into the proposed BCs named the simplified generalized hydrodynamic (SGH) BCs as a correction parameter. Compared with the classical Maxwell/Smoluchowski (MS) BCs, the SGH BCs may be more sensitive to capture the nonequilibrium information of flows adaptively and produce physically consistent solutions near the wall. Subsequently, the SGH BCs are implemented in the NSF equations for planar micro-Couette gas flows over a wide range of Knudsen numbers. The results indicate that the SGH BCs make impressive improvements against the MS BCs for diatomic and monatomic gases at the slip region and early transition regime, particularly in terms of capturing precisely the temperature and normal heat flux profiles in the flow and the temperature jump on the wall. More importantly, the SGH BCs conducted in NSF equations with less computational cost still can obtain well-pleased results comparable to the non-Newton–Fourier equations, such as several Burnett-type equations and regularized 13-moment equations, and even perform better than these models near the wall compared with direct simulation Monte Carlo data for the Couette flows to some extent.

The outflow of gas mixture into vacuum, periodically interrupted by bodies moving towards the jet

The problem about the unsteady outflow of a mixture of gases into a vacuum, periodically interrupted by bodies moving toward the jet, was studied. This problem was considered as a possible gas separation technique. The unsteady flow of the gas mixture was simulated using the event-driven molecular dynamics (EDMD) approach under the following conditions: (1) moving boundaries, (2) a wide range of Knudsen number, (3) a wide range of Mach number, (4) violation of equilibrium between rotational and translational degrees of freedom. The problem of periodically interrupted flow was solved as close to reality as possible, taking into account the finite gap between the stator and the moving runner blades. The dependences of the molar fraction and flow rate of the resulting mixture on the system parameters were obtained. As a result, a number of practically important recommendations for the design of such systems have been formulated.

Application of Gas-Kinetic Scheme for Continuum and Near-Continuum Flow on Unstructured Mesh

A gas-kinetic scheme (GKS) with kinetic boundary condition based on unstructured mesh is present here. In the GKS method, the solid wall boundary conditions can be constructed by virtue of the gas distribution function, which is similar to the diffuse-scattering rule used in the other kinetic schemes. The kinetic boundary condition has a concise form and easy to implement. The use of unstructured mesh expands the adaptability of GKS to simulate the flows with complex geometry. The kinetic boundary condition can recover to the non-slip boundary condition in the continuum regime. In the slip regime, the slip velocity can be accurately predicted by kinetic boundary condition, which turns into the slip boundary condition. The use of kinetic boundary condition improves the calculation results of GKS in near-continuum flow. A series of test cases, from incompressible to compressible flow with a wide range of Knudsen number, are investigated to demonstrate the performance of kinetic boundary condition in near-continuum flow, which can provide a reference for the construction and optimisation for GKS-based multi-scale hybrid algorithms.

A novel transient-adaptive subcell algorithm with a hybrid application of different collision techniques in direct simulation Monte Carlo (DSMC)

A novel hybrid transient adaptive subcell (TAS) direct simulation Monte Carlo (DSMC) algorithm is proposed to simulate rarefied gas flows in a wide range of Knudsen numbers. It is derived and analyzed by using a time and spatial discrete operator approach based on the non-homogeneous, local N-particle kinetic equation, first proposed by Stefanov. The novel algorithm is considered together with the standard and hybrid collision algorithms built on uniform grids. The standard collision algorithm uses only one single scheme—the NoTime Counter (NTC), or the Generalized or Simplified Bernoulli trials (GBT, SBT). The hybrid algorithm employs NTC, GBT, or SBT depending on the instantaneous number of particles in the considered cell. The novel hybrid TAS algorithm benefits from both the hybrid collision approach and the transient adaptive subcell grid covering each collision cell to achieve a uniform accuracy of order O(Δt, Δr) independently of the number of particles in the cells. To this aim, a local time step is defined as coherent with the TAS grid covering the corresponding collision cell. The novel hybrid TAS algorithm is tested on two-dimensional benchmark problems: supersonic rarefied gas flow past of a flat plate under an angle of incidence and pressure-driven gas flow in a microchannel. The results obtained by the hybrid TAS algorithm are compared to those obtained by the standard algorithms and the available Bird's DS2V code using nearest neighbor collision and open-source OpenFOAM code. The comparison shows an excellent accuracy of the suggested algorithm in predicting the flow field.

Nanoconfined Bulk-Gas Transport Behavior over a Wide Range of Knudsen Number

Shale matrix, located at Guizhou Province, is rich in nanopores, and gas slippage takes place during shale gas development, resulting in inapplicability of the classical Navier-Stokes equation. Investigation of gas transmission mechanism in nanoscale is helpful to reach a clear understanding of shale gas production performance. On the basis of the Knudsen number, the gas flow mechanism is divided into continuous flow, slippage flow, transition flow, and Knudsen diffusion. Notably, the accurate characterization of transition flow is still challenging up to date. Although there are many established models, they either fail to cover all the flow mechanisms or contain many fitting parameters, the determination of which requires a large number of experimental and molecular simulation data, limiting application of the existing models. Therefore, establishment of bulk-gas transport model over a wide range of Knudsen number without fitting parameters is urgent. First of all, existed theoretical models are compared, and the advantages and disadvantages of previous contributions are analyzed. Weight factors of Knudsen diffusion and slippage flow are obtained, respectively, according to Knudsen’s model and Wu’s model. Then, a model, free of empirical coefficients, is proposed. After that, effects of pore size, reservoir pressure, and temperature on transmission capacity are investigated. As the proposed model does not contain fitting parameters and remains high accuracy over a wide range of Knudsen number, it shares broad application potential, like shale gas production prediction, reserve estimation in Guizhou Province.

HALO3D: An All-Mach Approach to Hypersonic Flows Simulation

This paper presents HALO3D, a CFD code simulating the wide range of Mach numbers a hypervelocity vehicle experiences. HALO3D is composed of (a) a RANS edge-based Finite Element (FE) solver modelling high-velocity thermo-chemical non-equilibrium flows, (b) an electromagnetic interactions solver, (c) an ablation module, (d) a Direct Simulation Monte Carlo (DSMC) solver for the rarefied regime, seamlessly linked to the continuum one, and (e) a powerful automatic mesh optimiser finely capturing the many singular hypersonic flow phenomena. Verification and validation simulations for two- and three-dimensional hypersonic flows over a flat plate, a cylinder, a blunt conical body, re-entry capsules, and a waverider geometry demonstrate accurate aerothermodynamic predictions in a wide range of Knudsen numbers, complex geometries and multiphysics.

Conductive Heat Transfer Through Nanoconfined Gas: From Continuum to Free-Molecular Regime

In the past few decades, great efforts have been devoted to studying heat transfer on the nanoscale due to its importance in multiple technologies such as thermal control and sensing applications. Heat conduction through the nanoconfined gas medium differs from macroscopic predictions due to several reasons. The continuum assumption is broken down; the surface forces which extend deeper through the gas medium become prominent due to the large surface-to-volume ratio, and, finally, the gas molecules are accumulated nonuniformly on the solid surfaces. In this work, to better understand the combination of these phenomena on the heat conduction through the nanoconfined gas medium, we present a series of molecular dynamics simulations of argon gas confined between either metals or silicon walls. The gas density is set so that gas experiences a wide range of Knudsen numbers from continuum to the free molecular regime. It is observed that the intrinsic characteristics of the solid determine the gas density distribution near the walls and consequently in the bulk region, and these distributions control the heat conduction through the gas medium. While the nanochannel walls have their most significant impact on the density and temperature distributions of the rarefied gas, the pressure and the heat flux across the gas domain converge toward a plateau as the gas becomes denser. We propose new analytical formulas for calculating the gas pressure, induced heat flux, and effective thermal conductivity through the strongly nanoconfined gas, which incorporates the wall force field impacts on the gas transport characteristics for the Knudsen number in the range of 0.05 to 20.

Inversion of the transverse force on a spinning sphere moving in a rarefied gas

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

Self-diffusivity of dense confined fluids

Molecular transport through tight porous media is crucial to shale gas exploration, but deeper insights of the elemental physics are still required, particularly under high pressures and nanoscale confinements, where Navier–Stokes and Boltzmann solutions are no longer valid. In this work, we carry out a fundamental and systematic study of self-diffusion using event-driven molecular dynamics simulations, varying fluid rarefaction, confinement, and surface friction. We differentiate between fluid–fluid and fluid-wall collisions to identify the interplay of the underpinning diffusive mechanisms, namely, molecular and Knudsen diffusion. We find that the Bosanquet formula, which has been used for describing rarefied gases, is also able to provide a good semi-analytical description of self-diffusivities in confined dense fluids, as long as the pore height is not smaller than five molecular diameters. Importantly, this allows us to predict the self-diffusion coefficient, regardless of the fluid rarefaction, confinement state, and surface roughness, in a wide range of Knudsen numbers that were not possible before. Often as a source of debate, we prove here that despite strong fluid inhomogeneities arising in these conditions, the Einstein self-diffusivity can still be used within Fick's law, provided boundary effects are considered when using Fick's setup. Finally, we notice that a previously identified linear scaling of self-diffusivities with confinement is only valid in the limit of low densities and frictionless walls, which is not representative of shale reservoirs. This work will serve as a foundation for investigating the anomalous gas transport behavior observed in the recent work of dense, confined fluids.

Learning Thermodynamically Stable and Galilean Invariant Partial Differential Equations for Non-Equilibrium Flows

Abstract In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the conservation-dissipation formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod’s shock tube problem although it is trained only with smooth initial data.

Surface Roughness Effects on Molecular Gas Lubrication

The effects of surface roughness on the molecular gas lubrication characteristics have been studied and it has become clear that the transverse flow, which is neglected in conventional molecular gas lubrication (MGL) equation, appears in the lubrication film as the roughness slope increases. In this paper, the detail of the transverse flow in the wide range of Knudsen number was analyzed by using direct simulation Monte-Carlo (DSMC) method. It was clarified that the transverse flow is caused by the slip flow on the slope of rough surface．The transverse flow exists only around the rough surface in the case of small Knudsen number, though the flow extends in whole area of lubrication film against large Knudsen number. The flow through the lubrication film complementary decreases as the transverse flow increases.

Efficient DSBGK simulations of the low speed thermal transpiration gas flows through micro-channels

Low speed thermal transpiration gas flows through micro-channels are simulated by the direct simulation BGK (DSBGK) method that is based on the Bhatnagar-Gross-Krook model equation. Different from the ordinary application of BGK equation having the viscosity matched, we selected its relaxation time according to the thermal conductivity coefficient and thus significantly improved the accuracy. In comparing the detailed flow field solutions of a 2D Knudsen pump, the validation by the standard direct simulation Monte Carlo (DSMC) method shows that the improved BGK equation has about the same accuracy as the Shakhov equation that is solved by the well-developed gas kinetic unified algorithm (GKUA). We also present the ordinary application of BGK equation with noticeable error to clearly show the difference made in our study. Then, the efficient DSBGK method is applied to simulate the low-speed rarefied gas flows in 3D real-size thermal transpiration problem to show its applicability and its validity is verified by comparing with the experimental measurement of mass flow rates. The DSBGK simulation results have excellent agreement with experimental data for three types of gases over a wide range of Knudsen number. Additionally, the channel end effect is also studied and discussed as a prerequisite for high-fidelity modeling, otherwise it will be mistakenly interpreted as the variation of accommodation coefficients of the molecular reflection process on the solid boundary.

Multiscale simulation of molecular gas flows by the general synthetic iterative scheme

The in-depth knowledge of rarefied gas dynamics is crucial to address challenges in a wide range of engineering problems, where gas flows are usually multiscale, i.e., covering a wide range of Knudsen numbers. As the traditional Navier–Stokes equations fail, gas kinetic equations are required to model the flows. So far, very few numerical methods are designed to efficiently solve the multiscale gas dynamics and reveal the role of internal degrees of freedom of gas molecules. In this work, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the gas kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state. In the macroscopic synthetic equations, while the momentum equation is the same as that used in the GSIS for monatomic gas, two energy equations are introduced here for polyatomic gases: one is for the translational energy and the other for the internal energy; these equations are derived exactly from the gas kinetic equations hence no approximation is made in final solutions. The Fourier stability analysis is performed to show that the GSIS permits fast convergence to steady-state solutions in the entire flow regime; meanwhile the asymptotic analysis shows that the GSIS recovers the Navier–Stokes equations when the Knudsen number is small, even on the spatial grid with cell size much larger than the molecular mean free path. With all these unique features, several challenging numerical examples are given to show that the proposed GSIS is a promising tool to simulate multiscale molecular gas flows and investigate the effects of internal degrees of freedom.

Numerical analysis of nonlinear thermal stress flow between concentric elliptical cylinders

The present study numerically investigates nonlinear thermal stress in a rarefied gas flow between two coaxial elliptic cylinders with a hot outer wall. Monatomic argon is considered, and isothermal boundary conditions are implemented on the walls. Three different numerical methods are used to solve this problem, i.e., first, the direct simulation Monte Carlo (DSMC) method is used and validated against the numerical solution of the Boltzmann equation available in the literature. The results of the DSMC algorithm are compared with the results of the discrete unified gas kinetic scheme and the continuum set of equations of slow non-isothermal flows, which includes nonlinear thermal stress terms compared to the Navier–Stokes equations. For the first time, the flow is examined in a wide range of Knudsen numbers in the slip regime from the Knudsen number (Kn) of Kn = 0.005 to Kn = 0.1. At a minimal Knudsen number, there is just one vortex due to the nonlinear thermal stress flow; however, as Kn increases, the velocity peak near the inner cylinder is replaced with another vortex, which is created due to the second-order thermal creep effects on the inner wall of the cylinder. We show that as Kn increases, the inner vortex increases in size, while the outer vortex shrinks. The mechanisms of the formation of both vortices are described in detail. Other flow characteristics, including walls’ velocity slip, shear stress, and vorticity, are discussed.

Driving mechanism of thermal transpiration pump with porous material

The behavior of the rarefied gas in the thermal transpiration pump with the porous material is investigated numerically by the direct simulation Monte Carlo method. The mass flux achieved by the pump is analyzed for a wide range of Knudsen numbers and ratios of the pore length to the pore diameter. The results show that the thermal edge flows around the ends of the pore play an essential role in determining the maximum performance. The effect of the thermal edge flow leads to a qualitative difference in the driving mechanism from a similar thermal transpiration pump by Knudsen. The mass flow takes a maximum value at a considerably large Knudsen number when the pore length is much larger than the pore diameter. The numerical tests show that a larger mass flux is possible when the edge flow is suppressed. The mass flux is investigated for several values of accommodation coefficient and complex pore geometries. The present results show that only the latter leads to the reduction in the mass flux. The compression ratio, including the performance curve of the pump, is also analyzed for several cases. The results show that the small accommodation coefficient decreases the compression ratio of the pump.