Direct Simulation Monte Carlo Data Research Articles

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

Effect of near-wall distance on velocity slip and temperature jump conditions in hypersonic rarefied gas flows

In nonequilibrium slip and jump conditions, normal gas velocity and temperature gradients are used to calculate the gas slip velocity and temperature at the surface, respectively. Gökçen et al. (Computational fluid dynamics near the continuum limit, AIAA Paper No. 87-1115, 1987, and Gökçen and MacCormack, Nonequilibrium effects for hypersonic transitional flows using continuum approach, AIAA Paper No. 89-0461, 1989) stated that the tangential velocity and temperature of the gas molecules before a collision with the surface could be interpreted as the macroscopic tangential velocity and temperature of the gas molecules at the so-called near-wall distances auλ and aTλT away from the surface, respectively. The coefficients au and aT are the order of unity. In the present work, new forms of the slip and jump conditions are proposed by modifying the Gökçen slip and jump conditions to include the coefficients (au, aT). Numerical investigations are comprehensively conducted to determine the numerically proper values (au, aT) for the hypersonic rarefied gas flows. Cases such as the circular cylinder in cross-flow and sharp and blunted leading edge wedge are considered in the present work, with nitrogen as the working gas. The simulation results show the significant effects of the coefficients (au, aT) on the accuracy of the slip velocity and surface gas temperature predictions, and the values of au = 1.2 and aT = 1.1 show good agreement with the direct simulation Monte Carlo data.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Capturing non-equilibrium effects in cylindrical Couette flow problem using the Extended Onsager-13 moment equations

In the present work, cylindrical Couette flow is analyzed using the recently derived third-order accurate 13-moment transport equations by transforming them into cylindrical coordinates. Assuming the Mach number and normalized temperature difference between the cylinders to be relatively small, closed-form expressions for all relevant quantities, velocity, pressure, temperature, heat flux, and stresses, are obtained from the semi-linearized form of the equations. These closed-form expressions from the present study have been validated against the corresponding linearized Grad 13 moment (G13) equation solutions. It is further demonstrated that contrary to the G13 equations, the pressure in the cylinder annulus is not constant, while the temperature depends upon the magnitude of viscous heating apart from non-isothermal boundary conditions. The coupling among velocity, temperature, heat flux, and stress and its effect on the variation of various physical quantities across the annulus has been discussed. The obtained analytical solution shows that the equations correctly capture most known non-equilibrium effects, such as the presence of a Knudsen layer, non-Newtonian stresses, and non-Fourier heat flux for Knudsen numbers falling well into the transition regime through a quantitative agreement with direct simulation Monte Carlo data, G13, and regularized 13-moment equations. The presence of non-zero radial and tangential heat fluxes, even when both the cylinders are at the same temperature, has been observed. The analysis helps us to demonstrate the ability of the recently derived equations in accurately solving complex rarefied flow problems. Moreover, understanding of higher-order rarefaction effects should greatly improve with the availability of closed-form analytical expressions of all physical quantities obtained here.

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.

A finite difference scheme for non-Cartesian mesh: Applications to rarefied gas flows

A novel numerical scheme based on the finite-difference framework is developed, which allows us to model moderately rarefied gas flows in irregular geometries. The major hurdle in constructing numerical methods for rarefied gas flows is the prescription of the velocity-slip and temperature-jump boundary conditions as well as the discretization of an intricate set of partial differential equations. The proposed scheme is demonstrated to solve the non-linear coupled constitutive relations model along with the corresponding non-linear slip and jump boundary conditions. The computation of the discretized weights is proposed using two approaches: (i) polynomial shape functions and (ii) a generalized inverse distance approach. The non-linear terms are discretized using the fixed-point iteration method. The numerical method is validated for the Laplace equation over an annulus, and results are presented for a lid-driven curved cavity and a triangular lid-driven cavity, which delineates its performance on a skewed non-Cartesian grid. The results are validated with direct simulation Monte Carlo data from the literature, and a robust convergence for the solutions is demonstrated.

Using gene expression programming to discover macroscopic governing equations hidden in the data of molecular simulations

The unprecedented amount of data and the advancement of machine learning methods are driving the rapid development of data-driven modeling in the community of fluid mechanics. In this work, a data-driven strategy is developed by the combination of the direct simulation Monte Carlo (DSMC) method and the gene expression programming (GEP) method. DSMC is a molecular simulation method without any assumed macroscopic governing equations a priori and is employed to generate data of flow fields, while the enhanced GEP method is leveraged to discover governing equations. We first validate our idea using two benchmarks, such as the Burgers equation and Sine–Gordon equation. Then, we apply the strategy to discover governing equations hidden in the complex fluid dynamics. Our results demonstrate that in the continuum regime, the discovered equations are consistent with the traditional ones with linear constitutive relations, while in the non-continuum regime such as shock wave, the discovered equation comprises of high-order constitutive relations, which are similar to those in the Burnett equation but with modified coefficients. Compared to the Navier–Stokes–Fourier equations and the Burnett equation, the prediction of the viscous stress and heat flux in the shock wave via the presented data-driven model has the best match to the DSMC data. It is promising to extend the proposed data-driven strategy to more complex problems and discover hidden governing equations which may be unknown so far.

Linear stability analysis of hypersonic boundary layers computed by a kinetic approach: a semi-infinite flat plate at $$\varvec{4.5\le \mathrm{M}_\infty \le 9}$$

Linear stability analysis is performed using a combination of two-dimensional Direct Simulation Monte Carlo (DSMC) method for the computation of the basic state and solution of the pertinent eigenvalue problem, as applied to the canonical boundary layer on a semi-infinite flat plate. Three different gases are monitored, namely nitrogen, argon and air, the latter as a mixture of 79\% Nitrogen and 21\% Oxygen at a range of free-stream Mach numbers corresponding to flight at an altitude of 55km. A neural network has been utilised to predict and smooth the raw DSMC data; the steady laminar profiles obtained are in very good agreement with those computed by (self-similar) boundary layer theory, under isothermal or adiabatic wall conditions, subject to the appropriate slip corrections computed in the DSMC method. The leading eigenmode results pertaining to the unsmoothed DSMC profiles are compared against those of the classic boundary layer theory. Small quantitative, but no significant qualitative differences between the results of the two classes of steady base flows have been found at all parameters examined. The frequencies of the leading eigenmodes at all conditions examined are practically identical, while perturbations corresponding to the DSMC profiles are found to be systematically more damped than their counterparts arising in the boundary layer at the conditions examined, when the correct velocity slip and temperature jump boundary conditions are imposed in the base flow profiles; by contrast, when the classic no-slip boundary conditions are used, less damped/more unstable profiles are obtained, which would lead the flow to earlier transition. On the other hand, the DSMC profiles smoothed by the neural network are marginally more stable than their unsmoothed counterparts.

Modelling and simulation of atmosphere-breathing electric propulsion intakes via direct simulation Monte Carlo

The Air-Breathing Ion Engine (ABIE) is an electric propulsion system capable of compensating for drag at low altitudes by ingesting the surrounding atmospheric particles to be utilized as propellant. The current state of the art of the ABIE performance is evaluated via Direct Simulation Monte Carlo (DSMC), due to the rarefied nature of the atmosphere in Very-Low Earth Orbit (VLEO). Nevertheless, the scarce availability of relevant simulation methodologies in the literature limits the repeatability of such numerical studies. Therefore, this paper proposes an independent methodology applicable to the modelling and simulation of Atmosphere-Breathing Electric Propulsion (ABEP) intakes that aims to validate the ABIE DSMC results retrieved from the literature. This is achieved by investigating the ABIE intake collection efficiency and compression ratio through the open-source solver dsmcFoam+ and by assessing the results against the available RARAC-3D DSMC data. First, the variation of grid transparency is discussed and compared between both solvers, yielding a mean percentage error of 2.97% for the compression ratio and 2.06% for the collection efficiency. Second, the absence of intermolecular collisions is verified for which errors of 1.61% for collection efficiency and 3.49% for compression ratio are observed. Then, the variation of flow incidence angle is simulated between 0^{circ } and 15^{circ }, yielding differences lower than 1.80%. Consecutively, the intake aspect ratio is varied between 10 and 40, for which a maximum discrepancy of 1.83% is measured and, finally, the drag coefficient of the intake is obtained in dsmcFoam+ to define the power density requirements.

A unified theory for modelling nonequilibrium and equilibrium gas flows

For over 50 years, gas flows have usually been investigated in computational fluid dynamics by using the Navier–Stokes–Fourier framework. Unfortunately, Navier–Stokes–Fourier fails to describe non-equilibrium states. To fill this gap, we give a thermodynamically consistent generalized hydrodynamic theory to develop a unified framework for both equilibrium and nonequilibrium gas flows. The critical step is that we extend the linear Rayleigh–Onsager dissipation to highly nonlinear dissipation to treat nonlinear irreversible thermodynamics. This extension fulfils the positive entropy generation criterion. The solutions of the generalized hydrodynamic theory are compared with experimental and/or particle-based direct simulation Monte Carlo data of the characteristic equilibrium and nonequilibrium gas flows. Good agreement is observed, as expected. The significance of this study is that it provides a unified framework for modelling nonequilibrium and equilibrium gas flows.

Pulsed laser ablation of binary compounds: effect of time delay in component evaporation on ablation plume expansion

Pulsed laser ablation of compound materials often occurs with delayed evaporation of a less volatile component; however, the effect of the delay on ablation plume expansion remains virtually unexplored. Here, we have performed an experimental and theoretical study of the delayed evaporation effect using an example of a plume produced by nanosecond laser ablation of a gold–silver alloy in a vacuum and comparing it with ablation of pure gold and silver targets. The plume expansion dynamics are investigated by time-of-flight (TOF) mass spectrometry and direct simulation Monte Carlo (DSMC), while the laser-induced target evaporation is analyzed using a thermal model. A dramatic effect of the delay time on the average kinetic energy of the plume particles, especially for the less volatile gold, is demonstrated and the main collisional processes governing the two-component plume expansion under the conditions of delayed evaporation are revealed. Based on comparison of experimental and DSMC data, the delay of the gold evaporation onset is estimated as approximately 0.6 ns. The delayed evaporation is therefore an important factor for correct interpretation of TOF measurements in ablation plumes with components of different volatilities.

Boltzmann-based second-order constitutive models of diatomic and polyatomic gases including the vibrational mode

Describing diatomic and polyatomic gases at high temperatures requires a deep understanding of the excitation of molecules to a higher vibrational level. We developed new second-order constitutive models for diatomic and polyatomic gases with vibrational degrees of freedom, starting from the modified Boltzmann–Curtiss kinetic equation. The closing-last balanced closure and cumulant expansion of the calortropy production associated with the Boltzmann collision term are key to the derivation of the second-order models, compatible with the second law of thermodynamics. The topology of the constitutive models showed the presence of highly nonlinear and coupled protruding or sunken regions in the compression branch. It was also shown that the vibrational mode reduces the level of nonlinearity in the topology. In addition, analysis of a strong shock structure highlighted the interplay between the second-order effects in the constitutive relations and the vibrational–translational relaxation. Finally, the analysis showed that the results of the second-order models were in better agreement with the direct simulation Monte Carlo data, when compared with the results of the first-order models, especially in the profiles and slopes of density, velocity, and vibrational temperatures.

Nonlinear transport of rarefied Couette flows from low speed to high speed

The nonlinear transport properties and macroscopic flow features of rarefied plane Couette flows from low speed to high speed for a monatomic gas are investigated in detail using the direct simulation Monte Carlo (DSMC) method. The effective viscosity and thermal conductivity are directly computed from the DSMC results according to the linear constitutive relations. The detailed structure of the Knudsen layer (KL) and the functional dependence of the effective transport coefficients on local Knudsen numbers in the whole system are presented and compared with existing theoretical models. The results show that the effective viscosity and thermal conductivity distributions in the KL for different Mach number flows can be recast into the same profile (i.e., isothermal scaling function) in terms of a scaled wall distance η=∫0y1/λ(y)dy, though the local flow is nonisothermal. For all cases, the shear-stress Knudsen number distributions across the channel show a well opposite trend to the effective transport coefficient profiles. The functional dependence between them in the bulk region always coincides with the normal solution that is derived from the Boltzmann model equations for unbounded shear flows, while that in the KL for low-speed cases shows a large difference with the normal solution. As the Mach number increases, the DSMC data in the KL can also agree approximately with the normal solution at a large shear-stress Knudsen number. These results can be very useful for developing phenomenological models to describe a wall-bounded rarefied shear flow, showing a good prospect in both microflow and high-altitude applications.

Modified Nonlinear Coupled Constitutive Relations Model for Hypersonic Nonequilibrium Flows

To obtain accurate physical description of nonequilibrium flows, Eu proposed a set of generalized hydrodynamic equations (GHEs) using a nonequilibrium ensemble method. However, Eu’s equations are very difficult to solve by employing the well-established numerical schemes for hyperbolic partial differential equations. Recently, a simplified nonlinear model based on Eu’s equations has been proposed by Myong for GHEs in conjunction with hyperbolic conservation laws, which is called the nonlinear coupled constitutive relations (NCCR) model. However, in the NCCR model, two specific terms in the evolution equation of heat flux are omitted for providing simplification, which reduces the accuracy of GHEs to some degree especially for hypersonic flows. To address this deficiency in NCCR, a modified NCCR+ model is proposed in this paper, which includes the two omitted terms in the NCCR model. NCCR+ model is applied to compute the hypersonic flow past an Apollo capsule at different altitudes and in expansion corners in a tube with various expansion angles. It is found that the computed NCCR+ solutions are almost identical to the Navier–Stokes (NS) solutions at low Knudsen numbers, but as the Knudsen number increases, the difference between the NCCR+ model solutions and the NS results increases rapidly and the NCCR+ solutions become closer in agreement with the direct simulation Monte Carlo data. By adding the two omitted terms in the NCCR model, the NCCR+ model also improves the overall accuracy of the original NCCR model. The results presented in this paper demonstrate the potential of the NCCR+ model in simulation of nonequilibrium hypersonic flows in both compression and expansion regions at moderate Knudsen numbers.

New slip boundary condition in high-speed rarefied gas flow simulations

In this paper, we propose a new slip boundary condition in hypersonic gas flow simulations. It is derived by considering the Langmuir isotherm adsorption into the Kaniadarkis et al. model of the kinetic theory of gas. Moreover, the motion of the adsorbed molecules over the surface (i.e. surface diffusion) is considered for the calculation of the mean free path in new slip condition. Three aerodynamic configurations are selected for evaluating new slip condition such as (1) the sharp-leading-edge flat plate, (2) circular cylinder in cross-flow, and (3) the sharp 25°–55° biconic cases. Hypersonic gas flows have the Mach number ranging from 6.1 to 15.6, and the working gases are argon and nitrogen. The simulation results show that new slip condition predicts better slip velocity than the Maxwell slip condition and gives good agreement with the direct simulation Monte-Carlo data for all cases considered in the present work.

Computation of Hypersonic Flows over Flying Configurations Using a Nonlinear Constitutive Model

The linear Navier–Stokes–Fourier (NSF) constitutive relations are derived on the assumption of the small deviation from local thermodynamic equilibrium, and consequently they may fail in describing flows removed far from local equilibrium, like rarefied hypersonic flows. In this paper, a nonlinear constitutive model of diatomic gases named as “nonlinear coupled constitutive relations” (NCCR) is presented. The model conceptually starts from Eu’s generalized hydrodynamics and is developed for simulating rarefied hypersonic gas flows with a goal of recovering NSF’s solutions in near-continuum regime and more accurate in transition regime. To enhance stable computation, an undecomposed algorithm is further developed for the nonlinear constitutive model within finite volume framework. An analysis is carried out to compare the algorithm with Myong’s original decomposed algorithm. Local nonequilibrium flow regions are also investigated for rarefied hypersonic flows over a cone tip, a hollow cylinder-flare, and a hypersonic technology vehicle–type flying vehicle. The convergent solutions of NCCR model are compared with NSF, direct simulation Monte Carlo (DSMC) calculations, and experimental data. It is demonstrated from results of general flow-field and surface properties that the NCCR model is as computationally efficient as the NSF model in continuum regime, and, at the same time, more accurate in comparison with DSMC and experimental data than NSF in nonequilibrium flows.

Large Mpemba-like effect in a gas of inelastic rough hard spheres.

We report the emergence of a giant Mpemba effect in the uniformly heated gas of inelastic rough hard spheres: The initially hotter sample may cool sooner than the colder one, even when the initial temperatures differ by more than one order of magnitude. In order to understand this behavior, it suffices to consider the simplest Maxwellian approximation for the velocity distribution in a kinetic approach. The largeness of the effect stems from the fact that the rotational and translational temperatures, which obey two coupled evolution equations, are comparable. Our theoretical predictions agree very well with molecular dynamics and direct simulation Monte Carlo data.

Characterization of a Knudsen force based vacuum sensor for N2[sbnd]H2O gas mixtures

Radiometric phenomena can arise in gas flows in which the local molecular mean free path is in the same order as the temperature gradient length scale. The forces exerted on immersed bodies by such rarefied flows are known as Knudsen (or thermophoretic) forces and can play a major role in nonisothermal microflows. In this work, the direct simulation Monte Carlo (DSMC) method is used to numerically characterize the force and thermal mechanisms on a second generation Microscale In-Plane Knudsen Radiometric Actuator (MIKRA) designed for gas pressure and composition sensing. First, a brief verification analysis of the present DSMC solver in reproducing the thermal conductivity for the gas species of interest – nitrogen and water vapor – is carried out. The geometric impact of the gap between MIKRA beams is then investigated for pure N2 environments. In addition, the beam heat transfer coefficients for N2H2O binary mixtures are extracted from the DSMC data and a dimensionless correlation is proposed for predicting the corresponding Knudsen forces. These results are intended to assist in the operation and design of the MIKRA device.

Simulation of a temperature drop for the flow of rarefied gases in microchannels

ABSTRACTThe Direct Simulation Monte Carlo (DSMC) method is widely used to study the flow characteristics of subsonic flow in 2-D microchannels. It is seen from existing numerical studies that the temperature of the flow decreases towards the exit of the microchannel whereas such a drop has not been reported in experimental studies. To resolve this discrepancy, effect of flow parameters such as Knudsen number, aspect ratio and pressure ratio on temperature change is studied systematically using an in-house developed DSMC code. Based on the parametric analysis, a correlation for temperature drop is proposed which predicts the DSMC data within ±15%. A control volume analysis is further carried out to understand the reason for temperature drop in microchannels. The effect of mass diffusion is also modeled. It is found that accounting for mass diffusion improves prediction of mass flow rate but not temperature drop across the microchannel.

New Type of Smoluchowski Temperature Jump Condition Considering the Viscous Heat Generation

The effect of the sliding friction has been important in calculating the heat flux of gas flow from the surface because there is some slip over the surface. There is not any temperature jump condition including the sliding friction part (viscous heat generation) in its derivation so far. In this paper, the Smoluchowski temperature jump condition is modified by considering viscous heat generation. This new temperature jump condition will be evaluated for three configurations in high-speed rarefied gas flow simulations using the computational fluid dynamics method, which solves the Navier–Stokes equations within the OpenFOAM framework: 1) the flat plate, with Mach number 2) the NACA0012 microairfoil, with Mach number ; and 3) the sharp 25–55 deg biconic case, with Mach number . The working gases are air and nitrogen. The airfoil case is simulated with various Knudsen numbers from 0.026 to 0.26, and the angles of attack are from 10 to 20 deg. The surface gas temperatures predicted by the new temperature jump condition give good agreement with the direct simulation Monte Carlo data, especially the NACA0012 microairfoil cases with the high Knudsen numbers ( and ) with an angle of attack of 20 deg.