Continuum Breakdown Research Articles

Kinetic Simulation of Nozzle Flow in a Micronewton-Class Cold Gas Thruster

Many scientific space missions need highly precise attitude and orbit control or ultrafine drag compensation, which relies on the variable-thrust propulsion technology operating at the micronewton level. Cold gas thruster (CGT) is a very promising solution, mainly because of its high reliability. One of the keys to the success of micronewton variable-thrust CGT is to understand the flow in its nozzle, whose configuration is much more complex than traditional CGT nozzles. This paper applies kinetic-based multiscale models to investigate the gas flow in the complex nozzle of a micronewton variable-thrust CGT. The simulations reveal that the flow simultaneously experiences all kinds of regimes from continuum to free molecular. The continuum breakdown is likely to occur near the throat region due to large gradients of flow variables and in the expander due to low gas density. Frictional choking is observed, and the nozzle length can be optimized to improve the thruster performance. Nozzle performance measures such as thrust, discharge coefficient, and thrust efficiency are found to change only with the throat Knudsen number Knt. The performance curves can be divided into two sections at Knt≃0.1, and thereby an empirical piecewise formula for thrust prediction is proposed.

Studies on Continuum Breakdown and Water Transport Behavior of Nanotubes for Water Purification: Impact of Temperature, Nanotube Material, and Diameter

Studies on Continuum Breakdown and Water Transport Behavior of Nanotubes for Water Purification: Impact of Temperature, Nanotube Material, and Diameter

Feasibility of macroscopic parameters for NS to DSMC solver switching in micronozzle simulations

Enhancing the design and performance of micronozzles could lead to novel applications and advancements in propulsion systems, making the exploration of micronozzles crucial for the future. This paper critically examines the feasibility of utilizing macroscopic property-based Kn as indicator for defining the breakdown region during the transition from the NS solver to the DSMC solver in micronozzle simulations. The aim is to specify a parameter that can be calculated from both NS and DSMC simulations, making it suitable for implementation in hybrid simulations that dynamically switch between the two solvers. The results show that the density-based Kn accurately represents the continuum breakdown, and it exhibits an earlier breakdown compared to pressure and temperature-based Kn values. The study also analyzes the rarefaction effects and introduces the rarefaction parameter (RP), quantifying the increase in Kn for a unit change in the non-dimensionalized distance. The findings demonstrate that at very low exit pressures, the rarefaction effects increase rapidly as the flow moves towards the nozzle exit, leading to a transition from the continuum to the rarefied regime. The hybrid NS-DSMC simulations show good agreement with experimental data, validating the proposed approach. Additionally, the research examines the effect of back pressure on the RP and identifies the transition regime based on the slope of the RP curve. Therefore, the manuscript provides detailed insights into novel elements, such as the quantification of rarefaction within the nozzle using the RP, the classification of the nozzle into different regimes (continuum, slip, and transition), the definition of an easily obtainable parameter for switching between NS and DSMC methods, and an examination of the contributions of the shear stress term and heat addition term to non-equilibrium conditions.

Instability of mixing layers: Momentum and thermal transport in the continuum breakdown regime.

We examine the momentum and thermal transport in the continuum breakdown regime of a mixing layer flow, which exhibits Kelvin-Helmholtz instability under ideal continuum conditions. The Grad 13 moment model is used as it provides an adequate description of the flow physics (second-order accurate in Knudsen number) in the transition regime. Analytical solutions are developed under breakdown conditions for two-dimensional, compressible, parallel shear flows. It is shown that the deviation of viscous stress and heat flux from the Navier-Stokes-Fourier system follows two different scaling regimes depending upon the Mach number. At low Mach numbers, the departure of all stress and heat-flux components depends only upon the Knudsen number. At high Mach number, the scaling of shear stress and transverse heat flux depends on the product of the Knudsen and Mach numbers. The normal stresses depend individually on the Knudsen and Mach number. The scaling results are verified against numerical simulations of compressible mixing layers performed using the unified gas kinetic scheme for various degrees of rarefaction.

Non-continuum effects on the sound of a heated line source: from a monopole to non-isotropic radiation

We study the effect of continuum breakdown on the two-dimensional thermoacoustic radiation of a thin plate set in a perfect monatomic gas. The plate is heated harmonically in time and the acoustic field is investigated in the entire range of gas rarefaction rates. Analytical approximations are obtained in the limits of high (free-molecular) and low (continuum limit) gas rarefaction, accompanied by direct simulation Monte Carlo calculations at intermediate flow conditions. While the source acoustic field is of a monopole type in the continuum regime, it turns non-isotropic at non-continuum conditions, exhibiting dipole directivity and exponential decay rate in the collisionless limit. The combined effects of source heating frequency and gas Knudsen number on the far-field acoustic radiation are illustrated and rationalized.

Study of Side-Jet Interactions over a Hypersonic Cone Flow Using Kinetic Methods

The interaction between the hypersonic flow over a cone and a side jet creates complex multiple recirculation regions as well as separation, reattachment, and bow shocks known as Edney type-IV shock/shock interactions. The interactions between these multiscale structures are shown to be well captured with the direct simulation Monte Carlo (DSMC) method, which is a particle-kinetic approach. This computational technique was used because even though the freestream conditions were continuumlike, sufficiently strong gradients in the shock and jet expansion were observed to cause continuum breakdown. The time-accurate DSMC approach has been shown in previous work to capture the multiscale structures and unsteadiness in the shock structure that occurs in such flows, as was also observed in this work. These sources of unsteadiness in the shock system are shown to locally affect surface parameters, such as heat transfer and skin friction, as well as predict a low–frequency unsteadiness consistent with earlier values in the literature.

Predicting continuum breakdown with deep neural networks

The multi-scale nature of gaseous flows poses tremendous difficulties for theoretical and numerical analysis. The Boltzmann equation, while possessing a wider applicability than hydrodynamic equations, requires significantly more computational resources due to the increased degrees of freedom in the model. The success of a hybrid fluid-kinetic flow solver for the study of multi-scale flows relies on accurate prediction of flow regimes. In this paper, we draw on binary classification in machine learning and propose the first neural network classifier to detect near-equilibrium and non-equilibrium flow regimes based on local flow conditions. Compared with classical semi-empirical criteria of continuum breakdown, the current method provides a data-driven alternative where the parameterized implicit function is trained by solutions of the Boltzmann equation. The ground-truth labels are derived rigorously from the deviation of particle distribution functions and the approximations based on the Chapman-Enskog ansatz. Therefore, no tunable parameter is needed in the criterion. Following the entropy closure of the Boltzmann moment system, a data generation strategy is developed to produce training and test sets. Numerical analysis shows its superiority over simulation-based samplings. A hybrid Boltzmann-Navier-Stokes flow solver is built correspondingly with an adaptive partition of local flow regimes. Numerical experiments including the one-dimensional Riemann problem, shear flow layer, and hypersonic flow around a circular cylinder are presented to validate the current scheme for simulating cross-scale and non-equilibrium flow physics. The quantitative comparison with a semi-empirical criterion and benchmark results demonstrates the capability of the current neural classifier to accurately predict continuum breakdown. The code for the data generator, hybrid solver, and neural network implementation is available in the open source repositories [1,2].

Continuum breakdown in compressible mixing layers.

Gas-kinetic simulations of rarefied and compressible mixing layers are performed to characterize continuum breakdown and the effect on the Kelvin-Helmholtz instability. The unified gas-kinetic scheme (UGKS) is used to perform the simulations at different Mach and Knudsen numbers. The UGKS stress tensor and heat-flux vector fields are compared against those given by the Navier-Stokes-Fourier constitutive equations. The most significant difference is seen in the shear stress and transverse heat flux. The study demonstrates the existence of two distinct continuum breakdown regimes, one at low and the other at high convective Mach numbers. Overall, at low convective Mach numbers, the deviation from continuum stress and heat flux appears to scale exclusively with the micro-macro length scale ratio given by the Knudsen number. On the other hand, at high convective Mach numbers, the deviation depends on the global micro-macro timescale ratio given by the product of Mach and Knudsen numbers. We further demonstrate that, unlike shear stresses and transverse heat flux, the deviations in normal stresses and the streamwise heat flux depend separately on Knudsen and Mach numbers. A local parameter called the gradient Knudsen number is proposed to characterize the rarefaction effects on the local momentum and thermal transport. Noncontinuum aspects of gas-kinetic stress-tensor and heat-flux behavior that Grad's 13-moment equationmodel reasonably captures are identified.

Quantitative uncertainty metric to assess continuum breakdown for nonequilibrium hydrodynamics

Multiscale problems such as hypersonic flows with strong nonequilibrium due to strong shocks and expansions result in flow physics which is no longer accurately described by the Navier-Stokes equations (NSE). Similarly, the NSE break down in rarefied (low density) gas flows. Therefore, hybrid methods, which can combine the continuum description using NSE and the kinetic description (KD), are necessary for efficient high-fidelity numerical simulations. A key input to hybrid methods is a metric to identify regions in the flow-field where the NSE breaks down and the KD should be used. In this Letter, starting from kinetic theory, we develop a rigorous metric to assess where the NSE breaks down.

Geometry effects on flow characteristics of micro-scale planar nozzles

A numerical investigation of geometry effects on the flow characteristics associated with converging-diverging 2D planar micronozzles is reported. The classical N–S with linear slip model, DSMC method, and a hybrid N–S/DSMC based on the continuum breakdown concept are employed for the simulations. The various methods adopted are validated with available experimental data. The study addresses the issue of heterogeneous findings on optimized half divergence angle recommended by previous studies. The analysis is conducted for various throat dimensions (2–200 µm thoat height), divergence angles (5∘–30∘), and divergence length. The performance and flow characteristics are compared at two distinct operating conditions, i.e. when the flow is mostly subsonic and supersonic in the divergent section. Results show an optimum divergence angle and length, maximizing the performance for a specific operating condition and nozzle size. Literature data suggest various divergence angles for optimum performance. This is due to the differences in operating conditions and the size of the micronozzles used. The current study shows that differences in flow behavior are mainly due to the pressure difference across the nozzle. When the micronozzle is operating under low-pressure differences, the flow is primarily subsonic in the divergent section. However, it accelerates due to the effect created by the growing thick boundary layer. The flow at higher pressure differences is characterized by clear flow separation and the occupation of the separated flow to a considerable portion of the divergent section. The separated phenomenon becomes less pronounced as the pressure difference and nozzle size decreases. As the nozzle size decreases to the nanoscale, Re is very low, and the subsonic layer fully occupies the divergent nozzle section even at very high-pressure differences. The results show that the performance is significantly influenced by the wall thermal conditions. The study highlights the differences in flow behaviour between micro and nano-scale nozzles at various operating conditions. Based on numerical data obtained in this present work, a new correlation is proposed to predict thrust per unit width for micronozzles.

Coupling kinetic and continuum using data-driven maximum entropy distribution

An important class of multi-scale flow scenarios deals with an interplay between kinetic and continuum phenomena. While hybrid solvers provide a natural way to cope with these settings, two issues restrict their performance. Foremost, the inverse problem implied by estimating distributions has to be addressed, to provide boundary conditions for the kinetic solver. The next issue comes from defining a robust yet accurate switching criterion between the two solvers. This study introduces a data-driven kinetic-continuum coupling, where the Maximum-Entropy-Distribution (MED) is employed to parametrize distributions arising from continuum field variables. Two regression methodologies of Gaussian-Processes (GPs) and Artificial-Neural-Networks (ANNs) are utilized to predict MEDs efficiently. Hence the MED estimates are employed to carry out the coupling, besides providing a switching criterion. To achieve the latter, a continuum breakdown parameter is defined by means of the Fisher information distance computed from the MED estimates. We test the performance of our devised MED estimators by recovering bi-modal densities. Next, MED estimates are integrated into a hybrid kinetic-continuum solution algorithm. Here Direct Simulation Monte-Carlo (DSMC) and Smoothed-Particle Hydrodynamics (SPH) are chosen as kinetic and continuum solvers, respectively. The problem of monatomic gas inside Sod's shock tube is investigated, where DSMC-SPH coupling is realized by applying the devised MED estimates. Very good agreements with respect to benchmark solutions along with a promising speed-up are observed in our reported test cases.

Velocity-Slip and Temperature-Jump Effects in Near-Continuum Hypersonic Flows

Velocity-slip and temperature-jump effects on sharp leading-edge geometries are studied for three canonical hypersonic flows: the hollow-cylinder-flare, the double-cone, and the double-wedge. Simulation results are compared between the direct simulation Monte Carlo (DSMC) method that naturally captures slip effects and computational fluid dynamics (CFD) using an appropriate slip model. Only the leading-edge portions of these flow configurations are studied. The simulations are verified to be converged in terms of grid and particle resolution. Overall, excellent agreement is found between DSMC and CFD predictions of the leading-edge boundary layer, the magnitude of velocity-slip and temperature-jump, and surface heat flux for all cases considered, with the exception of the double-cone case due to the relatively high Knudsen number of the flow. The largest discrepancies were found for the double-cone (transitional flow conditions), and near-exact agreement was found for the double-wedge (continuum flow conditions). It was determined that slip effects cannot account for the discrepancy between CFD predictions and experimentally measured heat flux for the hollow-cylinder flare. Furthermore, it was determined that the previously recommended threshold for the continuum breakdown parameter might be overly conservative in the near-wall region if appropriate slip models are used in CFD.

Navier–Stokes equations with rovibrational state-resolved transport flux closure

In the time since the Navier–Stokes equations were introduced more than two centuries ago, their application to problems involving real gas effects has relied on appropriate closure for the mass, momentum, and energy transport fluxes via the constitutive laws. Determination of the corresponding transport coefficients, most readily obtained through generalized Chapman–Enskog theory, requires knowledge of the intermolecular potentials for rotationally and vibrationally excited molecules. Recent advances in computational chemistry provide extraordinary detail of interactions involving rovibrationally excited molecules, offering a means for transport flux closure with unprecedented accuracy. Here, the bracket integrals for rovibrationally resolved molecular states are developed, and the resulting transport flux closure is presented for the rovibrationally resolved Navier–Stokes equations. The accompanying continuum breakdown parameters are also derived as a rigorous metric to establish the range of applicability of the aforementioned equations in flow conditions approaching the rarefied regime.

Collision rate of bidisperse, hydrodynamically interacting spheres settling in a turbulent flow

Abstract

Effect of heat transfer and geometry on micro-thruster performance

Coupled Navier-Stokes and Direct Simulation Monte Carlo (NS-DSMC) simulations of gas flow in micro-nozzles for various wall thermal conditions and geometrical aspects are presented. Micro-thrusters employed in miniature spacecraft and microsatellites experience substantial changes in wall thermal conditions. This can influence the internal boundary layer development and the exit plume structure of a micro-nozzle. These changes in flow physics differ with the nozzle divergence angle and the proximity of a similar nozzle in the cluster. Continuum solvers often fail to analyze the micro-nozzles operating in vacuum conditions as the flow in micro-nozzles experiences continuum, transitional, and rarefied regimes. Coupled NS-DSMC solver is an effective alternative that can simulate the non-equilibrium effects in a micro-nozzle flow field. A steady solution of the entire flow field has been obtained using a non-linear Harten-Lax-van Leer-Contact (HLLC) scheme based finite volume solver with a higher order slip boundary condition. Continuum breakdown regions are identified based on the gradient-length local (GLL) Knudsen number condition. This initial steady solution on the flow transition boundary is implemented in the DSMC solver as a Dirichlet boundary condition. The present computations are useful in calibrating micro-propulsion controllers to adapt to the substantial momentum changes associated with various nozzle wall thermal conditions and the proximity of similar nozzles in the cluster.

The sound of a pulsating sphere in a rarefied gas: continuum breakdown at short length and time scales

The pressure field of a pulsating sphere is a canonical problem in classical acoustics, used to illustrate the acoustic efficiency of a monopole source at continuum conditions. We consider the counterpart vibroacoustic and thermoacoustic problems in a rarefied gas, to investigate the effect of continuum breakdown on monopole radiation. Focusing on small-amplitude normal-to-boundary mechanical and heat-flux excitations, the perturbation field is analysed in the entire range of gas rarefaction and input frequencies. Numerical calculations are carried out via the direct simulation Monte Carlo method, and are used to validate analytical predictions in the free-molecular and near-continuum regimes. In the latter, the regularized thirteen moments model (R13) is applied, to capture the system response at states where the Navier–Stokes–Fourier (NSF) description breaks down. Comparing with the continuum inviscid solution, the results quantitate the dampening effect of gas rarefaction on source point-wise strength and acoustic power. At near-continuum conditions, the acoustic field is composed of exponentially decaying ‘compression’, ‘thermal’ and ‘Knudsen-layer’ modes, reflecting thermoviscous and higher-order rarefaction effects. With reducing rarefaction, the contributions of the latter two modes vanish, and the former degenerates into the ideal-flow inverse-to-distance decaying wave. Stronger attenuation is obtained with increasing rarefaction, where boundary sphericity results in a ‘geometric reduction’ of the molecular layer affected by the source. Notably, while the R13 model at low frequencies appears valid up to moderate gas rarefaction rates, both the NSF and R13 descriptions break down at common low Knudsen numbers at higher frequencies. Further study should therefore be carried out to extend the applicability of moment models to unsteady flows with short time scales.

Continuum breakdown and surface catalysis effects in NASA arc jet testing at SCIROCCO

A facility characterization test campaign was performed for NASA in SCIROCCO Plasma Wind Tunnel, with increasing enthalpy and constant reservoir pressure. In the frame of numerical rebuilding of the experimental data, an important gap among measured and numerically predicted values of heat flux led to deep analyse the numerical methods and the models usually employed for test rebuilding. The Navier–Stokes model with chemical reacting non-equilibrium flows, denoted a lack of physical accuracy due to local rarefaction effects, as certified by means of the continuum breakdown parameter. Furthermore, the low stagnation pressure environment could influence the surface catalytic behaviour of the hemisphere copper probe, and the chemical contribution to the stagnation heat flux. At the end, a set of direct simulation Monte Carlo with partial catalytic behaviour of the probe was performed, in order to address both critical phenomena highlighted and close the gap with the measured heat fluxes, understanding the actual test chamber environment.

Assessment of continuum breakdown for chemically reacting wake flows

Enabled by a newly developed species perturbation parameter, an analysis identifies forebody surface chemistry, among all other competing thermophysical processes, as the key contributor to continuum breakdown in hypersonic reacting wake flows.

Investigation on the Interface Smoothing of Coupled N–S/DSMC Method Using Image Processing Filters

Gas flow problems involving both continuum and rarefied regimes are common in many scientific and engineering applications. Coupled N–S/DSMC method is a major solution to simulate the continuum–rarefied transitional flow. Interface determination is one of the key aspects in developing coupled N–S/DSMC solver, which is often implemented using continuum breakdown parameters to indicate the rarefication level. Due to the statistics characteristic of DSMC method, distribution of the continuum breakdown parameters usually fluctuates, resulting in difficulty in locating the interface. Considering the similarity between continuum breakdown parameter smoothing and noise reduction in image processing, a general smoothing method is proposed in this paper, which employs the image filters to smoothen the distribution of continuum breakdown parameters. Two commonly used image filters, the mean filter and the median filter, are investigated. Several comparison studies are implemented by simulating a typical vacuum plume impingement flow problem. The median filter with 5 × 5 mask provides the best performance. The flow problem of flow over a 2D cylinder is used to validate the coupled N–S/DSMC solver using median filter to smoothen continuum breakdown parameter, which proves the accuracy of the coupled solver and validity of the smoothing method.

Development of a coupled NS-DSMC method for the simulation of plume impingement effects of space thrusters

A coupled NS-DSMC method possessing adapted-interface and two-way coupling features is studied to simulate the plume impingement effects of space thrusters. The continuum-rarefied interface is determined by combining KnGL and Ptne continuum breakdown parameters. State-based coupling scheme is adopted to transfer information between continuum and particle solvers, and an overlapping grid technique is investigated to combine structured-grid NS code and Cartesian-grid DSMC code to form the coupled solver. Flow problem of a conical thruster plume impinging on a cone surface is simulated using the coupled solver, and the simulation result is compared with experimental data, which proves the validity of the proposed method. Plume flow while the ascent stage of lunar module lifting off in lunar environment is also computed by using the present coupled NS-DSMC method to demonstrate its capability. The whole flow field from combustion chamber to the vacuum environment is obtained, and the result reveals that special attention should be paid to the plume aerodynamic force at the early stage of launching process.