Continuum Regime Research Articles

Drag and thermophoresis on a sphere in a rarefied gas based on the Cercignani–Lampis model of gas–surface interaction

Abstract

Simulations of Unsteady Three-Dimensional Hypersonic Double-Wedge Flow Experiments

Hypersonic flow over a three-dimensional 30–55 deg double wedge is investigated for three different conditions of varying enthalpy and Reynolds number with nitrogen as the test gas. Streamwise and spanwise wall heat flux are compared to the experimental data that were obtained at the University of Illinois at Urbana–Champaign and the California Institute of Technology. In previous studies, there were concerns that portions of the flowfield were rarefied; however, after an in-depth analysis, we find the flowfield to be firmly in the continuum regime. Comparisons of heat flux at an instant of time in the simulations show good agreement with the experiments. However, averaged heat flux over a long period of time relative to the experimental test time are underpredicted and develop low-frequency periodic motion. To better understand the unsteadiness and three-dimensionality of the flowfield, we analyze the mass flux through a spanwise plane, and we perform sparsity-promoting dynamic mode decomposition for the wall heat flux and midplane pressure. Finally, we analyze the effect of vibrational nonequilibrium on the flowfield. It is found that the flowfield dynamics vary with increasing enthalpy and Reynolds number. Furthermore, the flowfield is found to be three dimensional, unsteady, and asymmetric for all cases.

Multiple-flow-regime models for real gas transport in fractal porous media at high pressure

A Mathematical model for total mass flow rate per unit area and total permeability of real gas flow in fractal porous media was proposed on the basis of fractal theory. Proposed models in this paper paid attention to the influence of real gas effect on gas, taking transition regime as the coupling of slip regime and Knudsen diffusion, considering multi-flow-regime coexist, multi-flow-regime is continuum flow regime, slip flow regime, transition flow regime and Knudsen diffusion regime., pore which pore diameter is λ mass flow rate per unit area multiply corresponding pore number dN obtain mass flow rate(dq). Integrate dq within corresponding diameter range and divide fractal porous media work out corresponding flow regime the mass flow rate per unit area. The total mass flow rate per unit area was then calculated through integrating over the mass flow rate deduced, where different flow regimes have been engaged, including continuum regime, slip regime, transition regime and Knudsen diffusion. Based on the Darcy theory, the total permeability was then calculated. In order to verify correctness of model, compare proposed model to Wang's model and Fan's model. Calculate correlation coefficient of proposed model and Wang's model, R2=0.9912; Calculate correlation coefficient of proposed model and Wang's model, R2=0.9997. Indicate that proposed model can fit well with Wang's model and Fan's model, proposed model can correctly predict total permeability. And calculate correlation coefficient of D.B.Bennion experiment data and three model.In the last part, the paper analyzed the influence factor of total mass flow rate per unit area and total permeabilities and mass flow rate per unit area and permeability of different flow regimes. The concludes as follow: (1) total mass flow rate per unit area, total permeability and each flow regime mass flow rate per unit area and permeability increase with fractal dimension(Dp) increasing and tortuous fractal dimension(Dt) decreasing. Variation of Dp and Dt has different influence on each flow regime. Dp and Dt has most influence on Knudsen diffusion flow regime, least influence on continuum flow regime. (2) Real gas effect affects the mass flow rate per unit area and permeability in different flow regime by affecting the mean free path of gas molecule. Whether consider real gas effect would cause great difference, maximum difference is 232.35%. (3) Flow regime that have the lowest Knudsen number play leading role in variation of total mass flow rate and total permeability. (4) The more maximum pore diameter of fractal porous media, the more significant influence of total mass flow rate per unit area and total permeability. Maximum pore diameter has different influence on each flow regime, has most influence on Knudsen diffusion flow regime, least influence on continuum flow regime.

Simplified unified wave-particle method with quantified model-competition mechanism for numerical calculation of multiscale flows.

A quantified model-competition (QMC) mechanism for multiscale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic and hydrodynamic) model are quantified. Then, a simplified unified wave-particle method (SUWP) is constructed based on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cell of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high nonequilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an antidissipation effect in the continuum mechanism is also discussed.

Modelling and Simulation of Transpiration Cooling Systems for Atmospheric Re-Entry

Aerothermodynamic heating is one of the primary challenges faced in progressing towards reliable hypersonic transportation. In the present study, the transpiration cooling method applied to the thermal protection system of re-entry vehicles is investigated. The complexity in analysing the incoming heat flux for re-entry lies not only in the extreme conditions of the flow but also in the fact that the coolant flow through the porous medium needs to be treated appropriately. While the re-entering spacecraft passes through various flow regimes, the peak conditions are faced only near continuum regime. Focusing on these conditions, traditional computational fluid dynamics techniques are used to model transpiration cooling for re-entry vehicles. In the current work, the open source CFD framework OpenFOAM is used to couple two different solvers iteratively and then analyse the thermal response for flow speed conditions typical of re-entry vehicles. Independent computations are performed using the explicit, loosely coupled procedure for high speed argon flow over a 2D axi-symmetrical cylindrical vehicle. The results presented indicate distinct heat flux drop along the surface of the cylindrical vehicle as a function of parameters such as coolant pressure and wall temperature.

Numerical Study on Gas Transport in Shale Kerogen with Adsorption Effect Using LBM

Gas transport and storage mechanism in gas reservoirs with complex pore size distribution play a key role in gas production. In the nano-scale, during the process of density disturbance caused by depressurization development, notable output gas collides with the wall as they convey along the channel. The continuum assumption of fluid flow is therefore invalid, and taking the influence of particles to collide with solid walls into account is necessary. The effective viscosity and slip boundary conditions were introduced in the simulation. Meanwhile, because of the adsorption effect is one of the most important intrinsic properties of shale, the impact of surface adsorption on the solid wall by using Langmuir adsorption kinetics was also involved in our investigation. In this work, a unified LB model was developed for gas multiple transports and surface adsorption in nanopores. In the model, the double distribution function was employed for a coupled bulk flow and gas diffusion model. The model is valid within the range of the continuum regime to the transition regime and has been verified by the analytical solution and the results of some previously published works. The results show that (1) the driven force of the fluid, the combination coefficient of slippage and the Knudsen number of the flow have an impact on the adsorption rate, (2) adsorption effect only directly influence the local small region of the solid surface, and the induced concentration gradient makes the global concentration variation, (3) once acting the surface adsorption on the boundary, the flux falls more significant as Knudsen number increases.

A novel solver for simulation of flows from continuum regime to rarefied regime at moderate Knudsen number

A novel and simple solver for simulation of flows from continuum regime to rarefied regime at moderate Knudsen number is developed in this work. The present solver combines good features of gas kinetic flux solver (GKFS), discrete velocity method (DVM) and the moment method. Like the GKFS, in the present solver, the macroscopic governing equations are discretized by finite volume method and the numerical fluxes at cell interfaces are evaluated by the local solution of Boltzmann equation. To get the local solution of Boltzmann equation, the initial distribution function is reconstructed with the help of Grad's distribution function, which is inspired from the moment method. For the high order moments in Grad's distribution function, they are computed directly by the moments of distribution function, which is inspired from DVM. In principle, the present solver only needs to discretize the physical space and solves the governing equations resulted from conservation laws of mass, momentum and energy. Thus, its governing equations are much simpler than those of the moment method. To validate the proposed solver, some test examples covering continuum regime and rarefied regime are simulated. Numerical results showed that the present solver can give accurate prediction in the continuum regime and reasonable results as the traditional moment method in the rarefied regime at moderate Knudsen number.

Thermodynamic and experimental evaluation of a cloud chamber for ultrafine particle detection

Particle sensing based on condensational growth has long been the basis for robust nanoparticle measurement. Increasingly cloud chamber devices offer the potential for low-cost and portable measurement when operated semi-continuously with relatively small system volumes. Models based on isentropic and isenthalpic expansion are derived to predict the time evolution of temperature, saturation ratio, particle growth, and resultant light extinction in cloud chambers. A laboratory cloud chamber is fabricated and experiments using NaCl aerosol particles as the condensation nucleus are conducted to verify the models. The isentropic model, suggests that the temperature drops 0.6 ℃ within 40 ms, and accordingly, the saturation ratio reaches 1.04. For an aerosol with lognormal distribution, the predicted geometric mean diameter grows more than 5 times while the distribution narrows due to ∝1/dp growth in the continuum regime. The performance of the cloud chamber agrees with the system physics and reference instruments, with relative error in measured extinction coefficient and signal intensities of ±5%. Detailed error propagation shows that the measured number concentrations agree well with reference instruments and the underlying theory. The lower limit of detection (∼4 × 106 cm−3) for the device is suitable for fire detection and emissions characterization.

An analytical solution of the population balance equation for simultaneous Brownian and shear coagulation in the continuum regime

Brownian coagulation of aerosol particles can take place in both laminar and turbulent flows. Thus, the simultaneous Brownian and shear coagulation will always occur in practical applications. This study presents an analytical solution to describe the size evolution of polydisperse particles undergoing simultaneous Brownian and shear coagulation. The analytical solution is derived using the log-normal method of moments (LNMOM) with some approximations. Then, the analytical solution is validated by comparing with previous analytical solutions derived for limiting cases. The results show that the present analytical solution is consistent with previous analytical solutions for these limiting cases. Further, the time trajectories of the total particle number concentration, the geometric standard deviation and the geometric number mean particle volume predicted by the present analytical solution are compared with those of a numerical LNMOM model. The results show that the present analytical solution gives good predictions of the total particle number concentration but less accurate predictions of the geometric standard deviation and the geometric number mean particle volume. A dimensionless analysis shows that the coagulation rate ratio and the initial polydispersity are two important factors for characterizing the size evolution of simultaneous Brownian and shear coagulation.

Luminosity calculation of meteor entry based on detailed flow simulations in the continuum regime

Context. Composition, mass, and trajectory parameters of meteors can be derived by combining observations with the meteor physics equations. The fidelity of these equations, which rely on heuristic coefficients, significantly affects the accuracy of the properties inferred. Aims. Our objective is to present a methodology that can be used to compute the luminosity of meteor entry based on detailed flow simulations in the continuum regime. Methods. The methodology consists in solving the Navier–Stokes equations using state-of-the-art physico-chemical models for hypersonic flows. It includes accurate boundary conditions to simulate the surface evaporation of the molten material and coupled flow-radiation effects. Such detailed simulations allow for the calculation of heat-transfer coefficients and luminous efficiency, which can be incorporated into the meteor physics equations. Finally, we integrate the radiative transfer equation over a line of sight from the ground to the meteor to derive the luminosity magnitude. Results. We use the developed methodology to simulate the Lost City bolide and to derive the luminosity magnitude, obtaining good agreement between numerical results and observations. The computed color index is more prominent than the observations. This is attributed to a lack of refractory elements such as Ca in the modeled flow that might originate from the vaporization of droplets in the trail, a phenomenon currently not included in the model.

Prediction of nano/micro aluminum particles ignition in oxygen atmosphere

Ignition prediction of aluminum particle is of great significance for a variety of propulsion and power systems to achieve optimal energy release within a limited residence time. In this study a heat transfer model employing temperature dependent coefficients was developed and validated to describe the heat exchange between quiescent/flow gas and aluminum particles from nano- to micro-size, covering the free-molecular to continuum regimes. By coupling heat transfer and aluminum oxidation, a theoretical model has been proposed to accurately capture ignition properties of both aluminum nanoparticle and microparticle (ANP and AMP) burning in hot oxygen atmosphere. Two formulas were obtained to predict the ignition temperature and ignition delay time for nano/micro particles, which show good agreements with experimental results, providing a convenient and accurate method for practical application. A parametric study illustrates that AMP ignition is affected by bulk flow velocity, radiation and oxygen concentration, particularly for AMP over 100 μm in diameter; in contrast, ANP is more sensitive to alumina thickness which generally raises both ignition temperature and ignition delay time. The present study not only deepens the fundamental understanding of aluminum combustion but also provides a guideline for prompting ignition.

Q/V enhancement of micropillar resonator in bound states in the continuum regime

We show how the optical quasi-bound states in the continuum, recently predicted and observed in dielectric nanoantennae, can be utilized to substantially enhance quality factor to mode volume ratio in pillar microcavities.

Temperature measurement of a heated rod array within a square cross section enclosure filled with dry rarefied helium

The goal of this work is to obtain experimental data that can be used to validate computational models that quantify the effect of gas rarefaction on heat transfer during vacuum drying of used nuclear fuel transport/storage canisters. An experimental apparatus is constructed, consisting of a 7 × 7 array of electrically heated rods held by spacer plates near their ends and contained within a square cross-section helium-filled pressure vessel. Thermocouples are used to measure heater rod, spacer plate, and enclosure temperatures for a range of rod heat generation rates, helium pressures in the continuum and rarefied-gas slip regimes, and different thicknesses of insulation outside the enclosure, which increases the apparatus temperatures. The results shows that the temperature difference between the enclosure and the rods increases by less than 4% when the pressure decreases within the continuum regime (from ~ 105 to 5700 Pa). However, it increases by up to 78% in the slip regime (~5700 to 65 Pa), due to the temperature-jump thermal resistance at the gas/solid interfaces. Random variation in the measured temperatures, caused by configuration and measurement errors, is less than 1.1℃, which makes this data well suited for benchmarking computational methods for calculating heat transfer and temperatures in used nuclear fuel canisters under vacuum drying conditions.

Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow

The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by the evolution of macroscopic variables, and guarantees the conservation of the scheme in the final wave-particle formulation. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced. Multiscale preserving means UGKWP method preserves the flow regime from collisionless regime to hydrodynamic regime without requiring the cell size and time step to be less than the mean free path and collision time. Multi-efficiency preserving means the computational cost of the scheme including the computational time and memory cost is on the same level as the most efficient method in the corresponding regime, such as the particle methods in the rarefied regime and hydrodynamic solvers in continuum regime. The UGKWP method is shown to satisfy the MMP requirement. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include the hypersonic flow passing a circular cylinder at Mach numbers 20 and 30 and Knudsen numbers 1 and 10−4, low speed lid-driven cavity flow, laminar boundary layer, shock structure, and shock tube problems. These results validate the accuracy, efficiency, and multiscale and multi-efficiency property of UGKWP method.

A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions with second-order accuracy. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

Influence of three-dimensional transverse micro-ridges on the Poiseuille number in a gaseous slip flow

Micro-ridges are regular micro-engineering structures protruding outward from the microchannel wall surface. The role of these micro-ridges in changing the friction factor (f) of a microchannel with internal rarefied gas flows is not explored in the literature. In this work, we present three-dimensional numerical simulations in a microchannel containing ridges arranged transverse to the flow direction. The length of the ridges is varied with respect to the pitch of the ridges; this ratio being christened as ridge fraction (δ). The effect of δ, ridge height ratio (h/H), Reynolds number (Re), Knudsen number (Kn) and Tangential momentum accommodation coefficient (αv) on the flow friction in terms of Poiseuille number (fRe) is investigated. In the continuum and slip regimes, it is found that fRe decreases with an increase in δ and h and a significant reduction in flow friction was shown with respect to a straight microchannel. There exists a critical height after which decrease in fRe does not occur. Further, fRe was observed to be a strong function of Re at high δ, which is attributed to the intensity of acceleration–deceleration experienced by the gas during the flow. Also, fRe is established to be directly proportional to αv. The various flow characteristics are scrutinised where vortices are found trapped inside the ridge affecting the dynamics of flow in the ridges, whose size increases on increasing the gas rarefaction. This comprehensive investigation of behaviour of slip flow in complex microchannels will be useful in designing micro-devices with reduced friction.

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.

Hypersonic aerodynamics of a deformed aeroshell in continuum and near-continuum regimes

The flexible inflatable aeroshell naturally deforms under aerodynamic loads during atmospheric entry. In this paper, we investigate the hypersonic aerodynamics of an inflatable aeroshell forebody with undulated surface deformation in the continuum and near-continuum flow regimes, where flow physics is complex involving rarefaction and thermochemical nonequilibrium effect. The aeroshell forebody shape is originated from a model of 8.3 m diameter stacked-tori hypersonic inflatable aerodynamic decelerator, and a series of surface deflections is generated by a parametric method based on the consistency of undulation with underlying structure. The trajectory points of 90, 80, and 68 km altitudes in a low Earth orbit reentry are considered. The thermochemical nonequilibrium Navier-Stokes equations with slip and no-slip boundary conditions are numerically solved to characterize flowfields and provide aerodynamic predictions. The results demonstrate that the undulated surface deflection induces local fluctuations of pressure and local Knudsen number, while the local variations of aerodynamic quantities create minor difference in total drag. The surface friction and drag are dependent on boundary conditions, i.e., either slip or no-slip model. The results also indicate that it is required to employ slip boundary conditions for the accurate prediction of friction and drag in the near continuum regime.

Effect of nozzle spacing in the formation of primary and secondary deposits in multi-nozzle inertial impactors part II: Numerical study.

In this study CFD models are introduced and validated to describe the flow field on the experimental impactor (3 impinging jets) and to determine the particle deposition mechanisms (primary and secondary deposits) for each experimental condition carried out in Part I. For the determination of particle trajectories (dp between 1.5 and 6.0 μm), a steady state numerical analysis was carried out for laminar and turbulent flow continuum regimes. In addition to that, the effect of gravity and the so-called stochastic turbulent models on particle trajectories was also studied.Regarding the effects of the jet interaction, the simulation results show that:- The half-moon shaped premature primary deposits are formed due to the deviation of the jet caused by the recirculation of the jet-to-jet interaction.- The straight lines deposits are formed by particles that are not able to follow the high velocities generated in the recirculation zone between pairs of jets and are separated inertially in the line of stagnation between them.- The nozzle plate backside deposits are formed with particles of the interaction region that ascend to the nozzle plate in the area in which the three jets converge.- With respect to the halo, from the combined analysis of the numerical and experimental results, it is possible to distinguish two types of halo occurring at low and at larger Reynolds numbers. Therefore, the analysis of the formation of the halo must take into account both inertial and gravitational effects and the balance between them.The impactor CFD models have been validated with the experimental data obtained for the same operating parameters, with good agreement for both the √Stk50 and the steepness of the efficiency curves. These accurate simulations are obtained using a geometry of multiple nozzles, with an impaction plate sufficiently large that allowed the formation of the halo, and by using a two-equation turbulence model that included gravity and turbulent dispersion to calculate the particle trajectories.

Flowfield structure characteristics of the hypersonic flow over a cavity: From the continuum to the transition flow regimes

The cavity configurations produced by discontinuities at thermal protection system coated at surface of reentry space vehicles can lead to separation, attachment, and even transition of gas flows around the vehicle's surface, which can bring serious aerodynamic heating during the space vehicles reentering the Earth's atmosphere. In this paper, the continuum, slip, and transition flow regimes are chosen to represent reentry atmosphere environment and then a numerical investigation of hypersonic gas flows over a family of cavities is performed by using the direct simulation Monte Carlo method to reveal flow structures in the cavity flowfield under different length-to-depth ratios and freestream incoming Mach numbers. In particular, when the freestream varies from the continuum to the transition flow regimes, the minimal length-to-depth ratios required for cavity flow becoming the closed type are 8, 6, and 5, respectively. The effect of the freestream incoming Mach number on the cavity flow characteristics is getting weaker when the freestream varies from the continuum to the transition flow regime. The two significant findings are that the cavity flow characteristics under hypersonic conditions are primarily determined by cavity geometry rather than freestream incoming Mach number and secondly, effect of the freestream incoming Mach number on cavity flow type is very slight if the freestream is in the transition flow regime.