# Physics of Fluids

Table of Contents for Physics of Fluids. List of articles from both the latest and ahead of print issues.

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### Assessment of effectiveness of optimum physical distancing phenomena for COVID-19

Physics of Fluids, Volume 33, Issue 5, May 2021.

Currently, COVID-19 is a global pandemic that scientists and engineers around the world are aiming to understand further through rigorous testing and observation. This paper aims to provide safe distance recommendations among individuals and minimize the spread of COVID-19, as well as examine the efficacy of face coverings as a tool to slow the spread of respiratory droplets. These studies are conducted using computational fluid dynamics analyses, where the infected person breathes, coughs, and sneezes at various distances and environmental wind conditions and while wearing a face-covering (mask or face shield). In cases where there were no wind conditions, the breathing and coughing simulations display 1–2 m physical distancing to be effective. However, when sneezing was introduced, the physical distancing recommendation of 2 m was deemed not effective; instead, a distance of 2.8 m and greater was found to be more effective in reducing the exposure to respiratory droplets. The evaluation of environmental wind conditions necessitated an increase in physical distancing measures in all cases. The case where breathing was measured with a gentle breeze resulted in a physical distancing recommendation of 1.1 m, while coughing caused a change from the previous recommendation of 2 m to a distance of 4.5 m or greater. Sneezing in the presence of a gentle breeze was deemed to be the most impactful, with a recommendation for physical distancing of 5.8 m or more. It was determined that face coverings can potentially provide protection to an uninfected person in static air conditions. However, the uninfected person's protection can be compromised even in gentle wind conditions.

Currently, COVID-19 is a global pandemic that scientists and engineers around the world are aiming to understand further through rigorous testing and observation. This paper aims to provide safe distance recommendations among individuals and minimize the spread of COVID-19, as well as examine the efficacy of face coverings as a tool to slow the spread of respiratory droplets. These studies are conducted using computational fluid dynamics analyses, where the infected person breathes, coughs, and sneezes at various distances and environmental wind conditions and while wearing a face-covering (mask or face shield). In cases where there were no wind conditions, the breathing and coughing simulations display 1–2 m physical distancing to be effective. However, when sneezing was introduced, the physical distancing recommendation of 2 m was deemed not effective; instead, a distance of 2.8 m and greater was found to be more effective in reducing the exposure to respiratory droplets. The evaluation of environmental wind conditions necessitated an increase in physical distancing measures in all cases. The case where breathing was measured with a gentle breeze resulted in a physical distancing recommendation of 1.1 m, while coughing caused a change from the previous recommendation of 2 m to a distance of 4.5 m or greater. Sneezing in the presence of a gentle breeze was deemed to be the most impactful, with a recommendation for physical distancing of 5.8 m or more. It was determined that face coverings can potentially provide protection to an uninfected person in static air conditions. However, the uninfected person's protection can be compromised even in gentle wind conditions.

Categories: Latest papers in fluid mechanics

### Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Physics of Fluids, Volume 33, Issue 5, May 2021.

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.

Categories: Latest papers in fluid mechanics

### A real-time flow forecasting with deep convolutional generative adversarial network: Application to flooding event in Denmark

Physics of Fluids, Volume 33, Issue 5, May 2021.

Real-time flood forecasting is crucial for supporting emergency responses to inundation-prone regions. Due to uncertainties in the future (e.g., meteorological conditions and model parameter inputs), it is challenging to make accurate forecasts of spatiotemporal floods. In this paper, a real-time predictive deep convolutional generative adversarial network (DCGAN) is developed for flooding forecasting. The proposed methodology consists of a two-stage process: (1) dynamic flow learning and (2) real-time forecasting. In dynamic flow learning, the deep convolutional neural networks are trained to capture the underlying flow patterns of spatiotemporal flow fields. In real-time forecasting, the DCGAN adopts a cascade predictive procedure. The last one-time step-ahead forecast from the DCGAN can act as a new input for the next time step-ahead forecast, which forms a long lead-time forecast in a recursive way. The model capability is assessed using a 100-year return period extreme flood event occurred in Greve, Denmark. The results indicate that the predictive fluid flows from the DCGAN and the high fidelity model are in a good agreement (the correlation coefficient [math] and the mean absolute error [math]) for a lead-900 time step forecast. This is an important step toward real-time flow forecasting although further evaluation of the DCGAN performance is required in complex realistic cases in the future.

Real-time flood forecasting is crucial for supporting emergency responses to inundation-prone regions. Due to uncertainties in the future (e.g., meteorological conditions and model parameter inputs), it is challenging to make accurate forecasts of spatiotemporal floods. In this paper, a real-time predictive deep convolutional generative adversarial network (DCGAN) is developed for flooding forecasting. The proposed methodology consists of a two-stage process: (1) dynamic flow learning and (2) real-time forecasting. In dynamic flow learning, the deep convolutional neural networks are trained to capture the underlying flow patterns of spatiotemporal flow fields. In real-time forecasting, the DCGAN adopts a cascade predictive procedure. The last one-time step-ahead forecast from the DCGAN can act as a new input for the next time step-ahead forecast, which forms a long lead-time forecast in a recursive way. The model capability is assessed using a 100-year return period extreme flood event occurred in Greve, Denmark. The results indicate that the predictive fluid flows from the DCGAN and the high fidelity model are in a good agreement (the correlation coefficient [math] and the mean absolute error [math]) for a lead-900 time step forecast. This is an important step toward real-time flow forecasting although further evaluation of the DCGAN performance is required in complex realistic cases in the future.

Categories: Latest papers in fluid mechanics

### Maximum spreading and energy analysis of ellipsoidal impact droplets

Physics of Fluids, Volume 33, Issue 5, May 2021.

Droplet impacts on solid surfaces are ubiquitous in nature and industry. Before impact, the droplet shape may be affected by gravity, shear flow, and the electric and magnetic fields, inducing non-spherical droplets. However, most previous studies focused on the impact dynamics of spherical droplets. In this study, we conduct experiments, simulations, and theoretical analyses to investigate the impact behaviors of ellipsoidal water droplets whose symmetry axis is perpendicular to the surface. In particular, we explore the maximum spreading and energy evolution during impact. A numerical model adopting the Volume of Fluid method and Kistler's dynamic contact angle model achieves good agreement with the experimental results for both the temporal droplet profile and spreading factor. The effects of Weber number, contact angle, and aspect ratio on the impact dynamics are systematically investigated, and the outcomes show that both the maximum spreading time and factor enlarge with the increasing aspect ratio. Their relations approximately follow the 2/3-power and 1/6-power laws, respectively. Reducing the aspect ratio enhances the viscous dissipation during impact. Based on the theoretical analyses of above results, we modify the viscous dissipation in the conventional energy balance model to include the effects of aspect ratio on the maximum spreading factor. The modified theoretical model reduces the deviations from −23%–51% to −5%–25% and elucidates the scaling law between the maximum spreading factor and aspect ratio. This work deepens our understanding of the interaction between non-spherical impact droplets and surfaces and may contribute to associated applications.

Droplet impacts on solid surfaces are ubiquitous in nature and industry. Before impact, the droplet shape may be affected by gravity, shear flow, and the electric and magnetic fields, inducing non-spherical droplets. However, most previous studies focused on the impact dynamics of spherical droplets. In this study, we conduct experiments, simulations, and theoretical analyses to investigate the impact behaviors of ellipsoidal water droplets whose symmetry axis is perpendicular to the surface. In particular, we explore the maximum spreading and energy evolution during impact. A numerical model adopting the Volume of Fluid method and Kistler's dynamic contact angle model achieves good agreement with the experimental results for both the temporal droplet profile and spreading factor. The effects of Weber number, contact angle, and aspect ratio on the impact dynamics are systematically investigated, and the outcomes show that both the maximum spreading time and factor enlarge with the increasing aspect ratio. Their relations approximately follow the 2/3-power and 1/6-power laws, respectively. Reducing the aspect ratio enhances the viscous dissipation during impact. Based on the theoretical analyses of above results, we modify the viscous dissipation in the conventional energy balance model to include the effects of aspect ratio on the maximum spreading factor. The modified theoretical model reduces the deviations from −23%–51% to −5%–25% and elucidates the scaling law between the maximum spreading factor and aspect ratio. This work deepens our understanding of the interaction between non-spherical impact droplets and surfaces and may contribute to associated applications.

Categories: Latest papers in fluid mechanics

### Coupled vibration-dissociation time-histories and rate measurements in shock-heated, nondilute O2 and O2–Ar mixtures from 6000 to 14 000 K

Physics of Fluids, Volume 33, Issue 5, May 2021.

Validation of high-fidelity models for high-temperature hypersonic flows requires high-accuracy kinetics data for oxygen (O2) reactions, including time-histories and rate parameter measurements. Consequently, shock-tube experiments with ultraviolet (UV) laser absorption were performed to measure quantum-state-specific time-histories and coupled vibration-dissociation (CVDV) rate parameters in shock-heated, nondilute O2 and oxygen–argon (O2–Ar) mixtures. Experiments probed mixtures of 20% O2–Ar, 50% O2–Ar, and 100% O2 for initial post-reflected-shock conditions from 6000 to 14 000 K and 26–210 Torr. Two UV lasers—one continuous-wave laser and one pulsed laser—measured absorbance time-histories from the fifth and sixth vibrational levels of the electronic ground state of O2, respectively. The absorbance time-histories subsequently yielded time-histories for vibrational temperature (Tv) from the absorbance ratio, translational/rotational temperature (Ttr) from energy conservation, total O2 number density ([math]) from the individual absorbances, and vibrational-state-specific number density ([math]) from the Boltzmann population fractions. These state-specific temperature and number density time-histories demonstrate the low uncertainty necessary for high-temperature model validation and provide data to higher temperature than previous experiments. Additional analysis of the temperature and number density time-histories allowed inference of rate parameters in the Marrone and Treanor CVDV model, including vibrational relaxation time ([math]), average vibrational energy loss (ε), vibrational coupling factor (Z), and dissociation rate constants ([math] and [math]). The results for each of these five parameters show reasonable consistency across the range of temperatures, pressures, and mixtures and generally agree with a modified Marrone and Treanor model by Chaudhry et al. [“Implementation of a chemical kinetics model for hypersonic flows in air for high-performance CFD,” in Proceedings of AIAA Scitech Forum (2020)]. Finally, the results for [math], [math], and [math] exhibit much lower scatter than previous experimental studies.

Validation of high-fidelity models for high-temperature hypersonic flows requires high-accuracy kinetics data for oxygen (O2) reactions, including time-histories and rate parameter measurements. Consequently, shock-tube experiments with ultraviolet (UV) laser absorption were performed to measure quantum-state-specific time-histories and coupled vibration-dissociation (CVDV) rate parameters in shock-heated, nondilute O2 and oxygen–argon (O2–Ar) mixtures. Experiments probed mixtures of 20% O2–Ar, 50% O2–Ar, and 100% O2 for initial post-reflected-shock conditions from 6000 to 14 000 K and 26–210 Torr. Two UV lasers—one continuous-wave laser and one pulsed laser—measured absorbance time-histories from the fifth and sixth vibrational levels of the electronic ground state of O2, respectively. The absorbance time-histories subsequently yielded time-histories for vibrational temperature (Tv) from the absorbance ratio, translational/rotational temperature (Ttr) from energy conservation, total O2 number density ([math]) from the individual absorbances, and vibrational-state-specific number density ([math]) from the Boltzmann population fractions. These state-specific temperature and number density time-histories demonstrate the low uncertainty necessary for high-temperature model validation and provide data to higher temperature than previous experiments. Additional analysis of the temperature and number density time-histories allowed inference of rate parameters in the Marrone and Treanor CVDV model, including vibrational relaxation time ([math]), average vibrational energy loss (ε), vibrational coupling factor (Z), and dissociation rate constants ([math] and [math]). The results for each of these five parameters show reasonable consistency across the range of temperatures, pressures, and mixtures and generally agree with a modified Marrone and Treanor model by Chaudhry et al. [“Implementation of a chemical kinetics model for hypersonic flows in air for high-performance CFD,” in Proceedings of AIAA Scitech Forum (2020)]. Finally, the results for [math], [math], and [math] exhibit much lower scatter than previous experimental studies.

Categories: Latest papers in fluid mechanics

### Effect of the inclination angle on the transient melting dynamics and heat transfer of a phase change material

Physics of Fluids, Volume 33, Issue 5, May 2021.

We report two-dimensional simulations and analytic results on the effect of the inclination on the transient heat transfer, flow, and melting dynamics of a phase change material within a square domain heated from one side. The liquid phase has Prandtl number Pr = 60.8, Stefan number Ste = 0.49, and Rayleigh numbers extend over eight orders of magnitude [math] for the largest geometry studied. The tilt determines the stability threshold of the base state. Above a critical inclination, there exists only a laminar flow at the melted phase, irrespective of the Rayleigh number. Below that inclination, the base state destabilizes following two paths according to the inclination: either leading to a turbulent state for angles near the critical inclination or passing through a regime of plume coarsening before reaching the turbulent state for smaller angles. We find that the Nusselt and Reynolds numbers follow a power law as [math] in the turbulent regime. Small inclinations reduce very slightly α and strongly β. The inclination leads to subduction of the kinematic boundary layer into the thermal boundary layer. The scaling laws of the Nusselt and Reynolds numbers and boundary layers are in agreement with different results at high Rayleigh convection. However, some striking differences appear as the stabilization of turbulent states with further increasing of the Rayleigh number. We find as well that the turbulent regime exhibits a higher dispersion in quantities related to heat transfer and flow dynamics on smaller domains.

We report two-dimensional simulations and analytic results on the effect of the inclination on the transient heat transfer, flow, and melting dynamics of a phase change material within a square domain heated from one side. The liquid phase has Prandtl number Pr = 60.8, Stefan number Ste = 0.49, and Rayleigh numbers extend over eight orders of magnitude [math] for the largest geometry studied. The tilt determines the stability threshold of the base state. Above a critical inclination, there exists only a laminar flow at the melted phase, irrespective of the Rayleigh number. Below that inclination, the base state destabilizes following two paths according to the inclination: either leading to a turbulent state for angles near the critical inclination or passing through a regime of plume coarsening before reaching the turbulent state for smaller angles. We find that the Nusselt and Reynolds numbers follow a power law as [math] in the turbulent regime. Small inclinations reduce very slightly α and strongly β. The inclination leads to subduction of the kinematic boundary layer into the thermal boundary layer. The scaling laws of the Nusselt and Reynolds numbers and boundary layers are in agreement with different results at high Rayleigh convection. However, some striking differences appear as the stabilization of turbulent states with further increasing of the Rayleigh number. We find as well that the turbulent regime exhibits a higher dispersion in quantities related to heat transfer and flow dynamics on smaller domains.

Categories: Latest papers in fluid mechanics

### Confinement and complex viscosity

Physics of Fluids, Volume 33, Issue 5, May 2021.

Whereas much is known about the complex viscosity of polymeric liquids, far less is understood about the behavior of this material function when macromolecules are confined. By confined, we mean that the gap along the velocity gradient is small enough to reorient the polymers. We examine classical analytical solutions [O. O. Park and G. G. Fuller, “Dynamics of rigid and flexible polymer chains in confined geometries. II. Time-dependent shear flow,” J. Non-Newtonian Fluid Mech. 18, 111–122 (1985)] for a confined rigid dumbbell suspension in small-amplitude oscillatory shear flow. We test these analytical solutions against the measured effects of confinement on both parts of the complex viscosity of a carbopol suspension and three polystyrene solutions. From these comparisons, we find that both parts of the complex viscosity decrease with confinement and that macromolecular orientation explains this. We find the persistence length of macromolecular confinement, [math], to be independent of both [math] and [math].

Whereas much is known about the complex viscosity of polymeric liquids, far less is understood about the behavior of this material function when macromolecules are confined. By confined, we mean that the gap along the velocity gradient is small enough to reorient the polymers. We examine classical analytical solutions [O. O. Park and G. G. Fuller, “Dynamics of rigid and flexible polymer chains in confined geometries. II. Time-dependent shear flow,” J. Non-Newtonian Fluid Mech. 18, 111–122 (1985)] for a confined rigid dumbbell suspension in small-amplitude oscillatory shear flow. We test these analytical solutions against the measured effects of confinement on both parts of the complex viscosity of a carbopol suspension and three polystyrene solutions. From these comparisons, we find that both parts of the complex viscosity decrease with confinement and that macromolecular orientation explains this. We find the persistence length of macromolecular confinement, [math], to be independent of both [math] and [math].

Categories: Latest papers in fluid mechanics

### A Lagrangian probability-density-function model for turbulent particle-laden channel flow in the dense regime

Physics of Fluids, Volume 33, Issue 5, May 2021.

Modeling particle-laden turbulent flows at high volume fractions requires accounting for the coupling between phases. The latter is often a sensitive point, and proper closure of the exchange and production terms due to the presence of particles is not straightforward. In the present work, a Lagrangian probability-density-function model developed for homogeneous cluster-induced turbulence is extended to a channel flow. The key features are consistent two-way coupling and the decomposition of the particle velocity into spatially correlated and uncorrelated components, which is crucial for dense flows and which allows dealing with collisions from a statistical point of view. A numerical scheme for the coupled solution of the stochastic differential equations for the particles and a Reynolds-stress model for the fluid is developed. Tests with tracer particles without two-way coupling are done to assess the validity and the consistency of the numerical scheme. Finally, two sets of numerical simulations with particles with different diameters in a turbulent channel flow at a shear Reynolds of [math] are reported. The effect of two-way coupling by varying the mass loading of the dispersed phase in the mass-loading range [math] 0–2 is analyzed, and the results are compared to previous Eulerian–Lagrangian and Eulerian–Eulerian direct-numerical simulation (DNS) studies. Mean velocities and turbulent kinetic energy show good agreement with DNS, especially regarding the trend with respect to mass loading. Consistent with prior work, increased mass loading causes a drastic reduction of turbulent kinetic energy in the range [math] 0–2.

Modeling particle-laden turbulent flows at high volume fractions requires accounting for the coupling between phases. The latter is often a sensitive point, and proper closure of the exchange and production terms due to the presence of particles is not straightforward. In the present work, a Lagrangian probability-density-function model developed for homogeneous cluster-induced turbulence is extended to a channel flow. The key features are consistent two-way coupling and the decomposition of the particle velocity into spatially correlated and uncorrelated components, which is crucial for dense flows and which allows dealing with collisions from a statistical point of view. A numerical scheme for the coupled solution of the stochastic differential equations for the particles and a Reynolds-stress model for the fluid is developed. Tests with tracer particles without two-way coupling are done to assess the validity and the consistency of the numerical scheme. Finally, two sets of numerical simulations with particles with different diameters in a turbulent channel flow at a shear Reynolds of [math] are reported. The effect of two-way coupling by varying the mass loading of the dispersed phase in the mass-loading range [math] 0–2 is analyzed, and the results are compared to previous Eulerian–Lagrangian and Eulerian–Eulerian direct-numerical simulation (DNS) studies. Mean velocities and turbulent kinetic energy show good agreement with DNS, especially regarding the trend with respect to mass loading. Consistent with prior work, increased mass loading causes a drastic reduction of turbulent kinetic energy in the range [math] 0–2.

Categories: Latest papers in fluid mechanics

### Numerical investigation of conjugate heat transfer in a microchannel with a hydrophobic surface utilizing nanofluids under a magnetic field

Physics of Fluids, Volume 33, Issue 5, May 2021.

Conjugate heat transfer in a microchannel with a slip boundary condition imposed on the channel's walls by a uniform magnetic field is studied. The working fluid consists of a Water/Ag mixture nanofluid. A preconditioned lattice Boltzmann method (LBM), formed by combining the incompressible LBM with the regular LBM, is applied to the velocity field and temperature field, respectively. The microchannel's upper wall is thermally isolated when a constant heat flux is imposed on the basin of the microchannel. The simulations are carried out under a variety of different conditions, e.g., various Reynold numbers, Re = 50 and 150, nanoparticle concentrations (φ = 0, 3%), slip coefficients (0 ≤ B ≤ 0.03), and Hartmann numbers (0 ≤ Ha ≤ 30). Surface hydrophobicity results in a reduction of surface friction of up to 46% at B = 0.03 and Ha = 30. The surface friction reductions at Ha = 0, 10, and 20 are 15%, 27%, and 38%, respectively. These results indicate that as the surface slip increases, the drag resisting the fluid dynamics decreases. Moreover, adding the nanoparticles to the base flow can improve the heat transfer by 50%. Besides, using the magnetic field increase the shear stress and, consequently, the drag force dramatically (up 340%). On the other hand, the magnetic field enhances the heat transfer by improving the fluid velocity near the wall, while its effect on the Nu number improvement is not more than 20%. As a result, the magnetic power should be controlled to achieve the best heat transfer performance with the lowest pumping energy consumption.

Conjugate heat transfer in a microchannel with a slip boundary condition imposed on the channel's walls by a uniform magnetic field is studied. The working fluid consists of a Water/Ag mixture nanofluid. A preconditioned lattice Boltzmann method (LBM), formed by combining the incompressible LBM with the regular LBM, is applied to the velocity field and temperature field, respectively. The microchannel's upper wall is thermally isolated when a constant heat flux is imposed on the basin of the microchannel. The simulations are carried out under a variety of different conditions, e.g., various Reynold numbers, Re = 50 and 150, nanoparticle concentrations (φ = 0, 3%), slip coefficients (0 ≤ B ≤ 0.03), and Hartmann numbers (0 ≤ Ha ≤ 30). Surface hydrophobicity results in a reduction of surface friction of up to 46% at B = 0.03 and Ha = 30. The surface friction reductions at Ha = 0, 10, and 20 are 15%, 27%, and 38%, respectively. These results indicate that as the surface slip increases, the drag resisting the fluid dynamics decreases. Moreover, adding the nanoparticles to the base flow can improve the heat transfer by 50%. Besides, using the magnetic field increase the shear stress and, consequently, the drag force dramatically (up 340%). On the other hand, the magnetic field enhances the heat transfer by improving the fluid velocity near the wall, while its effect on the Nu number improvement is not more than 20%. As a result, the magnetic power should be controlled to achieve the best heat transfer performance with the lowest pumping energy consumption.

Categories: Latest papers in fluid mechanics

### Atomistic-scale investigations of hyperthermal oxygen–graphene interactions via reactive molecular dynamics simulation: The gas effect

Physics of Fluids, Volume 33, Issue 5, May 2021.

Hyperthermal atomic oxygen (AO) bombardment to thermal protection system surface has been identified to impact the aerodynamic heating significantly, due to complex chemical reactions at the gas–solid interface, e.g., surface catalysis recombination, oxidation, and ablation. Previous investigations have focused on the surface effects of the AO collision process, while the influence of impacting gas characteristics remains unclear under various non-equilibrium aerodynamic conditions. This work conducts a reactive molecular dynamics (RMD) study of AO collisions over graphene surface, by considering the incoming gas at different translational energies (0.1 ≤ Ek ≤ 10 eV), incident angles (θ = 15°, 30°, 45°, 60°, 75°, and 90°), and O/O2 ratios (χO2 = 0.00, 0.25, 0.50, 0.75, and 1.00). The RMD results indicate that for AO normal incidence, the predominant reactive products of O2, CO, and CO2 molecules are produced due to the synergistic catalytic recombination and surface ablation reaction effects. A maximum recombination performance is identified around 5-eV AO incidence. For off-normal AO incidence, the recombination coefficient increases with the increase in incidence angle from 15° to 60° due to the larger perpendicular components of translational energy and then decreases smoothly. With the increase in O2 mole fraction, the surface reflection probabilities increase, which result in the decrease in both catalytic recombination and ablation activities. Via revealing the atomistic-scale mechanism of gas effects on the surface under hypersonic non-equilibrium conditions, this work sheds light for the future design and optimization of thermal protection materials.

Hyperthermal atomic oxygen (AO) bombardment to thermal protection system surface has been identified to impact the aerodynamic heating significantly, due to complex chemical reactions at the gas–solid interface, e.g., surface catalysis recombination, oxidation, and ablation. Previous investigations have focused on the surface effects of the AO collision process, while the influence of impacting gas characteristics remains unclear under various non-equilibrium aerodynamic conditions. This work conducts a reactive molecular dynamics (RMD) study of AO collisions over graphene surface, by considering the incoming gas at different translational energies (0.1 ≤ Ek ≤ 10 eV), incident angles (θ = 15°, 30°, 45°, 60°, 75°, and 90°), and O/O2 ratios (χO2 = 0.00, 0.25, 0.50, 0.75, and 1.00). The RMD results indicate that for AO normal incidence, the predominant reactive products of O2, CO, and CO2 molecules are produced due to the synergistic catalytic recombination and surface ablation reaction effects. A maximum recombination performance is identified around 5-eV AO incidence. For off-normal AO incidence, the recombination coefficient increases with the increase in incidence angle from 15° to 60° due to the larger perpendicular components of translational energy and then decreases smoothly. With the increase in O2 mole fraction, the surface reflection probabilities increase, which result in the decrease in both catalytic recombination and ablation activities. Via revealing the atomistic-scale mechanism of gas effects on the surface under hypersonic non-equilibrium conditions, this work sheds light for the future design and optimization of thermal protection materials.

Categories: Latest papers in fluid mechanics

### Quasi-stationarity of scalar turbulent mixing statistics in a non-symmetric case

Physics of Fluids, Volume 33, Issue 5, May 2021.

The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.

The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.

Categories: Latest papers in fluid mechanics

### Modeling of dusty gas flows due to plume impingement on a lunar surface

Physics of Fluids, Volume 33, Issue 5, May 2021.

A novel, two-way coupled, dusty-gas flow model has been developed in the direct simulation Monte Carlo (DSMC) framework and employed for the dust-dispersion study on lunar surface. In this model, the gas–gas collisions are modeled probabilistically, whereas, grain–grain interactions are computed deterministically. Most importantly, the gas–grain interactions are modeled in a two-way coupled manner through the consideration of momentum and energy exchange between the two phases. The proposed model is validated against the two-phase theoretical relations for a zero-dimensional simulation. The computational model is used to study the dust dispersion problem due to plume impingement on lunar surface. The influence of particle diameter and hovering altitudes on gas and grain phases, and dust transportation are analyzed in the modified DSMC framework. Furthermore, the sensitivity of the two-way coupled gas–grain interaction model is discussed in relation to the one-way coupled model.

A novel, two-way coupled, dusty-gas flow model has been developed in the direct simulation Monte Carlo (DSMC) framework and employed for the dust-dispersion study on lunar surface. In this model, the gas–gas collisions are modeled probabilistically, whereas, grain–grain interactions are computed deterministically. Most importantly, the gas–grain interactions are modeled in a two-way coupled manner through the consideration of momentum and energy exchange between the two phases. The proposed model is validated against the two-phase theoretical relations for a zero-dimensional simulation. The computational model is used to study the dust dispersion problem due to plume impingement on lunar surface. The influence of particle diameter and hovering altitudes on gas and grain phases, and dust transportation are analyzed in the modified DSMC framework. Furthermore, the sensitivity of the two-way coupled gas–grain interaction model is discussed in relation to the one-way coupled model.

Categories: Latest papers in fluid mechanics

### Robust and unstable axisymmetric vortices, including neutral vortices, of a new two-dimensional vortex family

Physics of Fluids, Volume 33, Issue 5, May 2021.

Solutions of robust axisymmetric neutral vortices, that is, vortices with zero amount of vorticity, in two-dimensional (2D) Euler flows with distributed vorticity are obtained. These solutions are particular linear combinations of vorticity layer-modes, which are defined as truncated, shifted, and conveniently normalized Bessel functions of order-0, each one occupying a circular layer defined by a zero of the Bessel function of order-1. It is found that some linear combinations of these modes have a vanishing net amount of vorticity and remain axysimmetrically robust to small amplitude vorticity perturbations. These neutral vortices are quiescent and remain steady in the presence of similar vortices. Other linear combinations of these vorticity layer-modes give rise to unstable neutral vortices that develop into neutral tripoles, pentapoles, etc. It is found numerically that the robustness of these neutral vortices is related to the spiralization and axisymmetrization of the initially growing vorticity disturbances as are advected by a convex azimuthal velocity distribution beyond its first inflection point. In particular, it is found that two co-rotating neutral tripoles attract due to the phase synchronization of their respective octupolar potential flow but repel when touched due to vorticity exchange. This interaction mechanism makes possible equilibrium states for sets of a large number of neutral tripoles. Other linear combinations of these vorticity layer-modes give rise to non-neutral shielded vortices which interact and may form coherent vortex structures as pairs of co-rotating shielded vortices sharing their outermost vorticity layer or counter-rotating shielded vortices translating with uniform speed as vortex dipoles.

Solutions of robust axisymmetric neutral vortices, that is, vortices with zero amount of vorticity, in two-dimensional (2D) Euler flows with distributed vorticity are obtained. These solutions are particular linear combinations of vorticity layer-modes, which are defined as truncated, shifted, and conveniently normalized Bessel functions of order-0, each one occupying a circular layer defined by a zero of the Bessel function of order-1. It is found that some linear combinations of these modes have a vanishing net amount of vorticity and remain axysimmetrically robust to small amplitude vorticity perturbations. These neutral vortices are quiescent and remain steady in the presence of similar vortices. Other linear combinations of these vorticity layer-modes give rise to unstable neutral vortices that develop into neutral tripoles, pentapoles, etc. It is found numerically that the robustness of these neutral vortices is related to the spiralization and axisymmetrization of the initially growing vorticity disturbances as are advected by a convex azimuthal velocity distribution beyond its first inflection point. In particular, it is found that two co-rotating neutral tripoles attract due to the phase synchronization of their respective octupolar potential flow but repel when touched due to vorticity exchange. This interaction mechanism makes possible equilibrium states for sets of a large number of neutral tripoles. Other linear combinations of these vorticity layer-modes give rise to non-neutral shielded vortices which interact and may form coherent vortex structures as pairs of co-rotating shielded vortices sharing their outermost vorticity layer or counter-rotating shielded vortices translating with uniform speed as vortex dipoles.

Categories: Latest papers in fluid mechanics

### A prediction model for vertical turbulence momentum flux above infinite wind farms

Physics of Fluids, Volume 33, Issue 5, May 2021.

Large wind farms can significantly change the vertical layered structures and some of the statistical characteristics of the atmospheric boundary layer (ABL). The vertical turbulence momentum flux (VTMF) above a wind farm, which quantifies the vertical transport of the ABL, is important to meteorological simulation and power absorption of the wind farm. However, we still lack a fast prediction model for the VTMF. To this end, a suite of large-eddy simulations (LESs) is performed for infinite wind farms with various turbine positionings. We show that, in the outer layer above a wind farm, the VTMF normalized by the wind farm's equivalent frictional velocity exhibits a linear relationship with height, which agrees well with the linear law for the canonical rough wall. In contrast, in both the wake layer and the inner layer, the VTMF is significantly dependent on the turbine positionings. Consequently, a prediction model for the VTMF in the outer layer of the ABL is proposed only using the mean velocity in the inner layer of the ABL (below the wind rotors). The kinetic energy transport downward to wind farms is also calculated using the proposed model.

Large wind farms can significantly change the vertical layered structures and some of the statistical characteristics of the atmospheric boundary layer (ABL). The vertical turbulence momentum flux (VTMF) above a wind farm, which quantifies the vertical transport of the ABL, is important to meteorological simulation and power absorption of the wind farm. However, we still lack a fast prediction model for the VTMF. To this end, a suite of large-eddy simulations (LESs) is performed for infinite wind farms with various turbine positionings. We show that, in the outer layer above a wind farm, the VTMF normalized by the wind farm's equivalent frictional velocity exhibits a linear relationship with height, which agrees well with the linear law for the canonical rough wall. In contrast, in both the wake layer and the inner layer, the VTMF is significantly dependent on the turbine positionings. Consequently, a prediction model for the VTMF in the outer layer of the ABL is proposed only using the mean velocity in the inner layer of the ABL (below the wind rotors). The kinetic energy transport downward to wind farms is also calculated using the proposed model.

Categories: Latest papers in fluid mechanics

### Manipulation of toroidal-spiral particles internal structure by fluid flow

Physics of Fluids, Volume 33, Issue 5, May 2021.

We report on the precise manipulation of the fine structures of toroidal-spiral particles (TSPs) generated by a self-assembly process of droplet sedimentation at low Reynolds numbers in a miscible bulk solution followed by solidification. The biocompatible polymeric TSP can serve as a device for drug delivery and in vivo therapeutic cell expansion, activation, and delivery, for which highly tunable and reproducible structures are essential to design dosages and release kinetics. TSP formation can be divided into two stages: initial infusion of the drop vs its subsequent sedimentation, deformation, and entrainment of the surrounding bulk solution. The infusion rate affects the drop shape and tail length. These two features represent crucial initial conditions for subsequent shape evolution, which determines the overall morphology of the TSP and fine structure of the internal channel. Our computer simulations of drop dynamics add a new capability to the swarm-of-Stokeslets technique: unequal viscosities of the drop and bulk phases (i.e., non-unit viscosity ratio). During sedimentation, the density difference between the droplet and the bulk solution played a more pronounced role than the viscosity ratio, which was revealed both by experimental observations and numerical simulations. Understanding the fundamental hydrodynamics and developing a flow map will ultimately aid in the design of TSPs with tunable empty channels toward drug delivery and cell encapsulation.

We report on the precise manipulation of the fine structures of toroidal-spiral particles (TSPs) generated by a self-assembly process of droplet sedimentation at low Reynolds numbers in a miscible bulk solution followed by solidification. The biocompatible polymeric TSP can serve as a device for drug delivery and in vivo therapeutic cell expansion, activation, and delivery, for which highly tunable and reproducible structures are essential to design dosages and release kinetics. TSP formation can be divided into two stages: initial infusion of the drop vs its subsequent sedimentation, deformation, and entrainment of the surrounding bulk solution. The infusion rate affects the drop shape and tail length. These two features represent crucial initial conditions for subsequent shape evolution, which determines the overall morphology of the TSP and fine structure of the internal channel. Our computer simulations of drop dynamics add a new capability to the swarm-of-Stokeslets technique: unequal viscosities of the drop and bulk phases (i.e., non-unit viscosity ratio). During sedimentation, the density difference between the droplet and the bulk solution played a more pronounced role than the viscosity ratio, which was revealed both by experimental observations and numerical simulations. Understanding the fundamental hydrodynamics and developing a flow map will ultimately aid in the design of TSPs with tunable empty channels toward drug delivery and cell encapsulation.

Categories: Latest papers in fluid mechanics

### Boltzmann's colloidal transport in porous media with velocity-dependent capture probability

Physics of Fluids, Volume 33, Issue 5, May 2021.

Mathematical modeling of suspension-colloidal-nano transport in porous media at different scales has long been a fascinating topic of fluid mechanics. In this study, we discuss the multi-pore scale, where Boltzmann's approach of distributed velocities is valid, and average (homogenize) the micro-scale equation up to the core scale. The focus is on the filtration function (particle capture probability per unity trajectory length) that highly depends on the carrier fluid velocity. We develop a modified form of the Boltzmann equation for micro-scale particle capture and diffusion. An equivalent sink term is introduced into the kinetic equation instead of non-zero initial data, resulting in the solution of an operator equation in the Fourier space and an exact homogenization. The upper scale transport equation is obtained in closed form. The upscaled model contains the dimensionless delay number and large-scale dispersion and filtration coefficients. The explicit formulas for the large-scale model coefficients are derived in terms of the micro-scale parameters for any arbitrary velocity-dependent filtration function. We focus on three micro-scale models for the velocity-dependent particle capture rate corresponding to various retention mechanisms, i.e., straining, attachment, and inertial capture. The explicit formulas for large-scale transport coefficients reveal their typical dependencies of velocity and the micro-scale parameters. Treatment of several laboratory tests reveals close match with the modeling-based predictions.

Mathematical modeling of suspension-colloidal-nano transport in porous media at different scales has long been a fascinating topic of fluid mechanics. In this study, we discuss the multi-pore scale, where Boltzmann's approach of distributed velocities is valid, and average (homogenize) the micro-scale equation up to the core scale. The focus is on the filtration function (particle capture probability per unity trajectory length) that highly depends on the carrier fluid velocity. We develop a modified form of the Boltzmann equation for micro-scale particle capture and diffusion. An equivalent sink term is introduced into the kinetic equation instead of non-zero initial data, resulting in the solution of an operator equation in the Fourier space and an exact homogenization. The upper scale transport equation is obtained in closed form. The upscaled model contains the dimensionless delay number and large-scale dispersion and filtration coefficients. The explicit formulas for the large-scale model coefficients are derived in terms of the micro-scale parameters for any arbitrary velocity-dependent filtration function. We focus on three micro-scale models for the velocity-dependent particle capture rate corresponding to various retention mechanisms, i.e., straining, attachment, and inertial capture. The explicit formulas for large-scale transport coefficients reveal their typical dependencies of velocity and the micro-scale parameters. Treatment of several laboratory tests reveals close match with the modeling-based predictions.

Categories: Latest papers in fluid mechanics

### Growth of barchan dunes of bidispersed granular mixtures

Physics of Fluids, Volume 33, Issue 5, May 2021.

Barchans are dunes of crescentic shape found on Earth, Mars, and other celestial bodies, growing usually on polydisperse granular beds. In this Letter, we investigate experimentally the growth of subaqueous barchans consisting of bidisperse grains. We found that the grain distribution within the dune changes with the employed pair, and that a transient stripe appears on the dune surface. We propose that observed patterns result from the competition between fluid entrainment and easiness of rolling for each grain type, and that grains segregate with a diffusion-like mechanism. Our results provide new insights into barchan structures found in other environments.

Barchans are dunes of crescentic shape found on Earth, Mars, and other celestial bodies, growing usually on polydisperse granular beds. In this Letter, we investigate experimentally the growth of subaqueous barchans consisting of bidisperse grains. We found that the grain distribution within the dune changes with the employed pair, and that a transient stripe appears on the dune surface. We propose that observed patterns result from the competition between fluid entrainment and easiness of rolling for each grain type, and that grains segregate with a diffusion-like mechanism. Our results provide new insights into barchan structures found in other environments.

Categories: Latest papers in fluid mechanics

### An efficient deep learning framework to reconstruct the flow field sequences of the supersonic cascade channel

Physics of Fluids, Volume 33, Issue 5, May 2021.

Accurate and comprehensive flow field reconstruction is essential for promptly monitoring the flow state of the supersonic cascade. This paper proposes a novel data-driven method for reconstructing the slices of the two-dimensional (2D) pressure field in three-dimensional (3D) flow of the supersonic cascade by using deep neural networks. Considering the complicated spatial effects of 2D pressure field slices, the architecture embeds the convolution into the long short-term memory (LSTM) network to realize the purpose of using the upstream pressure to reconstruct downstream pressure. Numerical simulations of the supersonic cascade under different back pressures are performed to establish the database capturing the complex relationship between the upstream and downstream flow. The pressure of different upstream slices can be used as a spatial-dependent sequence as the input of the model to reconstruct the pressure of different downstream slices. A deep neural network including special convolutional LSTM layers and convolutional layers is designed. The trained model is then tested under different back pressures. The reconstruction results are in good agreement with the computational fluid dynamics, especially for the identification of shock wave position changes and the recognition of complex curved shock waves in 3D flow with high accuracy. Moreover, analyzing the frequency distribution of reconstructed pressure at different positions can clearly distinguish the flow separated zone, which will further improve the accuracy of the state monitoring. Specifically, it is of great significance for identifying the stall of the flow field promptly.

Accurate and comprehensive flow field reconstruction is essential for promptly monitoring the flow state of the supersonic cascade. This paper proposes a novel data-driven method for reconstructing the slices of the two-dimensional (2D) pressure field in three-dimensional (3D) flow of the supersonic cascade by using deep neural networks. Considering the complicated spatial effects of 2D pressure field slices, the architecture embeds the convolution into the long short-term memory (LSTM) network to realize the purpose of using the upstream pressure to reconstruct downstream pressure. Numerical simulations of the supersonic cascade under different back pressures are performed to establish the database capturing the complex relationship between the upstream and downstream flow. The pressure of different upstream slices can be used as a spatial-dependent sequence as the input of the model to reconstruct the pressure of different downstream slices. A deep neural network including special convolutional LSTM layers and convolutional layers is designed. The trained model is then tested under different back pressures. The reconstruction results are in good agreement with the computational fluid dynamics, especially for the identification of shock wave position changes and the recognition of complex curved shock waves in 3D flow with high accuracy. Moreover, analyzing the frequency distribution of reconstructed pressure at different positions can clearly distinguish the flow separated zone, which will further improve the accuracy of the state monitoring. Specifically, it is of great significance for identifying the stall of the flow field promptly.

Categories: Latest papers in fluid mechanics

### Statistical properties of a model of a turbulent patch arising from a breaking internal wave

Physics of Fluids, Volume 33, Issue 5, May 2021.

The turbulent patch arising from internal gravity wave breaking is investigated with direct numerical simulation of a stably stratified flow over a two-dimensional hill. The turbulent patch is distinguished from the non-turbulent wave region with potential vorticity. The turbulent patch is highly intermittent, and its location fluctuates with space and time. The buoyancy Reynolds number slowly decays with time in the turbulent patch and the mixing efficiency stays around 0.2. The turbulent patch is separated from the non-turbulent wave region by a turbulent/non-turbulent interfacial (TNTI) layer, whose thickness is about five times the Kolmogorov scale. The kinetic energy dissipation rate also sharply decreases from the turbulent to the wave region while the potential energy dissipation rate has a large peak within the TNTI layer. Both shear and stable stratification are strong in the upper area of the turbulent patch. On the other hand, the lower area has a small mean density gradient, i.e., weak stratification, which is related to the strong intermittency of the turbulent patch in the lower area. Furthermore, weak stratification in the lower area results in a low gradient Richardson number, which is below the critical value for the shear instability, and the roller vortex appears. The outer edge of the turbulent patch aligns with the perimeter of the roller vortex, and the vortex affects the spatial distribution of the turbulent patch.

The turbulent patch arising from internal gravity wave breaking is investigated with direct numerical simulation of a stably stratified flow over a two-dimensional hill. The turbulent patch is distinguished from the non-turbulent wave region with potential vorticity. The turbulent patch is highly intermittent, and its location fluctuates with space and time. The buoyancy Reynolds number slowly decays with time in the turbulent patch and the mixing efficiency stays around 0.2. The turbulent patch is separated from the non-turbulent wave region by a turbulent/non-turbulent interfacial (TNTI) layer, whose thickness is about five times the Kolmogorov scale. The kinetic energy dissipation rate also sharply decreases from the turbulent to the wave region while the potential energy dissipation rate has a large peak within the TNTI layer. Both shear and stable stratification are strong in the upper area of the turbulent patch. On the other hand, the lower area has a small mean density gradient, i.e., weak stratification, which is related to the strong intermittency of the turbulent patch in the lower area. Furthermore, weak stratification in the lower area results in a low gradient Richardson number, which is below the critical value for the shear instability, and the roller vortex appears. The outer edge of the turbulent patch aligns with the perimeter of the roller vortex, and the vortex affects the spatial distribution of the turbulent patch.

Categories: Latest papers in fluid mechanics

### Mass-balance and locality versus accuracy with the new boundary and interface-conjugate approaches in advection-diffusion lattice Boltzmann method

Physics of Fluids, Volume 33, Issue 5, May 2021.

We introduce two new approaches, called A-LSOB and N-MR, for boundary and interface-conjugate conditions on flat or curved surface shapes in the advection-diffusion lattice Boltzmann method (LBM). The Local Second-Order, single-node A-LSOB enhances the existing Dirichlet and Neumann normal boundary treatments with respect to locality, accuracy, and Péclet parametrization. The normal-multi-reflection (N-MR) improves the directional flux schemes via a local release of their nonphysical tangential constraints. The A-LSOB and N-MR restore all first- and second-order derivatives from the nodal non-equilibrium solution, and they are conditioned to be exact on a piece-wise parabolic profile in a uniform arbitrary-oriented tangential velocity field. Additionally, the most compact and accurate single-node parabolic schemes for diffusion and flow in grid-inclined pipes are introduced. In simulations, the global mass-conservation solvability condition of the steady-state, two-relaxation-time (S-TRT) formulation is adjusted with either (i) a uniform mass-source or (ii) a corrective surface-flux. We conclude that (i) the surface-flux counterbalance is more accurate than the bulk one, (ii) the A-LSOB Dirichlet schemes are more accurate than the directional ones in the high Péclet regime, (iii) the directional Neumann advective-diffusive flux scheme shows the best conservation properties and then the best performance both in the tangential no-slip and interface-perpendicular flow, and (iv) the directional non-equilibrium diffusive flux extrapolation is the least conserving and accurate. The error Péclet dependency, Neumann invariance over an additive constant, and truncation isotropy guide this analysis. Our methodology extends from the d2q9 isotropic S-TRT to 3D anisotropic matrix collisions, Robin boundary condition, and the transient LBM.

We introduce two new approaches, called A-LSOB and N-MR, for boundary and interface-conjugate conditions on flat or curved surface shapes in the advection-diffusion lattice Boltzmann method (LBM). The Local Second-Order, single-node A-LSOB enhances the existing Dirichlet and Neumann normal boundary treatments with respect to locality, accuracy, and Péclet parametrization. The normal-multi-reflection (N-MR) improves the directional flux schemes via a local release of their nonphysical tangential constraints. The A-LSOB and N-MR restore all first- and second-order derivatives from the nodal non-equilibrium solution, and they are conditioned to be exact on a piece-wise parabolic profile in a uniform arbitrary-oriented tangential velocity field. Additionally, the most compact and accurate single-node parabolic schemes for diffusion and flow in grid-inclined pipes are introduced. In simulations, the global mass-conservation solvability condition of the steady-state, two-relaxation-time (S-TRT) formulation is adjusted with either (i) a uniform mass-source or (ii) a corrective surface-flux. We conclude that (i) the surface-flux counterbalance is more accurate than the bulk one, (ii) the A-LSOB Dirichlet schemes are more accurate than the directional ones in the high Péclet regime, (iii) the directional Neumann advective-diffusive flux scheme shows the best conservation properties and then the best performance both in the tangential no-slip and interface-perpendicular flow, and (iv) the directional non-equilibrium diffusive flux extrapolation is the least conserving and accurate. The error Péclet dependency, Neumann invariance over an additive constant, and truncation isotropy guide this analysis. Our methodology extends from the d2q9 isotropic S-TRT to 3D anisotropic matrix collisions, Robin boundary condition, and the transient LBM.

Categories: Latest papers in fluid mechanics