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Li, Xiang-Yu, Axel Brandenburg, Gunilla Svensson, Nils E. L. Haugen, Bernhard Mehlig, and Igor Rogachevskii. "Condensational and Collisional Growth of Cloud Droplets in a Turbulent Environment." Journal of the Atmospheric Sciences 77, no. 1 (December 26, 2019): 337–53. http://dx.doi.org/10.1175/jas-d-19-0107.1.
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Abstract We investigate the effect of turbulence on the combined condensational and collisional growth of cloud droplets by means of high-resolution direct numerical simulations of turbulence and a superparticle approximation for droplet dynamics and collisions. The droplets are subject to turbulence as well as gravity, and their collision and coalescence efficiencies are taken to be unity. We solve the thermodynamic equations governing temperature, water vapor mixing ratio, and the resulting supersaturation fields together with the Navier–Stokes equation. We find that the droplet size distribution broadens with increasing Reynolds number and/or mean energy dissipation rate. Turbulence affects the condensational growth directly through supersaturation fluctuations, and it influences collisional growth indirectly through condensation. Our simulations show for the first time that, in the absence of the mean updraft cooling, supersaturation-fluctuation-induced broadening of droplet size distributions enhances the collisional growth. This is contrary to classical (nonturbulent) condensational growth, which leads to a growing mean droplet size, but a narrower droplet size distribution. Our findings, instead, show that condensational growth facilitates collisional growth by broadening the size distribution in the tails at an early stage of rain formation. With increasing Reynolds numbers, evaporation becomes stronger. This counteracts the broadening effect due to condensation at late stages of rain formation. Our conclusions are consistent with results of laboratory experiments and field observations, and show that supersaturation fluctuations are important for precipitation.
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BARANGER, C. "MODELLING OF OSCILLATIONS, BREAKUP AND COLLISIONS FOR DROPLETS: THE ESTABLISHMENT OF KERNELS FOR THE T.A.B. MODEL." Mathematical Models and Methods in Applied Sciences 14, no. 05 (May 2004): 775–94. http://dx.doi.org/10.1142/s0218202504003441.
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In this work, we consider a spray consisting of droplets surrounded by a gas. The droplets are described by a kinetic equation and the gas verifies an equation of fluid dynamics such as Navier–Stokes. We write down the kernels corresponding to complex phenomena such as oscillations, breakup and collisions/coalescences. We use for that the T.A.B. model of oscillations introduced in particular in the KIVA code of combustion of Los Alamos, and the collision model introduced by Villedieu. We briefly explain the numerical method for solving such equations, and present results.
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INOUE, O., Y. HATTORI, and T. SASAKI. "Sound generation by coaxial collision of two vortex rings." Journal of Fluid Mechanics 424 (November 16, 2000): 327–65. http://dx.doi.org/10.1017/s0022112000002123.
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Sound pressure fields generated by coaxial collisions of two vortex rings with equal/unequal strengths are simulated numerically. The axisymmetric, unsteady, compressible Navier–Stokes equations are solved by a finite difference method, not only for a near field but also for a far field. The sixth-order-accurate compact Padé scheme is used for spatial derivatives, together with the fourth-order-accurate Runge–Kutta scheme for time integration. The results show that the generation of sound is closely related to the change of direction of the vortex ring motion induced by the mutual interaction of the two vortex rings. For the case of equal strength (head-on collision), the change of direction is associated with stretching of the vortex rings. Generated sound waves consist of compression parts and rarefaction parts, and have a quadrupolar nature. For the case of unequal strengths, the two vortex rings pass through each other; the weaker vortex ring moves outside the stronger vortex ring which shows a loop motion. The number of generated waves depends on the relative strength of the two vortex rings. The sound pressure includes dipolar and octupolar components, in addition to monopolar and quadrupolar components which are observed for the case of a head-on collision.
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Lohrasbi, Alireza, and Moharram D. Pirooz. "Navier Stokes model of solitary wave collision." Chaos, Solitons & Fractals 68 (November 2014): 139–50. http://dx.doi.org/10.1016/j.chaos.2014.08.003.
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Almady, Wasif. "Analytical Solution for Boltzmann Collision Operator for the1-D Diffusion equation." International Journal for Research in Applied Science and Engineering Technology 9, no. 9 (September 30, 2021): 1514–17. http://dx.doi.org/10.22214/ijraset.2021.38189.
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Abstract: In this paper, we have presented the analytical solution of the collision operator for the Boltzmann equation of onedimensional diffusion equation using the analytical solution of the one-dimensional Navier Stoke diffusion equation. Keywords: Boltzmann equation; analytical collision operator; one-dimensional diffusion equation.
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Naso, Aurore, Jennifer Jucha, Emmanuel Lévêque, and Alain Pumir. "Collision rate of ice crystals with water droplets in turbulent flows." Journal of Fluid Mechanics 845 (April 27, 2018): 615–41. http://dx.doi.org/10.1017/jfm.2018.238.
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Riming, the process whereby ice crystals get coated by impacting supercooled liquid droplets, is one of the dominant processes leading to precipitation in mixed-phase clouds. How a settling crystal collides with very small water droplets has been mostly studied in laminar conditions. The present numerical study aims at providing further insight on how turbulent flow motion affects the riming of ice crystals. We model the crystals as narrow oblate ellipsoids, smaller than the Kolmogorov elementary scale. By neglecting the effect of fluid inertia on the motion of the crystals and droplets, and using direct numerical simulations of the Navier–Stokes equations in a moderately turbulent regime, over a range of kinetic energy dissipation $1~\text{cm}^{2}~\text{s}^{-3}\lesssim \unicode[STIX]{x1D700}\lesssim 256~\text{cm}^{2}~\text{s}^{-3}$, we determine the collision rate between disk-shaped ice crystals and very small liquid water droplets. Whereas differential settling plays the dominant role in determining the collision rate at small turbulence intensity, the role of turbulence becomes more important at the large values of $\unicode[STIX]{x1D700}$ simulated, an effect that can be partly attributed to the increased role of inertia. We always find that collisions occur with a large probability on the rim of the ellipsoids, a phenomenon that can be explained to a large extent by kinematic considerations. The difference in the settling velocity of crystals and droplets induces a strong asymmetry in the probability of collision between the faces of the ellipsoids. Our results shed light on the physical mechanisms involved in the riming of ice crystals in clouds.
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Lin, S. C., T. C. Kuo, and C. C. Chieng. "Particle Trajectories Around a Flying Slider." Journal of Tribology 120, no. 1 (January 1, 1998): 69–74. http://dx.doi.org/10.1115/1.2834192.
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The Eulerian-Lagrangian approach is employed to simulate droplet trajectories due to the large-velocity gradient between two solid surfaces: a stationery block (slider) and a rotating plane (disk). Sudden expansion after the extremely small spacing will trap the particles in the open spaces. The fluid phase flowfield is obtained by solving Navier-Stokes equations with slip boundary correction in the Eulerian approach, and the droplet trajectories are calculated by integrating equations of motion with slip correction in the Lagrangian approach. Because of the extremely small spacing and the droplet size, Brownian motion effectively increases the probability of slider-head collisions, especially for extremely small particles. This study demonstrates that the effect due to particle size is the dominant factor in determining the probability of particle-slider collision, especially for particle sizes comparable with the air mean free path and the flowfield immediately adjacent to the solid surfaces. The results also show that lowering the flying height of the slider and increasing the disk velocity attracts the particles toward the gap between the disk and the slider.
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XU, KUN, and ZHAOLI GUO. "GENERALIZED GAS DYNAMIC EQUATIONS WITH MULTIPLE TRANSLATIONAL TEMPERATURES." Modern Physics Letters B 23, no. 03 (January 30, 2009): 237–40. http://dx.doi.org/10.1142/s0217984909018096.
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Based on a multiple stage BGK-type collision model and the Chapman–Enskog expansion, the corresponding macroscopic gas dynamics equations in three-dimensional space will be derived. The new gas dynamic equations have the same structure as the Navier–Stokes equations, but the stress strain relationship in the Navier–Stokes equations is replaced by an algebraic equation with temperature differences. In the continuum flow regime, the new gas dynamic equations automatically recover the standard Navier–Stokes equations. The current gas dynamic equations are natural extension of the Navier–Stokes equations to the near continuum flow regime and can be used for near continuum flow study.
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Mayhew, Kent W. "Illusions of Elastic Collisions in the Sciences:." European Journal of Engineering Research and Science 5, no. 1 (January 23, 2020): 87–90. http://dx.doi.org/10.24018/ejers.2020.5.1.1693.
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Employing elastic collisions rather than the reality of inelastic collisions simplifies much of the theoretical sciences. The consequences of such simplification is completely ignored/unrealized by the majority, hence must be addressed. At the crux of the problem is arguably the illusion of elastic collisions in kinetic theory, but this extends to other realms of physics including statistical theory, Lagrangian mechanics and the Navier-Stokes equations.
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Mayhew, Kent W. "Illusions of Elastic Collisions in the Sciences:." European Journal of Engineering and Technology Research 5, no. 1 (January 23, 2020): 87–90. http://dx.doi.org/10.24018/ejeng.2020.5.1.1693.
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Employing elastic collisions rather than the reality of inelastic collisions simplifies much of the theoretical sciences. The consequences of such simplification is completely ignored/unrealized by the majority, hence must be addressed. At the crux of the problem is arguably the illusion of elastic collisions in kinetic theory, but this extends to other realms of physics including statistical theory, Lagrangian mechanics and the Navier-Stokes equations.
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Gonzalez-Ondina, Jose M., Luigi Fraccarollo, and Philip L. F. Liu. "Two-level, two-phase model for intense, turbulent sediment transport." Journal of Fluid Mechanics 839 (January 26, 2018): 198–238. http://dx.doi.org/10.1017/jfm.2017.920.
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The study of sediment transport requires in-depth investigation of the complex effects of sediment particles in fluid turbulence. In this paper we focus on intense sediment transport flows. None of the existing two-phase models in the literature properly replicates the liquid and solid stresses in the near bed region of high concentration of sediment. The reason for this shortcoming is that the physical processes occurring at the length scale of the particle collisions are different from those occurring at larger length scales and therefore, they must be modelled independently. We present here a two-level theoretical derivation of two-phase, Favre averaged Navier–Stokes equations (FANS). This approach treats two levels of energy fluctuations independently, those associated with a granular spatial scale (granular temperature and small-scale fluid turbulence) and those associated with the ensemble average (turbulent kinetic energy for the two phases). Although similar attempts have been made by other researchers, the two level approach ensures that the two relevant length scales are included independently in a more consistent manner. The model is endowed with a semi-empirical formulation for the granular scale fluid turbulence, which is important even in the dense collisional shear layer, as has been recently recognized. As a result of the large and small scale modelling of the liquid and solid fluctuations, predictions are promising to be reliable in a wide range of flow conditions, from collisional to turbulent suspensions. This model has been validated for steady state flows with intense, collisional or mixed collisional–turbulent sediment transport, using various sources of detailed experimental data. It compares well with the experimental results in the whole experimental range of Shields parameters, better than previous models, although at the cost of increased complexity in the equations. Further experiments on turbulent suspensions would be necessary to definitely assess the model capabilities.
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Smida, K., H. Lamloumi, K. Maalel, and Z. Hafsia. "CFD Analysis of Water Solitary Wave Reflection." Journal of Engineering Research [TJER] 8, no. 2 (December 1, 2011): 10. http://dx.doi.org/10.24200/tjer.vol8iss2pp10-18.
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A new numerical wave generation method is used to investigate the head-on collision of two solitary waves. The reflection at vertical wall of a solitary wave is also presented. The originality of this model, based on the Navier-Stokes equations, is the specification of an internal inlet velocity, defined as a source line within the computational domain for the generation of these non linear waves. This model was successfully implemented in the PHOENICS (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series) code. The collision of two counter-propagating solitary waves is similar to the interaction of a soliton with a vertical wall. This wave generation method allows the saving of considerable time for this collision process since the counter-propagating wave is generated directly without reflection at vertical wall. For the collision of two solitary waves, numerical results show that the run-up phenomenon can be well explained, the solution of the maximum wave run-up is almost equal to experimental measurement. The simulated wave profiles during the collision are in good agreement with experimental results. For the reflection at vertical wall, the spatial profiles of the wave at fixed instants show that this problem is equivalent to the collision process.
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Degond, Pierre, Amic Frouvelle, and Jian-Guo Liu. "From kinetic to fluid models of liquid crystals by the moment method." Kinetic and Related Models 15, no. 3 (2022): 417. http://dx.doi.org/10.3934/krm.2021047.
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<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>
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Mácha, Václav, and Šárka Nečasová. "Self-propelled motion in a viscous compressible fluid." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 2 (January 19, 2016): 415–33. http://dx.doi.org/10.1017/s0308210515000487.
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In this paper we focus on the existence of a weak solution to a system describing a self-propelled motion of a single deformable body in a viscous compressible fluid that occupies a bounded domain in the three-dimensional Euclidean space. The governing system considered for the fluid is the isentropic compressible Navier–Stokes equation. We prove the existence of a weak solution up to a collision.
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VÁZQUEZ, JUAN LUIS, and ENRIQUE ZUAZUA. "LACK OF COLLISION IN A SIMPLIFIED 1D MODEL FOR FLUID–SOLID INTERACTION." Mathematical Models and Methods in Applied Sciences 16, no. 05 (May 2006): 637–78. http://dx.doi.org/10.1142/s0218202506001303.
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In this paper we consider a simplified model for fluid–solid interaction in one space dimension. The fluid is assumed to be governed by the viscous Burgers equation. It is coupled with a finite number of solid masses in the form of point particles, which share the velocity of the fluid and are accelerated by the jump in velocity gradient of the fluid on both sides, which replaces here the standard pressure jump of Navier–Stokes models. We prove global existence and uniqueness of solutions. This requires proving that the solid particles never collide in finite time, a key fact that follows from suitable a priori estimates together with uniqueness results for ordinary differential equations. We also describe the asymptotic behavior of solutions as t → ∞, extending previous results established for a single solid mass. The evolution of the relative position of the particles is examined in terms of the strength of the convection term. The possible 2D analogues of these results in the context of Navier–Stokes equations are open problems.
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Golse, François, and Laure Saint-Raymond. "The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels." Inventiones mathematicae 155, no. 1 (September 9, 2003): 81–161. http://dx.doi.org/10.1007/s00222-003-0316-5.
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GRAILLE, BENJAMIN, THIERRY E. MAGIN, and MARC MASSOT. "KINETIC THEORY OF PLASMAS: TRANSLATIONAL ENERGY." Mathematical Models and Methods in Applied Sciences 19, no. 04 (April 2009): 527–99. http://dx.doi.org/10.1142/s021820250900353x.
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In the present study, we derive from kinetic theory a unified fluid model for multicomponent plasmas by accounting for the electromagnetic field influence. We deal with a possible thermal nonequilibrium of the translational energy of the particles, neglecting their internal energy and reactive collisions. Given the strong disparity of mass between the electrons and heavy particles, such as molecules, atoms, and ions, we conduct a dimensional analysis of the Boltzmann equation and introduce a scaling based on a multiscale perturbation parameter equal to the square root of the ratio of the electron mass to a characteristic heavy-particle mass. We then generalize the Chapman–Enskog method, emphasizing the role of the perturbation parameter on the collisional operator, the streaming operator, and the collisional invariants of the Boltzmann equation. The system is examined at successive orders of approximation, each corresponding to a physical timescale. At the highest approximation order investigated, the multicomponent Navier–Stokes regime is reached for the heavy particles and is coupled to first-order drift-diffusion equations for the electrons. The transport coefficients are then calculated in terms of bracket operators whose mathematical structure allows for positivity properties to be determined and Onsager's reciprocal relations to hold. The transport coefficients exhibit an anisotropic behavior when the magnetic field is strong enough. We also give a complete description of the Kolesnikov effect, i.e. the crossed contributions to the mass and energy transport fluxes coupling the electrons and heavy particles. Finally, the first and second laws of thermodynamics are proved to be satisfied by deriving a total energy equation and an entropy equation. Moreover, the purely convective system of equations is shown to be hyperbolic.
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Yacoubi, Acmae El, Sheng Xu, and Z. Jane Wang. "A New method for computing particle collisions in Navier-Stokes flows." Journal of Computational Physics 399 (December 2019): 108919. http://dx.doi.org/10.1016/j.jcp.2019.108919.
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Shan, Xiaowen, Xuhui Li, and Yangyang Shi. "A multiple-relaxation-time collision model by Hermite expansion." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2208 (August 30, 2021): 20200406. http://dx.doi.org/10.1098/rsta.2020.0406.
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The Bhatnagar–Gross–Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier–Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.
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Kida, S., M. Takaoka, and F. Hussain. "Collision of two vortex rings." Journal of Fluid Mechanics 230 (September 1991): 583–646. http://dx.doi.org/10.1017/s0022112091000903.
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The interaction of two identical circular viscous vortex rings starting in a side-by-side configuration is investigated by solving the Navier–Stokes equation using a spectral method with 643 grid points. This study covers initial Reynolds numbers (ratio of circulation to viscosity) up to 1153. The vortices undergo two successive reconnections, fusion and fission, as has been visualized experimentally, but the simulation shows topological details not observed in experiments. The shapes of the evolving vortex rings are different for different initial conditions, but the mechanism of the reconnection is explained by bridging (Melander & Hussain 1988) except that the bridges are created on the front of the dipole close to the position of the maximum strain rate. Spatial structures of various field quantities are compared. It is found that domains of high energy dissipation and high enstrophy production overlap, and that they are highly localized in space compared with the regions of concentrated vorticity. The kinetic energy decays according to the same power laws as found in fully developed turbulence, consistent with concentrated regions of energy dissipation. The main vortex cores survive for a relatively long time. On the other hand, the helicity density which is higher in roots of bridges and threads (or legs) changes rapidly in time. The high-helicity-density and high-energy-dissipation regions overlap significantly although their peaks do not always do so. Thus a long-lived structure may carry high-vorticity rather than necessarily high-helicity density. It is shown that the time evolution of concentration of a passive scalar is quite different from that of the vorticity field, confirming our longstanding warning against relying too heavily on flow visualization in laboratory experiments for studying vortex dynamics and coherent structures.
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Reshetova, Anna, and Tatyana Poplavskaya. "Numerical Investigation Of The Evolution Of Disturbances On A Flat Plate In A Hypersonic Flow Of A Mixture Of Vibrationally Excited Gases." Siberian Journal of Physics 12, no. 2 (June 1, 2017): 11–19. http://dx.doi.org/10.54362/1818-7919-2017-12-2-11-19.
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The problem of the evolution of disturbances in a hypersonic viscous shock layer on a plate in the flow of a mixture of vibrationally excited carbon dioxide and nitrogen is considered by solving of the Navier – Stokes equations. Two channels of vibrational relaxation of CO2 molecules in collisions with CO2 and in collisions with N2 were taken into account by using a two-temperature model of relaxation flows in modeling the thermal nonequilibrium. The data on the dynamics of the evolution of disturbances on f flat plate in a wide range of determining parameters (attack angle α = 5÷20°, concentration of CO2 in mixture, braking temperature T0 = 2000÷4000 K) are presented in this paper.
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Doroshenko, Yaroslav, Julia Doroshenko, Vasyl Zapukhliak, Lyubomyr Poberezhny, and Pavlo Maruschak. "MODELING COMPUTATIONAL FLUID DYNAMICS OF MULTIPHASE FLOWS IN ELBOW AND T-JUNCTION OF THE MAIN GAS PIPELINE." Transport 34, no. 1 (January 16, 2019): 19–29. http://dx.doi.org/10.3846/transport.2019.7441.
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The research was performed in order to obtain the physical picture of the movement of condensed droplets and solid particles in the flow of natural gas in elbows and T-junctions of the linear part of the main gas pipeline. 3D modeling of the elbow and T-junction was performed in the linear part of the gas main, in particular, in places where a complex movement of multiphase flows occurs and changes its direction. In these places also occur swirls, collisions of discrete phases in the pipeline wall, and erosive wear of the pipe wall. Based on Lagrangian approach (Discrete Phase Model – DPM), methods of computer modeling were developed to simulate multiphase flow movement in the elbow and T-junction of the linear part of the gas main using software package ANSYS Fluent R17.0 Academic. The mathematical model is based on solving the Navier–Stokes equations, and the equations of continuity and discrete phase movement closed with Launder–Sharma (k–e) two-parameter turbulence model with appropriate initial and boundary conditions. In T-junction, we simulated gas movement in the run-pipe, and the passage of the part of flow into the branch. The simulation results were visualized in postprocessor ANSYS Fluent R17.0 Academic and ANSYS CFD-Post R17.0 Academic by building trajectories of the motion of condensed droplets and solid particles in the elbow and T-junction of the linear part of the gas main in the flow of natural gas. The trajectories were painted in colors that match the velocity and diameter of droplets and particles according to the scale of values. After studying the trajectories of discrete phases, the locations of their heavy collision with the pipeline walls were found, as well as the places of turbulence of condensed droplets and solid particles. The velocity of liquid and solid particles was determined, and the impact angles, diameters of condensed droplets and solid particles in the place of collision were found. Such results provide possibilities for a full and comprehensive investigation of erosive wear of the elbow and T-junction of the linear part of the gas main and adjacent sections of the pipeline, and for the assessment of their strength and residual life.
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Idrisov, Edvin G., Eddwi H. Hasdeo, Byjesh N. Radhakrishnan, and Thomas L. Schmidt. "Hydrodynamic Navier-Stokes equations in two-dimensional systems with Rashba spin-orbit coupling." Low Temperature Physics 49, no. 12 (December 1, 2023): 1385–97. http://dx.doi.org/10.1063/10.0022364.
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We study a two-dimensional (2D) electron system with a linear spectrum in the presence of Rashba spin-orbit (RSO) coupling in the hydrodynamic regime. We derive a semiclassical Boltzmann equation with a collision integral due to Coulomb interactions on the basis of the eigenstates of the system with RSO coupling. Using the local equilibrium distribution functions, we obtain a generalized hydrodynamic Navier–Stokes equation for electronic systems with RSO coupling. In particular, we discuss the influence of the spin-orbit coupling on the viscosity and the enthalpy of the system and present some of its observable effects in hydrodynamic transport.
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Mužík, Juraj. "Lattice Boltzmann Method for Two-Dimensional Unsteady Incompressible Flow." Civil and Environmental Engineering 12, no. 2 (December 1, 2016): 122–27. http://dx.doi.org/10.1515/cee-2016-0017.
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Abstract A Lattice Boltzmann method is used to analyse incompressible fluid flow in a two-dimensional cavity and flow in the channel past cylindrical obstacle. The method solves the Boltzmann’s transport equation using simple computational grid - lattice. With the proper choice of the collision operator, the Boltzmann’s equation can be converted into incompressible Navier-Stokes equation. Lid-driven cavity benchmark case for various Reynolds numbers and flow past cylinder is presented in the article. The method produces stable solutions with results comparable to those in literature and is very easy to implement.
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Jin, Yuzhen, Huang Zhou, Linhang Zhu, and Zeqing Li. "Dynamics of Single Droplet Splashing on Liquid Film by Coupling FVM with VOF." Processes 9, no. 5 (May 11, 2021): 841. http://dx.doi.org/10.3390/pr9050841.
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A three-dimensional numerical study of a single droplet splashing vertically on a liquid film is presented. The numerical method is based on the finite volume method (FVM) of Navier–Stokes equations coupled with the volume of fluid (VOF) method, and the adaptive local mesh refinement technology is adopted. It enables the liquid–gas interface to be tracked more accurately, and to be less computationally expensive. The relationship between the diameter of the free rim, the height of the crown with different numbers of collision Weber, and the thickness of the liquid film is explored. The results indicate that the crown height increases as the Weber number increases, and the diameter of the crown rim is inversely proportional to the collision Weber number. It can also be concluded that the dimensionless height of the crown decreases with the increase in the thickness of the dimensionless liquid film, which has little effect on the diameter of the crown rim during its growth.
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MONACO, R., M. PANDOLFI BIANCHI, and A. ROSSANI. "CHAPMAN-ENSKOG EXPANSION FOR A DISCRETE VELOCITY MODEL OF A GAS MIXTURE WITH BI-MOLECULAR CHEMICAL REACTIONS." Mathematical Models and Methods in Applied Sciences 04, no. 03 (June 1994): 355–72. http://dx.doi.org/10.1142/s0218202594000212.
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We propose a new discrete velocity model of the Boltzmann equation for a mixture of four gases admitting particle elastic collisions and bi-molecular chemical reactions. We first prove an H-theorem and determine the thermodynamical equilibrium state. A Chapman-Enskog expansion on the kinetic equations is then performed, deriving both the Euler and the Navier-Stokes equations of the model. Finally the transport coefficients of diffusivity and viscosity are provided as well.
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Maderich, Vladimir, Kyung Tae Jung, Kateryna Terletska, and Kyeong Ok Kim. "Head-on collision of internal waves with trapped cores." Nonlinear Processes in Geophysics 24, no. 4 (December 22, 2017): 751–62. http://dx.doi.org/10.5194/npg-24-751-2017.
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Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in a thin pycnocline were studied numerically within the framework of the Navier–Stokes equations for a stratified fluid. The peculiarity of this collision is that it involves trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly non-linear waves without trapped cores, (ii) stable strongly non-linear waves with trapped cores, and (iii) shear unstable strongly non-linear waves. The wave phase shift of the colliding waves with equal amplitude grows as the amplitudes increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum run-up amplitude, normalized by the amplitude of the waves, over the sum of the amplitudes of the equal colliding waves increases almost linearly with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The colliding waves of class (ii) lose fluid trapped by the wave cores when amplitudes normalized by the thickness of the pycnocline are in the range of approximately between 1 and 1.75. The interacting stable waves of higher amplitude capture cores and carry trapped fluid in opposite directions with little mass loss. The collision of locally shear unstable waves of class (iii) is accompanied by the development of instability. The dependence of loss of energy on the wave amplitude is not monotonic. Initially, the energy loss due to the interaction increases as the wave amplitude increases. Then, the energy losses reach a maximum due to the loss of potential energy of the cores upon collision and then start to decrease. With further amplitude growth, collision is accompanied by the development of instability and an increase in the loss of energy. The collision process is modified for waves of different amplitudes because of the exchange of trapped fluid between colliding waves due to the conservation of momentum.
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Wang, Xiaodong, Kai Chen, Ting Kang, and Jie Ouyang. "A Dynamic Coarse Grain Discrete Element Method for Gas-Solid Fluidized Beds by Considering Particle-Group Crushing and Polymerization." Applied Sciences 10, no. 6 (March 12, 2020): 1943. http://dx.doi.org/10.3390/app10061943.
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The discrete element method (DEM) coupled with computational fluid dynamics (CFD) is used extensively for the numerical simulation of gas-solid fluidized beds. In order to improve the efficiency of this approach, a coarse grain model of the DEM was proposed in the literature. In this model, a group of original particles are treated as a large-sized particle based on the initial particle distribution, and during the whole simulation process the number and components of these particle-groups remain unchanged. However, collisions between particles can lead to frequent crushing and polymerization of particle-groups. This fact has typically been ignored, so the purpose of this paper is to rationalize the coarse grain DEM-CFD model by considering the dynamic particle-group crushing and polymerization. In particular, the effective size of each particle-group is measured by a quantity called equivalent particle-group diameter, whose definition references the equivalent cluster diameter used by the energy-minimization multi-scale (EMMS) model. Then a particle-group crushing criterion is presented based on the mismatch between the equivalent diameter and actual diameter of a particle-group. As to the polymerization of two colliding particle-groups, their velocity difference after collision is chosen as a criterion. Moreover, considering the flow heterogeneity induced by the particle cluster formation, the EMMS drag force model is adopted in this work. Simulations are carried out by using a finite volume method (FVM) with non-staggered grids. For decoupling the Navier-Stokes equations, the semi-implicit method for pressure linked equations revised (SIMPLER) algorithm is used. The simulation results show that the proposed dynamic coarse grain DEM-CFD method has better performance than the original one.
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Voßkuhle, Michel, Alain Pumir, Emmanuel Lévêque, and Michael Wilkinson. "Collision rate for suspensions at large Stokes numbers – comparing Navier–Stokes and synthetic turbulence." Journal of Turbulence 16, no. 1 (August 30, 2014): 15–25. http://dx.doi.org/10.1080/14685248.2014.948628.
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Tong, Ying, and Jian Xia. "The hydrodynamic FORCE of fluid–structure interaction interface in lattice Boltzmann simulations." International Journal of Modern Physics B 34, no. 14n16 (May 30, 2020): 2040085. http://dx.doi.org/10.1142/s0217979220400858.
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The hydrodynamic force (HF) evaluation plays a critical role in the numerical simulation of fluid–structure interaction (FSI). By directly using the distribution functions of lattice Boltzmann equation (LBE) to evaluate the HF, the momentum exchange algorithm (MEA) has excellent features. Particularly, it is independent of boundary geometry and avoids integration on the complex boundary. In this work, the HF of lattice Boltzmann simulation (LBS) is evaluated by using the MEA. We conduct a comparative study to evaluate two lattice Boltzmann models for constructing the flow solvers, including the LBE with single-relaxation-time (SRT) and multiple-relaxation-time (MRT) collision operators. The second-order boundary condition schemes are used to address the curve boundary. The test case of flow past a cylinder asymmetrically placed in a channel is simulated. Comparing the numerical solutions of Lattice Boltzmann method (LBM) with those of Navier–Stokes equations in the literature, the influence of collision relaxation model, boundary conditions and lattice resolution is investigated. The results demonstrate that the MRT-LB improves the numerical stability of the LBM and the accuracy of HF.
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VEGA REYES, FRANCISCO, and JEFFREY S. URBACH. "Steady base states for Navier–Stokes granular hydrodynamics with boundary heating and shear." Journal of Fluid Mechanics 636 (September 25, 2009): 279–93. http://dx.doi.org/10.1017/s0022112009007800.
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We study the Navier–Stokes steady states for a low density monodisperse hard sphere granular gas (i.e a hard sphere ideal monatomic gas with inelastic inter-particle collisions). We present a classification of the uniform steady states that can arise from shear and temperature (or energy input) applied at the boundaries (parallel walls). We consider both symmetric and asymmetric boundary conditions and find steady states not previously reported, including sheared states with linear temperature profiles. We provide explicit expressions for the hydrodynamic profiles for all these steady states. Our results are validated by the numerical solution of the Boltzmann kinetic equation for the granular gas obtained by the direct simulation Monte Carlo method, and by molecular dynamics simulations. We discuss the physical origin of the new steady states and derive conditions for the validity of Navier–Stokes hydrodynamics.
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Chen, Hudong, Ilya Staroselsky, Katepalli R. Sreenivasan, and Victor Yakhot. "Average Turbulence Dynamics from a One-Parameter Kinetic Theory." Atmosphere 14, no. 7 (July 4, 2023): 1109. http://dx.doi.org/10.3390/atmos14071109.
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We show theoretically that the mean turbulent dynamics can be described by a kinetic theory representation with a single free relaxation time that depends on space and time. A proper kinetic equation is constructed from the Klimontovich-type kinetic equation for fluid elements, which satisfies the Navier–Stokes hydrodynamics exactly. In a suitably averaged form, the turbulent kinetic energy plays the role of temperature in standard molecular thermodynamics. We show that the dynamics of turbulent fluctuations resembles a collision process that asymptotically drives the mean distribution towards a Gaussian (Maxwell–Boltzmann) equilibrium form. Non-Gaussianity arises directly from non-equilibrium shear effects. The present framework overcomes the bane of most conventional turbulence models and theoretical frameworks arising from the lack of scale separation between the mean and fluctuating scales of the Navier-Stokes equation with an eddy viscous term. An averaged turbulent flow in the present framework behaves more like a flow of finite Knudsen number with finite relaxation time, and is thus more suitably described in a kinetic theory representation.
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Garg, Deepak, Antonella Longo, and Paolo Papale. "Modeling Free Surface Flows Using Stabilized Finite Element Method." Mathematical Problems in Engineering 2018 (June 11, 2018): 1–9. http://dx.doi.org/10.1155/2018/6154251.
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This work aims to develop a numerical wave tank for viscous and inviscid flows. The Navier-Stokes equations are solved by time-discontinuous stabilized space-time finite element method. The numerical scheme tracks the free surface location using fluid velocity. A segregated algorithm is proposed to iteratively couple the fluid flow and mesh deformation problems. The numerical scheme and the developed computer code are validated over three free surface problems: solitary wave propagation, the collision between two counter moving waves, and wave damping in a viscous fluid. The benchmark tests demonstrate that the numerical approach is effective and an attractive tool for simulating viscous and inviscid free surface flows.
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BREVDO, LEONID, PATRICE LAURE, FREDERIC DIAS, and THOMAS J. BRIDGES. "Linear pulse structure and signalling in a film flow on an inclined plane." Journal of Fluid Mechanics 396 (October 10, 1999): 37–71. http://dx.doi.org/10.1017/s0022112099005790.
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The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function.The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.
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Rieber, M., and A. Frohn. "Three-dimensional Navier-Stokes simulation of binary collisions between droplets of equal size." Journal of Aerosol Science 26 (September 1995): S929—S930. http://dx.doi.org/10.1016/0021-8502(95)97372-l.
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Fotakis, Jan A., Moritz Greif, Gabriel S. Denicol, and Carsten Greiner. "Diffusion of Conserved Charges in Relativistic Heavy Ion Collisions." Proceedings 10, no. 1 (April 17, 2019): 31. http://dx.doi.org/10.3390/proceedings2019010031.
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We discuss the diffusion currents occurring in a dilute system and show that the charge currents do not only depend on gradients in the corresponding charge density, but also on the other conserved charges in the system—the diffusion currents are therefore coupled. Gradients in one charge thus generate dissipative currents in a different charge. In this approach, we model the Navier-Stokes term of the generated currents to consist of a diffusion coefficient matrix, in which the diagonal entries are the usual diffusion coefficients and the off-diagonal entries correspond to the coupling of different diffusion currents. We evaluate the complete diffusion matrix for a specific hadron gas and for a simplified quark-gluon gas, including baryon, electric and strangeness charge. Our findings are that the off-diagonal entries can range within the same magnitude as the diagonal ones.
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Schullian, O., H. S. Antila, and B. R. Heazlewood. "A variable time step self-consistent mean field DSMC model for three-dimensional environments." Journal of Chemical Physics 156, no. 12 (March 28, 2022): 124309. http://dx.doi.org/10.1063/5.0083033.
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A self-consistent mean field direct simulation Monte Carlo (SCMFD) algorithm was recently proposed for simulating collision environments for a range of one-dimensional model systems. This work extends the one-dimensional SCMFD approach to three dimensions and introduces a variable time step (3D-vt-SCMFD), enabling the modeling of a considerably wider range of different collision environments. We demonstrate the performance of the augmented method by modeling a varied set of test systems: ideal gas mixtures, Poiseuille flow of argon, and expansion of gas into high vacuum. For the gas mixtures, the 3D-vt-SCMFD method reproduces the properties (mean free path, mean free time, collision frequency, and temperature) in excellent agreement with theoretical predictions. From the Poiseuille flow simulations, we extract flow profiles that agree with the solution to the Navier–Stokes equations in the high-density limit and resemble free molecular flow at low densities, as expected. The measured viscosity from 3D-vt-SCMF is ∼15% lower than the theoretical prediction from Chapman–Enskog theory. The expansion of gas into vacuum is examined in the effusive regime and at the hydrodynamic limit. In both cases, 3D-vt-SCMDF simulations produce gas beam density, velocity, and temperature profiles in excellent agreement with analytical models. In summary, our tests show that 3D-vt-SCMFD is robust and computationally efficient, while also illustrating the diversity of systems the SCMFD model can be successfully applied to.
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Sambath, Krishnaraj, Vishrut Garg, Sumeet S. Thete, Hariprasad J. Subramani, and Osman A. Basaran. "Inertial impedance of coalescence during collision of liquid drops." Journal of Fluid Mechanics 876 (August 1, 2019): 449–80. http://dx.doi.org/10.1017/jfm.2019.498.
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The fluid dynamics of the collision and coalescence of liquid drops has intrigued scientists and engineers for more than a century owing to its ubiquitousness in nature, e.g. raindrop coalescence, and industrial applications, e.g. breaking of emulsions in the oil and gas industry. The complexity of the underlying dynamics, which includes occurrence of hydrodynamic singularities, has required study of the problem at different scales – macroscopic, mesoscopic and molecular – using stochastic and deterministic methods. In this work, a multi-scale, deterministic method is adopted to simulate the approach, collision, and eventual coalescence of two drops where the drops as well as the ambient fluid are incompressible, Newtonian fluids. The free boundary problem governing the dynamics consists of the Navier–Stokes system and associated initial and boundary conditions that have been augmented to account for the effects of disjoining pressure as the separation between the drops becomes of the order of a few hundred nanometres. This free boundary problem is solved by a Galerkin finite element-based algorithm. The interplay of inertial, viscous, capillary and van der Waals forces on the coalescence dynamics is investigated. It is shown that, in certain situations, because of inertia two drops that are driven together can first bounce before ultimately coalescing. This bounce delays coalescence and can result in the computed value of the film drainage time departing significantly from that predicted from existing scaling theories.
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Sun, Guanwen, Hanyin Cui, Chao Li, Weijun Lin, and Chang Su. "Experimental and theoretical investigations of dispersion of ultrasonic waves in the low-temperature and low-pressure nitrogen gas." Journal of the Acoustical Society of America 153, no. 2 (February 2023): 821–34. http://dx.doi.org/10.1121/10.0017097.
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Temperature has a complex effect on acoustic dispersion in dilute gases. In this paper, the effect of temperature on the acoustic dispersion of dilute gases is analyzed theoretically and experimentally. Theoretically, the Navier–Stokes (NS) equation and the Greenspan's theory, which includes the rotational-relaxation correction, are applied to calculate the dispersive sound speed. It is concluded that the temperature affects the molecular translational relaxation and the rotational relaxation by influencing the average molecular collision frequency and the relaxation collision number, respectively, and thus, change the amplitude of the acoustic dispersion. Numerical calculations led to the conclusion that both translational and rotational dispersions weakened as the temperature decreased. Experimentally, sound speed measurements of 21–40 kHz acoustic waves were also carried out in gaseous nitrogen at temperatures ranging from −70 °C to 20 °C and pressures of 150–105 Pa. Theoretical predications indicate that the speed of sound should increase with decreasing pressure at all temperatures, and the degree of dispersion should diminish at lower temperatures. The experimental observation of dispersion is consistent with theory within experimental error (1%) but was not able to distinguish the small (0.01%) increase in sound speed expected at 150 Pa.
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ISHIWATA, TATSUYA, TERUYOSHI MURAKAMI, SATOSHI YUKAWA, and NOBUYASU ITO. "PARTICLE DYNAMICS SIMULATIONS OF THE NAVIER–STOKES FLOW WITH HARD DISKS." International Journal of Modern Physics C 15, no. 10 (December 2004): 1413–24. http://dx.doi.org/10.1142/s0129183104006820.
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Flow simulation with a particle dynamics method is studied. The fluid is made of hard particles which obey the Newtonian equations of motion and the collisions between particles are elastic, that is, energy and momentum are conserved. The viscosity appears autonomously together with the local equilibrium state. When a particle collides with a nonslip boundary, a new velocity is given randomly from the thermal distribution if the wall is isothermal, or a random reflection angle is selected if the wall is adiabatic. Shear viscosity is estimated from simulations of plane Poiseuille flow together with the confirmation that the system obeys the Navier–Stokes equation. Flows past a cylinder are also simulated. Depending on the Reynolds number up to 106, flow patterns are properly reproduced, and Kármán vortex shedding is observed. The estimated values of drag coefficient show quantitative agreement with experiments.
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Li, Yanbing, and Xiaowen Shan. "Lattice Boltzmann method for adiabatic acoustics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1944 (June 13, 2011): 2371–80. http://dx.doi.org/10.1098/rsta.2011.0109.
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The lattice Boltzmann method (LBM) has been proved to be a useful tool in many areas of computational fluid dynamics, including computational aero-acoustics (CAA). However, for historical reasons, its applications in CAA have been largely restricted to simulations of isothermal (Newtonian) sound waves. As the recent kinetic theory-based reformulation establishes a theoretical framework in which LBM can be extended to recover the full Navier–Stokes–Fourier (NS) equations and beyond, in this paper, we show that, at least at the low-frequency limit (sound frequency much less than molecular collision frequency), adiabatic sound waves can be accurately simulated by the LBM provided that the lattice and the distribution function ensure adequate recovery of the full NS equations.
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Mizuno, Yusuke, Shun Takahashi, Kota Fukuda, and Shigeru Obayashi. "Direct Numerical Simulation of Gas–Particle Flows with Particle–Wall Collisions Using the Immersed Boundary Method." Applied Sciences 8, no. 12 (November 26, 2018): 2387. http://dx.doi.org/10.3390/app8122387.
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We investigated particulate flows by coupling simulations of the three-dimensional incompressible Navier–Stokes equation with the immersed boundary method (IBM). The results obtained from the two-way coupled simulation were compared with those of the one-way simulation, which is generally applied for clarifying the particle kinematics in industry. In the present flow simulation, the IBM was solved using a ghost–cell approach and the particles and walls were defined by a level set function. Using proposed algorithms, particle–particle and particle–wall collisions were implemented simply; the subsequent coupling simulations were conducted stably. Additionally, the wake structures of the moving, colliding and rebounding particles were comprehensively compared with previous numerical and experimental results. In simulations of 50, 100, 200 and 500 particles, particle–wall collisions were more frequent in the one–way scheme than in the two-way scheme. This difference was linked to differences in losses in energy and momentum.
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Akkaya, Volkan Ramazan, and Ilyas Kandemir. "Event-Driven Molecular Dynamics Simulation of Hard-Sphere Gas Flows in Microchannels." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/842837.
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Classical solution of Navier-Stokes equations with nonslip boundary condition leads to inaccurate predictions of flow characteristics of rarefied gases confined in micro/nanochannels. Therefore, molecular interaction based simulations are often used to properly express velocity and temperature slips at high Knudsen numbers (Kn) seen at dilute gases or narrow channels. In this study, an event-driven molecular dynamics (EDMD) simulation is proposed to estimate properties of hard-sphere gas flows. Considering molecules as hard-spheres, trajectories of the molecules, collision partners, corresponding interaction times, and postcollision velocities are computed deterministically using discrete interaction potentials. On the other hand, boundary interactions are handled stochastically. Added to that, in order to create a pressure gradient along the channel, an implicit treatment for flow boundaries is adapted for EDMD simulations. Shear-Driven (Couette) and Pressure-Driven flows for various channel configurations are simulated to demonstrate the validity of suggested treatment. Results agree well with DSMC method and solution of linearized Boltzmann equation. At low Kn, EDMD produces similar velocity profiles with Navier-Stokes (N-S) equations and slip boundary conditions, but as Kn increases, N-S slip models overestimate slip velocities.
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Lubin, Pierre, Stéphane Vincent, and Jean-Paul Caltagirone. "On the Navier–Stokes equations simulation of the head-on collision between two surface solitary waves." Comptes Rendus Mécanique 333, no. 4 (April 2005): 351–57. http://dx.doi.org/10.1016/j.crme.2005.02.005.
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Stanković, Nikola. "Modelling the closure of narrow oceanic basins by means of numerical simulations." Tehnika 76, no. 6 (2021): 741–46. http://dx.doi.org/10.5937/tehnika2106741s.
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Numerical simulations represent an established approach in investigation of geodynamic processes in the Earth's interior and it is applied to a vast variety of geological problems. The closure of a narrow oceanic realm is modelled in this paper. Firstly, a simplified numerical model of a narrow ocean is established. The thermomechanical system is then simulated by solving Navier-Stokes and temperature equations. By imposing boundary conditions for velocity the convergent regime is simulated which leads to a subduction process along a predefined weak shear zone mantle. The subduction subsequently results in development of magmatism, the final closure of the oceanic realm and the onset of continental collision. Considerable amount of ophiolites were successfully obducted onto the overriding continent during the course of the simulation.
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Li, Weidong, and Li-Shi Luo. "Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes." Communications in Computational Physics 20, no. 2 (July 21, 2016): 301–24. http://dx.doi.org/10.4208/cicp.211015.040316a.
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AbstractA genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent, i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.
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NASR, HOJJAT, GOODARZ AHMADI, and JOHN B. MCLAUGHLIN. "A DNS study of effects of particle–particle collisions and two-way coupling on particle deposition and phasic fluctuations." Journal of Fluid Mechanics 640 (November 13, 2009): 507–36. http://dx.doi.org/10.1017/s0022112009992011.
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This study is concerned with the effects of particle–particle collisions and the two-way coupling on the dispersed and carrier phase turbulence fluctuations in a channel flow. The time history of the instantaneous turbulent velocity vector was generated by the two-way coupled direct numerical simulation of the Navier–Stokes equations via a pseudo-spectral method. The particle equation of motion included the wall-corrected nonlinear drag force and the wall-induced and shear-induced lift force. The effect of particles on the flow was included in the analysis via a feedback force that acted on the computational grid points. Several simulations for different particle relaxation times and particle mass loadings were performed, and the effects of particle–particle collisions, particle feedback force and inter-particle interactions on the particle deposition velocity, fluid and particle fluctuating velocities, and particle concentration profiles were determined. The effect of particle aerodynamic interactions was also examined for certain cases.The simulation results indicated that when particle–particle collisions were included in the computation but two-way coupling effects were ignored, the particle normal fluctuating velocity increased in the wall region causing an increase in the particle deposition velocity. When the particle collisions were neglected but the particle–fluid two-way coupling effects were accounted for, the two-way coupling and the particle normal fluctuating velocity decreased near the wall causing a decrease in the particle deposition velocity. In the case of the four-way coupling in which both inter-particle collisions and two-way coupling effects were present, it was found that the particle deposition velocity increased compared with the one-way coupling case. When the particle aerodynamic interactions were added to the four-way coupled case (termed six-way coupled case), no significant changes in the mean fluid and particle velocities and the fluid and particle fluctuating velocities were obtained.The results for the particle concentration profile indicated that the inclusion of two-way coupling or inter-particle collisions into the computation reduced the accumulation of particles near the wall. It was also observed that particle–particle collisions and two-way coupling weakened the preferential distribution of particles.
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Chinappi, M., and E. De Angelis. "Confined dynamics of a single DNA molecule." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1944 (June 13, 2011): 2329–36. http://dx.doi.org/10.1098/rsta.2011.0096.
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The effect of a slit-like confinement on the relaxation dynamics of DNA is studied via a mesoscale model in which a bead and spring model for the polymer is coupled to a particle-based Navier–Stokes solver (multi-particle collision dynamics). The confinement is found to affect the equilibrium stretch of the chain when the bulk gyration radius is comparable to or smaller than the channel height and our data are in agreement with the ( R g,bulk / h ) 1/4 scaling of the polymer extension in the wall tangential direction. Relaxation simulation at different confinements indicates that, while the overall behaviour of the relaxation dynamics is similar for low and strong confinements, a small, but significant, slowing of the far-equilibrium relaxation is found as the confinement increases.
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Neustupa, Jiří, and Patrick Penel. "The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions." Comptes Rendus Mathematique 347, no. 11-12 (June 2009): 685–90. http://dx.doi.org/10.1016/j.crma.2009.03.021.
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Zhu, Yajun, Chengwen Zhong, and Kun Xu. "GKS and UGKS for High-Speed Flows." Aerospace 8, no. 5 (May 19, 2021): 141. http://dx.doi.org/10.3390/aerospace8050141.
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The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation.