From Kinetic Theory to Navier–Stokes Hydrodynamics

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Kinetic Theory ◽  
Knudsen Number ◽  
Asymptotic Expansions ◽  
Large Scale ◽  
Local Equilibrium ◽  
Navier Stokes ◽  
Fluid Equations ◽  
Scale Limit

This Chapter illustrates the derivation of the macroscopic fluid equations, starting from Boltzmann’s kinetic theory. Two routes are presented, the heuristic derivation based on the enslaving of fast modes to slow ones, and the Hilbert–Chapman–Enskog procedure, based on low-Knudsen number asymptotic expansions. The former is handier but mathematically less rigorous than the latter. Either ways, the assumption of weak departure from local equilibrium proves crucial in recovering hydrodynamics as a large-scale limit of kinetic theory.

Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

2013 ◽  
Vol 738 ◽  
Author(s):  
Peter P. Mitrano ◽  
John R. Zenk ◽  
Sofiane Benyahia ◽  
Janine E. Galvin ◽  
Steven R. Dahl ◽  
...  
Keyword(s):  
Kinetic Theory ◽  
Knudsen Number ◽  
Continuum Model ◽  
Cooling System ◽  
Critical Length ◽  
Md Simulations ◽  
Navier Stokes ◽  
Molecular Gases ◽  
Velocity Gradients ◽  
Transient Simulations

AbstractIn this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small-Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ($K{n}_{u} $) at the time of cluster detection reveals $K{n}_{u} \gg 1$ due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small-$Kn$ (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small-$Kn$) theories are surprisingly accurate outside their expected range of validity.

scholarly journals Assessment of Numerical Methods for Plunging Breaking Wave Predictions

2021 ◽  
Vol 9 (3) ◽  
pp. 264
Author(s):  
Shanti Bhushan ◽  
Oumnia El Fajri ◽  
Graham Hubbard ◽  
Bradley Chambers ◽  
Christopher Kees
Keyword(s):  
Solitary Wave ◽  
Wave Breaking ◽  
Large Scale ◽  
Turbulence Models ◽  
Navier Stokes ◽  
Wave Crest ◽  
Breaking Wave ◽  
Rans Turbulence Models ◽  
Run Up ◽  
Better Than

This study evaluates the capability of Navier–Stokes solvers in predicting forward and backward plunging breaking, including assessment of the effect of grid resolution, turbulence model, and VoF, CLSVoF interface models on predictions. For this purpose, 2D simulations are performed for four test cases: dam break, solitary wave run up on a slope, flow over a submerged bump, and solitary wave over a submerged rectangular obstacle. Plunging wave breaking involves high wave crest, plunger formation, and splash up, followed by second plunger, and chaotic water motions. Coarser grids reasonably predict the wave breaking features, but finer grids are required for accurate prediction of the splash up events. However, instabilities are triggered at the air–water interface (primarily for the air flow) on very fine grids, which induces surface peel-off or kinks and roll-up of the plunger tips. Reynolds averaged Navier–Stokes (RANS) turbulence models result in high eddy-viscosity in the air–water region which decays the fluid momentum and adversely affects the predictions. Both VoF and CLSVoF methods predict the large-scale plunging breaking characteristics well; however, they vary in the prediction of the finer details. The CLSVoF solver predicts the splash-up event and secondary plunger better than the VoF solver; however, the latter predicts the plunger shape better than the former for the solitary wave run-up on a slope case.

Unsteady Three-Dimensional Analysis of Inlet Distortion in a Transonic Fan

2006 ◽  
Author(s):  
Peng Sun ◽  
Guotal Feng
Keyword(s):  
Shock Wave ◽  
Total Pressure ◽  
Large Scale ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Inlet Distortion ◽  
Flow Loss ◽  
Large Scale Vortex ◽  
Total Temperature ◽  
Transonic Fan

A time-accurate three-dimensional Navier-Stokes solver of the unsteady flow field in a transonic fan was carried out using "Fluent-parallel" in a parallel supercomputer. The numerical simulation focused on a transonic fan with inlet square wave total pressure distortion and the analysis of result consisted of three aspects. The first was about inlet parameters redistribution and outlet total temperature distortion induced by inlet total pressure distortion. The pattern and causation of flow loss caused by pressure distortion in rotor were analyzed secondly. It was found that the influence of distortion was different at different radial positions. In hub area, transportation-loss and mixing-loss were the main loss patterns. Distortion not only complicated them but enhanced them. Especially in stator, inlet total pressure distortion induced large-scale vortex, which produced backflow and increased the loss. While in casing area, distortion changed the format of shock wave and increased the shock loss. Finally, the format of shock wave and the hysteresis of rotor to distortion were analyzed in detail.

Macroscopic description of arbitrary Knudsen number flow using Boltzmann–BGK kinetic theory. Part 2

2010 ◽  
Vol 658 ◽  
pp. 294-309 ◽  
Author(s):  
HUDONG CHEN ◽  
STEVEN A. ORSZAG ◽  
ILYA STAROSELSKY
Keyword(s):  
Relaxation Time ◽  
Kinetic Theory ◽  
Closed Form ◽  
Knudsen Number ◽  
Constitutive Models ◽  
Spatial Dimension ◽  
Scalar Transport ◽  
Constant Relaxation Time ◽  
Number Flow ◽  
Short Times

We extend our previous analysis of closed-form equations for finite Knudsen number flow and scalar transport that result from the Boltzmann–Bhatnagar–Gross–Krook (BGK) kinetic theory with constant relaxation time. Without approximation, we obtain closed-form equations for arbitrary spatial dimension and flow directionality which are local differential equations in space and integral equations in time. These equations are further simplified for incompressible flow and scalars. The particular case of no-flow scalar transport admits analytical solutions that exhibit ballistic behaviour at short times while behaving diffusively at long times. It is noteworthy that, even with constant relaxation time BGK microphysics, quite complex macroscopic descriptions result that would be difficult to obtain using classical constitutive models or continuum averaging.

Coriolis force influence on the AKA effect

2021 ◽  
Author(s):  
Peter Rutkevich ◽  
Georgy Golitsyn ◽  
Anatoly Tur
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Nonlinear Equations ◽  
Coriolis Force ◽  
Large Scale ◽  
Fluid Velocity ◽  
Navier Stokes ◽  
Basic Solution ◽  
Navier Stokes Equation ◽  
Alpha Effect

<p>Large-scale instability in incompressible fluid driven by the so called Anisotropic Kinetic Alpha (AKA) effect satisfying the incompressible Navier-Stokes equation with Coriolis force is considered. The external force is periodic; this allows applying an unusual for turbulence calculations mathematical method developed by Frisch et al [1]. The method provides the orders for nonlinear equations and obtaining large scale equations from the corresponding secular relations that appear at different orders of expansions. This method allows obtaining not only corrections to the basic solutions of the linear problem but also provides the large-scale solution of the nonlinear equations with the amplitude exceeding that of the basic solution. The fluid velocity is obtained by numerical integration of the large-scale equations. The solution without the Coriolis force leads to constant velocities at the steady-state, which agrees with the full solution of the Navier-Stokes equation reported previously. The time-invariant solution contains three families of solutions, however, only one of these families contains stable solutions. The final values of the steady-state fluid velocity are determined by the initial conditions. After account of the Coriolis force the solutions become periodic in time and the family of solutions collapses to a unique solution. On the other hand, even with the Coriolis force the fluid motion remains two-dimensional in space and depends on a single spatial variable. The latter fact limits the scope of the AKA method to applications with pronounced 2D nature. In application to 3D models the method must be used with caution.</p><p>[1] U. Frisch, Z.S. She and P. L. Sulem, “Large-Scale Flow Driven by the Anisotropic Kinetic Alpha Effect,” Physica D, Vol. 28, No. 3, 1987, pp. 382-392.</p>

scholarly journals Study of Vortex Systems as a Method to Weakening the Urban Heat Islands within the Financial District in Large Cities

2021 ◽  
Vol 13 (23) ◽  
pp. 13206
Author(s):  
Luis Rodriguez-Lucas ◽  
Chen Ning ◽  
Marcelo Fajardo-Pruna ◽  
Yugui Yang
Keyword(s):  
Rural Areas ◽  
Large Scale ◽  
Flow Characteristics ◽  
Maximum Velocity ◽  
Vortex Generator ◽  
Urban Heat Islands ◽  
Navier Stokes ◽  
Specific Work ◽  
Heat Islands ◽  
Large Cities

This paper presents a new concept called the urban vortex system (UVS). The UVS couples a vortex generator (V.G.) that produces updraft by artificial vortex and a vortex stability zone (VSZ) consisting of an assembly of four buildings acting as a chimney. Through this system, a stable, upward vortex flow can be generated. The Reynolds Averaged Navier–Stokes (RANS) simulation was carried out to investigate the flow field in the UVS. The Renormalized Group (RNG) k–ε turbulent model was selected to solve the complex turbulent flow. Validation of the numerical results was achieved by making a comparison with the large-size experimental model. The results reported that a steady-state vortex could be formed when a vapor-air mixture at 2 m/s and 450 K enters the vortex generator. This vortex presented a maximum negative central pressure of −6.81 Pa and a maximum velocity of 5.47 (m/s). Finally, the similarity method found four dimensionless parameters, which allowed all the flow characteristics to be transported on a large scale. The proposed large-scale UVS application is predicted to be capable, with have a maximum power of 2 M.W., a specific work of 3 kJ/kg, buildings 200-m high, and the ability to generate winds of 6.1 m/s (20 km/h) at 200 m up to winds of 1.5 m/s (5 km/h) at 400 m. These winds would cause the rupture of the gas capsule of the heat island phenomenon. Therefore, the city would balance its temperature with that of the surrounding rural areas.

Shock waves in microchannels

2013 ◽  
Vol 724 ◽  
pp. 259-283 ◽  
Author(s):  
G. Mirshekari ◽  
M. Brouillette ◽  
J. Giordano ◽  
C. Hébert ◽  
J.-D. Parisse ◽  
...  
Keyword(s):  
Shock Wave ◽  
Shock Tube ◽  
Large Scale ◽  
Numerical Models ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Navier Stokes ◽  
Shock Attenuation ◽  
Pressure Time ◽  
Time Histories

AbstractA fully instrumented microscale shock tube, believed to be the smallest to date, has been fabricated and tested. This facility is used to study the transmission of a shock wave, produced in a large (37 mm) shock tube, into a 34 $\mathrm{\mu} \mathrm{m} $ hydraulic diameter and 2 mm long microchannel. Pressure microsensors of a novel design, with gigahertz bandwidth, are used to obtain pressure–time histories of the microchannel shock wave at five axial stations. In all cases the transmitted shock wave is found to be weaker than the incident shock wave, and is observed to decay both in pressure and velocity as it propagates down the microchannel. These results are compared with various analytical and numerical models, and the best agreement is obtained with a Navier–Stokes computational fluid dynamics computation, which assumes a no-slip isothermal wall boundary condition; good agreement is also obtained with a simple shock tube laminar boundary layer model. It is also found that the flow developing within the microchannel is highly dependent on conditions at the microchannel entrance, which control the mass flux entering into the device. Regardless of the micrometre dimensions of the present facility, shock wave propagation in a microchannel of that scale exhibits a behaviour similar to that observed in large-scale facilities operated at low pressures, and the shock attenuation can be explained in terms of accepted laminar boundary models.

scholarly journals Gyrotactic swimmers in turbulence: shape effects and role of the large-scale flow

2018 ◽  
Vol 856 ◽  
Author(s):  
M. Borgnino ◽  
G. Boffetta ◽  
F. De Lillo ◽  
M. Cencini
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Scale ◽  
Navier Stokes Equations ◽  
Preferential Sampling ◽  
Wide Range ◽  
Shape Effects ◽  
Dilute Suspensions ◽  
Large Scale Flow

We study the dynamics and the statistics of dilute suspensions of gyrotactic swimmers, a model for many aquatic motile microorganisms. By means of extensive numerical simulations of the Navier–Stokes equations at different Reynolds numbers, we investigate preferential sampling and small-scale clustering as a function of the swimming (stability and speed) and shape parameters, considering in particular the limits of spherical and rod-like particles. While spherical swimmers preferentially sample local downwelling flow, for elongated swimmers we observe a transition from downwelling to upwelling regions at sufficiently high swimming speed. The spatial distribution of both spherical and elongated swimmers is found to be fractal at small scales in a wide range of swimming parameters. The direct comparison between the different shapes shows that spherical swimmers are more clusterized at small stability and speed numbers, while for large values of the parameters elongated cells concentrate more. The relevance of our results for phytoplankton swimming in the ocean is briefly discussed.

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