Refined Navier-Stokes-Fourier Equations for Rarefied Polyatomic Gases

2014 ◽  
Author(s):  
Behnam Rahimi ◽  
Henning Struchtrup
Keyword(s):  
Moment Method ◽  
Macroscopic Model ◽  
Navier Stokes ◽  
Gas Flows ◽  
Order Of Accuracy ◽  
Closed Set ◽  
Rarefied Polyatomic Gases ◽  
Order Of Magnitude ◽  
Independent Variable ◽  
Refined Version

A macroscopic model for the description of rarefied polyatomic gas flows is introduced. Grad’s moment method is used to construct a closed set of equations for 36 primary moments. The order of magnitude method is then applied to acquire optimized moment definitions. The appropriate sets of equations corresponding to the desired order of accuracy in the Knudsen number are derived by reducing the equations. A refined version of the Navier-Stokes-Fourier (NSF) equations is obtained where the dynamic temperature is an additional independent variable.

Non-stationary process characteristics of the gas outflow into a liquid

2019 ◽  
Vol 14 (2) ◽  
pp. 82-88
Author(s):  
M.V. Alekseev ◽  
I.S. Vozhakov ◽  
S.I. Lezhnin
Keyword(s):  
Numerical Simulation ◽  
Liquid Column ◽  
Gas Content ◽  
Liquid Lead ◽  
Gas Flows ◽  
Asymptotic Model ◽  
Process Characteristics ◽  
Significant Difference ◽  
Order Of Magnitude ◽  
Volume Method

A numerical simulation of the process of the outflow of gas under pressure into a closed container partially filled with liquid was carried out. For comparative theoretical analysis, an asymptotic model was used with assumptions about the adiabaticity of the gas outflow process and the ideality of the liquid during the oscillatory one-dimensional motion of the liquid column. In this case, the motion of the liquid column and the evolution of pressure in the gas are determined by the equation of dynamics and the balance of enthalpy. Numerical simulation was performed in the OpenFOAM package using the fluid volume method (VOF method) and the standard k-e turbulence model. The evolution of the fields of volumetric gas content, velocity, and pressure during the flow of gas from the high-pressure chamber into a closed channel filled with liquid in the presence of a ”gas blanket“ at the upper end of the channel is obtained. It was shown that the dynamics of pulsations in the gas cavity that occurs when the gas flows into the closed region substantially depends on the physical properties of the liquid in the volume, especially the density. Numerical modeling showed that the injection of gas into water occurs in the form of a jet outflow of gas, and for the outflow into liquid lead, a gas slug is formed at the bottom of the channel. Satisfactory agreement was obtained between the numerical calculation and the calculation according to the asymptotic model for pressure pulsations in a gas projectile in liquid lead. For water, the results of calculations using the asymptotic model give a significant difference from the results of numerical calculations. In all cases, the velocity of the medium obtained by numerical simulation and when using the asymptotic model differ by an order of magnitude or more.

scholarly journals Statistical Uncertainty of DNS in Geometries without Homogeneous Directions

2021 ◽  
Vol 11 (4) ◽  
pp. 1399
Author(s):  
Jure Oder ◽  
Cédric Flageul ◽  
Iztok Tiselj
Keyword(s):  
Channel Flow ◽  
A Priori ◽  
Time Integration ◽  
Navier Stokes ◽  
Statistical Uncertainty ◽  
Time Step ◽  
Order Of Magnitude ◽  
Averaging Time ◽  
The Individual ◽  
Time Required

In this paper, we present uncertainties of statistical quantities of direct numerical simulations (DNS) with small numerical errors. The uncertainties are analysed for channel flow and a flow separation case in a confined backward facing step (BFS) geometry. The infinite channel flow case has two homogeneous directions and this is usually exploited to speed-up the convergence of the results. As we show, such a procedure reduces statistical uncertainties of the results by up to an order of magnitude. This effect is strongest in the near wall regions. In the case of flow over a confined BFS, there are no such directions and thus very long integration times are required. The individual statistical quantities converge with the square root of time integration so, in order to improve the uncertainty by a factor of two, the simulation has to be prolonged by a factor of four. We provide an estimator that can be used to evaluate a priori the DNS relative statistical uncertainties from results obtained with a Reynolds Averaged Navier Stokes simulation. In the DNS, the estimator can be used to predict the averaging time and with it the simulation time required to achieve a certain relative statistical uncertainty of results. For accurate evaluation of averages and their uncertainties, it is not required to use every time step of the DNS. We observe that statistical uncertainty of the results is uninfluenced by reducing the number of samples to the point where the period between two consecutive samples measured in Courant–Friedrichss–Levy (CFL) condition units is below one. Nevertheless, crossing this limit, the estimates of uncertainties start to exhibit significant growth.

scholarly journals FIRST PROJECTION OF THE EQUATION OF MOTION IN THE CYLINDRICAL COORDINATE SYSTEM

2016 ◽  
pp. 90-92
Author(s):  
A. G. Obukhov ◽  
R. E. Volkov
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Equation Of Motion ◽  
Cylindrical Coordinate System ◽  
Navier Stokes ◽  
Gas Flows ◽  
Heat Conducting ◽  
Complex Flows ◽  
Navier Stokes Equations ◽  
Cylindrical Coordinate

It is proved that complex flows of the viscous compressible heat-conducting gas, arising during heating the vertical field, have a pronounced axial symmetry. Therefore, for the numerical solution of the full Navier-Stokes equations for description of such gas flows it are advisable to use a cylindrical coordinate system. This paper describes the transformation of the first projection of the equation of motion of the full Navier-Stokes equations system. The result of the transformation is a record of the first projection of the equation of a continuous medium motion in the cylindrical coordinate system.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

Flow induced by jets and plumes

1981 ◽  
Vol 108 ◽  
pp. 55-65 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Stokes Equations ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Jet ◽  
Order Of Magnitude ◽  
Valid Solution ◽  
Solid Walls

The order of magnitude of the flow velocity due to the entrainment into an axisymmetric, laminar or turbulent jet and an axisymmetric laminar plume, respectively, indicates that viscosity and non-slip of the fluid at solid walls are essential effects even for large Reynolds numbers of the jet or plume. An exact similarity solution of the Navier-Stokes equations is determined such that both the non-slip condition at circular-conical walls (including a plane wall) and the entrainment condition at the jet (or plume) axis are satisfied. A uniformly valid solution for large Reynolds numbers, describing the flow in the laminar jet region as well as in the outer region, is also given. Comparisons show that neither potential flow theory (Taylor 1958) nor viscous flow theories that disregard the non-slip condition (Squire 1952; Morgan 1956) provide correct results if the flow is bounded by solid walls.

scholarly journals Numerical modelling of bank failures during reservoir draw-down

2018 ◽  
Vol 40 ◽  
pp. 03001 ◽  
Author(s):  
Nils Reidar B. Olsen ◽  
Stefan Haun
Keyword(s):  
Stokes Equations ◽  
Limit Equilibrium ◽  
Numerical Algorithms ◽  
High Viscosity ◽  
Navier Stokes ◽  
Bank Failures ◽  
Navier Stokes Equations ◽  
Order Of Magnitude ◽  
Sediment Layers ◽  
Hydropower Reservoir

Numerical algorithms are presented for modeling bank failures during reservoir flushing. The algorithms are based on geotechnical theory and the limit equilibrium approach to find the location and the depth of the slides. The actual movements of the slides are based on the solution of the Navier-Stokes equations for laminar flow with high viscosity. The models are implemented in the SSIIM computer program, which also can be used for modelling erosion of sediments from reservoirs. The bank failure algorithms are tested on the Bodendorf hydropower reservoir in Austria. Comparisons with measurements show that the resulting slides were in the same order of magnitude as the observed ones. However, some scatter on the locations were observed. The algorithms were stable for thick sediment layers, but instabilities were observed for thin sediment layers.

scholarly journals Event-Driven Molecular Dynamics Simulation of Hard-Sphere Gas Flows in Microchannels

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Volkan Ramazan Akkaya ◽  
Ilyas Kandemir
Keyword(s):  
Molecular Dynamics ◽  
Hard Sphere ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Flow Characteristics ◽  
Dynamics Simulation ◽  
Navier Stokes ◽  
Gas Flows ◽  
Linearized Boltzmann Equation ◽  
Event Driven

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.

An extended Navier-Stokes formulation for gas flows in the Knudsen layer near a wall

2007 ◽  
Vol 80 (2) ◽  
pp. 24001 ◽  
Author(s):  
Z. L. Guo ◽  
B. C. Shi ◽  
C. G. Zheng
Keyword(s):  
Knudsen Layer ◽  
Navier Stokes ◽  
Gas Flows

An upwind parabolized Navier-Stokes code for real gas flows

1988 ◽  
Author(s):  
JOHN TANNEHILL ◽  
JOHN IEVALTS ◽  
SCOTT LAWRENCE
Keyword(s):  
Navier Stokes ◽  
Gas Flows ◽  
Real Gas
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