Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number

The Couette flow of binary gaseous mixtures is studied on the basis of the McCormack model of the Boltzmann equation, which was solved numerically by the discrete velocity method. The calculations were carried out for three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. The stress tensor and bulk velocity of both species were calculated for several values of the gas rarefaction in the range from 0.01 to 40 for three values of the molar concentrations: 0.1,0.5 and 0.9. The numerical solution together with an analytical solution based on the slip boundary condition cover the whole range of the gas rarefaction. It was showed that the Couette flow is weakly affected by the intermolecular interaction law.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described, e.g. by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to closure problems in for the obtention of effective properties; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in space with a finite- volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.     

Rarefied Pure Gas Transport in Non-isothermal Porous Media: Effective Transport Properties from Homogenization of the Kinetic Equation

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This article develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e.g., by a digitized 3D image. The homogenization procedure—neglecting the effect of gas density gradients on heat transfer through the solid—leads to macroscopic transfer relations, and to closure problems in for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor, which is distinct from the permeability and Knudsen diffusion tensors.     

Simultaneous entrainment of oil and water droplets in high Reynolds number gas turbulence in horizontal pipe flow

We develop a 1D cross sectional concentration profile model for oil and water droplets that coexist in the turbulent gas phase (of Re ∼ 10⁶) in near horizontal stratified pipe flows. Entrainment of the oil and water mixture from a liquid film near the bottom of the pipe into the gas is modeled based on earlier single-fluid entrainment correlations. A Gamma distribution for the droplet sizes based on the breakup of liquid filaments, is adopted. An explicit algebraic–exponential formula for the total concentration profile for either phase can then be derived.     

Unsteady pressure in the annular flow between two concentric cylinders,one of which is oscillating: Experiment and theory

The first objective of this paper is to present a series of accurate experimental measurements of the unsteady pressure in the annulus between two concentric cylinders, the outer one of which executes a harmonic planar motion, either transverse translational or rocking motion about a hinge, with and without annular flow. The second objective is the solution of the unsteady Navier–Stokes and continuity equations for the same annular geometry under the same boundary conditions for an incompressible fluid in the laminar regime. The solutions are obtained with a three-time-level implicit integration method in a fixed computational domain by assuming small amplitudes of oscillation of the outer cylinder. A pseudo-time integration method with artificial compressibility is used to advance the solution between consecutive real time levels. The finite difference method is used for spatial discretization on a stretched staggered grid. The problem is reduced to a scalar tridiagonal system, solved by a decoupling procedure which is based on a factored Alternating Direction Implicit (ADI) scheme with lagged nonlinearities. The third objective is the comparison of the experimental results with the theoretical ones. This comparison shows that the two are in good agreement in the case of translational motion, and in excellent agreement in the case of rocking motion. The experimental and theoretical work presented in this paper is useful for fluid–structure interaction and flow-induced vibration analyses in such geometries.     

Recent developments in the computation of compressible low Mach number flows

The computation of low speed flows can usualy be performed by incompressible models or various Low Mach number approximations (Boussinesq, Anelastic, etc). However, there is a large number of flows where although the velocities are small, compressible effects cannot be ignored and the use of full compressible models is required.Unfortunately, standard finite element or finite volume method experiences various difficulties in the computation of these low Mach number flows. This talk will explain the origin of these difficulties and describe some techniques to circumvent them.     

Investigation on nonlinear multi-scale effects of unsteady flow in hydraulic fractured horizontal shale gas wells

A unified mathematical model is established to simulate the nonlinear unsteady percolation of shale gas with the consideration of the nonlinear multi-scale effects such as slippage, diffusion, and desorption. The continuous inhomogeneous models of equivalent porosity and permeability are proposed for the whole shale gas reservoir including the hydraulic fracture, the micro-fracture, and the matrix regions. The corresponding semi-analytical method is developed by transforming the nonlinear partial differential governing equation into the integral equation and the numerical discretization. The non-linear multi-scale effects of slippage and diffusion and the pressure dependent effect of desorption on the shale gas production are investigated.     

Extension of the volume-of-fluid method for analysis of free surface viscous flow in an ideal gas

The volume-of-fluid (VOF) method is a simple and robust technique for simulating free surface flows with large deformations and intersecting free surfaces. Earlier implementations used Laplace's formula for the normal stress boundary condition at the interface between the liquid and vapour phases. We have expanded the interfacial boundary conditions to include the viscous component of the normal stress in the liquid phase and, in a limited manner, to allow the pressure in the vapour phase to vary. Included are sample computations that show the accuracy of added third-order-accurate differencing schemes for the convective terms in the Navier-Stokes equation (NSE), the viscous terms in the normal stress at the interface and the solution of potential flow in the vapour phase coupled with the solution of the NSE in the liquid phase. With these modifications we show that the VOF method can accurately predict the instability of a thin viscous sheet flowing through a stagnant vapour phase.     

Suspensions in turbulent liquid pipe flow: Kinetic modelling and added mass effects

We develop a Eulerian model for a particle suspension in fully developed turbulent liquid in a horizontal pipe. Virtual mass effects and hydrodynamic interactions are accounted for by an extension of the kinetic theory of Reeks and Swailes.     

瓦斯爆炸压力与波前瞬态流速演化特征及其定量关系

为了建立爆炸波前的瞬态流速和超压的定量关系,采用数值模拟的方法分别研究了开口型方管内瓦斯爆炸超压和瞬态流速传播特征。研究结果表明:开口型方管内,波前瞬态流速和超压的波形曲线的峰值个数不一样,而且超压峰值总是早于波前瞬态流速峰值出现。大部分情况下,方管横截面边长越大,其超压峰值相对较小,并且超压峰值沿传播方向呈现降低趋势。波前瞬态流速峰值沿传播方向呈不断增长趋势,而且方管横截面边长越大,其峰值也相对较小。长径比(方管长度与横截面边长之比)小于125时,超压峰值与波前瞬态流速峰值的定量关系始终呈现线性反比关系;大于或等于125时,超压峰值和波前瞬态流速峰值的定量关系呈分段关系。研究结果可为爆炸冲击波扬尘的研究提供基础数据。     

On the formulation of the problem of strip oscillations and stability in supersonic gas flow

The formulation of the strip flutter problem based on a refined expression for the excess pressure is proposed. Two cases are considered: almost transverse and almost longitudinal flow past the strip. In the first case, an exact expression for the excess pressure, fundamentally different from the formulas of piston theory, which leads to novel, little studied eigenvalue problems is obtained. In the second case, an exact expression is obtained for the flow potential and, for purely longitudinal flow, for the excess pressure also. It is shown that for purely longitudinal flow at high supersonic velocities the critical flutter velocity is equal to the phase velocity of perturbation propagation along the strip, which coincides with the results of piston theory.     

Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

Stokes eigenfunctions and Galerkin projection of the disturbance equations in plane Poiseuille flow: a systematic analytical approach

In the present paper, the L ²-normalized Stokes eigenfunctions for plane Poiseuille flow, which form an orthonormal functional basis for the space of disturbances, are written in a general exponential form. Then, the evolution equations for the disturbances are Galerkin-projected on the considered basis functions, and all the terms of the resulting dynamical system are expressed systematically in analytical form. Finally, a numerical example is given in which the proposed basis functions are used for the simulation of the time evolution of the critical disturbance predicted by the energetic stability theory.     

A first assessment of the NEPTUNE_CFD code: Instabilities in a stratified flow comparison between the VOF method and a two-field approach

This paper presents some results concerning a first benchmark for the new European research code for thermal hydraulics computations: NEPTUNE_CFD. This benchmark relies on the Thorpe experiment to model the occurrence of instabilities in a stratified two-phase flow. The first part of this work is to create a numerical trial case with the VOF approach. The results, in terms of time of onset of the instability, critical wave-number or wave phase speed, are rather good compared to linear inviscid theory and experimental data. Additional numerical tests showed the effect of the surface tension and density ratio on the growing dynamics of the instability and the structure of the waves. In the second part, a code to code (VOF/multi-field) comparison is performed for a case with zero surface tension. The results showed some discrepancies in terms of wave amplitudes, growing rates and a time shifting in the global dynamics. Afterward, two surface tension formulations are proposed in the multi-field approach. Both formulations provided similar results. The time for onset of the instability, the most amplified wave-number and its amplitude were in rather good agreement with the linear analysis and VOF results. However, the time-shifted dynamics was still observed.     

Nonmonotonic relationship between the Reynolds number and the lift coefficient of a plate in a hypersonic rarefied gas flow

The aerodynamic coefficients of a plate in a hypersonic stream at low Reynolds numbers are studied over a wide range of similarity parameters. The dependence of the lift coefficientC _Y on the tangential force coefficient, the finite Mach number at the outer edge of the boundary layer and the velocity-slip and temperature-jump boundary conditions is taken into consideration. The nonmonotonic character of the relationship betweenC _Y and the Reynolds number, revealed previously in experiments, is explained within the framework of the viscous hypersonic interaction model.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 186–189, January–February, 1996.     

An implementation of the Spalart–Allmaras DES model in an implicit unstructured hybrid finite volume/element solver for incompressible turbulent flow

In this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible Navier–Stokes solver for turbulent flows based on the Spalart–Allmaras detached eddy simulation (SA‐DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered‐mesh scheme. The SA‐DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix‐free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA‐DES model in a hybrid flow solver. This paper will address important implementation issues for high‐Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA‐DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high‐Reynolds number flow around a high angle‐of‐attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation. Copyright © 2008 John Wiley & Sons, Ltd.     

On the role of droplets in cocurrent annular and churn-annular pipe flow

Decreasing the gas flow-rate in an initially vertical upward annular dispersed pipe-flow, will eventually lead to a down-flow of liquid. The onset of this down-flow has been related in the literature to the presence of the dispersed phase and the instability of the liquid film. Here we investigate how the dispersed-phase may influence the down-flow, performing detailed PDA-measurements in a 5 cm vertical air–water annular-flow. It is shown that the dispersed-phase does not cause the liquid down-flow, but that it delays the onset of liquid down-flow. In cocurrent annular flow the dispersed phase seems to stabilise the film flow, whereas in churn-annular flow the opposite seems to be true.     

Effects of aspect ratio and orientation on the wake characteristics of low Reynolds number flow over a triangular prism

The wake characteristics of unconfined flows over triangular prisms of different aspect ratios have been numerically analysed in the present work. For this purpose, a fixed Cartesian-grid based numerical technique that involves the porous medium approach to mimic the effect of solid blockage has been utilised. Correspondingly, laminar flow simulations ranging from the sub-critical regime (before the onset of vortex shedding) to the super-critical regime have been considered here within the limits of two-dimensionality. In the sub-critical regime, correlations relating the wake bubble length with Reynolds number (Re) have been proposed for various aspect ratios. Also, the effects of aspect ratio and Reynolds Number on the drag force coefficient (C_D) have been characterised for two different geometrical orientations of the prism (base or apex facing the flow). Subsequently, the critical Reynolds number at the onset of vortex shedding has been predicted for each of the aspect ratio considered, by an extrapolation procedure. The unsteady flow characteristics of the super-critical regime are finally highlighted for different aspect ratios and triangular orientations considered in the study.     

Gas-dynamic approach in the nonlinear theory of ion acoustic waves in a plasma: An exact solution

An exact solution of the problem of the acoustic wave structure in a plasma is obtained. Both plasma component are treated as gases with specified initial temperatures and adiabatic exponents. The system of equations describing the wave profile is solved using an original method consisting of reducing the system to the Bernoulli equation. A numerical example of the obtained general solution of the problem of the wave profile for arbitrary parameters is given. Curves are constructed that bound the region of existence of a stationary solitary ion acoustic wave in the parameter space. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 3–11, September–October, 2007.