Flows of rarefied gaseous mixtures in networks of long channels

A method is presented to calculate the steady flows of rarefied gaseous mixtures in networks of long channels. The approach is based on the kinetic level. First, the McCormack linearized kinetic model is solved to obtain the local flow properties in the channels in a wide range of gaseous rarefaction and mole fraction. Second, the global flow properties including the flow rates and the distribution of the pressure and the mole fraction are deduced. An integral equation is introduced in order to determine the flow rates as functions of the differences of the partial pressures between the two ends of each channel. The conservation of mass at the nodes of the network results into a system of linear algebraic equations. The overall mathematical problem is solved iteratively. Pressure driven flows of He/Xe and He/Ar through an example network of circular tubes are calculated at intermediate values of the gaseous rarefaction. The results of the flow rates and the pressures and the mole fractions at the nodes in the whole system and the representative distributions of the pressure and the mole fraction along the channels are presented and commented on.     

Gas kinetic algorithm using Boltzmann model equation

Based on the Boltzmann-BGK model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented by the aid of the basic characteristic on molecular movement and collision approaching to equilibrium. The optimum Golden Section principle is extended and applied to the discrete velocity ordinate method in order to discretize the corresponding velocity components, and then the molecular velocity distribution function equation will be cast into hyperbolic conservation laws form with non-linear source terms. In view of the unsteady characteristic of molecular convective movement and colliding relaxation, the time-splitting method is applied to decompose the velocity distribution function equations into the colliding relaxation equations with non-linear source terms and the convective motion equations. Based on the second-order Runge-Kutta method and the non-oscillatory, containing no free parameters, and dissipative (NND) finite difference method, the gas kinetic finite difference second-order scheme is constructed to directly solve the discrete velocity distribution functions. Four types of discrete velocity quadrature rules, such as the modified Gauss-Hermite formula and the Golden Section number-theoretic integral method, are developed and applied to evaluate the macroscopic flow moments of the distribution functions over the velocity space. As a result, a unified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime. To test the reliability of the present method, the one-dimensional shock wave structures, the flows past two-dimensional circular cylinder and the three-dimensional flows over sphere with various Knudsen numbers are simulated. The computational results are found in high resolution of the flow fields and good agreement with the theoretical, DSMC, and experimental results.     

Comparative study between computational and experimental results for binary rarefied gas flows through long microchannels

A comparative study between computational and experimental results for pressure-driven binary gas flows through long microchannels is performed. The theoretical formulation is based on the McCormack kinetic model and the computational results are valid in the whole range of the Knudsen number. Diffusion effects are taken into consideration. The experimental work is based on the Constant Volume Method, and the results are in the slip and transition regime. Using both approaches, the molar flow rates of the He–Ar gas mixture flowing through a rectangular microchannel are estimated for a wide range of pressure drops between the upstream and downstream reservoirs and several mixture concentrations varying from pure He to pure Ar. In all cases, a very good agreement is found, within the margins of the introduced modeling and measurement uncertainties. In addition, computational results for the pressure and concentration distributions along the channel are provided. As far as the authors are aware of, this is the first detailed and complete comparative study between theory and experiment for gaseous flows through long microchannels in the case of binary mixtures.     

Gas-kinetic numerical study of complex flow problems covering various flow regimes

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the Boltzmann–Shakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation.     

A novel experimental setup for gas microflows

A new experimental setup for flow rate measurement of gases through microsystems is presented. Its principle is based on two complementary techniques, called droplet tracking method and constant-volume method. Experimental data on helium and argon isothermal flows through rectangular microchannels are presented and compared with computational results based on a continuum model with second-order boundary conditions and on the linearized kinetic BGK equation. A very good agreement is found between theory and experiment for both gases, assuming purely diffuse accommodation at the walls. Also, some experimental data for a binary mixture of monatomic gases are presented and compared with kinetic theory based on the McCormack model.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

Transitional cylindrical swirling flow in presence of a flat free surface

This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type—no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier-Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.     

Numerical modeling of rarefied gas flow through a slit into vacuum based on the kinetic equation

A rarefied gas flow through a thin slit into vacuum is calculated on the basis of the kinetic model equations applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature along the symmetry axis are reported. A comparison with the corresponding results obtained previously by the direct simulation Monte Carlo method is performed. A good agreement between these results shows a reliability of the model equations which require less computational effort than the Monte Carlo method.     

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads to a large reduction of the computational scale. Under some realistic assumptions, the unconditional stability of the fully discrete scheme is proved. Moreover, the optimal error estimates are obtained, which are better than the existing results. Finally, some numerical results are given to verify the theoretical analysis. The difference between the motion of the Newton and non-Newton fluid is also observed.     

Numerical simulation for gas microflows using Boltzmann equation

A computational method based on a kinetic model Boltzmann equation has been developed for microscale low speed flows. The results obtained with the method are compared with those of the direct simulation Monte Carlo method and experiments for supersonic flows. Numerical results for low speed flows over a microcircular cylinder and a microsphere are also obtained with the method, while it is difficult to obtain the low speed flow results with the direct simulation Monte Carlo method. Results of the Navier–Stokes equations with slip boundary conditions generally agree with those of the kinetic model Boltzmann equation if the Knudsen number is less than 0.1. A kinetic/continuum hybrid method has also been developed. The hybrid method may be a promising tool for analyzing whole flow regimes from free molecule to continuum flows.     

A RANS flamelet-progress-variable method for computing reacting flows of real-gas mixtures

This paper provides an efficient numerical method for solving reacting flows of industrial interest in the presence of significant real-gas effects. The method combines a state-of-the-art solver of the Reynolds-averaged Navier-Stokes equations - equipped with the low-Reynolds number k-ω turbulence closure - with a combustion flamelet-progress-variable approach. A real-gas model as well as a detailed kinetic scheme are used to generate the flamelet library. The method is tested versus several applications chosen to demonstrate the importance of the real-gas effects and of the kinetic scheme for computing high-pressure combustion. The major contribution of the paper is to provide a single-phase approach which solves turbulent reacting real-gas flows at a computational cost comparable with that of the simulation of a non-reacting flow thanks to the use of the flamelet library.     

A fine grid finite element computation of two-dimensional high Reynolds number flows

A velocity—pressure integrated, mixed interpolation, Galerkin finite element computation of the Navier-Stokes equations using fine grids, is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions: and the pressure was interpolated using linear shape functions defined on a triangular element, which is contained inside the quadratic element for velocity variables. Comprehensive computational results for a cavity flow for Reynolds number of 400 through 10,000 and a laminar backward-facing step flow for Reynolds number of 100 through 900 are presented in this paper. Many high Reynolds number flows involve convection dominated motion as well as diffusion dominated motion (such as the fluid motion inside the subtle pressure driven recirculation zones where the local Reynolds number may become vanishingly small) in the flow domain. The computational results for both of the fluid motions compared favorably with the high accuracy finite difference computational results and/or experimental data available.     

A Spectral Element Projection Scheme for Incompressible Flow with Application to the Unsteady Axisymmetric Stokes Problem

This paper presents a modified Goda scheme in the simulation of unsteady incompressible Navier–Stokes flows in cylindrical geometries. The study is restricted to the case of axisymmetric flows. For the justification of the robustness of our scheme some computational test cases are investigated. It turns out that by adopting the new approach, a significant accuracy improvement on both pressure and velocity can be obtained relative to the classical Goda scheme.     

Boundary domain integral method for high Reynolds viscous fluid flows in complex planar geometries

The article presents new developments in boundary domain integral method (BDIM) for computation of viscous fluid flows, governed by the Navier–Stokes equations. The BDIM algorithm uses velocity–vorticity formulation and is based on Poisson velocity equation for flow kinematics. This results in accurate determination of boundary vorticity values, a crucial step in constructing an accurate numerical algorithm for computation of flows in complex geometries, i.e. geometries with sharp corners. The domain velocity computations are done by the segmentation technique using large segments. After solving the kinematics equation the vorticity transport equation is solved using macro-element approach. This enables the use of macro-element based diffusion–convection fundamental solution, a key factor in assuring accuracy of computations for high Reynolds value laminar flows. The versatility and accuracy of the proposed numerical algorithm is shown for several test problems, including the standard driven cavity together with the driven cavity flow in an L shaped cavity and flow in a Z shaped channel. The values of Reynolds number reach 10,000 for driven cavity and 7500 for L shaped driven cavity, whereas the Z shaped channel flow is computed up to Re = 400. The comparison of computational results shows that the developed algorithm is capable of accurate resolution of flow fields in complex geometries.     

Non-Newtonian behaviour of the gyroid mesophase: A lattice-Boltzmann study

We use a lattice-Boltzmann (LB) kinetic scheme for modelling amphiphilic mixtures to impose Couette flow on self-assembled gyroid mesophase. We report evidence of non-Newtonian behaviour (“shear thinning”) and we show that the particular lack of inversion symmetry which characterizes the gyroid unit cell does not quantitatively affect the flow dynamics beyond a certain predictable shear velocity threshold. This purely kinetic approach is of general importance for the modelling and simulation of complex fluid flows in situations when rheological properties cannot be predicted a priori.     

An upwind compact difference scheme for solving the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equation

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both $\left(n\right)$th and $(n+1)$th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

Multi-block lattice Boltzmann simulations of subcritical flow in open channel junctions

A two-dimensional lattice Boltzmann model (LBM) for subcritical flows in open channel junctions is developed. Shallow water equations coupled with the large eddy simulation model is numerically simulated by the lattice Boltzmann method, so that the turbulence, caused by the combination of the main channel and tributary flows, can be taken into account and modeled efficiently. In order to obtain more detailed and accurate results, a multi-block lattice scheme is designed and applied at the area of combining flows. The model is first verified by experimental data for a 90° junction flow, then is used to investigate the effect of the junction angle on flow characteristics, such as velocity field, water depth and separation zone. The objectives of this study are to validate the two-dimensional LBM in junction flow simulation and compare the results with available experimental data and classical analytical solutions in the separation zone.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

Electroosmotic flow control in complex microgeometries

Numerical simulation results for pure electroosmotic and combined electroosmotic/pressure driven Stokes flows are presented in the cross-flow and Y-split junctions. The numerical algorithm is based on a mixed structured/unstructured spectral element formulation, which results in high-order accurate resolution of thin electric double layers with discretization flexibility for complex engineering geometries. The results for pure electroosmotic flows in cross-flow junctions under multiple electric fields show similarities between the electric and velocity fields. The combined electroosmotic/pressure driven flows are also simulated by regulating the flowrate in different branches of the cross-flow junctions. Flow control in the Stokes flow regime is shown to have linear dependence on the magnitude of the externally applied electric field, both for pure electroosmotic and combined flows. This linear behavior enables utilization of electroosmotic forces as nonmechanical means of flow control for microfluidic applications