Micro diffuser flow modeling for cold gas propulsion systems

The use of highly diluted and hypersonic gas flow is in the scope of application of cold gas thrusters for space applications. Satellites and small spacecrafts are navigated to their orbital trajectory with these nozzles. Inside these propulsion systems high density gradients are dominating the efficiency and the thrust steering behavior of the propulsion systems. Micro flows in the transient regime between free molecular flow and continuous flow are not able to be computed with trustworthy results by using a continuous model with no-slip boundary conditions. Therefore boundary slip-velocity models are used for modeling the reduced wall shear stress. Molecular shear stresses decrease the molecular mean velocity near the wall. With a Knudsen number depending slip-velocity model the effective shear stress is computed by the mean gradient of the velocity profile near the wall. In the present study a trans-sonic nozzle flow is computed by using a calibrated velocity slip model what depends on the Knudsen number. The Knudsen numbers are lower the Kn=1 at the nozzle neck of the propulsion system. The results are compared with simulation results of a uniform channel flow and computations of the corresponding no-slip approach. The differences in the hypersonic region are following discussed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the thermal-creep flow of a binary gas mixture over a plane wall with imperfect accommodation

The thermal-creep flow of a binary gas mixture over a plane wall is investigated analytically on the basis of the linearized Boltzmann equation of BGK type under the boundary condition of Maxwell's type or diffuse-specular reflection type. By an accurate analysis of the Knudsen layer formed near the wall, the Knudsen-layer structure of the velocity field has been clarified and, hence, the velocity distribution over the whole flow region is given explicitly together with the macroscopic slip coefficient. For future comparison with experimental data which may become available, the values of the slip coefficient of thermal-creep flow for several pairs of gases, Ne-Ar, He-Ne, He-Ar, N₂-Ar and N₂-O₂ are also given and listed in Table together with the values calculated based on the result given by other authors.     

Gas Near a Wall: Shortened Mean Free Path,Reduced Viscosity,and the Manifestation of the Knudsen Layer in the Navier–Stokes Solution of a Shear Flow

For the gas near a solid planar wall, we propose a scaling formula for the mean free path of a molecule as a function of the distance from the wall, under the assumption of a uniform distribution of the incident directions of the molecular free flight. We subsequently impose the same scaling onto the viscosity of the gas near the wall and compute the Navier–Stokes solution of the velocity of a shear flow parallel to the wall. Under the simplifying assumption of constant temperature of the gas, the velocity profile becomes an explicit nonlinear function of the distance from the wall and exhibits a Knudsen boundary layer near the wall. To verify the validity of the obtained formula, we perform the Direct Simulation Monte Carlo computations for the shear flow of argon and nitrogen at normal density and temperature. We find excellent agreement between our velocity approximation and the computed DSMC velocity profiles both within the Knudsen boundary layer and away from it.     

An asymptotic theory of the sound dispersion in a binary mixture of gases

The sound propagation in a binary mixture of multiatomic non-relaxing gases is considered using Burnett's equations (for the case of rapid exchanges of the internal and translational energies of the molecules). Asymptotic expressions (for small Knudsen numbers) are obtained for the absorption and dispersion coefficients, which are expressed in terms of the Navier–Stokes and the Navier–Stokes and the Burnett transport coefficients respectively. “Working” expressions of different levels of accuracy for these coefficients are known for the case of a binary mixture of monatomic gases. For this case, the results obtained are compared with known ones and the drawbacks and errors of previous papers are pointed out.     

Numerical study of the Couette flow of a diatomic rarefied gas

The Couette flow is numerically studied using a model kinetic equation for a diatomic rarefied gas (nitrogen). The boundary condition set on the wall takes into account that the molecular rotational energy passes into translational energy when the molecule interacts with the wall. For comparison purposes, the Couette flow is computed using the classical diffuse model of the gas-wall interaction. A comparison of the results obtained with both types of boundary conditions shows that the computed parameters of the Couette flow coincide only for sufficiently low Knudsen numbers. This suggests that transitions between rotational and translational energy in the gas-wall interaction have to be taken into account in the boundary condition.     

Computation of rarefied diatomic gas flows through a plane microchannel

A numerical method based on a model kinetic equation was developed for computing diatomic rarefied gas flows in two dimensions. Nitrogen flows through a plane microchannel were computed, and the gas flow rate was constructed as a function of the Knudsen number for various channel lengths.     

High-order accurate conservative method for computing the poiseuille rarefied gas flow in a channel of arbitrary cross section

A high-order accurate method is proposed for analyzing the isothermal rarefied gas flow in an infinitely long channel with an arbitrarily shaped cross section (Poiseuille flow). The basic idea behind the method is the use of hybrid unstructured meshes in physical space and the application of a conservative technique for computing the gas velocity. Examples of calculations are provided for channels of various cross sections in a wide range of Knudsen numbers. Schemes of the first-, second-, and third orders of accuracy in space are compared.     

Semiclassical lattice hydrodynamics of rarefied channel flows

The two-dimensional channel flows of gas of arbitrary statistics in the slip and transition regimes as characterized by the Knudsen number are studied using a newly developed semiclassical lattice Boltzmann method. The method is directly derived by projecting the Uehling-Uhlenbeck Boltzmann-BGK equations onto the tensor Hermite polynomials using moment expansion method. The intrinsic discrete nodes of the Gauss-Hermite quadrature provide the natural lattice velocities for the semiclassical lattice Boltzmann method. The mass flow rates and the velocity profiles are calculated for the three particle statistics over wide range of Knudsen numbers and the Knudsen minimum can be captured. The results indicate distinct characteristics of the effects of quantum statistics.     

Heat transfer in a gas mixture between parallel plates

The one-dimensional steady-state heat flux and the temperature distribution in a rarefied gas mixture between two parallel plates with different temperatures are studied using the kinetic theory. The Boltzmann equation is solved by the projection method assuming that the gas consists of elastic hard spheres and the reflection from the surfaces is diffuse. The flow features are analyzed for a wide range of the Knudsen number. The molecular numerical densities of the components, the total temperature of the mixture, and the mixture heat flux are obtained. The behavior of the distribution functions for the components is discussed. A comparison with other authors’ results shows that the accuracy of the given method is good.     

Thermodynamics / turbulence analogy modelling dissipating molecular gas flows for high Knudsen numbers

Modelling micro channel flows momentum and heat diffusion / convection are recent parameters modelling the molecule velocity distribution. Macroscopic models describe velocity and energy / enthalpie with integrals of mass increments. Using microscopic models motion and forces of a molecular flow have to be computed by models of physical properties, whose are described by statistical power moments of the molecule velocity. Therefore dilute flows have to be investigated in small channels with a mean free path length of molecules higher than the channel width of the the micro channel itself (λ₀≥H₀). Modelling this process by a continuous flow the boundary conditions have to be modified (e.g. [6]). The present model uses the statistical approximation of the molecule velocity distribution to simulate the behaviour of this discrete flow with a weighted averaged molecule velocity ∼ξ_i, its standard deviation σ and the characterisic molecule collision rate z. The number density N per volume V near one position is used for the weighting factor averaging method describing the mean molecule velocity. The present model is validated computing Poiseuille and Couette flows with different Knudsen numbers. Showing the advantages of the present model the simulation results are compared with simulation results of the wall–distance depending diffusivity model of Lockerby and Reese [4] and BGK results of a Lattice–Boltzmann simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of moment equations to the mathematical simulation of gas microflows

An approach to the simulation of moderately rarefied gas flows in a transition zone is developed. The applicability of the regularized Grad 13-moment (R13) equations to the numerical simulation of a transition flow between the continual and free-molecular gas flow regimes is explored. For the R13 equations, a numerical method is proposed that is a higher order accurate version of Godunov’s explicit method. A numerical procedure for implementing solid-wall boundary conditions is developed. One- and two-dimensional test problems are solved, including the shock wave structure and the Poiseuille flow in a plane channel. The possibility of applying the R13 equations to the simulation of plane channel and jet flows in a transition regime is explored. To this end, the flow in a square cavity generated by the motion of one of the walls is studied and the operation of the Knudsen pump is analyzed.     

The applicability of continuum models in the transitional regime of hypersonic flow over blunt bodies

Hypersonic rarefied gas flow over blunt bodies in the transitional flow regime (from continuum to free-molecule) is investigated. Asymptotically correct boundary conditions on the body surface are derived for the full and thin viscous shock layer models. The effect of taking into account the slip velocity and the temperature jump in the boundary condition along the surface on the extension of the limits of applicability of continuum models to high free-stream Knudsen numbers is investigated. Analytic relations are obtained, by an asymptotic method, for the heat transfer coefficient, the skin friction coefficient and the pressure as functions of the free-stream parameters and the geometry of the body in the flow field at low Reynolds number; the values of these coefficients approach their values in free-molecule flow (for unit accommodation coefficient) as the Reynolds number approaches zero. Numerical solutions of the thin viscous shock layer and full viscous shock layer equations, both with the no-slip boundary conditions and with boundary conditions taking into account the effects slip on the surface are obtained by the implicit finite-difference marching method of high accuracy of approximation. The asymptotic and numerical solutions are compared with the results of calculations by the Direct Simulation Monte Carlo method for flow over bodies of different shape and for the free-stream conditions corresponding to altitudes of 75–150 km of the trajectory of the Space Shuttle, and also with the known solutions for the free-molecule flow regine. The areas of applicability of the thin and full viscous shock layer models for calculating the pressure, skin friction and heat transfer on blunt bodies, in the hypersonic gas flow are estimated for various free-stream Knudsen numbers.     

Diffusion coefficients in a nonideal lattice gas with high concentration gradients

Expressions for the flow of migrating particles and for the diffusion coefficient in a nonideal lattice gas are derived from the equations of chemical kinetics. The diffusion coefficients depent not only on the concentration of the chemical components the gas, but also on their gradients and surface reactions. The diffusion coefficients are linear operators: the direction of flow may deviate from the gradient even for a one-component gas. These factors are shown to be essential at high gradients. Translated from Prikladnaya Matematika i Informatika, No. 1, pp. 51–57, 1999.     

Simulation of rarefied gas flows on the basis of the Boltzmann kinetic equation solved by applying a conservative projection method

Flows of a simple rarefied gas and gas mixtures are computed on the basis of the Boltzmann kinetic equation, which is solved by applying various versions of the conservative projection method, namely, a two-point method for a simple gas and gas mixtures with a small difference between the molecular masses and a multipoint method in the case of a large mass difference. Examples of steady and unsteady flows are computed in a wide range of Mach and Knudsen numbers.     

Properties of Solutions for the Problem of a Joint Slow Motion of a Liquid and a Binary Mixture in a Two-Dimensional Channel

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.     

Advanced continuum modelling of gas-particle flows beyond the hydrodynamic limit

The accurate prediction of dilute gas-particle flows using Euler–Euler models is challenging because particle–particle collisions are usually not dominant in such flows. In other words, in dilute flows the particle Knudsen number is not small enough to justify a Chapman–Enskog expansion about the collision-dominated near-equilibrium limit. Moreover, due to the fluid drag and inelastic collisions, the granular temperature in gas-particle flows is often small compared to the mean particle kinetic energy, implying that the particle-phase Mach number can be very large. In analogy to rarefied gas flows, it is thus not surprising that two-fluid models fail for gas-particle flows with moderate Knudsen and Mach numbers. In this work, a third-order quadrature-based moment method, valid for arbitrary Knudsen number, coupled with a fluid solver has been applied to simulate dilute gas-particle flow in a vertical channel with particle-phase volume fractions between 0.0001 and 0.01. In order to isolate the instabilities that arise due to fluid-particle coupling, a fluid mass flow rate that ensures that turbulence would not develop in a single phase flow (Re = 1380) is employed. Results are compared with the predictions of a two-fluid model with standard kinetic theory based closures for the particle phase. The effect of the particle-phase volume fraction on flow instabilities leading to particle segregation is investigated, and differences with respect to the two-fluid model predictions are examined. The influence of the discretization on the solution of both models is investigated using three different grid resolutions. Radial profiles of phase velocities and particle concentration are shown for the case with an average particle volume fraction of 0.01, showing the flow is in the core-annular regime.     

Kinetic analysis of the isothermal flow in a long rectangular microchannel

The linearized kinetic BGK model is used to study the steady Poiseuille flow of a rarefied gas in a long channel of rectangular cross section. The solution is constructed using the finite-volume method based on a TVD scheme. The basic computed characteristic is the mass flow rate through the channel. The effect of the relative width of the cross section is examined, and the difference of the solution from the one-dimensional flow between infinite parallel plates is analyzed. The numerical solution is compared to available results and to the analytical solution of the Navier-Stokes equations with no-slip and slip boundary conditions. The limits of applicability of the hydrodynamic solution are established depending on the degree of rarefaction of the flow and on the ratio of the side lengths of the channel cross section.     

Monte Carlo Microflow Simulation for the Gas Film Lubrication in Modern Magnetic Disk Storage

A two‐dimensional unsteady problem of gas flow in an extremely narrow channel with an inclined upper wall and moving lower one is studied by the DSMC method. This is a model of gas film lubrication which occurs in modern magnetic disk storage, that is now under development. Far from the magnetic head the flow produced by the disk motion could be described by solution of the Rayleigh problem. Space and time distributions of the pressure on the upper wall as well as density and average velocity inside and outside of the channel were obtained. They show that as a result of the flow slowing‐down by the front wall of the magnetic head the region with an increased density is formed there. At the same time marked non‐homogeneity of gas velocity before the inlet of the channel is observed.     

Particular solutions of the linearized Boltzmann equation for a binary mixture of rigid spheres

Particular solutions that correspond to inhomogeneous driving terms in the linearized Boltzmann equation for the case of a binary mixture of rigid spheres are reported. For flow problems (in a plane channel) driven by pressure, temperature, and density gradients, inhomogeneous terms appear in the Boltzmann equation, and it is for these inhomogeneous terms that the particular solutions are developed. The required solutions for temperature and density driven problems are expressed in terms of previously reported generalized (vector-valued) Chapman–Enskog functions. However, for the pressure-driven problem (Poiseuille flow) the required particular solution is expressed in terms of two generalized Burnett functions defined by linear integral equations in which the driving terms are given in terms of the Chapman–Enskog functions. To complete this work, expansions in terms of Hermite cubic splines and a collocation scheme are used to establish numerical solutions for the generalized (vector-valued) Burnett functions.