حجم و گسترش تجزیه و تحلیل نسبت از جریان گاز نازل میکرو

Size and expansion ratio effects on the flowfield are investigated for micro converging-diverging nozzles. Numerical computations are conducted by using two dimensional augmented Burnett equations and Navier-Stokes equations that were derived from the Boltzmann equation. The Maxwell-Smoluchowski slip boundary condition is used for adiabatic walls, and Steger-Warming flux vector splitting scheme is applied to the convective inviscid flux terms. The results from the augmented Burnett equation are compared with Navier-Stokes and Direct Simulation Monte Carlo (DSMC) results. Then, nozzle-size analysis is conducted for between 2 µm and 100 µm throat width. Influence of the Knudsen number is investigated, and temperature and Mach number variations are presented. In addition, the influence of the expansion ratio is studied with three (1.7:1, 3.4:1, and 6.8:1) different configurations. The results are compared with each other and an experimental data in the literature.

Recent developments in micro electrical mechanical systems (MEMS) have guided the construction of many small-sized devices. Aerospace industry has benefited from these devices particularly in development of new micro-size spacecraft and satellites. However, design and development of such products have brought new challenges especially related to the aero-propulsion area. Micro nozzle flow is one of the key topics in the aero-propulsion development that needs to be studied to design the next generation micro-size aerospace vehicles. The objective of the present study comes from the effort in simulating the flowfield and heat transfer characteristics in MEMS using computational fluid dynamics (CFD) techniques. The common CFD codes, which have been developed from Navier Stokes equation becomes especially inaccurate for microfluidic flows, because the local Knudsen number (Kn) lies beyond the continuum regime [1]. Similar challenges of such flowfield problems are also presented in the recent literature [2], [3], [4] and [5]. Typically, a flowfield with Kn ≤ 0.001 is called as the continuum regime. The continuum-transition regime or slip flow regime is 0.001 ≤ Kn ≤ 0.1, and the transition regime is between Knudsen number of 0.1 and 10. The flowfield with higher Knudsen number than 10 is generally identified as the free molecular flow. The gas flow in micro scale devices is usually categorized in the continuum-transition regime, which is neither completely in the continuum regime nor in the rarefied (free molecular flow) regime [6], and the flowfield in the nano scale devices is categorized in the transition regime [7]. Along with the direct simulation Monte Carlo (DSMC) approach, several extended numerical models have been recently introduced to model the flowfield in continuum-transition regime. Since statistical DSMC methods solve directly the Boltzmann equations with very high computational costs, more efficient numerical methods got additional consideration in the MEMS field. One of these methods is the Burnett equations that are obtained from Chapman-Enskog expansion of the Boltzmann equations with the parameter of Knudsen number [8]. There are various Burnett approximations in the literature such as conventional Burnett equations, augmented Burnett equations and BGK-Burnett equations [9] and [10]. Performance analysis for micro nozzle flow has become more important with the development of micro aero-propulsion systems. First experimental study on small-scale nozzle performance was performed by Rothe [11] with throat of 2.5–5 mm nozzle and a chamber pressure less than 1 atm, where the Reynolds number was in range between 55 and 550. The subsonic and supersonic parts of the Rothe's nozzle are cones with half angle of 30 and 20°, respectively, with longitudinal radius of curvature of the throat equal half of the throat radius. The test gas in experiment was nitrogen at the temperature of 300 K. Later several authors simulated the Rothe nozzle by using both the continuum based Navier Stokes equations [12] and [13] and the statistical based DSMC methods [14]. Rae [12] indicated that viscous boundary layer fills the nozzle for small divergence angle at low Reynolds numbers and there is a transition without shock. This is accomplished by a thermalization of the flow energy. Kim [13] developed a finite volume code using the complete forms of Navier Stokes equations. He stated the trade-off between viscous losses due to the merging boundary layers and the divergence losses due to the radial component of thrust nulling with its symmetric counterpart. However, these continuum based studies did not consider the slip wall velocity effects. An experimental micro nozzle analysis with throat of 650 µm was investigated by Grisnik et al. [15], and Zelesnik et al. [16] compared their results obtained from direct simulation Monte Carlo methods with Grisnik's experiment. They stated that boundary layer growth can result in a significant viscous dissipation and raise the temperature of the flow expander. Further investigations of the viscous effects in supersonic MEMS-fabricated micro nozzles with minimum throat widths averaging 19 µm and 35 µm have been performed by Bait and Breuer demonstrating the capability of fabricating and testing cold gas micro nozzle [17]. Later, Liou et al. [18] studied a rectangular cross-section convergent–divergent micro nozzle with throat width of 20 µm, depth of 120 µm and the expansion area ratio of 1.7:1, which is one of the ratios studied in the present study. However, Liou et al. performed the numerical section of the study using two dimensional compressible Navier Stokes equations. In the present study, the differences between Navier Stokes equations and augmented Burnett equations are analyzed, slip and no-slip boundary conditions are studied. Size and expansion ratio effects are investigated by computing both sets of equations in convergent divergent nozzles. In order to analyze the flow behavior for micro scale gas flows in wide range of nozzles several different geometries are selected. The throat width for the numerical simulations changes between 2 µm and 20 µm, and expansion ratio (ER = De:Dt) is between 1.7:1 and 6.8:1. Different outlet pressure boundary conditions are applied while the inlet pressure is kept at 100 kPa. Based on the case, the upper and lower boundaries are modeled as slip or non-slip walls.

A two-dimensional augmented Burnett equation solver is developed using Steger and Warming flux-vector splitting scheme. The first order Maxwell-Smoluchowski slip boundary condition is treated in order to obtain correct model for flow behavior on the wall surface. The solver is validated with experimental data for various Kn numbers. The grid dependency study is conducted to ensure computational accuracy and efficiency. The augmented Burnet equation results are compared with Navier-Stokes and Direct Simulation Monte Carlo (DSMC) solvers. The results show that there is very small difference in the slip and no slip models for low Knudsen number. The augmented Burnett and the Navier Stokes equations nearly overlap when the same type of boundary condition is used on the solid surface. Because of increasing rarefied effects, the differences between both sets of equations increase with the decrease in the geometry size. Using the second order constitutive relations with higher order terms for stability enhancement, the augmented Burnett equations extend the region to whole continuum-transition regime. The size analysis portion of the present work shows that the streamwise velocity decreases while the size of nozzle becomes smaller. In addition, the sonic region moves away from the throat to the outlet because of the higher viscous dissipation in micro nozzle. The expansion ratio analysis shows that the flow separation with high expansion ratio diminishes with the decrease in size due to the lower Reynolds number.