Background

Fluid dynamics is one of fields in which computer simulation shows a great deal of promise, and interest in the development of better algorithms is strong. In a broad sense, there are two methods by which fluid flows are simulated: continuum methods and particle method. The former relies on the numerical solution of a set of partial differential equations, for example Navier-Stokes equations, describing the fluid flow, with proper boundary conditions applied. On the other hand, particle method is the other way to attempts to model a fluid flow by simulating the interactions of particles themselves and the interactions between particles and body boundaries. Despite of its requirement of tremendous computational effort, particle-based method can handle gas flows involving rarefaction and non-equilibrium phenomena, while the generally more efficient continuum method like Navier-Stokes solver becomes invalid. The understanding of rarefied gas dynamics (high-Knudsen number flows) plays an important role in several research disciplines, e.g.

space flight research and semiconductor processing, and many related important flow problems of interest often are involving both of continuum and rarefied regions.

Several important flow problems often involve continuum and rarefied regions in the

flow fields. Examples include, but are not limited to, expanding reaction control system plumes [Taniguchi et al., 2004 and Ivanov et al., 2004], expanding plumes from a flying projectile at high altitude [Wilmoth et al., 2004], spiral-grooved turbo booster pump with high compression ratio [Cheng et al., 1999] and jet-type (Chemical Vapor Deposition) [Versteeg et

al., 1994] and will be introduced in the following. Fig. 1.1 is the sketch of expanding

reaction control system plumes. When spacecraft or satellite flies through an altitude of near vacuum environment, the RCS thrusts are used to provide the power and control the spacecraft or satellite. In this flow field, the flow is continuum flow from the throat and becomes transition flow and rarefied as the flow pass through the nozzle to the space.

Usually, the thrusters are made up by a lot of small nozzles. Exhaust jets issuing from thrusts of spacecraft produce a huge jet plume. Various jet plumes cause many types of plume impingement on the associated or neighboring surfaces. When any part of surface suffers impingement of the plume, undesirable effects such as contamination, heating, disturbance torque and erosion will damage to the safety of spacemen and cause an anomalous behavior of the instruments. Thus, simulation of the plume impingement is very important.

The second example is expanding plumes from a flying projectile at high altitude shown in Fig. 1.2. There are many complicating phenomena in the upper continuum boost phase regime between 40 and 70 km the including turbulent transition onset, body flow separation,

body heating, angle of attack effects, thermal non-equilibrium, dispersion of particulates and flow interactions from complex flying object geometries. The more important thing is plumes issuing from those flying objects usually are high temperature gases, thus, reversed flow with radiation effect may damage the tail of those high speed objects at high altitude. Accurate predictive tools for the modeling of flying object exhausting plume flows are critical to the development of high speed flying object related technologies.

Fig. 1.3 shows a sketch of spiral-grooved turbo booster pump (TBP). Spiral-grooved turbo booster pump has both volume type and momentum transfer type vacuum pump functions, and is capable of operating at optimum discharge under pressures from approximately 1000 Pa to a high vacuum. Pumping performance is usually predicted and examined via traditional CFD methods. However, such the numerical tools are unsuitable for calculating such rarefied gas region in the highly vacuum chamber. Meanwhile the computational cost by using a DSMC method in the operation condition with high foreline pressure (1000 Pa) will be extremely high. Only a hybrid particle-continuum method could meet both of the above issues with computational efficiency and physical accuracy in both of the continuum and rarefied gas region.

Configuration of a jet-type CVD reactor is shown in Fig. 1.4. Pulsed Pressure CVD (PP-CVD) is one of jet-type CVD technique that has demonstrated improved performance

over traditional CVD technologies and the precursor is delivered in timed pulses into a continuously evacuated reactor volume. Experimental studies and a phenomenological model of PP-CVD have shown that during the deposition of titania, the conversion efficiency of the TTIP precursor into solid film with highly uniform thicknesses exceeds 90% under certain operational conditions. The unsteady under-expanded jet which forms during the injection phase contains high property gradients in the shock structure. Consequently during a pulse cycle the flow contains regions in the continuum, transition and rarefied regimes.

This makes modelling the process extremely challenging, since the validity of continuum techniques is questionable and particle based techniques such as DSMC are extremely computationally expensive. In order to continue to develop this promising technology, a greater understanding of the flow dynamics of the unsteady pulsed pressure regime in PP-CVD is required.

These above problems can not solved only by either particle method or continuum method. Thus, it is necessary to develop a proper simulation tool or practical strategy while considering both of solution accuracy and computational efficiency while involving continuum and rarefied regions.

1.1.1 Classification of Gas flows

Knudsen number (Kn=λ/L) is usually used to denote the degree of rarefaction where λ is

the mean free path traveled by molecules before collision and L is the flow characteristic length. Flows are divided into four regimes as follows in general: Kn <0.01 (continuum), 0.01<Kn<0.1 (slip flow), 0.1<Kn<3 (transitional flow) and Kn>3 (free molecular flow).

Figure 1.5 shows the various flow regions based on Knudsen number and their corresponding solution methods in a dilute gas. The local Kn number is defined with L as the scale length of the macroscopic gradient; that is,

dx L d

ρ

= ρ . As shown in the lower bar, when the local

Kn number approaches zero, the flow reaches inviscid limit and can be solved by Euler equation. Molecular nature of the gas becomes dominated in the flow of interest with the increase of local Kn increases. When the Kn larger than 0.1, continuum assumption begins to break down and the particle-based method is necessary. That’s why that the Navier-Stoke equation based computational fluid dynamics (CFD) techniques will not be adopted while the Kn are greater that 0.1. The top bar in this figure also shows the validity of the molecular modeling in the microscopic formulation. It indicates the Boltzmann equation is valid for all flow regimes. It is well known that Boltzmann equation is more appropriate for all flow regimes; it is, however, rarely used to numerically solve the practical problems because of two major difficulties: 1) Higher dimensionality (up to seven) of the Boltzmann equation and 2) difficulties of modeling the integral collision term.

Direct Simulation Monte Carlo (DSMC), proposed by Bird, is an alternative method to solve the Boltzmann equation using direct simulation of particle collision kinetics, and the

associated monograph was published in [Bird, 1976] and [Bird, 1994]. It is demonstrated that the DSMC method is equivalent to solving the Boltzmann equation as the simulated particle numbers become large by both Nanbu [1986] and Wagner [1992]. This method has been widely used for gas flow simulations in which molecular effects become important.

The advantage of using particle method under these circumstances is that molecular model can be implemented directly to the calculation of particle collisions without the macroscopic continuum assumption. Most importantly, DSMC is the only practical way to deal with flows in the transitional regime, without resorting to the difficult Boltzmann equation, which requires modeling an integral-differential (collision) term.

1.1.2 Challenge to Particle-Continuum Flow Simulation

For the realistic flows of interest having continuum and continuum breakdown regions, the direct simulation Monte Carlo (DSMC) method can provide more physically accurate results in flows having rarefied and non-equilibrium regions than continuum flow models.

However, the DSMC method is extremely computational expensive especially in the near-continuum region, which prohibits its applications to practical problems with complex geometries and large domains. In contrast, the computational fluid dynamics (CFD) method, employed to solve the Navier-Stokes (NS) or Euler equation for continuum flows, is computationally efficient in simulating a wide variety of flow problems. But the use of continuum theories for the flow problems involving the rarefied gas or very small length

scales (equivalently large Knudsen numbers) can produce inaccurate results due to the breakdown of continuum assumption or thermal equilibrium. A practical approach for solving the flow fields having near-continuum to rarefied gas is to develop a numerical model combining the CFD method for the continuum regime with the DSMC method for the rarefied and thermal non-equilibrium regime. A well-designed hybrid scheme is expected to take advantage of both the computational efficiency and accuracy of the NS solver in the continuum regime and the physical accuracy of the DSMC method in the rarefied or thermal non-equilibrium regime.