Numerical Solution of the Taylor-Quette Flow Problem: A Commodious Statistical Approach

The Boltzmann equation is the governing equation for rarefied gas dynamics, which is relevant to the aerodynamic design of rarefied hypersonic vehicles. Direct solution of the Boltzmann equation for realistic gas flows using either analytic or computational techniques is almost impossible. However, the Direct Simulation Monte Carlo (DSMC) method has been developed to provide realistic solutions for rarefied gas flows. Bird formally introduced this method hi 1969, and ever since several arguments have developed as to whether the DSMC method provides solutions of the Boltzmann equation. This paper demonstrates that the Boltzmann equation and the DSMC method have a common origin which rests on the fundamental tenets of kinetic theory. These fundamental tenets are mathematically condensed into the Boltzmann equation. On the other hand, the DSMC method is a numerical implementation of these same fundamental tenets. Thus, the Boltzmann equation and the DSMC method express the same idea but use a different language. The Boltzmann equation uses the language of mathematics whereas the DSMC method uses the language of physics. It is explained how both approaches are related and it is concluded that the DSMC method does indeed provide solutions of the Boltzmann equation. It is also shown that both approaches can be easily extended to a wider range of problems than the literature has recognised (such as to high-density gases and liquids). One important issue is how to set the range of the molecular potential used in deflection angle formulae. It is shown that, contrary to traditional expectations, within a macroscopic volume the range of the potential is always finite. Finally, the thorny issue of molecular chaos is examined and it is concluded that it is not as limiting as commonly feared.

Results of numerical simulations of high-altitude aerothermodynamics of the EXPERT reentry capsule along its descent trajectory are presented. Aerodynamic characteristics for different angles of attack and rolling of the capsule at altitude of 150 down to 20 km are studied. An engineering local bridging method is used in computations. The uncertainty of the engineering method in the transitional regime is determined by comparisons with results obtained by DSMC simulations. I. Introduction During disorbiting, the spacecraft experiences the influence of the atmosphere with significantly varying parameters. To estimate the aerodynamic loads and to predict the landing area, one should know, already at the stage of design, the coefficients of aerodynamic forces and moments of the spacecraft for varying temperature, density, flight velocity, and angles of attack and sideslip for numerous possible descent trajectories. Thousands of various variants have to be computed. At the beginning of descent, at high altitudes where the gas is strongly rarefied, numerical simulations do not involve many difficulties, because theoretical approaches have been developed for simple shapes and the Test Particle Monte Carlo method can be readily used for more complicated shapes. On the other hand, various engineering approaches have been developed for calculating aerodynamic characteristics of various bodies at the final segment of the descent trajectory, in the continuum regime, which are fairly effective. The main problem is the analysis of spacecraft aerodynamics in the transitional regimes between the free-molecular and continuum regimes. The Navier-Stokes equations yield, strictly speaking, incorrect results in the transitional regime and require special modifications for taking into account flow slipping. The Direct Simulation Monte Carlo (DSMC) method provides rather accurate values of aerodynamic characteristics with allowance for physical and chemical processes but requires large amounts of computer memory and performance. Application of the software based on the DSMC method is unreasonably expensive at the initial stage of spacecraft design and trajectory analysis. A possible solution of this problem is the approximate engineering methods. These methods allow calculating aerodynamic characteristics of bodies of arbitrary geometry for multiple variants of free-stream parameters within a reasonable time and then to refine the results at the most important segments of the flight trajectory by the DSMC method. The use of fast approximate engineering methods is the most acceptable approach to solving problems of spacecraft aerodynamics at the stage of conceptual design. The present paper describes the computations of aerodynamics of the EXPERT capsule by an engineering local bridging method at altitudes of 150 down to 20 km with flow parameters corresponding to the expected descent trajectory. The results in the transitional regime are refined by the DSMC method. The errors of the engineering method are studied in detail, and the contribution of various elements of the EXPERT capsule to aerodynamic characteristics is analyzed. II. EXPERT geometry The KHEOPS configuration (Fig. 1) is a blunted pyramidal shape featuring four flaps. The computational model was constructed by the RuSat software system. The total number of panels is approximately 36 thousand. The nose is made in the form of an ellipse with a misalignment of 0.9165. The local radius of the nose part is 0.6 m. The ellipse-cone junction is described by a clothoid. The conical body has a cone angle of 12.5°. The cone is truncated by planes at an angle of 8.35° to the axis of symmetry. Four control flaps are deflected by 20 °.

Adaptive domain decomposition for unstructured meshes applied to the direct simulation monte carlo method

The direct simulation Monte Carlo (DSMC) method is used to solve the three-dimensional nano-scale gas film lubrication problem between a gas bearing slider and a rotating disk, and this solution is compared to the numerical solution of the compressible Reynolds equations with the slip flow correction based on the linearized Boltzmann equation as presented by Fukui and Kaneko [molecular gas film lubrication (MGL) method] [ASME J. Tribol. 110, 253 (1988)]. In the DSMC method, hundreds of thousands of simulated particles are used and their three velocity components and three spatial coordinates are calculated and recorded by using a hard-sphere collision model. Two-dimensional pressure profiles are obtained across the film thickness direction. The results obtained from the two methods agree well with each other for Knudsen numbers as large as 35 which corresponds to a minimum spacing of 2 nm. The result for contact slider is also obtained by the DSMC simulation and presented in this paper.

This paper describes the basic study of the simulation of chemical vapor deposition (CVD) step coverage that is used to estimate the deposition of thin films in LSI production. Given the trend toward extremely high density in the semiconductor manufacturing process, such as that of LSIs, the deposition process of silicon thin film on the circuit pattern needs to be accurate. To optimize the deposition of thin film around the trench of the order of less than a micron, computer simulations of the step coverage and of the film growth rate in the process are very important. In this simulation reported in this paper, the thin film deposition process, whose reaction is modeled as silicon from silane (SiH4) and silylene (SiH2), is solved using the direct simulation Monte Carlo (DSMC) method. The DSMC method is usually adopted in this case, but, since 1) this method solves the Boltzmann equation with a huge amount of molecular collisions and free motions and 2) sticking probability of molecules is of the order of 10-5 [1], even a high performance computer needs much time for this calculation. Although the performance of supercomputers and other types of computers is improving, the DSMC method as conventionally applied seems to be reaching its limit in terms of practicality.

In every system, if Knudsen number is larger than 0.1, the Navier-Stokes equation will not be satisfied for investigation of flow patterns. In this condition, the Boltzmann equation, presented by Ludwig Boltzmann in 1872, can be useful. The conditions that this equation can be used were investigted by Cercignani in 1969. The most successful method for solving Boltzmann equation for a rarefied gas system is Direct Simulation Monte Carlo (DSMC) method. This method was suggested by Bird in 1974. The cylindrical Couette flow and occurrence of secondry flow (Taylor vortex flow) in a annular domin of two coaxial rotating cylinders is a classical problem in fluid mechanics. Because this type of gas flow can occur in many industrical types of equipment used in chemical industries, Chemical engineers are interested in this problem. In 2000, De and Marino studied the effect of Knudsen number on flow patterns and in 2006 the effect of temperature gradient between two cylinders was investigated by Yoshio and his co-workers. The aim of the present paper is investigation of understanding of the effect of different conditions of rotation of the cylinders on the vortex flow and flow patterns.

Parallel simulation on rayleigh-bénard convection in 2D by the direct simulation monte carlo method

The implementation of an adaptive mesh‐embedding (h‐refinement) scheme using unstructured grid in two‐dimensional direct simulation Monte Carlo (DSMC) method is reported. In this technique, local isotropic refinement is used to introduce new mesh where the local cell Knudsen number is less than some preset value. This simple scheme, however, has several severe consequences affecting the performance of the DSMC method. Thus, we have applied a technique to remove the hanging node, by introducing the an‐isotropic refinement in the interfacial cells between refined and non‐refined cells. Not only does this remedy increase a negligible amount of work, but it also removes all the difficulties presented in the originals scheme. We have tested the proposed scheme for argon gas in a high‐speed driven cavity flow. The results show an improved flow resolution as compared with that of un‐adaptive mesh. Finally, we have used triangular adaptive mesh to compute a near‐continuum gas flow, a hypersonic flow over a cylinder. The results show fairly good agreement with previous studies. In summary, the proposed simple mesh adaptation is very useful in computing rarefied gas flows, which involve both complicated geometry and highly non‐uniform density variations throughout the flow field. Copyright © 2002 John Wiley & Sons, Ltd.

A Turbomolecular pump (TMP) is one of the key apparatus to produce high and ultrahigh vacuum. It works mainly in the conditions of free molecular and transitional regimes, where the Navier-Stokes equations of continuum gas dynamics can not be correctly applied. In this study, the flow field in single blade row of one stage TMP is investigated by direct simulation Monte Carlo (DSMC) method with a 3D analysis in a rotating reference frame. Considering the Coriolis and centrifugal accelerations, the equations about the molecular velocities and position are deduced on this frame. The VSS model and NTC collision schemes are used to calculate the intermolecular collisions. The diffuse reflection is employed on the molecular reflection from the surfaces of boundary. The transmission probabilities are calculated and applied to analyze the relationship between the outlet pressure and the maximum pressure ratio. The pumping performances between H2 and N2 on the same blade speed and same blade speed ratio are compared and analyzed carefully. The maximum pumping efficiencies on the different blade angles are also calculated. Numerical results show good quantitative agreement with existing experiment data.

The variable sphere (VS) molecular model for the inverse power law (IPL) potential is combined with the statistical inelastic cross section (SICS) and Borgnakke-Larsen (BL) models and the transport coefficients are estimated using the Monte Carlo integration technique for Wang-Chang-Uhlenbeck (WCU) theory. The VS model with SICS (VS-SICS) and BL (VS-BL) models are applied to the simulation of the normal shock wave structure in a nitrogen gas using the direct simulation Monte Carlo (DSMC) method, and then compared with the results of VSS model combined with the SICS model (VSS-SICS) and BL model (VSS-BL). Structures of number density, rotational temperature profiles, and rotational energy distribution in the VS-SICS and VS-BL models shock wave agree well with those of the VSS-SICS and VSS-BL models. The VS model demonstrated applicability to the inelastic molecular collision model.

Large-scale simulations on multiple Graphics Processing Units (GPUs) for the direct simulation Monte Carlo method

Molecular models for simulation of rarefied gas flows using direct simulation Monte Carlo method

A direct simulation Monte Carlo (DSMC) method is implemented on graphical processing unit (GPU) architecture to simulate the deposition phenomena that occur on a glass panel during modern organic light emitting diodes (OLED) manufacturing processes with a huge number of simulated particles because a fully three-dimensional (3D) particle simulation is required to accurately predict deposition thickness and uniformity on the substrate in a low-pressure environment. To examine the accuracy and efficiency of the newly developed DSMC code running on GPU architecture, verification and validation tests were performed by solving several benchmarking problems. Under three conditions which have been applied in previous OLED deposition process test experiments, the new DSMC model successfully simulates the transport of organic material evaporated from the heated nozzle array and its deposition onto a glass panel inside a low-pressure chamber. The uniformity and normalized thickness between the DSMC and experimental results for the three test cases are then compared. The normalized thickness of the organic material deposited to the large glass panel calculated from the DSMC is found to have an error of less than 4%, and the uniformity values calculated from the absolute deposition thickness show good agreement with those obtained from experimental data in all test cases.

Scalar and Parallel Optimized Implementation of the Direct Simulation Monte Carlo Method

The Meshless Direct Simulation Monte Carlo method