Direct Simulation Monte Carlo for Atmospheric Entry.docx

1、Direct Simulation Monte Carlo for Atmospheric EntryDirect Simulation Monte Carlo for Atmospheric Entry2. Code Development and Application ResultsIain D. BoydDepartment of Aerospace EngineeringUniversity of MichiganAnn Arbor, Michigan, USA*AbstractThe direct simulation Monte Carlo method (DSMC) has e

2、volved over 40 years into a powerful numerical technique for the computation of complex, nonequilibrium gas flows. In this context, nonequilibrium means that the velocity distribution function is not in an equilibrium form due to a low number of intermolecular collisions within a fluid element. In a

3、tmospheric entry, nonequilibrium conditions occur at high altitude and in regions of flow fields with small length scales. In this second article of two parts, several different implementations of the DSMC technique in various, widely used codes are described. Validation of the DSMC technique for hy

4、personic flows using data measured in the laboratory is discussed. A review is then provided of the application of the DSMC technique to atmospheric entry flows. Illustrations of DSMC analyses are provided for slender and blunt body vehicles for entry into Earth, followed by examples of DSMC modelin

5、g of planetary entry flows.1.0 IntroductionThe direct simulation Monte Carlo (DSMC) method was first introduced by Graeme Bird in 1961 1. Since that time, Bird has written two books on the method 2,3 and thousands of research papers have been published that report on development and application of t

6、he technique. The DSMC method is most useful for analysis of kinetic nonequilibrium gas flows. In this context, nonequilibrium indicates that the velocity distribution function (VDF) of the gas molecules is not in the well-understood, Maxwellian, equilibrium form. The physical mechanism that pushes

7、the VDF towards equilibrium is inter-molecular collisions, and so a gas falls into a nonequilibrium state under conditions where there is not a large enough number of collisions occurring to maintain equilibrium. The two main physical flow conditions that lead to nonequilibrium are low density and s

8、mall length scales. A low density leads to a reduced collision rate while a small length scale reduces the size of a fluid element. The usual metric for determining whether a particular gas flow is in a state of nonequilibrium is the Knudsen number defined as follows: (1.1)where is the mean free pat

9、h of the gas and L is the characteristic length scale. The mean free path is the average distance traveled by each particle between collisions and is given for a hard sphere by (1.2)where n is the number density, and is the hard sphere collision cross section. Thus, at low density, the mean free pat

10、h (and therefore Kn) becomes large. Similarly, for small length scales, L becomes small and again Kn becomes large. As a guiding rule, it is generally accepted that kinetic nonequilibrium effects become important when Kn 0.01.Atmospheric entry flow conditions may fall into the kinetic nonequilibrium

11、 regime at sufficiently low density (that occurs at high altitude in a planets atmosphere) and for very small entry shapes (e.g. meteoroids that have a diameter on the order of a centimeter 4). In addition, situations arise where localized regions of a flow may contain low density (e.g. the wake beh

12、ind a capsule) or small length scales (e.g. sharp leading edges on a vehicle, or shock waves and boundary layers that may have very steep spatial gradients in flow field properties). Analysis of high Knudsen number flows could in principle be performed through solution of the Boltzmann equation, tha

13、t is the fundamental mathematical model of dilute gas dynamics 3. However, development of robust and general numerical solution schemes for the Boltzmann equation has proved a significant challenge. The DSMC technique emulates the same physics as the Boltzmann equation without providing a direct sol

14、ution. The DSMC method follows a representative set of particles as they collide and move in physical space. It has been demonstrated that DSMC converges to solution of the Boltzmann equation in the limit of a very large number of particles 3.In part one of this article 5, the fundamental aspects of

15、 the DSMC technique are first described with an emphasis on physical modeling issues related to its application to hypersonic, atmospheric entry problems. In this second article, the capabilities of the DSMC method are illustrated with regards to simulation of hypersonic, laboratory experiments and

16、then to several different vehicle entry applications: (1) Earth entry of slender vehicles; (2) Earth entry of blunt vehicles; and (3) entry into planetary atmospheres. These studies will illustrate that the DSMC technique has been verified using several different sets of entry flight data. The level

17、 of confidence in the accuracy of the DSMC technique has reached the stage where it is now routinely employed for pre-mission design and post-mission data analysis of atmospheric entry flows.2.0 Fundamental Aspects of the DSMC Technique2.1 General FeaturesThe DSMC technique emulates the physics of t

18、he Boltzmann equation by following the motions and collisions of a large number of model particles. Each particle possesses molecular level information including a position vector, a velocity vector, and physical information such as mass and size. Particle motion and collisions are decoupled over a

19、time step t that is smaller than the local mean free time. During the movement of particles, boundary conditions such as reflection from solid surfaces are applied. The physical domain to be simulated in a DSMC computation is covered by a mesh of cells. These cells are used to collect together parti

20、cles that may collide. There are a number of DSMC schemes for simulating collisions and all of them achieve a faster numerical performance than the molecular dynamics (MD) method 6 by ignoring the influence of the relative positions of particles within a cell in determining particles that collide. T

21、his simplification requires that the size of each cell be less than the local mean free path of the flow. Birds No Time Counter (NTC) scheme 3 is the most widely used collision scheme in which a number of particle pairs in a cell are formed. Each of the pairs of particles is formed at random regardl

22、ess of position within the cell, and then a probability of collision for each pair is evaluated using the product of the collision cross section and the relative velocity of the particle pair. This procedure reproduces the expected equilibrium collision rate under conditions of equilibrium. A number

23、 of collision cross section models have been develop for DSMC, with the most widely used forms being the Variable Hard Sphere (VHS) 7 and the Variable Soft Sphere (VSS) 8. For hypersonic flow, VHS is considered sufficiently accurate. Values of the VHS and VSS collision parameters for many common spe

24、cies are provided in Bird 3. It is determined whether the particle pair actually collides by comparing the collision probability to a random number. When a collision occurs, post-collision velocities are calculated using conservation of momentum and energy. For the VHS model, isotropic scattering is

25、 assumed in which the unit vector of the relative velocity is assigned at random on the unit sphere. The cells employed for simulating collisions are also often used for the sampling of macroscopic flow properties such as density, velocity, and temperature. There is no necessity to have the collisio

26、n and sampling cells be identical, however, and sometimes a coarser mesh is used for sampling.The basic steps in each iteration of the DSMC method are: (1) move particles over the time step t; (2) apply boundary conditions such as introducing new particles at inflow boundaries, removing particles at

27、 outflow boundaries, and processing reflections at solid boundaries; (3) sort particles into cells and calculate collisions; and (4) sample average particle information. A simulation will begin from some initial condition, and it will require a finite number of iterations for the flow to reach a ste

28、ady state. Generally, steady state is detected as a leveling off of the total number of particles in the simulation. After steady state is reached, the simulation is continued a further number of iterations in order to reduce the statistical noise in the sampled information to an acceptable level. A

29、 typical DSMC computation may employ one million particles, reach steady state after 50,000 iterations, and continue sampling for a further 50,000 iterations. On a modern desktop computer, such a simulation should take about 3 hours.While the ideas behind the DSMC technique are simple, implementatio

30、n in an algorithm takes on many different forms. Specific DSMC algorithms have been developed for vector computers 9 and parallel computers 10,11. Bird has focused work on customizing the algorithm to achieve efficient performance on single processor machines 12.Having provided a general overview of

31、 the basic elements of the DSMC method, the reader is referred to the first article 5 where a more detailed review is provided of the physical models used in DSMC that are most critical to the analysis of hypersonic entry flows.2.2 DSMC CodesUnlike CFD, there are a relatively small number of differe

32、nt implementations of the above fundamental DSMC ideas in numerical codes. Among the most widely used DSMC codes for hypersonic entry analysis are DS2V/3V 13, DAC 11, SMILE 14, and MONACO 10. These codes vary mainly in the treatment of collision selection methods and mesh topology (from orthogonal cut-cells to body-fitted unstructured cells). Most of the codes are parallelized and three-dimensional. Results from all of these codes for analysis of hypersonic entry flows are included in the following section.DS2V/3V continues to undergo development by B