Sean's Research

DSMC in J 1

Nov
27

Background & Motivation

Some of my research involves flow having very low densities, or flows of moderate densities in very confined spaces, such as capillary tubes. These flows are known as rarefied flows. Rarefied flows have some interesting properties: in particular, flow can slip past boundaries, whereas at higher densities the flow would have zero velocity at a surface relative to the velocity at that surface (why cars remain dusty even when you drive them fast). In hypersonics, rarefied flows are interesting because as the flow is fast and the density is low, then fluid mechanical or chemical reaction changes can take place over similar time scales as the flow time past the object of interest. This leads to what are called nonequilibrium processes, where time scales can have an important effect on flow or on chemical reaction processes.

The DSMC Method

Because the computational solution of the fluid flow equations that treat fluids as a continuous medium assume no flow slip at surfaces, those models begin to fail when the flow gets rarefied. One very successful method for simulating rarefied flow is the direct simulation Monte Carlo (DSMC) method. This is a statistical model of flow, and was invented in the 1960s by the Australian aeronautical engineer Professor Graham Bird. I was fortunate to meet Professor Bird on several occasions later in his life (he died in 2018) and he was a very inspiring scientist. He continued to make significant contributions to rarefied flows right up to the end of his life.

DSMC is a fairly straightforward simulation method based upon a statistical treatment of collisions in rarefied gases, and because it models collisions with walls, then the slip phenomenon automatically drops out of the simulation, provided the modelling of collisions of gas atoms or molecules with the wall are physically correct. Now the most obvious way to simulate the flow of a low density gas using a computer would be to keep track of the position and momentum of every nitrogen and oxygen molecule in an air flow, step forwards in time in very small increments, calculates trajectories for each molecule during this time step to determine whether individual molecules collide with each other or with surfaces and then implement some sort of model (like Newton’s laws) for how energy and momentum are transferred for whatever collisions that occur during the time step. This idea is called Molecular Dynamics (MD). It is very useful because of its simplicity and the transparency of the physical modelling involved, but it involves too much computation for all but the smallest flow volumes and all but the lowest densities – the flow domain has to be checked at each time step for collisions between each molecule and every other molecules.

DSMC overcomes much of the problems of MD for larger flow volumes by making two very clever assumptions. The first is that the behaviour of very large collections of molecules can be simulated by considering the behaviour of a much smaller number of molecules, appropriately scaled. This alleviates the problem of dealing with astronomical numbers of physical molecules. The second assumption is that on average the behaviour of collisions between molecules can be explained by statistical means based upon knowledge of the relative speeds of the potential collision partners. Thus instead of having to solve the conservation of momentum and energy equations for each collision, the probability of collision and the post-collision trajectories can be calculated using random number generation, with the overall behaviour of the flow averaging out to the correct values. These two assumptions make DSMC capable of accurately simulating rarefied flow behaviour for a range of practical engineering problems.

I will try to explain how I split the problem up and tested the components independently as I go through the design and implementation of DSMC. The implementation will be split among various posts as I go along (and as it’s being done in my spare time, this might take a while), and I’ll discuss how I decided to attempt different solutions that didn’t work, and the struggles to get an elegant and functional loopless code. I’m not an expert in either J or DSMC, just an enthusiastic amateur. But I hope that some people will find it interesting enough to begin their own investigation of this simulation method, or be inspired to take a closer look at the frankly beautiful J programming language.

I teach a very basic version of the DSMC method to my hypersonics class, and this year I asked them to code up a simple solution as an assignment question. For this blog, I thought I would go through my method of implementing the simulation, so that those unfamiliar with molecular simulation may be tempted to try it out. To make the problem more fun for me, and because it’s a good match, I have decided to implement my solution in the J programming language. J is an array-based programming language, created by Kenneth Iverson and descended from APL, the first of the array-based languages. I chose J because, being an array-based language, it is very well suited to operation on arrays of large numbers of particles that are all doing similar things. Given that very few people know the J language, and that the intersection of those who are interested in J and are interested in DSMC has probably only one member (me) then this can be seen either as a tutorial for the J programming language that uses DSMC as an example application, or an explanation of simple DSMC that uses J as a method for describing the algorithms. My other goal in putting this together was to see if I could write some nice functional code in J that helps me cement my understanding of the DSMC method.

DSMCFlowchart.png

Figure 1: DSMC Flowchart

The flowchart in Fig. 1 shows the algorithm at its simplest. A representative set of molecules is uniformly distributed across the flowfield, with randomised thermal velocities superimposed upon a bulk velocity. The thermal velocities are determined using a Boltzmann distribution as a function of the initial temperature of the gas. The flowfield itself is broken up into a group of collision cells that can be of arbitrary shape or size, with a fixed number of particles within the cell. The size of the cell is therefore dependent on the local density of the flow. These particles (which I will, for convenience, call molecules regardless of whether they are molecules or atoms) are each representative of millions to billions of actual particles in the flow. A very small time step, which depends on the size of the cell, is used to determine the motion of the molecules and, along with the relative velocities of pairs of molecules, the probability of collisions. Collisions with walls and with other molecules are simulated, and the particles are re-assigned to their new cells if they move from one cell to another. After this movement, the velocities of the particles in each cell are used to determine the flow properties in the cell, and the time is incremented by Δt again. This continues until either the flow has reached a steady state or a pre-assigned time has elapsed, at which point the code stops.

As Fig. 1 and the description above indicates, the DSMC algorithm at heart is not complicated, although it should be pointed out that state-of-the-art DSMC implementations have many improvements to this simple model, to ensure that the code runs quickly and that cells or simulations are parallelised. Some real DSMC codes that you can download and use include Bird’s DS2V code or Michael Gallis’ SPARTA code, amongst others. I mention these ones specifically because they are directly downloadable so you can play with them.

There is a lot of information available in books about the DSMC method. Some references that you should look at include Prof. Bird’s original text bird1995molecular and his more recent implementation book bird2013dsmc. Other more recent texts that cover the DSMC algorithm include those by Sharipov sharipov2015rarefied and by Boyd and Schwartzentruber boyd2017nonequilibrium. We have used DSMC for validation of a number of hypersonic separated flow configurations. Some references to this work include gai2019large, le2019rotational, prakash2018direct and hruschka2011comparison.

Most of the textbooks mentioned above contain a fair amount of kinetic theory, because there is a lot of physics to be considered when dealing with collisions of molecules with each other and with surfaces. For the purposes of explaining the process of the DSMC algorithm, that extra detail is not necessary. Instead, I have used the paper by Alexander and Garcia alexander1997direct as the model for the DSMC method used here. The paper itself can be found on Prof. Alejandro Garcia’s web site (Paper), along with a lot of other interesting simulation papers. He has also written a book on numerical methods in physics which has one chapter dedicated to implementation of DSMC in Matlab or Python (or C++ or Fortran) if you prefer to avoid J or my explanations.

Over time on this blog, the plan is to implement a simple DSMC simulation in J of the flow of low density gas through a heated capillary tube of square cross-section. Each one or two steps in Fig. 1 will be implemented and described in individual blog posts, building up to the final program.

Bibliography

  • [bird1995molecular] Bird, Molecular gas dynamics and the direct simulation of gas flows, Oxford, United Kingdom: Clarendon Press(Oxford Engineering Science Series,, (42), (1995).
  • [bird2013dsmc] Bird, The DSMC method, CreateSpace Independent Publishing Platform (2013).
  • [sharipov2015rarefied] Sharipov, Rarefied gas dynamics: fundamentals for research and practice, John Wiley & Sons (2015).
  • [boyd2017nonequilibrium] Boyd & Schwartzentruber, Nonequilibrium Gas Dynamics and Molecular Simulation, Cambridge University Press (2017).
  • [gai2019large] Gai, Prakash, Khraibut, Le Page & O’Byrne, Large scale hypersonic separated flows at moderate Reynolds numbers and moderate density, 100005, in in: AIP Conference Proceedings, edited by (2019)
  • [le2019rotational] Le Page, Barrett, O’Byrne & Gai, Rotational temperature imaging of a leading-edge separation in hypervelocity flow, 110001, in in: AIP Conference Proceedings, edited by (2019)
  • [prakash2018direct] Prakash, Gai & O’Byrne, A direct simulation Monte Carlo study of hypersonic leading-edge separation with rarefaction effects, Physics of Fluids, 30(6), 063602 (2018).
  • [hruschka2011comparison] Hruschka, O’Byrne & Kleine, Comparison of velocity and temperature measurements with simulations in a hypersonic wake flow, Experiments in fluids, 51(2), 407-421 (2011).
  • [alexander1997direct] Alexander & Garcia, The direct simulation Monte Carlo method, Computers in Physics, 11(6), 588-593 (1997).

Hello, Org

Nov
27

This is a test blog post for writing wordpress blog items using org-mode. I’m hoping this will let me populate my blog in a more seamless manner than editing it via the WordPress web interface.

Introduction

When Ι started this blog, I got an account with an ISP and installed WordPress using some kind of automatic template. The good thing about this is that I could start a blog without knowing what I was doing. The unfortunate part about it is that when I need to configure the blog, I have no idea whatsoever about how things are set up. Also, I would need to log in to the WordPress site and use their editor to build the pages.

But what I really want to do is use org-mode directly to upload posts to my blog. As I do most of my work documentation it would be awesome to convert some of that to my web site. Fortunately some awesome person has already done the work to make this possible. It’s called Org2Blog, created by Puneeth Chaganti, and currently maintained by Grant Rettke. It’s an org-mode package that allows you to edit posts using org-mode markup and extensions. You can edit and upload the post without leaving emacs, which is great!

The remainder of this post contains some fairly useless examples of this packages capabilities.

Examples

Figure 1 is an example of html import for images. One of the nice things about org-mode is you can specify different output parameters for images in pdf and html export formats. This can be handy for making things work in HTML.

avatar image

Here is an equation, in case you ever need to know the solution to a quadratic equation:

\[ x = -b \pm \frac{\sqrt{b^2-4 a c}}{2a} \]

  • Here are some points
    • \(\LaTeX\) is great to put inline into your blog, like \(\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\)
    • Centered equations use two dollar signs, as opposed to inline equations like that above, which use only one

    \[\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\]

I have not yet done much to see what can be done with org-mode and wordpress. There are several things I don’t know how to do. For example, to center the image above I had to set margins, and if I were to caption it, the caption would be left aligned. I also don’t know how to number equations. References would be the next interesting thing to try hruschka2010two. Well, it seems that citations from a bibtex file will export to WordPress, but the citation style, based on the key, does not seem to change (ie I can’t get numbered or superscript citations). Nonetheless, I’m declaring the export sufficiently capable to be useful for most blog posting purposes.

Bibliography

  • [hruschka2010two] Hruschka, O’Byrne & Kleine, Two-component Doppler-shift fluorescence velocimetry applied to a generic planetary entry probe model, Experiments in Fluids, 48(6), 1109-1120 (2010).

Reminiscences on programming

Nov
20

I’ve just been thinking: programming computers has been a big part of my life ever since I was a child.  The first computer I ever got to touch was a microbee computer in year 7.  Before that I had seen some Atari 400s in Myer, and a TRS-80 at Tandy Electronics (Australia’s version of Radio Shack), but I was not allowed to play on them. Microbees were an Australian-made computer built specifically for the education market.  For some reason our school had two of these.  We were allowed to use some kind of word processor on it, but nobody at school seemed to know what to do with it.  I don’t remember much about it other than using it made me want a computer.  At around the same time as I was using the microbee at school, my father took early retirement from his job on the waterfront and we moved from Western Sydney to a very small town.  I was about 12.    My parents had never had much money, but they were keen on my education, and I was most likely constantly banging on about computers so one day to my amazement my Dad put down the unthinkable amount of $950 on a Commodore 64 with disk drive and 1 floppy disk, and a programmer’s reference guide and we walked out of the store with it.  I guess that once we had brought the (relatively inexpensive) house he had one shot at buying something for each of us:  Mum got an olive-green IBM Selectric typewriter, Dad got an air conditioner and a monstrous Kreisler TV with roll-out wooden doors and I got the Commodore.  Then we were poor again!

I would set the computer up in front of the big TV and spend hours with it and the programmer’s reference guide.  My father refused to buy games (having spent so much on the machine) so any games were going to have to be made by me.  My limit on the computer was around 4 hours sitting cross-legged on the floor until I would lose all feeling in my legs or Dad would make me do some outside work (the house was a fixer-upper).  I had no idea what I was doing, but I read everything I could.  During those times nothing else mattered to me.  I learned to program in BASIC, and read all the copies of Byte Magazine and Creative Computing trying (usually without success) to make the type-in basic programs work.  On the odd occasions I’d save up to purchase Australian Personal Computer or one of the many British Commodore magazines to find out more about programming.  But my town was no silicon valley, and there was no internet.  The only person I knew who knew how computers worked was the Hungarian TV repair man, who had to come in to fix a problem with Dad’s pride and joy (which I’m sure he thought my computer was responsible for creating…).  This fellow explained to me how flip-flops worked, which was to me like learning the mysteries of some secret society.

At my school we also had computers: Apple II/es in the maths department and a 32 kB BBC micro in the library.  Again, nobody seemed to know what to do with them, but I was happy to fill the void.  I think I was the only student who ever got to use the BBC micro.  It was hooked up to a Telstra-run bulletin board service called VIATEL, if I remember correctly, which was a very primitive text-based centrally served prototype for the internet.  You could check the weather, share prices and other things.  I didn’t get much of a chance to look at it, though, because they charged about 10 cents per page of text downloaded, and in the first day of playing with it, I managed to rack up a bill of around $65 in about an hour.  Before I had another chance to check my portfolio and play online poker I got called into the Vice Principal’s office and it was explained to me that I would no longer be using that computer…

We kept using the Apples, which were really nice machines.  Very sturdy construction, but only 8 colours on the display, compared to the amazing graphics and sound capability of the Commodore at home.  The sprite graphics were great.  I remember trying to program graphics on the Apple but getting nowhere.  I didn’t understand about things like memory mapping that would have really helped.  There was one older kid at the school (by the last name of Ward) who would make graphics of a sort by using lots of print statements strung together to make eg a car race track scroll down the page.  We looked down on these  ‘Wardy graphics’ programs as primitive, but at least he was able to make things that looked like animations, and he put so much time into it.

On the C64 you were limited to BASIC, although I knew that assembly language or a compiler was what all the professional programmers used.  But that stuff was certainly not readily available to me.  I think I will take to my grave the knowledge that POKE 53281,0 turned the background from the default blue to black, and that putting different numbers using the POKE command at location 53280 would do the same for the screen border.

Then one fateful day in 1985 I found a cartridge in the K-mart Albury bargain bin.  It was HES Forth, an implementation of the Forth programming language for the C-64.  Not only was it a much better language than Commodore basic, with a built-in line editor, but it also had a (somewhat unusual) assembler, so that now I could finally take full advantage of a computer by converting code to machine language!  I didn’t accomplish a lot, but I learnt a lot about computers and the Forth language paradigm.  In fact, I still use that language when teaching microcontroller-based instrumentation.    I remember trying to decipher the terse user manual with all its assumed knowledge, and trying to reverse engineer a prime number generating program that they included as a demo.  That really flicked a switch somewhere in my head and programming has varied somewhere between an obsession and an interest ever since.

The line editor from HES Forth for the c64

All the programming since has been trying to recreate the thrill of making a computer do something I want it to do, delighting in working out some trick to use the computer to model something.  In hindsight I’m glad my Dad refused to buy me any games, though some of those Infocom titles did look pretty sweet at the time.  Instead I got hooked on a pastime I still get great pleasure from some 35 years later.

Over the next indefinite period when I can find the time, I’ll write up a little programming project I’ve started to model the Direct Simulation Monte Carlo method for simulating rarefied flows, using my current favourite programming language: J.  I’ll try to split the task into little parts and explain what I’m doing at each step, in case the 13-year-old version of me is listening.  I guess that’s what made me think of writing this post.