“Math rules”?

My work (and working) specialization today is computational science and engineering. I have taught FEM, and am currently teaching both FEM and CFD.

However, as it so happens, all my learning of FEM and CFD has been through self-studies. I have never sat in a class-room and learnt these topics that way. Naturally, there were, and are, gaps in my knowledge.


The most efficient way of learning any subject matter is through traditional formal learning—I mean to say, our usual university system. The reason is not just that a teacher is available to teach you the material; even books can do that (and often times, books are actually better than teachers). The real advantage of the usual university education is the existence of those class-mates of yours.

Your class-mates indirectly help you in many ways.

Come the week of those in-semester unit tests, at least in the hostels of Indian engineering schools, every one suddenly goes in the studies mode. In the hostel hallways, you casually pass someone by, and he puts a simple question to you. It is, perhaps, his genuine difficulty. You try to explain it to him, find that there are some gaps in your own knowledge, too. After a bit of a discussion, some one else joins the discussion, and then you all have to sheepishly go back to the notes or books or solve a problem together. It helps all of you.

Sometimes, the friend could be even just showing off to you—he wouldn’t ask you a question if he knew you could answer it. You begin answering in your usual magnificently nonchalant manner, and soon reach the end of your wits. (A XI standard example: If the gravitational potential inside the earth is constant, how come a ball dropped in a well falls down? [That is your friend’s question, just to tempt you in the wrong direction all the way through.]… And what would happen if there is a bore all through the earth’s volume, assuming that the earth’s core is solid all the way through?) His showing off helps you.

No, not every one works in this friendly a mode. But enough of them do that one gets [too] used to this way of studying/learning.

And, it is this way of studying which is absent not only in the learning by pure self-studies alone, but also in those online/MOOC courses. That is the reason why NPTEL videos, even if downloaded and available on the local college LAN, never get referred to by individual students working in complete isolation. Students more or less always browse them in groups even if sitting on different terminals (and they watch those videos only during the examination weeks!)

Personally, I had got [perhaps excessively] used to this mode of learning. [Since my Objectivist learning has begun interfering here, let me spell the matter out completely: It’s a mix of two modes: your own studies done in isolation, and also, as an essential second ingredient, your interaction with your class-mates (which, once again, does require the exercise of your individual mind, sure, but the point is: there are others, and the interaction is exposing the holes in your individual understanding).]

It is to this mix that I have got too used to. That’s why, I have acutely felt the absence of the second ingredient, during my studies of FEM and CFD. Of course, blogging fora like iMechanica did help me a lot when it came to FEM, but for CFD, I was more or less purely on my own.

That’s the reason why, even if I am a professor today and am teaching CFD not just to UG but also to PG students, I still don’t expect my knowledge to be as seamlessly integrated as for the other things that I know.

In particular, one such a gap got to the light recently, and I am going to share my fall—and rise—with you. In all its gloriously stupid details. (Feel absolutely free to leave reading this post—and indeed this blog—any time.)


In CFD, the way I happened to learn it, I first went through the initial parts (the derivations part) in John Anderson, Jr.’s text. Then, skipping the application of FDM in Anderson’s text more or less in its entirety, I went straight to Versteeg and Malasekara. Also Jayathi Murthy’s notes at Purdue. As is my habit, I was also flipping through Ferziger+Peric, and also some other books in the process, but it was to a minor extent. The reason for skipping the rest of the portion in Anderson was, I had gathered that FVM is the in-thing these days. OpenFOAM was already available, and its literature was all couched in terms of FVM, and so it was important to know FVM. Further, I could also see the limitations of FDM (like requirement of a structured Cartesian mesh, or special mesh mappings, etc.)

Overall, then, I had never read through the FDM modeling of Navier-Stokes until very recent times. The Pune University syllabus still requires you to do FDM, and I thus began reading through the FDM parts of Anderson’s text only a couple of months ago.

It is when I ran into having to run the FDM Python code for the convection-diffusion equation that a certain lacuna in my understanding became clear to me.


Consider the convection-diffusion equation, as given in Prof. Lorena Barba’s Step No.8, here [^]:

\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right)  \\  \dfrac{\partial v}{\partial t} + u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)

I had never before actually gone through these equations until last week. Really.

That’s because I had always approached the convection-diffusion system via FVM, where the equation would be put using the Eulerian frame, and it therefore would read something like the following (using the compact vector/tensor notation):

\dfrac{\partial}{\partial t}(\rho \phi) +  \nabla \cdot (\rho \vec{u} \phi)  = \nabla \cdot (\Gamma \nabla \phi) + S
for the generic quantity \phi.

For the momentum equations, we substitute \vec{u} in place of \phi, \mu in place of \Gamma, and S_u - \nabla P in place of S, and the equation begins to read:
\dfrac{\partial}{\partial t}(\rho \vec{u}) +  \nabla \cdot (\rho \vec{u} \otimes \vec{u})  = \nabla \cdot (\mu \nabla \vec{u}) - \nabla P + S_u

For an incompressible flow of a Newtonian fluid, the equation reduces to:

\dfrac{\partial}{\partial t}(\vec{u}) +  \nabla \cdot (\vec{u} \otimes \vec{u})  = \nu \nabla^2 \vec{u} - \dfrac{1}{\rho} \nabla P + \dfrac{1}{\rho} S_u

This was the framework—the Eulerian framework—which I had worked with.

Whenever I went through the literature mentioning FDM for NS equations (e.g. the computer graphics papers on fluids), I more or less used to skip looking at the maths sections, simply because there is such a variety of reporting NS, and those initial sections of the papers all are covering the same background material. (Ferziger and Peric, I recall off-hand, mention of some 72 ways of writing down the NS equations.)  The meat of the paper comes only later.


The trouble occurred when, last week, I began really reading through (as in contrast to rapidly glancing over) Barba’s Step No. 8 as mentioned above. Let me copy-paste the convection-diffusion equations once again here, for ease of reference.

\dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right)  \\  \dfrac{\partial v}{\partial t} + u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)

Look at the left hand-side (LHS for short). What do you see?

What I saw was an application of the following operator—an operator that appears only in the Lagrangian framework:

\dfrac{\partial}{\partial t} + (\vec{u} \cdot \nabla)

Clearly, according to what I saw, the left hand-side of the convection-diffusion equation, as written above, is nothing but this operator, as applied to \vec{u}.

And with that “vision,” began my fall.

“How can she use the Lagrangian expression if she is going to use a fixed Cartesian—i.e. Eulerian—grid? After all, she is doing FDM here, isn’t she?” I wondered.

If it were to be a computer graphics paper using FDM, I would have skipped over it, presuming that they would sure transform this equation to the Eulerian form some time later on. But, here, I was dealing with a resource for the core engineering branches (like mech/aero/met./chem./etc.), and I also had a lab right this week to cover this topic. I couldn’t skip over it; I had to read it in detail. I knew that Prof. Barba couldn’t possibly make a mistake like that. But, in this lesson, even right up to the Python code (which I read for the first time only last week), there wasn’t even a hint of a transformation to the Eulerian frame. (Initially, I even did a search on the string: “Euler” on that page; no beans.)

There must be some reason to it, I thought. Still stuck with reading a Lagrangian frame for the equation, I then tried to imagine a reasonable interpretation:

Suppose there is one material particle at each of the FDM grid nodes? What would happen with time? Simplify the problem all the way down. Suppose the velocity field is already specified at each node as the initial condition, and we are concerned only with its time-evolution. What would happen with time? The particles would leave their initial nodal positions, and get advected/diffused away. In a single time-step, they would reach their new spatial positions. If the problem data are arbitrary, their positions at the end of the first time-step wouldn’t necessarily coincide with grid points. If so, how can she begin her next time iteration starting from the same grid points?

I had got stuck.

I thought through it twice, but with the same result. I searched through her other steps. (Idly browsing, I even looked up her CV: PhD from CalTech. “No, she couldn’t possibly be skipping over the transformation,” I distinctly remember telling myself for the nth time.)

Faced with a seemingly unyielding obstacle, I had to fall back on to my default mode of learning—viz., the “mix.” In other words, I had to talk about it with someone—any one—any one, who would have enough context. But no one was available. The past couple of days being holidays at our college, I was at home, and thus couldn’t even catch hold of my poor UG students.

But talking, I had to do. Finally, I decided to ask someone about it by email, and so, looked up the email ID of a CFD expert, and asked him if he could help me with something that is [and I quote] “seemingly very, very simple (conceptual) matter” which “stumps me. It is concerned with the application of Lagrangian vs. Eulerian frameworks. It seems that the answer must be very simple, but somehow the issue is not clicking-in together or falling together in place in the right way, for me.” That was yesterday morning.

It being a week-end, his reply came fairly rapidly, by the yesterday afternoon (I re-checked emails at around 1:30 PM); he had graciously agreed to help me. And so, I rapidly wrote up a LaTeX document (for equations) and sent it to him as soon as I could. That was yesterday, around 3:00 PM. Satisfied that finally I am talking to someone, I had a late lunch, and then crashed for a nice ciesta. … Holidays are niiiiiiiceeeee….

Waking up at around 5:00 PM, the first thing I did, while sipping a cup of tea, was to check up on the emails: no reply from him. Not expected this soon anyway.

Still lingering in the daze of that late lunch and the ciesta, idly, I had a second look at the attached document which I had sent. In that problem-document, I had tried to make the comparison as easy for the receiver to see, and so, I had taken care to write down the particular form of the equation that I was looking for:

\dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right)  \\  \dfrac{\partial v}{\partial t} + \dfrac{\partial uv}{\partial x} + \dfrac{\partial v^2}{\partial y} = \nu \; \left(\dfrac{\partial ^2 v}{\partial x^2} + \dfrac{\partial ^2 v}{\partial y^2}\right)

“Uh… But why would I keep the product terms u^2 inside the finite difference operator?” I now asked myself, still in the lingering haze of the ciesta. “Wouldn’t it complicate, say, specifying boundary conditions and all?” I was trying to pick up my thinking speed. Still yawning, I idly took a piece of paper, and began jotting down the equations.

And suddenly, way before writing down the very brief working-out by hand, the issue had become clear to me.

Immediately, I made me another cup of tea, and while still sipping it, launched TexMaker, wrote another document explaining the nature of my mistake, and attached it to a new email to the expert. “I got it” was the subject line of the new email I wrote. Hitting the “Send” button, I noticed what time it was: around 7 PM.

Here is the “development” I had noted in that document:

Start with the equation for momentum along the x-axis, expressed in the Eulerian (conservation) form:

\dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y} = \nu \; \left(\dfrac{\partial ^2 u}{\partial x^2} + \dfrac{\partial ^2 u}{\partial y^2}\right)

Consider only the left hand-side (LHS for short). Instead of treating the product terms $u^2$ and $uv$ as final variables to be discretized immediately, use the product rule of calculus in the same Eulerian frame, rearrange, and apply the zero-divergence property for the incompressible flow:

\text{LHS} = \dfrac{\partial u}{\partial t} + \dfrac{\partial u^2}{\partial x} + \dfrac{\partial uv}{\partial y}  \\  = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u\dfrac{\partial u}{\partial x} + u \dfrac{\partial v}{\partial y} + v \dfrac{\partial u}{\partial y}  \\  = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u \left[\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} \right] + v \dfrac{\partial u}{\partial y}  \\  = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + u \left[ 0 \right] + v \dfrac{\partial u}{\partial y}; \qquad\qquad \because \nabla \cdot \vec{u} = 0 \text{~if~} \rho = \text{~constant}  \\  = \dfrac{\partial u}{\partial t} + u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y}

We have remained in the Eulerian frame throughout these steps, but the final equation which we got in the end, happens to be identical in its terms to that for the Lagrangian frame—when the flow is incompressible.

For a compressible flow, the equations should continue looking different, because \rho would be a variable, and so would have to be accounted for with a further application of the product rule, in evaluating \frac{\partial}{\partial t}(\rho u), \frac{\partial}{\partial x}(\rho u^2) and \frac{\partial}{\partial x}(\rho uv) etc.

But as it so happens, for the current case, even if the final equations look exactly the same, we should not supply the same physical imagination. We don’t imagine the Lagrangian particles at nodes. Our imagination continues remaining Eulerian throughout the development, with our focus not on the advected particles’ positions but on the flow variables u and v at the definite (fixed) points in space.


Sometimes, just expressing your problem to someone else itself pulls you out of your previous mental frame, and that by itself makes the problem disappear—in other words, the problem gets solved without your “solving” it. But to do that, you need someone else to talk to!


But how could I make such stupid and simple a mistake, you ask? This is something even a UG student at an IIT would be expected to know! [Whether they always do, or not, is a separate issue.]

Two reasons:

First: As I said, there are gaps in my knowledge of computational mechanics. More gaps than you would otherwise expect, simply because I had never had class-mates with whom to discuss my learning of computational  mechanics, esp., CFD.

Second: I was getting deeper into the SPH in the recent weeks, and thus was biased to read only the Lagrangian framework if I saw that expression.

And a third, more minor reason: One tends to be casual with the online resources. “Hey it is available online already. I could reuse it in a jiffy, if I want.” Saying that always, and indefinitely postponing actually reading through it. That’s the third reason.


And if I could make so stupid a mistake, and hold it for such a long time (a day or so), how could I then see through it, even if only eventually?

One reason: Crucial to that development is the observation that the divergence of velocity is zero for an incompressible flow. My mind was trained to look for it because even if the Pune University syllabus explicitly states that derivations will not be asked on the examinations, just for the sake of solidity in students’ understanding, I had worked through all the details of all the derivations in my class. During those routine derivations, you do use this crucial property in simplifying the NS equations, but on the right hand-side, i.e., for the surface forces term, in simplifying for the Newtonian fluid. Anderson does not work it out fully [see his p. 66] nor do Versteeg and Malasekara, but I anyway had, in my class… It was easy enough to spot the same pattern—even before jotting it down on paper—once it began appearing on the left hand-side of the same equation.

Hard-work pays off—if not today, tomorrow.


CFD books always emphasize the idea that the 4 combinations produced by (i) differential-vs-integral forms and (ii) Lagrangian-vs-Eulerian forms all look different, and yet, they still are the same. Books like Anderson’s take special pains to emphasize this point. Yes, in a way, all equations are the same: all the four mathematical forms express the same physical principle.

But seen from another perspective, here is an example of two equations which look exactly the same in every respect, but in fact aren’t to be viewed as such. One way of reading this equation is to imagine inter-connected material particles getting advected according to that equation in their local framework. Another way of reading exactly the same equation is to imagine a fluid flowing past those fixed FDM nodes, with only the nodal flow properties changing according to that equation.

Exactly the same maths (i.e. the same equation), and of course, also the same physical principle, but a different physical imagination.

And you want to tell me “math [sic] rules?”


I Song I Like:

(Hindi) “jaag dil-e-deewaanaa, rut jaagee…”
Singer: Mohamad Rafi
Music: Chitragupt
Lyrics: Majrooh Sultanpuri

[As usual, may be another editing pass…]

[E&OE]

Errors in my CFD notes for the lecture no. 3

I have taken off the slides for the Lecture # 3 of my introductory course on CFD, because it contained some serious errors. (….Yes, those were the serious errors—not just typos.)

I had caught the errors right the next day after posting it (about 10 days ago), but it so happened that in the meanwhile, I ended up (literally) wasting my time (and money) attending to certain highly promising promises of interviews for a professor’s job, as well as also actually attending a couple of highly promising interviews for promising positions of a professor, and subsequently, also trying to interact with the  “management”s of these colleges.

I therefore couldn’t find the frame of the mind (and in fact also the necessary time at one stretch) to write down those lengthy equations in LaTeX, by way of rectifying the mistakes. I in fact was also traveling for these job-related matters. During these last 10 days, I visited some 4 places out of town, with two of them being more than a hundred km away. … No, I haven’t landed a professor’s job yet.

Anyway, back to the mistakes in the uploaded notes.

The mistakes in particular were concerned with the tensor calculus part of the slides. I caught the mistakes when I went a bit further in my notes, to the point of preparing the slides for the next couple of lectures—which would be: on the Navier-Stokes equations.

I had never worked out the full derivation of the Navier-Stokes equations in this way, i.e., for an infinitesimal CV, but using the Eulerian approach right from the beginning. … In the past, I had always relied on books, and never worked out my own derivations without referring to the proofs given in them. Many of these books are excellent. However, for the infinitesmal CV, they all always derive it only in the Lagrangian frame, and only later on do they use the vector calculus manipulations (or identities) to map the end-result to the Eulerian frame. Every one proceeds only that way. None does what I had unwittingly ended up attempting. …

… In fact, most of them use only the Reynolds’ Transport Theorem or RTT for short. (BTW, Reynolds had never himself stated this theorem in his entire life-time; the entire RTT movement was started only later, by an MIT professor.) Now, RTT is an integral approach, not differential. Usually, the books do the derivation using the RTT, and then proceed to get the differential form from this initial integral form. In the rare cases that they at all try to use an infinitesimal CV in an ab initio manner, they invariably use only the Lagrangian i.e. the non-conservative form.

Indeed, see the unanswered query on the Physics StackExchange here: [^]. … The first part of the question has gone unanswered for 3.5 years by now, after 10,000+ views. So, you know what I was getting at, here. And how, my errors, caught by me before engaging a single class based on these notes, therefore, might perhaps be excusable.

Anyway, what is more important is to note down the references which I found useful in working out this entire issue. These are the following two. (No, they of course don’t give you the derivation; they just deal with the basics of tensors and their calculus):

  • “A brief introduction to tensors and their properties,” by Prof. Allan Bower of Brown (a fellow iMechanician!) [^]
  • “Tensor derivative (continuum mechanics,” Wiki, section on divergence of a tensor [^] (and Prof. Piaras Kelly’s notes that it refers to, here [ (.PDF) ^] )

I then worked out the tensors appearing in the Navier-Stokes equations, in fully expanded components form. In this way, my path-way to the final Navier-Stokes equations now seems OK.

In other words, yes, I am now getting ready to answer that Physics StackExchange question, in my upcoming notes. … Give me a few days’ time, and both the components-wise worked out results, as well as the relevant portion excerpted for the slides of the Lecture # 3 of my CFD course, should be online.

But, also, please note, I haven’t run my work by anyone so far. So, it’s still an easy possibility that there are some elementary mistakes in it, too. At least, it would be easy enough for some unwarranted assumptions to creep in. (For instance, it was only during this recent phase of working out these things that I gathered for the first time in my life that there are some subtle pre-suppositions going into the Helmholtz decomposition theorem for the vector fields, too—assumptions like the field having to approach zero as distance tends to infinity—assumptions that I wasn’t at all aware of….) Therefore, I do plan to privately run my notes through a few mechanician friends/blogging acquaintances—even as I simultaneously post them here, within a few days’ time.

BTW, no, coming to those earlier errors in the Lecture #3 slides, even if someone had caught my errors (IMO, a low probability), none had pointed it out to me. None. I found it on my own. But only after publishing something else, in the first place!!

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A Song I Like:

You are (or at least should be) well-familiar with the well-worn out story by now.

[E&OE]

An introductory course on CFD—3

The Beamer slides for Lecture # 3 are uploaded; see here [^].

This lecture is not directly concerned with CFD, or for that matter, even with fluid dynamics proper. It instead concerns itself with some material that is pre-requisite to both. Namely, the topic of tensors. However, Indian universities don’t cover this topic very well during the earlier, pre-requisite courses—esp. at the UG level. So, I decided to throw in some formulae and a few points concerning vectors and tensors, that’s all.

The material for this lecture, thus, is totally “extra” (at least in a course on CFD). The treatment here is, therefore, very cursory. The idea was to give students at least some background into these topics, by way of a rapid review.

As usual, feel free to point out errors and offer criticism.

Further, this topic being challenging to present to a newcomer in a brief manner, this lecture is the one where I am confident in the least. (At least, it’s the first lecture of this kind.) So, any suggestions for a better presentation would be highly appreciated (though, given my experience of this blog, really speaking, I don’t expect any comments to come in, anyway.)

All the same, I am happy that so much of typing in of these equations is, finally, out of the way!

“Enjoy”!

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A Song I Like:

I still have not yet reviewed and taken a decision as to the song I like, also for this time round.

So, this section will continue to remain suspended.

[E&OE]

 

An introductory course on CFD—2

The Beamer slides for Lecture # 2 are uploaded, here [^].

As usual, feel free to point out errors and offer criticism.

[I did spot a few typos in the Lecture # 1 (see previous post). These are being corrected, and an improved version will be uploaded some time later.]

Update on 2015.07.16:

For the convenience in updating, I have created a separate page at my personal Web site; it holds all the links to the material for this CFD course.

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A Song I Like:

I have not yet reviewed and taken a decision as to the song I like.

So, this section will remain suspended, for the time being.

[E&OE]