What is Method of Characteristics?

[HollyFlame]

(I am new to this forum so please do forgive me if I am posting at a wrong place.)

Hello,
I am a final year student of BS Mechanical Engineering and method of characteristics is not a part of our curriculum. In-fact I heard of it first time after finally picking my FYP. My final year project is to design analyze and manufacture a CD Nozzle which would achieve Mach 3. I was told by my supervisor that I would have to build the profile of the nozzle using method of characteristics because he wants the profile to be generalized so that every time the inlet condition changes, the profile changes automatically. My supervisor who himself is only a doctorate has never studied MOC himself so he can not help me. I want to know what it is and how could I start practicing it. Moreover what are the pre-requisites to start learning it. Any help would highly be appreciated.
Kind regards.

 

[MisterX]

I think you don't need much besides multi-variable calculus and some exposure to ODEs. You can find out what it is by a google search or by reading a PDE textbook, such as the thorough but somewhat formidable one by Lawrence Evans. In the simplest case, we'd have a first order PDE of the form
$$\sum a_i(\mathbf{x})\partial_{x_i}u(\mathbf{x}) = 0$$
Then we consider the curves of constant $u$, parameterized by $t$. On the level curve $\partial_t u = \sum \partial_{x_i}u \,\partial_{t} x_i = 0$. By the first equation this is satisfied if $$ \partial_{t} x_i \propto a_i(\mathbf{x}) $$. So this gives us a way to potentially solve for the curves of constant $u$, and thus find the solution to the PDE. This is only a simple case, which is generalized for quasi-linear first order PDEs.

 

[HollyFlame]

I think you don't need much besides multi-variable calculus and some exposure to ODEs. You can find out what it is by a google search or by reading a PDE textbook, such as the thorough but somewhat formidable one by Lawrence Evans. In the simplest case, we'd have a first order PDE of the form
$$\sum a_i(\mathbf{x})\partial_{x_i}u(\mathbf{x}) = 0$$
Then we consider the curves of constant $u$, parameterized by $t$. On the level curve $\partial_t u = \sum \partial_{x_i}u \,\partial_{t} x_i = 0$. By the first equation this is satisfied if $$ \partial_{t} x_i \propto a_i(\mathbf{x}) $$. So this gives us a way to potentially solve for the curves of constant $u$, and thus find the solution to the PDE. This is only a simple case, which is generalized for quasi-linear first order PDEs.

Thank you for your reply. I have not studied multi-variable calculus, and I am not very good at differential equations. It seems I have to start fresh. Can you please recommend an outline for how should I begin this.

 

[MisterX]

MIT OCW has a multivariable course you can watch online.
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/
You could try that or consider taking that course at your school.

 

[HollyFlame]

MIT OCW has a multivariable course you can watch online.
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/
You could try that or consider taking that course at your school.

I actually meant, what should I do after I have gone though Multi variable Calculas and DE. I have sniffed through various books on fluid dynamics in engineering library of our campus, and tragically none had any reference of MOC. Is there any book that specifically deals with this topic?

 

[MisterX]

The only book I know which has it is the one I already mentioned, by Lawrence Evans.
https://www.amazon.com/dp/0821849743/?tag=pfamazon01-20

 

[HollyFlame]

The only book I know which has it is the one I already mentioned, by Lawrence Evans.
https://www.amazon.com/dp/0821849743/?tag=pfamazon01-20

Thanks a lot for your help, I will try to get my hands on this book.

 

[Demystifier]

Thanks a lot for your help, I will try to get my hands on this book.

Forget Evans, this book is for advanced pure mathematicians, not for someone who only recently learned mutli-variable calculus. Since you are an engineer, I recommend you the book
J.D. Hoffman, Numerical Methods for Engineers and Scientists (2nd edition)
https://www.amazon.com/dp/0824704436/?tag=pfamazon01-20
Despite the title, the book explains also some analytical methods. In particular, the method of characteristics is explained in simple terms, starting from the bottom of page 505.

 

[Demystifier]

My final year project is to design analyze and manufacture a CD Nozzle which would achieve Mach 3.

Since you want to solve a problem in fluid mechanics, perhaps even more useful for you is the book
E.B. Wylie, V.L. Streeter, Fluid Transients
which contains a whole chapter on the characteristics method.

 

[the_wolfman]

The method of characteristics is frequently taught as a method to solve the Burgers equation. The equation is a simple model wave equation. There is a ton of literature on the subject and a Google search of method of characteristics with burgers equation should produce a few introductory lecture notes.