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 Frank Wedburn September 25, 2005 10:54

I have recently began studying CFD as part of my Masters degree and have been reading through several textbooks on the subject of finite differencing.

In many of the books an equation is presented and the text refers to an "advection" term within the equation. THe books also mention the "disperstion" terms.

I understand that advection deals with some property moving with the flow and that dispersion occurs in every direction (for example brownian motion). I am unable to determine which terms in the equation are the advection term or the dispersion term.

I am led to believe that the two different terms are to be dealt with in slightly different ways to ensure the accuracy and stability of the scheme. I cannot look at an equation, however, and identify the advection terms nor the dispersion term. I imagine things will get even more complicated when trying to identify non-linear advection terms etc.

Could someone please give me an idea how I can identify advection & dispersion terms in a general equation?

Cheers, F

 diaw September 25, 2005 11:10

Very roughly, to put you in the ballpark,

Diffusion = Conduction = a*(d2T/dx2)

I have shown portions of the Energy Eqn in 1D. For 2d & 3d problems, all derivatives are in partial form.

The u in the convection term is the fluid velocity. In the momentum eqn, the convection term may appear as u*(du/dx), leading to a non-linear equation of motion.

diaw...

 Frank Wedburn September 25, 2005 19:16

du/dt - c*(du/dx) = d*(d^2u/dx^2) [c&d constants]

The second term is the advection term but it is not multiplied by the velocity. The problem I'm having is identifying the advection terms in general equations. This would really help me understand much of the literature.

 Frank Wedburn September 25, 2005 19:20

{Sorry the notation might not have been clear enough there regarding the constant 'd' - my mistake :o) }

du/dt - C*(du/dx) = E*(d^2u/dx^2) [C&E constants]

The second term is the advection term but it is not multiplied by the velocity. The problem I'm having is identifying the advection terms in general equations. This would really help me understand much of the literature."

 diaw September 25, 2005 20:44

Term 1 = transient (rate) term Term 3 = diffusion term in energy eqn (could be stress term in momentum eqn)

Term 2 - for it to represent a convection term, C would need to be -u (1D), otherwise it represents an additional term which is not convection.

My suggestion is for you to read the books by Patankar & Versteeg - these will set you on the correct path. Study the 'general transport equation' & then apply it to your particular problem.

Rate + convection = diffusion + source

diaw...

 Renato N. Elias September 25, 2005 21:27

I think, you should try to memorize some standard forms of the diferential equations, p. ex.:

NAVIER STOKES (MOMENTUM CONSERVATION): u,t + u grad(u) + mu * div(sigma) = f

u is the velocity, t is the time, mu is the viscosity and sigma the stress tensor, f are external forces (gravity, coriolis, etc...), fi is the mass or temperature being transported, nu is the difusivity (for mass transport)

For both equation we have: 1st term: temporal derivative 2nd term: advection/convection 3rd term: difusion

SOME REMARKS:

- Sometimes you'll see the difusion term being presented through its normal and shear contributions.

- Both equations have the same structure and roughly speaking they differ only in the parameter being "transported" (velocity: Navier Stokes, mass or temperature: advection/difusion)

- The nonlinearity appears only in the Navier-Stokes equation due the convective term (u grad(u) )

- The viscosity (mu) is the parameter responsible for the dispersion (difusion) of the inertia through the "layers" of the fluid.

- The difusivity (nu) is the similar parameter responsible for the dispersion of the mass or temperature in the advection/difusion equation.

- In general, the transport of mass and/or temperature is preceeded by the solution of the Navier-Stokes equations to supply the velocity field necessary. Sometimes they are solved in a coupled manner.

Some very basic concepts associated with advection and difusion could be reached with simple examples: pure advection: a plastic being transported in a fluid flow (it will never spread in the fluid flow) pure difusion: something being spreaded in a static fluid like a drop of oil in a pool.

That's it... Have I cleared up anything?!

I leave it open for further discussion.

Regards

Renato

 diaw September 26, 2005 01:40

Very well put... I hope you are on the Wiki team... :)

 zxaar September 26, 2005 02:35

Or to get better idea you can read : An introduction to Computationa Fluid Dynamics H K Versteeg W Malalasekera Thsi books deals each term in detail in its chapters. very good book to build concepts.

 Frank Wedburn September 26, 2005 03:57

Thanks very much - that makes things much clearer!

I agree with one of the later comments - something like this should be added to the wikipedia site. I have tried looking through several introductory courses on CFD but wasn't able to clear this up until now.

Thanks again :o) F

 zxaar September 26, 2005 04:19

on wiki pedia, at the moment a samll group of people are working (small compared to people who visit this site daily).

You can also make some contributions, in fact since you are reading it will be easier to make contributions. it really does not matter how small your contribustion is, it might help someone. (as you also wish that you might get things to read at one place). give it a try.

 Renato N. Elias September 26, 2005 12:59

I'm collecting some material related with CFD and I hope to contribute with the wikipedia's site in a short time.

regards

Renato N. Elias

 M September 26, 2005 13:31