[Patryk Stawicki]

How I can write,as a fvScheme, laplace equation?

I always find a syntax like this laplacian(nu,u), where u is the function under the differentiation. What I don't know is how to find "nu", I know it's a diffusion term, but I don't know how It should it be calculated so the equation holds for my case.

[Patryk Stawicki]

what i meant is that, I always thought of laplacian operator to be: ^2

but now i see some different version

(v*)

I don't see how to change the equation to the second form and to introduce(and to determine the valeu) of this "v" - diffusion term

[dlahaye]

Laplacian(u) = div( grad(u) ).

To verify this, it suffices to use the definition. Indeed,

LHS = Laplacian(u) = \sum_i \partial^2 u / partial^2 x_i

while grad(u) = Vector( \partial u / partial x_1, \partial u / partial x_2, \partial u / partial x_3 )

thus RHS = div( grad(u) ) = \sum_i \partial^2 u / partial^2 x_i

and thus LHS = RHS.

Does this help?

[Patryk Stawicki]

Yes it helped a lot. Thank you very much.

Now what I still don't know how I should define the value or a function "nu"

In syntax of the laplacian is for example laplacian(nu,u).

I have no idea what's exactly is this nu term and if it's needed always.

For example in a problem i'm right now solving, I have just a laplace equation, there is no diffusion term(that's what it's called I believe). In such case should i put it as some constant like nu=1?

Thanks in advance!

[dlahaye]

Yes, true.

For nu = 1, div( nu grad(u) ) = div( 1 grad(u) ) = Laplacian(u)

Diffusion is direction-independent and spatial-independent.

For nu <> 1, div( nu grad(u) ) is a "generalised Laplacian" that can be direction and/or spatial dependent.

Think of diffusion of heat. In regions in space with small (large) temperature gradient, diffusion will be small (large).