[cjgn44]

Hello! I'm comparatively new to this whole world of CFD. I've looked at the examples in "12 Steps to navier stokes" and would say I understand that okay.

(http://lorenabarba.com/blog/cfd-pyth...navier-stokes/)

I've now gone back to 1D, and I'm trying to do some computations on the 1D heat equation: http://en.wikipedia.org/wiki/Heat_equation

My case is for example this: I've got a rod that's insulated around the sides such that the 1D case makes sense to use, the right side of the rod is connected to something such that the temperature is a constant T_1 and the left side a constant T_2. (similar to step 3 in the link) If the thermal diffusivity is constant throught the rod, then this problem is something I can find solved on the internet everywhere (using a finite difference scheme). But what if the thermal diffusivity is nonuniform, for instance, if the rod is splitted up into 2 equally big parts but with a different diffusivity for each part (ie, made of different materials).

Can I then simply make the thermal diffusivity a function of x_i in the equation? Ie, write:

du/dt = a(x)*(d^2 u/dx^2)

instead of

du/dt = a*(d^2 u/dx^2)

If yes, why? if no, why not ?

[tas38]

Originally Posted by cjgn44

Hello! I'm comparatively new to this whole world of CFD. I've looked at the examples in "12 Steps to navier stokes" and would say I understand that okay.

(http://lorenabarba.com/blog/cfd-pyth...navier-stokes/)

I've now gone back to 1D, and I'm trying to do some computations on the 1D heat equation: http://en.wikipedia.org/wiki/Heat_equation

My case is for example this: I've got a rod that's insulated around the sides such that the 1D case makes sense to use, the right side of the rod is connected to something such that the temperature is a constant T_1 and the left side a constant T_2. (similar to step 3 in the link) If the thermal diffusivity is constant throught the rod, then this problem is something I can find solved on the internet everywhere (using a finite difference scheme). But what if the thermal diffusivity is nonuniform, for instance, if the rod is splitted up into 2 equally big parts but with a different diffusivity for each part (ie, made of different materials).

Can I then simply make the thermal diffusivity a function of x_i in the equation? Ie, write:

du/dt = a(x)*(d^2 u/dx^2)

instead of

du/dt = a*(d^2 u/dx^2)

If yes, why? if no, why not ?

You can solve the problem analytically by considering a two domain problem (each with constant conductivity) and match the temperatures and heat fluxes at the domain interface.

If you solve the problem numerically, the variable conductivity does not present a problem. You simply discretize the conductivity.

For example, a finite difference discretization of the diffusion term would be:

d/dx ( a*du/dx ) = ( (a*du/dx)_i+1 - (a*du/dx)_i ) / ( x_i+1 - x_i ) = ...
a_i+1*( u_i+1 - +u_i ) / ( x_i+1 - x_i )^2 - a_i*( u_i - u_i-1 ) / ( ( x_i+1 - x_i)*( x_i - x_i-1 ) )

In the case of non-spatially varying conductivity and constant grid spacing, this reduces the second-order, central difference formula for second derivative.