Question

How is an insulated boundary handled in the finite difference formulation of a problem? How does a symmetry line differ from an insulated boundary in the finite difference formulation?

Short answer

Answer: In the finite difference formulation, the treatment of heat flux and temperature distribution differs for insulated boundary conditions and symmetry lines. Insulated boundary conditions involve approximating spatial derivatives and setting them equal to zero, corresponding to zero heat flux at the boundary. On the other hand, symmetry lines imply that the temperature distribution is symmetric with respect to that line, which allows using the symmetric temperature distribution to reduce problem size or fix the boundary condition for the temperature at the symmetry line.

Step-by-step solution

Insulated Boundary Condition

An insulated boundary condition, also called a Neumann boundary condition, is when the heat transfer through a boundary is zero, which means there is no heat flux at the boundary. In a 1D problem, the heat flux is the derivative of the temperature with respect to the space coordinate. In the finite difference formulation, we approximate this derivative by calculating the difference in the temperature between neighboring nodes. So, to apply an insulated boundary condition, we will need to express the derivative at the boundary using a finite difference approximation. Consider the following steps for a 1D problem: 1. Write down the finite difference equation using forward, backward, or central differences. 2. For the insulated boundary, express the first derivative as a finite difference approximation. 3. Set the heat flux at the boundary equal to zero. 4. Solve the resulting system of finite difference equations for the unknown temperatures.

Symmetry Line

A symmetry line is a boundary where the temperature distribution is symmetric. It means that the temperature distribution will be the same on both sides of the symmetry line, but with opposite spatial gradients due to the symmetry. In the finite difference formulation, we do not impose symmetry explicitly as in an insulated boundary. Instead, we can take advantage of the symmetric temperature distribution to reduce the problem's size and decrease the computational effort. Consider the following steps for a 1D problem with a symmetry line: 1. Write down the finite difference equation using forward, backward, or central differences. 2. Identify the symmetry line and then find its corresponding node index or location in the finite difference mesh. 3. Since, due to symmetry, the temperature distribution is identical on both sides of the symmetry line, you can either remove the symmetry line node and only work with half of the mesh or use the symmetry line temperature value in the finite difference equations as a fixed boundary condition. 4. Solve the resulting system of finite difference equations for the unknown temperatures.

Comparison of Insulated Boundary and Symmetry Line

The primary difference in applying insulated boundary and symmetry line in finite difference formulation is how heat flux and temperature distribution are treated. Insulated boundary refers to zero heat flux at the boundary, whereas symmetry line implies that the temperature distribution is symmetric with respect to that line. Insulated boundary conditions involve approximating spatial derivatives and setting them equal to zero, whereas symmetry lines involve employing the symmetric temperature distribution to reduce problem size or fix the boundary condition for the temperature at the symmetry line.