Steps to Apply the (FVM)

Steps to Apply the Finite Volume Method (FVM) for a One-Dimensional Convection-Diffusion Problem

Step 1: Define the Physical Problem Start by considering a one-dimensional domain where a scalar property, ϕ, is transported through the domain by convection and diffusion. The domain could represent a physical scenario such as heat conduction in a rod or the concentration of a chemical species in a flow field.

Step 2: Governing Equation and Boundary Conditions The governing equation for steady-state convection and diffusion of the property ϕ in a one-dimensional flow field is given by the convection-diffusion equation.

Step 3: Discretize the Domain Divide the one-dimensional domain into N control volumes (cells). The control volumes should be centered around grid points where the value of ϕ will be calculated. The central differencing scheme can be used to approximate the convective and diffusive fluxes at the faces of these control volumes.

For example, if the domain is a pipe, you would divide it into several control volumes, each representing a segment of the pipe.

Step 4: Discretize the Governing Equation Integrate the governing equation over each control volume to apply the finite volume approach. This results in a balance equation for each control volume, where the flux of ϕ into the volume must equal the flux out of the volume plus any source terms.

Using the central differencing scheme, the discretized form of the convection-diffusion equation for each control volume can be written as:
$ a_pϕ_p= a_wϕ_w + a_Eϕ_E + S_p $

Step 5: Assemble the Coefficients and Source Terms For each control volume, determine the coefficients aw, aE, and aϕ based on the discretized convection and diffusion terms. Ensure that the coefficients satisfy the requirements for conservativeness, boundedness, and transportiveness to maintain stability and accuracy in the solution.

Step 6: Apply Boundary Conditions Incorporate the boundary conditions into the discretized equations. This could involve modifying the coefficients or source terms at the boundaries to account for specified values of ϕ or fluxes.

Step 7: Solve the System of Equations After assembling the system of linear equations for all control volumes, solve the resulting matrix equation for the unknown values of ϕ at each grid point. This can be done using numerical techniques such as Gauss-Seidel iteration or other linear solvers.

Step 8: Analyze and Validate the Results Once the solution is obtained, analyze the distribution of ϕ across the domain. Check for consistency with physical expectations and validate the results against analytical solutions or experimental data, if available.