CONVECTION DIFFUSION PROBLEM

**This project is a part of an Assignment submitted at Flowthermolab.**

OVERVIEW

1. Problem Statement:
  Using n equally spaced cells and the central differencing scheme for convection and diffusion, calculate the distribution of φ as a function of x for:

• Case 1: u = 0.1 m/s,
• Case 2: u = 2.5 m/s, and compare the results with the analytical solution
• Case 3: recalculate the solution for u = 2.5 m/s with 20 grid nodes and compare the results with the analytical solution

Note:
1. Unique Work: Prepared Algorithm, Coded in MATLAB and Verification from the Analytical Calculations. 2. The problem statement mentioned has been taken from textbook, ' An Introduction to Computational Fluid Dynamics by Versteeg_Malalasekera_2edition'.

METHODOLOGY

Tabel 1: Methodology Adopted

Layout	Details
1. Schematic Diagram	Figure 1: Diagram specifying the values Note: The inlet & outlet were extended to prevent errors during solving like backflow problem.
2. Defining Governing Equation	Figure 2: Governing Equation for diffusion / heat conduction problem Note:    - No Source term    - No Unsteady term
3. Discretize Geometry	Figure 3: Meshing Geometry and applying initial boundary conditions
4. Algorithm	1. Define the geometry: Length (L) [m], density [kg/m^3] 2. Discretize the geometry:   - Define Number of Grids (N)   - Grid size (𝛥𝑥) = Length / Number of grids = L / N 3. Define Boundary Conditions and Initialize   - Initialize the Variable matrix with zeros function of dimensions (1, N)   - Define the values of constants separately for internal and boundary nodes at base and tip, Matrix of constants is of size (N,N)   - Define unknown matrix (1, N) 4. Solve the matrix 5. Make data visually understandable and clear to first visual users
5. Results: Verification/Validation & Case	Figure 4: Computed Values of Matrices Note: The values were verified from the textbook ,’ An Introduction to Computational Fluid Dynamics by Versteeg_Malalasekera_2edition. Figure 5: Validation of Graph (Graphical Representation) for Case1, Case2, & Case3

DISCUSSION & CONCLUSION

The Code was verified through analytical calculations and from the solution provided in the book. Prepared a generic code that has the capability to generate graph for values of the variable in the code as per user’s choice.
   In Conclusion code work well with the analytical results. While it is important to note that on increasing the velocity number of nodes need to increased as required by the user for better accuracy but there is a threshold below which results will diverge to greater errors, set the values such that the error is minimal for maximum of the values as can also be interpreted from the values provided in the table.