Project Overview: 2D Finite-Volume Euler Solver for Supersonic Flow

This project aims to develop a 2D finite-volume solver for the Euler equations, solving supersonic flow over a diamond-shaped airfoil using different numerical schemes. The primary methods include Steger-Warming Flux Vector Splitting (FVS). The solver should initially implement a first-order accurate scheme. 

File Structure and Code Organization

The code is structured around a finite-volume discretization of the Euler equations. Each file contributes to different aspects of the numerical solver:

1. Initialization and Boundary Conditions

• initializeQ & stegerINITIALIZE:

  • Set initial flow conditions using given density, velocity, and pressure values.
  • Compute flux vectors E and F for numerical flux evaluations.
  • stegerINITIALIZE specifically implements the Steger-Warming flux splitting method.

• firstcolumn:

  • Initializes the first column of the computational domain, setting the state vector Q and flux terms.
  • Uses ghost cells to enforce boundary conditions.

• slipwall2:

  • Implements adiabatic slip-wall boundary conditions.
  • Adjusts velocity components to maintain tangential flow along walls.

2. Core Computational Functions

• Fluxcalc:

  • Computes the fluxes E and F at each cell face.
  • Uses conservative variables: Q=[ρ,ρu,ρv,ρ_et]
  • Pressure is recovered using:
    • P=0.4×(ρ_et−0.5ρ(u_2+v_2))P

• iter & stegIter:

  • Main iteration loop for solving the Euler equations.
  • Computes the updated state Q at each time step using flux differences.

• timestep:

  • Determines adaptive time step based on the Courant-Friedrichs-Lewy (CFL) condition.
  • Ensures numerical stability by computing:
    • Δt=CFL×min⁡( Δx/(U+c), Δy/(V+c) )
  • Speed of sound is computed using:
    • c=γRTc​

3. Visualization and Output

• makeIT:

  • Extracts and stores flow variables (pressure, velocity, density, temperature).
  • Used for post-processing.

• PLOTiter:

  • Generates plots of velocity contours.
  • Uses color mapping based on normalized velocity values.

Code Flow and Execution

1. Initialize Grid & Conditions (initializeQ, stegerINITIALIZE)
2. Apply Boundary Conditions (firstcolumn, slipwall2)
3. Compute Fluxes (Fluxcalc)
4. Update Solution (iter, stegIter)
5. Adjust Time Step (timestep)
6. Iterate Until Convergence
7. Post-Processing and Plotting (makeIT, PLOTiter)

Final Objective

The goal is to compute steady-state supersonic flow over the airfoil, verifying results against oblique shock relations. Convergence is measured using L2-norms of residuals, and solutions are validated by comparing them to theoretical shock wave properties.