Hi. I'm Opprah Manyika.

1. Governing equations

For CFD: incompressible RANS with the standard $k-\epsilon$ closure.

$$\frac{\partial}{\partial x_i}(\rho u_i u_j) = -\frac{\partial p}{\partial x_j} + \frac{\partial}{\partial x_i}\left[\left(\mu + \mu_t\right)\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j}\right)\right]$$

For the heat solver: 2D transient heat conduction on a square plate.

$$\frac{\partial T}{\partial t} = \alpha \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right), \quad \alpha = \frac{k}{\rho c_p}$$

2. Discretisation

CFD: second-order upwind, SIMPLE pressure-velocity coupling, residuals to $10^{-5}$.

Heat: explicit FTCS in space and time.

$$T_{i,j}^{n+1} = T_{i,j}^{n} + r\left(T_{i+1,j}^{n} + T_{i-1,j}^{n} + T_{i,j+1}^{n} + T_{i,j-1}^{n} - 4T_{i,j}^{n}\right), \quad r = \frac{\alpha \Delta t}{h^2}$$

Aerodynamics: polynomial fits to the wind-tunnel data, lift-curve slope extracted by linear regression on the linear region.

3. Stability & convergence

The von Neumann stability criterion for explicit FTCS on a 2D uniform grid is

$$r = \frac{\alpha \Delta t}{h^2} \le \frac{1}{4} \quad \Longrightarrow \quad \Delta t \le \frac{h^2}{4\alpha} = 0.25\,\text{s}$$

The MATLAB solver checks this assertion at run-time and refuses to advance if the criterion is violated.

4. Validation

CFD: validated against the wind-tunnel data within $\pm 3\%$.

Aerodynamics: cylinder pressure distribution compared to the theoretical potential-flow solution.

Heat: hand-computed corner temperatures compared to the FDM solution at steady state.

Bellmouth: total-pressure recovery $\eta_p > 0.99$, compared to the von Karman integral boundary-layer method.