The uploaded document contains a detailed numerical and analytical project involving transient nonlinear flow. Here's a breakdown of its requirements and tasks:

Core Equations and Requirements

The study revolves around solving a set of partial differential equations governing transient nonlinear flow:

1. Momentum Equation ($v$) - Equation (1)
2. Vorticity Equation ($\eta$) - Equation (2)
3. Stream Function Equation ($\psi$) - Equation (3)

The system is solved with boundary conditions and initial conditions specified, using:

• Second-order centered finite differences for spatial discretization.
• Heun’s method for temporal integration (predictor-corrector).
• Parameters $Re = 1, 10, 100, 1000$, and $\Gamma = 1.5$.

Tasks and Goals

1. Solve and compare nonlinear results with transient linear results for $v$ from previous work (HW9).
2. Compile results into a write-up that includes:
   • Starting from the Navier-Stokes equations and deriving simplifications.
   • Steady and transient solutions (linear and nonlinear).
   • Comparison of L2-norm convergence for linear steady analytic and numerical solutions.
   • Investigation of steady-state behavior with varying $Re$.

Steps to Approach

1. Analytical Work:

   • Start with Navier-Stokes equations.
   • Discuss assumptions leading to the reduced system of equations.

2. Numerical Implementation:

   • Use second-order finite differences with $\delta r = \delta z = 1/100$.
   • Implement Heun’s method for time-stepping.

3. Analysis of Results:

   • Convergence behavior of steady linear solutions.
   • Time to reach steady state for unsteady problems with varying $Re$.
   • Compare nonlinear steady results for $Re = 10, 100, 1000$ to linear steady solutions ($Re \to 0$).

4. Write-Up:

   • Include the narrative.
   • Provide numerical codes in appendices.
   • Discuss all convergence metrics, comparisons, and time scales.

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Would you like help with any specific part of this project (e.g., implementing finite differences, Heun’s method, or analyzing the L2-norm)? Here are some potential areas to explore:

1. Explanation and derivation of numerical schemes for finite differences and Heun's method.
2. Implementation details in Python or MATLAB for solving these equations.
3. Strategies for analyzing convergence using L2-norms.
4. Discussion on the impact of varying $Re$ on transient and steady-state solutions.
5. Guidance on compiling the results into a coherent narrative.

Tip: Always validate the numerical code with simpler cases (e.g., linear steady-state solutions) to ensure correctness before tackling nonlinear problems.