Using Method of Characteristics to Design Nozzle Wall

How the code works

1. main.cpp (Entry Point)

  • This is the main file that initializes the engine and equations classes, processes command-line arguments, and executes the Method of Characteristics (MOC) computation.
  • Reads parameters like the number of lines (-nlines), heat capacity ratio (-gamma), exit Mach number (-exitmach), throat radius (-rthroat), and output file name (-out).
  • Calls MOC(Engine, eq) to execute the method of characteristics.
  • Writes the results to a file using outputToFile().

2. MOC.cpp (Core MOC Computation)

  • Implements the Method of Characteristics (MOC) for supersonic nozzle design.
  • Calls Engine.initializePoints() to set up characteristic points.
  • Computes Prandtl-Meyer angles using eq.getPMAngle().
  • Interpolates angles for characteristic line divisions.
  • Initializes points along the wall and centerline.
  • Iteratively calculates Mach number, flow angles, and characteristic line intersections.
  • Computes xy-coordinates for characteristic points based on previous points.

3. engine.cpp (Engine Class – Manages Points & Flowfield)

  • initializePoints():
    • Defines the characteristic points (internal points, wall points, and centerline points).
    • Assigns flow angles and y-coordinates.
  • reset(): Clears stored characteristic points before recomputation.

4. equations.cpp (Mathematical & Thermodynamic Equations)

  • returnOutsideTemperatureAndPressure(): Estimates temperature and pressure at different altitudes.
  • getExitMach(): Computes exit Mach number from stagnation properties.
  • getPMAngle(): Returns Prandtl-Meyer angle.
  • getMachFromPM(): Calculates Mach number from Prandtl-Meyer angle using an iterative solver.
  • interpolateAngle(): Generates an array of angles between 0 and a given maximum angle.
  • returnXYIntersectionPoint(): Finds the intersection of two characteristic lines.

5. cPoint.cpp (Defines Characteristic Point Class)

  • Implements the cPoint class, which represents a characteristic point.
  • Each point stores:
    • Index (m_index)
    • Whether it is a wall or centerline point
    • Position (m_x, m_y), Mach number, Prandtl-Meyer angle, and flow angle.

6. outputToFile.cpp (Handles Writing Results to File)

  • Writes characteristic points to an output file.
  • The output includes:
    • Index, X-coordinate, Y-coordinate, Mach number, Flow angle, and Prandtl-Meyer angle.
  • The output file is later used by the Python script for plotting.

7. PLOTresults.py (Plots the Nozzle Shape)

  • Reads the output file (output.txt).
  • Extracts X and Y coordinates of characteristic points.
  • Plots nozzle contours using matplotlib:
    • Plots the top and mirrored bottom profile.
    • Allows customizing color and output PNG filename. 

8. run_exp.csh (Shell Script)

  • Likely an execution script (CSH shell script).
  • Typically used to run the main program, set environment variables, or execute multiple runs with different parameters.

The Method of Characteristics (MOC) is a mathematical technique used to design supersonic rocket nozzles, ensuring smooth, shock-free expansion of exhaust gases to maximize thrust efficiency.

Key Steps in Using MOC for Nozzle Design:

  1. Define the Nozzle Exit Conditions:

    • Start with the desired exit Mach number based on mission requirements.
    • Typically, high expansion ratios are used for optimal exhaust velocity.

  2. Generate the Characteristic Mesh:

    • Solve the governing equations of gas dynamics, which include Mach number, pressure, temperature, and density variations.
    • The method calculates characteristic lines that define the nozzle shape, ensuring a gradual and smooth expansion.

  3. Design the Expansion Section:

    • A Prandtl-Meyer expansion fan is introduced at the nozzle throat to facilitate smooth supersonic flow expansion.
    • The wall contour is shaped using characteristic lines to prevent shocks and minimize losses.

  4. Optimize the Nozzle Contour:

    • The final diverging section is refined to ensure flow uniformity at the exit, maximizing specific impulse (Isp).
    • Adjustments can be made for over-expanded or under-expanded flow depending on altitude conditions.

Using MOC allows engineers to design high-performance, efficient nozzles with minimal flow disturbances, crucial for maximizing thrust in chemical rocket engines. 🚀

Using MOC to design a Nozzle profile

Video tutorial I used