Developing a representative aero thermal simulation system for analysing the nose cone tip will be important in ensuring the tip is designed to be able to survive launch loads. 

The analysis is expected to be performed in the following steps:

  1. Estimate the flow properties (Mach, Velocities, Pressure, Temperature, Density) between the tip surface and the oblique shock coming off the nose cone tip. 
  2. Determine the expected convective heat flux into the side walls of the tip, and the stagnation point convective heat flux
  3. Perform an unsteady 1D heat simulation where we estimate the temperature profile along the cone axis that balances the convective heat flux in, the conduction through the metal, and the radiative heat flux. 
  4. Enjoy plots, or stress out, and redesign the nose cone tip. 

 

To perform step 1, section 13.6 of Anderson's Fundamentals of Aerodynamics (5th Edition) is used. 

Supersonic theory around a perfect cone can be written in a much simpler form than the fully continuity, momentum and energy equations. This is due to the symmetry of the problem, where the 5 coupled differential equations can be simplified to 1 ordinary differential equation and solved using boundary conditions. From this one ODE, the rest of the flow properties can be reconstructed.

The crucial equation (derived in section 13.6) is of the radial flow velocity between the cone surface and the oblique shock, is below. 

because of the symmetry of the problem, the flow properties along ANY ray from the cone vertex has to have constant values. This greatly simplifies all the equations. 



Equation 13.78 is the key equation that must be solved. Non-dimensionalize by Vmax = sqrt(h0) where h0 is the total enthalpy and you get:

where eqn 13.80 is only a function of gamma and theta. 

This can't be solved by starting at the cone wall, and then integrating out. It can however be solved by:

  1.  Assume a shock angle theta_s
  2. Determine the flow properties right behind the shock using standard oblique shock theory
  3. Solve eqn 13.80 by integrating from theta=theta_s towards theta = 0
  4. at each step check if V_theta = 0. If it is, there is no normal flow, and this must happen at a wall. Therefore, you have determined the cone angle. 
  5. Reconstruct the dimensional Vr, Vtheta and Mach numbers for all theta between cone and shock
  6. Determine the pressure temperature and density at each theta.
That gives you all the information you could want on the flow properties around the cone. If the cone was not a perfect cone, as it often is for rocket team, none of these equations apply, and a axisymmetric 2D flow solver would be needed to do this through something like finite elements. However if we can approximate our tip as a cone, the equations are far, far simpler. 

In fact we can easily determine the shock angle from tables:

For Hermes II, the cone angle approximately is 9.5 degrees, and so the range of shock angles goes from approximately 42 degrees at M=1.5, to 15 degrees at M=5. 


This is a significantly large range of values, and incorporating this into the simulator will not be trivial.



References:
Anderson John, D. "Fundamentals of aerodynamics." University of Maryland. New York, Mc Grawhill. America(2011).



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