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What are the theoretical condition for a shape to be more aerodynamic?

How can you design an aerodynamic shape without any experiment,just with equations?

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    $\begingroup$ Hi Stefan, you are asking questions, which are not unsuitable for the site, imo, but what have you looked up yourself, on wikipedia for example? $\endgroup$ – user140606 Feb 19 '17 at 17:52
  • $\begingroup$ "to be more aerodynamic" I believe you mean to minimize opposing forces (drag) $\endgroup$ – docscience Feb 19 '17 at 18:38
  • $\begingroup$ Since you provide no constraints as to the minimum thickness or length of the body in question, the most aerodynamic theoretical shape is no shape at all. See the problem with an ill posed question? $\endgroup$ – TRF Feb 20 '17 at 3:37
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There are no first principles that I can think of that lead to simple analytical solutions to determine aerodynamic geometries.

Until the availability of practical Computational Fluid Dynamics (CFD) which are the main method of design today, and are based on finite element modeling of the Navier Stokes Equations, engineers used methods like lofting to design the geometry of ship hulls. The main problem is that 3 dimensional fluid flow is very nonlinear and defies analytical solution. To predict you need to simulate, and often that only gets you near-term solutions because the nonlinear behavior can be sensitive to initial conditions.

Any method that can predict the shape of streamlines in fluid flow, even in an incremental or iterative manner can be used to determine a suitable boundary which can become a defined surface of the inside of a nozzle or nose cone of a rocket. In other words define the fluid behavior, then design a surface that contains it. Constraints often are minimum energy loss and prevention or minimization of discontinuities like shocks.

Airfoils can be more easily analyzed using conformal transformations that map for example the cross section of a wing into a circle. Here is a good tutorial from SIAM that describes the method.

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