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Tony's Intertwingled Memex • Reference Manual

Supersonic Nose Cone Aerodynamic Pressure (Aerospace & Mechanical Engineering)

Formulas & Definitions

An aerospace aerodynamicist evaluates the supersonic airflow across the nose cone of a high-speed atmospheric flight test vehicle. In wind tunnel validation studies, the upper curved aerodynamic profile of the fuselage nose cone is modeled by the paraboloid of revolution:

z=20−0.05x2−0.1y2z = 20 - 0.05x^2 - 0.1y^2
where coordinates x,y,zx, y, z are measured in centimeters (cm\text{cm}).

During supersonic Mach 2 flight, a pitot-static pressure sensor probe is installed at the surface location (x0,y0)=(10,5) cm(x_0, y_0) = (10, 5)\text{ cm}.

  1. Compute the exact height z0=f(10,5)z_0 = f(10, 5) of the fuselage at the sensor installation point.
  2. Determine the general analytical expressions for the partial derivatives ∂z∂x\frac{\partial z}{\partial x} and ∂z∂y\frac{\partial z}{\partial y}.
  3. Evaluate the partial derivatives at (10,5)(10, 5) and state the equation of the tangent plane to the fuselage surface at this location in standard form.
  4. Extract the outward-pointing unit normal vector n^\hat{n} to the fuselage surface at (10,5)(10, 5), and explain why aerodynamicists use n^\hat{n} to compute the local static pressure force vector F⃗p=−pn^ dA\vec{F}_p = -p \hat{n} \, dA.