Pressure Coefficient

Pressure coefficient is the nondimensional difference between a local pressure and the freestream pressure. The idea of finding the velocity distribution is to find the pressure distribution and then integrate it to get lift, moment, and pressure drag. For three-dimensional flows, the pressure coefficient Cp given by (Equation (2.54) of Reference 1) is:

where M» and V» are the freestream Mach number and velocity, respectively, u, v and w are the x, y and z-components of perturbation velocity and y is the ratio of specific heats. Expanding the right-hand side of this equation binomially and neglecting the third and higher-order terms of the perturbation velocity components, we get:

For two-dimensional or planar bodies, the Cp simplifies further, resulting in:

This is a fundamental equation applicable to three-dimensional compressible (subsonic and supersonic) flows, as well as for low speed two-dimensional flows.

9.11.1 Bodies of Revolution

For bodies of revolution, by small perturbation assumption, we have u ^ У», but v and w are not negligible. Therefore, Equation (9.73) simplifies to:

u v2 + w2 C — —2 ‘

= 2 V V 2

V oo V ^

The above equation, which is in Cartesian coordinates, may also be expressed as:

„ u f v/l 2 Cp = -2^ – .

Combining Equations (9.72) and (9.75), we get:

where R is the expression for the body contour.