Pressure coefficient

							                       Pressure coefficient

A non-dimensional aerodynamic quantity that is used widely and hence is very
important is the pressure coefficient, Cp :


                              p - p∞        p - p∞
                     Cp   =            =           2
                                q∞         0.5 ρ∞ V∞




Cp basically describes how the pressure on the surface of the wing deviates from
the freestream pressure, and how large that deviation is relative to the dynamic
pressure.
          p(x)



p


p∞

                   Pressure on the
                   upper surface



          x


Cp
                  Pressure coefficient
-ve
                  on the upper surface


+ve           x

      1
At the leading edge, the pressure is the total pressure po, since the flow has
stagnated there. In fact, for incompressible flow, po = p∞ + q∞ and so Cp = 1 at
the leading edge. As the fluid accelerates over the wing, the pressure drops
quite low, so that Cp quickly becomes zero and then a reasonably large negative
value. The flow then decelerates, the pressure rises, and the magnitude of Cp
shrinks again.

Cp is usually plotted upside down, with the negative axis pointing up.

At low speed (M∞ < 0.3), Cp at a given point on the wing essentially does not
vary with Mach number, and is the incompressible pressure coefficient Cp,o at
that point :
                          Cp (M∞ < 0.3) ≈ Cp,o.




However, as the freestream Mach number rises, compressibility becomes
important, and the magnitude of the pressure coefficient begins to rise. In other
words, the deviation of surface pressure from the freestream pressure becomes
a bigger and bigger multiple of the dynamic pressure.
For Mach numbers 0.3 < M∞ < 0.7, the variation of Cp is given approximately by
the Prandtl-Glauert rule.

                                  C p, o
                          Cp =
                                 1 - M2
                                      ∞



(this equation becomes very inaccurate as M∞ approaches 1). The Prandtl-Glauert
rule provides a compressibility correction for the incompressible pressure
coefficient.


             Cp




                           0.3               0.7    M∞
One of the useful features of the pressure coefficient is that it can be integrated
over the wing surface to give the lift coefficient.

                                               Consider this segment of unit span
                                               of an infinite wing. At each point a
                                               on the surface, there is a pressure p
                                               acting normal to the surface (along
                                               ac). p is constant along the slice
                                               parallel to the leading and trailing
                                               edges. The normal force on that slice
                                               is pressure x area, or

                                                 dF = pl dA = pl ds (lower surface)
                                                    = - pu dA = - pu ds (upper surface)




The component of force normal to the chord line is therefore

                 dN = pl cosθ ds (lower surface) or - pu cosθ ds (upper surface)

since the surface lies at angle θ to the chord line.
The total force normal to the chord line, N, is found by integrating dN over the
top and bottom of the wing :


                   TE       TE                               TE
             N = ∫ LE dN = ∫LE plcosθ ds +                  ∫LE -p ucosθds


(note : this ignores any component of wall shear stress normal to the chord line,
since such components are usually small).


                           c
                     N = ∫ 0 ( pl - p u ) dx
                           c
                        = ∫0 [( pl - p ∞ ) - ( pu - p∞ ) ]dx
                               c
                        = q ∞ ∫0
                                   ( pl   - p∞ )
                                                 -
                                                      ( pu    - p∞ )
                                                                     dx
                                          q∞                 q∞
                               c
                        = q ∞ ∫0 C p, l - C p,
                                   (              u   )dx
                                                       N            N
The normal force coefficient is            cn =            =
                                                      q ∞S         q ∞c
Hence                              1 c
                         cn    =     ∫0 C p, l - C p, u dx
                                       (             )
                                   c

                                                                        N
                                                                    L
If the airfoil is in general flying at angle of attack α, then                R

                  L = N cosα - A sinα
                                                                         A
                                                               α



where A is the axial force along the chord line (for which shear stress cannot be
neglected). Hence
                          cl = cn cosα - ca sinα

For low angles of attack, cosα ≈ 1 and sinα ≈ 0, and so we can neglect ca (and the
shear stress). In that case, cl ≈ cn, or


                                     1 c
                          cl   ≈       ∫0 C p, l - C p, u dx
                                           (             )
                                     c