Pressure coefficient

The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow field has its own unique pressure coefficient, formula_1.
In many situations in aerodynamics and hydrodynamics, the pressure coefficient at a point near a body is independent of body size. Consequently, an engineering model can be tested in a wind tunnel or water tunnel, pressure coefficients can be determined at critical locations around the model, and these pressure coefficients can be used with confidence to predict the fluid pressure at those critical locations around a full-size aircraft or boat.
Definition.
The pressure coefficient is a parameter for studying both incompressible/compressible fluids such as water and air. The relationship between the dimensionless coefficient and the dimensional numbers is
where:
Incompressible flow.
Using Bernoulli's Equation, the pressure coefficient can be further simplified for potential flows (inviscid, and steady):
where u is the flow speed at the point at which pressure coefficient is being evaluated, and Ma is the Mach number: the flow speed is negligible in comparison with the speed of sound. For a case of an incompressible but viscous fluid, this represents the "profile" pressure coefficient, since it is associated with the pressure hydrodynamic forces rather than the viscous ones.
This relationship is valid for the flow of incompressible fluids where variations in speed and pressure are sufficiently small that variations in fluid density can be neglected. This is a reasonable assumption when the Mach Number is less than about 0.3.
formula_1 of minus one is significant in the design of gliders because this indicates a perfect location for a "Total energy" port for supply of signal pressure to the Variometer, a special Vertical Speed Indicator which reacts to vertical movements of the atmosphere but does not react to vertical maneuvering of the glider.
In the fluid flow field around a body there will be points having positive pressure coefficients up to one, and negative pressure coefficients including coefficients less than minus one, but nowhere will the coefficient exceed plus one because the highest pressure that can be achieved is the stagnation pressure.
Compressible flow.
In the flow of compressible fluids such as air, and particularly the high-speed flow of compressible fluids, formula_14 (the dynamic pressure) is no longer an accurate measure of the difference between stagnation pressure and static pressure. Also, the familiar relationship that stagnation pressure is equal to "total pressure" does not always hold true. (It is always true in isentropic flow but the presence of shock waves can cause the flow to depart from isentropic.) As a result, pressure coefficients can be greater than one in compressible flow.
Perturbation theory.
The pressure coefficient formula_1 can be estimated for irrotational and isentropic flow by introducing the potential formula_17 and the perturbation potential formula_18, normalized by the free-stream velocity formula_19
formula_20
Using Bernoulli's Equation,
formula_21
which can be rewritten as 
formula_22
here formula_23 is the sound speed.
The pressure coefficient becomes
formula_24
here formula_25 is the far-field sound speed.
Local piston theory.
The classical piston theory is a powerful aerodynamic tool. From the use of the momentum equation and the assumption of isentropic perturbations, one obtains the following basic piston theory formula for the surface pressure:
formula_26
here formula_27 is the downwash speed and formula_23 is the sound speed.
formula_29
The surface is defined as 
formula_30
The slip velocity boundary condition leads to 
formula_31
The downwash speed formula_27 is approximated as 
formula_33
Pressure distribution.
An airfoil at a given angle of attack will have what is called a pressure distribution. This pressure distribution is simply the pressure at all points around an airfoil. Typically, graphs of these distributions are drawn so that negative numbers are higher on the graph, as the formula_1 for the upper surface of the airfoil will usually be farther below zero and will hence be the top line on the graph.
Relationship with aerodynamic coefficients.
All the three aerodynamic coefficients are integrals of the pressure coefficient curve along the chord.
The coefficient of lift for a two-dimensional airfoil section with strictly horizontal surfaces can be calculated from the coefficient of pressure distribution by integration, or calculating the area between the lines on the distribution. This expression is not suitable for direct numeric integration using the panel method of lift approximation, as it does not take into account the direction of pressure-induced lift. This equation is true only for zero angle of attack.
where:
When the lower surface formula_1 is higher (more negative) on the distribution it counts as a negative area as this will be producing down force rather than lift.