In fluid dynamics, the pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field.
Every point in a fluid flow field has its own unique pressure coefficient, Cp.
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.
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 [1][2] where: Using Bernoulli's equation, the pressure coefficient can be further simplified for potential flows (inviscid, and steady):[3] where: 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 assumption is commonly made in engineering practice when the Mach number is less than about 0.3.
are significant in the design of gliders because this indicates a suitable 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.
As a result, pressure coefficients can be greater than one in compressible flow.
can be estimated for irrotational and isentropic flow by introducing the potential
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: where
The surface is defined as The slip velocity boundary condition leads to The downwash speed
is approximated as In hypersonic flow, the pressure coefficient can be accurately calculated for a vehicle using Newton's corpuscular theory of fluid motion, which is inaccurate for low-speed flow and relies on three assumptions:[5] For a freestream velocity
relative to the freestream, the change in normal momentum is
Then the momentum flux, equal to the force exerted on the surface
, from Newton's second law is equal to: Dividing by the surface area, it is clear that the force per unit area is equal to the pressure difference between the surface pressure
, leading to the relation: The last equation may be identified as the pressure coefficient, meaning that Newtonian theory predicts that the pressure coefficient in hypersonic flow is: For very high speed flows, and vehicles with sharp surfaces, the Newtonian theory works very well.
A modification to the Newtonian theory, specifically for blunt bodies, was proposed by Lester Lees:[6] where
is the maximum value of the pressure coefficient at the stagnation point behind a normal shock wave: where
The last relation is obtained from the ideal gas law
The Rayleigh pitot tube formula for a calorically perfect normal shock says that the ratio of the stagnation and freestream pressure is: Therefore, it follows that the maximum pressure coefficient for the Modified Newtonian law is: In the limit when
, recovering the pressure coefficient from Newtonian theory at very high speeds.
The modified Newtonian theory is substantially more accurate than the Newtonian model for calculating the pressure distribution over blunt bodies.
[5] An airfoil at a given angle of attack will have what is called a pressure distribution.
for the upper surface of the airfoil will usually be farther below zero and will hence be the top line on the graph.
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.