Pressure coefficient

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In fluid dynamics, the pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field. The pressure coefficient is used in aerodynamics and hydrodynamics. 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]


is the static pressure at the point at which pressure coefficient is being evaluated
is the static pressure in the freestream (i.e. remote from any disturbance)
is the freestream fluid density (Air at sea level and 15 °C is 1.225 )
is the freestream velocity of the fluid, or the velocity of the body through the fluid

Incompressible flow[edit]

Using Bernoulli's equation, the pressure coefficient can be further simplified for potential flows (inviscid, and steady):[3]


is the flow speed at the point at which pressure coefficient is being evaluated
is the Mach number, which is taken in the limit of zero
is the flow's stagnation pressure

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.

  • of zero indicates the pressure is the same as the freestream pressure.
  • of one corresponds to the stagnation pressure and indicates a stagnation point.
  • the most negative values of in a liquid flow can be summed to the cavitation number to give the cavitation margin. If this margin is positive, the flow is locally fully liquid, while if it is zero or negative the flow is cavitating or gas.

Locations where 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.

In an incompressible 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.

Compressible flow[edit]

In the flow of compressible fluids such as air, and particularly the high-speed flow of compressible fluids, (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.[4]

Perturbation theory[edit]

The pressure coefficient can be estimated for irrotational and isentropic flow by introducing the potential and the perturbation potential , normalized by the free-stream velocity

Using Bernoulli's equation,

which can be rewritten as

where is the sound speed.

The pressure coefficient becomes

where is the far-field sound speed.

Local piston theory[edit]

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 is the downwash speed and is the sound speed.

The surface is defined as

The slip velocity boundary condition leads to

The downwash speed is approximated as

Pressure distribution[edit]

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 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[edit]

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.


is pressure coefficient on the lower surface
is pressure coefficient on the upper surface
is the leading edge location
is the trailing edge location

When the lower surface is higher (more negative) on the distribution it counts as a negative area as this will be producing down force rather than lift.

See also[edit]


  1. ^ L. J. Clancy (1975) Aerodynamics, § 3.6, Pitman Publishing Limited, London. ISBN 0-273-01120-0
  2. ^ Abbott and Von Doenhoff, Theory of Wing Sections, equation 2.24
  3. ^ Anderson, John D. Fundamentals of Aerodynamics. 4th ed. New York: McGraw Hill, 2007. 219.
  4. ^ [bare URL PDF]

Further reading[edit]

  • Abbott, I.H. and Von Doenhoff, A.E. (1959) Theory of Wing Sections, Dover Publications, Inc. New York, Standard Book No. 486-60586-8
  • Anderson, John D (2001) Fundamentals of Aerodynamic 3rd Edition, McGraw-Hill. ISBN 0-07-237335-0