Pressure coefficient

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, $C p$ .

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 [1][2]

C_{p}={p-p_{\infty } \over {\frac {1}{2}}\rho _{\infty }V_{\infty }^{2}}={p-p_{\infty } \over p_{0}-p_{\infty }}

where:

p

is the static pressure at the point at which pressure coefficient is being evaluated

p_{\infty }

is the static pressure in the freestream (i.e. remote from any disturbance)

p_{0}

is the stagnation pressure in the freestream (i.e. remote from any disturbance)

\rho _{\infty }

is the freestream fluid density (Air at sea level and 15 °C is 1.225

{\rm {kg/m^{3}}}

)

V_{\infty }

is the freestream velocity of the fluid, or the velocity of the body through the fluid

Incompressible flow

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

C_{p}|_{Ma\,\approx \,0}={1-{\bigg (}{\frac {u}{u_{\infty }}}{\bigg )}^{2}}

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.

$C_{p}$ of zero indicates the pressure is the same as the free stream pressure.
$C_{p}$ of one corresponds to the stagnation pressure and indicates a stagnation point.
the most negative values of $C_{p}$ 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.

$C_{p}$ 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, ${\rho v^{2}}/2$ (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]

$C_{p}$ greater than one indicates the freestream flow is compressible.

Perturbation theory

The pressure coefficient $C_{p}$ can be estimated for irrotational and isentropic flow by introducing the potential $\Phi$ and the perturbation potential $\phi$ , normalized by the free-stream velocity $u_{\infty }$

\Phi =u_{\infty }x+\phi (x,y,z)

Using Bernoulli's equation,

{\frac {\partial \Phi }{\partial t}}+{\frac {\nabla \Phi \cdot \nabla \Phi }{2}}+{\frac {\gamma }{\gamma -1}}{\frac {p}{\rho }}={\text{constant}}

which can be rewritten as

{\frac {\partial \Phi }{\partial t}}+{\frac {\nabla \Phi \cdot \nabla \Phi }{2}}+{\frac {a^{2}}{\gamma -1}}={\text{constant}}

where $a$ is the sound speed.

The pressure coefficient becomes

{\begin{aligned}C_{p}&={\frac {p-p_{\infty }}{{\frac {\gamma }{2}}p_{\infty }M^{2}}}={\frac {2}{\gamma M^{2}}}\left[\left({\frac {a}{a_{\infty }}}\right)^{\frac {2\gamma }{\gamma -1}}-1\right]\\&={\frac {2}{\gamma M^{2}}}\left[\left({\frac {\gamma -1}{a_{\infty }^{2}}}({\frac {u_{\infty }^{2}}{2}}-\Phi _{t}-{\frac {\nabla \Phi \cdot \nabla \Phi }{2}})+1\right)^{\frac {\gamma }{\gamma -1}}-1\right]\\&\approx {\frac {2}{\gamma M^{2}}}\left[\left(1-{\frac {\gamma -1}{a_{\infty }^{2}}}(\phi _{t}+u_{\infty }\phi _{x})\right)^{\frac {\gamma }{\gamma -1}}-1\right]\\&\approx -{\frac {2\phi _{t}}{u_{\infty }^{2}}}-{\frac {2\phi _{x}}{u_{\infty }}}\end{aligned}}

where $a_{\infty }$ 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:

p=p_{\infty }\left(1+{\frac {\gamma -1}{2}}{\frac {w}{a}}\right)^{\frac {2\gamma }{\gamma -1}}

where $w$ is the downwash speed and $a$ is the sound speed.

C_{p}={\frac {p-p_{\infty }}{{\frac {\gamma }{2}}p_{\infty }M^{2}}}={\frac {2}{\gamma M^{2}}}\left[\left(1+{\frac {\gamma -1}{2}}{\frac {w}{a}}\right)^{\frac {2\gamma }{\gamma -1}}-1\right]

The surface is defined as

F(x,y,z,t)=z-f(x,y,t)=0

The slip velocity boundary condition leads to

{\frac {\nabla F}{|\nabla F|}}(u_{\infty }+\phi _{x},\phi _{y},\phi _{z})=V_{\text{wall}}\cdot {\frac {\nabla F}{|\nabla F|}}=-{\frac {\partial F}{\partial t}}{\frac {1}{|\nabla F|}}

The downwash speed $w$ is approximated as

w={\frac {\partial f}{\partial t}}+u_{\infty }{\frac {\partial f}{\partial x}}

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 $C_{p}$ 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.

C_{l}={\frac {1}{x_{TE}-x_{LE}}}\int \limits _{x_{LE}}^{x_{TE}}\left(C_{p_{l}}(x)-C_{p_{u}}(x)\right)dx

where:

C_{p_{l}}

is pressure coefficient on the lower surface

C_{p_{u}}

is pressure coefficient on the upper surface

x_{LE}

is the leading edge location

x_{TE}

is the trailing edge location

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

References

L. J. Clancy (1975) Aerodynamics, § 3.6, Pitman Publishing Limited, London. ISBN 0-273-01120-0
Abbott and Von Doenhoff, Theory of Wing Sections, equation 2.24
Anderson, John D. Fundamentals of Aerodynamics. 4th ed. New York: McGraw Hill, 2007. 219.
https://thesis.library.caltech.edu/608/1/Scherer_lr_1950.pdf