Pressure coefficient

Incompressible flow

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

${\displaystyle C_p{|}_{M\approx 0}=\frac{p-p_{\mathrm{\infty }}}{p_0-p_{\mathrm{\infty }}}=1-(\frac{u}{u_{\mathrm{\infty }}}{)}^2}$

where:

${\displaystyle u}$ is the flow speed at the point at which pressure coefficient is being evaluated

${\displaystyle M}$ is the Mach number, which is taken in the limit of zero

${\displaystyle p_0}$ 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.

• ${\displaystyle C_p}$ of zero indicates the pressure is the same as the freestream pressure.
• ${\displaystyle C_p}$ of one corresponds to the stagnation pressure and indicates a stagnation point.
• the most negative values of ${\displaystyle C_p}$ 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.

Locations where ${\displaystyle C_p=-1}$ 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

In the flow of compressible fluids such as air, and particularly the high-speed flow of compressible fluids, ${\displaystyle \frac{1}{2}\rho v^2}$ (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.

Perturbation theory

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

${\displaystyle \mathrm{\Phi }=u_{\mathrm{\infty }}x+\varphi (x,y,z)}$

Using Bernoulli's equation,

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

which can be rewritten as

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

where ${\displaystyle a}$ is the sound speed.

The pressure coefficient becomes

${\displaystyle \begin{array}{cc}{C}_p& =\frac{p-p_{\mathrm{\infty }}}{\frac{\gamma }{2}{p}_{\mathrm{\infty }}{M}^2}=\frac{2}{\gamma M^2}\left[{\left(\frac{a}{a_{\mathrm{\infty }}}\right)}^{\frac{2\gamma }{\gamma -1}}-1\right]\\ & =\frac{2}{\gamma M^2}\left[{\left(\frac{\gamma -1}{a_{\mathrm{\infty }}^2}(\frac{u_{\mathrm{\infty }}^2}{2}-{\mathrm{\Phi }}_t-\frac{\mathrm{\nabla }\mathrm{\Phi }\cdot \mathrm{\nabla }\mathrm{\Phi }}{2})+1\right)}^{\frac{\gamma }{\gamma -1}}-1\right]\\ & \approx \frac{2}{\gamma M^2}\left[{\left(1-\frac{\gamma -1}{a_{\mathrm{\infty }}^2}({\varphi }_t+u_{\mathrm{\infty }}{\varphi }_x)\right)}^{\frac{\gamma }{\gamma -1}}-1\right]\\ & \approx -\frac{2{\varphi }_t}{u_{\mathrm{\infty }}^2}-\frac{2{\varphi }_x}{u_{\mathrm{\infty }}}\end{array}}$

where ${\displaystyle a_{\mathrm{\infty }}}$ 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:

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

where ${\displaystyle w}$ is the downwash speed and ${\displaystyle a}$ is the sound speed.

${\displaystyle C_p=\frac{p-p_{\mathrm{\infty }}}{\frac{\gamma }{2}{p}_{\mathrm{\infty }}{M}^2}=\frac{2}{\gamma M^2}\left[{\left(1+\frac{\gamma -1}{2}\frac{w}{a}\right)}^{\frac{2\gamma }{\gamma -1}}-1\right]}$

The surface is defined as

${\displaystyle F(x,y,z,t)=z-f(x,y,t)=0}$

The slip velocity boundary condition leads to

${\displaystyle \frac{\mathrm{\nabla }F}{|\mathrm{\nabla }F|}(u_{\mathrm{\infty }}+{\varphi }_x,{\varphi }_y,{\varphi }_z)=V_{\text{wall}}\cdot \frac{\mathrm{\nabla }F}{|\mathrm{\nabla }F|}=-\frac{\mathrm{\partial }F}{\mathrm{\partial }t}\frac{1}{|\mathrm{\nabla }F|}}$

The downwash speed ${\displaystyle w}$ is approximated as

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

Hypersonic flow

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:

1. The flow can be modeled as a stream of particles in rectilinear motion
2. Upon impact with a surface, all normal momentum is lost
3. All tangential momentum is conserved, and flow follows the body

For a freestream velocity ${\displaystyle V_{\mathrm{\infty }}}$ impacting a surface of area ${\displaystyle A}$, which is inclined at an angle ${\displaystyle \theta }$ relative to the freestream, the change in normal momentum is ${\displaystyle V_{\mathrm{\infty }}\sin \theta }$ and the mass flux incident on the surface is ${\displaystyle {\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}A\sin \theta }$, with ${\displaystyle {\rho }_{\mathrm{\infty }}}$ being the freestream air density. Then the momentum flux, equal to the force exerted on the surface ${\displaystyle F}$, from Newton's second law is equal to:

${\displaystyle F=({\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}A\sin \theta )(V_{\mathrm{\infty }}\sin \theta )={\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^{2}A{\sin }^2\theta }$

Dividing by the surface area, it is clear that the force per unit area is equal to the pressure difference between the surface pressure ${\displaystyle p}$ and the freestream pressure ${\displaystyle p_{\mathrm{\infty }}}$, leading to the relation:

${\displaystyle \frac{F}{A}=p-p_{\mathrm{\infty }}={\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2{\sin }^2\theta ⟹\frac{p-p_{\mathrm{\infty }}}{\frac{1}{2}{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}=2{\sin }^2\theta }$

The last equation may be identified as the pressure coefficient, meaning that Newtonian theory predicts that the pressure coefficient in hypersonic flow is:

${\displaystyle C_p=2{\sin }^2\theta }$

For very high speed flows, and vehicles with sharp surfaces, the Newtonian theory works very well.

Modified Newtonian law

A modification to the Newtonian theory, specifically for blunt bodies, was proposed by Lester Lees:

${\displaystyle C_p=C_{p,max}{\sin }^2\theta }$

where ${\displaystyle C_{p,max}}$ is the maximum value of the pressure coefficient at the stagnation point behind a normal shock wave:

${\displaystyle C_{p,max}=\frac{p_o-p_{\mathrm{\infty }}}{\frac{1}{2}{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}=\frac{p_{\mathrm{\infty }}}{\frac{1}{2}{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}\left(\frac{p_o}{p_{\mathrm{\infty }}}-1\right)=\frac{2}{\gamma M_{\mathrm{\infty }}^2}\left(\frac{p_o}{p_{\mathrm{\infty }}}-1\right)}$

where ${\displaystyle p_o}$ is the stagnation pressure and ${\displaystyle \gamma }$ is the ratio of specific heats. The last relation is obtained from the ideal gas law ${\displaystyle p=\rho RT}$, Mach number ${\displaystyle M=V/a}$, and speed of sound ${\displaystyle a=\sqrt{\gamma RT}}$. The Rayleigh pitot tube formula for a calorically perfect normal shock says that the ratio of the stagnation and freestream pressure is:

${\displaystyle \frac{p_o}{p_{\mathrm{\infty }}}={\left[\frac{(\gamma +1{)}^2{M}_{\mathrm{\infty }}^2}{4\gamma M_{\mathrm{\infty }}^2-2(\gamma -1)}\right]}^{\gamma /(\gamma -1)}\left[\frac{\gamma (2M_{\mathrm{\infty }}^2-1)+1}{\gamma +1}\right]}$

Therefore, it follows that the maximum pressure coefficient for the Modified Newtonian law is:

${\displaystyle C_{p,max}=\frac{2}{\gamma M_{\mathrm{\infty }}^2}\left\{{\left[\frac{(\gamma +1{)}^2{M}_{\mathrm{\infty }}^2}{4\gamma M_{\mathrm{\infty }}^2-2(\gamma -1)}\right]}^{\gamma /(\gamma -1)}\left[\frac{\gamma (2M_{\mathrm{\infty }}^2-1)+1}{\gamma +1}\right]-1\right\}}$

In the limit when ${\displaystyle M_{\mathrm{\infty }}\to \mathrm{\infty }}$, the maximum pressure coefficient becomes:

${\displaystyle C_{p,max}={\left[\frac{(\gamma +1{)}^2}{4\gamma }\right]}^{\gamma /(\gamma -1)}\left(\frac{4}{\gamma +1}\right)}$

And as ${\displaystyle \gamma \to 1}$, ${\displaystyle C_{p,max}=2}$, 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.

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 ${\displaystyle C_p}$ 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.

${\displaystyle C_l=\frac{1}{x_{TE}-x_{LE}}\underset{x_{LE}}{\overset{x_{TE}}{\int }}\left(C_{p_l}(x)-C_{p_u}(x)\right)dx}$

where:

${\displaystyle C_{p_l}}$ is pressure coefficient on the lower surface

${\displaystyle C_{p_u}}$ is pressure coefficient on the upper surface

${\displaystyle x_{LE}}$ is the leading edge location

${\displaystyle x_{TE}}$ is the trailing edge location

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