= Pressure coefficient
{title2=$C_p=(p-p_\infty)/q_\infty$}
The <pressure coefficient> compares a local <static pressure> with a specified reference <static pressure>, using a fixed reference <dynamic pressure>:
$$
C_p(\mathbf x)=\frac{p(\mathbf x)-p_\infty}{q_\infty},\qquad
q_\infty=\frac12\rho_\infty U_\infty^2>0.
$$
The reference <mass density>, <speed>, <pressure> and location must accompany a reported <pressure coefficient>. Keep the same $q_\infty$ throughout a surface distribution; replacing it by local $\rho U^2/2$ gives a different quantity and fails at a <stagnation point>. The definition itself requires neither <Bernoulli's principle> nor <inviscid flow>, so measured <pressure coefficients> remain useful when those idealizations fail.
Positive $C_p$ means $p>p_\infty$; zero means equality; negative $C_p$ means $p<p_\infty$. This sign does not specify the sign of <absolute pressure>. For stipulated $p_\infty=101325\,\mathrm{Pa}$ absolute and $q_\infty=240\,\mathrm{Pa}$, $C_p=-3$ gives $p=100605\,\mathrm{Pa}$ absolute: a $720\,\mathrm{Pa}$ deficit, not a negative <absolute pressure>. If $p_\infty$ equals <atmospheric pressure>, the numerator is <gauge pressure>; a <wind tunnel>'s reference <static pressure> need not equal room <atmospheric pressure>. These are reference choices, not different physical <pressures>.
Under the same-height, constant-<mass density> $\rho=\rho_\infty$, <steady flow> and <inviscid flow> hypotheses of the <Bernoulli equation>, and with the same <Bernoulli invariant> as the reference flow,
$$
p+\frac12\rho U^2=p_\infty+\frac12\rho U_\infty^2,
\qquad C_p=1-\left(\frac{U}{U_\infty}\right)^2.
$$
The first equation immediately gives the second after subtracting $p_\infty$ and dividing by $q_\infty$. Hence a lossless <stagnation point> has $C_p=1$, $U=U_\infty$ gives $C_p=0$, and $U=2U_\infty$ gives $C_p=-3$. For a real stationary wall satisfying the <no-slip boundary condition>, the <velocity> is zero right at the wall; inserting that wall <velocity> in an <inviscid flow> formula would wrongly predict $C_p=1$ everywhere. A thin <boundary layer> approximation instead relates the wall <pressure> to the outer-flow <pressure> and uses the outer <velocity> where the <Bernoulli equation> applies.
\Image[/cylinder-pressure-taps-ideal-coefficient.png]
{title=Ideal cylinder <pressure coefficients>, <pressure tapping> and cancellation of pressure <drag>}
{description=Original <potential flow around a circular cylinder> with zero <circulation>, not experimental data or a fit. <Angle> is measured counterclockwise from the downstream direction. Hollow circles mark sampling locations; arrows show only the reference-subtracted <pressure> <force> per unit <area>. A separate <pressure tapping> measures <static pressure>, rather than <stagnation pressure>. The front and rear ideal <pressure> <forces> cancel, so this model predicts zero <drag>.}
For <potential flow around a circular cylinder> of radius $a$ with zero <circulation>, the surface tangential <velocity> is $u_\theta=-2U_\infty\sin\theta$. Thus the ideal <pressure coefficient> is
$$
C_p(\theta)=1-4\sin^2\theta.
$$
With $\theta$ measured from downstream, the front is at $\pi$ and the rear at zero. Both ideal <stagnation points> have $C_p=1$, while the sides at $\pi/2$ and $3\pi/2$ have $C_p=-3$. This exact model is a useful baseline for an experiment; its complete rearward <pressure> recovery is not a prediction for a separated real <fluid wake>.
To obtain a <force>, integrate the local normal contributions with their directions. Let $\mathbf n$ be the unit <normal vector> from the body into the <fluid> and $\Sigma$ its closed wetted surface:
$$
\mathbf F_p=-\int_\Sigma p\mathbf n\,dA
=-q_\infty\int_\Sigma C_p\mathbf n\,dA,
\qquad \int_\Sigma\mathbf n\,dA=\mathbf0.
$$
The second equality removes a spatially constant reference <pressure> over the closed surface; it does not remove a nonuniform <hydrostatic pressure> or justify dropping the reference term on an isolated open patch. A negative $C_p$ contributes outwards in this reference-subtracted calculation, even though a positive <absolute pressure> still pushes inwards locally. Project $\mathbf F_p$ onto the chosen <lift> and <drag> directions. The full <fluid> <force> also includes $\int_\Sigma\boldsymbol\sigma_{\rm visc}\mathbf n\,dA$ from the <viscous stress tensor>, including tangential <shear stress>; <pressure tapping> does not measure that contribution.
For the two-dimensional cylinder, <force> per unit span follows from $dA=a\,d\theta$ per unit span and $\mathbf n=(\cos\theta,\sin\theta)$:
$$
(D'_p,L'_p)=-q_\infty a\int_0^{2\pi}C_p(\theta)(\cos\theta,\sin\theta)\,d\theta.
$$
Substituting $C_p=1-4\sin^2\theta$ gives $(0,0)$: opposite points have identical <pressure coefficients> and opposite <normal vectors>. Zero <drag> despite a strongly nonuniform <pressure> is the instructive limitation associated with <D'Alembert's paradox>. Real <viscosity>, <boundary layers> and <flow separation> can prevent rearward <pressure> recovery and produce <pressure drag>. Thus the measured curve can test the model and estimate a <pressure> contribution without proving that total <drag> vanishes.
A motivating measurement is Virginia Tech's <AOE 3054 laboratory manual/Experiment 3 - Flow Past a Circular Cylinder>. Its open-jet <wind tunnel> holds a rotatable $140\,\mathrm{mm}$ cylinder with end plates and 36 one-millimetre <pressure tappings> at ten-degree intervals. A reference <Pitot-static tube> supplies $p_{0\infty}$ and $p_\infty$ to one digital <manometer>; a second connects a surface tap to its positive port and reference <static pressure> to its negative port. Record the simultaneous readings:
$$
\Delta p_{\rm ref}=p_{0\infty}-p_\infty,\qquad
\Delta p_i=p_i-p_\infty,\qquad
C_{p,i}=\frac{\Delta p_i}{\Delta p_{\rm ref}}.
$$
The last equality uses the low-speed stopping relation, not an additional <speed> measurement. The manual requires sealed tubing and zero checks with the tunnel off. It also provides a water-column <manometer>: read each column's initial level, and subtract the independently measured reference-to-atmosphere <pressure> from the tap-to-atmosphere readings. Record tap positions after rotation. Compare the two instruments and check spanwise uniformity; end plates do not guarantee two-dimensional flow. <AOE 3054 laboratory manual/Experiment 3 - Flow Past a Circular Cylinder>{ref}
The inferential distinction matters: the <manometers> directly measure <pressure> differences, the ratio estimates <pressure coefficients>, and a <surface integral> estimates the pressure part of the <force>. Measure room <absolute pressure> and <temperature> if converting the reference difference to <speed> through the <ideal gas law> $\rho=p_{\rm abs}/(RT)$; use <thermodynamic temperature> in that equation. For <force> reconstruction, cover the complete contour and use <numerical integration> with the tap geometry. Finite tap spacing, unknown end effects, probe disturbance and fluctuations limit that reconstruction; tubes and instruments also average or delay changing <pressures>. Retain the signed raw readings, repeated readings and reference readings so that these limits can be assessed. A small reference difference magnifies any zero error in the ratio, rather than producing a reliably large $C_p$.
Back to article page
