= Dynamic pressure
{title2=$q=\rho U^2/2$}
<Dynamic pressure> is the bulk <kinetic energy> per unit <volume> associated with <fluid> motion at <speed> $U$ in a specified <reference frame>:
$$
q=\frac12\rho U^2,\qquad
\frac{\tfrac12mU^2}{V}=\frac12\rho U^2,\qquad
\mathrm{J\,m^{-3}}=\mathrm{Pa}.
$$
Here $\rho=m/V$ is <mass density>. The word "pressure" identifies its units and its use as a <pressure> scale; $q$ is not an extra isotropic <stress> to add to the local <static pressure> in every surface <traction>. A stationary <fluid> can have large <static pressure> with $q=0$. Conversely, knowing $U$ and $\rho$ fixes $q$ without fixing <static pressure>. The definition remains useful in <viscous fluid flow> and <compressible flow>, even when the simplest <Bernoulli equation> is inapplicable. <Dynamic Pressure by NASA Glenn Research Center>{ref}
For stipulated <mass density> $\rho_\infty=1.20\,\mathrm{kg\,m^{-3}}$ and reference <speed> $U_\infty=20\,\mathrm{m\,s^{-1}}$, $q_\infty=240\,\mathrm{Pa}$. At $40\,\mathrm{m\,s^{-1}}$ with the same <mass density>, $q_\infty=960\,\mathrm{Pa}$. This quadratic scaling makes <dynamic pressure> a useful normalization for the <lift coefficient>, <drag coefficient> and <pressure coefficient>; it predicts four times a particular <force> only if its corresponding coefficient remains unchanged.
\Image[/pitot-static-pressure-difference.png]
{title=A <Pitot-static tube> compares <stagnation pressure> with <static pressure>}
{description=Original schematic with separate <pressure> passages and a differential readout. The right panel uses stipulated values, not measurements: ideal stopping in <incompressible flow> at unchanged height raises <static pressure> by 240 Pa as the <kinetic energy> per unit <volume> falls by 240 Pa. The diagram is not a probe design or a claim that real <static pressure> ports are error free.}
The connection with <stagnation pressure> requires hypotheses. For <steady flow>, constant <mass density>, <inviscid flow>, and <gravity> as the only <body force>, the <Bernoulli equation> gives, along one <streamline>,
$$
p+\frac12\rho U^2+\rho gz=B.
$$
Here $p$ is <static pressure>, $g$ is <gravitational acceleration>, and $z$ is height. Bring that flow to rest without losses at the same height and the limiting <static pressure> is the <stagnation pressure> $p_0$:
$$
p_0=p+q,\qquad p_0-p=\frac12\rho U^2.
$$
There must be no intervening pump supplying <work> or other energy input. Constancy along one <streamline> does not assert equal $B$ on all <streamlines>; <irrotational flow> in a connected region supplies that additional conclusion. Changing height requires retaining $\rho gz$. In <compressible flow> the stopping <pressure> difference generally differs from $\rho U^2/2$, and <viscous dissipation> can reduce recovered <stagnation pressure>. <Bernoulli’s Equation by NASA Glenn Research Center>{ref}
A <Pitot-static tube> turns this relation into a measurement: its upstream-facing opening senses <stagnation pressure>, while separate side openings sample <static pressure>. Connect these to the positive and negative ports of a differential <pressure transducer> or <manometer>. Under the hypotheses above, the measured difference $\Delta p$ gives
$$
U=\sqrt{\frac{2\Delta p}{\rho}}.
$$
Thus the stipulated $240\,\mathrm{Pa}$ and $1.20\,\mathrm{kg\,m^{-3}}$ imply $20\,\mathrm{m\,s^{-1}}$, not an independently measured <speed>. Align the probe with the incoming flow and keep its passages clear: blockage and pinched tubing invalidate the comparison. At very low <speed>, the difference can be smaller than instrument error; at high <Mach number>, <compressibility> and possibly a <shock wave> require a different reduction. <Pitot – Static Tube – Speedometer by NASA Glenn Research Center>{ref}
A useful control is <differential-pressure calibration>. For a digital <manometer>, zeroing establishes the reading at equal port <pressures>; it does not establish the gain away from zero. The manufacturer's <Series 475 Handheld Digital Manometer: Specifications - Installation and Operating Instructions by Dwyer Instruments, Inc.> directs the user to vent both ports to the atmosphere before zeroing. Its stated accuracy is a fraction of full scale, not of the reading, and depends on instrument <temperature>. Select a suitable range and record the instrument variant and units; the manual covers several ranges.<Series 475 Handheld Digital Manometer: Specifications - Installation and Operating Instructions by Dwyer Instruments, Inc.>[p. 1: Specifications, Model Chart and ZERO instructions]{ref}
An independent physical check can use a liquid-column <manometer>: apply a stable <pressure> difference to both instruments and compare the digital reading with the <hydrostatic pressure> inferred from the vertical level difference. For equal-height connections carrying a gas of <mass density> $\rho_g$ above a manometric liquid of <mass density> $\rho_m$,
$$
|\Delta p|=(\rho_m-\rho_g)g|h|.
$$
This follows by balancing <hydrostatic pressure> down and up the two connected columns. Use vertical height, even if a tube is inclined. Known <mass densities>, measured height and <gravitational acceleration> set the comparison scale. Check several differences across the working range and reverse the port connections to check the sign. This is a proposed independent check, not a claim that the cited teaching experiment performed a full instrument calibration. Recheck zero after reconnecting the sealed tubing; a zero adjustment alone cannot detect a blocked passage or an incorrect scale.
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