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In incompressible [[fluid dynamics]] '''dynamic pressure''' (indicated with ''q'', or ''Q'', and sometimes called '''velocity pressure''') is the quantity defined by:<ref name=LJC3.5>Clancy, L.J., ''Aerodynamics'', Section 3.5</ref>

:<math>q = \tfrac12\, \rho\, v^{2},</math>

where (using [[International System of Units|SI]] units):

:{| border="0"
|-
| <math>q\;</math>    || = dynamic [[pressure]] in [[Pascal (unit)|pascals]],
|-
| <math>\rho\;</math> || = fluid [[density]] in kg/m<sup>3</sup> (e.g. [[density of air]]),
|-
| <math>v\;</math>    || = fluid [[velocity]] in m/s.
|}

== Physical meaning ==
Dynamic pressure is the [[kinetic energy]] per unit volume of a fluid particle. Dynamic pressure is in fact one of the terms of [[Bernoulli's equation#Bernoulli_equations|Bernoulli's equation]], which can be derived from the [[conservation of energy]] for a fluid in motion. In simplified cases, the dynamic pressure is equal to the difference between the [[stagnation pressure]] and the [[static pressure]].<ref name=LJC3.5/>

Another important aspect of dynamic pressure is that, as [[dimensional analysis]] shows, the [[aerodynamics|aerodynamic]] stress (i.e. [[Stress (physics)|stress]] within a structure subject to aerodynamic forces) experienced by an aircraft traveling at speed <math>v</math> is proportional to the air density and square of <math>v</math>, i.e. proportional to <math>q</math>.
Therefore, by looking at the variation of <math>q</math> during flight, it is possible to determine how the stress will vary and in particular when it will reach its maximum value. The point of maximum aerodynamic load is often referred to as ''[[max Q]]'' and it is a critical parameter in many applications, such as during spacecraft launch.

==Uses==
The dynamic pressure, along with the [[static pressure]] and the pressure due to elevation, is used in [[Bernoulli's principle]] as an [[First law of thermodynamics|energy balance]] on a [[closed system]]. The three terms are used to define the state of a closed system of an [[incompressible]], constant-density fluid.

If we were to divide the dynamic pressure by the product of fluid density and [[Standard gravity|acceleration due to gravity, g]], the result is called [[velocity head]], which is used in head equations like the one used for [[hydraulic head]].

== Compressible flow ==
Many authors define ''dynamic pressure'' only for incompressible flows. (For compressible flows, these authors use the concept of [[impact pressure]].)  However, some British authors extend their definition of ''dynamic pressure'' to include compressible flows.<ref>Clancy, L.J., ''Aerodynamics'', Section 3.12 and 3.13</ref><ref>"the dynamic pressure is equal to ''half rho vee squared'' only in incompressible flow."<br />Houghton, E.L. and Carpenter, P.W. (1993), ''Aerodynamics for Engineering Students'', Section 2.3.1</ref>

If the fluid in question can be considered an [[ideal gas]] (which is generally the case for air), the dynamic pressure can be expressed as a function of fluid pressure and [[Mach number]].

By applying the [[ideal gas law]]:<ref>Clancy, L.J., ''Aerodynamics'', Section 10.4</ref>

:<math>p_s = \rho_m\, R\, T,\,</math>

the definition of [[speed of sound]] <math>a</math> and of Mach number <math>M</math>:<ref>Clancy, L.J., ''Aerodynamics'', Section 10.2</ref>

:<math>a = \sqrt{\gamma\, R\, T \over m_m}</math> {{pad|3em}} and {{pad|3em}} <math>M = \frac{v}{a},</math>

and also <math>q = \tfrac12\, \rho\, v^2 </math>, dynamic pressure can be rewritten as:<ref>Liepmann & Roshko, ''Elements of Gas Dynamics'', p. 55.</ref>

:<math>q = \tfrac12\, \gamma\, p_{s}\, M^{2},</math>

where (using SI units):
:{| border="0"
|-
| <math>p_{s}\;</math>  || = static pressure in [[Pascal (unit)|Pascals]], Is also the basic SI unit of Pressure
|-
| <math>\rho_m\;</math> || = molar density of the ideal gas in mol/m<sup>3</sup>
|-
| <math>m_m\;</math>    || = mass of a mole of the ideal gas in kg/mol
|-
| <math>\rho\    = \rho_m m_m\;</math> || = density of the ideal  gas in kg/m<sup>3</sup>
|-
| <math>R\;</math>      || = [[gas constant]] (8.3144 J/(mol·K)),
|-
| <math>T\;</math>      || = [[absolute temperature]] in  [[Kelvin]] (K),
|-
| <math>M\;</math>      || = Mach number (non-dimensional),
|-
| <math>\gamma\;</math> || = [[ratio of specific heats]] (non-dimensional) (1.4 for air at sea level conditions),
|-
| <math>v\;</math>      || = fluid velocity in m/s</sup>,
|-
| <math>a\;</math>      || = speed of sound in m/s</sup>
|}

== See also ==
*[[Pressure]]
*[[Hydraulic head]]
*[[Total dynamic head]]
*[[Bernoulli's_principle#Derivations_of_Bernoulli_equation|Derivations of Bernoulli equation]]

== References ==
* Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. ISBN 0-273-01120-0
* Houghton, E.L. and Carpenter, P.W. (1993), ''Aerodynamics for Engineering Students'', Butterworth and Heinemann, Oxford UK.  ISBN 0-340-54847-9
* {{citation | title=Elements of Gas Dynamics | first1=Hans Wolfgang | last1=Liepmann | first2=Anatol | last2=Roshko | publisher=Courier Dover Publications | year=1993 | isbn=0-486-41963-0 }}

===Notes===
{{reflist}}

== External links ==
* Definition of dynamic pressure on [http://scienceworld.wolfram.com/physics/DynamicPressure.html ''Eric Weisstein's World of Science'']

[[Category:Fluid dynamics]]