Dynamic pressure

In incompressible fluid dynamics dynamic pressure (indicated with q, or Q, and sometimes called velocity pressure) is the quantity defined by:^[1]

$q = \tfrac12\, \rho\, v^{2},$

where (using SI units):

$q\;$	= dynamic pressure in pascals,
$\rho\;$	= fluid density in kg/m3 (e.g. density of air),
$v\;$	= fluid velocity in m/s.

Physical meaning

Dynamic pressure is closely related to the kinetic energy of a fluid particle, since both quantities are proportional to the particle's mass (through the density, in the case of dynamic pressure) and square of the velocity. Dynamic pressure is in fact one of the terms of Bernoulli's equation, which is essentially an equation of energy conservation for a fluid in motion. The dynamic pressure is equal to the difference between the stagnation pressure and the static pressure.^[1]

Another important aspect of dynamic pressure is that, as dimensional analysis shows, the aerodynamic stress (i.e. stress within a structure subject to aerodynamic forces) experienced by an aircraft traveling at speed v is proportional to the air density and square of v, i.e. proportional to q. Therefore, by looking at the variation of q 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 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 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.^[2][3]

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:^[4]

$p = \rho\, R\, T,\,$

the definition of speed of sound a and of Mach number M:^[5]

$a = \sqrt{\gamma\, R\, T}$   and   $M = \frac{v}{a},$

dynamic pressure can be rewritten as:^[6]

$q = \tfrac12\, \gamma\, p_{s}\, M^{2},$

where (using SI units):

$p_{s}\;$	= static pressure in pascals,
$\rho\;$	= density in kg·m−3, or kg/kL
$R\;$	= specific gas constant (287.05 J·kg−1·K−1 for air),
$T\;$	= absolute temperature in kelvin,
$M\;$	= Mach number (non-dimensional),
$\gamma\;$	= ratio of specific heats (non-dimensional) (1.4 for air at sea level conditions),
$v\;$	= fluid velocity in m·s−1,
$a\;$	= speed of sound in m·s−1

See also

• Pressure
• Hydraulic head
• Total dynamic head
• 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
• Liepmann, Hans Wolfgang; Roshko, Anatol (1993), Elements of Gas Dynamics, Courier Dover Publications, ISBN 0486419630 

Notes

1. ^ ^{a b} Clancy, L.J., Aerodynamics, Section 3.5
2. ^ Clancy, L.J., Aerodynamics, Section 3.12 and 3.13
3. ^ "the dynamic pressure is equal to half rho vee squared only in incompressible flow."
   Houghton, E.L. and Carpenter, P.W. (1993), Aerodynamics for Engineering Students, Section 2.3.1
4. ^ Clancy, L.J., Aerodynamics, Section 10.4
5. ^ Clancy, L.J., Aerodynamics, Section 10.2
6. ^ Liepmann & Roshko, Elements of Gas Dynamics, p. 55.

External links

• Definition of dynamic pressure on Eric Weisstein's World of Science