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 the kinetic energy per unit volume of a fluid particle. Dynamic pressure is in fact one of the terms of 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.[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.

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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.

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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]

$p_s = \rho_m\, R\, T,\,$

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

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

and also $q = \tfrac12\, \rho\, v^2$, dynamic pressure can be rewritten as:[6]

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

where (using SI units):

 $p_{s}\;$ = static pressure in Pascals, $\rho_m\;$ = molar density of the ideal gas in mol/m3 $m_m\;$ = mass of a mole of the ideal gas in kg/mol $\rho\ = \rho_m m_m\;$ = density of the ideal gas in kg/m3 $R\;$ = gas constant (8.3144 J/(mol·K)), $T\;$ = absolute temperature in Kelvin (K), $M\;$ = 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, $a\;$ = speed of sound in m/s
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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 0-486-41963-0

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.
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