Dynamic pressure

 ( Fluid Dynamics ) 

At a stagnation point the dynamic pressure is equal to the difference between the stagnation pressure and the static pressure, so the dynamic pressure in a flow field can be measured at a stagnation point.

Another important aspect of dynamic pressure is that, as dimensional analysis shows, the aerodynamics stress (i.e. stress within a structure subject to aerodynamic forces) experienced by an aircraft travelling 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 launch vehicles.

Dynamic pressure can also appear as a term in the incompressible Navier-Stokes equation which may be written:

$\backslash rho\backslash frac\{\backslash partial\; \backslash mathbf\{u\}\}\{\backslash partial\; t\}\; +\; \backslash rho(\backslash mathbf\{u\}\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\{u\}\; -\; \backslash rho\backslash nu\; \backslash ,\backslash nabla^2\; \backslash mathbf\{u\}\; =\; -\; \backslash nabla\; p\; +\; \backslash rho\backslash mathbf\{g\}$

By a vector calculus identity ($u=|\; \backslash mathbf\{u\}\; |$)

$\backslash nabla\; (u^2/2)=(\backslash mathbf\{u\}\backslash cdot\; \backslash nabla)\; \backslash mathbf\{u\}\; +\; \backslash mathbf\{u\}\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf\{u\})$

so that for incompressible, irrotational flow ($\backslash nabla\; \backslash times\; \backslash mathbf\{u\}=0$), the second term on the left in the Navier-Stokes equation is just the gradient of the dynamic pressure. In hydraulics, the term $u^2/2g$ is known as the Hydraulic head (h_v) so that the dynamic pressure is equal to $\backslash rho\; g\; h\_v$.

When the dynamic pressure is divided by the product of fluid density and Standard gravity, the result is called velocity head, which is used in head equations like the one used for pressure head and hydraulic head. In a venturi flow meter, the differential pressure head can be used to calculate the differential velocity head, which are equivalent in the adjacent picture. An alternative to velocity head is dynamic head.

For compressible flow the isentropic relations can be used (also valid for incompressible flow):

$q=p\_s\backslash left(1+\backslash frac\{\backslash gamma-1\}\{2\}M^2\backslash right)^\{\backslash frac\{\backslash gamma\}\{\backslash gamma-1\}\}-p\_s$

Where:

>

dynamic pressure,

static pressure

Mach number (non-dimensional),

ratio of specific heats (non-dimensional; 1.4 for air at sea-level conditions),

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See also

• Pressure
• Pressure head
• Hydraulic head
• Total dynamic head
• Drag coefficient, Lift coefficient and pitching moment coefficients
• Derivations of Bernoulli equation

• L. J. Clancy (1975), Aerodynamics, Pitman Publishing Limited, London.
• Houghton, E.L. and Carpenter, P.W. (1993), Aerodynamics for Engineering Students, Butterworth and Heinemann, Oxford UK.

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Notes

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External links

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

Categories: Fluid Dynamics