# Dynamic pressure

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

$q={\frac {1}{2}}\rho \,u^{2}$ where (using SI units):

 $q,\;$ dynamic pressure in pascals (i.e., kg/m⋅s2), $\rho ,\;$ fluid mass density (e.g. in kg/m3, in SI units), $u,\;$ flow speed in m/s.

It can be thought of as the fluid's kinetic energy per unit volume.

For incompressible flow, the dynamic pressure of a fluid is the difference between its total pressure and static pressure. From Bernoulli's law, dynamic pressure is given by

$p_{0}-p_{\text{s}}={\frac {1}{2}}\rho \,u^{2}$ where $p_{0}$ and $p_{\text{s}}$ are the total and static pressures, respectively.

## Physical meaning

Dynamic pressure is the kinetic energy per unit volume of a fluid. Dynamic pressure is one of the terms of Bernoulli's equation, which can be derived from the conservation of energy for a fluid in motion.

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

$\rho {\frac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla )\mathbf {u} -\rho \nu \,\nabla ^{2}\mathbf {u} =-\nabla p+\rho \mathbf {g}$ By a vector calculus identity ($u=|\mathbf {u} |$ )

$\nabla (u^{2}/2)=(\mathbf {u} \cdot \nabla )\mathbf {u} +\mathbf {u} \times (\nabla \times \mathbf {u} )$ so that for incompressible, irrotational flow ($\nabla \times \mathbf {u} =0$ ), 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 velocity head (hv) so that the dynamic pressure is equal to $\rho gh_{v}$ .

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 aerodynamic 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.

## Uses A flow of air through a venturi meter, showing the columns connected in a U-shape (a manometer) and partially filled with water. The meter is "read" as a differential pressure head in cm or inches of water and is equivalent to the difference in velocity head.

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.

When the dynamic pressure is divided 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 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.

## Compressible flow

Many authors define dynamic pressure only for incompressible flows. (For compressible flows, these authors use the concept of impact pressure.) However, the definition of dynamic pressure can be extended to include compressible flows.

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.

Using the definition of the speed of sound $a$ and of Mach number $M$ :

$a={\sqrt {\gamma p \over \rho }}$ and   $M={\frac {u}{a}},$ and also ${\textstyle q={\frac {1}{2}}\rho \,u^{2}}$ , dynamic pressure can be rewritten as:

$q=M^{2}{\frac {1}{2}}\,\gamma \,p\,,$ where:

 $p,$ gas (static) pressure (expressed in pascals, in the SI system) $\rho =mn,\;$ mass density (in kg/m3) is always the product between number density and the gas average molecular mass $M,\;$ Mach number (non-dimensional), $\gamma ,\;$ ratio of specific heats (non-dimensional; 1.4 for air at sea-level conditions), $u,\;$ flow speed in m/s, $a,\;$ speed of sound in m/s