Total Pressure

total_pressure.jpg

The differences between static and dynamic pressures are shown in the figure.  As the velocity increases the static pressure decreases. The total pressure is constant at each station.

By User:Rex the first (User:Rex the first) [Public domain or Public domain], via Wikimedia Commons

Total pressure is an important concept in fluid dynamics and applies to both liquids and gasses. The total pressure is based on a stagnation state that is achieved by bringing the flow to a state of zero velocity. A stagnation point may or may not actually exist in a particular flow field. However, the concept of a stagnation point and the associated total pressure is still very useful in the analysis of fluid dynamics problems.


Bernoulli’s equation gives rise to the concept of total pressure.  The figure shows streamlines for flow over a stationary cylinder.  Note that there is a center streamline for which the flow stops at the front of the cylinder.  All flow above this streamline will go over the cylinder, while all flow below this streamline will go under the cylinder.  The velocity of the fluid on the center streamline must equal zero since the cylinder is stationary.  Application of Bernoulli’s equation to the center streamline gives the equation

Note that all terms in this equation have units of pressure (lb/in2 or N/m2).  The term P1 is called the static pressure because it is the actual pressure measured if there is zero flow rate (i.e. ambient pressure).  It is often given the notation, Ps.  The second term is called the dynamic pressure, because it is due to the flowing fluid.  It does not exist if there is no fluid velocity.  The right hand side term, P2, is called the total pressure and is given the notation, Pt.  Using this notation the equation can be written as


Pressure distribution acting on a plane that slices through a sphere.  Note the high pressure (red area) at the nose or stagnation point on the sphere.  Low pressure regions (blue) can also be seen at the top and bottom of the sphere.

Results computed using Simulation CFD

The total pressure was derived based on the concept that at some point the fluid velocity along a streamline is brought to a stop.  If there is no elevation change, then Bernoulli’s equation written between two points is

This equation states that the total pressure is constant along a streamline that has no elevation change. 

 

Learning Objectives:

Understand the concept of a stagnation point and the total pressure associated with it.