Static Stagnation Pressure

Concept of Static Pressure

• The thermodynamic or hydrostatic pressure caused by molecular collisions is known as static pressure in a fluid flow and is usually referred to as the pressure p.

• When the fluid is at rest, this pressure p is the same in all directions and is categorically known as the hydrostatic pressure.

• For the flow of a real and Stoking fluid (the fluid which obeys Stokes’s law) the static or thermodynamic pressure becomes equal to the arithmetic average of the normal stresses at a point. The static pressure is a parameter to describe the state of a flowing fluid.

Let us consider the flow of a fluid through a closed passage as shown in Fig. 1a.
 


• If a hole is made at the wall and is connected to any pressure measuring device, it will then sense the static pressure at the wall. This type of hole at the wall is known as a wall tap.

• The fact that a wall tap actually senses the static pressure can be appreciated by noticing that there is no component of velocity along the axis of the hole.

• In most circumstances, for example, in case of parallel flows, the static pressure at a cross-section remains the same. The wall tap under this situation registers the static pressure at that cross-section.

• In practice, instead of a single wall tap, a number of taps along the periphery of the wall are made and are mutually connected by flexible tubes (Fig. 1b) in order to register the static pressure more accurately.

Hydrostatic, Hydrodynamic, Static and Total Pressure

Let us consider a fluid flowing through a pipe of varying cross sectional area. Considering two points A and B as shown in Figure 1(c), such that A and B are at a height Z_A and Z_B respectively from the datum.
 

If we consider the fluid to be stationary, then, where the subscript ‘h_s’ represents the hydrostatic case.

So,   p_Ahs - p_Bhs = ρ_g( Z_B–Z_A)    (1)
                  
where p_Ahs is the hydrostatic pressure at A and pBhs is the hydrostatic pressure at B.

Thus, from above we can conclude that the Hydrostatic pressure at a point in a fluid is the pressure acting at the point when the fluid is at rest or pressure at the point due to weight of the fluid above it.

Now if we consider the fluid to be moving, the pressure at a point can be written as a sum of two components, Hydrodynamic and Hydrostatic.

p_A = p_Ahs + p_Ahd                  (2)
             
Where p_Ahs is the hydrostatic pressure at A and pAhd is the hydrodynamic pressure at A.

Using equation (2) in Bernoulli's equation between points A and B.



From equation (1), the terms within the square bracket cancel each other.

Hence,


                                                
Equations (4) and (5) convey the following. The pressure at a location has both hydrostatic and hydrodynamic components. The difference in kinetic energy arises due to hydrodynamic components only.

• In a frictionless flow, the sum of flow work due to hydrodynamic pressure and the kinetic energy is conserved. Such conservation shall apply to the entire flow field if the flow is irrotational.

• The hydrodynamic component is often called static pressure and the velocity term, dynamic pressure. The sum of two, p₀ is known as total pressure. This is conserved in isentropic, irrotational flow.

Stagnation Pressure

• The stagnation pressure at a point in a fluid flow is the pressure which could result if the fluid were brought to rest isentropically.

• The word isentropically implies the sense that the entire kinetic energy of a fluid particle is utilized to increase its pressure only. This is possible only in a reversible adiabatic process known as isentropic process.
 


Let us consider the flow of fluid through a closed passage (Fig. 2). At Sec. l-l let the velocity and static pressure of the fluid be uniform. Consider a point A on that section just in front of which a right angled tube with one end facing the flow and the other end closed is placed.

When equilibrium is attained, the fluid in the tube will be at rest, and the pressure at any point in the tube including the point B will be more than that at A where the flow velocity exists.

By the application of Bernoulli’s equation between the points B and A, in consideration of the flow to be inviscid and incompressible, we have,


            
where p and V are the pressure and velocity respectively at the point A at Sec. I-I, and p₀ is the pressure at B which, according to the definition, refers to the stagnation pressure at point A.

It is found from Eq. (6) that the stagnation pressure p₀ consists of two terms, the static pressure, p and the term ρV²/2 which is known as dynamic pressure. Therefore Eq. (6) can be written for a better understanding as


       
Therefore, it appears from Eq.(7), that from a measurement of both static and stagnation pressure in a flowing fluid, the velocity of flow can be determined.

But it is difficult to measure the stagnation pressure in practice for a real fluid due to friction. The pressure p’₀  in the stagnation tube indicated by any pressure measuring device (Fig. 2) will always be less than p0, since a part of the kinetic energy will be converted into intermolecular energy due to fluid friction). This is taken care of by an empirical factor C in determining the velocity from Eq. (7) as

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