Bernoulli Equation

Ghosh - 550 Page 1 4/25/2018

Bernoulli Equation

The procedure given in the example for Euler’s equation is simple but lengthy. It will be nice if we can develop a formula, which will yield the pressure field but will not have to involve the usual integration process. For this, we will integrate the Euler’s equation by making some simplifications. The resulting equation is called the inviscid Bernoulli equation (or simply the Bernoulli equation). It is very popular because of its simplicity and ease of use. However, there are restrictions to use this equation as will be apparent during the derivation.

Given below is the inviscid Bernoulli equation. As we have realized during the derivation, it is applicable for incompressible, steady, frictionless flows along a streamline. p V 2  s  gz  const  2

The inviscid Bernoulli equation has many uses. Construction of Pitot tubes, or pitot- static probes utilize this equation. The concept of stagnation pressure can be defined right from this equation.

Imagine a flow, which fits the requirements of the inviscid Bernoulli equation. Let us choose two points on the same streamline, (1) and (2).

z (1) (2)

Streamline

The points are close by such that z2  z1  0. p V 2 p V 2 Thus, 1  1  2  2  2  2 Now, if we make the point (1) general point by dropping the subscript. On the other hand, we make the point (2) a special point where the fluid velocity is zero, (i.e., V2 = 0). Then we call the second point a stagnation point and define the pressure then by a special symbol, po. p V 2 p Then,   o  2  or, 1 p  p  V 2 0 2 Ghosh - 550 Page 2 4/25/2018

The boxed expression gives us the measure of stagnation pressure. It is the sum total of 1 the static pressure, p, and another term, V2 , which we call the dynamic pressure. 2 We know that static pressure, p, is the pressure measured in a static fluid. For a flowing fluid, this pressure may be measured in the direction perpendicular to the flow direction. On the other hand, both dynamic and stagnation pressures must be measured along the flow direction. Dynamic pressure is obtained by moving with the fluid velocity, whereas the stagnation pressure is obtained if you decelerate the fluid particles to a zero velocity isentropically. The construction of a pitot tube illustrates the use of stagnation pressure. By bending the tube 90 at the neck, fluid particles are brought to a halt in the flow direction.

Static Holes Neck

Flow Direction

Static Channel

Pitot Channel Pitot-Static Tube

On the other hand, a Pitot-static tube uses two channels, one facing the flow and the other perpendicular to the flow.

An interesting theorem by Crocco gives: 1   p  V    o

This theorem provides a measure of the changes in stagnation pressure in an incompressible flow. It can be related to the vector (cross) product of velocity and  vorticity vectors. In an irrotational flow,   0. Thus, po  0 in an irrotational flow. Since the gradient operator involves spatial derivatives, we may write: Ghosh - 550 Page 3 4/25/2018

p p p  0,  0,  0 x y z p p p dp  dx  dy  dz  0 Or, p  const x y z o

Therefore for irrotational flows, the stagnation pressure, po, holds constant not only between two points on a streamline but between any two points in the fluid flow. Thus, the use of the inviscid Bernoulli equation becomes less restrictive. For rotational flows, however, Bernoulli equation can only be applied between any two points on a streamline.

Examples Continue