Text Lecture 1c - Bernoulli’s Principle

In this lecture I would like to treat a simple, yet powerful fundamental aerodynamic equation, which enables us to calculate pressures and velocities in incompressible flows: The Bernoulli equation.

In the previous lecture we derived the Euler equation for fluid flow from Newton’s second law, force is mass times acceleration, remember?

We found the differential equation: dp=-rhoVdV.

The assumptions in the derivation of this relation where:

Steady flow and Neglect of gravity and viscosity forces.

Now let us integrate the Euler equation dp+rhoVdV = 0 along a streamline between the points, 1 and 2.

So we get: the integral of dp from p1 to p2 plus the integral from v1 to v2 of rhoVdV is zero.

This develops into p2-p1+rho times (1/2 V2 squared minus 1/2 V1 squared) is zero or if we rearrange this: p1 +1/2rhoV1 squared =p2 + ½ rhoV2 squared.

So, along a streamline we have: p+1/2 rhoV squared is constant.

This constant is called the total pressure pt. p is the static pressure and] ½ rho V^2 is the dynamic pressure.

This is known as Bernoulli’s principle, or Bernoulli’s equation. Who actually was this Daniel Bernoulli?

Daniel Bernoulli was born in January 1700 in Groningen in the Netherlands. His father Johann was professor in mathematics at the university of Groningen. The family came from Basel in Switzerland and after 10 years abroad they returned when Daniel was 5 years old. His father’s brother Jacob was holding a chair of mathematics at Basel University. And we he died Johann was offered to fill the vacancy. Daniel had a younger brother Johann jr. and an older brother Nicolaus, and all three were heavily interested in mathematics. However, his father did not allow Daniel to study this, since he had the conviction that there was no money in mathematics. He wanted Daniel to become a merchant. That’s why Daniel studied philosophy and logic and later on medicine in which he completed his doctorate at the age of 20.

In the meantime he did receive lessons from his father and his older brother in mathematics and he studied his father’s theories on kinetic energy. Still he pursued an academic career and after several unsuccessful applications for chairs at Basel University in anatomy, botany and logic, in 1725 he was appointed to the chair of mathematics of the University of St. Petersburg in Russia, together with his older brother Nicolaus whom also was offered a position. Sadly, Nicolaus died of fever soon after his arrival. In 1727 this vacancy was filled by. Leonard Euler, one of his father’s brightest students. It was in St. Petersburg that Daniel Bernoulli laid the foundation of the equation we just derived. He reported about it in his work Hydrodynamica, which was published in 1738, 4 years after his return to Basel. Here we see the cover of this document. As you can see it is in Latin. According to the stamp this copy comes from the library of the Groningen University where Daniel’s father Johann was a professor during 10 years. On the cover there is a remarkable addition to Daniels name, meaning that he was the son of Johann. The relation between the two was a bit troublesome, so with his father being a renowned mathematician, Daniel either put it there to give himself more standing, or he wanted to show his good intentions toward his father.

The equation we now attribute to Bernoulli can, however, not be found in his book. Although the basis for it is discussed in his Hydrodynamica, the derivation as shown comes from Euler.

So, although the equation bares Bernoulli’s name we have to thank Euler as well for his contribution to one of the simplest but widely applicable equations in fluid dynamics.

After having taught botany and physiology, in 1750 Daniel was appointed to the chair of physics to which he devoted his time until he retired in 1776. He contributed to science with outstanding work on a great variety of topics in fluid dynamics and physics. When he died in 1784 he had won the most prestigious Prize of the Paris Academy of Sciences 10 times.

Signed: Daniel Bernoulli.

One of the applications of Bernoulli’s principle is the pressure distribution along an airfoil. Since the pressure is constant perpendicular to the airfoil surface, by measuring the pressure distribution you can derive the velocity at some distance around the airfoil.

Bernoulli’s equation can also be used to determine the flight speed in incompressible flow. For that we need a pitot tube, named after the French physicist Henri Pitot who developed it in 1732 to determine the velocity of the water in rivers. It measures the total pressure. To calculate the speed, we also need the static pressure, which is measured by a static pressure port in the fuselage. Here you see pitot tubes installed on an aircraft. In many cases these two are combined in one instrument, a pitot-static tube, or Prandtl tube. Such an instrument measures the dynamic pressure directly, from which we can derive the flight speed with V=sqrt(2 times q divided by rho). We will hear more about Prandtl in following lectures.

This concludes the lecture on Bernoulli’s equation. It was derived for an incompressible flow neglecting viscosity. In the next lecture we will set some steps on the slippery path of compressible flows.