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 were: 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, Daniels father did not allow him 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 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 (2q/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.