To understand why the Bernoulli equation
will kill you, you have to see what it's
made of. And to do this we begin with the
energy balance equation. The energy
balance equation tells you all that can
happen to the different forms of energy
as the fluid transits through the
control volume. This equation here is
valid for two conditions: one is that the
flow is steady, which means as you take
different pictures of the flow at
different time intervals all the
pictures are the same, and the second
condition is that the control volume is
not inflating or deflating. Otherwise
this is always true, and this equation
tells us everything can happen to the
different forms of energy. Let's take a
look. You have here on the left the power
added from outside as heat, the power
added from outside as work, and on the
right side you have the difference
between the inlet and the outlet in
terms of energy flow. And energy can take
different forms inside the inlet and
inside the outlet. Here you have the mass
flow and in parentheses here you have a
sum of different terms. And these terms
are the internal energy, which is carried
as heat, goes to increased temperature
inside the fluid,
pressure divided by density, kinetic
energy, one-half of V squared, and then g z
which is altitude potential energy that
being positive positive upwards. So these
are different forms of energy at
inlet, and then you have the same
forms at outlet. This is super good
because we can see all the different
shapes that energy can take at the inlet
and the outlet, and we can see all the
different exchanges that can happen in
between the inlet and the outlet. The
problem with this equation is that it
has a lot of terms. Now let's count. you
have here at inlet one two three four
five then you have five more at the
outlet and then you have those two here
which are the net transfers from the
outside. So that's twelve terms, if you
are looking for one value in there you
need to put in eleven other values.
And this is quite a lot of frustration
for fluid dynamicists.And so it's
tempting to take this equation and then
remove everything you don't like about
it, yeah? So what we would do then, is you
just take all the things that you feel
uncomfortable with in this equation and
you just remove them from the equation.
If you did this you would make this
equation pretty useless, and you would
land on the Bernoulli equation. The
Bernoulli equation is an energy balance
equation that has additional
restrictions to it. What what are
these restrictions? You have first steady
flow, and that's fair enough
because we started with steady flow
already, you know we had this already in
this steady flow energy equation before,
so that's fine.
An added restriction is incompressible flow.
This means that Rho remains constant.
Another added restriction is that there
is no heat and no work transfer. So the
two terms on the left side of the
equation, those two terms go away. There's
no friction, because why would you want
to have friction? Without friction it's
pretty nice.
No heat transfer, no [work] power no
friction means i, the term of the
internal energy, this remains constant as fluid flows. And finally you have
one dimensional flow which means you
fluid flows and finally you have only
one dimensional flow which means you
only one-dimensional flow,  only have one inlet and one outlet, and you
are certain that as the flow goes from the inlet
to the outlet, it follows only one line
in between those two those two places.
If you add all those restrictions then that
equation which we had before becomes
just this equation now here, and you have
zero on the left terms, on the left side
sorry, and then on the other side you have
this. i becomes a constant, is the same
between the inlet and the outlet, p may
change, but Rho does not change, and then
you have kinetic energy and potential
energy here. And so this gives us a very
very well-known result, which is the
Bernoulli equation, which tells you that
pressure plus one-half of Rho V squared
plus Rho g z, the sum of this,
that's one number, and if you go at the
outlet of your control volume, then it is
equal to the same number, the sum of
those terms is equal to
the same number, yeah? And this describes the
flow of one particle in a steady
incompressible frictionless flow with no
energy transfer. So very very very
restrictive. Let’s have a look at what
this means now in practice.  It’s enough
for the math, let's have a look at what it
means for the engineer in practice. First
constriction, a restriction of steady
flow is okay. It prevents you from doing
things like studying flow in the veins
inside the human body, where the heart
would just pump the flow and give some
unsteadiness to the flow yeah? So no
transient effect, no unsteadiness, but
that's still fair enough.
Second restriction is restriction of
incompressible flow. There are two
things with this.
One is that you cannot go extremely fast,
so you cannot have compressible flow
means you cannot have high Mach numbers.
A Mach of 0.5 or 0.6, typically in air you
are limited to 500 kilometers an hour
approximately. And that's very nice
because most flows of practical interest
have low speeds. But you also cannot have
any compressor, no turbine, no diffuser,
no nozzle, no rocket engine, nothing where
the fluid will store energy and expend
energy as internal compression effects.
So that's quite a big restriction. Third
restriction is: no heat and no work
transfer. Okay, so no machine of any kind,
yes?  No pump, no turbine, no combustion
chamber, no cooler, no hairdryer, nothing
that adds or subtracts energy to the
flow will work with the Bernoulli
equation, okay? Fourth restriction is no
friction. And this is quite astonishing,
to be honest. No turbulence, no shear, no
viscous effects, the flow just glides
along in one perfect steady smooth way
throughout your control volume. And that's
quite a large restriciton in engineering.
One implication of this is that you
cannot come close to any object. So every
time you have a steady fixed object with
a flow going around it,
near the skin of that solid object the
velocity is zero and so as the fluid goes
over it there is a lot of shear. There is
a shear layer, which we call the
boundary layer, that applies around the
skin of that object. So the Bernoulli
equation does not apply anywhere close
to objects because of that shear layer.
An the fifth restriction is
that it's one-dimensional flow. So you
need to be certain that the particle
[going in] at inlet is also the same particle
that will go at the outlet, and that it
follows a line in between those. You cannot
predict what is going to happen with the
Bernoulli equation, yeah? You cannot use
it to predict what the flow will be. You
have to observe only one certain given
flow that is in front of your eyes and
then it will work. Okay, so, so much for
the talking and all the lists of things
why it will not be very very useful for
engineers. Now let's take a look at a
practical example. Let's say you are
going to the airport and you're sitting
there in the grass, it's a summer day and
you see airplanes flying about. And this
is a very standard airplane, this is a Dash 8 commuter airplane. This is a
machine that's based entirely on fluid
mechanics, yes?
So you see this airplane, you snap a
picture and you ask yourself where can I
use the Bernoulli equation around this machine
to predict how it works in general?
Well let's take a look. First you can't use the
Bernoulli equation inside the engines.
There's a compressor with a combustion
chamber, turbines, exhaust nozzles, all
those things in there add and subtract
energy to the flow [or] have a very
compressible flow in there, so no Bernoulli
equation in here. Those engines will emit
a plume of hot air which will dissipate,
and this plume is very turbulent, so lots
of shear in there, and it also loses a
lot of heat to the atmosphere, so no
Bernoulli in that zone either. You can’t
come close to the propellers because
propellers add energy to the flow, a
great deal of it, and so don't come
anywhere close to the blades
of the propeller. Once the propellers
have turned the flow and pushed it
backwards, the flow will transform into
one big rotating cylinder of very
turbulent air, so you cannot go into
there and apply the Bernoulli equation.
This is a dead zone for the Bernoulli
equation in there, in the wake of those
propellers. Okay remember that next to
solid surfaces the velocity is zero so
there's a lot of shear next to
surfaces, so we don't want to come close
to the skin of the airplane. Anywhere
close to the airplane, lots of shear,
no Bernoulli equation. And finally all this
shear will just be trailed away behind
the airplane into one very turbulent,
very messy, very dissipative sheet of air
that just follows behind the airplane
and so the Bernoulli equation does not
apply there either! So where can you use
the Bernoulli equation? Well I guess I found
a couple of places around the airplane
where I guess it could work, as long as
you don't come close to anything
interesting then the Bernoulli equation
will apply, yeah? so stay safe and be very
careful for this equation when you use
it. In fact you will find if you just
play with a hose and try to reduce the
area and you play with a velocity of the
water coming out of a hose of water, you
try to predict this with a Bernoulli
equation, you will fall flat on your face.
The Bernoulli equation will predict the same outlet
velocity regardless of the outlet
area, so play a bit with this equation,
beware of its limitations, before you
use it.
Now, a lot of people come to me then and
say "well Olivier obviously you use the
Bernoulli equation without losses but I use
a special version where we are taking into
account everything that you were talking
about, and I take it this into account as
one additional term here called Delta
p loss, and this makes the equation
true". And I agree it's magic, very nice,
good for you. If I told you that 2 plus 3
is equal to 4 and you would look at this
and tell me "I’m pretty sure this is
not true" and I would then tell you "oh of
course
you just have to add Delta p loss over
here to make the equation true" then
certainly you would feel like
that's magic. You know that's just one
added term on top of this. We /will/ use
the Bernoulli equation with losses in
very specific cases, and in those cases
we are under control of what happens
inside our flow. Namely those cases are
flows inside pipes and we can calculate
this Delta P loss using interpolation
tables, using experimental data, very well.
And so it is useful in practice. But do
not go about studying the flow around an
airplane and then guessing the value of
this Delta p loss, you will get
completely wrong results. So before you
use the Benroulli equation, are you /sure/
that the five conditions apply? if you
cannot remember those five conditions,
just don't use it. Go back to the energy
equation. If you have a delta P loss term
that you add in there, are you /sure/ that
is this delta p loss can be predicted very
well, safely yeah, very predictably, or
are you just guessing the value of this
Delta P loss? Because if any of those two
conditions are not met then just go back
to the energy equation. It will force you
to question your choices. It will force
you to cross out terms and wonder is
really internal energy the same at the
inlet, in the inlet as the outlet? And
this will force you to question your
assumptions about the flow. Now I'll
finish with just historical
information for fluid dynamics geeks who
are wondering, "if the Bernoulli equation
is so useless then why do fluid dynamicists
use it? Well Daniel Bernoulli wrote this
equation, and it's not his fault that the
equation is useless. Hydrodynamica,
the book in which Daniel Bernoulli
wrote this equation, or suggested
this equation, was a seminal piece. It was
the first time we really had a
relationship between flow properties
inside the fluid flow, not just in
statics but I mean in the fluid flow, and
that's quite, that was quite impressive.
The concept of energy didn't show up for
another century, yeah? So it’s easy to
say that the Bernoulli equation is just a
simplified version of the energy
equation, but the energy equation just
came a lot, a lot later. And just finally
just for context, historical context,
it's easy to take a picture of an
airplane, cross out the zones where the
Bernoulli equation doesn't apply, there
were no airplanes in 1738, yeah?
The fastest thing on the planet over
there, at that time, was a horse! So
this is for context. And just for the
nerds, I will show you the cover page of
the book, Hydrodynamica, 1738,
and just read the title and the subtitle
to you because I think this sounds so
cool. This is:
"Hydrodynamica, sive de viribus et motibus
fluidorum comentarii, opus academicum as auctore,
dum petropoli ageret, congestum." How cool does this sound?
I was thinking about putting such a
title like this on my PhD Thesis but I
think I'm really just chickening out.
So again coming back to the start,
to this energy equation here.
This is safe to use. There are lots of
terms in there, if you don't have them
all; do not worry, and do not give in to
the little voice behind the back
of your head that tells you "just use the
Bernoulli equation" because the Bernoulli
equation will give you a very nice
result that's easy to calculate and
it's wrong, yeah? So take care and see you later.
