in this chapter we want to compare the
influence of different influences of
different parameters inside the dynamics
of fluids and we do this with a very
specific kind of application in mind
which is making miniature or enlarged
versions of fluid flows so let me show
you why this is important and what we
want to do the principle is this we're
engineers and we want to investigate
flows that are difficult to reproduce so
for example you could be working for an
airplane manufacturer and you have an
airplane which is 100 meters long and
you want to improve the airplane so you
have six different ideas for how to
improve the plane and you want to try
those out how do you do this without
building six times an airplane that is
just gigantic or you're
interested in seeing how a mosquito
flies there are lots of mosquitoes in
this room and you want to study how the
fluid flows around the wings of a
mosquito and you certainly
have to have an enlarged version of a
model of a mosquito at hand to study
this or you're interested in the fluid
flow of liquid metal inside a furnace
and so any instrument you put into the
furnace just melts away with the heat
how do you investigate this making a
model making a model in which you can
have fluids that are much more friendly
to your instruments than liquid metal or
you want to study fluid flow in a
beating heart and so it turns out not
too many people are happy for you just
open their heart and stick instruments
into it just to study the fluid flow in
there so you want to make models of that
and once you make all of those models
then the question is how do you adapt
the flow parameters for example your
airplane model that you built is much
cheaper than the real sized airplane
this airplane model is 20 times smaller
or 100 times smaller and the large
airplane so what do you do with the
velocity do you make it 100 times
smaller - or or should that be a hundred
times larger this is the question we try
to answer and so what we want to do is
this we want to have flows that are
dynamically similar
which means they are the exact
miniature or enlarged version one of the
other and the flows are dynamically
similar when the flow parameters are the
same so what we're trying to define here
is what those flow parameters with this
kind of mystifying term means and and
how to make those the same okay let's
take a look what I want to do is this we
want to write an equation this is very
well known very useful which is the
non-dimensional version of the
navier-stokes equation so what I'm gonna
do for this is is we're gonna start with
the navier-stokes equation the Navier Stokes
equation says mass times acceleration
which is this part here this is the
change in time of the velocity field and
this is the acceleration field this here
mass times acceleration is due to
gravity pressure and shear and we want
to rewrite this equation in a
non-dimensional way so for this we're
going to create non-dimensional terms
and we start with time which we're going
to rewrite as T star instead of T we're
on T star and I'm going to define this
as a combination of factors on the right
and I'm gonna explain those I'm gonna do
the same thing for velocity for pressure
for gravity and for the mathematics of
vector E operation is the
non-dimensional del operator so a
non-dimensional term is is basically a
way of saying every time you have a
vector which is an arrow it has three dimensions you split
this into two components and one is the
length of that vector and the other is
the geometry the direction in which is
pointing and when every time write this
vector with a star meaning it's a unit
vector that points in the same direction
as the original vector but has only
length one and the length itself will be
embedded inside the scalar term over
here so we replace the vector by a
scalar multiplied by unit vector okay
and we do this for many physical
dimensions and what we do is time time
first is we take time we say this time
multiplied by the frequency at which the
flow repeats itself so we're going to run
time non-dimensional time from zero to
one
every time we're going to study a flow
instead of studying it for 30 seconds we
study it for
from 0 to 1 1 corresponds to 30 seconds
0 corresponds to 0 and so this happens
in within the period 1 over F or with
the frequency F over here yeah so very
high frequency flow is highly unsteady
changes all the time while a low
frequency flow we have a very long
repeating period or an infinitely long
period and is then quasi steady
this non-dimensional velocity field is
the same thing we take a velocity field
here and we divide it by the length of
the velocity field everywhere yeah so we
get this sea of little vectors this
entire field of little vectors which are
oh length 1 and are pointing in the
direction where the fluid flow is going
every time so V Star is a unit vector
field and we do the same thing for
pressure pressure we take pressure as
being we subtract from pressure a P
infinity a reference value if you want
and we divide this pressure by a
reference standard pressure over here
now we get P star and this P star if you
choose your reference value is p 0 and P
infinity wisely this P star goes from 0
to 1 so you have all the pressures
inside your field are normalized down
from 0 to 1 then we have non dimensional
gravity which is again gravity divided
by the norm of gravity and then we have
non dimensional math which we need to
manipulate those unit vector fields and
this is going to be non-dimensional del
which is some reference length
multiplied by the del operator over here
and you put those now into the normal
terms so you force those into the normal
terms you have
those things so we're going to replace
every time time by non-dimensional time
divided by frequency we're gonna replace
velocity by the length of velocity
multiplied by unit vectors and so on so
forth so we replace normal fully
physical terms with non-dimensional
versions of those and we put this now
into the navier-stokes equation and this is
very tedious to go through and I'll let
you go through if you want looking at
the notes but I'm gonna skip through all
the algebra through the math so we just
pretend we can read this very fast
you take the navier-stokes equation here
you insert those non
terms over there and of course you get
this which becomes that which becomes
this which becomes that again and this
again and that again and then you write
this equation this is now, the principle
here is important, we have an equation
with terms where there are scalars
that have physical values and those
terms are in front of unit vectors and
this equation looks like this this is it
this is the navier-stokes equation it is
not more or less than the usual
every Stokes equation it says exactly
the same thing there's not more or less
information in there it is just
rewritten in a different way and the way
it is written is that we have every time
in brackets the length of the vectors
and every time in green here those terms
in green are all non dimensional vectors
which are vectors they're pointing in
the way that the vector would point but
they all have a length one and it's
kind of cool because now you can rewrite
again this equation and reorganize those
terms here until you have terms here
that all have no dimension no dimension
at all and so you land on
this equation here which I just wrote
previously this equation there and in
those we like to give those blue terms
here we'd like to give those names and
so we define by convention for
historical reasons the Strouhal number as
being the frequency times the length
divided by velocity then the Euler
number as a difference in pressures
divided by Rho V squared then the Froude
number has become as velocity divided by
the spirit of gravity and the reference
length and the Reynolds number as being
Rho times V times L divided by mu and
then once you insert those into the
navier-stokes equation you get this this
is the supercool glorious non
dimensional incompressible navier stokes
equation this is the navier stokes equation
but written in a way that those vectors
here that are all little star vectors
they all have length 1 and in front of
those vectors you have numbers scalar
fields for numbers which are the values
the links of every time the the local
vector and those lengths here have been
given names
have a Strouhal number here you have one
over the square of the Froude number we
could have defined… we could have given
a name to one over Froude squared but for
historical reasons it just happens to be
like so you have minus the Euler number
here and you have one over the Reynolds
number there and all of those are
compared to this reference value which
is one here which is sometimes
implicitly written it's not even written
and so you can see now that the
magnitude of the pressure term here this
term this is compared to the magnitude
of the acceleration field yeah the
magnitude of the convective of the
velocity and so you can see that those
terms here give you the relative weight
of different terms inside Navier Stokes
equation compared to how much the flow
is accelerating locally this is super
cool it's super cool for two reasons the
first reason is that we quantify the
relative weight of the terms and this is
not because we are physicists or we're
very curious about this but typically
because we want to neglect those we want
to remove some of the terms and so
finding out which ones we can neglect is
easy now because all we have to do is is
quantify those one two three and
four numbers here compare them to one
yes and so if one over the Reynolds
number is is very close to zero
then we can just dump this term here
from the equation yes if the Euler number
is very close to zero then we can just
dump the pressure term form our
equations and this saves us a lot of
time when we compute flows especially
with computers and the second reason why
this is great is that we can know now
how to obtain dynamic similarity between
two scales this is what we started this
presentation with dynamic similarity
between two scales means if you have a
miniature version of a full scale flow
you want to have the same unit vectors
oscillating and vibrating than in the
original form then you need to put the
same numbers here yes if you want to
have the same non-dimensional velocity
field then you need to create in your
miniature version of the flow
you need to create the
same Strouhal number the same Euler
number the same Froude number and the
same Reynolds number which it turns out
in practice is very difficult but at
least from a physical point of view we
know what we need to do in order to
create miniature or in large versions of
certain flows so this is called the
super cool super duper Navier Stokes
equation non dimensional navier-stokes
equation or short for tattoo-it-on-your-arm
non dimensional incompressible
navier stokes equation yeah it's cool
enough actually to tattoo it on your arm and
to study and in fact typically in fluid
mechanics you may call this even the
Sphinx because SPHINX is the acronym of
the of equation it turns out if you look
at what it's called you could say it's
some pretty important equation no
exaggeration so it's definitely worth
your time studying and understanding the
meaning of this equation.
