PROFESSOR: So as a technical aside,
let me go through and sketch the derivation of the structure
of a turbulent jet, in particular the conical shape
that we have when the flow is turbulent.
So in order to study the mean flow profile
we began with the Navier-Stokes equations
which describe the momentum conservation and mass
conservation or continuity of an incomprehensible so-called
Newtonian fluid.
So this is a complicated set of equations.
In particular, we have this nonlinear term here,
which is the inertial term.
And we've already said that we had our high Reynolds number
and turbulence results because the inertia is
very strong compared to the viscous term which is here.
So that's the divergence of the viscous
or the viscous forces on the fluid.
So these two terms we know are important.
They have to balance and the inertia is particularly strong
and it is what leads to the very complicated flows that we see.
So you can solve these equations numerically on a computer
and generate simulations that look a lot like experiments
on turbulent jets.
What I'd like to do here is just to derive by simple scaling
arguments what sort of the structure of the solutions
could look like.
So these two terms, as we just indicated,
are the ones that are most likely to balance in the time
average flow.
So let's consider a time averaged
steady flow which has a velocity component of v_z
that depends on r and z.
So it's basically something like this,
which is basically expanding, but has
a certain sort of localization of the flow in the middle.
And it's smooth because we're averaging over all
the complexity of the jets.
So the jet looks something like this with all kinds of vortices
and eddies that are getting bigger
as it goes as you're entraining more
and more air from the outside.
So we're going to look at the time average flow.
And we're also going to, importantly,
assume that we have an eddy viscosity.
So the kinematic viscosity, nu in the equations
as I've written them here, represents the diffusion
momentum.
If a parcel of fluid is moving with a certain momentum,
it has a chance of passing that momentum
to the neighboring fluid and moving it along with it.
And that is accomplished through viscous stresses.
So the eddy viscosity basically assumes
that that diffusion process from momentum
happens at the scale of the largest eddy in the flow.
And so we've talked about the assumption of eddy diffusivity.
But for eddy viscosity, what I'll write
is the eddy viscosity is a typical velocity
which is v_z times a length scale which is delta.
So what I'm saying here with this
is that if I go out to a certain position z and ask myself,
what is the sort of width of the jet at site z,
then there's all kinds of eddies but the largest eddy
is kind of at that scale.
And so if I write down an eddy viscosity
it's going to be these sort of average velocity
there are times that scale.
So that's going to be the eddy velocity.
And I'm going to replace--
so when I do my time averaging, I'm
going to replace the microscopic viscosity
of the fluid, kinematic viscosity,
with the eddy viscosity.
So that's an important modification.
And so if I do that.
If I do this time averaging and look at the eddy viscosity,
then I take these two terms and balance them,
I'm going to get v_z bar, so that's my average v_z.
I'm looking at the z component of momentum
here of that first Navier-Stokes equation.
And I get v_z dot derivative of v_z with respect to z plus v_r.
And there's also an r component of velocity.
So there is also some velocity in fact
which is coming in from the sides.
But I'm just going be interested in this term here v_z dr.
And I'm going to balance this against the eddy viscosity,
the eddy--
or nu eddy I should say, sorry, nu eddy is eddy viscosity--
times and then the Laplacian in is 1 r d/dr r dv_z dr. So
that's just the Laplacian in its cylindrical coordinates.
And now I'm going to make the assumption that this ve
scales as v bar z times delta.
And so now I'm going to do a scaling
analysis on this equation.
And so what we see is we have v_z over z times--
and then at least for that first so--
I should say these two terms will
be of comparable size because of incompressibility--
the second equation.
I won't go through the details of that.
And we'll just do a scaling argument
balancing these two terms.
So if I look at v_z divided by z times v_z
so that's a scaling of those two terms,
I can balance that against v_z delta times--
and the scale for r is delta.
So I have 1/delta for the 1/r,
1/delta for the derivative times
delta * (1/delta) * v_z.
So there's a lot there.
But notice the v_z's all cancel.
And we're left with a bunch of deltas here.
And how many, because of the eddy viscosity,
we are left with--
all of this is just 1 over delta.
This is 1 over z.
And so we find here the delta scales as z.
So in other words, we have a conical shape.
So the boundary of their thickness
is a constant times z.
And what we write is that delta is equal to alpha z
specifically.
And we define the turbulent entrainment coefficient alpha
that way.
And then once we've done that, we've
already shown that from the momentum flux
that v_z scales as the square root of k/rho_air
times 1 over delta.
So this basically now gives me the scaling of the problem.
In fact, there is a similarity solution
for the shape this profile that one could solve for.
And it has the form that, for example, the v_z
is square root of--
because delta is proportional to z.
So it's the square root of k over rho
a z times some function of r over alpha z.
And then there's a similar expression
for the other velocity component.
And the function F looks very much as I've sketched here.
It's essentially a Gaussian type profile or a bell curve
that kind of localizes the velocity across this distance
delta.
The second thing that we're interested in
is the mean concentration.
And that would be a concentration of, let's say,
virions contained in infectious aerosol droplets.
So there's a mean concentration profile
in the jet assuming that we're injecting
a fluid of a constant concentration
at the source of the jet.
So again, we can do some scaling arguments here.
So if we ask ourselves, what is the mass flow
rate through a slice or actually the volumetric flow rate,
use me, that is what is called a Q and it'll just be an average.
This will be the average velocity
times the cross-sectional area at a given position.
So this is scaling like--
so area scaled is like delta squared.
And then the velocity scales in this way is 1 over delta.
So this ends up scaling as k over rho a times just delta.
So the volumetric flow rate is increasing
with r and that's a sign that we are actually in training fluid
as I indicated.
This is not just the fluid we're injecting
but it's moving forward and it's sucking more fluid in.
And all that fluid is kind of becoming
part of the turbulent jet as it grows.
Now if we ask-- so this is our flow rate, volumetric flow
rate, but we can also ask ourselves,
what is the flux of concentration
of virions per unit volume.
Well, that would be the average concentration
times the average flow rate because flow rate
is volume per time and concentration
is number per volume.
So this is a total number per time.
And this we will assume should be a constant
because as you can see from this picture,
if we're injecting a bunch of concentration of let's say
droplets here, they will spread out in the turbulent flow
but they don't really have a good mechanism to get out
of the turbulent flow.
The turbulent flow is sucking fluid into the plume
and so the particles are just kind of well
mixed in that plume and we could assume they have a roughly
constant concentration.
And so, if that's the case, and in fact, this constant
would be lambda Q if we're thinking of, for example,
infect--
c is the concentration infection quanta.
Then lambda Q is the rate of admission of infection quanta
from the mouth.
We've already talked about that quantity.
And this is now telling me how the concentration infection
quanta decays with time.
And so we find if we substitute now
is that the concentration infection quanta at a position
z scales as so I have to divide by Q so I
get the inverse of this.
So I get square root of rho a over k.
And then I have lambda Q over alpha z.
So this tells me that if I plot as a function of distance
from the mouth in the direction of the jet,
the concentration of infection quanta
that are carried by virions in aerosol droplets,
then somewhere here I have, let's say, at z equals 0,
is the mouth where I'm exhaling.
And the concentration there is actually cQ.
In fact that is something we've talked
about before, which is that's the key disease parameter--
the concentration of infection quanta in the exhaled breath
of an infected person.
So we know at the mouth, that's what we start with.
And what the turbulent theory is telling us
is how that concentration is decaying with time
and is decaying like 1 over z.
And so that tells us sort of our relative risk of infection
in different positions relative to being mouth to mouth
with the infected person.