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Hello.

First, I want to be sure that you are awake,

because the main purpose of this session is quite technical.

We will apply the Reynolds decomposition to the Navier-Stokes equations

in order to quantify the turbulent dissipation in a given boundary layer.

Here is a typical velocity profile for a boundary layer.

It corresponds to a 10-minute temporal averaging of the horizontal winds

in the first 300 meters above the ground.

Superimposed to this mean velocity profile, we should add turbulent velocity fluctuations.

The spectrum gap between the meteorological winds

and the rapid turbulent fluctuations allows us to perform a Reynolds decomposition,

and assumes that the temporal leveraging is identical to a statistical mean.

Hence, we could apply the following properties of the statistical mean.

The mean of the turbulent fluctuation that adhere by â is equal to zero.

The mean of the product of the mean variables is equal to the product of the means.

The mean of a mean variable times a turbulence field is equal to zero.

The mean of the products of two turbulent variables is not zero.

And finally,

the mean of any derivatives will be equal to the derivatives of the mean variable.

Let's now apply the Reynolds decomposition to the Navier-Stokes equation.

The last term in the Navier-Stokes equation is related to the molecular

or viscous dissipation.

And the dimensional analysis tells us that this last term scales as

viscosity new times the velocity V divided by L square.

We usually compare this term to the advection term in the left-hand side

and the ratio of these two terms leads to the Reynolds number,

which quantify the turbulent flow regime.

For both atmospheric or marine boundary layer, the molecular velocity is very small.

And for typical velocities of just one meter per second or more,

the Reynolds number will be very large.

Atmospheric and oceanic flows are highly turbulent.

And we generally use the inviscid Euler equation to describe these turbulent motions.

We will then use the Reynolds decomposition on the horizontal component

of motion along the x-axis and we combine this equation

to the incompressibility condition.

We then multiply supply the latter by Vx, and we get the following relation.

If we add again the Euler equation,

we formally re-write the Euler equation along the x-axis in the following form.

We know that we can take all the derivatives out of the mean - the brackets - and we get.

Then, we need to explicit the statistical mean of the squared terms.

For instance, the mean of the x square will be equal to the square of the mean of the x,

plus two times the mean of x times û, plus the mean of û square.

According to the properties of the statistical mean, the middle term is equal to zero,

and we get the sum of the square of the mean flow,

plus the mean of the square of the turbulent velocity fluctuations.

If now, we apply the Reynolds decomposition to the project Vx times Vz, we get

the mean of Vx times Vz is equal to the sum of four terms,

and the two in the middle vanishes.

Which leads to the products of the mean velocity,

plus the mean of the products of the turbulent components û times ŵ.

If we apply the same decomposition to the last term,

we get the same formal result on the mean of Vy times Vx.

Hence, three new terms,

which depends on the turbulent velocities, will appear in the Euler equation.

We can repeat this procedure on the y-axis,

and we get the Reynolds average Navier-Stokes equations, usually called RANS.

The right-hand side describes the impact of turbulent velocity fluctuations.

The statistical mean of nonlinear turbulent fluctuations correspond

to the turbulent dissipation, while the left-hand side terms correspond

to the Euler equation applied to the mean flow.

The RANS provides here dynamical relation between the mean flow

and some statistical properties of the turbulent fluctuations.

However, we introduced here new variables,

and in order to solve the dynamical system, we need to provide another set of equations.

We need to close the system.

The closing relations, who express the statistical mean

of the fluctuation as a function of the mean velocity fields

and/or their derivatives, will correspond to a given turbulence theory.

We will see, in the next sessions,

simple closure relations and the corresponding boundary layer profiles.

Thank you.