Introduction
How to describe Turbulence
Chaos and random are not accurate words to describe turbulence.
However, turbulence is unsteady, 3d, random-like (but not really - structures are coherent).
It increases mixing and flux, and has irregular swirls of motion (called eddies) with a continuous spectrum of
sizes.


The largest eddy is assumed to be the size of the chamber.
Eddies dissipates downstream by the viscosity.

If the geometry is fixed, if you increase the Reynold's number, the smaller the eddies.

𝐿
𝜂


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= 𝑅𝑒 4 , where L is the size of the largest eddy (the dimension of the structure/chamber/pipe),
and 𝜂 is the size of the smallest eddy.
The smaller the eddies, the higher the Reynold's number.
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Dynamical systems and chaos
𝑑
𝐱(𝑡) = 𝐟(𝐱(t), t; 𝛃)
𝑑𝑡
Where 𝐱 is the state of the system, and 𝐟 is a vector field that possibly depends on the state 𝐱, time 𝑡,
and a set of parameters 𝜷.
Lorenz problem



𝑥̇ = 𝜎(𝑦 − 𝑥)
𝑦̇ = 𝑥(ρ − 𝑧) − 𝑦
𝑧̇ = 𝑥𝑦 − 𝛽𝑧
with parameters: 𝜎 = 10, 𝜌 = 28, and 𝛽 = 8/3.


1st set of initial conditions: 𝑥(0) , 𝑦(0) , 𝑧(0) = [0.1,0.1,0.1]
2nd set of initial conditions: 𝑥(0) , 𝑦(0) , 𝑧(0) = [0.100001,0.1,0.1]
The problem is very sensitive to boundary/initial conditions.
Energy cascade
Atmospheric boundary layer is at L=1 km, as things happening within that distance of the ground are the
most interesting.
Q: What do we need to make a model (numerical analysis) for this?
A: We need to estimate the Re, and then we find an estimate of the smallest eddy.
Framework in the course
 Newtonian fluids.
 Incompressible flows.
 Uniform properties of the fluid throughout the domain.
 𝜌 and 𝜇 are constant.
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Notation
𝜕

𝜕𝑗 = 𝜕𝑥

𝜕
𝜕𝑡
𝜕𝑡 =
𝑗
Navier-Stokes
In conservative form:
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𝐷𝑡 𝑢𝑖 = 𝜕𝑡 𝑢𝑖 + 𝜕𝑗 𝑢𝑖 + 𝜕𝑗 (𝑢𝑖 𝑢𝑗 ) = − 𝜕𝑖 𝑝 + 𝜈𝜕𝑗 𝜕𝑗 𝑢𝑖
𝜌
𝜇
With 𝜈 = 𝜌 , and initial/boundary conditions at a fixed solid wall 𝐮= 0: no-slip condition.
The equations of fluid motion
Advection: transport of a substance by the movement of a fluid determined by the direction and rate of
the fluid flow.
Continuity
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Repeated indices imply summation over all possible values of j (i.e. 1, 2 and 3), which is why the continuity
eq is one equation.
𝜕𝑡 𝜌 + 𝜕𝑗 (𝜕𝑢𝑗 ) = 𝜕𝑡 𝜌[𝜕1 (𝜌𝑢1 ) + 𝜕2 (𝜌𝑢2 ) + 𝜕3 (𝜌𝑢3 )]
Statistics
 The mean is denoted with an angled bracket.
𝑁
1
𝑚𝑒𝑎𝑛(𝐹(𝑢)) = ⟨𝐹(𝑢)⟩ = ∑ 𝐹(𝑢𝑖 )
𝑁
𝑖=1

Variance is denoted:
𝑁
𝑣𝑎𝑟(𝑢) =
⟨𝑢′2 ⟩
1
= ∑ 𝑢𝑖′2
𝑁
𝑖=1

Standard deviation is root mean square.
𝑠𝑑𝑒𝑣(𝑢) = √𝑣𝑎𝑟(𝑢) = ⟨𝑢′2 ⟩1/2
Final notes
 Stationarity: the statistical characteristics of turbulence is independent of time.
 Homogeneity: in 𝐲 direction: the statistical characteristics of turbulence is independent of
location of space.
 Isotropy: the statistical characteristics of turbulence is independent of the direction of space.
The statistics (mean, var, sdev) are the same if you change the basis of the coordinate system.
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