1. Turbulence: a computational
overview & CO2 transfer
Fabio Fonseca

2. Overview
 Turbulent flow
◦ Randomicity vs. Chaos
◦ The Navier-Stokes equation
 Computational overview
◦ Discretizing and solving the fluid mechanics
equations
◦ Sample applications
◦ Understanding the computational problem
 CO2 transfer at the equatorial Atlantic ocean
◦ CO2 on the oceans
◦ Turbulent transfer
◦ Heat and momentum transfer

3. Turbulent vs. smooth flow
Turbulent flow
Smooth flow

4. Turbulence
“Turbulence is composed of
eddies: patches of
zigzagging, often swirling
fluid, moving randomly around the
overall direction of motion.” source: SCIAM
Source: McDonnough 2004, 2007

5. Why study turbulence?
 What if we could...
◦ Predict the weather?
◦ Simulate the human heart?
◦ Simulate the galaxies flow through the
space?
◦ Create air and sea transportation relying
only on computer simulations (i.e., not
relying on air tunnels)?

6. Turbulence: The problem
“The most important unsolved problem of classical physics.”
Richard Feynman
“I am an old man now, and when I die and go to heaven
there are two matters on which I hope for enlightenment.
One is quantum electrodynamics, and the other is the
turbulent motion of fluids. And about the former I am rather
optimistic.”
Sir Horace Lamb
(1932)

7. Turbulence: The problem (as of
today)
source: http//www.claymath.org/millennium

8. A brief history of turbulence
“Observe the motion of the surface of the
water, which resembles that of hair, which has
two motions, of which one is caused by the
weight of the hair, the other by the direction of
the curls; thus the water has eddying
motions, one part of which is due to the
principal current, the other to the random and
reverse motion.”
- Leonardo da Vinci, circa 1500

9. A brief history of turbulence
 The Navier-Stokes equation (Early
19th century):
• Believed to embody the physics of all fluid flows
(turbulent or otherwise)
•Accounts for:
The rate of change of momentum at each
point in a viscous fluid, u is the fluid velocity
field; pressure variations, P is the pressure

10. A brief history of turbulence
 This equation cannot be analytically
“solved”:
◦ “Two realizations of the flow with
infinitesimally different initial conditions
may be complete unrelated to each other”
– Ecke (2005)

11. A brief history of turbulence
 Late 19th century:
◦ In 1987 Boussinesq hypothesis tell us that
the turbulent motions are proportional to
strains in the flow.
◦ In 1894 Reynolds states that turbulence is
far too complicated to permit a detailed
understanding and simplifies the problem
using an statistical approach,
decomposing the flow in its mean and
fluctuating parts.

12. A brief history of turbulence
 1960‟s, the onset of the digital
computer:
◦ MIT‟s E. Lorenz presented a numerical
solution to the Navier-Stokes equation
 Lorenz‟s model could:
◦ be sensible to variations to the initial
conditions
◦ offer „turbulence like‟ structures
◦ be non-repeatable or chaotic
◦ offer a deterministic solution to the N-S
equations

13. Interval: randomicty vs. chaos
 Lorenz‟s solutions were chaotic

14. A brief history of turbulence
 Today:
◦ The statistical approach to the turbulence
has become the standard
◦ Experimental studies advanced and
helped to define and pinpoint the physical
structures of the turbulent flow
◦ Mathematical advances on the solution of
the N-S equation
◦ Boom of computational techniques born in
the 70‟s and 80‟s

15. Computational overview
 What are computers really doing?
Simulated fire-plume

16. Computational overview
 They‟re solving the N-S equation
◦ At least a specialized, statistically based,
version of them
◦ Using simple geometry
◦ Using simplified and/or parameterized
physical effects such as chemistry,
radiation and others

17. Estimating the N-S equation
 The N-S equation is a partial
differential equation:
◦ Discretize the partial differential equation
by choosing a numerical method
◦ Obtain initial conditions for each variable
at every spatial coordinate (and boundary
conditions for the simulation)
◦ Choose an adequate time/space grid
◦ Improve it to be feasible to solve at an
adequate time (or have a supercomputer
and then wait)

18. Estimating the N-S equation
 Partial differential equation

19. Estimating the N-S equation
 A (very) simple discretization of a
partial differential equation
Classical explicit method

20. Estimating the N-S equation
 Initial condition
 Boundary condition

21. Estimating the N-S equation
 Visualizing the conditions Boundary values
given to the problem
beforehand
This shape was given
to the problem as a
initial
condition
Numerical solution to the wave equation – Fonseca (20

22. Estimating the N-S equation
 The computational grid
Adaptative grid following a shock solution – Fonseca (20

23. 2 sample applications
 Aircraft engineering

24. 2 sample applications
 Aircraft engineering
Initial conditions: wind velocity & atmospheric disturbances
Boundary conditions: the precise airplane shape
Grid: The mesh over the airplane
Numerical Model: Dependent on the physical properties you want
to simulate: drag, lift, momentum, heat diffusion, etc.

25. 2 sample applications
 Air pollution

26. 2 sample applications
 Air pollution
Source: http://www.me.jhu.edu/eddy-simulation.htm
Initial conditions: Wind velocity, pressure distribution, pollutant
source
Boundary conditions: The cityscape
Grid: The mesh over the cityscape
Numerical Model: N coupled types:
particle, turbulence, chemistry, etc

27. Computer simulations: the nitty-
gritty
 What kind of computer can solve the
turbulent flow?
ANY
One dimensional,
parameterized
Hundred points can Can be intractable even in
be solved in 20s in a today‟s supercomputer;
Intel‟s CORE 2 quad e.g., DNS

28. Computer simulations: the nitty-
gritty
 Simulations done by the IAG-USP
micrometeorology laboratory
◦ ~ 2003: 3D, serial LES. 8 nodes. 5 days
to simulate 1 h using a CRAY
supercomputer
◦ ~ 2008: 3D, parallelized LES. 8 nodes. 4
days to simulate 10 h using an Intel
cluster
◦ ~ 2009: 3D, parallelized LES. 8 nodes.
1.5 day to simulate 12 h using an Intel
cluster
◦ ~ 2011: 3D, parallelized LES. 1024 nodes.
Sources: Codato (2008), Marques Filho, (2004); Oliveira et al., (2002); Barbaro (2010)
8 hours to simulate an hour* using an IBM
* Personal letter to Barbaro, 2011, SARA: http://www.sara.nl/systems/huygens. The LES model used computed several different physical
effects, from cloud microphysics to chemistry. In short, a much more complex and superior model than the ones listed before

29. Computer simulations: the nitty-
gritty
 Why does it take so long to compute?
◦ Each point in the computation grid
depends on the calculated values of its
neighbors and itself …
◦ … and values for the same points
obtained at previous iteration steps
◦ Roughly speaking, the more physical
properties you retain, the more
“communication” between points and
physical parameterizations you have.

30. Computer simulations: the nitty-
gritty
 Computational grid
Discretization for velocities and viscous stresses around a cell cut by an interface, extracted from
Tryggvason G., Scardovelli R. and Zaleski S., Direct Numerical Simulations of Gas-Liquid Multiphase
Flows, Cambrigge University Press, to appear (February 2011) .
Source: http://www.lmm.jussieu.fr/~zaleski/drops2.html

31. Computer simulations: the nitty-
gritty
 In a nutshell
◦ The maths that allow the N-S equations to
be discretized at all are crude
representations of the complete flow.
◦ The more physical effects you retain in
the problem, greater is the effort to
compute it.
◦ The more realistic the geometry of the
problem, the more intensive is the use of
computer resources.

32. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 Objectives
◦ Investigate the CO2 transfer on the
equatorial Atlantic Ocean
 Gather CO2 and other meteorological data
 Estimate the heat fluxes
 Estimate the CO2 flux
◦ Provide a methodology for further studies
◦ Couple the CO2 transfer algorithm to a 1D
turbulence model

33. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 CO2 over the oceans
Atmospheric CO2 concentration at Ascension Island (8°S, 14°W)
Fonseca (2011)
Atmospheric CO2 concentration over the oceans Source: NOAA

34. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 CO2 transfer on the oceans

35. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 What is flux? Flux is the amount of “something”
passing through a surface
http://betterexplained.com/articles/flu
x/
Source: http://isites.harvard.edu/icb/icb.do?keyword=k41471&pageid=icb.page194990
Source: wikipedia

36. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 Why the heat flux?
The latent and
sensible heat
fluxes are
estimated by the
algorithm and can
be used as initial
and boundary
condition to a
turbulence model.
Solar radiation powers the Earth system
Source: http://www.answers.com/topic/planetary-boundary-layer

37. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 Why the heat flux?
Values of the
flux of Latent
and sensible
heat and the
shortwave and
longwave
radiation are
used by the
algorithm to
estimate the
CO2 transfer
Source: NASA; The Earth Observer November-December, 2006

38. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 Wind speed & momentum flux
Transfer of wind
momentum to
the ocean
drives the CO2
transfer on the
equatorial
Atlantic ocean.

39. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 The CO2 fluxes can be obtained from the
heat and momentum fluxes
 The equatorial Atlantic ocean act as a
CO2 source to the atmosphere

40. Turbulent CO2 transfer on the
equatorial Atlantic ocean
 Implications:
◦ The CO2 transfer algorithm can now be
coupled to ocean/atmosphere turbulence
models
 It can act as an upper boundary
condition, forcing the water column/atmosphere
 That, in turn, can force the algorithm, resulting
in physically sound estimations of the upper
layers of the ocean and the atmosphere.

41. Summary
 Turbulent vs. Smooth flow
 History and theory development
 Randomicity vs. chaos
 The N-S equations
◦ Describe all flow state, turbulent or
otherwise
◦ Described by a non-linear partial
differential equation
◦ Discretized by numerical methods

42. Summary
 Numerical methods
◦ Initial and boundary conditions
◦ The numerical grid
 Sample applications
◦ Aircraft engineering
◦ Air pollution
 Computer simulations: The nitty-gritty
◦ Dependent on the numerical model and
the parameterized physical processes
◦ Dependent on the geometry

43. Summary
 Turbulent CO2 transfer
◦ CO2 over the oceans
◦ Flux
 Heat & Momentum
◦ CO2 over the equatorial Atlantic ocean
 Gas flux
 Turbulent mixing
◦ Coupling of the algorithm to a turbulence
model

44. Obrigado!
Fabio Fonseca