Turbulence: a computational overview & CO2 transferFabio Fonseca

OverviewTurbulent 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

Turbulent vs. smooth flow

Turbulent flow

Smooth flow

Turbulence

Source: McDonnough 2004, 2007

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

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)?

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)

Turbulence: The problem (as of today)

source: http//www.claymath.org/millennium

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

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 field

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)

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.

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 equationLorenz’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

Interval: randomicty vs. chaosLorenz’s solutions were chaotic

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

Computational overviewWhat are computers really doing?

Simulated fire-plume

Computational overviewThey’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

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)

Estimating the N-S equationPartial differential equation

Estimating the N-S equationA (very) simple discretization of a

partial differential equation

Classical explicit method

Estimating the N-S equationInitial condition

Boundary condition

Estimating the N-S equationVisualizing the conditionsBoundary values

given to the problem

beforehand

This shape was given to the problem as a initial condition

Numerical solution to the wave equation – Fonseca (2008)

Estimating the N-S equationThe computational grid

Adaptative grid following a shock solution – Fonseca (2008)

2 sample applicationsAircraft engineering

2 sample applicationsAircraft engineering

Initial conditions: wind velocity & atmospheric disturbancesBoundary conditions: the precise airplane shapeGrid: The mesh over the airplaneNumerical Model: Dependent on the physical properties you want to simulate: drag, lift, momentum, heat diffusion, etc.

2 sample applicationsAir pollution

2 sample applicationsAir pollution

Initial conditions: Wind velocity, pressure distribution, pollutant sourceBoundary conditions: The cityscapeGrid: The mesh over the cityscapeNumerical Model: N coupled types: particle, turbulence, chemistry, etc

Source: http://www.me.jhu.edu/eddy-simulation.html

Computer simulations: the nitty-grittyWhat kind of computer can solve

the turbulent flow?

ANY

One dimensional,parameterized

Hundred points can be solved in 20s in a Intel’s CORE 2 quad

Can be intractable even in today’s supercomputer; e.g., DNS

Computer simulations: the nitty-grittySimulations 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. 8 hours to simulate an hour* using an IBM power6 (Huygens) supercomputer.

Sources: Codato (2008), Marques Filho, (2004); Oliveira et al., (2002); Barbaro (2010) * 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

Computer simulations: the nitty-grittyWhy 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.

Computer simulations: the nitty-grittyComputational 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

Computer simulations: the nitty-grittyIn 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.

Turbulent CO2 transfer on the equatorial Atlantic oceanObjectives

◦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

Turbulent CO2 transfer on the equatorial Atlantic oceanCO2 over the oceans

Source: NOAA

Fonseca (2011)

Atmospheric CO2 concentration at Ascension Island (8°S, 14°W)

Atmospheric CO2 concentration over the oceans

Turbulent CO2 transfer on the equatorial Atlantic oceanCO2 transfer on the oceans

Turbulent CO2 transfer on the equatorial Atlantic oceanWhat is flux?

Source: wikipedia

Source: http://isites.harvard.edu/icb/icb.do?keyword=k41471&pageid=icb.page194990

Flux is the amount of “something” passing through a surface http://betterexplained.com/articles/flux/

Turbulent CO2 transfer on the equatorial Atlantic oceanWhy the heat flux?

Solar radiation powers the Earth systemSource: http://www.answers.com/topic/planetary-boundary-layer

The latent and sensible heat fluxes are estimated by the algorithm and can be used as initial and boundary condition to a turbulence model.

Turbulent CO2 transfer on the equatorial Atlantic oceanWhy 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

Turbulent CO2 transfer on the equatorial Atlantic oceanWind speed & momentum flux

Transfer of wind momentum to the ocean drives the CO2 transfer on the equatorial Atlantic ocean.

Turbulent CO2 transfer on the equatorial Atlantic oceanThe CO2 fluxes can be obtained from the

heat and momentum fluxes

The equatorial Atlantic ocean act as a CO2

source to the atmosphere

Turbulent CO2 transfer on the equatorial Atlantic oceanImplications:

◦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.

SummaryTurbulent vs. Smooth flow

History and theory developmentRandomicity 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

SummaryNumerical 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

SummaryTurbulent 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

Obrigado!

Fabio Fonseca