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An Overview of Computational FluidDynamics and Uncertainty Quantification

September 4, 2015

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Introduction to CFD

Justification of CFD

I Fluid flows are governed by the Navier-Stokes equations, aset of coupled and nonlinear partial differential equations(PDE) that enforce the basic laws of conservation of mass,momentum and energy.

I The coupling ties the dynamic variables, usually thevelocity, the pressure and the density (for stratified fluids)and some tracers like temperature and salinity, which mustbe computed simultaneously. Moreover, the nonlinearitiesprevent the construction of complex solutions to the PDEby linearly combining simpler solutions.

I Computational Fluid Dynamics is the science and art ofapproximating the PDEs by an algebraic system ofequations that can be solved on a computer.

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Introduction to CFD

Fluid Dynamics: the Science of Fluid MotionInvestigation of Fluid Flow can be categories according to

I Experimental/ObservationalDeploy instruments to measure flow variablesBuild replicas of the flow in a lab

I TheoreticalRepresent the physical laws with mathematical equationsSolve the equations when possible (requires simplifications)

I ComputationalAttempts to solve the PDEs without simplifications usingcomputersIntroduces approximations to PDEs that must be analyzed

CFD allows scientists to perform numerical experiments usingcomputers. Realistic experiments (still) require sophisticatedalgorithms and very big computers.

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Introduction to CFD

Type of Fluid Flows

I Atmospheric: winds, jet stream, winds, hurricanes,tornadoes, clouds

I Oceanic: World Ocean circulation, Coastal Flows, SurfaceWaves

I Geology: Magma flows, flow through porous mediaI Climate: Earth system model

(atmosphere-ocean-ice-land-waves)I Engineering: Flow around airplanes, combustionI Biomedical: Blood flow (hemodynamics)

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Introduction to CFD

Atmospheric Flows

Figure: Left: Satellite image of Hurricane Katrina on August 28, 2010.NOAA, NASA, SSEC. Right:An F5 tornado that struck Elie, Manitoba,Canada, (40 kilometers west of Winnipeg) on June 22, 2007. Thelarge cloud of flying debris at the base is a major cause of injury intornadoes. Photo: Justin Hobson

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Introduction to CFD

Oceanic Flows

Figure: Left: Sea Surface Temperature image of the Gulf Stream.Middle: Sea Surface Height in the Gulf of Mexico revealing the LoopCurrent. Right: Large swells in the Pacific.

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Introduction to CFD

Experiments vs SimulationsI Experiment

I Quantitative description of flow using measurementsI Space-time scarcity of measurementsI Limited range of flow conditions and regimesI Measurement errors and flow distubance by sensors

I SimulationI Quantitative prediction of flow using softwareI High resolution data (too much actually)I Limited by complexity of model equationsI Modeling errors and input data uncertainties

I Experiment and Simulation are complementaryI Experiments are used to validate modelsI Models can be used to anticipate or explain measurements.I Simulation can mitigate the cost of observational

experiments.

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Introduction to CFD

Typical CFD Applications

I Forecast of atmospheric and oceanic weathersI Climate predictionsI Hindcast analyses to fill in gaps in observationsI Improve efficiencies of combustion in cars and jet enginesI Design of better heart pumps and drug delivery systemsI Risk analysis when evaluating environmental impact of

potential environmental disasters

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Introduction to CFD

CFD is an interdisciplinary scienceI The physics

I State the fluid flow problem that needs analysisI identify the physical processes influencing the dynamicsI Identify the relevant space-time scales of the problemI Identify the input data neededI Validate the results

I The MathematicsI Decide on the appropriate PDEs for the dynamics

Compressible, inviscid, hydrostatic, etc.I Discretize the flow domain and PDEI Identify the optimal solution algorithm

I The Computer ScienceI Implement (or borrow) the actual code needed for the

discrete algorithmI Solve the problem using available computer resourcesI Explore the parameter space of the input data

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Introduction to CFD

CFD analysis steps

I Problem statement: Gather information about the flowI Mathematical model: IBVP = PDEs + IC + BC + ForcingI Mesh generation: Discretize domain geometryI Discretize the PDE : obtain ODEsI Time Discretization: ODEs become an algebraic system

Ax = b. Costly solution if A is not easy invertable anddim(A) is large.

I CFD software: Turn algorithm into codeI Simulation: Calculate solution on computersI Postprocessing: visualization and analysis of dataI Verification: model validation/adjustment.

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Introduction to CFD

Spatial DiscretizationComputational Methods are categorized according to theirspatial discretization and their discretization of the PDEs

I Finite Difference Method:I Earliest of the techniquesI Lacks geometric flexibility because of its structured gridsI Enforces the PDEs at the nodes of the grid.

I Finite Volume MethodsI Enforces conservations at cell levelsI Can use structured or unstructured grids

I Spectral MethodsI High order/accuracyI Costly per grid pointI Geometrically simple domains

I Spectral/Finite Element MethodsI Relies on unstructured gridsI Minimizes residual error over elements

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Introduction to CFD

Structured ocean meshes in Atlantic Ocean

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Introduction to CFD

Hexagonal Finite Volume Mesh

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time steps could be used, at great expense, tomaintain computational stability near the poles,but the high resolution in the east–west direc-tion near them would be wasted because themodel uses lower resolution elsewhere. Solu-tions to the pole problem are sometimes basedon longitudinal filtering, but these methods havesome unfortunate side effects and scale badly asmodel resolution increases.

OGCMs also use finite-difference methods.1In the Parallel Ocean Program (POP),2 stretch-ing the spherical coordinates so that the coordi-nate system’s north pole is on a landmass (for ex-ample, Greenland) rather than in the ArcticOcean eliminates the pole problem. Althoughthis ad hoc approach alleviates the problem in theOGCM, the peculiarities of the stretched oceanmodel grid further complicate the coupling ofthe atmosphere, ocean, and land surface sub-models. Moreover, the stretched grid is tailoredfor the particular arrangement of the continentsand oceans on Earth and thus lacks generality.

Spectral AGCMs are based on spherical har-monic expansions.3 This approach also avoidsthe pole problem but has problems with almost-discontinuous functions such as steep terrain andthe sharp boundaries of clouds. Moreover, spec-tral models poorly simulate the transports ofhighly variable nonnegative scalars, such as the

concentration of moisture.4 This weakness hascaused most spectral AGCMs to evolve into hy-brid models in which grids represent most or allof the advective processes.

The spectral method (which is based on spher-ical harmonics) is not applicable to ocean mod-eling because of the ocean basins’ complicatedgeometry. Researchers are exploring spectral el-ements based on local expansions, generally ofhigh-order accuracy, as an alternative approachand for possible use in atmosphere modeling.5

We found that the best way to avoid the poleproblem is to use quasi-uniform spherical geo-desic grids—tessellations of the sphere gener-ated from icosahedra or other Platonic solids.6Icosahedral grids, first tried in the 1960s, givealmost homogeneous and quasi-isotropic cov-erage of the sphere.7,8 Modern numerical algo-rithms including fast elliptic solvers andmimetic discretization schemes have madethese grids more attractive, renewing interestin the idea.9,10

Grid generationTo construct an icosahedral grid, we begin

with an icosahedron (see Figure 3), which hastriangular faces and vertices. Euler’s famous the-orem tells us that the number of edges is

Figure 1. An example of a geodesic grid with acolor-coded plot of the observed sea-surface temperature distribution. The continents are depicted in white. This grid has 10,242 cells, eachof which is roughly 240 km across. Twelve of thecells are pentagons; the rest are hexagons.

Figure 2. An example of a latitude–longitude grid.A pole is at the top. The black dots represent gridcell centers that are equally spaced in longitude(west to east) and latitude (south to north). Thered lines are the Equator and two lines ofconstant longitude.

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Introduction to CFD

Multi-Resolution Unstructured Triangular FiniteElement Mesh of N. Atlantic

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Introduction to CFD

CFD: subgrid scale features

I Models have inherently limited resolution.I Example: Impractical to resolve particle-sized processes in

clouds and compute on climate length and time scales.I Scales below a certain resolution are badly represented on

computational grids. Their effects on the large scale has todepend on subgrid models like turbulence models.

I Otherwise the small scale features have to be filtered outto prevent noise build-up and catastrophic failure.

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Introduction to CFD

Computing

I Exponential growth in computing power (speed andstorage) allowed CFD to grow and flourish.

I Small problems can be solved on laptops and workstationsI 3D-time dependent realistic simulations require

supercomputers even in 2015I Choose algorithms that map well onto supercomputing

architecture,I Codes must be parallel to take advantage of today’s

computersGraphical Processing Units are now very trendy.

I Code portability is an issue.I Most supercomputing codes in US are in Fortran

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Introduction to CFD

CFD example:Gravitational adjustment

Figure: Simulating the gravitational adjustment problem at highresolution

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Introduction to CFD

CFD example: Hurrican Sandy

I Miami Coupled Atmosphere Wave Ocean model ofhurricane Sandy (2012)

I Each frame is a 10-minute simulation snapshot of windspeed before and during landfall.

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Introduction to CFD

CFD example: HYCOM Simulations with DataAssimilation

http://viewer.coaps.fsu.edu/GlobalHycom/mapviewer

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Introduction to CFD

Error sources for CFD

I Conceptual: Misspecifying the PDEs for the problemI Discretization: Errors in numerical algorithm (numerical

instabilities)I Programming: Computer BugsI Data Errors: Input data (initial and boundary conditions,

forcing) is uncertainI Epistemic uncertainty: Lack of fundamental knoweldge.

Can be reduced by acquiring more data.I Aleatoric uncertainty: Data is inherently uncertain (e.g.

nozzle inflow is variable).I User must explore uncertainty space to gain confidence in

model resultsI Uncertainty quantification seeks to approach this problem

using probabilities