Immersed Boundary Technique

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Immersed Boundary Technique for Turbulent Flows
with Industrial Applications
Gianluca Iaccarino Center for Turbulence
Research Stanford University
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Computational Fluid Dynamics (CFD)
CFD Study of flow physics using computers CFD
is becoming a fundamental tool for design,
engineering and for scientific discoveries
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Computational Fluid Dynamics (CFD)
Pollutant dispersion in urban environment
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Computational Fluid Dynamics (CFD)
Design of cooling passages for turbine blades
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Some of the Challenges for Computational Fluid
Dynamics
Accurate physical modeling Multi-physics
simulations Rapid turn-around time for complex,
realistic industrial applications Automation for
industrial design and optimization Error
estimation and uncertainty analysis
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Classic CFD Approach
The physical problem is converted in a
mathematical description (a set of PDEs) The
computational domain is contained within physical
boundaries A body-fitted mesh is generated
within the domain Solution algorithms handle
polygonal (polyhedral) cells
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Industrial Configurations
Extreme Geometrical Complexity100 parts and 30
(Moving) Components Flow and conjugate heat
transfer Conventional CFD analysis might
require up to 6 months of CAD Clean-Up and
Grid Generation! Not suited for design changes
given typical project timing (12 months) and
budget (the motor sells for 120)
Electric motor, courtesy of
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Cartesian Methods
Grids are NOT Conforming with the boundaries of
the CFD domain
A special treatment is necessary for the cells
in the vicinity of the immersed boundary
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Cartesian Methods
Options
Stair-Step Approach a Cell is either
considered inside or outside. The geometry Is
approximated
Immersed Boundary The governing equations
are modified in the cells cut by the interface by
adding a source term (Indirect BC or forcing)
Cell-Cut Approach Computational stencils
are modified to use information on the
boundaries (Direct BC)
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Motivation for the Present IB Method
High Reynolds number industrial problems Complex
physics (heat transfer, multiphase flows)
Reduce time-to-solution for realistic
problems Ingredients Numerical algorithms,
computational geometry, turbulence modeling
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Reynolds Averaged Navier-Stokes
Turbulent flows are usually not captured
directly, i.e. by solving the NS
equations because of the huge computational
requirements Derive equations for the mean flow
(RANS) and model the turbulent fluctuations using
additional PDEs
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Classification of IB Methods
Continuous forcing (Peskin approach) Indirect
boundary condition enforcement Source/sink term
in the momentum equations Diffused
interface Discrete forcing (Present
approach) Direct boundary condition
enforcement Reconstruction in the vicinity of
the IB Sharp interface
Mittal Iaccarino Immersed Boundary Methods,
Ann. Rev. Fluid Mech., V. 37, 2005
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Peskins IB Method
Introduced in the 70s to simulate the flow in the
human heart Navier-Stokes equations are solved
on a Cartesian grid. Heart walls are modeled as
elastic membranes. The interaction is
modeled using a source term added to the
governing equations
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Limitations of the Forcing Approach
Peskins IB approach is sound and well suited for
elastic boundaries Standard forcing terms become
ill-behaved in the rigid limit Ad-hoc forcing
terms (i.e. porosity) tend to be inaccurate and
unstable Definition of the forcing terms for
turbulent quantities (as in RANS) is extremely
challenging The smoothing of the forcing
functions implies a non-sharp representation of
the boundary which prevent the correct capturing
of boundary layers at high Reynolds
numbers Solution is required inside solid bodies
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Outline of my presentation

• Treatment of the immersed boundary conditions
• Handling complex geometries and anisotropic grid
  refinement
• RANS turbulence modeling and adaptive wall
  functions
• Validation of the approach
• Current research projects

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Treatment of the boundary conditions on the
Immersed Boundary
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Discrete Forcing Concept
The immersed boundary forcing is defined at the
discretization step It is desirable that the
accuracy and stability of the numerical scheme
affect directly the forcing terms It is crucial
to eliminate ANY user-specified parameter from
the definition of the forcing The approach must
be suited for any physical quantity (mean
velocity, turbulent scalars, temperature, etc.)
Iaccarino Verzicco,Turbulent flow simulations
using the immersed boundary technique, Applied
Mech. Review, V. 121, 2004
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Discrete Forcing Concept
In the continuous forcing approach the equations
are discretized directly together with the
forcing term
In the discrete forcing approach the equations
are discretized without accounting for the
immersed boundary conditions
The discretization stencil is then modified at
the IB cells (RHS contains known information at
the boundaries)
An this is equivalent to solving the system
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Modified Stencils at the IB Surfaces
Linear Reconstruction Quadratic
Reconstruction
F fluid points B boundary point G ghost point
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Non-Aligned Boundary Layers
First test boundary layers oblique to grid lines
Velocity Magnitude
Re 5,000 Linear reconstruction
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Physics-based Reconstruction Schemes
The reconstruction schemes presented are purely
based on geometrical information It is possible
to build reconstruction schemes that are based on
certain (expected) physical/mathematical
properties of the solutions asymptotic behavior
(especially for turbulent quantities) conservatio
n properties governing equations
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Momentum-Based Reconstruction Schemes
The quadratic reconstruction scheme requires 4
fluid nodes BC
n
F fluid points B boundary point G ghost
point I image of the ghost point
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Momentum-Based Reconstruction Schemes
The quadratic reconstruction scheme requires 4
fluid nodes BC We can build a better
reconstruction by using the conservation of
momentum in a ghost control volume
F fluid points B boundary point G ghost
point I image of the ghost point
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Reconstruction Accuracy
Decaying vortex problem (exact solution of the
Navier Stokes equations)
The problem is solved on a Cartesian grid but the
boundary conditions are applied on a
non-conformal surface
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Reconstruction Accuracy
Taylor Vortex Decaying Problem
Kang, Iaccarino, Moin Accuracy of reconstruction
techniques for the immersed boundary method.
Submitted to J. Comp. Physics, 2004
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Handling complex geometries and anisotropic grid
refinement
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Handling complex geometries
A major advantage of the IB method is the
automation of the grid generation
process Immersed surface representations are
disconnected from the volume mesh. The STL
format (one of the standard for rapid
prototyping) is used We have developed
computational geometry tools to robustly
identify fluid/solid/interface cells and compute
all geometrical Information required by the
reconstruction algorithm (ray tracing)
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Anisotropic Local Grid Refinement
Pure Cartesian grids CANNOT be used for high
Reynolds numbers Local grid refinement must be
employed to increase resolution within boundary
layers Anisotropy can provide substantial grid
saving for (nearly) aligned immersed surfaces
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LGR Only Way to High Re Numbers
3D Ellipsoid (D/L0.1)
Uniform Isotropic LGR Anisotropic LGR
Specified normal and tangential resolution (?n/L
and ?t/L10?n/L)
?n/L
Anisotropic LGR leads to one order of magnitude
saving over isotropic LGR
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Novel Approach to Local Grid Refinement
Present approach is Semi-Structured an
underlying continuous grid is defined and actual
cells are built using an agglomeration operation
(i2,j2,k2)
The ONLY connectivity stored for each cell is
the bounding box
(i1,j1,k1)
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Example of Applications of LGR
Laminar driven cavity Re 1K
Turbulent backstep Re 5K
Durbin Iaccarino, An approach to local
refinement of structured grids, J. Comp. Phys,
V. 181, 2002
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Example of automatic grid generation
Geometry courtesy of
Fully dressed GM vehicle underhood,
underbody, external surfaces
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Example of automatic grid generation
Geometry courtesy of
Fully dressed Valeo motor rotor, stator,
electronic components, cooling system
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RANS turbulence models and adaptive wall functions
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Turbulence Modeling
Initially the approach has been to adopt RANS
models that are well suited for the IB
approach As an example we use turbulent scalars
that vary linearly close to walls starting from
the well-known k-? model and introducing
g(1/??0.5 the k-g model is obtained Recent
work has focused on incorporating wall models
within the IB reconstruction
Linear Reconstruction
Quadratic Reconstruction
Wall-Function Reconstruction
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Wall Functions Background
Motivated by the universal nature of the flat
plate boundary layer
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Wall Functions Background
The universal profile is function of the
turbulence model!
One important component of accurate wall
functions is the consistent capturing of the
intermediate region 5ltylt20
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A Novel Adaptive Wall Model
Consistency the model provides exactly the same
solution as the grid approaches y1 (wall
integration solution) for mean velocity AND
turbulence quantities Accuracy the numerical
errors due to the coarse meshes are clearly
identified and possibly reduced Efficiency
analytical (explicit) solutions or look-up tables
are used
Kalitzin, Medic, Iaccarino, Durbin Near wall
behavior of RANS turbulence models and
Implication for wall functions, J. Comp. Phys.,
V. 204, 2005
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Application to Recirculating Flows

• Separation on a flat plate via suction/blowing
• (ReL 3x107)

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Wall functions as IB reconstructions
Objective define IB reconstruction for the
location B (ghost G)

• Compute friction velocity based on wall model in
  cells F1 and F2

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Wall functions as IB reconstructions
Objective define IB reconstruction for the
location B (ghost G)

• Compute friction velocity based on wall model in
  cells F1 and F2
• Map the friction velocity on the IB surface
• Interpolate friction velocity corresponding to
  the B location

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Wall functions as IB reconstructions
Objective define IB reconstruction for the
location B (ghost G)

• Compute friction velocity based on wall model in
  cells F1 and F2
• Map the friction velocity on the IB surface
• Interpolate friction velocity corresponding to
  the B location
• Extract mean velocity and turbulence quantity
  from tables to generate information in I

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Wall functions as IB reconstructions
Objective define IB reconstruction for the
location B (ghost G)

• Compute friction velocity based on wall model in
  cells F1 and F2
• Map the friction velocity on the IB surface
• Interpolate friction velocity corresponding to
  the B location
• Extract mean velocity and turbulence quantity
  from tables to generate information in I
• Use computed friction value to obtain ghost value
  G

Interpolation is ONLY used for friction velocity
on the surface
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Validation Study
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Pick-Up Truck
Reynolds number 288,000 PIV Experiments
performed by U. of Michigan (L. Bernal) and GM
(B. Khalighi)
Computational Grid
Iaccarino, Kalitzin Khalighi, AIAA Paper
2003-0775 Jindal, Iaccarino, Khalighi, SAE Paper
2005-B419
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CFD vs. Experiments
Exp. CFD
Symmetry Plane Off-Symmetry Plane
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Computed Friction Lines
KG Model
SA Model
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CFD vs. Experiments
Pressure Coefficient Distribution on the Symmetry
Plane
Cartesian Grid
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CFD vs. Experiments
Velocity profiles in the Symmetry Plane
Experiments
Structured Grid
LGR grid
Location of PIV measurement sheet
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Summary

• A novel IB/RANS approach based on discrete
  forcing has been developed
• The computational method is based on
• efficient computational geometry tools to handle
  complex geometry
• anisotropic grid generation
• accurate and stable IB reconstructions
• use of adaptive wall models and IB-friendly
  turbulence models
• Applications have illustrated the predictive
  capability of the present approach

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Summary
The analysis of a realistic industrial
configuration can be accomplished in less than 1
day with the IB technique as opposed to about 6
months with conventional CFD methods
Velocity Magnitude at two Cross Sections
Flow Rate vs. Pressure Drop
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Applications of the Present IB Technique
Thesis work of Fabio Damiani, Politecnico di
Bari Aerodynamic analysis of front and rear
wings in F1 cars
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Applications of the Present IB Technique
Thesis work of Simona Nardulli, Politecnico di
Bari Drag Reduction for Truck-Trailer
Configurations
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Future Research Directions

• Include complex physics in the IB formulation
• Heat transfer
• Multiphase flows
• Combustion
• Compressible flows
• Moving boundaries
• Develop accurate IB/LES framework
• Reconstruction (energy conservation, acoustics,
  etc.)
• Numerical algorithms
• Applications
• Design and optimization
• Rapid prototyping using CFD

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Hydrodynamics in Coral Colonies
Magnetic Resonance Velocimetry
Geometries are generated by scanning
real skeletons
Coral growth and morphology is strongly affected
by flow
Immersed Boundary Simulations
With S. Chang, C. Elkins, S. Monismith, J. Eaton
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Hydrodynamics in Coral Colonies
With S. Chang, C. Elkins, S. Monismith, J. Eaton
Magnetic Resonance Velocimetry
Immersed Boundary Simulations
(-) Streamwise velocity
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In collaboration with
IB for multiphase flows
Realistic Topography and accurate capturing of
the Interface dynamics using a Level set method
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In collaboration with
Tsunami modeling
Stanford calculations
Ward Day (2001)
Stanford calculations
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Acknowledgments
Prof. Roberto Verzicco and Prof. Rajat
Mittal Collegues and coworkers (in alhabetical
order) Fabio Damiani, Paul Durbin, Chris Elkins,
Massimiliano Fatica, Frank Ham, Georgi Kalitzin,
Seongwon Kang, Bahram Khalighi, Sekhar Majumdar,
Rajat Mittal, Stephane Moreau, Andrea Pascarelli,
Fabrizio Tessicini and Meng Wang. Collegio dei
Docenti del Dottorato di Ricerca in Ingegneria
Meccanica del Politecnico di Bari
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Peskins IB Method
Eulerian fluid governing equations
Lagrangian fiber tracking
Coupling term
Key ingredients
and
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Discrete Forcing Concept
In the continuous forcing approach the equations
are discretized directly together with the
forcing term
In the discrete forcing approach the equations
are discretized without accounting for the
immersed boundary conditions
The discretization stencil is then modified at
the IB cells (RHS contains known information at
the boundaries)
An this is equivalent to solving the system