Diapositiva 1

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Wall modeling challenges for the immersed
boundary method
G. Pascazio
pascazio_at_poliba.it
M. D. de Tullio, P. De Palma, M. Napolitano G.
Iaccarino, R. Verzicco G. Adriani, P. Decuzzi
.
Workshop Num. Methods non-body fitted grids -
Maratea, May 13-15 2010
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OUTLINE

• Immersed Boundary technique
• Tagging, Forcing, Near-wall reconstruction
• High-Re turbulent flows wall modeling
• Tables, Analytical, Numerical
• Preconditioned compressible-flow solver
• Results
• Arbitrarily shaped particle transport in an
  incompressible flow
• Fluid-structure interaction solver
• Results

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IMMERSED BOUNDARY TECHNIQUE
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TAGGING
Cartesian Grid
Geometry
fluid cells
ray tracing
solid cells
interface cells
A special treatment is needed for the cells close
to the immersed boundary
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FORCING
During the computation, the flow variables at the
center of the fluid cells are the unknowns, the
solid cells do not influence the flow field at
all, and at the interface cells the forcing is
applied

• Direct forcing (Mohd-Yusof)
• - The governing equations are not modified
• - The boundary conditions are enforced directly
• Sharp interface

The boundary condition has to be imposed at the
interface cells, which do not coincide with the
body.
Thus, a local reconstruction of the solution
close to the immersed boundary is needed.
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Procedure (de Tullio et al., JCP 2007)
Generation of a first uniform mesh (Input Ximin
, Ximax , DXi )
Refinement of prescribed selected regions (
Input Ximin , Ximax , DXi )
Automatic refinement along the immersed surface (
Input Dn, Dt ) Iterative
Automatic refinement along prescribed surface (
Input Dn, Dt ) Iterative
Coarsening
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Multi-dimensional linear recontruction
(2D) (e.g., Yang and Balaras, JCP 2006)
Dirichlet boundary condition
Neumann bounary condition
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DISTANCE-WEIGHTED RECONSTRUCTION (LIN) (de Tullio
et al., JCP 2007)
Dirichlet boundary conditions
Neumann boundary conditions
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COMPRESSIBLE SOLVER (RANS)
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NUMERICAL METHOD

• Reynolds Averaged Navier-Stokes equations
  (RANS)
• k-? turbulence model (Wilcox, 1998)

• Pseudo-time derivative term added to the LHS to
  use a time marching approach for steady and
  unsteady problems (Venkateswaran and Merkle,
  1995)
• Preconditioning matrix G to improve the
  efficiency for a wide range of the Mach number
  (Merkle, 1995)

• Euler implicit scheme discretization in the
  pseudo-time
• 2nd order accurate three point backward
  discretization in the physical time
• Diagonalization procedure (Pulliam and Chausee,
  1981)
• Factorization of the LHS

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NUMERICAL METHOD

• BiCGStab solver to solve the three sparse
  matrices

• Colocated cell-centred finite-volume space
  discretization
• Convective terms 1st, 2nd and 3rd order
  accurate flux difference splitting scheme or 2nd
  order accurate centred scheme
• Viscous terms 2nd order accurate centred
  scheme
• Minmod limiter in presence of shocks
• Semi-structured Cartesian grids

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FLUX EVALUATION
- For each face, the contributions of the
neighbour cells are collected to build the
corresponding operators (convective/diffusive)
for the cell
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RESULTS
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• NACA-0012
• M0.8, Re20, angle of attack 10
• Space domain -10 c , 11 c -10 c , 10 c
• 5 meshes

• Exact solution obtained by means of a
  Richardson extrapolation employing the two finest
  meshes

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• M0.03 Re100,120,140 T1.0, 1.1, 1.5, 1.8
  (T Tw/Tinf)
• (-10,40) D (-15, 15) D
• - Mesh 41509 cells, 293647 faces
• - T (Energy equation) is crucial for Tgt1
• unsteady periodic flow

Temperature contours (Re100, T1.8)
Exp (Wang et al., Phys. Fluids, 2000)
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Flusso supersonico su cilindro

• M1.7 Re2.e5
• Domain (-10,15) D (-10, 10) D

Locally refined mesh 75556 cells
545700 faces
q 113 q 112 (exp.) CD 1.41 CD 1.43
(exp.)
Mach number contours
Pressure coefficient
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- Re500, M0.003
Imposed cross-flow frequency
Natural shedding frequency (fixed cylinder)
a(t)y(t)/D
vorticity (F0.875)
Ref.1 Blackburn, J.Fluid Mech, 1999
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WALL MODELING
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WALL MODELING

• Linear interpolation is adequate for laminar
  flows or when the interface point is within the
  viscous sublayer
• Brute-force grid refinement is not efficient in
  a Cartesian grid framework
• Local grid refinement alleviates the resolution
  requirements, but still it is not an adequate
  solution for very high Reynolds number flows

Wall functions motivated by the universal nature
of the flat plate boundary layer
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WALL FUNCTIONS

• The Navier Stokes equations are solved down to
  the fluid point P1

• Flow variables at the interface point I are
  imposed solving a two-point boundary value
  problem

P1
F1
I
Turbulence model equations
W
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WALL FUNCTIONS (TAB)
(Kalitzin et al. J. Comput. Phys. 2005)
It is possible to define a local Reynolds number,
based on y and U. The following is a universal
function
This function is evaluated once and for all using
a wall resolved, grid-converged numerical
solution and stored in a table along with its
inverse (look-up tables)
Rey y u k w


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WALL FUNCTIONS (TAB)
Compute velocity in F1
Compute friction velocity corresponding to IB
surface (W), based on wall model
uF1 , yF1 , nF1 ? ReF1 ReF1 , tables ? yF1
ut (yF1 nF1) / yF1
F1
Extract mean velocity and turbulence quantities
in I
I
W
ut , yI , nI ? yI yI , tables ? ui , ki,
wi ut , yI , ui , ki, wi ? ui , ki, wi
F1-W is equal to twice the largest distance from
the wall of the interface cells
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WALL FUNCTIONS (ANALYTICAL)
(Craft et al. Int. J. of Heat Fluid Flow, 2002)
To simplify integration, rather than a
conventional damping function, a shift of the
turbulent flow origin from the wall to the edge
of the viscous layer is modeled.
Molecular and turbulent viscosity variations
viscous sublayer
where
(variation of fluid properties in the viscous
sublayer is neglected)
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WALL FUNCTIONS (ANALYTICAL)
(Craft et al. Int. J. of Heat Fluid Flow, 2002)
Velocity variation in the near-wall region
The equation is integrated separately across the
viscous and fully turbulent regions, resulting in
analytical formulations for U, given the value of
UN
Shear stress
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TWO-LAYER WALL MODELING
Point F1 is found, along the normal-to-the-wall
direction, at twice the largest distance from the
wall of the interface cells
A virtual refined mesh is embedded between the
wall point W and F1 in the normal direction
P1

• The Navier Stokes equations are solved down to
  the fluid point P1

F1

• Velocity at F1 is interpolated using the
  surrounding cells

h

• Simplified turbulent boundary layer equations
  are solved at the virtual grid points

I

• Velocity at the interface point I is interpolated

W
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WALL FUNCTIONS (NUMERICAL, NWF)
Momentum equation (Balaras et al. 1996 Wang and
Moin, 2002)
y normal direction x tangential direction
The eddy viscosity is obtained from a simple
mixing length model with near wall damping (Cabot
and Moin, FTC 1999)
k 0.4 A16
An iterative procedure has been implemented to
solve the equations simultaneously

• Boundary conditions
• velocity at point F1 (interpolated from
  neighbours fluid nodes)
• velocity at the wall (zero).

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WALL FUNCTIONS (THIN BOUNDARY LAYER, TBLE)
Momentum equation
y normal direction x tangential direction
Turbulence model equations
An iterative procedure has been implemented to
solve the equations simultaneously

• Boundary conditions
• at point F1 (interpolated from neighbours fluid
  nodes)
• at the wall (zero velocity and k, and Menter for
  w).

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FLAT PLATE

• n 1.6 x 10-5 m2/s
• Uinf 90 m/s
• ReL1 6 x 106

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FLAT PLATE
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FLAT PLATE
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RECIRCULATING FLOW

• ReL 3.6 x 107
• L 6 m
• A 0.35 x Uinf
• Uinf 90 m/s

x/L 0.16
x/L 0.58
x/L 0.75
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RECIRCULATING FLOW
x/L 0.16
x/L 0.58
x/L 0.75
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RECIRCULATING FLOW (A 0.35 Uinf)
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RECIRCULATING FLOW (A 0.35 Uinf)
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RECIRCULATING FLOW (A 0.35 Uinf)
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RECIRCULATING FLOW (A 0.27 Uinf)
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M2,is 0.81, 1.0, 1.1, 1.2
Re 8.22x105, 7.44x105, 7.00x105, 6.63x105

• Locally refined mesh 33301 cells
• wall functions (Tables)

M2,is1.2
M2,is1.2
Mis along the blade
Mach number contours
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RAE-2822 AIRFOIL
Wall resolved reference solution 700000 cells.
IB grid 20000 cells
Local view of the grid
Local view of the grid
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RAE-2822 AIRFOIL
Pressure coefficient distribution
Mach number contours (NWF)
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Conjugate heat transfer T106 LP turbine
Temperature contours
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CONCLUSIONS (1) High Reynolds number turbulent
flows

• Wall modelling appears to be an efficient tool
  for computation of high-Re flows
• Different approaches have been investigated to
  model the flow behaviour normal to the wall (a)
  look-up tables (b) analytical wall functions
  (c) numerical wall functions (NWF TBLE)
• Wall functions provide good results for attached
  flows
• Encouraging results for separated flows in
  particular NWF and TBLE with embedded
  one-dimensional grids

Work in progress and future developments

• Study the influence of source terms
• Investigate in details the robustness and
  efficiency issues
• Include an accurate thermal wall model

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Arbitrarily shaped particle transport in an
incompressible flow
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Use of micro/nano-particles for drug delivery and
imaging. Properly designed micro/nano-particles,
once administered at the systemic level and
transported by the blood flow along the
circulatory system, are expected to improve the
efficiency of molecule-based therapy and imaging
by increasing the mass fraction of therapeutic
molecules and tracers that are able to reach
their targets
ligands
PEG
Ferrari, Nat Can Rev, 2005
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Particles are transported by the blood flow and
interact specifically (ligand-receptor bonds) and
non-specifically (e.g., van der Waals,
electrostatic interactions) with the blood vessel
walls, seeking for their target (diseased
endothelium). The intravascular journey of the
particle can be broken down into two events
margination dynamics and firm adhesion.
ligands
PEG
Ferrari, Nat Can Rev, 2005
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The margination is a well-known term in
physiology conventionally used to describe the
lateral drift of leukocytes and platelets from
the core blood vessel towards the endothelial
walls.
The observation of inhomogeneous radial
distributions of particles in tube flow dates
from the work of Poiseuille (1836) who was mainly
concerned by the flow of blood and the behavior
of the red and white corpuscles it carries.
Experimental results (Segré Silberberg, JFM
1962) show the radial migration develops in a
pipe from a uniform concentration at the
entrance. Equilibrium position r/R 0.62
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Matas, Morris, Guazzelli, 2004
Experimental distribution of particle position
(particle diameter 900 µm) over a cross section
of the flow observed for two values of the
Reynolds number Re 60 (left) and Re 350
(right).
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• Micro/nano-particle with different
• size from few tens of nm to few µm
• composition gold- and iron-oxide, silicon
• shape spherical, conical, discoidal, .
• surface physico-chemical properties charge,
  ligants

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• Design parameters
• Particle size and shape
• Reynolds number based on the channel diameter
• Particle density (particle-fluid density ratio)
• Number of particles in the bolus
• An accurate model predicting the behavior of
  intravascularly injectable particles can lead to
  a dramatic reduction of the bench-to-bed time
  for the development of innovative MNP-based
  therapeutic and imaging agents.

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Governing equations Navier-Stokes equations for
a 3D unsteady incompressible flow solved on a
Cartesian grid
Rigid body dynamic equations
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• Flow solver (Verzicco, Orlandi, J. Comput.
  Phys., 1996)
• staggered-grid
• second-order-accurate space discretization
• fractional-step method
• non-linear terms explicit Adam-Bashford scheme
• linear terms implicit Crank-Nicholson scheme
• immersed boundary with 1D reconstruction
• (Fadlun et al., J. Comput. Phys., 2000)

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Implicit coupled approach
PREDICTOR
Flow equations
F and T exerted by the fluid on the particle
Rigid-body dynamic equations
New particle configuration
CORRECTOR
F and T exerted by the fluid on the particle
Flow equations
Rigid-body dynamic equations
New particle configuration
YES
NO
NEW TIME LEVEL
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Predictor-corrector scheme
Predictor second-order-accurate Adam-Bashford
scheme
Corrector iterative second-order-accurate
implicit scheme with under-relaxation
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Sedimentation of a circular particle in a channel

• W/d 4
• Re 200
• 202x1002 cells
• ?s/ ?f 1.1
• Fr 6.366

Yu, Z., and Shao, X., 2007. A direct-forcing
fictitious domain method for particulate flows.
Journal of Computational Physics (227), pp.
292314.
56
Sedimentation of a circular particle in a channel

• W/d 4
• Re 0.1
• 202x302 cells
• ?s/ ?f 1.2
• Fr 1398

Yu, Z., and Shao, X., 2007. A direct-forcing
fictitious domain method for particulate flows.
Journal of Computational Physics (227), pp.
292314.
57
Sedimentation of a sphere in a channel settling
velocity

• W/d 7
• Re 1.5
• 150x150x192 cells
• ?s/ ?f 1.155
• Fr 101.9

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
58
Sedimentation of a sphere in a channel sphere
trajectory

• W/d 7
• Re 1.5
• 150x150x192 cells
• ?s/ ?f 1.155
• Fr 101.9

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
59
Sedimentation of a sphere in a channel settling
velocity

• W/d 7
• Re 11.6
• 150x150x192 cells
• ?s/ ?f 1.164
• Fr 17.77

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
60
Sedimentation of a sphere in a channel sphere
trajectory

• W/d 7
• Re 11.6
• 150x150x192 cells
• ?s/ ?f 1.164
• Fr 17.77

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
61
Sedimentation of a sphere in a channel settling
velocity

• W/d 7
• Re 31.9
• 150x150x192 cells
• ?s/ ?f 1.167
• Fr 8.98

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
62
Sedimentation of a sphere in a channel sphere
trajectory

• W/d 7
• Re 31.9
• 150x150x192 cells
• ?s/ ?f 1.167
• Fr 8.98

ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and
Van den Akker, H.E.A., 2002. Particle imaging
velocimetry experiments and lattice-Boltzmann
simulations on a single sphere settling under
gravity. Physics of fluid Vol.14 (11), pp.
40124025.
63
Sedimentation of a triangular particle in a
channel

• W/d 7
• Re 100
• 300x602 cells
• ?s/ ?f 1.5
• Fr 50

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Sedimentation of an elliptical particle in a
channel

• W/d 4
• Re 12.6
• 161x402 cells
• ?s/ ?f 1.1
• Fr 62.78
• ?x 45 a/b 2

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Sedimentation of an elliptical particle in a
channel
Xia, Z., W. Connington, K., Rapaka, S., Yue, P.,
Feng, J. and Chen, S., 2009. Flow patterns in
the sedimentation of an elliptical particle. J.
Fluid Mech. Vol.625, pp. 249-272.
66
Sedimentation of an elliptical particle in a
channel
Xia, Z., W. Connington, K., Rapaka, S., Yue, P.,
Feng, J. and Chen, S., 2009. Flow patterns in
the sedimentation of an elliptical particle. J.
Fluid Mech. Vol.625, pp. 249-272.
67
Sedimentation of an elliptical particle in a
channel
68
Transport dynamics of a triangular particle in a
plane Poiseuille flow

• W/d 7
• Re 50

• 161x402 cells
• ?s/ ?f 1.1

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CONCLUSIONS (2) Arbitrarily shaped particle
transport

• Fluid-structure interaction solver is effective
  in the simulation of the transport dynamics of
  particles in an incompressible flow
• Particles with arbitrary shape can be handled
• Transport of bolus of particles is feasible.

Work in progress

• Selection of the particle shape for optimal
  margination
• Interaction models particle-wall
  particle-particle

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Sedimentation of cylindrical and spherical
particles
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VALIDATION
Pressure gradient influence
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Conjugate heat transfer T106 LP turbine
33000 cells
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Conjugate heat transfer T106 LP turbine
Mach number contours