POLITECNICO DI BARI
DIMeG & CEMeC
Via Re David 200, 70125, Bari, ITALY
Wall modeling challenges for the immersed
boundary method
G. Pascazio
pascazio@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
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
IMMERSED BOUNDARY TECHNIQUE
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
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.
Locally refined grids: automatic generation
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
RECONSTRUCTION
One-dimensional approach (Fadlun et al., JCP 2000)
At each interface cell, a linear interpolation is employed along the
Cartesian direction closest to the normal to the immersed surface
RECONSTRUCTION
Multi-dimensional linear recontruction (2D)
(e.g., Yang and Balaras, JCP 2006)
Dirichlet boundary condition:
Neumann bounary condition:
DISTANCE-WEIGHTED RECONSTRUCTION (LIN)
(de Tullio et al., JCP 2007)
Dirichlet boundary conditions:
Neumann boundary conditions:
COMPRESSIBLE SOLVER (RANS)
NUMERICAL METHOD
• Reynolds Averaged Navier-Stokes equations (RANS)
• k-ω 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 Γ 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
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
RESULTS
MESH-REFINEMENT STUDY
• NACA-0012
• M=0.8, Re=20, 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:
MESH-REFINEMENT STUDY
velocity,
pressure
u component
velocity,
temperature
v component
Heated circular cylinder
- M=0.03; Re=100,120,140; T*=1.0, 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 T*>1
- unsteady periodic flow
Temperature contours
(Re=100, T*=1.8)
Exp: (Wang et al., Phys. Fluids, 2000)
Supersonic
flow pastsu
a cylinder
Flusso supersonico
cilindro
- M=1.7; Re=2.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
Cross-flow oscillating cylinder
- Re=500, M=0.003
Imposed cross-flow frequency
Natural shedding frequency (fixed cylinder)
α(t)=y(t)/D
vorticity (F=0.875)
Ref.[1]: Blackburn, J.Fluid Mech, 1999
WALL MODELING
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
viscous sublayer:
U = y
logarithmic layer:
1
U  = logy   B

t =  y

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
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, gridconverged numerical solution and stored in a table along with its
inverse (look-up tables)
Rey
y+
u+
k+
w+
WALL FUNCTIONS (TAB)
Compute velocity in F1
Compute friction velocity
corresponding to IB surface (W),
based on wall model
uF1 , yF1 , F1  ReF1
F1
ReF1 , [tables]  y+F1
ut = (y+F1 F1) / yF1
Extract mean velocity and
turbulence quantities in I
I
W
ut , yI , I  y+I
y+I , [tables]  u+i , k+i, wi
ut , yI , u+i , k+i, w+i  ui , ki, wi
F1-W is equal to twice the largest distance from the wall of the interface cells
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:
where:
viscous sublayer
(variation of fluid properties in the viscous
sublayer is neglected)
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:
TWO-LAYER WALL MODELING
Point F1 is found, along the normal-to-thewall 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
F1
solved down to the fluid point P1
• Velocity at F1 is interpolated using
h
I
the surrounding cells
• Simplified turbulent boundary layer
W
equations are solved at the virtual grid
points
• Velocity at the interface point I is
interpolated
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
A=16
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).
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).
FLAT PLATE
•  = 1.6 x 10-5 m2/s
• Uinf = 90 m/s
• ReL=1 = 6 x 106
FLAT PLATE
FLAT PLATE
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
RECIRCULATING FLOW
x/L = 0.16
x/L = 0.75
x/L = 0.58
RECIRCULATING FLOW (A = 0.35 Uinf)

max
y
100
RECIRCULATING FLOW (A = 0.35 Uinf)

ymax
100
RECIRCULATING FLOW (A = 0.27 Uinf)

max
y
 20
VKI-LS59 Turbine cascade
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,is=1.2
M2,is=1.2
Mach number contours
Mis along the blade
RAE-2822 AIRFOIL
M  = 0.754, Re = 6.2 10 ,  = 2.57
6

Wall resolved reference solution: 700000 cells.
IB grid: 20000 cells;

ymax
100
Local view of the grid
Local view of the grid
RAE-2822 AIRFOIL
Pressure coefficient distribution
Mach number contours (NWF)
Conjugate heat transfer: T106 LP turbine
Temperature contours
CONCLUSIONS (1)
High Reynolds number turbulent flows
• Wall modelling appears to be an efficient tool for computation of highRe 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
Arbitrarily shaped particle transport
in an incompressible flow
MOTIVATION and BACKGROUND
ligands
PEG
Ferrari, Nat Can Rev, 2005
Use of micro/nano-particles for
drug delivery and imaging.
Properly designed micro/nanoparticles, 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
MOTIVATION and BACKGROUND
ligands
PEG
Ferrari, Nat Can Rev, 2005
Particles are transported by the blood
flow and interact specifically (ligandreceptor 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.
MOTIVATION and BACKGROUND
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
MOTIVATION and BACKGROUND
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).
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
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.
Governing equations
Navier-Stokes equations for a 3D unsteady incompressible
flow solved on a Cartesian grid:
 u = 0
 p  2
Du
=     u  f
Dt
 
Rigid body dynamic equations:
dq
I 2 =T
dt
2
d2x
M 2 =F
dt
T =  (t  n  pn r dS
s
F =  (t  n  pndS
s
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)
Implicit coupled approach
Flow equations
Rigid-body dynamic
equations
Flow equations
Rigid-body dynamic
equations
NO
F and T exerted by the fluid on the
particle
New particle configuration
F and T exerted by the fluid on the
particle
New particle configuration
x k  x k 1
e=
 emin
k 1
x
YES
NEW TIME
LEVEL
P
R
E
D
I
C
T
O
R
C
O
R
R
E
C
T
O
R
Predictor-corrector scheme
Predictor: second-order-accurate Adam-Bashford scheme
v
n 1
 3 n 1 n 1 
= v   a  a  Dt
2
2

n
Corrector: iterative second-order-accurate implicit scheme
with under-relaxation
v
n 1
k 1
(

1
= v  0.9 akn11  0.1 akn 1  a n Dt
2
n
Sedimentation of a circular particle in a channel
 W/d = 4
 Re = 200
 202x1002 cells
 ρs/ ρf = 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. 292–314.
Sedimentation of a circular particle in a channel
 W/d = 4
 Re = 0.1
 202x302 cells
 ρs/ ρf = 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. 292–314.
Sedimentation of a sphere in a channel:
settling velocity
 W/d = 7
 Re = 1.5
 150x150x192 cells
 ρs/ ρf = 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. 4012–4025.
Sedimentation of a sphere in a channel:
sphere trajectory
 W/d = 7
 Re = 1.5
 150x150x192 cells
 ρs/ ρf = 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. 4012–4025.
Sedimentation of a sphere in a channel:
settling velocity
 W/d = 7
 Re = 31.9
 150x150x192 cells
 ρs/ ρf = 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. 4012–4025.
Sedimentation of a sphere in a channel:
sphere trajectory
 W/d = 7
 Re = 31.9
 150x150x192 cells
 ρs/ ρf = 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. 4012–4025.
Sedimentation of a triangular particle in a channel
 W/d = 7
 Re = 100
 300x602 cells
 ρs/ ρf = 1.5
 Fr = 50
Sedimentation of an elliptical particle in a channel
 W/d = 4
 Re = 12.6
 161x402 cells
 ρs/ ρf = 1.1
 Fr = 62.78
 θx = 45°; a/b = 2
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.
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.
Transport dynamics of a triangular particle
in a plane Poiseuille flow
 W/d = 7
161x402 cells
 Re = 50
 ρs/ ρf = 1.1
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
Sedimentation of cylindrical and spherical particles