1. Wall modeling challenges for the immersed boundary method
POLITECNICO DI BARI
DIMeG & CEMeC
Via Re David 200, 70125, Bari, ITALY
Wall modeling challenges for the immersed boundary method
G. Pascazio
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

2. 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

3. IMMERSED BOUNDARY TECHNIQUE

4. TAGGING “ray tracing” Cartesian Grid Geometry “fluid” cells
“solid” cells
interface cells
A special treatment is needed for the cells close to the immersed boundary

5. FORCING Direct forcing (Mohd-Yusof)
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.

6. 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

7. One-dimensional approach (Fadlun et al., JCP 2000)
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

8. Multi-dimensional linear recontruction (2D)
RECONSTRUCTION
Multi-dimensional linear recontruction (2D)
(e.g., Yang and Balaras, JCP 2006)
Dirichlet boundary condition:
Neumann bounary condition:

9. DISTANCE-WEIGHTED RECONSTRUCTION (LIN)
(de Tullio et al., JCP 2007)
Dirichlet boundary conditions:
Neumann boundary conditions:

10. COMPRESSIBLE SOLVER (RANS)

11. 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

12. 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

13. FLUX EVALUATION
- For each face, the contributions of the neighbour cells are collected to build the corresponding operators (convective/diffusive) for the cell

14. RESULTS

15. 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:

16. MESH-REFINEMENT STUDY
pressure
temperature
velocity, u component
velocity, v component

17. 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: cells, 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)

18. Supersonic flow past a cylinder
Flusso supersonico su cilindro
M=1.7; Re=2.e5
Domain: (-10,15) D – (-10, 10) D
Locally refined mesh: cells
faces
q =113°
q =112° (exp.)
CD= 1.41
CD= 1.43 (exp.)
Mach number contours
Pressure coefficient

19. 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

20. WALL MODELING

21. 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:
logarithmic layer:

22. 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

23. 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+

24. 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]  y+F1
ut = (y+F1 nF1) / yF1 F1
Extract mean velocity and turbulence quantities in I I W
ut , yI , nI  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

25. 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)

26. 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:

27. 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

28. 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).

29. 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).

30. FLAT PLATE
n = 1.6 x m2/s
Uinf = 90 m/s
ReL=1 = 6 x 106

31. FLAT PLATE

32. FLAT PLATE

33. 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

34. RECIRCULATING FLOW
x/L = 0.16
x/L = 0.58
x/L = 0.75

35. RECIRCULATING FLOW (A = 0.35 Uinf)

36. RECIRCULATING FLOW (A = 0.35 Uinf)

37. RECIRCULATING FLOW (A = 0.35 Uinf)

38. RECIRCULATING FLOW (A = 0.27 Uinf)

39. 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: cells
wall functions (Tables)
M2,is=1.2
M2,is=1.2
Mis along the blade
Mach number contours

40. RAE-2822 AIRFOIL Local view of the grid Local view of the grid
Wall resolved reference solution: cells.
IB grid: cells;
Local view of the grid
Local view of the grid

41. RAE-2822 AIRFOIL Pressure coefficient distribution
Mach number contours (NWF)

42. Conjugate heat transfer: T106 LP turbine
Temperature contours

43. High Reynolds number turbulent flows
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

44. Arbitrarily shaped particle transport in an incompressible flow

45. MOTIVATION and BACKGROUND
ligands
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
PEG
Ferrari, Nat Can Rev, 2005

46. MOTIVATION and BACKGROUND
ligands
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.
PEG
Ferrari, Nat Can Rev, 2005

47. 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

48. 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).

49. 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

50. Particle density (particle-fluid density ratio)
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.

51. Governing equations
Navier-Stokes equations for a 3D unsteady incompressible flow solved on a Cartesian grid:
Rigid body dynamic equations:

52. 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)

53. 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

54. Predictor-corrector scheme
Predictor: second-order-accurate Adam-Bashford scheme
Corrector: iterative second-order-accurate implicit scheme
with under-relaxation

55. 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., “A direct-forcing fictitious domain method for particulate flows”. Journal of Computational Physics (227), pp. 292–314.

56. 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., “A direct-forcing fictitious domain method for particulate flows”. Journal of Computational Physics (227), pp. 292–314.

57. 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., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

58. 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., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

59. Sedimentation of a sphere in a channel:
settling velocity
W/d = 7
Re = 11.6
150x150x192 cells
ρs/ ρf = 1.164
Fr = 17.77
ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

60. Sedimentation of a sphere in a channel:
sphere trajectory
W/d = 7
Re = 11.6
150x150x192 cells
ρs/ ρf = 1.164
Fr = 17.77
ten Cate, A., Nieuwstad, C.H., Derksen, J.J., and Van den Akker, H.E.A., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

61. 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., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

62. 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., “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity”. Physics of fluid Vol.14 (11), pp. 4012–4025.

63. Sedimentation of a triangular particle in a channel
W/d = 7
Re = 100
300x602 cells
ρs/ ρf = 1.5
Fr = 50

64. 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

65. Sedimentation of an elliptical particle in a channel
Xia, Z., W. Connington, K., Rapaka, S., Yue, P., Feng, J. and Chen, S., “Flow patterns in the sedimentation of an elliptical particle”. J. Fluid Mech. Vol.625, pp

66. Sedimentation of an elliptical particle in a channel
Xia, Z., W. Connington, K., Rapaka, S., Yue, P., Feng, J. and Chen, S., “Flow patterns in the sedimentation of an elliptical particle”. J. Fluid Mech. Vol.625, pp

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
ρs/ ρf = 1.1

69. Arbitrarily shaped particle transport
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

70. Sedimentation of cylindrical and spherical particles

71. VALIDATION
Pressure gradient influence:

72. Conjugate heat transfer: T106 LP turbine
33000 cells

73. Conjugate heat transfer: T106 LP turbine
Mach number contours