Wall modeling challenges for the immersed boundary method G. Pascazio Workshop Num. Methods non-body...

Wall modeling challenges for the immersed boundary method

G. Pascazio

Workshop Num. Methods non-body fitted grids - Maratea, May 13-15 2010

POLITECNICO DI BARIDIMeG & CEMeC

Via Re David 200, 70125, Bari, ITALY

[email protected]

M. D. de Tullio, P. De Palma, M. Napolitano;

G. Iaccarino, R. Verzicco;

G. Adriani, P. Decuzzi …….

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

Geometry

Cartesian Grid

“ray tracing”

TAGGING

A special treatment is needed for the cells close to the immersed boundary

“fluid” cells

“solid” cells

interface cells

FORCING

Direct forcing (Mohd-Yusof)

- The governing equations are not modified- The boundary conditions are enforced directly- Sharp interface

Thus, a local reconstruction of the solution close to the immersed boundary is needed.

The boundary condition has to be imposed at the interface cells, which do not coincide with the body.

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

Procedure (de Tullio et al., JCP 2007)

Generation of a first uniform mesh (Input: Xi

min , Ximax , Xi )

Refinement of prescribed selected regions ( Input: Xi

min , Ximax , Xi )

Automatic refinement along the immersed surface ( Input: n, t ) Iterative

Automatic refinement along prescribed surface ( Input: Dn, Dt ) Iterative

Coarsening

Locally refined grids: automatic generation

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

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)

• 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)

NUMERICAL METHOD

• Reynolds Averaged Navier-Stokes equations (RANS)

• k-ω turbulence model (Wilcox, 1998):

• 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

• BiCGStab solver to solve the three sparse matrices:

NUMERICAL METHOD

• 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

• 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

pressure temperature

MESH-REFINEMENT STUDY

velocity, u component velocity, v component

Exp: (Wang et al., Phys. Fluids, 2000)

- 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)

Heated circular cylinder

Flusso supersonico su cilindro

Mach number contours

Pressure coefficient

- M=1.7; Re=2.e5- Domain: (-10,15) D – (-10, 10) D

Locally refined mesh: 75556 cells 545700 faces

=112° (exp.)

CD= 1.41 CD= 1.43 (exp.)

Supersonic flow past a cylinder

Ref.[1]: Blackburn, J.Fluid Mech, 1999

- Re=500, M=0.003

vorticity (F=0.875)

α(t)=y(t)/D

Cross-flow oscillating cylinder

Imposed cross-flow frequency

Natural shedding frequency (fixed cylinder)

WALL MODELING

U y

U 1

log y B

t y

viscous sublayer: logarithmic layer:

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

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

W

I

P1

• 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:

WALL FUNCTIONS

Turbulence model equations

F1

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)

WALL FUNCTIONS (TAB)

Rey y+ u+ k+ +

(Kalitzin et al. J. Comput. Phys. 2005)

W

F1

Compute friction velocity corresponding to IB surface (W), based on wall model

I Extract mean velocity and turbulence quantities in I

Compute velocity in F1

F1-W is equal to twice the largest distance from the wall of the interface cells

WALL FUNCTIONS (TAB)

uF1 , yF1 , F1 ReF1

ReF1 , [tables] y+F1

uy+F1 F1) / yF1

u, yI , I y+I

y+I , [tables] u+

i , k+i,

i

u , yI , u+i , k+

i, +i ui , ki, i

Molecular and turbulent viscosity variations:

WALL FUNCTIONS (ANALYTICAL)

viscous sublayer

(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.

where:

(variation of fluid properties in the viscous sublayer is neglected)

(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 :

WALL FUNCTIONS (ANALYTICAL)

Shear stress:

TWO-LAYER WALL MODELING

W

F1

I

A virtual refined mesh is embedded between the wall point W and F1 in the normal direction

P1

h

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

Point F1 is found, along the normal-to-the-wall direction, at twice the largest distance from the wall of the interface cells

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

• Velocity at F1 is interpolated using the surrounding cells

• Velocity at the interface point I is interpolated

y = normal direction

x = tangential direction

Momentum equation (Balaras et al. 1996; Wang and Moin, 2002) :

An iterative procedure has been implemented to solve the equations simultaneously

k = 0.4

A=16

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

The eddy viscosity is obtained from a simple mixing length model with near wall damping (Cabot and Moin, FTC 1999) :

WALL FUNCTIONS (NUMERICAL, NWF)

y = normal direction

x = tangential direction

Momentum equation:

WALL FUNCTIONS (THIN BOUNDARY LAYER, TBLE)

Turbulence model equations:

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

An iterative procedure has been implemented to solve the equations simultaneously

FLAT PLATE

• = 1.6 x 10-5 m2/s

• Uinf = 90 m/s

• ReL=1 = 6 x 106

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 = 0.58

x/L = 0.75

RECIRCULATING FLOW (A = 0.35 Uinf)

100max y

RECIRCULATING FLOW (A = 0.27 Uinf)

20max y

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)

Mach number contours Mis along the blade

M2,is=1.2M2,is=1.2

VKI-LS59 Turbine cascade

RAE-2822 AIRFOIL

Local view of the gridLocal view of the grid

57.2,102.6Re,754.0 6 M

Wall resolved reference solution: 700000 cells.

IB grid: 20000 cells; 100max y

Mach number contours (NWF)Pressure coefficient distribution

RAE-2822 AIRFOIL

Conjugate heat transfer: T106 LP turbine

Temperature contours

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

• 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

CONCLUSIONS (1)High Reynolds number turbulent flows

Work in progress and future developments

Arbitrarily shaped particle transport

in an incompressible flow

MOTIVATION and BACKGROUND

Ferrari, Nat Can Rev, 2005

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

ligands


Ferrari, Nat Can Rev, 2005

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

ligands


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


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).

Matas, Morris, Guazzelli, 2004

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:

0 u

fuu

2

p

Dt

D

Rigid body dynamic equations:

dSrpnndt

d

s

TTI2

2

dSpnndt

dM

s FF

x2

2

Flow solver (Verzicco, Orlandi, J. Comput. Phys., 1996)

• staggered-grid

• second-order-accurate space discretization

• fractional-step method: non-linear terms: explicit Adam-Bashford schemelinear terms: implicit Crank-Nicholson scheme

• immersed boundary with 1D reconstruction(Fadlun et al., J. Comput. Phys., 2000)

Implicit coupled approach

PPRREEDDIICCTTOORR

Flow equations F and T exerted by the fluid on the particle

Rigid-body dynamicequations

New particle configuration

Flow equations F and T exerted by the fluid on the particle

Rigid-body dynamicequations New particle configuration

CCOORRRREECCTTOORR

min1

1

ex

xxe

k

kk

YESNEW TIME

LEVEL

NO

Predictor-corrector scheme

Predictor: second-order-accurate Adam-Bashford scheme

Corrector: iterative second-order-accurate implicit scheme with under-relaxation

taavv nnnn

11

2

1

2

3

taaavv nnk

nk

nnk

111

11 1.09.0

2

1

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 circular particle in a channel

W/d = 7

Re = 1.5

150x150x192 cells

ρs/ ρf = 1.155

Fr = 101.9

Sedimentation of a sphere in a channel: settling velocity

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.


W/d = 7

Re = 1.5

150x150x192 cells

ρs/ ρf = 1.155

Fr = 101.9

Sedimentation of a sphere in a channel: sphere trajectory

W/d = 7

Re = 31.9

150x150x192 cells

ρs/ ρf = 1.167

Fr = 8.98


Sedimentation of a sphere in a channel: settling velocity


W/d = 7

Re = 31.9

150x150x192 cells

ρs/ ρf = 1.167

Fr = 8.98

Sedimentation of a sphere in a channel: sphere trajectory

W/d = 7

Re = 100

300x602 cells

ρs/ ρf = 1.5

Fr = 50

Sedimentation of a triangular 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

Transport dynamics of a triangular particle in a plane Poiseuille flow

W/d = 7

Re = 50

161x402 cells

ρs/ ρf = 1.1

• Selection of the particle shape for “optimal” margination• Interaction models: particle-wall & particle-particle

• 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.

CONCLUSIONS (2)Arbitrarily shaped particle transport

Work in progress

Sedimentation of cylindrical and spherical particles

Wall modeling challenges for the immersed boundary method G. Pascazio Workshop Num. Methods non-body...

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