Fluid-structure-interaction with zero-thickness bodies

Fluid-structure-interaction with zero-thickness bodies

M.D. de Tullio1, G. Pascazio1, R. Verzicco2,3

1 Exellence Centre of Computational Mechanics, Politecnico di Bari, Italy 2 Mechanical Engineering Department, University of Rome “Tor Vergata”, Italy

3 Physics of Fluids, University of Twente, Enschede, The Netherlands

EUROMECH Colloquium 549, June 2013, Leiden, The Netherlands

Many fluid/structure interaction problems involve shells or membranes whose thickness is negligible with respect the other dimensions (zero-thickness structures)

Motivation

Simulating their dynamics by “brute-force” resolution is unpractical

IB procedures need ad hoc treatment

In the IB context zero-thickness structures are also the bodies that do not have inner points

Fluid: Navier-Stokes equations for a 3D unsteady incompressible flow solved on a Cartesian grid:

!"u = 0 DuDt

= !"p!

#

$%

&

'(+

µ!"2u+ f

Structure: deformable shells/membranes

I d2!dt2

=T T = r! ! "n# pn( )$% &'s( dSM d 2x

dt2= F F = ! !n" pn( )

s! dS

The Problem

!"! s = !s#2ds#t2

! s =CEs Es =12!"ds + (!"ds )

T + (!"ds )T!"ds#$ %&

… or rigid body dynamics

A flow and a structure with coupled dynamics

PREDICTOR

Flow equations Fluid Loads on the structure

Structure equations New structure configuration

CORRECTOR

e = v pj ! v p

j!1 < eminYES NEW TIME

STEP NO

Fluid-structure-interaction Strongly coupled implicit approach (needed for added mass dominated problems)

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

Flow equations Structure equations

Fluid Loads on the structure New structure configuration

The iterations are beneficial to the “pressure inconsistency” of the direct forcing (as already mentioned in Fadlun et al. 2000)

!

!

Fluid Interface Solid

Old immersed boundary treatment Domain is discretized by (non-uniform) Cartesian grid

Immersed bodies are discretized by triangular elements, independently of the underlying fluid mesh

Fadlun et al. Journal of Computational Physics, 2000

“Old” direct-forcing method:

Each cell is tagged as “fluid”, “interface” or “solid”; The velocity field is reconstructed at the interface cells, using different interpolations

In case of moving bodies, one has oscillatory hydrodynamic forces that are potential source of instabilities.

Immersed boundary treatment “Lagrangian” direct-forcing method:

The forcing is computed on Lagrangian markers (laying on the immersed surface) so as to satisfy the boundary condition, and then is transferred to the Eulerian grid points

Vannella and Balaras, Journal of Computational Physics, 2009

ne Eulerian grid points associated to the Lagrangian marker (9 in 2D, 27 in 3D)

L a g r a n g i a n m a r k e r (centroid of the triangle)

Support domain (±1.6Dxi)

Immersed surface

Immersed boundary treatment •  Compute the intermediate velocity (non-solenoidal) in all Eulerian

points •  Compute the corresponding velocity at the Lagrangian points:

•  Calculate the volume force at all Lagrangian points in order to get the desired velocity on the boundary, Ub:

•  Transfer the forcing to the Eulerian grid points associated with Lagrangian markers, properly scaling by a factor cl such that the total forcing is not altered by the transfer:


Immersed boundary treatment •  Correct the intermediate velocity by means of the forcing:

•  This velocity field is not divergence-free and is projected into a divergence-free space by applying the pressure correction.

The transfer operator f containing the shape functions is obtained by a versatile Moving Least Square approximation, minimizing with respect to a(x) the weighted L2-norm defined as:

The weight function used is:

width and rw the size of the support domain.

•  The above step (slightly) changes the velocity at the boundary but few iterations (needed anyway for FSI) are enough to converge

Surface loads evaluation For each Lagrangian marker a probe is created along its normal direction, at a distance, hl , equal to the local averaged grid size:

Using the same MLS formulation, the pressure and velocity derivatives are evaluated at the probe location. The velocity derivatives on the body are assumed to be equal to that on the probe location (linear velocity profile).

pressure at probe

marker acceleration


Surface forces evaluation (zero-thickness) For each Lagrangian marker TWO probes are created along the normal direction, at a distance, hl :

U s i n g t h e s a m e M L S formulation, the pressure and v e l o c i t y d e r i v a t i v e s a r e evaluated at the probe location on both sides, obtaining the total force acting on the marker.

Oscillating circular cylinder in a cross-flow Ø  Re = 185

Ø  Domain: 50d x 40d

Ø  DxE=0.01d

Ø DxL=0.007d

Ø  y(t)=0.2d*sin(2pfet)

Ø  fe/f0=1, 1.2 (f0 natural shedding freq.)

pressure streamwise velocity

vorticity

Oscillating circular cylinder in a cross-flow

Pressure and skin-friction coefficients, case with fe/f0=1, cylinder at the extreme upper position o = body fitted results by Guilmineau and Queutey, Journal of Fluids and Structures, 2002

force coefficients

fe/f0=1 fe/f0=1.2

Oscillating circular cylinder in a cross-flow

Comparison between the drag coefficient calculated using the present method (black line) and that obtained using the forcing of Fadlun et al. Journal of Computational Physics, 2000

Oscillating loads (in space and time) If the body motion is imposed the oscillations can be cancelled out by local time averages (filtering)

If the body is rigid and massive its inertia filters out the force oscillations

If the motions is constrained (for example hinges, springs …) the effects are limited

BUT If the body is freely moving (objects free fall) the error accumulation is catastrophic

Deformable bodies react to local loads and oscillations quickly bring to divergence

Single sphere settling under gravity Ø  Closed container: 6.67d x6.67d x 10.67d

Ø  γ = ρs/ ρf

Ø  ΔxE = 0.028d

Ø ΔxL= 0.02d

Sedimenta)on velocity ten Cate et al. Physics of Fluids, 2002

Sphere trajectory

Sedimentation of an elliptical particle

Ø  Frt = Ut/(ga)0.5 = 0.126

Ø  Ret = Uta/n = 12.5

Ø  L/a = 4; a/b = 2 Ø  rs/ rf = 1.1

Ø  q0 = 45°

Ø  DxE=0.013a

Ø DxL=0.01a

Xia et al. Journal of Fluid Mechanics, 2009

Sedimentation velocity

Xia et al. Journal of Fluid Mechanics, 2009 Particle orientation

Particle trajectory (location of center of mass)

Sedimentation of an elliptical particle

L= length of the plate h= thickness g= gravity acceleration M= mass of the plate rf= fluid density rs= solid density

Field et al., Nature, 1997; Belmonte et al., Physical Review Letters, 1998

Reynolds number:

Froude Number:

Thickness ratio:

Freely falling plates

Steady fall (very low Re) Flutter (high Re, small Fr) Chaotic (high Re, intermediate Fr) Tumble (high Re, high Fr)

The plate will have different types of motion:

Fluttering of a single elliptical plate

Ø  Fr = 0.45 Ø  Re = 140

Ø  h/L=0.125 Ø  rs/ rf = 2.12

Ø  q0 = - 45°

Ø  DxE=0.01L

Ø DxL=0.007L

Horizontal force coef. Ver)cal force coef. Moment coef.

vor)city

Fluttering of a single elliptical plate

Tumbling of a single elliptical plate

Ø  Fr = 0.89 Ø  Re = 420

Ø  h/L=0.125 Ø  rs/ rf = 8.31

Ø  q0 = 75°

Ø  DxE=0.01L

Ø DxL=0.007L

Tumbling of a single elliptical plate

Horizontal force coef. Vertical force coef. Moment coef.

q0 = 75°

q0 = - 45°

Cases with different initial angle:

Bio-prosthetic heart valve

St Jude Medical – Trifecta™ Valve

•  valve frame made from a titanium stent covered by polyester tissue; •  the leaflets are made of pericardial tissue and are connected to the outside of the stent;

This mimics the hemodynamic performance of a healthy aortic heart valve.

The leaflets are modeled as zero-thickness, deformable bodies.

Opening phase (left=experiments)


Re=6800 72 bpm

Exp. from St. Jude web site

Closing phase (left=experiments)


Re=6800 72 bpm

(symmetry is imposed on the contact line)

Exp. from St. Jude web site


Non-dimensional velocity (left) and pressure (right) at different times during the cardiac cycle


Non-dimensional velocity during the cardiac cycle

Jellyfish

Herschlag and Miller, Journal of Theoretical Biology, 2011

The cross section of the jellyfish is modeled as an ellipse which is deleted below some lower bound.

Jellyfish

Non-dimensional position

Vorticity contours (oblate jellyfish)

Jellyfish

Non-dimensional total force

Non-dimensional velocity

Jellyfish

Vorticity contours (oblate jellyfish)

Jellyfish

Vorticity contours (prolate jellyfish)

Closing Remarks The moving least square IB procedure (Vannella & Balaras JCP 2009) gives reliable and smooth fluid dynamic loads that allow for the computation of fluid structure interactions even for free-evolving and/or zero-thickness bodies.

The old direct forcing method (Fadlun et al. 2000) worked only for “massive” and/or constrained bodies

On the other hand, the computation of the forcings (fluid and structure) on Lagrangian markers required a body description (triangulation …) as fine as the local mesh size.

This might become a bottleneck already at moderately high Re

More time consuming than than the old procedure

Wall modeling becomes mandatory for IB at high-Re flows

•  Fractional-step method:

- non-linear terms: explicit Adams-Bashforth scheme - linear terms: implicit Crank-Nicholson scheme

Numerical method

Verzicco and Orlandi, J. Comput. Phys., 1996

•  Second-order-accurate space discretization

•  Staggered grid

Fixed circular cylinder

Ø  Re = 20,40,100,200

Ø  Domain: 50d x 40d

Ø  ΔxE=0.01d

Ø ΔxL=0.007d

Fixed circular cylinder

Force coefficients, Re=200

viscous

pressure

Fluid-structure-interaction with zero-thickness bodies

Documents