CLC number: TU375

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Received: 2004-10-07

Revision Accepted: 2004-10-20

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PAN Tsorng-Whay, GLOWINSKI Roland, JOSEPH Daniel D.. Simulating the dynamics of fluid-cylinder interactions[J]. Journal of Zhejiang University Science A, 2005, 6(2): 97~109.

@article{title="Simulating the dynamics of fluid-cylinder interactions",

author="PAN Tsorng-Whay, GLOWINSKI Roland, JOSEPH Daniel D.",

journal="Journal of Zhejiang University Science A",

volume="6",

number="2",

pages="97~109",

year="2005",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.2005.A0097"

}

%0 Journal Article

%T Simulating the dynamics of fluid-cylinder interactions

%A PAN Tsorng-Whay

%A GLOWINSKI Roland

%A JOSEPH Daniel D.

%J Journal of Zhejiang University SCIENCE A

%V 6

%N 2

%P 97~109

%@ 1673-565X

%D 2005

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.2005.A0097

TY - JOUR

T1 - Simulating the dynamics of fluid-cylinder interactions

A1 - PAN Tsorng-Whay

A1 - GLOWINSKI Roland

A1 - JOSEPH Daniel D.

J0 - Journal of Zhejiang University Science A

VL - 6

IS - 2

SP - 97

EP - 109

%@ 1673-565X

Y1 - 2005

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.2005.A0097

**Abstract: **We present the simulation of the dynamics of fluid-cylinder interactions in a narrow three-dimensional channel filled with a Newtonian fluid, using a Lagrange multiplier based fictitious domain methodology combined with a finite element method and an operator splitting technique. As expected, a settling truncated cylinder turns its broadside perpendicular to the main stream direction and the center of mass moves to the central axis of the channel. In the case of two truncated cylinders, they first move around each other for a while and then stay together in a “T” shape. After the “T” shape has been formed for a long enough time, we found no vortex shedding behind the cylinders. When simulating the fluidization of 60 truncated cylinders, we captured the features of interactions among fluidized cylinders as observed in experiments.

**
**

. INTRODUCTION

. A MODEL PROBLEM AND A FICTITIOUS DOMAIN FORMULATION FOR 3-D PARTICULATE FLOW

with the resultant and torque of the hydrodynamical forces given respectively by,

and by the no-slip boundary conditions

To solve system Eqs.(

The principle of the fictitious domain method that we employ is rather simple. It consists in

(1) Filling the particles with a fluid having the same density and viscosity as the surrounding one.

(2) Compensating the above step by introducing, in some sense, an anti-particle of mass (−1)

(3) Finally, assuming the fluid contained in

via a Lagrange multiplier

We obtain then an equivalent formulation of Eqs.(

For a.e.

and

with the following functional spaces

In practice we track two orthogonal normalized vectors rigidly attached to the body

. TIME AND SPACE DISCRETIZATION

for

and set

Next, compute

and set

Then, compute

and set

Now predict the motion of the center of mass and the angular velocity of the particle via the solution of

and set

(

and set

(

Correct the motion of the center of mass and the angular velocity of the particle via the solution of

Then set

(

A typical choice of points for defining Eq.(

Using the above finite dimensional spaces and the backward Euler’s method for most of the subproblems in scheme (19)−(34), we obtain the following scheme after dropping some of the subscripts

For

Next, compute

and set

Now predict the motion of the center of mass and the angular velocity of the particle via the solution of

Then set

Correct the motion of the center of mass and the angular velocity of the particle via the solution of

Then set

(

Systems (44)−(48) and (51)−(55) are systems of ordinary differential equations thanks to operator splitting. For their solutions one can choose a time step smaller than Δ

1. Translate

for

The rigid body motion is enforced in

We need to have

. NUMERICAL EXPERIMENTS

The snapshots of the cylinder position and its orientation for {

For

Snapshots of the positions and orientations of two cylinders for {

To see the effect of the length and the density of the cylinder, we considered the two following tests. We only either increase the density of cylinders to 1.25 or reduce the lengths to 0.4 and keep all other parameters the same as above with {

The memory used in the simulation was about 71 MB (resp., 190 MB) for

* Project supported by NSF (Nos. ECS-9527123, CTS-9873236, DMS-9973318, CCR-9902035, DMS-0209066), and DOE/LASCI (No. R71700K-292-000-99), USA

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