# Amazing simulations 2:

Urban and Civil

Nov 16, 2013 (4 years and 5 months ago)

106 views

Lisa J. Fauci

Tulane University, Math Dept.

New Orleans, Louisiana, USA

Amazing simulations 2:

Capturing

the

fluid

dynamics

of

phytopl
ankton:

active

and

passive

structures.

Collaborators:

Hoa Nguyen
Tulane University

Lee Karp
-
Boss

University of Maine

Pete Jumars

University of Maine

Ricardo Ortiz
University of North Carolina

Ricardo Cortez

Tulane University

Diatoms, dinoflagellates

Plankton are the foundation of the oceanic food chain and are responsible
for much of the oxygen present in the Earth’s atmosphere.

Thalassiosira nordenskioeldii

Institute of Ontario

Thalassiosira punctigera

image by Ashley Young,
University of Maine

4

Pfiesteria piscicida

Delaware Biotechnology
Institute

Goal: Use CFD to model flows around or generated by phytoplankton.

How do spines alter the
rotational period of diatoms in
shear flow? .

Thalassiosira nordenskioeldii

Copyright of the Biodiversity Institute of Ontario

Thalassiosira punctigera

5

Discretization:

Spherical Centroidal Voronoi Tessellation

The triangulation on the unit sphere is the dual mesh of the Spherical Centroidal Voronoi Tessellation (SCVT), as coded
by Lili Ju [6]. We map this triangulation to a surface (such as an ellipsoid, a flat disc or a plankter’s cell body) to creat
e a

discretization of the structure.

6

Ellipsoid in Shear Flow

Variation of
φ

with time (where
φ

= rotation angle relative to the initial
position).

The period from the simulation is about
1.55 s
, compared with the
theoretical
period

T = 1.59 s
.

7

Diatom in Shear Flow

The cell body has

diameter
4.25x10
-
3

cm
and
height
1.77x10
-
3

cm
.

The spine length is

0.49x10
-
3

cm.

Thalassiosira punctigera

Our Model

(Re = 0.0181)

Shear Rate = 10.0 s
-
1

T = 1.323 s

8

Are full 3D CFD calculations necessary?

Motion of spined cells can be predicted
from simple theory by examining the
smallest spheroid that inscribes the cell.

Spines thus can achieve motion
associated with shape change that greatly
alters rotational frequency with
substantially less material than would be
needed to fill the inscribing spheroid.

Hydrodynamics of spines: a different spin.
Limnology & Oceanograpy :Fluids, Nguyen,
Karp
-
Boss, Jumars, Fauci 2011, Vol. 1
.

Grid

free numerical method for

zero Reynolds number

Method of regularized Stokeslets

(R. Cortez, SIAM SISC 2001;

Cortez, Fauci,Medovikov, Phys. Fluids, 2004)

Forces are spread over a small ball
--

in the case x
k
=0:

Grid

free numerical method for

zero Reynolds number

Method of regularized Stokeslets

(R. Cortez, SIAM SISC 2001;

Cortez, Fauci,Medovikov, Phys. Fluids, 2004)

Forces are spread over a small ball
--

in the case x
k
=0:

For the choice
:

the resulting velocity field is:

Note:

u(x) is defined everywhere

u(x) is an exact solution to the Stokes equations, and is incompressible

If regularized forces are exerted at “N” points, the velocities

at these points can be computed by superposition of

Regularized Stokeslets

u

= A
g

Here
A

is a 3n by 3n matrix that depends upon the

geometry
.

or

S. Goldstein

Univ. Minnesota

Cell body: right handed helix

Anterior helix: left handed

Posterior hook

Leptonema illini

Body length: 11.93 microns

How many rotations required

to swim one body length?

rigid body
and use computed ‘resistance matrices’

F =
T
U +
P
W

L =
P
T

U +
R

W

Where F is the total hydrodynamic force,

L is the total hydrodynamic torque,

T, P, R

are resistance matrices acting on

velocity U and angular velocity
W

Linear relationship:

D
)
(
ds
)
(
x
x
f
F

D
)
(
ds
)
(
x
x
f
x
L
We systematically assemble these resistance matrices

by applying velocity (or angular velocity) in

each direction, and integrating the resulting forces and torques…

0=
T
U +
P
W

Where F is the total hydrodynamic force,

L is the total hydrodynamic torque,

T, P, R

are resistance matrices acting on

velocity U and angular velocity
W

D
)
(
ds
)
(
x
x
f
F

D
)
(
ds
)
(
x
x
f
x
L
Cell body: right handed helix

Anterior helix: left handed

Posterior hook

What are dynamics of

a superhelix in a Stokes
fluid?

Experimental Setup

Counter
-
clockwise

Clockwise

Circles

Reg. Stokeslets

Squares

Resistive force theory

Triangles

Experiments

Transition from clockwise to counter
-
clockwise rotations is observed in
experiment and Reg. Stokeslet
calculations

but missed with
resistive force theory…

Jung, Mareck, Fauci

Shelley,
Phys. Fluids
,

2007

Motivation: Dinoflagellates

Pfiesteria piscicida

Delaware Biotechnology Institute

Imbrickle.blogspot.com

Dinoflagellates have 2 flagella

transverse and
longitudinal

Tom Fenchel,
How Dinoflagellates Swim
, Protist , Vol. 152:329
-
338, 2001

“The transversal flagellum causes the cell to rotate around its length axis. The

trailing flagellum is responsible for the translation of the cell; ” Fenchel 2001

“The transversal flagellum causes the cell to rotate around its length axis. The

trailing flagellum is responsible for the translation of the cell; ” Fenchel 2001

“The transverse flagellum works as a propelling device that provides the

main driving force or thrust to move the cell along the longitudinal axis of its

swimming path. The longitudinal flagellum works as a rudder, giving a

lateral force to the cell…”

Miyasaka, K. Nanba, K. Furuya, Y. Nimura, A. Azuma,
Functional roles of the
transverse and longitudinal flagella in the swimming motility of

Prorecentrum minimum (Dinophyceae
), J. Exp. Biol., 2004.

So, which is it?

Does the transverse, helically
-
beating

flagellum cause rotational or longitudinal

motion, or both?

So, which is it?

Does the transverse, helically
-
beating

flagellum cause rotational or longitudinal

motion, or both?

Classical fluid dynamics examined swimming of
helices with a straight axis…

Cortez, Cowen, Dillon, Fauci

Comp. Sci. Engr., 2004

X(s,t) = [r

R sin (2 π s / λ

ω t)] cos (s/r)

Y(s,t) = [r

R sin (2 π s / λ

ω t)] sin (s/r)

Z(s,t) = R cos (2 π s / λ

ω t)

Dual approach

Solve the kinematic problem using
Lighthill’s slender body theory and
regularized Stokeslets.

Solve the full Stokes equations coupled to a
ring that is actuated by elastic links whose
rest lengths change dynamically over the
wave period.

The action of waving cylindrical rings in a viscous fluid
.

J. Fluid Mech. 2010 Nguyen, Ortiz, Cortez, Fauci

Wave moving counterclockwise

viewed from above

Material points of

ring progress in opposite

direction .

Tangential and longitudinal velocity as a
function of amplitude R

Lighthill’s slender body theory
gives an excellent approximation
for the longitudinal velocity.

For small R, tangential velocity is O(R
2
).
.

For all R, longitudinal velocity is O(R
2
).
.

Change number of pitches on ring

Change number of pitches on ring

What if there was a cell body?

Interactions between a helical ring and spherical cell
using
IBAMR

top view

side view

Sphere:

VB =
2.71x10
-
3

Helical ring:

VB =
1.04x10
-
3

Colliding rings?

Conclusions

Undulating helical rings exhibit both

rotational and translational velocity in a

Stokes fluid.

These helical rings provide an interesting

kinematic problem to validate the method

of regularized Stokeslets used for

complex fluid
-
structure interaction problems.