Adapting the 2.29 FV Framework forSimple Non-Newtonian Fluid Flows

2.29 Numerical Fluid Mechanics

Spring 2018

Anoop Rajappan

May 16, 2018

Non-Newtonian Fluids

2

Many non-Newtonian fluids consist of dissolved polymer molecules – very long chains of high

molecular weight. These act as an “entropic spring” that gives rise to elasticity.

Molecules are initially

coiled …… but are stretched and

deformed by flow …

… and relax back to

the coiled state by

thermal motion.

t They can store energy via their

configuration.

Viscoelasticity Normal stresses under shear Shear thinning

“Open Siphon” effect

0.5% Polyethylene oxide

A few distinctively non-Newtonian phenomena:

“Rod climbing” effect

Credits: Ewoldt group, UIUCMcKinley group, MIT

Constitutive Equations for Polymeric Liquids

3

Newtonian fluid

S

S S

Purely viscous

A simple model for a

polymer solution

S P

PGP

S

Viscoelastic fluid

“Maxwell liquid”

P PG

P P

G

PP

d

dt

P

PG

Relaxation

time

S P S P P P

Upper Convected Maxwell (Oldroyd-B) Model

Differential Constitutive Equations

Tu u

T

P P P P Pu u ut

Shear rate tensor

Upper convected (contravariant) derivative:

Giesekus Model

Nonlinear term removes stress singularity due to unbounded extension of the polymer “spring”.

P P P P PP

1

02

Constitutive Equations for Polymeric Liquids

4

Newtonian fluid

S

S S

Purely viscous

A simple model for a

polymer solution

S P

PGP

S

Viscoelastic fluid

“Maxwell liquid”

P PG

P P

G

PP

d

dt

P

PG

Relaxation

time

Conformational Constitutive Equations

Utilizes a conformation tensor that tracks “how deformed polymer chains are” on average.

FENE-CR Model (Chilcott and Rallison, 1988)

Specifically formulated for

computations to remove

unphysical infinite stresses.

1P

P

A

2

tr1

PP

GA I

A

L

FENE-P Model (Peterlin, 1966)

Derived from a molecular

dumbbell model.

1

2

tr1P P

AG A I

L

1P

P

A

Numerical Implementation – Differential Models

5

The incremental projection method with rotational correction is used.

No changes needed for (Newtonian) solvent stress.

Additional polymer stress term added as source term in the predictor step.

2

11 11* 2 *3 4

2

n nn nn

n PS

n

x

u u u u puu

t x x x

The corrector steps are unchanged.

Grad2D_T.m

22xx P xx xy

xx xxx x u u

yu v

x y

u

x xt

22

yy Pyy xy

yy yyy y v v

xu v

x y

v

y yt

Three stress components in 2D evolved as unknowns at the pressure node locations.

xy Pyy

xy xy

xx

xy u v u v

y x yvu

y xxt

Initially zero polymer stress at all locations: true if material is fully

relaxed at time zero.

Although a single evolution equation for extra stress tensor can be

derived, the solvent and polymeric stresses were kept separate.

This “Elasto-Viscous Stress Splitting” (EVSS) helps in stability.

S P

S S

11 3

2

n nn n P PP P

t

t t

AB2 Time Marching

Advection using mexed scalar

advection routines

Grad2D_uv2.m

Red terms not directly

available on a staggered grid.

Numerical Implementation – Conformational Models

6

111 12

11 1 11 2xx

P

uA

A Au v

uA

tA

y xx y

222 12

22 2 22 2yy

P

vA

A Au v

vA

tA

y yx x

212 121

2 112 xy

P

A

t y

uA Au v

v

y xA A

x

Three conformation tensor components are evolved as unknowns at the pressure node locations.

Initially, conformation at all locations set to identity.

1

1 32

n n

ij ijn n

ij ij

A AtA A

t t

AB2 Time Marching

Advection using mexed scalar

advection routines

Grad2D_uv2.m

Red terms not directly

available on a staggered grid.

Stresses are computed from the conformation tensor, and inserted into the predictor step as

a source term.

1

1 11 111 22

1121 1

n nn nPxx

A AA

L

1

1 11 111 22

2221 1

n nn nPyy

A AA

L

11 1

1 111 22122

1n n

n nPxy

A AA

L

2

11 11* 2 *3 4

2

n nn nn

n PS

n

x

u u u u puu

t x x x

Grad2D_T.m

Convergence: LDC

7

22( ) 16 1u x x x

0u

0u

0u

1a

Re 27S P

Ua

8Wi 0.533

15

U

a

Reynolds number Weissenberg number

Time step:

160x yN N 25T 32 10t

Reference solution:

Viscoelastic parameters:

0.01S P 1 3 2 100L 1 1

Slope -1

Slope -2

Test Case 1: Sudden Expansion

8

a 4aU

l

ReS P

Ua

Wi

U

a

Reynolds number Weissenberg number

The Oldroyd-B solver was used with UW advection for stability.

Geometry parameters:

0.5a

125xN

10xL

25yN

50T 35 10t

Viscoelastic parameters:

0.01S P 1 3 2 100L 1

1

1U

The recirculation length decreases due to viscoelasticity, except for Giesekus model.

Test Case II: Flow Past Cylinder

9

Re 50S P

Ua

Wi 4

U

a

Reynolds number Weissenberg number Geometry parameters:

0.5a

200xN

20xL

30yN

100T 0.01t

2aU 6a

Viscoelastic parameters:

0.01S P 1 3 2 100L 1 1

The Oldroyd-B solver was

unstable even with UW.

Newtonian

Giesekus

Test Case II: Flow Past Cylinder

10

Re 50S P

Ua

Wi 4

U

a

Reynolds number Weissenberg number Geometry parameters:

0.5a

200xN

20xL

30yN

100T 0.01t

2aU 6a

Viscoelastic parameters:

0.01S P 1 3 2 100L 1 1

The Oldroyd-B solver was

unstable even with UW.

FENE-P

FENE-CR

Test Case III: Rising Buoyant Bubble

11

Viscoelastic

parameters:

0.01S P

1 3 2 100L

Geometry

parameters:

0.25a

60xN

1xL

300yN

400T

0.1t

5yL

Buoyancy and

Diffusivity:

0

0.01

1g

Newtonian Oldroyd-B Giesekus FENE-P FENE-CR

Test Case IV: Lock Exchange

12

Newtonian FENE-PFENE-CR

Geometry parameters:

1xL

200xN

1xL

200yN

8T 41 10t

Viscoelastic parameters:

41 10S P

10 2 100L

Density and Diffusivity:

0

1

1g

High Weissenberg Number Problem

13

Many FE, FV and FM methods fail at modest

Wi values – between 1 and 10.

Called the High Weissenberg Number

Problem or HWNP.

Due to the inability of simple polynomial

approximations to capture exponential

profiles in polymer stress.

Even a linear velocity profile can lead to

exponential stress profiles in viscoelastic fluids!

( )u x x

( ) (0) tl t l e

Stagnation points, sharp corners, discontinuous

BC’s are all potentially problematic.

Solution is to evolve the logarithm of the conformation tensor, rather than the tensor itself.

Log Conformation Formulation (Fattal and Kupferman, 2004).

Involves additional cost of diagonalizing the conformation tensor, and decomposing

the velocity gradient along the principal directions of the conformation tensor.

Summary

14

Modified the 2.29 FV Code to implement two differential

and two conformational viscoelastic constitutive equations:

Oldroyd B

Giesekus model

FENE-P

FENE-CR

Simulated simple test cases of steady and unsteady flows.

Can be easily extended to include polymer diffusion and

spatial variation in viscoelastic properties.

Overcoming the HWNP would probably require using a log

conformation approach.