Eindhoven University of Technology MASTER Viscous mixing of particles with inertia van den Broek, B.C.

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Eindhoven University of Technology

MASTER

Viscous mixing of particles with inertia

van den Broek, B.C.

Award date:

2010

Link to publication

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Eindhoven University of Technology M echanical Engineering TU/e

Graduation report

Viscous mixing of particles with inertia.

Author: B.C. van den Broek, 0569185

Supervisors: prof.dr. H.J.H. Clercx dr.ir. M.F.M. Speetjens Report number: WET 2010.04.

Eindhoven, July 28, 2010

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Ben van den Broek page 1

Abstract

In many processes in nature and in industry viscous mixing of fluids in three- dimensional (3D) is an important element. While it is applied in many industrial processes still there is a poor understanding of the underlying mechanisms. Babiano³ has made a start for a two-dimensional (2D) flow, first fora steady flow and second fora time depending flow, Cartwright⁶has done the same but than for a 3D steady flow. In both neutrally buoyant particles, meaning that the particles have the same density as the fluid, are released and the Basset history force is neglected. In these cases the particles settle in a vortex.

In this research this behavior is validated and expended with the Basset force and for particles heavier than the fluid. For the case that the particles are heavier the particles are not able to settle and for the case that the Basset force is added the particle accelerations are damped with respect to the case where the Basset force is neglected.

The pressure gradient is the force that makes it able for particles to settle, this happens when the particles are neutrally buoyant and to be driven inwards, what happens when particles are lighter than the fluid.

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Nomenclature

a

=

Particle radius [m]

Dev

=

Deviation number [-]

g

=

Gravitational acceleration [~]

k

=

Density ratio [-]

m1

=

Mass of the fluid displaced by the particle [kg]

mp

=

Particle mass [kg]

Re = Reynolds number [-]

St

=

Stokes number [-]

t

=

Time [s]

W

=

Dimensionless velocity difference [s]

u

=

Fluid velocity

[r:i]

Up = Particle velocity [~]

x Particle location [m]

a

=

Angle between particle trajectory and streamline [m]

PJ

=

Fluid density [~]

PJ

=

Particle density [~]

µ

=

Dynamic viscosity [~]

v

=

Kinematic viscosity [~²^]

page 4

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Chapter 1 Introduction

In many processes in nature and in industry viscous mixing of fluids in three- dimensional (3D) is an important element. While it is applied in many industrial processes still there is a poor understanding of the underlying mechanisms. In this research the goal is to get a better understanding of these underlying mechanisms. Babiano³ has made a start for a two-dimensional (2D) flow, first for a steady flow and second for a time depending flow, Cartwright⁶has done the same but than fora 3D steady flow. In both neutrally buoyant particles, meaning that the particles have the same density as the fluid, are released and the Basset history force is neglected. In this research the effect of releasing denser particles in the flow will be examined and adding the Basset History force, chapter 5 (2D steady flow), chapter 6 (2D unsteady flow) and chapter 7 (3D steady flow). To be able to do these simulations the Maxey-Riley¹equation is modified and scaled, chapter 2. These scaled equation is implemented with a three step Taylor Galerkin prediction correction method. The Basset history force is the most challenging term, which is in chapter 3 explained how to do this. In chapter 4 an analysis is done to get a general idea of the effect of adding the different forces.

5

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Chapter 2 Model and analysis

2.1 Flow Model

To describe the motion, when assumed there is an one way coupling, of a small rigid spherical particle, with radius a and mass mp, moving in an incompressible fluid, with velocity up, the equation proposed by Maxey and Riley¹is:

1 ₂ ₂ Du

6Jraµ(u - Up+

6

^a ^V ^u)

+

m1 Dt

( ) 1

d(

¹ ² ² ⁾

+

^mp-mfg+

2

^m1dt ^u-up+10^a Vu

1

^t^{..4.. (}^{u - u}

⁺

^.!a²^\}²^u)

+

61ra2 µ ^dT ^P ⁶ ⁱ dT

o [nv(t - T)] 2 ^(2.1)

The undisturbed flow field is u(x, t), m1 is the mass of fluid displaced by the sphere, and dynamic viscosity and kinematic viscosity are µ and v, respectively.

The terms on the right-hand side of (2.1) then correspond in turn to the effects of viscous Stokes drag, the force exerted by the undisturbed flow on the particle, commonly called pressure gradient, buoyancy, added mass and augmented viscous drag from the Basset history force. The terms in a²

v

u are the Faxèn corrections. The pressure gradient term in (2.1) was written in this form on the assumption that the undisturbed flow is incompressible and satisfies

Du 1 ₂

- = -- VP+ v V u

Dt p (2.2)

6

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Ben van den Broek Flow Model page 7

~ _Up

u

Figure 2.1: A particle moving with velocity ^Upin the flow which is streaming with velocity u.

In (2.1) the added mass term should be

~mf (Du_ duP)

2 Dt dt (2.3)

as was pointed out by Auton et al.²and used by Babiano et al.³It is important to note the distinction between the two different time derivatives.

The derivative D /Dt is used to denote the time derivative following a fluid element, so that

D <5

- = -+u•v

Dt öt ^(2.4)

is the fluid acceleration as observed of the fluid velocity. The derivative d/dt is used to denote a time derivative following the moving sphere, so that

d <5

- = - + u ·'v

dt öt P (2.5)

is the time derivative at the instantaneous center of the sphere.

The particle velocity ^Upis equal to the time derivative of the place of the particle, in formula:

dx .

U p = - = X

dt (2.6)

When (2.6) and the correct form of the added mass term are substituted in (2.1) with the assumption that the particles are spherical results in:

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Ben van den Broek Dimensional Analysis page 8

(2.7)

2.2 Dimensional Analysis

The appropriate non-dimensional formulation of the equation of motion, (2.7), can be found through an order of magnitude assessment for each indi- vidual contribution, resulting in

x Lx*

u Uu*

t

g (2.8)

The typical time scale used is the advective time scale,

Ta=b-

Substituting (2.8) into (2. 7) yields

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Ben van den Broek Dimensional Analysis page 9

Dividing by PP11ra³

,K

on both sides (2.9) can be written as:

.. *

X

(2.10)

The non-dimensional parameters associated with the problem at hand then are

k

=

PP PJ ' ^S

_ 2a²kU

t - 9vL

2kRe a²U _ (1 - t)goL

- 9 -, Re= vL ' Ilg -

u2 '

^(2.11)

with Re and St the Reynolds and Stokes number, respectively, as dynamica!

parameters. The flow related parameters k and TI₉reflect the density ratio and the dimensionless gravitational acceleration of the particle due to a density difference between the fluid and the particle. When the assumption is made that a

«

^Lthe Faxèn corrections can be neglected and dropping the asterisks yields

1 ( . ) 1 Du TI 1 ( Du .. )

x

=

St u - x

+ k

Dt

+

gg

+

^2k Dt - x

1

+ (

9 ) ² tdu/dT-~(T)dT

2k1rSt }0 (t - 7)2 ^(2.12)

Rearranging (2.12) leads to:

(2.13)

Ai are dimensionless groups, reading

1

A - l A - ³ A - TI~ A -

(~)2

l - St

+

^{Re ,} ²^- 2k

+

1, ³^- St

+

^Re^, ⁴^- St

+

^Re

9 9 9

(2.14)

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Ben van den Broek Genera! dynamica! characteristics page 10

with TI~

=

~ and T₉

=

₂

!r~

₀ the settling time scale. When (2.13) is written in terms of the velocity difference, W, which is defined as

W = U-Up (2.15)

(2.13) becomes

dW (

Du du) 1t

^dW

- = -Ai

w -

A2- - - - A3g - A4 dt 1 dT (2.16)

dt Dt dt o ( t - T) ²

2.3 Relevant Time Scales

Performing a dimensional analysis to the above kinematic equation (2.6) advances the advective time scale Ta =

b

as characteristic time of the tracer dynamics. This suggest the existence of three time scales in the problem of interest, viz. the viscous, advective and the settling time scales Tv a;, Ta

= b

^and^Tg

⁼ 2:r~o,

respectively, relating through

Tv _ R _ 9St Ta

=

Il' Ta - e - 2k ' T

9 g· ^(2.17)

The advective time scale Ta forms the principal time scale for the tracer dynamics. The dimensionless time t/Ta namely acts as a univeral time parameter for the tracer advection and allows for a direct comparison of the temporal evolution of different flow configurations. The viscous time scale relates to the advective time scale through (2.17) via the non-dimensional parameter Re, which defines how steady the flow is. The settling time scale relates to the advective time scale through (2.17) via the non-dimensional parameter TI~, which defines how fast a particle sinks in the fluid.

2.4 General dynamical characteristics

A way to get an idea about the response time of a particle a homogenous solution will be derived. This is done by bringing x and his derivatives to the left hand side of (2.13) and everything else to the other side:

(2.18)

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Ben van den Broek Particle dynamics in uniform flow field page 11

To derive the homogenous solution of (2.18) the right hand side of (2.18) is set to zero. When this is clone the added mass term drops out of the equation, what has as result that A1 reduces to

it

^so(2.18) becomes:

(2.19) This solution reads

(2.20) where a and b are integration constants, which can be determined by apply- ing the initial conditions

xh(O)

=

^Xo xh(O) = Uo.

Thus the homogeneous solution of (2.18) becomes:

(2.21)

(2.22) What can be concluded from this result is that the dimensionless relaxation time of the particle in a Stokes flow is equal to the Stokes number, St.

2. 5 Particle dynamics in uniform flow field

To develop a theoretical testcase, an analytical solution from (2.13) will be derived. This equation is a nonlinear differential and integral equation, in which the coefficients of derivatives and integral contain unknown variables and their first order derivative. Thus it is very difficult for the equation to be resolved analytically. Therefore an analytical solution will be derived for an uniform time dependent velocity field. The particle is released with a initial velocity up(O) at t=O. The velocity field is given by its Fourier expansion

00

u(t) = L

^Um^e(imt). ^(2.23)

m=-oo

Because the velocity field is uniform (2.16) can be simplified to

(2.24)

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Ben van den Broek

where

Particle dynamics in uniform flow field

A' _ 2(k - 1)

2 - 2k

+

1

page 12

First the equation of motion will be simplified by neglecting all the forces on the particle with exception of the Stokes drag, so (2.24) becomes:

dW W(t) du

dt = - ~ + dt

(2.25)

When (2.25) is transformed to the Laplace domain and rearranged it can be written as:

- Wo sü- uo

W = - - - + - - -

s+St-1 s+St-1 ^(2.26) When this is transformed back to physical space, the analytica! solution becomes:

t

ltdu

^{t - r}

W(t) = W₀e-st

+

- e-s,: dr

0 dr When u is constant

d:J: =

0 and (2.27) reduces to:

(2.27)

(2.28) From this solution can be concluded that when only Stokes drag is applied to the particle, the particle has a dimensionless relaxation time, St. This is in line with the homogeneous solution.

With this knowledge an analytica! solution will be derived for a time dependent homogeneous flow field from the complete equation of motion.

When

d:J:

is a continuous function at the region of [O,t], the equation's solution exists and is unique. To derive the analytica! solution (2.24) is Laplace transformed to

w

(2.29)

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Ben van den Broek Particle dynamics in uniform flow field page 13

where

(2.30)

When A₁ and A₄ are substituted in (2.30) and k >

i

c+ and c_ are two conjugate complexes. Inversive Laplace transformation of (2.29), which is clone in Appendix A, gives

W(t) _{- Ai}^A3g_- ⁽

_F

Wo ₊_AiFA3g ) ( _{c_e}^c2₊t _erfc( _- _{c+vt -}r;) _{c+e -}^c²t _erfc( _- _{c_vt}

r;))

~ imA;(A1

+

im) imt + m ~ = (c! - im)(c:_ - im) e Urn

+

~

^imA;^u^m⁽ ₂^{C+. ët}^terfc(-c+

^v't) ^-

₂^{c_. ë~}^terfc(-c_

vt))

6 F c -im c - im

m=-oo + -

00 m²A' A₄..fir .

+

L

² ^eimtu ^erf(~) ^(2.31)

m=-= (c! - im)(c:_ - im)0m ^m with

(2.32) When a closer look is taken at (2.31) it is clear that the Basset force increases the dimensional relaxation time, because A1 and A₄ have opposite signs in the exponent. The Stokes drag is decreasing the velocity difference between the particle and the fluid while the Basset force is damping this effect.

In the limt---too (2.31) becomes

W oo

=

m=- =

(i0m(A1+im)+m..firA4)mA; . t - - - u eim

(c! - im)(c:_ - im)0m ^m

This solution consists a term which causes a constant velocity difference, the terminal velocity caused by the gravity and a periodic velocity difference caused by the other forces.

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Chapter 3 N umerical model

The scaled equation of motion from the previous chapter will be implemented in Matlab. To check the reliability of the code the numerical method will be compared with the analytical solution from the previous chapter with a time dependent homogeneous flow field.

3.1 Method

Formula (2.13) is a second order differential equation which can be written as two first order differential equations when (2.6) is substituted in (2.13).

This results in the following set of equations:

When:

dx dt

dUp b .

cit can e wntten as:

G

F

(3.1)

[uv,x]

dG _dt

₌

[ _F^*

l

,Up (3.2)

14

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Ben van den Broek Method page 15

F* Du

A1(u - up)+ A2 Dt

+

A3g

+

A

41tdu/dT - d~p/dT dT

0 (t-7)2

Fst

+

Fp

+

F9

+

FB

(3.3) The added mass term is part of the other terms like derived in previous chapter. (3.3) is in principle suitable for integration by any standard method, e.g. three-step Taylor-Galerkin prediction correction method,

Gt

+

F(Gt)~t

3

Gt

+

F(Gt+t..t/3)

2

~t

Gt

+

F( Gt+t.t;2)~t, (3.4) which is second order accurate. With initial condition G(O)

=

^G0 . The Baset history force FB presents additional challenges. First, the evaluation of the Basset force can become extremely time consuming and memory demanding.

This is due to the fact that every time step an integral must be evaluated over the complete history of the particle. In simulations it is impossible to take the complete history of the particle, so a limited time window has to be taken from the history. There are two possibilities to handle the history which is not inside this time window. One is to assume that the contribution outside this window is equal to zero. By Hinsberg⁴another way, which is the method used, is derived to handle the contribution outside this window. The contribution outside this window is approximated by multiple exponential functions This has as effect that the time window can be taken smaller, because a smaller error is made. The choice is made to use one exponential function, but can be expanded with more, to approximate the tail of the Basset force, what results in:

Ktail(t)

= _V

^~_~^exp

(--t-),

_2twin ^(3.5)

with ^twinthe length of the window in time units.

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Ben van den Broek Discretization of the kernel of the integral of the Basset History Force page 16

3.2 Discretization of the kernel of the inte- gral of the Basset History Force

The kernel of the integral of the Basset history force will be discretized:

with:

Discretization of the time domain in /}.t:

so (3.6) can be written as:

The following step is to rewrite the integral:

Substituting this in (3.9):

tifb.t1b.t f(t- T' - (i - l)f}.t) ' F

=

~ - - - d T

B

f=:_

⁰ (T'

+

(i - l)f}.t)l/2

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

Approximate f(T) by a piecewise linear function between t- t1 and t. So by a linear function between t - i/}.t and t - ( i - 1) /}.t,

(3.12) which is visualized in fig. 3.1.

When (3.12) is substituted, (3.11) becomes

(3.13)

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Ben van den Broek Error estimation numerical method page 17

t

Figure 3.1: ai as function of t

When the integral is rewritten and calculated, see [B], (3.13) can be written as

(3.14)

With an error, which is calculated in [C], equal to

f"(t )6.t² error= ;

2

01 +

⁰

(f

¹¹¹(t₃)6.t³

0I),

(3.15) where O ~ {t2, t3} ~ t1.

So the Basset force is also second order accurate.

3.3 Error estimation numerical method

Now the numerical method and the way to implement the Basset force are known, the order of the method can be tested. This can be clone by imple- menting a one-dimensional homogeneous velocity field

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Ben van den Broek Error estimation numerical method

u

V

w

cos(2t) 0 0.

page 18

(3.16)

and calculate the difference in particle velocity between the analytica! solution and the numerical method, the error. The particle velocity calculated with the numerical method is done for different step sizes, .6.t. The error will be averaged over a certain time domain and the maximum error will be calculated so these can be compared with the step size. This is done for different forces applied to the particle. In the first situation only the Stokes drag is applied and from fig. 3.2a it can be concluded that the error is second order accurate. In the following cases the Stokes drag is applied with an additional force. For all the cases, with exception to the Basset force, the error remains second order accurate, fig. 3.2b-d. For the case that the Basset force is added, the error seems to become order 1.5, fig. 3.2-e. The reason that the order of the error is smaller than two sterns from the fact that error is made at the initial step. At t

=

0 the second derivative of

f (

T) goes to infinity, what results that the order of the initial step is one order smaller than all the other following steps. So the error goes for infinity long simulations to order two and for very short simulations to one. This means that the error made with the Basset force added is larger than when the Basset force is left out.

Another conclusion that can be drawn from fig. 3.2-e is that the method becomes unstable when the step size becomes to large. This is caused by the fact that an explicit method is used, the Basset force contains a term ^up.

In the last case all the forces will be applied to the particle and the error is order 1.5 accurate, fig. 3.2-f and also becomes unstable for a to large time step.

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Ben van den Broek Error estimation numerical method page 19

10· 2 10·2

10·3 10"'

10·' ,o ◄

10·• 10·6

~ 10·• ~ 10~

10·' 10·¹

10·• 10"'

10·10

10◄ 10"' 10·2 10·1 10·10

10·' 10·:I 10·' 10·1

d1 d1

(a) (b)

10·2 10·2

- mean

10·1 10"'

10·' ,o◄

10·• 10"'

i ^10·• 1 ^,o~

10·1 ,o·'

,o·• 10"'

10·• 10"'

1o•t0

10·' ,.

..

10·2 10·'

10·10

10·' 10·3 10·2 ,o·1

d1 d1

(c) (d)

10' 10'

~ ' ^10'

10'

10' 10'

è i ¹⁰⁰

~

10' d1

(e) (f)

Figure 3.2: Error as function of the stepsize for: (a) Stokes drag; (b) Pressure gradient; (c) Gravitation; (d) Added mass; (e) Basset history force; and, (f) All forces.

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Chapter 4 The influence of the different forces

In the simulation there are two independent dimensionless variables, the first one is the proportion between the densities, k, the other the Reynolds number, Re. The Stokes number is equally dependent of the Reynolds number and the density ratio, because St

=

2k:e. The other dimensionless variable IT~ is zero, because the gravity is neglected in the following simulations. A third dimensionless variable, from now on called deviation number, Dev, has to be introduced, which says something about how the particle deviates from a streamline where it is released. When the 2D velocity field is:

u sin(y)

V cos(x) (4.1)

The streamlines are circular near the center of the vortex, so when a particle follows a streamline, near the center of the vortex, the distance from the center is constant. This distance is the reference radius, R. When a particle is released at the streamline with radius R from the center, with an initial velocity equal to the fluid velocity, at that location. The location of the particle dependents on the density ratio, the Reynolds number and the simulation time

r

=

r(k, Re, t) The deviation number is defined as:

Dev= -r R

20

( 4.2)

( 4.3)

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Ben van den Broek Stokes drag page 21

In the following simulations the deviation number is calculated when different forces are applied to the particle for varying density ratio and Reynolds number. The dimensionless simulation time is held constant, for example 4, so the deviation number is not influenced by the time dependency, so

Dev

=

Dev(k, Re) (4.4)

The location at the end of the simulation is used to calculate the deviation number as visualized in fig. 4.1. The initial velocity of the particle is equal to the velocity of the fluid.

···

... . ...

.. . .

.. ·· ·· ...

~ ~

. .

...

---....

^···

^...

. - .

.

. _{. . .}

·· . ..

r(k,Re,t=4

··· ... ···

Figure 4.1: Deviation of a particle, with density ratio k, from a streamline, with radius R, in a flow with a Reynolds number Re. After a simulation of 4 dimensionless time units the particle is located at a radius r from the center of the vortex.

4.1 Stokes drag

In this group of simulations the only force acting on the particle released is the Stokes drag, so (2.13) reduces to:

.. 1 ( . )

X

= -

^{U - X}

St ^{( 4.5)}

What have to be acknowledge is that the deviation number is equally dependent of the density ratio and the Reynolds number because St

= 2k:,e,

so

Dev

=

Dev(St). ( 4.6)

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Ben van den Broek Stokes drag, Basset force and added mass page 22

To make it possible to compare the results with the simulations clone later in this chapter the density ratio and the Reynolds number will be varied. First a neutrally buoyant particle is released, so k=l, with a Reynolds number equal to one. The particle is driven outwards as can be seen in fig. 4.2. When the

0.25 0.2 0.15 0.1 0.05

>- 0

-0.05 -0.1 -0.15

> - - - ~

-0.2 - -1<=1 Re=1 - -k=2Re=1 -0.25 ··· k=1 Re=2

- -passive tracer I streamline

1.4 1.5 1.6 1.7 1.8 1.9

X

Figure 4.2: Particle trajectory, for a particle which is only forced by the Stokes drag. Three settings are used. The first one, k=l and Re=l, throws the particle from the streamline. The second, k=2 and Re=l, and the third, k=l and Re=2, have the same effect on the particle, which throws the particle harder from the streamline than the first one.

density ratio and Reynolds number are varied, from zero to ten, the deviation number can be calculated for all these cases, which are visualized in fig. 4.3.

From fig. 4.3 can be concluded that the particle is driven outwards from the streamline under every condition and that the deviation number is equally dependent of k and Re, which is also visualized in fig. 4.2, because the particle trajectories for the cases k=l, Re=2 and k=2, Re=l are the same.

4.2 Stokes drag, Basset force and added mass

The farces acting on the particle released in these group of simulations are the Stokes drag, added mass and the Basset force, so (2.13) reduces to:

.. -A ( ') A 1tdu/dr-x(r)d

X - 1 U - X

+

⁴ 1 7

0 (t-7)2 (4.7)

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Ben van den Broek Stokes drag, Basset force and added mass page 23

3.5

2.5

15

0o' - - ~OL..5 - - ' - - -,L..5 - -2'--- 2.L..5 _ _._ _ _._ _ _, k(•J

Figure 4.3: Contour plot of the deviation number when the Stokes drag is applied on the particle. The density ratio and Reynolds number are varied so the deviation number is derived after four time units.

First a neutrally buoyant particle is released, which is again driven outwards, fig. 4.4, but less than when only the Stokes drag is applied, fig. 4.2.

0.25 0.2 0.15

0.1 .·········• ...... .

0.05 ·.

\ ,

>- 0

-0.05 -0.1

-0.15

-0.2 - -1<=1 Re=1 - -k=2Re=1 -0.25 · ··· 1<=1 Re=2

- -passive tracer I stream!ine

1.4 1.5 1.6 1.7 1.8 1.9

Figure 4.4: Particle trajectory, fora particle forced by the Stokes drag, added mass and Basset force. Three settings are used. The three settings used all drive the particle outwards the vortex flow and in order the particles are driven outwards harder, the first case k=l and Re=l, the second k=l and Re=2 and the third k=2 and Re=l.

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Ben van den Broek

0.6 0.4 0.2

Stokes drag, Basset force, added mass and pressure gradient

0.5 1.5 2 2.5

k[-]

3.5

page 24

28 2.6

2.4 2.2

1.8

1.6

14

1 2

Figure 4.5: Contour plot of the deviation number when the Stokes drag, added mass and the Basset force are applied on the particle. The density ratio and Reynolds number are varied so the deviation number is derived after four time units.

When the density ratio and Reynolds number are varied the deviation number can be calculated for all these cases, fig. 4.5, and another conclusion can be drawn, which is that the deviation number is stronger dependent of the density ratio than the Reynolds number. Fig. 4.4 verifies this behavior for the case that k=l and Re=2, which is less driven outwards than the case that k=2 and Re=l.

4.3 Stokes drag, Basset force, added mass and pressure gradient

All the forces are now acting on the particle released in the two dimensional flow described by (3.16) with exception to the gravitational force, so (2.13) reduces to:

.. _ A ( ') A Du A 1tdu/dr - x(r)d

X - I U - X

+

^{2 -}

+

⁴ 1 T

Dt _O (t - r)2 (4.8)

When a neutrally buoyant particle is released the particle remains on the streamline independently of the Reynolds number, fig. 4.6 and fig. 4. 7.

When the density ratio is larger than one the particle is driven outwards, but less than without the pressure gradient. This can be concluded when fig.

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Ben van den Broek Stokes drag, Basset force, added mass and pressure gradient page 25

4.4 and fig. 4.6 are compared. The greatest difference in particle behavior is that a particle with a density ratio smaller than one is driven inwards while without the pressure gradient particles are always driven outwards, fig. 4. 7.

In (2.24) this behavior is also visualized because the term

A;

changes of sign for values of k larger and smaller than one. In (2.16) this changing of sign

1 h h ^DU ^dU

on y appens w en Dt

=

Tt·

0.25 0.2 0.15 0.1 0.05

-0.05 -0.1

-O.l 5 - - 1<=1 Re=1 -0.2 - -k=2 Re=1

···•·· k=1 Re=2 -0.25 · k=0.5 Re=1

- - passiva tracer/ streamline

1.4 1.5 1.6 1.7 1.8 1.9

X

Figure 4.6: Particle trajectory, for a particle forced by the Stokes drag, added mass, Basset force and pressure gradient. When the density ratio is larger than one particles are driven outwards, when the density ratio is smaller than one the particles are driven inwards and when the density ratio is equal to one particles remain on a streamline independently of the Reynolds number.

When the density ratio and Reynolds number are varied, from zero to ten, the deviation number can be calculated for all these cases and from fig. 4. 7 can be concluded that the deviation number is less sensitive to variation of the density ratio and Reynolds number. The deviation number is now for values of the Reynolds number larger than one almost only dependent of the density ratio. In fig. 4. 7 three regions can be located. For the first region, k < 1, the part iele is driven inwards, the second region, k> 1, the part iele is driven outwards and the last region, k=l, the particle stays at the streamline.

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Ben van den Broek Stokes drag, Basset force, added mass and pressure gradient page 26

~ ,.

l

18 ^d...

l ; .

^C.

1.6 ;;;

~

14 ;;

"'

12 .:

~ 1

. I

¹

"' ^:.. \

0.8

f~ )

..

^,.,.

0.8 ,

"

^...

04 / ,,.

.,o; 0.8

0.2 ^/ ,., ^f.3

1-2 06

1.1

00 0.5 1.5 2 2.5 3.5

kH

Figure 4.7: Contour plot of the deviation number when the Stokes drag, added mass, the Basset force and the pressure gradient are applied on the particle. The density ratio and Reynolds number are varied so the deviation number is derived after four time units.

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Chapter 5 Particle dynamics in 2D steady vortex flow

In this chapter the 2D steady velocity field applied will be the same as in Babiano et al. ³

u - - cos(x)sin(y)

v sin(x)cos(y) (5.1)

The flow field is periodic, so the fluid exiting the left of the domain, is entering the domain from the right side. The same holds the other way around and for the fluid at the upper and lower side of the domain. So fig. 5.1 contains four vortices, the one in the center and in the corners are positive orientated the other two are negative orientated. In the first simulation the particle is a neutrally buoyant spherical particle and the Stokes number will be equal to 20 as in Babiano et al. ³to check if the part iele behavior is as mentioned in Babiano et al. ³The forces applied are the Stokes drag, added mass and the pressure gradient. When this is verified the Stokes number will be set to one, because the flow equation is only valid for St < l. and the density ratio will be varied to check how the particle responds. This is done for two cases first one without the Basset force and second one with the Basset force. In all the simulations the gravitational force is neglected so A3 is equal to zero in (2.13).

27

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Ben van den Broek Varying the Stokes number page 28

Figure 5.1: Velocity field 2D steady vortex flow

5.1 Applied forces: Stokes drag, added mass and pressure gradient

First the Basset force will be neglected, so A₄ is equal to zero in (2.13).

The gravitational force as above mentioned is also neglected and (2.13) than reduces to

X (5.2)

5.2 Varying the Stokes number

The equation that describes the motion of the particle is only valid for Stokes numbers smaller than 1. To show that the model from chapter 3 holds the simulation from Babiano et al. ³^,where Stokes is 20, are imitated. According to Babiano et al.³ St = 0.20, but with a velocity field of order 100 the Stokes number becomes 20. So first a simulation is performed for a neutrally buoyant spherical particle with a St

=

20, see fig. 5.2(a). When the density ratio is one, A2 is also equal to one and up

=

u becomes a solution of (5.2) and the particle is able to settle at a streamline. When Up

=

u the two time derivative are also equal,

t1{ = d;::

and the Stokes drag drops out of the equation. To get the particle jumping from one vortex to another the initial velocity of the particle has to be a bit different from the velocity of the flow at the initial location. The particle is able to jump in the x-direction if there is a velocity difference in the x-direction and the same holds for the y-direction. From Babiano et al.³it is not clear what the initial conditions

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Ben van den Broek Varying the Stokes number

(a)

page 29

flO 10 100 120 140 1fl0 ,., Z>O

' (b)

Figure 5.2: ( a) Particle trajectory for Stokes = 20 and (6) The velocity difference Up:z:-Ua:between the particle and the flow against time for case (a), which goes to zero.

are, so the results are not exactly the same, but what can be said is that the particle is able the jump from vortices and settles finally. With this result the Stokes number is set to one with the same initial conditions as the simulation with Stokes equal to 20 and what can be concluded from fig. 5.3 is that the particle settles much faster and the particle is not able to jump to another vortex. up= u is still a solution of (5.2), but only the relaxation time is smaller, because the Stokes number is smaller. This is the result that could be expected, because the pressure gradient dominates the Stokes drag for large values of the Stokes number as can be concluded from (2.13). The Stokes drag is the force that damps the velocity difference while the pressure gradient amplifies the velocity difference. When the particle is settled at a streamline close to the center of the vortex, where the pressure field is low the particle is not able to jump over to other vortices, because the pressure gradient is only high at the hyperbolic points as depicted in Babiano et al.³^.

5.2.1 Varying the density ratio

Again a spherical particle is released in the 2D steady vortex flow as described in (5.1), but now the density ratio is increased. When the density ratio is larger than one the particle is swung outwards a vortex into another vortex and so on, fig. 5.4 in the middle. To be able to say something about how fast a particle is swirl of a streamline the angle between the streamline and the particle trajectory, a, is calculated with

(5.3)

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Ben van den Broek Varying the density ratio when the Basset force is added page 30

.

0.02

,:f

²

CJ

_u,.~.^0.015^O.O<I!^0.01_'

_·~

_'

"à "°"Il 20

.. ..

^., ^,.,₁ ¹²⁰ ^,

..

^,

..

^,

..

²⁰⁰

(a) (b)

Figure 5.3: ( a) Particle trajectory for Stokes

=

1 and (b) The velocity difference

Upx-Uxbetween the particle and the flow against time for case (a), which goes to zero.

and set out against the time, (5.4) the lower one. This is clone for different values of k. For each value of k the time signal of the angle is analyzed, by dividing the time signal, the lower one of fig. 5.4, in small time steps and taking the amplitude of the angle at this point. When all these values are known, the amplitude is set out against the counts a amplitude is represented in the time signal, from now on called the probability density function (pdf) of the angle between the particle trajectory and a streamline. Fig. 5.5 shows this result for three values of k. The x-axis represents the amplitude of the angle between the particle trajectory and a streamline and the y-axis the number of times this amplitude is represented in the time signal. In fig. 5.5 the peak becomes more smooth out over multiple amplitude for an increasing density ratio, so the conclusion that can be drawn from this is that particles released with a lower density ratio follow better the flow and for an increasing density ratio the particle gets thrown of a streamline more times and this happens with a larger amplitude. The particle always jumps to another vortex near the hyperbolic regions. This is because there the pressure gradient dominates the Stokes drag. The particle never settles because Up = u is not a solution of (5.2) for k

=J

l.

5.3 Varying the density ratio when the Bas- set force is added

The same simulations are clone but now the Basset force is added to the forces acting on the particle. So (2.13) becomes

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Ben van den Broek Varying the density ratio when the Basset force is added

,

; : ~

o w ^~ ^~ • ^~ m ro ~ ~ ~

'

· ! f

^o

~

^I ⁱ

^;

^,

:is

^• ^~

7

^e

~

¹⁰

~ ~

^W ^~

^'0.

^~

^'

^~

~

^'

^:

^M ¹⁰

^' ~ ~

^~ ^~

⁾

^~

page 31

Figure 5.4: The velocity difference Up:z:-U:z:between the particle and the flow against time for k=l.1 and the particle trajectory and the angle between the streamline and the particle trajectory.

90 80 70 60

30 20 10

l -

-

k

k=

= UXl1

101

r

- k:1.1

lL -__, \_- ----'---_ - _

^___._:== __,__- - - ' - - - ---'-- _ _ _ J

00 0.05 0.1 0.15

Amplll,je

0.2 0.25 03

Figure 5.5: Probability density function (pfd) for a 2D steady vortex flow, when the applied farces are Stokes drag, added mass and pressure gradient, for 3 cases, k=l.001, k=l.01 and k=l.l. As the density ratio, k, increases the peaks become smaller and the amplitude increases.

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Ben van den Broek Varying the density ratio when the Basset force is added page 32

(5.4)

So a spherical particle is released in the 2D steady vortex flow as described in (5.1) with a density ratio larger than one. When fig. 5.6 is compared with fig. 5.4 it can be concluded that the particle gets a smaller acceleration with the Basset force than without the Basset force. With as result that a particle remains longer in a vortex when the Basset force is applied. When fig. 5.5 and fig. 5.7 are compared the same can be concluded because the the peak is higher and the maximum amplitude is lower when the Basset force is applied. When the Basset force is applied the particle still doesn 't settle because up

=

u is not a solution of (5.4) for k =/- 1.

.

; ~

o w ~ ~ ~ ~ ~ ro ~ oo 100

'

Figure 5.6: The velocity difference Upx-Uxbetween the particle and the flow against time for k= 1.1 and the particle trajectory and the angle between the streamline and the particle trajectory when the Basset force is applied.

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Ben van den Broek

.,

Varying the density ratio when the Basset force is added

300,---,---.---r----.---,---.----,---;r: ^_=^{= k c}^_ ^{= ==1}^.00^=1:::;i

- k:1.01 k= 1.1 250

200

~ 150 u

008 0.1 0 12 0.14 0 16 0 18 02 Amoltude

page 33

Figure 5.7: pdf for a 2D steady vortex flow, when the applied forces are Stokes drag, added mass, pressure gradient and Basset force, for 3 cases, k=l.001, k=l.01 and k=l.l. As the density ratio, k, increases the peaks become smaller and the amplitude increases.

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Chapter 6 Particle dynamics in 2D unsteady vortex flow

In this chapter the velocity field from previous chapter will be extended with a time dependency in the x-direction. This velocity field is the same as the time dependent velocity field in Babiano et al. ³

u

V

-cos(x

+

Asin(t))sin(y)

sin(x

+

Asin(t))cos(y) (6.1) The velocity field is time dependent so fig. 6.1 is a Poincaré section of the 2D unsteady velocity field, with the constant A equal to 0.3. To get this Poincaré section the location of the particle is plotted after every oscillation,. Again the velocity field is periodic so the domain is reduced to 0-21r. The same approach as in the previous chapter will be used, so first the results from Babiano et al.³will be reproduced with neutrally buoyant particles and the Stokes number equal to 50. After this the Stokes number will be set to one, so the equation of motion becomes valid and the results will be compared.

When this is done the density ratio will be increased without and with the Basset force and the results will be compared.

34

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Ben van den Broek Applied forces: Stokes drag, added mass and pressure gradient page 35

Figure 6.1: Poincaré section of a 2D unsteady chaotic fluid trajectory with A=0.3.

6.1 Applied forces: Stokes drag, added mass and pressure gradient

6.1.1 Varying the Stokes number

When the Stokes number is set to fifty and the particle is released in the chaotic region the particle settles in a vortex, fig. 6.2, as in Babiano et al.3. Settling is possible, because up

=

^u is a solution of (5.2). This is not the only possible result. The equation of motion is only valid for Stokes number smaller than one, so if the result is a physical one is not clear. The other possible result is that the particle settles in a periodic point, fig. 6.3. What actually happens is that the particle flows periodic against the flow in and when this is shown in a Poincaré section it looks like the particle is settled. So when St

»

1, the pressure gradient dominates the Stokes drag and the Stokes drag can be neglected and up

=

-u becomes a second solution of (5.2).

When the Stokes number is set to one, with the density ratio still equal to one, the particle doesn't settle in a vortex as is shown in fig. 6.4, but follows the fluid lines almost immediately and is not thrown of the fluid line anymore.

Another location to release the neutrally buoyant particle is in a vortex, but than the particle always stays inside the same vortex, independently of the Stokes number. This result is in line with the conclusions drawn earlier in the 2D steady case from chapter 5. The particle has to be near a hyperbolic point to be able to get thrown of a streamline, so when a particle is released near an elliptic point, inside a vortex, the particle is never thrown of the streamline.

While when it is released in the chaotic region, where the particle is able to get near hyperbolic points, the particle can jump to other streamlines.

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Ben van den Broek Applied forces: Stokes drag, added mass and pressure gradient page 36

Figure 6.2: Poincaré section of the the motion of a neutrally buoyant particle with Stokes number St = 50 in the flow.

' ³

Figure 6.3: Poincaré section of the the motion of a neutrally buoyant particle with Stokes number St = 50 settling in a point, the red dot.

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Ben van den Broek Varying the density ratio when the Basset force is added page 37

When this streamline is inside a vortex the Stokes drag damps the velocity difference and the part iele doesn 't gets thrown of anymore. This behavior appears only when the pressure gradient dominates the Stokes drag otherwise the particle remains at the chaotic streamline.

'. ~. ^'·,. . . .

·/)-/ · _ :/: : ; _ /_<· \ :_ ..

. ,< '. ^I· ^{.~ :-··:}^~^{. ..}^~

~··· : : ......

Figure 6.4: Poincaré section of the the motion of a neutrally buoyant particle with Stokes number St = 1 in the flow.

6.1.2 Varying the density ratio

Now the density ratio will be increased to analyse what the effect is on the particle trajectory behavior. When a particle is released in a vortex it is thrown out this vortex and remains in the chaotic region as is visualized in fig. 6.5. This is because up

=

^u is not a solution of (5.2) for k =I= l. To be able to say something about the behavior of the particle in the chaotic region a pdf is made for the angle between the particle trajectory and the streamline where the particle is located, fig 6.6. What can be concluded from this is that the particle gets thrown of the streamline harder and more frequent for larger values of the density ratio, because the peak becomes smoother for an increasing density ratio.

6.2 Varying the density ratio when the Bas- set force is added

The same simulation is clone but now the Basset force is added, which results in fig. 6. 7. When a same type of pdf is made when the Basset force is added,

Eindhoven University of Technology MASTER Viscous mixing of particles with inertia van den Broek, B.C.

Eindhoven University of Technology M echanical Engineering TU/e

Graduation report

Viscous mixing of particles with inertia.

Abstract

Contents

Nomenclature

=

=

=

=

=

=

=

=

=

=

[r:i]

=

=

=

=

=

Chapter 1 Introduction

Chapter 2

Model and analysis

2.1 Flow Model

6

+

d(

+

2

1

+

+

v

~ Up

u

~mf (Du_ duP)

- = -+u•v

2.2 Dimensional Analysis

Ta=b-

,K

=

u2 '

«

=

+ k

+

+

+ (

(~)2

+

+

+

+

=

=

!r~

Du du) 1t

w -

2.3 Relevant Time Scales

b

= b

= 2:r~o,

=

2.4 General dynamical characteristics

it

=

2. 5 Particle dynamics in uniform flow field

u(t) = L

+

dt = - ~ + dt

ltdu

+

d:J: =

d:J:

w

i

F

⁺

~ _Up

⁼ 2:r~o,

_F

^v't) ^-

₌

= _V

^...

. _{. . .}