Conference of Global Chinese Scholars on Hydrodynamics

A FICTITIOUS DOMAIN METHOD FOR PARTICULATE FLOWS*

YU Zhaosheng Department of Mechanics, Zhejiang University, Hangzhou 310027, China, E-mail: yuzhaosheng@zju.edu.cn SHAO Xueming Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China WACHS Anthony Fluid Mechanics Department, Institut Francais du Pétrole, 1 & 4, avenue de Bois Préau, 92852 Rueil-Malmaison Cedex, France ABSTRACT: The distributed Lagrange multiplier based fictitious domain (DLM/FD) method was proposed by Glowinski and his coworkers for the simulation of particulate flows. We have recently extended the DLM/FD method to deal with the particle motion in a Bingham fluid and the particulate flow with heat transfer. The progresses are reported in this paper. KEY WORDS: fictitious domain method, Lagrange multiplier, particulate flows, Bingham, heat transfer. 1. Introduction

Particulate flows are widespread in nature and industrial applications. With the rapid development of computer power, the direct numerical simulation (DNS), based on the Navier-Stokes equations or the discrete lattice-Boltzmann equation for the solution of the fluid-flow problem, has become a practical and important tool to probe the mechanics in particulate flows. Over the past decade a variety of DNS methods have been proposed. They can be classified into two families: boundary-fitted methods[1] and non- boundary-fitted methods[2,3], according to whether or not the boundary-fitted mesh is used for the solution of the flow field. The DLM/FD method is one of the non-boundary-fitted methods and was developed by Glowinski et al.[3]. The key idea in this method is that the interior domains of the particles are filled with the same fluids as the surroundings and the Lagrange multiplier (physically a pseudo body force) is introduced to enforce the interior (fictitious) fluids to satisfy the constraint of rigid body motion. The method has been successfully applied to the simulation of particulate flows[4-6]. Here, we report our recent works regarding the extension of the DLM/FD method to deal with the particle motion in a Bingham

fluid and the particulate flow with heat transfer. 1.1 Backgrounds on particle motion in Bingham

fluid Various Non-Newtonian materials, such as paint,

toothpaste, blood, fresh concrete, magnetite dense medium in mineral industry and drilling fluids in petroleum industry, exhibit a yield stress. Such materials behave like plastic solids when the imposed stress is smaller than the yield stress, and can flow like fluids when yielded. Many industrial processes (such as the petroleum and mineral industries[7,8]) involve the sedimentation of particles in these viscoplastic materials. Because of practical importance, the drag coefficient for a sphere settling in a viscoplastic fluid has been investigated theoretically[9], numerically [10-15] and experimentally[16-18]. All numerical simulations for this problem, however, are static in the sense that the particle is fixed, and furthermore the codes are essentially two-dimensional. To fully understand the sedimentation of many particles, we need to resolve the particle configuration as a result of hydrodynamic interactions between the particles, and therefore, a three-dimensional dynamic simulation is required, in which the particles move freely under the hydro- dynamic forces. This is the aim of our first work.

1.2 Backgrounds on particulate flows with heat

transfer This work was motivated by the fact that in many

industrial applications the motion of the particles and the heat transfer are strongly coupled. For example, in the fluidized-bed reactor, highly active catalysts are used because they can make the processes for the

* Project supported by the National Natural Science Foundation of China (Grant No: 10472104). Biography: YU Zhao-sheng (1974-), Male, Doctor

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polymerization of olefins using “low pressure processes” very attractive, however, there is a danger that the catalysts might melt if the heat produced from the chemical reaction cannot be rapidly removed from the catalysts[19]. The interactions between catalyst particles were observed to affect the heat removal significantly[19], therefore, full resolution of the motion of catalysts and the heat transfer is definitely desirable. 2. Governing Equations and Numerical Methods 2.1 Bingham case

The dimensionless governing equations in weak form for Bingham fluid are composed of the following three parts:

Combined momentum equations:

2

( )

1( )Re Re,

b

P

dt

Bnp d

d

Ω

Ω

∂+ ⋅∇ ⋅ =

∂

−∇ + ∇ + ∇⋅ ⋅

+ ⋅

∫

∫∫

u u u v x

u λ v x

λ v x

(1)

* *( 1) ( )

( ) ,

r P

P

d dV Fr Jdt dt

d

ρ ⎡ ⎤− − ⋅ +⎢ ⎥⎣ ⎦

= − ⋅ + ×∫

U ωV ⋅ξ

λ V ξ r x

(2)

[ ( )] 0P

d− + × ⋅ =∫ u U ω r ς x . (3)

Continuity equation:

0.q dΩ

∇ ⋅ =∫ u x (4)

Constitutive equation:

( )Reb bBnP r rΛ= + ∀ >λ λ D 0 (5)

1( )

/ 1

ifP

ifΛ

⎧⎪= ⎨>⎪⎩

q qq

q q q

≤ (6)

In the above equations, , , , U ,ω are the fluid velocity, fluid pressure, distributed Lagrange multiplier, particle translational velocity and particle rotational velocity, respectively, and , q ,

u p λ

v ξ , and are their corresponding variations. denotes the fluid rate-of-strain tensor.

V ςD

rρ is the particle-fluid density ratio. denotes the Reynolds number defined by

ReRe /f c cU Lρ μ= , here fρ ,U , and c cL μ

being the fluid density, the characteristic velocity, the characteristic length and the fluid viscosity, respectively. Fr represents the Froude number defined by , being the gravitational

acceleration. is the Bingham number defined by

2/c cFr gL U= gBn

/( )y c cBn L Uτ μ= , yτ being the yield stress.

and

*pV

*J are the dimensionless particle volume (area in case of two-dimension) and moment of inertia, defined by and , here

* /( )dp sV M Lρ= c

* 2/( )ds cJ J Lρ +=

sρ , M , J and d are the dimensional particle density, mass and moment of inertia, and the dimensionality of the problem (d=2 for two dimensions and d=3 for three-dimensions) respectively. is the position vector with respect to the particle mass center.

r

We adopt the constitutive equation proposed by Dean and Glowinski[20] for modeling the Bingham material, which is equivalent to the classic constitutive equation but is easier to implement numerically. The fractional step time scheme is used to decouple the original system (1)-(6) into three sub-systems: Navier-Stokes problem:

*2 * 2

1

1 ( )2 Re

1 (3 )21

nn

n n nb

nd P

pt

G G

dh

−

−= −∇ + ∇ +∇

Δ

+ − +∇ ⋅

+ ⋅∫

u uu u

λ

λ v x

(7)

* 0∇⋅ =u (8) Plasticity problem:

# *1( n n

b bt+−

= ∇ ⋅ −Δ

u uλ λ ) (9)

1 1 #(Re

n nb b

BnP r+ +Λ= +λ λ D ) (10)

Rigid-body motion problem: 1 #

11 ( )n

n nd P

dt h

++−

= − ⋅Δ ∫

u uλ λ v x (11)

1 1* *

1

( 1) ( )

( ) ,

n n n n

r P

n

P

V Fr Jt t

d

ρ+ +

+

⎡ ⎤− −− − ⋅ + ⋅⎢ ⎥Δ Δ⎣ ⎦

= − ⋅ + ×∫

U U ω ωV ξ

λ V ξ r x

(12)

1 1 1[ ( )]n n n

Pd+ + + 0.− + × ⋅ =∫ u U ω r ς x (13)

The Navier-Stokes problem is solved with the projection method on a half-staggered grid and the resulted Poisson equation for the pressure is solved with a FFT-based fast method[21]. The plasticity problem is solved in an iterative manner. The rigid-body motion problem is solved with the Uzawa iterative method[3].

2.2 Heat transfer case The combined momentum equations for the flow based on Boussinesq approximation are

2

2

1( )Re

,Re fP

p dt

Grd dg

Ω

Ω

∂+ ⋅∇ +∇ − ∇ ⋅

∂

= ⋅ − Θ ⋅

∫

∫ ∫

u u u u v x

gλ v x v x (14)

483

* *

2

( 1) ( )

( )

( 1) ( )Re

r P

P

r r fP

d dV Fr Jdt g dt

d

Gr dg

ρ

ρ β

⎡ ⎤− − ⋅ +⎢ ⎥

⎣ ⎦

= − ⋅ + ×

− − Θ ⋅ + ×

∫

∫

U g ωV

,

⋅ξ

λ V ξ r x

g V r ξ x

(15)

[ ( )] 0P

d− + × ⋅ =∫ u U ω r ς x . (16)

The governing equations for the temperature are 1[ ( ) ]f

f f f T fP

dQ d d

dt Peλ

Ω

Θ−∇⋅ ∇Θ − ⋅ϒ = ϒ∫ x x ,∫ (17)

'[( 1) ( )]

( 1) ]

sr p r s f sP

r s s T sP

dc Q Q

dtk d

ρ

λΩ

Θ− − − ϒ

+ − ∇Θ ⋅∇ϒ =− ϒ

∫∫ ∫

x

x x ,

d

d

0.

(18)

[ ]f s fPdΘ −Θ ϒ =∫ x (19)

In the equations (14)-(19), fΘ and sΘ are the dimensionless fluid and solid temperatures, fQ and

sQ are the dimensionless fluid and solid heart sources, is the Grashof number, is the Peclet number,

and Gr Pe

rβ , and are the ratios of the heat expansion, heat capacity and heat conduction coefficient, respectively. The reader is referred to [21] for the numerical schemes.

'p rc rk

3. Results and Analyses 3.1 Bingham case Only spherical particles are considered in this problem. We take the particle diameter as the characteristic length. Depending on whether the inertial effect is strong or not, it is better to adopt different characteristic velocities.

D

For the case of small inertial effect, the Stokes velocity for a single sphere settling in an unbounded domain sU is taken as the characteristic velocity, i.e.

32

4 ( ) 2 ( )3 ,6 9

s fs f

c s

a g aU U

a

π ρ ρ π ρ ρπ μ μ

− −= = =

g (20)

where is the sphere radius. The drag coefficient asC is define by

3

*0

4 ( ) 13 ,6

s fc

sT T

a g UCa U U U

π ρ ρ

π η

−= =

T

= (21)

where represents the dimensionless terminal settling velocity of the particle.

*TU

When the effect of inertia is strong, it is better to define the characteristic velocity by

3

2

4 ( ) 8 ( 1)33

2

s fr

c I

f

a g aU Ua

π ρ ρ ρπ ρ

− −= = =

g , (22)

so that the standard drag coefficient can be evaluated

by 3

2 * 22

4 ( ) 13( )

2

s f

dT

f T

a gC

a UU

π ρ ρ

π ρ

−= =

T

, (23)

here we may refer to as an ‘inertial’ velocity. The

drag coefficient is unity for a sphere moving at

the velocity of .

IU

dC

IUNote that Re and Bn in the governing equations

(1)-(6) are based on the chosen characteristic velocity rather than the terminal settling velocity. To avoid misunderstanding, we write the ones based on the Stokes velocity (20) as ReS and BnS, and the ones based on the ‘inertial’ velocity (22) as ReI and BnI. The Reynolds and Bingham numbers based on the terminal velocity UT can be calculated from

* *Re Re ; /T T TU Bn Bn U= = , (24) irrespective of the definition of characteristic velocity. Clearly, the drag coefficient for a steady-settling particle depends on both ReT and BnT, however, it was observed experimentally that the drag coefficient can be well correlated with a modified (or effective) Reynolds number Rem, which is based on an effective viscosity eη involving the effect of yield stress and is defined by[8]

* 2

*

Re ( )Re/

f T f T I Tm

e y T T

U D U D UD U U BnI

ρ ρη μ τ

= = =+ +

. (25)

We first consider the sedimentation of a sphere settling in a vertical tube of radius 4a along the tube axis at a low Reynolds number. Blackery and Mitsoulis[11] computed the drag coefficient using the accurate static boundary-fitted method. We compare our results to theirs in Fig. 1. The two results agree well with each other, indicating that our non-boundary-fitted method is reasonably accurate.

BnT

.1 1 10 100

Cs

1

10

100

Blackery & Mitsoulis (1997)

Present result

Fig. 1 Comparison between the results of Blackery & Mitsoulis

and ours (with h = a/4) for the drag coefficients for a sphere settling in a vertical tube of radius 4a. Here a is the sphere radius. The solid line in (b) is the best fit given by Blackery & Mitsoulis.

We now consider the sedimentation of a sphere

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settling in a vertical tube of radius 4a along the tube axis at moderate Reynolds numbers. Fig. 2a and b shows the drag coefficients at RedC I = 100, 200 and 400 vs. the Bingham number BnI and the effective Reynolds number Rem, respectively. Fig. 2b confirms for the first time from the numerical aspect (to our knowledge) that the drag coefficient can be well correlated with Rem.

BnI

0 2 4 6 8 10 12 14

CD

1

10

ReI=100ReI=200ReI=400

(a)

Rem

.1 1 10 100

CD

1

10

ReI=100

ReI=200

ReI=400

(b)

Fig. 2 Drag coefficients for a sphere settling in a tube of 4a at different Reynolds numbers Re

dCI vs. (a) Bingham

number BnI and (b) effective Reynolds number Rem.

The steady-state velocity fields together with the yielded/unyielded zones at BnI = 4 and BnI = 11 for ReI = 400 are plotted in Fig. 3a and b respectively. For BnI = 4, the effective Reynolds number Rem is 63.5. Thus the inertial effect is strong and the wake structure is obvious, as shown in Fig. 3a. For BnI = 11,Rem decreases to 2.64, the flow and yield surface are reminiscent of those at low Reynolds numbers[11].

3.2 Heat transfer case We consider the motion a circular catalyst particle in a vertical channel of width 8a. It is assumed that the heat is generated inside the particle homogeneously at and there is no heat

source inside the fluid domain (i.e., ). Fig. 4

shows the isotherms and flow fields at different times for

1sQ =0fQ =

'(Re, , , , , , )r r p r rGr Pe k cρ β =(40, 1000, 28, 1.1, 5.0, 1.0, 0.0) . At early stage, the effect of the temperature on the flow is small and the particle falls downwards. With more and more heats being generated inside the particle, a pair of buoyance- induced circulations have appeared by t = 5 (Fig. 4a) and dominated the flow by t = 15 (Fig. 4b), leading the particle to rise up.

r

z

-2 -1 0 1 21

2

3

4

5

6

7

8

9

10

(a) r

z

-2 -1 0 1 21

2

3

4

5

6

7

8

9

10

(b) Fig. 3 Steady-state velocity fields together with the yielded/

unyielded zones for a sphere settling in a tube of radius 4a at ReI = 400, and (a) BnI = 4 and (b) BnI = 11

(a) t=5 (b) t=15

Fig. 4 Isotherms and velocity vectors for a catalyst particle in an enclosed box at different times.

'(Re, , , , , , )r r p r rGr Pe k cρ β =(40, 1000, 28, 1.1, 5.0,

1.0, 0.0)

485

The power of our DLM/FD method lies in the simulation of moderately to highly concentrated suspensions of many particles. As an example, we simulate the motion of 800 particles in a 128a×64a Couette cell (the velocities and temperatures of the two walls are different), which gives an intermediate area fraction ϕ = 0.307. The typical isotherms and the particle configuration are displayed in Fig. 5, and one can see the substantially inhomogeneous temperature fields as a result of strong thermal convection.

Fig. 5 Typical isotherms and particle configuration for a

sheared non-colloidal suspension of 800 circular particles in a 128a 64a Couette cell. ×

'(Re, , , , , , )r r p r rGr Pe k cρ β =(0.2, 0, 20, 1.001,

5.0, 1.0, 0.0).

4. Conclusions The DLM/FD method proposed by Glowinski

and his coworkers has been extended to handle the particle motion in a Bingham fluid in three dimensions and the particulate flow with heat transfer in two dimensions. Some preliminary results have been presented, demonstrating the accuracy and power of our method. It is shown that the drag coefficient for a sphere settling in a Bingham fluid can be well correlated with a modified Reynolds number. References [1] HU H H, PATANKAR A, ZHU M Y. Direct numerical

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