International Journal of Mechanical Engineering and Technology … BOLTZMANN... · 2014-08-30 · non-Newtonian fluids (shear-thinning and shear-thickening fluids) by considering

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

ISSN 0976 – 6359(Online), Volume 5, Issue 8, August (2014), pp. 20-33 © IAEME

20

LATTICE BOLTZMANN SIMULATION OF NON-NEWTONIAN FLUID

FLOW IN A LID DRIVEN CAVITY

M. Y. Gokhale1, Ignatius Fernandes

2

1Department of Mathematics, Maharashtra Institute of Technology, Pune, India

2Department of Mathematics, Rosary College, Navelim Goa, India

ABSTRACT

Lattice Boltzmann Method (LBM) is used to simulate the lid driven cavity flow to explore

the mechanism of non-Newtonian fluid flow. The power law model is used to represent the class of

non-Newtonian fluids (shear-thinning and shear-thickening fluids) by considering a range of 0.8 to

1.6. Investigation is carried out to study the influence of power law index and Reynolds number on

the variation of velocity profiles and streamlines plots. Velocity profiles and the streamline patterns

for various values of power law index at Reynolds numbers ranging 100 to 3200 are presented. Half

way bounce back boundary conditions are employed in the numerical method. The LBM code is

validated against the results taken from the published sources for flow in lid driven cavity and the

results show fine agreement with established theory and the rheological behavior of the fluids.

Keywords: Lid Driven Cavity, Non-Newtonian Fluids, Power Law, Lattice Boltzmann Method.

1. INTRODUCTION

Non-Newtonian fluid flow is an important subject in various natural and engineering

processes which include applications in packed beds, petroleum engineering and purification

processes. A wide range of research is available for Newtonian and non-Newtonian fluid flow in

these areas. In general, the hydrodynamics of non-Newtonian fluids is much complex compared to

that of their Newtonian counterpart because of the complex rheological properties. An important

factor in understanding the mechanism of non-Newtonian fluids is to identify a local profile of non-

Newtonian properties corresponding to the shear rate. Power law model is generally used to

represent a class of non-Newtonian fluids which are inelastic and exhibit time independent shear

stress. Though analytical solutions for the flow of power law fluids through simple geometry is

available, computational approach becomes unavoidable in most of the situations particularly if the

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)

ISSN 0976 – 6359 (Online)

Volume 5, Issue 8, August (2014), pp. 20-33

© IAEME: www.iaeme.com/IJMET.asp

Journal Impact Factor (2014): 7.5377 (Calculated by GISI)

www.jifactor.com

IJMET

© I A E M E



21

flow field is not one dimensional. Over the past several years, various computational methods have

been applied to simulate the power law fluid flows in different geometries. Among different

geometries, the lid driven cavity flow is considered to be one of the benchmark fluid problems in

computational fluid dynamics.

With recent advances in mathematical modeling and computer technology, lattice Boltzmann

method (LBM) has been developed as an alternative approach to common numerical methods which

are based on discretization of macroscopic continuum equations. LBM is effective for investigating

the local non-Newtonian properties since important information to non-Newtonian fluids can be

locally estimated. The fundamental idea of this method is to construct simplified kinetic models that

include the essential physics of mesoscopic processes so that the macroscopic averaged properties

obey the desired macroscopic properties [2].

In recent years, the problem of lid driven cavity flow has been widely used to understand the

behavior of non-Newtonian fluid flow using power law model. Patil et al [5] applied the LBM for

simulation of lid-driven flow in a two-dimensional, rectangular, deep cavity. They studied the

location and strength of the primary vortex, the corner-eddy dynamics and showed the existence of

corner eddies at the bottom, which come together to form a second primary-eddy as the cavity

aspect-ratio is increased above a critical value. Bhaumik et al. [6] investigated lid-driven swirling

flow in a confined cylindrical cavity using LBM by studying steady, 3-dimensional flow with respect

to height-to-radius ratios and Reynolds numbers using the multiple-relaxation-time method. Nemati

et al [7] applied LBM to investigate the mixed convection flows utilizing nanofluids in a lid-driven

cavity. They investigated a water-based nanofluid containing Cu, CuO or Al2O3 nanoparticles and

the effects of Reynolds number and solid volume fraction for different nanofluids on hydrodynamic

and thermal characteristics. Mendu et al. [9] used LBM to simulate non-Newtonian power law fluid

flows in a double sided lid driven cavity. They investigated two different cases-parallel wall motion

and anti-parallel wall motion of two sided lid driven cavity and studied the influence of power law

index )(n and Reynolds number (Re) on the variation of velocity and center of vortex location of

fluid with the help of velocity profiles and streamline plots. In another paper, Mendu et al. [10]

applied LBM to simulate two dimensional fluid flows in a square cavity driven by a periodically

oscillating lid. Yang et al. [25] investigated the flow pattern in a two-dimensional lid-driven semi-

circular cavity based on multiple relaxation time lattice Boltzmann method (MRT LBM) for

Reynolds number ranging from 5000 to 50000. They showed that, as Reynolds number increases, the

flow in the cavity undergoes a complex transition. Erturk [8] discussed, in detail, the 2-D driven

cavity flow problem by investigating the incompressible flow in a 2-D driven cavity in terms of

physical, mathematical and numerical aspects, together with a survey on experimental and numerical

studies. The paper also presented very fine grid steady solutions of the driven cavity flow at very

high Reynolds numbers.

The application of LBM to non-Newtonian fluid flow has been aggressively intensified in last

few decades. Though, the problem of fluid flow in lid driven cavity has been studied rigorously,

most of these studies have been confined to either laminar fluid flow or Newtonian fluids. The

present paper uses LBM to simulate the lid driven cavity flow to explore the mechanism of non-

Newtonian fluid flow which is laminar as well as under transition for a wide range of shear-thinning

and shear-thickening fluids. The power law model is used to represent the class of non-Newtonian

fluids (shear-thinning and shear-thickening). The influence of power law index )(n and Reynolds

number (Re) on the variation of velocity and center of vortex location of fluid with the help of

velocity profiles and streamline plots is studied.



22

Fig.1: Geometry of the lid driven cavity.

2. BACKGROUND AND PROBLEM FORMULATION

2.1. Governing equations In continuum domain, fluid flow is governed by Navier-Stokes (NS) equations along with the

continuity equation. For incompressible, two-dimensional flow, the conservative form of the NS

equations and the continuity equation can be written in Cartesian system as [1].

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂

y

u

yx

u

xx

p

y

vu

x

uu

t

uµµ

ρρ (1)

∂

∂

∂

∂+

∂

∂

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂

y

v

yx

v

xx

p

y

vv

x

uv

t

vµµ

ρρ (2)

0)()(

=∂

∂+

∂

∂

y

v

x

u ρρ (3)

The two-dimensional lid driven square cavity with the top wall moving from left to right with

a uniform velocity 1.00 == uU is considered, as shown in Fig 1. The left, right and bottom walls are

kept stationary i.e. velocities at all other nodes are set to zero. We consider fluid to be non-

Newtonian represented by power law model.

2.2. Physical boundary conditions

Top moving lid : 0),( uHxu = and 0),( =Hxv (4a)

Bottom stationary side : 0)0,( =xu and 0)0,( =xv (4b)

Left stationary side: 0),0( =yu and 0),0( =yv (4c)

Right stationary side: 0),( =yLu and 0),( =yLv (4d)



23

3. NUMERICAL METHOD AND FORMULATION

3.1. Lattice Boltzmann Method

In the present study, we cover incompressible fluid flows and a nine-velocity model on a two-

dimensional lattice (D2Q9). The lattice Boltzmann method can be used to model hydrodynamic or

mass transport phenomena by describing the particle distribution function ),( txf i giving the

probability that a fluid particle with velocity ie enters the lattice site x at a time t [1]. The subscript i

represents the number of lattice links and 0=i corresponds to the particle at rest residing at the

center. The evolution of the particle distribution function on the lattice resulting from the collision

processes and the particle propagation is governed by the discrete Boltzmann equation [11, 12, 13,

14].

),(),(),( txtxfdttdtexf iiii Ω=−++ 8,....,1,0=i (5)

where dt is the time step and iΩ is the collision operator which accounts for the change in the

distribution function due to the collisions. The Bhatnagar-Gross-Krook (BGK) model [19] is used for

the collision operator

[ ]),(),(1

),( txftxftxeq

iii −−=Ωτ

8,.....,1,0=i (6)

where τ is the relaxation time and is related to the kinematic viscosity υ by

−=

2

12 τυ dtcs (7)

Here sc is the sound speed expressed by ( ) 3/3/ cdtdxcs == ( c is the particle speed and dx is the

lattice spacing). ),( txfeq

i , in equation (6), is the corresponding equilibrium distribution function for

D2Q9 given by

( )

−

++= ),().,((

2

1),(.(

2

1),(.(

11,),(

2

2

42txutxu

ctxue

ctxue

ctxwtxf

s

i

s

i

s

ieq

i ρ (8a)

where ),( txu is the velocity and iw is the weight coefficient with values

=

=

=

=

8,7,6,536/1

4,3,2,19/1

09/4

i

i

i

wi (8b)

Local particle density ),( txρ and local particle momentum uρ are given by

∑=

=8

0

),(),(

i

i txftxρ and ∑=

=8

0

),(),(

i

ii txfetxuρ . (9)



24

3.2. Power law model

For a power law fluid, the apparent viscosity µ is found to vary with strain rate .

e (s-1

) by [15]

1.

0

−

=n

eµµ (10)

where n is the shear-thinning index and 0µ is a consistency constant. For 1<n , the fluid is shear-

thinning. Strain rate is related to the symmetric strain rate tensor,

αβαβ eee :2.

= (11a)

where αβe is the rate of deformation tensor which can be locally calculated by [11]

βααβτρ

ii

i

i eefdtC

e ∑=

−=8

0

)1(

2

1 (11b)

where ),(),(),()1(

txftxftxfeq

iii −= is the non-equilibrium part of the distribution function.

The Reynolds number for power law fluid is given by0

2

Reµ

ρ nn LU −

= , where U is the velocity

of the moving lid and L is the length of the cavity [9].

3.3. LBM boundary conditions

Boundary conditions play a crucial role in LBM simulations. In this paper, half-way bounce-

back conditions [17] are applied on the stationary wall. The particle distribution function at the wall

lattice node is assigned to be the particle distribution function of its opposite direction. At the lattice

nodes on the moving walls, boundary conditions are assumed as specified by Zou et al. [4]. Initially,

the equilibrium distribution function that corresponds to the flow-variables is assumed as the

unknown distribution function for all nodes at t = 0.

3.4. Numerical Implementation

A MATLAB code was developed for a 129129 × square cavity lattice grid. The velocity of

the moving lid was set to 0.1 tslu / and the velocity at all other nodes was set to zero. A uniform

density of 3/0.1 lumu=ρ is initially assumed for the entire flow field. The distribution function was

initialized with suitable values (here we assume that the fluid is initially stationary). The numerical

implementation of the LBM at each time step consists of collision, streaming, application of

boundary conditions, calculation of distribution functions and calculation of macroscopic flow

variables. Half way bounce back conditions were employed in simulation.

A range of 0.8 to 1.6 was taken for the power law index, so that, both the shear thinning and shear-

thickening fluid are considered.

4. RESULTS AND DISCUSSIONS

Investigation is performed to study the impact of Reynolds number and power index on

velocity profiles, vortex formation and the streamline patterns. The LBM was first validated by

comparing the results in the literature for power index 1=n , which corresponds to Newtonian fluids.



25

The circulation pattern and vortex formation is highly influenced by Reynolds number. The

investigation is performed for four values of Re (100, 400, 1000, 3200). Fig.2 presents velocity

profiles at the geometric center of the cavity for various values of n at 100Re = along with the

comparison of the results published in Ghia et al [3] for Newtonian fluids. The u-velocity profiles

along y-axis are presented in Fig.2a, which show almost a parabolic behavior. It is observed that u-

velocity starts increasing from zero at the bottom, continuously decreasing to the minimum negative

value and then increases to become zero at the center. The u-velocity then increases to attain the

maximum positive value at the top of the cavity. The minimum u- velocity value is observed to

decrease with an increase in n , whereas the maximum positive u-velocity increases with n . Though

the basic trend of the u-velocity profiles remain the same, the rheological behavior of the fluids

affects the minimum and the maximum values of the u-velocity profiles. Fig.2b presents the v-

velocity profile along x-axis through the geometric center of the cavity corresponding to the

conditions considered in Fig.2a. It is observed that the v-velocity starts from zero at the left side wall,

attains the maximum positive value, and then decreases to zero at the center of the cavity. The v-

velocity continuously decreases from the center and reaches the maximum negative value before it

again tends to zero at the right wall of the cavity. Lower values of n (shear-thinning fluid and

Newtonian) display lower values of v-velocity compared to that of shear-thickening fluids. Fig.4

presents the streamline plots for different values of power law index at Re=100. One primary vortex

is observed which moves towards the center of the cavity as n increases from 0.8 to 1.6. This is

because the viscosity of the fluid increases with increasing n . Secondary vortices are observed at the

top and bottom corners of the cavity. These vortices decrease marginally and almost vanish with an

increase in n , thus indicating that the primary circulation occupies almost whole of the cavity.

Primary circulation is concentrated towards the right of the cavity with streamlines parallel to the

right vertical side, indicating weak circulation in the region.

Fig.3 presents the velocity profiles at geometric center of the cavity for different values of n

at 400Re = . The velocity at the center of the cavity becomes almost linear with maximum and

minimum values attained at the top and bottom regions of the cavity, respectively. The streamline

plots in Fig. 5 for 400Re = show development of secondary vortices at the corners which keep

increasing in size with n , indicating the growth of secondary circulation in the regions. The primary

vortex is observed to shift towards the center as compared to the case for 100Re = . For 1000Re = , the

trend of the velocity profiles remains almost the same, but a significant variation is observed in

magnitude as indicated by Fig.7. This is due to the influence of moving lid on the fluid flow. The

secondary circulation increases as compared to lower values of Re. This is in theoretical agreement

of the properties of fluids, that the secondary circulation and formation of eddies closely related to

increase in Re. Fig.6 shows that the strength of the secondary circulation increases as n increases

from 0.8 to 1.6 It can be seen from Fig.3-7 that, for lower values of Re, the streamlines are

concentrated parallel to the right wall due to weak circulation in the region. At higher values of Re,

the primary circulation expands to cover whole of cavity indicating stronger circulation in the

regions.

The primary vortex shifts towards the center of the cavity for higher Re. The u-velocity and

v-velocity profiles for 3200Re = are presented in Fig.8. The profiles show a linear behavior in center

of the cavity, as seen from the figures. The u-velocity profiles for higher values of n show positive

u-velocities at the bottom of the cavity, as a result of strong secondary circulation in opposite

direction. This can be established from the streamline plots in Fig.9, which show formation of strong

secondary circulation as a result of transitional state of fluid. The velocity profiles show a linear

behavior in the centre of the cavity and this region (of linear behavior) increases as Re increases

from 100 to 3200. The magnitude of the u-velocity and v-velocity profiles increases, which is also

due to the impact of moving lid. Secondary circulation is more dominant compared to other values of

Re, indicating the influence of Re.



26

(a) (b)

Fig.2: Velocity profiles at the geometric center of the cavity for different values of n at Re=100. (a)

u-velocity profiles (b) v-velocity profiles

(a) (b)

Fig.3: Velocity profiles at center of cavity for varoius values of n at at Re=400.

(a) u-velocity profiles (b) v-velocity profiles

-0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u/U

y/L

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al.

0 0.2 0.4 0.6 0.8 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x/Lv

/U

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al.

-0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u/U

y/L

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al.

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

x/L

v/U

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al.



27

(a) (b)

(c) (d)

(e)

Fig.4: Streamline plots for various values of n at Re=100.

(a) n=0.8 (b) n=1.0 (c) n=1.2 (d) n=1.4 (e) n=1.6

-0.0

7

-0.0

7

-0.06

-0.06

-0. 0

6

-0.05

-0.0

5

-0.05

-0. 0

5

-0.04

-0.0

4

-0.04 -0.0

4

- 0.0

4

-0.03

-0.0

3

-0.03

-0.03

-0. 0

3

-0.0

3

-0.02

-0.02

-0.0

2

-0.02

-0.02

-0.0

2

-0.0

2

-0.01

-0.01

-0.0

1

-0.01

-0.01

-0.0

1

-0.0

1-0

.01

0 0

0

0

00 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.12

-0.1

2

-0.1

-0.1

-0.1

-0.08

-0.0

8-0.08

-0.0

8-0

.06-0.06

-0. 0

6

-0.06 -0.0

6

-0.0

4

-0.04

-0.0

4

-0.04

-0.04

-0.0

4

-0.0

4

-0.02

-0.02

-0.0

2

-0.02

-0.02

-0.02

-0.0

2- 0

.02

0

0

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1

4

-0.1

4

-0.1

2-0.1

2

-0.12

-0.1

-0.1

-0.1

-0.1

-0. 1

-0.0

8

-0.08

-0.0

8

-0.08

-0.08

-0.06

-0.06

-0.0

6

-0.06

-0.0

6

-0.0

6

-0.04

-0.04

-0.0

4

-0.04

-0.04 -0.0

4

-0.0

4

-0.02

-0.02

-0.02

-0.0

2

-0.02-0.02

-0.0

2

-0.0

2

00

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1

4

-0.1

4

-0.1

4

-0.12

-0.12

-0.12

- 0.1

2

-0.1

-0.1

-0.1

-0.1

-0. 1

-0.0

8-0.08

-0.0

8

-0.08

-0.08

-0.0

8-0

.06-0.06

-0.06

-0.06

-0.06

-0.0

6

-0.0

6- 0

.04

-0.04

-0.04

-0.0

4

-0.04

-0.04

-0.0

4

-0.0

4

-0.02

-0.02

-0.0

2

-0.0

2

-0.02-0.02

-0.02

-0.0

2-0

.02

00

00 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.16

-0.1

6

-0.14

-0.14

-0.1

4

-0.12

-0.1

2

-0.12

-0.1

2

-0.1

-0.1

-0.1

-0.1

-0.1

-0.08

-0.08

-0.0

8

-0.08-0.08

-0.0

8

-0.06

-0.06

-0.0

6

-0.06

-0.06

-0.0

6

-0.0

6-0

. 04

-0.04

-0.04

-0.0

4

-0.04-0.04

-0. 0

4

-0.0

4

-0.02

-0.02

-0.02

-0.0

2

-0.02-0.02

-0.02

-0.0

2-0

.02

00

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



28

(a) (b)

(c) (d)

(e)


(a) n=0.8 (b) n=1.0 (c) n=1.2 (d) n=1.4 (e) n=1.6

-0.8

-0.8

-0.7

-0.7

-0.7

-0.6

-0.6

-0.6

-0.6

-0.5

-0.5

-0.5-0.5

-0.5

-0.4

-0.4

-0.4

-0.4-0.4

-0. 4

-0.3

-0.3

-0.3

-0.3

-0.3

-0.3

-0.3

-0.2-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0. 2

-0.1-0.1

-0.1

-0.1

-0.1 -0.1

-0.1

-0.1

-0.1

0

0

0

00 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.2

-1.2

-1

-1

-1

-1

-0.8

-0.8

-0.8

-0.8

-0.8

-0.6

-0.6

-0.6

-0.6-0.6

-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0.2

0

0

0

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.4

-1.4

-1.4

-1.2

-1.2

-1.2

-1.2

-1

-1

-1

-1

-1

-0.8-0.8

-0.8

-0.8 -0.8

-0.8

-0.6-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.4-0.4

-0.4

-0.4

-0.4 -0.4

-0.4

-0.2-0.2

-0.2

-0.2

-0.2

-0.2 -0.2

-0.2

- 0.2

00

0

0

0

0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2

-2

-1.8

-1.8

-1.8

-1.6

-1.6

-1.6

-1.6

-1.4

-1.4

-1.4

-1.4

- 1.4

-1.2

-1.2

-1.2

-1.2

-1.2

-1

-1

-1

-1 -1

-1-0

.8

-0.8

-0.8

-0.8

-0.8-0.8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

- 0.6

-0.4

-0.4

-0.4

-0.4

-0.4 -0.4

-0.4

-0.4

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2 -0.2

-0. 2

-0. 2

0

0

0

0 0 0

0

0

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.4

-1.4

-1.2

-1.2

-1.2

-1

-1

-1

-1

-1-0

.8

-0.8

-0.8

-0.8-0.8

-0.6

-0.6

-0.6

-0.6 -0.6

-0.6

-0.4

-0.4

-0.4

-0.4-0.4

-0.4

- 0.4

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0.2

0

0

0

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



29

(a) (b)

(c) (d)

(e)


(a) n=0.8 (b) n=1.0 (c) n=1.2 (d) n=1.4 (e) n=1.6

-1

-1

-0.9

-0.9

-0.9

-0.8

-0.8

-0.8

-0.8

-0.7

-0.7

-0.7

-0.7

-0.7

-0.6-0.6

-0.6

-0.6 -0.6

-0. 6

-0.5

-0.5

-0.5

-0.5-0.5

-0.5

-0.4-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.3-0.3

-0.3

-0.3

-0.3 -0.3

-0.3

-0.3

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0.2

-0.1-0.1

-0.1

-0.1

-0.1 -0.1 -0.1

-0.1

-0.1

0

0

00

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.3

-1.3

-1.2

-1.2

-1.2

-1.1

-1.1

-1.1

-1.1

-1

-1

-1

-1

-1

-0.9

-0.9

-0.9

-0.9

-0.9

-0.8

-0.8

-0.8

-0.8-0.8

-0.8

-0.7

-0.7

-0.7

-0.7 -0.7

-0.7

-0.6-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

-0.4-0.4

-0.4

-0.4

-0.4 -0.4

-0.4

-0.4

-0.3-0.3

-0.3

-0.3

-0.3-0.3

-0.3

- 0.3

-0.2-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

- 0.2

-0.2

-0.1

-0.1

-0.1

-0.1

-0.1 -0.1-0.1

-0.1

-0.1

0

0

0

0

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.2

-1.1

-1.1

-1.1

-1

-1

-1

-1-0

.9

-0.9

-0.9

-0.9

-0.8

-0.8

-0.8

-0.8

-0.8

-0.7

-0.7

-0.7

-0.7-0.7

-0.7

-0.6

-0.6

-0.6

-0.6 -0.6

- 0.6

-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.3-0.3

-0.3

-0.3

-0.3 -0.3

-0.3

-0. 3

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

- 0. 2

-0. 2

-0.1-0.1

-0.1

-0.1

-0.1-0.1

-0.1

-0.1

-0.1

0 0

0

0

0

0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.8

-1.8

-1.6

-1.6

-1.6

-1.4

-1.4

-1.4

-1.4

-1.2

-1.2

-1.2

-1.2

-1.2

-1-1

-1

-1 -1

- 1

-0.8

-0.8

-0.8

-0.8-0

.8

-0.8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.4

-0.4

-0.4

- 0.4

-0.4-0.4

-0.4

-0.4

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0.2

-0. 2

0

0

0

0

0

0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2

-2

-2

-1.8

-1.8

-1.8

-1.6

-1.6

-1.6

-1.6

-1.4-1.4

-1.4

-1.4

-1.4

-1.2

-1.2

-1.2

-1.2 -1.2

-1

-1

-1

-1 -1

-1-0

.8

-0.8

-0.8

-0.8

-0.8 -0.8

-0.8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

- 0.6

-0.4

-0.4

-0.4

-0.4

-0.4-0.4

- 0.4

-0.4

-0.2

-0.2

-0.2

- 0.2

-0.2 -0.2

-0.2

-0.2

-0.2

0

0

0

0 0 0

0

0

0

0

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



30

(a) (b)

Fig.7: Velocity profiles at the center of the cavity for various values of n at Re=1000.


(a) (b)

Fig.8: Velocity profiles at center of the cavity for values of n at Re=3200


-0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u/U

y/L

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al.

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

x/Ly

/H

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u/U

y/L

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al

0 0.2 0.4 0.6 0.8 1

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x/L

v/U

n=0.8

n=1.0

n=1.2

n=1.4

n=1.6

Ghia et al



31

(a) (b)

(c) (d)

(e)

Fig.9: Streamline plots for various values of n at Re=3200. (a) n=0.8 (b) n=1 (c) n=1.2 (d) n=1.4 (e)

n=1.6

-3.5

-3.5

-3.5

-3

-3

-3

-3

-2.5

-2.5

-2.5

-2.5

-2.5

-2

-2

-2

-2-2

- 2

-1.5-1.5

-1.5

-1.5

-1.5

-1.5

-1.5

-1-1

-1

-1

-1 -1

-1

- 1

-0.5-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

-0.5

- 0.5

0

0

00

0 0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1

-1

-1

-0.8

-0.8

-0.8

-0.8

-0.8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

0

0

0

0

0 00

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.2

-1.2

-1

-1

-1

-1

-0.8

-0.8

-0.8

-0.8 -0.8

- 0.8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

- 0.6

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.4

-0.2-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

0

0

0

0

0 00

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.6

-1.6

-1.4

-1.4

-1.4

-1. 4

-1.2-1.2

-1.2

-1.2

-1. 2

-1

-1

-1

-1 -1

-1

-0.8

-0.8

-0.8

-0.8 -0.8

-0. 8

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.6

-0.4

-0.4

-0.4

-0.4

-0.4-0.4

-0.4

-0.4

-0.2-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

-0.2

0

0

0

0

0

0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.6

-1.6

-1.6

-1.4

-1.4

-1.4

-1.4

-1.2-1

.2

-1.2

-1.2

-1.2

-1

-1

-1

-1-1

-1

-0.8

-0.8

-0.8

-0.8-0.8

-0.8

-0.6

-0.6

-0.6

-0.6

-0.6 -0.6

-0. 6

-0.4

-0.4

-0.4

-0.4

-0.4 -0.4

-0.4

- 0. 4

-0.2

-0.2

-0.2

-0.2

-0.2-0.2

-0.2

-0.20

0 0

0

0

0 0

00

0

x/L

y/H

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



32

5. CONCLUSION

The paper presents numeric simulation of non-Newtonian (shear-thinning and shear-

thickening) fluid flow in a lid-driven square cavity. The influence of power law index and Reynolds

number on velocity profiles and the streamline plots is presented. The simulation is performed for

four values of Re (100, 400, 1000, 3200) and a range of 0.8 to 1.6 for power index. The Re has high

impact on the velocity profiles and the formation of the primary vortex. The u-velocity profiles

which show a parabolic behavior for smaller values of Re, become almost linear at the center of the

cavity for higher values of Re. The strength of the secondary vortices is also influenced by Re, the

strength of which increases with Re in agreement to the rheological behavior of the fluid. The

position of the primary vortex is highly influenced by the power law index and the Reynolds number,

which shifts towards the center of the cavity with increasing values of Re. The study also

demonstrates LBM to be an effective and alternative numerical method to simulate non-Newtonian

fluid flow.

REFERENCES

[1] A. Mohamad, Lattice Boltzmann method, Springer-Verlag London Limited 2011.

[2] Shiyi Chen, Gary D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid

Mech. 30 (1998) 329-364.

[3] U. Ghia, K. N. Ghia, C. T. Shin, High-Re Solutions for Incompressible Flow Using the

Navier-Stokes Equations and a Multigrid Method, Journal Of Computational Physics 48,

387-411 (1982).

[4] Qisu Zou, Xiaoyi He, On pressure and velocity boundary conditions for the lattice Boltzmann

BGK model, Phys. Fluids 9 (6), June 1997.

[5] D.V. Patil, K.N. Lakshmisha, B. Rogg, Lattice Boltzmann simulation of lid-driven flow in

deep cavities, Computers & Fluids 35 (2006) 1116–1125.

[6] S.K. Bhaumik, K.N. Lakshmisha, Lattice Boltzmann simulation of lid-driven swirling flow in

confined cylindrical cavity, Computers & Fluids 36 (2007) 1163–1173.

[7] H. Nemati, M. Farhadi, K. Sedighi, E. Fattahi, A.A.R. Darzi, Lattice Boltzmann simulation of

nanofluid in lid-driven cavity, International Communications in Heat and Mass Transfer 37

(2010) 1528–1534.

[8] Ercan Erturk, Discussions on Driven Cavity Flow, Int. J. Numer. Meth. Fluids 2009; Vol 60:

pp 275-294.

[9] Siva Subrahmanyam Mendu, P.K. Das, Flow of power-law fluids in a cavity driven by the

motion of two facing lids – A simulation by lattice Boltzmann method, Journal of Non-

Newtonian Fluid Mechanics 175–176 (2012) 10–24.

[10] Siva Subrahmanyam Mendu, P.K. Das, Fluid flow in a cavity driven by an oscillating lid—A

simulation by lattice Boltzmann method, European Journal of Mechanics B/Fluids 39 (2013)

59–70.

[11] Abdel Monim Artoli, Adelia Sequera, Mesoscopic Simulation of Unsteady shear-thinning

flows, Springer-Velag Berlin Heidelberg, 2006.

[12] Mitsuhira Ohta, Tatsuya Nakamura, Yutaka Yoshida, Yosuke Matsukuma, Lattice Boltzmann

simulation of viscoplastic fluid flows through complex flow channels, J. No-Newtonian Fluid

Mech. 166(2011) 404-412.

[13] C. R. Leonardy, D.R.J. Owen, Y.T.Feng, Numerical rheometry of bulk materials using power

law fluid and the lattice Boltzmann method, Journal of Non-Newtonian Fluid Mechanics

166(2011) 628-638.



33

[14] A. Nabovati, A. C. M. Sousa, Fluid flow simulation in random porous media at pore level

using the lattice Boltzmann method, Journal of Engineering Science and Technology, Vol. 2,

No. 3 (2007) 226-237.

[15] S.P. Sullivan, L. F. Gladden, M. L. Johns, Simulation of power-law flow through porous

media using lattice Boltzmann techniques, J. Non-Newtonian Fluid Mech. 133(2006) 91-98.

[16] D. Arumuga Perumal, Anoop K. Dass, Multiplicity of steady solutions in two-dimensional

lid-driven cavity flows by Lattice Boltzmann Method, Computers and Mathematics with

Applications 61 (2011) 3711–3721.

[17] A.A. Mohamad, Meso and Macro-Scales Fluid Flow Simulations with Lattice Boltzmann

Method, The 4th International Symposium on Fluid Machinery and Fluid Engineering

November 24-27, 2008, Beijing, China.

[18] Dirk Kehrwald, Lattice Boltzmann Simulation of Shear-Thinning Fluids, Journal of

Statistical Physics, Vol. 121, Nos. 1/2, October 2005.

[19] P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision process in gasses, Phys. Rev. 94

(1954) 511–525.

[20] S. Hou, Q. Zou, S. Chen, G. Doolen, A.C. Cogley, Simulation of cavity flow by the lattice

Boltzmann method, J. Comput. Phys. 118 (1995) 329–347.

[21] He X, Luo LS. Theory of the lattice Boltzmann equation: From the Boltzmann equation to

the lattice Boltzmann equation. Phys Rev E 1997;56:R6811.

[22] Succi S. The lattice Boltzmann equation for fluid dynamics and beyond. Oxford: Oxford

University Press; 2001.

[23] A. Artoli, Mesoscopic Computational Haemodynamic, Ph.D. Thesis, University of

Amsterdam, 2003.

[24] Romana Begum, M. Abdul Basit, Lattice Boltzmann method and its applications to fluid flow

problems, European Journal of Scientific Research, 22 (2008), 216-231.

[25] Fan Yang, Xuming Shi, Xueyan Guo, Qingyi Sai, MRT Lattice Boltzmann Schemes for High

Reynolds Number Flow in Two-Dimensional Lid-Driven Semi-Circular Cavity, Procedia

16(2012) 639-644.

Documents

International Journal of Mechanical Engineering and Technology … BOLTZMANN... · 2014-08-30 · non-Newtonian fluids (shear-thinning and shear-thickening fluids) by considering