Prediction of Transitional Lid-Driven Cavity Flow Using

the Lattice Boltzmann Method

Md. Shakhawath Hossain, D.J. Bergstrom§ and X.B. Chen

Dept. of Mechanical Engineering, University of Saskatchewan,

Saskatoon, SK, Canada S7N 5A9

§Correspondence author email: don.bergstrom@mail.usask.ca

ABSTRACT

Simulations of three-dimensional (3D) lid driven

cavity (LDC) flow have been performed using the

multiple relaxation time lattice Boltzmann method

(MRT LBM). The simulations considered flows at

Reynolds numbers of Re = 3200, 5000 and 7500,

which are all within the transitional flow regime. The

simulation results are compared with experimental

and numerical studies. In the transitional regime,

some of the unsteadiness and related fluctuations in

the velocity field are due to the presence of Taylor-

Gortler (T-G) vortices along the bottom wall of the

cavity. The results demonstrate that this methodology

is fundamentally able to capture a transitionally

turbulent flow, although a finer mesh may be

required to better resolve some features of the flow.

Keywords – MRT LBM, 3D LDC flow, transitional flow,

T-G vortices

1. INTRODUCTION

The Lattice Boltzmann method (LBM) has attracted

much attention in recent years as a numerical scheme

for simulating complex fluid flow problems. The

scheme is capable of modeling applications such as

multiphase flows and flow with complex boundaries.

The LBM evolves from the Lattice Gas Cellular

Automata (LGCA) approach [1, 2] and derives its

basis from kinetic theory [3]. The LBM models

capture the microscopic behavior of a flow and from

the microscopic properties calculate the macroscopic

properties [3]. Solution of the lattice Boltzmann

equation (LBE) uses a simple stream and collide

computational procedure. The LBM avoids solving a

Poission-type equation to obtain the pressure field

and instead uses an equation of state. The

implementation of boundary conditions is straight

forward, and due to its local nature parallelization of

the code is also relatively simple and effective [4].

The simplest LBE is the Boltzmann equation with the

so called BGK (Bhatnagar Gross Krook)

approximation based on the use of a single relaxation

time (SRT). Due to the simplicity of this equation, it

is the most popular lattice Boltzmann model.

However, recently it has been demonstrated that the

MRT LBM model is more advantageous in terms of

numerical stability for turbulent flow simulation [5,

6].

Multiple previous studies can be found in the

literature, where the reliability and accuracy of the

MRT LBM models for simulating different turbulent

benchmark problems have been extensively studied.

D’ Humieres et al. [5] performed a simulation of 3D

diagonally LDC flow at Reynolds numbers (Re) up to

4000 using the MRT LBM model and the result

clearly demonstrates the superior numerical stability

of the MRT LBM model. Yu et al. [7] combined the

MRT LBM with a Smagorinsky model for the

subgrid-scale stress to perform a large eddy

simulation (LES) of the near field of low aspect ratio

turbulent jets. Premnath et al. [4] performed a similar

LES of wall bounded turbulent flows, e.g a fully

developed turbulent channel flow and LDC flow,

using a generalized LBE which combines multiple

relaxation times with a forcing term. Superior

numerical stability compared to SRT LBM was again

obtained.

In this paper the simulation of a complex flow, i.e.

3D LDC in the transition regime, is performed using

a D3Q19 MRT LBM model. The LDC is a well-

known benchmark problem which is of particular

interest due to the richness of the flow physics, even

though the geometry of the flow is relatively simple.

Appearance of multiple complex counter-rotating

recirculation zones at the corners of the cavity

depending on the Reynolds number, bifurcation of

the flow from a steady regime to an unsteady regime

and transition to turbulence at high Re are some

important aspects of the flow physics [9, 10]. The

appearance of symmetric Taylor-Gortler (T-G) type

vortices within the transitional regime as confirmed

by Prasad et al.’s [11] experimental results is

demonstrated. The paper is organized as follows. In

section 2, the MRT LBM model is discussed.

Section 3 discusses the simulation results for a LDC

flow, and Section 4 draws some conclusions.

2. LATTICE BOLTZMANN EQUATION

In the MRT LBM model, unlike the SRT LBM, a set

of relaxation times is used. The MRT LBM model is

discussed here in the context of the D3Q19 lattice

model. In this model a cubic lattice with 19 discrete

lattice points is used to define the 3D space.

The discretized MRT LBM equation, which is the

same for all MRT LBM models, can be written as

[5]:

| ( )⟩ | ( )⟩ [| ( )⟩ | ( )⟩] (1)

In the equation above, the | ⟩ notation is used to

represent column vectors. The elements of | ⟩ are the

distribution function (DF) at each lattice point. The

19 discrete particle velocities | for the model are given by

{

( ) ( ) ( ) ( ) ( ) ( ) ( )

In equation (1), | ⟩ is the moment vector and the 19

moments of | ⟩ are arranged in the following order

[4, 5]:

| ⟩ (

) (2)

Here is the mass density, is the part of the kinetic

energy independent of density, is the part of

the kinetic energy squared independent of density and

kinetic energy, are the momentum

components, are the energy fluxes

independent of the mass flux, and

are the symmetric traceless

viscous shear tensor components [4, 5]. The moment

vector can be mapped as,

| ⟩ | ⟩ (3)

| ⟩ is the equilibrium moment vector and is the

transformation matrix. The diagonal collision matrix

is defined as

(

)

where, the ’s are the

relaxation parameters using various relaxation time

scales. The formulation of | ⟩ and the values of

and the relaxation parameters can be found in

reference [5].

The left hand side of the equation (1) represents the

streaming process while the right hand side is the

collision process. In the streaming process the

particle population streams to their adjacent location

from to with a velocity along each

characteristic direction. In the collision process

particles arrive at a node, interact with each other and

change their velocity directions.

The kinematic viscosity ν can be obtained using the

following relation [12],

(

)

(

) (4)

The viscosity value can be set by varying the

relaxation parameters. Using the parameter values

provided in the Ref. [5] for the D3Q19 model

introduces some constraints, i.e. the viscosity, should be greater than 2.54×10-3 and a maximum

speed of 0.19 (Mach number 0.33) can be used.

3. SIMULATION RESULTS AND DISCUSSION

For the LDC, when the Reynolds number based on

the cavity length is less than Re = 2000, the flow

field is laminar. Flow instabilities begin between

Reynolds numbers of Re = 2000 and 3000 at the

downstream corner eddies [4]. For Re ≥ 10000 the

flow becomes fully turbulent with turbulence being

generated near the cavity walls.

Figure 1 shows a schematic of the 3D LDC flow

together with the coordinate system. The primary

eddy, two secondary eddies and an upper upstream

eddy can be seen in the center plane which is normal

to the y-direction. The T-G vortices occur near the

lower cavity wall and extend a finite distance in the

X-direction. In the LDC simulation presented here,

the upper lid is moving in the X-direction at a

velocity U = 0.19 m/s. The corresponding Mach

number is Ma = ⁄ = 0.329, where √ ⁄ .

In this paper the flow is studied at three different

Reynolds numbers, Re = 3200, 5000 and 7500, all

within the transitional regime. The Reynolds number

in our computation Re = UL/ν is achieved by varying

the number of the nodes and the value of viscosity,

with the lid velocity being the same for all cases.

Fig.1 Configuration of 3D LDCF

The value of the viscosity depends on the relaxation

parameters. Relaxation parameters are varied to

achieve the desired Reynolds number: 813 lattice

nodes are used to achieve Re = 3200 and 5000, while

1133 lattice nodes are used to achieve Re = 7500.

3.1 Velocity Profiles

The one dimensional (1D) time averaged U-velocity

profile along the Z-axis at X = Y = 0.5L for Re =

3200 is shown in Figure 2. The U-velocity profile is

in relatively good agreement with the experimental

results of Prasad and Koseff [11], although small

differences in the peak values can be seen. The

numerical result of Kuo et al. [12] for the U-velocity

profile is also shown and closely matches the

experimental result. Figure 3 presents the 1D W-

velocity profile along the Y-axis at X = Z = 0.5L. In

contrast to the U-velocity profile, significant

discrepancies between the numerical and

experimental results can be seen in the W-velocity

profile, especially for the peak values.

Figure 4 shows the Urms profile obtained for the

simulation at Re = 3200. The rms velocities are

normalized using the lid velocity to compare with the

experimental results. The LBM compares favorably

with the experimental results, except for the peak

near the upper wall which is under resolved. At Re =

3200, the flow is not turbulent and the large peak

near the lower wall in the rms velocity profile is due

to unsteadiness associated with the T-G vortices.

Figure 5 shows the Wrms profile at Re = 3200. When

the profile is compared with the experimental results,

the LBM prediction is observed to be much smaller

than the experimental results. The reason for the fact

that the streamwise fluctuation is better predicted

than the vertical component is not yet understood.

Fig. 2 Mean streamwise velocity profile at

Re = 3200 at X = Y = 0.5L

Fig. 3 Mean vertical velocity profile at

Re = 3200 at X = Z = 0.5L

Fig. 4 Fluctuating streamwise velocity component for

Re = 3200 at X = Y = 0.5L

Fig. 5 Fluctuating vertical velocity component for

Re = 3200 at X = Z = 0.5L

Figure 6 shows the normalized correlation ⟨ ⟩ along the Z-axis at X = Y = 0.5L. From the

experimental ⟨ ⟩ profile, it is observed that the

value of ⟨ ⟩ remains positive over the entire

region of T-G vortices near the lower cavity wall.

The simulation result also shows some positive ⟨ ⟩ values in this region, although the magnitude

is much smaller than the experimental result. In

particular, the numerical result under-predicts the

strong peak near the bottom wall, which may indicate

that the averaging time is insufficient.

The 1-D time averaged U-velocity profile along the

Z-axis at X = Y = 0.5L for Re = 5000 is shown in

Figure 7. The profile has been plotted only for the

lower half of the center line to compare with the

experimental results. For the higher Re, the LBM

predicts a velocity profile with a smaller peak than is

evident in the experimental profile. This may indicate

the need for a more refined grid near the wall at

higher Reynolds numbers. For the U-velocity

profiles at Re = 3200 and Re = 5000, the magnitude

of the peak velocity near the wall is observed to

decrease with Reynolds number. The U-velocity

profile at Re = 7500 is similar to that of Re = 5000

with a lower peak value than the experimental profile

near the wall.

Fig. 6 Reynolds shear stress for Re = 3200

at X = Y = 0.5L

Fig. 7 Mean streamwise velocity profile

at Re = 5000 at X = Y = 0.5L

Based on the comparisons above, the MRT LBM

simulations show some discrepancies in the 1-D

velocity and rms profiles which warrant further

investigation. However, the MRT LBM demonstrated

much better stability than the SRT LBM method, and

was specifically able to capture the T-G vortices near

the lower cavity wall at higher Re, which is

considered in the next section.

3.2 Vector and Vorticity Fields

Figure 8 plots the velocity vectors on the mid X-

plane (Y-Z plane) and indicates the presence of three

pairs of T-G vortices near the bottom wall. The

vorticity and velocity components are normalized

using the lid velocity. The strong symmetry observed

in Figure 8 indicates that the flow is not yet turbulent.

Fig. 8 Velocity vectors and contour plot of X-

vorticity component on the Y-Z plane at X = 0.5L for

Re = 3200

The T-G vortices extend a finite distance along the

bottom wall with their axis aligned in the X-direction.

Their specific form varies in both space and time, and

also with Reynolds number. Figure 9 shows the flow

structures at specified planes for Re = 5000.

(a) X = 0.25L

(b) X = 0.5L

(c) X = 0.75L

Fig. 9 Velocity vectors and contour plot of X-

vorticity components on the Y-Z plane at different

locations for Re = 5000

In Figure 9, no distinct T-G vortices can be seen on

the X = 0.25L plane, whereas three pairs of

symmetrical T-G vortices are clearly seen at the mid-

plane and at X = 0.75L. From the plots, the

magnitude of the vorticity intensifies in the negative

X- direction.

Figure 10 shows the vector and vorticity fields on the

mid X-plane at different times for Re = 7500.

(a) t = 20s

(b) t = 25s

Fig. 10 Velocity vectors and contour plot of X-

vorticity components on the Y-Z plane at X = 0.5L

for Re = 7500 at two different times

In Figure 10, for the same plane the vortices occupy

different locations at different times. Again, from the

symmetry of the vorticity contours and vector

profiles, it is clear that the flow is not yet turbulent at

Re = 7500. At the same time, unsteady vortex

structures are present.

Figures 11 and 12 show time traces of the U and V

velocity components measured close to the lower

cavity wall and approximately between two T-G

vortices at Reynolds numbers of Re = 3200 and 7500,

respectively. The similarity in the occurrence of peak

values of U and V for the case of Re = 3200 is

evident in Figure 11. The velocity traces for the case

of Re = 7500 in Figure 12 show more small scale

variations than the traces at Re = 3200.

Fig. 11 Time traces of U and V velocity components

measured close to the lower wall within the zone of

influence of the T-G vortices at Re = 3200

Fig. 12 Time traces of U and V velocity components

measured close to the lower wall within the zone of

influence of the T-G vortices at Re = 7500

From the power spectra of the U-velocity component

measured over a period of 17 minutes, it was found

that the peak in the power spectrum occurs at 0.003

Hz (period ≈ 6 min) for Re = 3200 and at 0.006 Hz

(period ≈ 3 min) for Re = 7500. The time traces of

the velocity and the associated periods clearly

indicate that the unsteadiness of the flow increases

with increasing Reynolds number.

For the LDC within the transitional regime, multiple

vortices are also seen near the upstream and

downstream walls in the X-Y plane. Figure 13 plots

the velocity vectors and contours of Z-vorticity at Re

= 3200 for the mid Z-plane. Three pairs of

symmetric vortices can be seen near the upstream

wall and two symmetric vortices can be seen near the

downstream wall. The vorticity field is symmetric on

both sides of the centre line, and the vortices near the

downstream wall are stronger than the vortices near

the upstream wall.

These near-wall vortices also introduce unsteadiness

into the flow and the MRT LBM works well to

capture these vortices at higher Reynolds numbers.

Figure 14 shows the vector and vorticity fields on the

mid Z-plane at different times for Re = 7500. In

these figures, one observes two or three pairs of

symmetric vortices near the upstream wall and

multiple symmetric vortices near the downstream

side wall. The vortices interact with each other and

the relative movement of the vortices at different

times is evident from the figures. As before, the

vortex pattern is approximately symmetric, and both

the magnitude and location of the vortices changes

with time. The presence of the near-wall vortices

shown here agrees well with other numerical

simulations [12].

Fig. 13 Velocity vectors and contour plots of Z-

vorticity component on the X-Y plane at Z = 0.5L for

Re = 3200

(a) t = 30s

(b) t = 35s

(c) t = 40s

Fig. 14 Velocity vectors and contour plots of Z-

vorticity component on the X-Y plane at Z = 0.5L

and Re = 7500 for different times

CONCLUSIONS

The D3Q19 MRT LBM method appears to be a

reliable method for simulating complex flows such as

the 3-D LDC within the transitional regime.

Although the 1D time average and rms velocity

profiles deviate from the experimental results at

higher Reynolds number, the method does manage to

capture the T-G vortices near the lower cavity wall as

well as the vortices near the upstream and

downstream cavity side walls. Use of refined grids

near the walls and a sub-grid scale (SGS) model to

capture unresolved small-scale motions will likely

improve the model predictions at higher Reynolds

numbers. This should also enable the peak velocities

in the time average velocity profiles near the walls to

be better captured. In the future the authors plan to

extend the simulation into the fully turbulent regime

(Re > 10,000) to further assess the capability of MRT

LBM to simulate flows that are complex in time and

space.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the financial

support of the Natural Science and Engineering

Research Council of Canada (NSERC).

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