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: [email protected]
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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