QUASIEQUILIBRIUM LATTICE BOLTZMANN MODELS
WITH TUNABLE PRANDTL NUMBER FOR
INCOMPRESSIBLE HYDRODYNAMICS
CHAKRADHAR THANTANAPALLY*, SHIWANI SINGH†
and DHIRAJ V. PATIL‡
EMU, JNCASR, Jakkur
Bangalore 560064, India*[email protected]
†[email protected]‡[email protected]
SAURO SUCCI
Istituto Applicazioni Calcolo \Mauro Picone"
C.N.R., Via dei Taurini, 19, 00185
Rome, Italy
SANTOSH ANSUMALI
EMU, JNCASR, Jakkur, Bangalore 560064, [email protected]
Received 9 November 2012Accepted 27 July 2013
Published 9 September 2013
Recently, it was shown that energy conserving (EC) lattice Boltzmann (LB) model is moreaccurate than athermal LB model for high-resolution simulations of athermal °ows. However, in
the sub-grid (SG) domain, the behavior is found to be opposite. In this work, we show that via
multi-relaxation model, it is possible to preserve the accuracy of the EC LB for both SG and
direct numerical simulation (DNS) models. We show that by introducing a nonunit Prandtlnumber, under-resolved simulations can also be performed quite e±ciently, a property which we
attribute to the enhanced sound-relaxation.
Keywords: Turbulence; lattice Boltzmann; direct numerical simulation; multi-relaxation.
PACS Nos.: 47.27.ek, 47.11.+j, 05.20.Dd
1. Introduction
The Lattice Boltzmann (LB) method is an attractive option for the direct numerical
simulation (DNS) of turbulent °ows, due to its high parallel scalability and ease of
International Journal of Modern Physics C
Vol. 24, No. 12 (2013) 1340004 (12 pages)
#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0129183113400044
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application to complex geometries.1–6 However, unlike conventional methods, such
as pseudo-spectral (PS) for Navier–Stokes,7 grid-resolution requirements and accu-
racy of the LB method for DNS, have been thoroughly tested for a few setups only.8
Furthermore, it is empirically known that compressibility-related errors may lead to
poor comparison between PS and LB methods at moderate grid-sizes.9 At this point,
we would also like to remind that in sub-grid (SG) domain, multiple relaxation
models with high bulk viscosity often perform better than single relaxation models,
which exhibit no bulk viscosity. The typical interpretation for such a behavior is
that, in the SG domain, acoustic oscillations can be e®ectively suppressed by high
bulk viscosity.10
However, it was recently shown that for high resolution DNS, the energy con-
serving (EC) LB models deliver an order of magnitude increase in accuracy, as
compared to their athermal counterpart.11 This was attributed to the absence of
unphysical bulk viscosity in the thermal model. However, it was also found that in
the under-resolved regime, very often, the athermal model behaves better than the
EC model. It can be argued that in the under-resolved domain, high bulk viscosity
present in athermal model, helps damping down the acoustic oscillations. Thus, it is
natural to wonder whether it would be possible to keep the best of both approaches.
The aim of this work is to show that it is indeed possible to design an EC LB model
which is e±cient in damping acoustic oscillations and thus delivering better per-
formance in both resolved and under-resolved regimes.
Sections 2 and 3 give the details of the LB formulations. In Sec. 4, we show the
empirical observation that models with a higher bulk viscosity are better at
reducing acoustic perturbations, is consistent with the linearized hydrodynamics.
Furthermore, we argue that high energy dissipation introduced via low-Prandtl
number, is also an e®ective mechanism to damp down the acoustic perturbations
and that low-Prandtl number simulations are preferable to high bulk viscosity
simulations in the nonlinear °ow regime. Finally, we discuss the results of the
numerical simulations.
2. Isothermal and Thermal LB Formulation
In the LB formulation (for review, see Refs.1–3, 12) the °uid is represented by a set
of discrete populations f ¼ ffiðx; tÞg, describing the probability of ¯nding a particle
at position x in the lattice, at time t and with discrete velocity ci ði ¼ 1; . . . ;NÞ.Since the particles are grid-bound, the number of discrete velocities, N , also de¯nes
the connectivity of the lattice. Therefore, the basic variables in LB are the discrete
populations fi de¯ned for a set of discrete velocities ci (i ¼ 1; . . . ;N). The evolution
equations for these discrete populations are in the form
@tfi þ ci�fi ¼ �iðfÞ; ð1Þwhere �i is the collision term and it ensures that the dynamics goes to the chosen
equilibrium. In the entropic formulation of LB, in order to de¯ne equilibrium one
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begins by assuming the existence of a discrete H functional13 in the form
H ¼XNi¼1
fi lnfiwi
� �� 1
� �� �; wi > 0; ð2Þ
where wi are the normalization weights. Once the lattice H-function is known, the
equilibrium feq can be evaluated as a minimum of the H-function under constraints
of local conservation of macroscopic variables such as the mass density, �, the mo-
mentum density J� � �u� and the energy (trace of the pressure tensor
P � 12 �u
2 þ D2 ��), where � is temperature (in Boltzmann units), are de¯ned as linear
combinations of the discrete populations
� ¼XNi¼1
fi; �u� ¼XNi¼1
fici� P ¼ 1
2
XNi¼1
fic2i : ð3Þ
The only di®erence between isothermal and thermalmodels is that, in an isothermal
model, the constraint on energy conservation is relaxed by taking � ¼ 1=3. This for-
mulation of LB, which lacks energy conservation implies a ¯nite bulk viscosity.
In this work, we refer to the so-called D3Q27 model, for which mass, momentum
and energy conservation is taken into account. The explicit form of the equilibrium,
correct up to Oðu3Þ, reads as follows14:
f eqi ¼ �Wið�Þ 1þ u�ci�
�þ u�u�
2 �2ðci�ci� �Kið�Þ���Þ
h i; ð4Þ
where the temperature-dependent weights and factor Ki are
Wið�Þ ¼ ð1� �ÞD �
2ð1� �Þ� �ðci=cÞ2
; Kið�Þ ¼2D�2 þ ci
c
� �2ð1� 3�Þ
Dð1� �Þ : ð5Þ
The latter term reduces to the standard lattice sound speed squared, c2s ¼ 1=3, in the
limit � ! 1=3. Also, in the case of � ¼ 1=3, f eqi is given by
f eqi ¼ �Wið1=3Þ 1þ 3u�ci� þ 9
2u�u�ci�ci� �
3
2u�u�
� �: ð6Þ
The set of discrete populations obeys a discrete evolution equation based on two
basic steps: free-streaming and local collisions. This reads as follows
f �i ðx; tþ�tÞ ¼ fiðx; tÞ þ �iðfðx; tÞÞ;fiðx; tþ�tÞ ¼ f �
i ðx� ci�t; tÞ; ð7Þ
where the collision term �i, describes the collisions taking the system toward local
equilibrium. For most hydrodynamic purposes, this collision term can be modeled as
single-relaxation time with �i ¼ 1� ðf eq
i � fiÞ, known as a Bhatnagar–Gross–Krook
(BGK) model. In Sec. 3, we describe the entropic quasi-equilibrium procedure to
construct collision model with multiple-relaxation times, which o®ers a signi¯cant
improvement of the numerical stability of the method.
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3. Generalized Quasi-equilibrium LB Formulation
In this section, following the procedure outlined in Ref. 15, we brie°y remind how
Prandtl number as an independent parameter can be introduced in the LB frame-
work via quasi-equilibrium models. In this framework, multi-relaxation time can be
introduced, via the method of the so-called quasi-equilibrium distribution. As shown
in Fig. 1, BGK is a one-step relaxation to equilibrium, while the actual multi-re-
laxation formulation makes use of an intermediate quasi-equilibrium f� (found as a
minimum of H-function, under constraints on the additional quasi-conserved vari-
ables, such as the energy °ux). The two-step relaxation proceeds as follows. First
comes a fast relaxation to the quasi-equilibrium state f�, which is then followed by a
slow relaxation to the equilibrium state feq. Both relaxation mechanisms are taken in
the BGK form, with � � �1 (where � corresponds to the fast relaxation of f to f� and�1 corresponds to slow relaxation of f� to feq).16,15
As mentioned above, besides the usual conserved quantities, mass–momentum-
energy, a quasi-conserved one, namely the energy °ux (nonequilibrium part),
�q� ¼PNi¼1ðfi � f eq
i Þc2i ci�, is included in the relaxation process, which de¯nes quasi-
equilibrium.15 The resulting quasi-equilibrium distribution with accuracy OðMa2Þreads
f �i ¼ �Wið�Þ 1þ ci�
�u� � u 0
�
� �þ u�u�
2 �2ci�ci� �Kið�Þ���� ��
þ q�ðD� 1Þ�2ð1� �Þ ci�c
2i
�; ð8Þ
where we have set
u 0� ¼ q�
1þ ðD� 1Þ�ðD� 1Þ�ð1� �Þ : ð9Þ
Fig. 1. Approach to equilibrium in two stages. The curved path denotes the relaxation trajectory under
the e®ect of the actual Boltzmann collision integral.
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The two-time relaxation operator in equation Eq. (1) is given by
�i ¼1
�f �i �;u; �;qð Þ � fið Þ þ 1
�1f eqi �;u; �ð Þ � f �
i �;u; �;qð Þð Þ: ð10Þ
The relaxation time � ¯xes the dynamic viscosity � via usual relation, � ¼ � p,
while the second relaxation time controls thermal di®usion via the Prandtl number
Pr ¼ ð5=4Þð�=�1Þ. The kinetic equation (Eq. (1)) is integrated along the character-
istics, using the trapezoidal scheme to obtain the following evolution equation:
giðxþ c�t; tþ�tÞ ¼ giðx; tÞ 1� 2�ð Þ
þ 2� 1� �
�1
� �f �i �;u; �;qð Þ þ �
�1f eqi �;u; �ð Þ
� �; ð11Þ
where g is the auxiliary population de¯ned as,
gðx; tÞ ¼ fðx; tÞ � �t
2� fðx; tÞð Þ; � ¼ �t
2� þ�t: ð12Þ
The moments in terms of the auxiliary population g are given by:
�ðgÞ ¼ �ðfÞ; u�ðgÞ ¼ u�ðfÞ; T ðgÞ ¼ T ðfÞ and q�ðgÞ ¼ q�ðfÞ 1þ �t
2�1
� �:
ð13Þ
4. Analysis of Compressible Navier–Stokes
In LB simulations with initial conditions which are not consistent with incompres-
sibility, huge spurious acoustic disturbances may arise. These spurious waves also
arise whenever resolution is poor and there are sharp changes in space-time distri-
bution of the hydrodynamic ¯elds. Typically, one avoids such scenarios by enhancing
the resolution and carefully tuned initial conditions17. In this section, we propose an
alternate solution based on compressible Navier–Stokes–Fourier equations, with
delta perturbations in density as initial condition, as a model problem to describe
such behavior.
To this aim, we follow the usual prescription of decomposing the momentum j
into longitudinal and transverse components as
j ¼ jl þ jt; r� jl ¼ 0; r � jt ¼ 0: ð14ÞIn case of linear hydrodynamics, it is readily shown that the dynamics of the
transverse component is fully decoupled from the longitudinal component and obeys
the di®usion equation with � ¼ =� as di®usion coe±cient where is the shear
viscosity, � is density. Similarly, the longitudinal part of the dynamics can be written
as18
�@t2 þ 1
�
4
3 þ
� �@tr2
� ���ðx; tÞ þ @p
@�
����S
r2��þ 1
T�
@p
@~s
�����
r2~q ¼ 0; ð15Þ
Quasiequilibrium LB Models with Tunable Prandtl Number
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where is bulk viscosity, p is pressure, S is the entropy, ~s is entropy per unit mass, T
is the temperature, ~q is the heat per unit mass and @t~q ¼ �r2T ðx; tÞ where � is
thermal conductivity. In this case, it is known that any density perturbation decays
at a distance r from the disturbance source as,14,19
��ðr; tÞ ¼ ðLaLrÞ� 12 exp � ðr� cstÞ2
2LaL r
� �; ð16Þ
where, L is the characteristic length and La is the Landau number, de¯ned as
La ¼ Kn 2� 2
Dþ �
� �þ Knð � 1Þ
Pr; ð17Þ
with Knudsen number Kn as the ratio of Mach number Ma and Reynolds number Re,
Prandtl number Pr, spatial dimension D, � is the ratio of bulk to shear viscosity and
is the ratio of speci¯c heat at constant pressure and volume.19 From this solution,
we see that the density pro¯le becomes increasingly sharper as La is decreased and
thus a possible solution for low-Mach number simulations is to allow for large La,
which leads to smoother pro¯les (see Eq. (16)). Thus, it is not surprising that in LB
simulations, often a higher value of bulk viscosity is introduced via multi-relaxation
type collision models, to smoothen the solution.10,15 However, Eq. (17) suggests that,
an alternative way of increasing La is to perform simulations at very low-Prandtl
number. Here, we remind that the Prandtl number is largely irrelevant to low-Mach
number isothermal dynamics, and consequently, customizing it to e®ectively dampen
acoustic waves, appears to o®er a viable strategy. However, we must stress that
linear hydrodynamics cannot predict which one, whether low-Prandtl number or
high bulk viscosity, is to be preferred for numerical simulations. Only, the nonlinear
analysis can suggest which approach is to be favored. The nonlinear term in
momentum balance equation is given by
@�
j�j��
� �¼ @�
j l�jl� þ j t�j
t�
�
!þ j t�@�
j l��
� �þ j t�
�
� �@�j
l� þ j l�@�
j t��
� �
¼ @�
j l�jl� þ j t�j
t�
�
!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
A
þ ��� � ��@� j l�j t��
� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
B
þ j t�@�
j l��
� �þ j t�
�
!@�j
l� þ j l�@�
j t��
!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
C
; ð18Þ
where the term fBg does not contribute to the longitudinal dynamics and provides a
source for the transverse component evolution equation. Similarly, the term fCg,which is also linear in jl, shows that there is a contribution from the longitudinal to
the transverse part evolution. The choice between low-Prandtl number or high bulk
viscosity can be made by observing that the longitudinal and transverse components
are coupled in the nonlinear hydrodynamics. In the nonlinear regime, the arti¯cial
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(due to unphysical values of bulk viscosity) dynamics of longitudinal part also a®ects
the transverse part. Therefore, it may be argued that introducing a very high bulk
viscosity in the model may corrupt the numerical simulation. This was indeed
reported in the recent work, where it was observed that, when resolution is su±-
ciently high, models with bulk viscosity do not converge to the right solution.11
Based on the above, we believe that acoustic °uctuations are best controlled by
introducing an arti¯cially high thermal conductivity (low-Prandtl number). Here, we
remind that the error introduced because of temperature dynamics (driven solely by
viscous heating) is at least of second-order in momentum.
In the sequel, we shall present a series of simulations aimed at illustrating the
points discussed above. Since a detailed and thorough investigation of each of these
points would warrant a paper on its own, our analysis will often be con¯ned to a
semi-quantitative level.
5. Numerical Simulation of Acoustic Perturbations
In order to support this conjecture, we have carried out numerical simulations of
decay of acoustic perturbations with di®erent LB schemes. The initial condition is
that of °uid at rest with uniform density (� ¼ 1:0) throughout the domain except at
the center where the density is perturbed (�� ¼ 0:01). This initial condition is sim-
ulated with isothermal, thermal and quasi-equilibrium (with variable Pr) LB
methods with a grid size of 200� 200 and Re ¼ 100. To show the acoustic damping
behavior of various methods we show the L2 norm of the density perturbation in the
domain after 100 iterations with varying La (by varying Kn and Pr). Figure 2(a)
qualitatively shows the density °uctuations in the domain with initial condition
given above after 100 iterations. The La number (see Eq. (17)) for isothermal,
thermal and Prandtl models considering � ¼ f2=3; 0; 0g; ¼ 5=3 and Pr ¼ f4; 4; 0:1gare f11/6Kn, 7/6Kn, 46/6Kng respectively where Kn ¼ 1:73� 10�4. The ¯gure
0
0.2
0.4
-0.4 -0.2 0 0.2 0.4
δρ
x
x10-3
IsothermalThermalPrandtl
(a)
5
6
7
8
9
10
11
12
13
2 3 4 5 6 7 8
L2
norm
of
δρ
Kn x10-4
x10-6
IsothermalThermal
x10-4
x10-6
(b)
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
5 10 15 20 25 30 35 40
L2
norm
of
δρ
Pr x10-2
x10-6
Prandtl-model
(c)
Fig. 2. (Color online) (a) Variation of density °uctuation in x-direction, through the centre of the
domain, with same Kn for the three LB methods discussed. (b) Variation of L2 norm of density °uctuationwith Kn at a ¯xed Pr. (c) Variation of L2 norm of density °uctuation with Pr at a ¯xed Kn.
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shows that the density °uctuations indeed decrease with increasing La number. To
quantify the results obtained, these simulations are performed with varying Kn,
shown in Fig. 2 (b), to quantify the e®ect of La in damping the acoustic °uctuations.
As expected the acoustic °uctuations in the domain decrease with increasing Kn.
Further, the quasi-equilibrium (or Prandtl model) LB simulation is run with the least
of the Kn (1:73� 10�4) used in other two methods (i.e. with lowest damping of
oscillations), but with varying Pr e®ectively changing the La. The results of the
simulation are shown in Fig. 2 (c) which are supportive of the fact that density
perturbations decay faster with decreasing Pr number.
6. Two-dimensional Benchmark Simulations
6.1. Taylor–Green vortex
To further validate our method, we have chosen the Taylor–Green vortex simulation
as a model problem, in order to analyze the behavior of an isothermal (with a ¯nite
bulk viscosity), thermal (EC) and quasi-equilibrium model with tunable Prandtl
number. This setup was analyzed earlier, to compare isothermal and EC LB.11 In
order to compare the performance of the current model with that of Ref. 11, we
performed a grid resolution study for the velocity pro¯le, analytically given by
u ¼ sinx cos y e�2�t;
v ¼ �cosx sin y e�2�t;ð19Þ
along the x direction at a ¯xed Mach (Ma) and Reynolds (Re) numbers. Here, Mach
number is de¯ned as Ma ¼ U0=cs and Re is based on the characteristic length of the
°ow-¯eld, taken as 1 in a periodic-box of length 2�, thus Re ¼ U0=�. As the Prandtl
number is just a tunable parameter in the current study, all our simulations have
used Prandtl number Pr / ffiffiffiffiffiffiffiKn
pand Kn and La is completely dictated by behavior
of Pr. In the current simulations of Taylor–Green °ow, we have chosenffiffiffiffiffiffiffiKn
p ¼ 0:0014. As shown in Fig. 3, even at low grid resolution, the model with a
change in Pr, with di®erent �=�1 (here, Pr ¼ 4�=�1), provides a more accurate
simulation of the Taylor–Green vortex, in comparison to isothermal model (except at
a very poorly resolved regime). This result reinforces the fact varying Pr is a better
choice than changing the bulk viscosity for damping the acoustic °uctuations.
6.2. Double periodic shear layer
To demonstrate the method at least on semi-quantitative basis, we consider
the setup of doubly periodic shear layer in two dimensions, which is sensitive to the
computational method used because of the presence of large velocity gradients in the
domain. Thus, this initial condition permits a sensible comparison between various
methods, both in terms of stability and accuracy. The parameters used in simulation
are Re ¼ 30000, U0 ¼ 0:04 for isothermal as well as for simulations with Prandtl
correction. The ratio between two time scales in the latter, i.e. �=�1, is taken to be
0:005 and in this caseffiffiffiffiffiffiffiKn
p ¼ 0:00152. Figure 4 (top) shows that the Prandtl model
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0 500 1000 15000
0.002
0.004
0.006
0.008
0.01
N
L 1 nor
m
IsothermalECPr=0.4(τ/τ
1=0.1)
Pr=0.04(τ/τ1=0.01)
Pr=0.02(τ/τ1=0.005)
0 500 1000 15000
0.002
0.004
0.006
0.008
0.01
N L 2 n
orm
IsothermalECPr=0.4(τ/τ
1=0.1)
Pr=0.04(τ/τ1=0.01)
Pr=0.02(τ/τ1=0.005)
Fig. 3. (Color online) L1 and L2 norm for velocity in x direction at Re ¼ 4000 andMa ¼ 0:05 for Taylor–
Green vortex with di®erent grid size.
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−80
−60
−40
−20
0
20
40
60
80
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−60
−40
−20
0
20
40
60
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−60
−40
−20
0
20
40
60
X
Y
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
−60
−40
−20
0
20
40
60
Fig. 4. (Color online) Vorticity ¯eld at time t ¼ 1 on 200� 200 (top) and 312� 312 (bottom) grid for
athermal (left) and Prandtl (right) models at Re = 30000, Ma = 0.04.
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is stable, while the athermal model blows up at a given lower resolution setup. This
simple qualitative comparison shows that the range of applicability of Pr model is
broader than that of isothermal model, in the SG domain, in terms of stability. It can
be seen that at su±ciently high resolution, both collision models converge to the
same solution, as shown in Fig. 4 (bottom). However, at lower resolution the Prandtl
model leads to higher quality results, as shown in Fig. 4 (top).
7. Three-dimensional Benchmark Simulation
As a more stringent test for the current formulation, we simulate three-dimensional
decaying turbulence with a highly symmetric initial condition, proposed by Kida.20
These simulations are compared with the results obtained with a PS code,21,22. Kida–
Pelz initial conditions are a simple benchmark initial °ow conditions, that evolve to
turbulent °ows in time. Kida °ow has been used in Refs. 23 and 24 to study the
problem of Euler blowup and DNS of viscous turbulence was analyzed using this
initial °ow condition.25
In order to show the accuracy of the present formulation, we compare various
physical quantities, like enstrophy and maximum vorticity. As LB is a primitive
formulation of hydrodynamics, vorticity is not directly accessible to LB simulations.
However, the energy and enstrophy can be calculated on the °y, without taking any
spatial derivative. We remind that enstrophy can be computed from the symmetric
velocity gradient tensor,26 which is locally available in LB simulations locally via
S�� ¼ 2
�T ð2� þ�tÞXi
f eqi � fið Þci�ci�; � ¼ �
c2s: ð20Þ
Note that this does not imply the calculation of any explicit spatial derivative.
Enstrophy shown in Fig. 5, which is more sensitive to the numerical resolution
than the kinetic energy, is compared in both PS and LB. The ¯gure shows the
2 3 4 524
26
28
30
32
34
36
Time
Ens
trop
hy
2 3 4 524
26
28
30
32
34
36
Time
Ens
trop
hy
LB 5133 Pr
LB 5123 Thm
LB 5133 Iso
PS 5123
LB 7683 Pr
LB 7683 Thm
LB 7683 Iso
PS 5123
PS 7683
Fig. 5. (Color online) Comparison of Enstrophy for Kida °ow at Re = 1000 using D3Q27.
C. Thantanapally et al.
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behavior of various methods at di®erent resolutions. It is evident from the ¯gure that
the Pr model captures the peaks better than other LB models assuming that the PS
result is most accurate one among them. Maximum vorticity which is the in¯nite
norm of vorticity in the entire domain at a given instant is tracked in time as shown
in Fig. 6. This quantity is more sensitive to the grid resolution than the enstrophy,
which is an average quantity over the domain. From this ¯gure we infer that we are
still working in the SG domain and that the Pr model is competitive with other
methods.
8. Outlook
Summarizing, we have shown that it is possible to preserve the accuracy of the EC
LB for both SG and high resolution DNS. The Prandtl-corrected scheme is more
stable than the athermal model in SG domain, and it converges to the results of
thermal model at higher resolutions.
Acknowledgments
The project is funded by Department of Science and Technology via Ramanujan
fellowship grant.
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50
100
150
200
250
300
350
Time
Max
imum
Vor
tici
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PS 7683
Fig. 6. (Color online) Comparison of maximum vorticity for Kida °ow at Re = 1000.
Quasiequilibrium LB Models with Tunable Prandtl Number
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