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Synthetic turbulence generator for lattice Boltzmann method at the interface
between RANS and LES
Xiao Xue(薛骁),1, a) Hua-Dong Yao(姚华栋),1and Lars Davidson1
Division of Fluid Dynamics, Department of Mechanics and Maritime Sciences,
Chalmers University of Technology, 41296, Gothenburg,
Sweden
(Dated: 6 May 2022)
The paper presents a synthetic turbulence generator (STG) for the lattice Boltzmann
method (LBM) at the interface of the Reynolds averaged Naiver-Stokes (RANS)
equations and the LBM Large Eddy Simulation (LES). We first obtain the RANS
velocity field from a finite volume solver at the interface. Then, we apply a numerical
interpolation from the RANS velocity field to the LBM velocity field due to the
different grid types of RANS and LBM. The STG method generates the velocity
fluctuations, and the regularized LBM reconstructs the particle distribution functions
at the interface. We perform a turbulent channel flow simulation at Reτ= 180 with
the STG at the inlet and the pressure-free boundary condition at the outlet. The
velocity field is quantitatively compared with the periodic lattice Boltzmann based
LES (LES-LBM) channel flow and the direct numerical simulation (DNS) channel
flow. Both adaptation length and time for the STG method are evaluated. Also, we
compare the STG-LBM channel flow results with the existing LBM synthetic eddies
method (SEM-LBM) results. Our numerical investigations show good agreement with
the DNS and periodic LES-LBM channel flow within a short adaptation length. The
adaptation time for the turbulent channel flow is quantitatively analyzed and matches
the DNS around 1.5 to 3 domain flow-through time. Finally, we check the auto-
correlation for the velocity components at different cross-sections of the streamwise
direction. The proposed STG-LBM is observed to be both fast and robust. The
findings show good potential for the hybrid RANS/LES-LBM based solver on the
aerodynamics simulations and a broad spectrum of engineering applications. (This
paper has been accepted as a journal paper in Physics of Fluids)
a)Electronic mail: xiaox@chalmers.se
1
arXiv:2205.02774v1 [physics.flu-dyn] 5 May 2022
I. INTRODUCTION
In computational fluid dynamics (CFD), there are rising needs to understand the de-
tailed characteristics of turbulence ranging from fundamental to real-world applications. To
tackle these problems, direct numerical simulation (DNS) and large eddy simulation (LES)1,2
are frequently used to resolve flow structures. However, they are suffering from high com-
putational cost. Reynolds averaged Navier-Stokes (RANS) simulation gives hope of low
computational cost and reliable physics3. When detailed flow information is needed to re-
solve, LES is still necessary. Hybrid RANS/LES approach offers an excellent opportunity to
save considerable computational cost, whereas it preserves detailed flow information. This
requires correct modeling of the RANS/LES interface by capturing velocity fluctuations
with little spurious noise4. One approach is to use precursor DNS or LES data. However,
this approach is limited to fundamental flows studies with simple configurations and low
Reynolds numbers5. Another approach is ”recycling” the velocity field back to the inlet of
the simulation6–8. This method is still constrained by the flow complexity when a strong
pressure gradient occurs. The synthetic eddy method (SEM) can handle more general hy-
brid RANS/LES cases but suffers from a relatively long adaptation length for the fully
developed turbulence9–11. By introducing dynamic control forcing techniques (DCFT) to
the SEM framework, Roidl et al.12,13 showed good performance of the case for a transonic
airfoil with an adaptation region of merely 2-3 boundary-layer widths. However, the DCFT
is computationally expensive, which limits its application bandwidth. Synthetic turbulence
generators give opportunities to balance between accuracy, ease of implementation and com-
putationally relatively cheap14–19. Shur et al.19 highlighted the STG method in aerodynamics
and acoustics applications with a considerably short adaptation length.
Knowing that the LES is computationally expensive using conventional CFD due to the
data dependencies, the lattice Boltzmann method (LBM) can dramatically improve the com-
putational efficiency since the information on each lattice cell is stored and updated locally.
Unlike conventional CFD methods, each cell in LBM does not need to wait for its neighbors
to update during the algorithm’s evolution20–22. It is worth mentioning that LBM is not
solving the macroscopic scale quantities directly like Navier-Stokes equations. Governed by
the Boltzmann equation, LBM computes the particle’s distribution function in mesoscopic
scales. Through the Chapman-Enskog procedure, LBM can recover the Navier-Stokes equa-
2
tions. In the past decades, LBM has become widespread on account of its broad applicability
across a range of complex fluid dynamics problems ranging from micro-nano scales23–25 to
macroscopic scales26–29 at low Mach numbers. They have successfully studied complex fluid
phenomena including turbulence28,30,31 and non-ideal fluids with phase transition and/or
segregation25,32–36. Hou et al.26 brought the Smagorinsky LES into the lattice Boltzmann
models by introducing the effective viscosity. This contribution enables LBM to improve the
computational efficiency for LES computation. Hybrid RANS and LES-LBM method can
further reduce the computational cost of complex engineering applications. Thus, it is cru-
cial to impose turbulence in the LES-LBM framework accurately and efficiently. Although
it is an active area for the traditional CFD community, it is still an early-stage development
for the lattice Boltzmann method. Previous studies have used passive method by putting
obstacles in the flow field and removing obstacles after the turbulence is triggered37. How-
ever, this method has a relatively long adaptation time. Fan et al.38 have extended the SEM
method in the lattice Boltzmann method by reconstructing the force. However, it takes 15
boundary layer lengths to match DNS results fully. Buffa et al.39 have successfully adapted
the SEM method into the LBM framework by reconstructing the velocity field into particle
distribution functions. The above methods only consider the LES implementation without
considering the hybrid RANS input. In the present work, we integrate the synthetic tur-
bulent generator19 into the LBM framework(STG-LBM) so that it can be used at a hybrid
RANS/LES-LBM interface. The STG-LBM shows relatively short adaptation length with
only 2-4 boundary-layer thicknesses and fast converging speed with around 1.5-3 domain
flow-through time.
This paper is organized as follows: Section II introduces the lattice Boltzmann method
with the Bhatnagar–Gross–Kroog (BGK) collision operator and its Smagorinsky subgrid-
scale (SGS) LES modeling1,26. In Section III, we review the STG method. Then, we de-
scribe how to integrate the synthetic turbulent generator to the LBM framework at the
RANS/LES-LBM interface. Section IV shows the turbulent channel flow set up with the hy-
brid RANS/LES method. In Section V, we simulate the turbulent channel flow at Reτ= 180.
The results compare with the DNS, LES-LBM periodic channel flow, and the SEM-LBM
channel flow to show the convergence speed of the present method in both space and time.
Also, we examine the auto-correlations at the different cross-sections of the channel flow.
The conclusions are summarized in Section VI.
3
II. METHODOLOGY
In this section, we first present the D3Q19 lattice Boltzmann method with the single
relaxation time scheme. Then, the Smagorinsky subgrid-scale modeling for LES in the
lattice Boltzmann framework is presented.
A. The lattice Boltzmann method
The lattice Boltzmann method is a dimensionless model with all quantities formed in
the dimensionless lattice Boltzmann units(LBU). We apply a three dimensional(3D) lattice
model called the D3Q19 model which has 19 discretized velocity directions ci(i= 0...Q −
1). The particle’s probability distribution function, fi(x, t), denotes the i-th direction of a
lattice cell. The macro-scale quantities for the density, ρ(x, t), momentum, ρ(x, t)u(x, t),
and momentum flux tensors, Π(x, t), can be calculated from the distribution function, the
discrete velocities, and the volume force:
ρ(x, t) =
Q−1
X
i=0
fi(x, t),(1)
ρ(x, t)u(x, t) =
Q−1
X
i=0
fi(x, t)ci+1
2F∆t, (2)
Π(x, t) =
Q−1
X
i=0
fi(x, t)cici,(3)
where the momentum flux, Π, can be presented by the sum of the equilibrium and non-
equilibrium parts, Π(x, t) = Πeq(x, t) + Πneq (x, t). ∆tis the marching time step, which is
set to unity in the LBM algorithm. The lattice cell is located at position xat time t. The
evolution equation for the distribution functions, considering collision and forcing, can be
written as:
fi(x+ci∆t, t + ∆t) = fi(x, t)+Ωhfi(x, t)−feq
i(x, t)i+ ∆tFi(x, t),(4)
where Ω is a collision kernel, we use the Bhatnagar–Gross–Kroog (BGK) collision kernel
which has been widely adopted to various applications20, with Ω = ∆t
τ. The BGK collision
operator fixes the single relaxation time τfor the colliding process. The collision kernel
4
relaxes the distribution function towards the local Maxwellian distribution function feq
i20,21:
feq
i(x, t) = ωiρ(x, t)1 + ci·u(x, t)
c2
s
+[ci·u(x, t)]2
2c4
s−[u(x, t)·u(x, t)]
2c2
s,(5)
where csis the speed of the sound which is equal to cs= 1/√3 LBU. The kinematic viscosity
can be calculated by:
ν=c2
s(τ−1
2)∆t. (6)
Fiin Eq. (4) is the volume force acting on the fluid following the approach from Guo40 which
can be obtained by:
Fi(x, t) = (1 −1
2τ)ωici−u(x, t)
c2
s
+ci·u(x, t)
c4
s
ciF,(7)
where Fis the volume acceleration.
B. Smagorinsky subgrid-scale modeling
Below, we recall the basic formulation of the Smagrinsky SGS large-eddy simulations
techniques using the lattice Boltzmann method. Interested readers can refer to the litera-
ture1,26,37. The key of the Smagorinsky SGS modeling is to model the effective viscosity νeff,
which can be seen as the sum of the molecular viscosity ν0and the turbulent viscosity νt:
νeff =ν0+νt, νt=Cδ2¯
S,(8)
where Cis the Smagorinsky constant, δrepresents the filter size, and ¯
Sis the filtered strain
rate tensor:
¯
S=−τ0ρc∆x+q(τ0ρc∆x)2+ 18√2ρCδ2Q1/2
6ρCδ2,(9)
where τ0is the original relaxation time from the BGK collision operator, c= ∆x/∆t, and
Q1/2can be written as:
Q1/2=pΠneq :Πneq,(10)
where Πneq is the non-equilibrium part of the momentum flux tensor Πshown in Eq. (3).
With help of Eq. (6), we can transfer the effective viscosity into the effective relaxation time.
Following37, we can obtain the total relaxation time τeff, which is written as:
τeff =τ0
2+q(τ0ρc)2+ 18√2CQ1/2
2ρc .(11)
Finally, we replace τin the BGK collision operator with τeff to enclosure the lattice Boltz-
mann based LES system.
5
III. SYNTHETIC TURBULENCE GENERATOR IN THE LATTICE
BOLTZMANN FRAMEWORK
The synthetic turbulence generator (STG) has been used at the interface of RANS/LES
or RANS/IDDES19 by using the FVM. In the LBM community, previous studies have only
integrated the Synthetic Eddy Method (SEM)38,39 into the lattice-Boltzmann-based frame-
work. Notice that STG is different from the SEM method. In the SEM, coherent structures
are simulated by superimposing artificial eddies at the inlet plane. Each eddy is given vor-
ticity with a three-dimensional structure represented by a shape function describing the
spatial and temporal characteristics of the structure. As for the STG method, the syn-
thetic fluctuations are created using a Fourier series where the Fourier coefficient is given by
the −5/3 energy spectrum. In the present work, we mainly focus on integrating the STG
into the hybrid RANS/LES-LBM framework. In the following, we will briefly remind users
about the STG method and show how we couple RANS with the LES-LBM using the STG
method. Finally, the particle distribution functions at the interface is computed from the
velocity field via the regularized lattice Boltzmann method41.
A. Synthetic turbulence generator formulation
The synthetic turbulence generator at the interface between the RANS simulation and
the LES simulation reconstructs the velocity u(x, t) at the cell xat time t:
u(x, t) = u(x)RANS +u0(x, t),(12)
where u(x)RANS is the velocity vector obtained from the RANS results, and u0(x, t) is the
vector of the velocity fluctuations. The time-averaged fluctuation is zero, i.e. hu0(x, t)i= 0.
The velocity component from u0(x, t) can be expressed as uα(x, t):
u0(x, t) = aαβv0(x, t),(13)
where aαβ is the Cholesky decomposition of the Reynolds stress tensor:
{aαβ}=
√R11 0 0
R21/a11 pR22 −a2
21 0
R31/a11 (R32 −a21a31)/a22 pR33 −a2
31 −a2
32
,(14)
6
where Rαβ =u0
αu0
βis taken from Reynolds stresse tensor of the RANS simulation. The
auxiliary vector of the velocity fluctuations, v0(x, t), are constructed by superposition of N
weighted Fourier modes
v0(x, t) = √6
N
X
n=1
√qn[σncos (kndn·x0+φn)] ,(15)
where ndenotes the mode number, dnis the random unit vector that is uniformly located
over a sphere which is
dn=
sin(Θ)cos(Φ)
sin(Θ)sin(Φ)
cos(Θ)
,(16)
where Φ is uniformly distributed in the interval of [0,2π) and Θ = arccos(1 −γ/0.5) where
γis uniformly distributed in the interval of [0,2π). knis the amplitude of the vector dn,σn
represents the unit vector normal to the vector dn(σn·dn= 0)42,43 which can be written as
σn=
cos(Φ)cos(Θ)cos(η)−sin(Φ)sin(η)
sin(Φ)cos(Θ)cos(η) + cos(Φ)sin(η)
−sin(Θ)cos(η)
,(17)
where ηis random number that uniformly distributed in the interval of [0,2π). φnis random
number that uniformly distributed in the interval of [0,2π). qnrepresents the normalized
amplitude of a modified von Karman spectrum19:
qn=E(kn)∆kn
PN
n=1 E(kn)∆kn,
N
X
n=1
qn= 1,(18)
where E(kn) is the modified von Karman spectrum. x0in Eq. (15) is the pseudo-position
vector which can be defined as
x0=
2π
knmax{le(x)}(x−Ubt)
y
z
.(19)
In Eq. (19), Ubis the bulk velocity at the RANS/LES interface, and le(x) is the local scale
of the most energy-containing eddies. le(x) is given by le(x) = min(2dw(x), Cllt), where
dw(x) is the distance of the cell to the wall. Clis the empirical constant Cl= 3.0, ltis the
7
FIG. 1. The sketch of the D2Q9 lattice model at the inlet boundary. The dotted lines are unknowns.
The blue part is the inlet boundary, whereas the green part is the computational domain. nis the
unit normal vector that is perpendicular the to boundary pointing towards the inlet.
length-scale of the RANS model(lt=√kt/0.09ωt), where ktand ωtare given by the k−ω
RANS model44. In Eq. (15), Nmode is obtained by
N= ceil {ln(kmax/kmin)/ln(1 + α) + 1)},(20)
where α= 0.01, kmin =π/max {le(x)}, and kmax = 1.5max {2π/lcut}.lcut is given by
lcut = 2min {max {dx, dy,0.3dmax}+ 0.1dw, dmax},(21)
where dx, dy, dzare the local grid size of the RANS simulation at the RANS/LES-LBM
interface, dmax = max {dx, dy, dz}.v0is satisfying v0
αv0
β= 0, v0
αβ=δαβ (v0
αis the
component of v0). We have now reviewed the STG approach in the macroscopic CFD solver
for the RANS/LES interface. However, LBM is a dimensionless mesoscopic approach. Direct
coupling from RANS to LES-LBM is non-straightforward. Below, we demonstrate how to
impose the turbulent from the RANS/LES-LBM interface.
B. Boundary reconstruction with the regularized lattice Boltzmann method
The particle’s probability distribution function at the interface, fbc
i, can be defined as
fbc
i(x, t) = fbc(eq)
i(x, t) + fbc(neq)
i(x, t),(22)
8
where fbc(neq)
i(x, t) denotes the non-equilibrium part of the distribution function. The su-
perscript ”bc” denotes the boundary condition at the interface. The present work applies
the regularized scheme proposed by Latt, et al.45 to reconstruct the non-equlibrium part of
population which can be calculated via
fbc(neq)
i(x, t)≈ωi
c4
s
Qi:Πbc
neq,(23)
where Qi=cici−c2
sIwith Ibeing the identity matrix. Πbc
neq is the non-equilibrium part of
the moment flux tensor which is defined as
Πneq =
Q−1
X
i=0
Qifbc(neq)
i(x, t).(24)
Note that, at the boundary of the domain there are unknown variables in the ithe directions,
which can be calculated via the known direction following:
Qi=¯
Qinv(i), fbc(neq)
i(x, t) = ¯
fbc(neq)
inv(i)(x, t),(25)
where ¯
Qinv(i)and ¯
fbc(neq)
inv(i)(x, t) are the opposite direction of the unknown direction. For
example, Fig. 1 shows the case of the D2Q9 LBM model, where the unknown directions(1,
5 and 8) can be calculated by the opposite direction (3, 7 and 6) respectively; the same logic
applies to the D3Q19 lattice model. We would like to comment that this method could also
adapt to other 3D lattice models, for instance, D3Q15 and D3Q27. For the density of the
inlet boundary, we follow the idea from Zou and He46.
ρbc(x, t) = 1
1 + ˆubc(x, t)(2ρ⊥(x, t) + ρk(x, t)),(26)
where ˆubc is the cross product with the normal unit vector n at the boundary ˆubc =ubc
LB ·
n(|ˆubc|<0.3cs) and ubc
LB is the velocity of the lattice Boltzmann domain at the interface.
ρ⊥and ρkare the density calculated by
ρ⊥(x, t) = X
i∈{i|ci·n=0}
f⊥
i(x, t), ρk(x, t) = X
i∈{i|ci·n<0}
fk
i(x, t),(27)
where nis the normal vector pointing towards the boundary, f⊥
iand fk
iare the probabil-
ity density functions that point towards the boundary and are parallel to the boundary.
In Fig. 1, f⊥
iis represented by direction 3, 6, 7, and fk
iis represented by direction 2 and
9
4. Take Eq. (26) into Eq. (5), the equilibrium distribution function at the boundary can be
defined as
fbc(eq)
i(x, t) = ωiρbc(x, t)1 + ci·ubc
LB (x, t)
c2
s
+ci·ubc
LB (x, t)2
2c4
s−ubc
LB (x, t)·ubc
LB (x, t)
2c2
s,
(28)
In Section III C, we describe how to calculate the LBM velocity field, ubc
LB, from the RANS
velocity field, ubc
RANS.
FIG. 2. Overview of how to couple the RANS with the LES-LBM framework. The left panel is
the RANS computational domain. The colored layer is the velocity field at the RANS/LES-LBM
interface. The middle part is the interpolated velocity field obtained from the RANS data. The
right panel shows the sketch of the LES-LBM computational domain. The inlet velocity field
(colored light green) is reconstructed from the interpolated RANS mean profile with superimposed
synthetic fluctuations.
C. Integration of synthetic turbulence generator at RANS/LES-LBM
interface
At the hybrid RANS/LES-LBM interface, we first compute the RANS flow field in a
channel with the k−ωmodel44 using a finite-volume method (FVM). Thus, we can obtain
the physical macroscale flow quantities like the velocity, kand ω. The lattice Boltzmann
model is based on a uniform Cartesian grid and the RANS FVM calculation is based on
a non-uniform Cartesian grid. Therefore, it is necessary to interpolate the RANS velocity,
uRANS, to the uniform grid velocity, uint, at the interface which is the u(x)RANS in Eq. (12).
Second, we calculate the interpolated velocity fluctuations imposed at the interface following
ubc
phys(x, t) = uint(x) + u0(x, t),(29)
10
where u0(x, t) is generated by the STG appoarch and can be obtained by Eq. (13)-Eq. (19).
Now, we can calculate the dimensionless velocity field, ubc
LB(x, t), at the LES-LBM inlet by
ubc
LB(x, t) = ubc
phys(x, t)ct
cx
,(30)
where cxand ctare the conversion factor from the lattice Boltzmann simulation to physical
system which is defined as
cx=Lphys
LLB
, ct=tphys
tLB
,(31)
where Lphys and tphys represent the space and time unit from the physical system, LLB and
tLB are the space and time lattice Boltzmann unit from the lattice Boltzmann simulation.
Following Eq. (22)- Eq. (27), we use the regularized lattice Boltzmann method to reconstruct
the particle distribution function fi(x, t) at the inlet boundary with correct velocity and
density quantities. Finally, we apply the streaming and collision step to updating the fluid
field. Fig. 2 shows the general process of integrating the STG method at the RANS/LES-
LBM interface.
D. Implementation summary
Now, we have formulated all steps. The short summary of the implementation and the
formulas of the synthetic turbulence generator at the RANS/LES-LBM interface is presented
in the following pseudo-code.
11
Algorithm 1 Implementation of synthetic turbulence generator for the RANS/LES-LBM
interface (STG-LBM)
RANS section
1. Perform a RANS simulation with a finite volume method using the k−ωmodel.
2. Save the Reynolds stress tensor Rαβ, the velocity field uRANS, the bulk velocity Ub,kand ω
and other data needed for computing the velocity fluctuations u0(x, t).
LBM section
1. Obtain the RANS velocity at the interface u(x)RANS in Eq. (12).
2. Read the other saved value of Rαβ , the bulk velocity Ub,kfield, ωfield, and other data from
the RANS simulation.
3. Calculate aαβ using the Cholesky decomposition of the Reynolds stress tensor Eq. (14).
4. Calculate the normalized amplitude qnaccording to Eq. (18).
for all t= 0 till t=tend do
for all cells do
if cell xis at the RANS/LBM interface then
5. Update the pseudo-position x0according to Eq. (19).
6. Calculate v0(x, t) thanks to Eq. (15), Eq. (20) and Eq. (21).
7. Compute the fluctuating velocity vector u0(x, t) and ubc
phys following Eq. (13)
and Eq. (29) respectively.
8. Compute ubc
LB(x, t) with help of ct,cxfollowing Eq. (12).
9. Compute boundary density ρbc(x, t) thanks to Eq. (26).
10. Reconstruct the particle’s probability distribution function by combining Eq. (22),
Eq. (23) and Eq. (28):
fbc
i(x, t) = ωiρbc(x, t)1+ ci·ubc
LB (x, t)
c2
s
+ci·ubc
LB (x, t)2
2c4
s−ubc
LB (x, t)·ubc
LB (x, t)
2c2
s+ωi
c4
s
Qi:Πbc
neq
(32)
11. Update the LES-LBM relaxation time teff in Eq. (11).
end if
12. Apply stream and collide to update the fi(x, t) at each cell Eq. (4)
end for
end for 12
FIG. 3. 2D sketch of the RANS/LES-LBM simulation. The blue part is the RANS simulation,
and the green part is the LES-LBM simulation. The dark red dotted line is the RANS/LES-LBM
interface which applies the STG to generate turbulent fluctuations as inflow fluctuations in the
LES-LBM simulation. The outlet is imposed with the outflow pressure free boundary condition.
The sponge layer is the dark green region adjacent to the outflow boundary.
Note that the proposed integration methodology is designed to be used at any RANS/LES-
LBM interface for creating turbulence. The present work is a one-way coupling, which means
that the RANS simulation has to perform first, then the LES-LBM simulation can start.
Nonetheless, it is possible to dynamically couple the hybrid RANS/LES using FVM. One
interesting work can be found in47. Also, the present work uses the k−ωRANS model
and the Smagorinsky LES model, other RANS models and LES models2can also follow the
same logic presented in the above algorithm.
IV. NUMERICAL SETUP
This section will present the channel flow simulation for the hybrid RANS/LES-LBM
algorithm. The channel flow streamwise direction is the xdirection, the spanwise direction
is the zdirection, and the wall-normal direction is the ydirection. Figure 3 shows the 2D
sketch for the hybrid RANS/LES-LBM channel flow simulation. The blue region represents
the simulation with a finite-volume method48.
In the RANS simulation, we employ the Wilcox k−ωRANS model44.y+for the first
wall cell centers is 0.6. Then the cell size is increased by 4% toward the center of the channel
where ∆y+≈8. The total mesh in the ydirection has 96 cells. The extent of the domain
is 2m. One cell is used in the xand zdirection with homogeneous Neumann boundary
13
conditions. The kinematic viscosity is set to νphys = 1/180m2/s. A constant volume force of
1N/m3is applied in the streamwise direction.
Following the STG-LBM algorithm, the RANS results will be set up as the synthetic
turbulent generator inflow for the LES-LBM computational domain. The computational
domain for the physical system is Lx×Ly×Lz, where Lx= 19.2m, Ly= 2m and Lz= 1.6m.
Let’s set the height of the domain equal to 2δ, where δ= 1m. The red dotted line in
Fig. 3 represents the inlet STG boundary condition for the LBM. The green dotted line
represents the regularized outlet pressure-free boundary condition. The no-slip bounce-back
scheme is employed at the top and bottom walls in the ydirection. The periodic boundary
condition is set for the spanwise direction. First, RANS simulation part will be computed
with the friction Reynolds number Reτ= 180. The bulk velocity at the RANS/LES-LBM
interface is Ub= 15.37m/s in Eq. (19). Also, the maximum velocity at the interface from the
RANS simulation is Umax = 17.88m/s. For the LES-LBM part, to ensure the wall-resolved
turbulent channel flow (y+= 1), we need at least 180 lattice cells in y direction. Therefore,
the conversion factor of cxis set to cx= 1/90m. In this work, we use the uniform grid for
the lattice Boltzmann simulation, thus the computational domain is set to 1728×180 ×144
lattice Boltzmann units (LBU). The Mach number of the simulation can be defined as
Ma=uLB
cs
,(33)
where uLB is the maximum mean flow in the LBM simulation. To ensure the stability of
the BGK collision scheme of the LBM, the Mach number should be set to smaller than
0.1. We choose to set the BGK relaxation time equal to τ= 0.5025 with the Smagorinsky
constant equal to C= 0.01. From Eq. (6), we can obtain the kinematic viscosity in the
lattice Boltzmann simulation ν. Combining Eq. (31), the time conversion factor can be
rewrote as ct=ν/νphysc2
xwhich is ct= 1.85 ×10−5s. Therefore, we can calculate uLB from
Umax by uLB =Umaxct/cx. With help of Eq. (33), we obtain the Mach number which is equal
to Ma= 0.0516. The dark region in Fig. 3 is the sponge zone of the simulation, which is set
to 0.4δ. The sponge zone is set up as a damping region to reduce the velocity fluctuations
near the outlet38,49,50. In the sponge zone, we impose a higher kinematic viscosity, νsponge ,
as
νsponge =νeff Kx−xstart
xend −xstart p
+ 1,(34)
where Kand pare empirical constants, νeff is the effective viscosity calculated based on the
14
Velocity magnitude (LBU)
0.005 0.01 0.015 0.02 0.025 0.0330.0004
x
y
x
z
y
STG
RANS LES-LBM
x/δ = 2
x/δ = 4
19.2δ
2δ
1.6δ
x/δ = 8
x/δ = 16
(a)
(b)
FIG. 4. Instantaneous velocity magnitude. (a): the sketch of xy plane. (b): Changing cross-
sections along the streamwise direction for the LES-LBM simulation.
Smagorinsky LES lattice Boltzmann model, xstart and xend are the start point and end point
of the sponge zone, respectively. The empirical constants are set to K= 1000 and p= 3 to
ensure the stability of the simulation.
V. RESULTS
This section presents the results of the channel flow simulation using the synthetic turbu-
lent generator at the RANS/LES-LBM interface as inlet boundary condition. The dimen-
sionless time unit for the channel flow simulation can be defined as the time that the bulk
15
fluid passes the boundary layer thickness, which can be written as
t∗=δ
Ubulk
,(35)
where Ubulk is the bulk fluid velocity and δis the half-channel height. The shear Reynolds
number can be defined as
Reτ=uτδ
ν,(36)
where uτis the friction velocity and νis the kinematic viscosity. The friction velocity is
defined as
uτ=sν∂ustream
∂y ,(37)
where ustream is the ensemble average streamwise velocity at the first layer near the wall. The
differential is calculated with the first-order approximation. Fig. 4 shows the instantaneous
velocity fields in the xy and yz planes The velocity profiles in Fig. 4(b) are located at
different cross-sections x/δ = 2,4,8,16. Qualitatively speaking, the flow downstream of
the inlet quickly develops into turbulent flow downstream the inlet. The flow becomes less
turbulent near the outlet due to the influence of the sponge layer. In the following part, we
only focus on the LES-LBM part of the channel flow.
A. Comparison with periodic boundary condition
The initial conditions for the STG-LBM inlet-outlet simulations are u = v = w = 0,
where u, v, w are representing velocity component in streamwise, spanwise and the vertical
direction of the channel. For comparison, we set up a periodic channel flow as a reference
case (Periodic LES-LBM) by replacing the inlet/outlet boundary condition with the periodic
boundary condition with the same Ly= 2δand Lz= 1.6δ. For the streamwise direction, we
set Lx= 6.4σ. The computational domain of the periodic LES-LBM is set to 576 ×180×144
LBU with the uniform grid. Also, we keep the same parameters as the STG-LBM case for
the Smagorinsky constant C, the relaxation time τ, as well as conversion factors cxand ct.
We apply a volume force that is equal to F= 3.086 ×10−8LBU. For the periodic case,
the turbulence is triggered by placing a rectangular brick in the channel. After around 50
domain flow-through time, we removed the brick. We ran for another 130 domain flow-
through time (832t∗) to fully develop turbulent flow. Fig. 5 shows a qualitative comparison
16
x
y
0.0004 0.005 0.01 0.015 0.02 0.025 0.033
LES-LBM
Periodic
STG-LBM
-0.006 -0.002 0.01 0 0.002 0.0054 -0.0073 -0.004 -0.002 0 0.002 0.004 0.007
u(LBU) v(LBU) w(LBU)
FIG. 5. Instantaneous 2D fields of u, v, w at the section of fully developed turbulent region. The
top panel represents the streamwise periodic case for the LES-LBM simulation results. The bottom
panel represents the STG-generated LES-LBM results at the cross-section of x= 9.6δ.
between the streamwise periodic channel flow and the STG-LBM channel flow with inlet and
outlet. The snapshot for STG is taken at x= 8δat t= 58t∗from the abovementioned initial
condition. Even if the STG method starts from a fairly poor initial condition (velocity field
set to zero for the whole domain), the turbulence is well developed within a considerable
short time. In contrast, for the periodic case, the time needing to develop the turbulent flow
is highly dependent on the initial condition. Also, the periodic turbulent flow is limited to
fundamental studies due to the ”recycling” outflow.
Next, we will check the STG-LBM the adaptation length and the adaptation time, i.e.
when the flow is fully developed in space and time. First, we will check the adaptation
length by comparing the STG-LBM channel flow results with DNS, periodic LES-LBM and
17
synthetic eddy method in LBM38.
B. The adaptation length
In this subsection, all statistics for the STG-LBM channel flow simulations will be col-
lected after around 2T, where Trepresents the time for the bulk velocity passing domain,
which is T= 19.2t∗. Then, all statistics will be gathered for another 8T. Fig. 6 shows u+
as function of y+at different streamwise positions, where
u+=hui
uτ
, y+=ypτw/ρ
ν,(38)
where τwis the wall shear stress. In Fig. 6, mean velocity profiles at different cross-section at
the streamwise direction have been compared with the DNS data by Moser et al.51 and the
periodic LES-LBM results. For x/δ < 2, the STG-LBM captures the velocity near the wall
well but gives slightly too small values for y+>10. However, for x/δ ≥2 the STG-LBM
predicts a mean velocity field which agrees well with DNS data for all y+.
Figure 7 presents the u+
rms, v+
rms, w+
rms as function of y+for DNS, periodic LBM and STG-
LBM results. u+
rms, v+
rms, and w+
rms are the dimensionless root mean square(RMS) for three
velocity components that are normalized with the shear velocity uτ:
u+
rms =rPtend
i=tstart (ui−hui)2∆t
Ptend
i=tstart ∆t
uτ
,(39)
where h·i denotes the velocity field averaged in time. Similar to Fig. 6, STG-LBM failed
to predict the normalized RMS results for the cross section at x/δ = 0 and x/δ = 1. This
is reasonable since the inflow STG needs some space to develop into fully turbulent flow.
For x/δ ≥2, the STG-LBM LES simulations agree well with DNS data and the periodic
LES-LBM reference.
Figure 8 presents the normalized shear velocity uτ/utheory as function of x/δ along the
channel. utheory is obtained from a force balance between the driving volume force (pressure
gradient) and the wall shear stresses. The present STG-LBM results have been compared
with the SEM-LBM results38. The orange shadow is the trust region (TR) for the normalized
shear velocity. It has an error bar of ±3%. The green region is the sponge zone before the
outlet boundary condition. For the STG-LBM, the channel flow simulation ends at x/δ =
18
0
5
10
15
20
25
30
u+
x/δ = 0 x/δ = 1 x/δ = 2
1 2
10010 10
y+
1 2
10010 10
y+
1 2
10010 10
y+
0
5
10
15
20
25
30
u+
x/δ = 4 x/δ = 8 x/δ = 16
DNS
LES-LBM periodic
STG-LBM
FIG. 6. u+as function of y+. The black dotted lines are the DNS reference from Moser51. The
blue dots represent the LES-LBM periodic channel flow results.The red hollow round dots are the
present STG-LBM LES results.
19.2 and for the SEM-LBM simulation, the channel flow simulation ends at x/δ = 20.0.
For the STG-LBM case, the normalized shear velocity has a peak near the inlet and enters
the TR around x/δ = 1.7. The results in Fig. 8, Fig. 6 and Fig. 7 show that the STG-
LBM has an adaptation length of around x/δ = 2. Near the outlet (for x/δ > 18.0), the
normalized shear decreases due to the influence of the sponge zone. For SEM-LBM case,
the SEM forcing takes place in the region of x/δ = 3.0±0.8, and the normalized shear
velocity becomes trustworthy after x/δ = 7.5. As reported in38, the SEM-LBM is able to
fully converge to the DNS data of the mean velocity profile and the Reynolds shear stress
at x/δ = 18. Next, we are interested to exam the adaptation time, i.e. how long we can
get the high-quality turbulent results. Notice that we were not able to find the information
about the adaptation time in the lattice Boltzmann related turbulence generation method.
However, it is crucial to show this quantity to potential readers who are interested to check
how long it takes to generate the turbulent flows. According to the authors experience, it is
non-trivial to achieve high-quality turbulent flow within short simulation time.
19
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
x / δ= 0 x/ δ= 1 x/ δ= 2
0 20 40 60 80 100
y+
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
x/ δ= 4
0 20 40 60 80 100
y+
x/ δ= 8
0 20 40 60 80 100
y+
x/ δ= 16
u+,v +,w +
rms rms rms
u+,v +,w +
rms rms rms
DNS
DNS
DNS
STG-LBM
STG-LBM
STG-LBM
LES-LBM periodic
LES-LBM periodic
LES-LBM periodic
FIG. 7. u+
rms, v+
rms,w+
rms as function of y+at different cross-sections along the streamwise direction.
The dotted lines are the DNS reference from Moser51. The round dots are the present STG-LBM
LES results. The blue, red and green colors are representing u+
rms, w+
rms, v+
rms respectively.
C. The adaptation time
This subsection focuses on the time convergence study for turbulent channel flow using
the proposed STG-LBM LES method. Figure 9 shows the normalized ensemble average
velocity at the cross-section x/δ = 8 with y+= 100 as function of normalized time unit t/T .
The ensemble average starts from 1Twith interval of 1T. For instance, first point at 2T
in Fig. 9 represents the ensemble average velocity from 1Tto 2T. We can observe that the
ensemble average velocity converges to 1 within 1% of the difference to the mean velocity,
hUi, at y+= 100. A distinct advantage is that the present work shows a good convergence
rate even at 1T−2T, demonstrating the good potential for a speedy converging method.
We further check the adaptation time for both mean velocity profile and velocities RMS to
with quantitative analysis by comparing with DNS data.
According to Fig. 6, STG-LBM results match DNS data well for x/δ ≥2. In Fig. 10,
20
0 2 4 6 8 10 12 14 16 18 20
x/δ
0.7
0.8
0.9
1.0
1.1
1.2
u/utheory
SEM-LBM reference Present STG-LBM Theory
FIG. 8. Normalized uτas function of x/δ, the green region represents the sponge zone. The black
dotted line is the theoretical data from Reτ= 180. The yellow line is the STG-LBM results. The
grey line is the SEM-LBM reference38.
t/T
0 2 4 6 8 10 12 14
Normalized ensemble average u
Trend
1.000
0.995
0.990
1.005
FIG. 9. Normalized ensemble average velocity, hu(t/T )i/hUi, as function of t/T at the cross-
section of x/δ = 8 with y+= 100. hUidenotes the mean velocity at y+= 100. The statistics start
from 1T.
we present STG-LBM ensemble time and space average of u+at cross-sections along the
streamwise direction from x/δ = 2 to x/δ = 16 at various time intervals, namely, from
1Tto 2T, from 1.5Tto 3T, from 3Tto 6T. At t= 2T, the STG-LBM shows acceptable
discrepancies at x/δ = 8 and x/δ = 16 with DNS data. However, for t= 3Tand t= 6T
cases, both results almost overlap each other and show good agreement with the DNS data.
Thus, we need to further check for the velocity RMS of these cases.
Figure 11 shows the RMS of the three velocity components u+
rms, v+
rms, w+
rms as function
of y+at cross section of x/δ = 4 and x/δ = 8. As can be seen, for t= 3Tand t= 6T, all
21
0
5
10
15
20
25
30
u+
y+
0
5
10
15
20
25
30
u+
0120 1
10 10 10 10 10 102
y+
STG-LBM, from 1T to 2T
STG-LBM, from 1.5T to 3TDNS
STG-LBM, from 3T to 6T
x/δ = 16
x/δ = 8
x/δ= 2 x/δ = 4
FIG. 10. u+as function of y+at changing x/δ and at different simulation time. Ensemble time
and space average of u+at different cross-sections along the streamwise direction from x/δ = 2 to
x/δ = 16. The black dotted lines are the DNS reference from Moser51. The yellow triangles, the
blue round dots, and the red squares are the ensemble time average of u+from 1Tto 2T, from
1.5Tto 3T, and from 3Tto 6T, respectively.
stresses agree reasonably well with DNS data both at x/δ = 4 and x/δ = 8. For t= 2T
case, larger discrepancies have been observed for v+
rms and w+
rms cases.
The quantitative results illustrate that the STG-LBM framework presents a surprisingly
short adaptation time with around t= 1.5Tto t= 3Twhich is 28.8t∗to 57.6t∗.
D. Auto-correlation
Figure 12 shows auto-correlation, Buu, for the streamwise velocity as function of nor-
malized time separation ˆ
t/t∗, here ˆ
t=t−3T. We checked cells at different cross-sections
x/δ = 0,1,2,4,8,16 at y+= 100. The statistics starts to collect after t= 3T. For all cases,
the auto-correlation function drops quickly after the statistics start to be collected. Then,
they reach a minimum at around ˆ
t/t∗= 14 to ˆ
t/t∗= 16. After that, all curves oscillate
22
0 20 40 60 80 100
y+0 20 40 60 80 100
y+0 20 40 60 80 100
y+
STG-LBM, from 1T to 2T
STG-LBM, from 1.5T to 3TDNS
STG-LBM, from 3T to 6T
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
x/δ = 4 x/δ = 4 x/δ = 4
x/δ = 8 x/δ = 8 x/δ = 8
urms
+
,vrms
+
,w
rms
+
,
urms
+
,vrms
+
,w
rms
+
,
FIG. 11. u+
rms, v+
rms,w+
rms as function of y+at different simulation time T. The dotted lines are
the DNS reference from Moser51. The round dots are the present STG-LBM LES results. The
left, middle and right panels show results for u+
rms, v+
rms, and w+
rms. The yellow triangles, the blue
round dots, and the red squares are the ensemble time average of RMS velocties from 1Tto 2T,
from 1.5Tto 3T, and from 3Tto 6T, respectively.
around 0. Similar effects have also been observed in the literature52. From Fig. 12, we can
calculate the integral time length, Tt, as
Tt=Ztmin
0
Buu(ˆ
t)dˆ
t, (40)
where ˆ
tmin is the time separation when Buu reaches zero. Table I shows the integral time
length. Ttdecreases when x/δ reaches 8 and increases to 4.91 at x/δ = 16.An increase at
the last xstation is probably due to the sponge zone. Hence, we find that the integral time
scale increases for increasing xwhich indicates that we impose too large a time scale on the
STG fluctuations.
23
TABLE I. The integral time length at different cross-sections at y+= 100.
Cross section Tt(ˆ
t/t∗)
x/δ = 0 6.08ˆ
t/t∗
x/δ = 1 4.53ˆ
t/t∗
x/δ = 2 3.87ˆ
t/t∗
x/δ = 4 3.65ˆ
t/t∗
x/δ = 8 2.67ˆ
t/t∗
x/δ = 16 4.91ˆ
t/t∗
0 10 20 30 40 50
Buu
0.2
0.4
0.6
0.8
1.0
0.0
-0.2
-0.4
STG-LBM x/ = 0
STG-LBM x/ = 1
STG-LBM x/ = 2
STG-LBM x/ = 4
STG-LBM x/ = 8
STG-LBM x/ = 16
FIG. 12. The auto-correlation for the streamwise velocity, Buu, as function of the normalized time
separation ˆ
t/t∗at different cross-sections x/δ = 0,1,2,4,8,16 with y+= 100.
VI. CONCLUSIONS
This paper proposes a methodology to integrate the synthetic turbulent generator (STG)
into the lattice Boltzmann framework. The proposed method can be used to couple the
hybrid RANS/LES-LBM model with the STG interface. The present work uses the channel
flow at Reτ= 180 as a showcase, the RANS part is handled by a finite volume solver and
the LES part is calculated via the LBM framework. At the RANS/LES-LBM interface, the
24
STG serves as the inlet for the LBM channel flow. We qualitatively compare the STG-LBM
channel flow results with those of a periodic LES-LBM case. We find good agreement on the
turbulent structure. Compared with the SEM-LBM, periodic LES-LBM, and DNS channel
flow data, the STG-LBM framework reaches close-to-DNS turbulent flow already at x/δ = 2.
However, for the SEM-LBM case, it takes x/δ = 15 to get similar results. Moreover, we
check the time convergence speed by comparing DNS data and recording the mean velocity
and the velocity RMS at different time. Results show that the STG-LBM can reach a fully
developed turbulent state around t= 1.5Tto t= 3T. The STG-LBM shows delightful
potential in fluid/aero-dynamics-related applications, which requires fast convergence speed
for fully developed turbulent conditions.
VII. ACKNOWLEDGEMENTS
The authors kindly acknowledge the funding from Chalmers Transport Area of Advance.
The computations and data handling were enabled by resources provided by the Swedish
National Infrastructure for Computing (SNIC), partially funded by the Swedish Research
Council through grant agreement no. 2018-05973.
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