Improved treatment of the open
boundary in the method of lattice
Boltzmann equation
Dazhi Yu, Renwei Mei and Wei Shyy
Department of Mechanical and Aerospace Engineering, University of
Florida, Gainesville, FL 32611-6250 USA
Abstract: The method of lattice Boltzmann equation (LBE) is a kineticbased approach for fluid flow computations. In LBE, the distribution functions on various boundaries are often derived approximately. In this paper,
the pressure interaction between an inlet boundary and the interior of the
flow field is analysed when the bounce-back condition is specified at the
inlet. It is shown that this treatment reflects most of the pressure waves
back into the flow field and results in a poor convergence towards the steady
state or a noisy flow field. An improved open boundary condition is developed to reduce the inlet interaction. Test results show that the new
treatment greatly reduces the interaction and improves the computational
stability and the quality of the flow field.
Keywords: lattice Boltzmann equation; boundary condition; inlet;
bounce-back.
Reference to this paper should be made as follows: Yu, D., Mei, R. and
Shyy, W. (xxxx) ‘Improved treatment of the open boundary in the method
of lattice Boltzmann equation’, Progress in Computational Fluid Design,
Vol. x, No. x, pp.xxx–xxx.
Biographical notes: Dazhi Yu received his PhD in Aerospace from
University of Florida. He is now a Postdoctoral Research Associate in
Aerospace Department at Texas A&M University. His research interests
include lattice Boltzmann method, parallel computation, and turbulence.
Renwei Mei is Professor of the Department of Mechanical and Aerospace
Engineering at University of Florida. His research areas include particle
dispersion and collision in turbulence, boiling heat transfer and two-phase
flow, lattice Boltzmann method, and desalination. He is interested in applying computational methods and applied mathematics to solve practical
thermal fluid problem.
Wei Shyy is currently Professor and Chairman of the Department of Mechanical and Aerospace Engineering at University of Florida (UF). He is
the author or a coauthor of three books dealing with computational and
modelling techniques involving fluid flow, interfacial dynamics, and moving
boundary problems. He has written papers dealing with computational and
modelling issues related to fluid dynamics, heat/mass transfer, combustion,
material processing, parallel computing, biological flight and aerodynamics,
1
design optimisation, micro-scale and biofluid dynamics.
1 INTRODUCTION
1.1
General description of the method
Since the last two decades or so, there has been a
rapid progress in developing the method of the lattice
Boltzmann equation (LBE) as an alternative, computational technique for solving a variety of complex fluid dynamic systems [1–16]. Adopting the
macroscopic method for computational fluid dynamics (CFD), macroscopic variables of interest, such as
velocity u and pressure p, are usually obtained by
solving the Navier-Stokes (NS) equations (e.g., see
[17,18]). In the LBE approach, one solves the kinetic
equation for the particle velocity distribution function f (x, ξ, t), where ξ is the particle velocity vector,
x is the spatial position vector, and t is the time. The
macroscopic quantities (such as mass density ρ and
momentum density ρu) can then be obtained by evaluating the hydrodynamic moments of the distribution function f . Lattice gas automata, the predecessor of LBE, were first proposed by Frisch et al. [19].
The theoretical foundation of LBE was established
in the subsequent papers, notably in McNamara and
Zanetti [20], Higuera and Jimenez [21], Koelman [22]
and Qian et al. [23].
A popular kinetic model adopted in the literature
is the single relaxation time approximation, the socalled BGK model [24]:
∂f
1
+ ξ · ∇f = − (f − f (eq) )
∂t
λ
Figure 1: A 2-D, 9-velocity lattice (Q9D2) model
librium distribution function. The 9-bit square lattice model, which is often referred to as the D2Q9
model (Figure 1), has been successfully used for simulating 2-D flows. In the D2Q9 model, we use eα to
denote the discrete velocity set and we have
e0 = 0,
eα = c(cos((a − 1)p/4), sin((a − 1)p/4))
forα = 1, 3, 5, 7,
√
eα = 2c(cos((a − 1)p/4), sin((a − 1)p/4))
forα = 2, 4, 6, 8
(3)
(1)
where c = δx/δt, δx and δt are the lattice constant
and the time step size, respectively. The equilibrium
distribution for D2Q9 model is of the following form
[23]
3
9
3
(eq)
2
fa = wa ρ + 2 eα · u + 4 (eα · u) − 2 u · u
c
2c
2c
(4)
where f (eq) is the equilibrium distribution function
(the Maxwell-Boltzmann distribution function), and
λ is the relaxation time.
To solve for f numerically, equation (1) is first discretised in the velocity space using a finite set of velocities {ξα } without affecting the conservation laws
[15, 24,25],
where wα is the weighting factor given by

(2)
 4/9, α = 0
1/9, α = 1, 3, 5, 7
wα =
(5)

In the above equation, fα (x, t) ≡ f (x, ξα , t) is the
1/36, α = 2, 4, 6, 8.
distribution function associated with the α-th dis(eq)
crete velocity ξα and fα is the corresponding equi- With the discretised velocity space, the density and
1
∂fa
+ ξa · ∇fa = − (fa − fa(eq) )
∂t
λ
2
momentum fluxes can be evaluated as
the LBE method. Two classes of boundaries are often
encountered in computational fluid dynamics: open
8
8
(eq)
fα =
(6) boundaries and the solid wall. The open boundaries
fα
ρ=
typically include lines (or planes) of symmetry, perik=0
k=0
odic cross-sections, infinity, and inlet and outlet. At
and
these boundaries, velocity or pressure is usually specified in the macroscopic description of fluid flow. Un8
8
eα fα =
(7) like solving the NS equations where the macroscopic
eα fα(eq)
ρu =
variables, their derivatives, or a well established conk=1
k=1
straint (such as the mass continuity for pressure dis√
The speed of sound in this model is cs = c/ 3 and tributions) can often be explicitly specified at boundthe equation of state is that of an ideal gas,
ary [28], in the LBE method, these conditions need
to be converted into distribution function f .
2
(8)
p = ρcs
Equation (2) can be further discretised in space and
time. The completely discretised form of equation 2 COMPUTATIONAL ASSESSMENT
(1), with the time step δt and space step eα δt, is:
In the computational cases to be presented and disfα (xi + eα δt, t + δt) − fα (xi , t)
cussed, the following questions are addressed:
1
(9) • Will the interpolation-based superposition
= − [fα (xi , t) − fα(eq) (xi , t)]
τ
scheme give the same accuracy as those of
where τ = λ/δt, and xi is a point in the discretised
bounce back if both conditions lead to converged
physical space. The above equation is the discrete
results?
lattice Boltzmann equation [20,21,26] with BGK approximation [24]. Equation (9) is usually solved in • How does the inlet/free-stream boundary condition impact the convergence of the solution and
the following two steps where ∼ represents the postthe quality of the flow field?
collision state.
The viscosity in the NS equation derived from
At the symmetric and periodic boundaries, the
equation (9) is
conditions for fα s can be given exactly. At the outν = (τ − 1/2)c2s δt
(10) let of a computational domain, fα s can usually be
approximated by simple extrapolation. In general,
This modification of the viscosity formally makes the the boundary condition for fα s on a solid wall with
LBGK scheme a second-order method for solving in- a known velocity can only be given approximately.
compressible flows [25,27]. The positivity of the vis- A popular approach is to employ the bounce-back
scheme [29–31]. It is intuitively derived from lattice
cosity requires that τ > 1/2.
It is noted that the collision step is completely lo- gas automata and has been extensively applied in latcal and the streaming step takes very little computa- tice Boltzamnn equation simulations. The same idea
tional effort. Equation (9) is explicit, easy to imple- for the solid wall has been directly extended to the
inlet boundary with known velocity to provide the
ment, and straightforward to parallelise.
required inlet condition for fα s [31–34].
Let us consider a horizontal flow at an inlet (Figure
1.2 Issues in the boundary condition
2). The first vertical grid line is at xA . The inlet is
treatment
arbitrarily xI which is between lattice node A at xA
Like in any other fluid flow computations, the numer- and located at node B at xB . The node B at xB is the
ical boundary condition is a very important issue in first fluid node next to the inlet. Using the bounce
3
based inlet condition results in slower convergence
towards steady or dynamically steady state and
causes stronger interaction between the inlet and
interior field. Such interaction may affect the longterm behaviour of the solution, especially at higher
Re. The interpolation-based superposition scheme
can reduce the impact of the boundary-interior
interaction on the unsteady development of the flow
field, improves the convergence rate, improves the
computational stability, and improves the quality of
the overall solution.
ACKNOWLEDGEMENTS
Figure 2: Schematic of lattice nodes at inlet
The work reported in this paper has been partially
supported by NASA Langley Research Center, with
David Rudy as the project monitor. The authors
thank Dr. Li-Shi Luo for many helpful discussions.
The work reported in this paper has been partially
supported by NASA Langley Research Center, with
David Rudy as the project monitor. The authors
thank Dr. Li-Shi Luo for many helpful discussions.
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xA + 12 e1 dt with Ω = 0.5, where Ω is the fraction of a
link in the fluid region intersected by inlet boundary,
Ω=
|xB − xI |
0 ≤Ω≤1
|xB − xA |
(11)
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