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A new modification of the immersed boundaries method for fluid-solid flows: moderate Reynolds numbers.

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Short Note
A new modiﬁcation of the immersed boundaries methodfor ﬂuid–solid ﬂows: moderate Reynolds numbers
A. Vikhansky
*
Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK
Received 22 October 2002; received in revised form 4 June 2003; accepted 11 June 2003
Abstract
In this study we consider a combination of an interpolation technique with the D
Õ
Alambert principle that allows thedirect numerical simulation of ﬂuid-solid ﬂows on a regular rectangular grid. The method models a no-slip boundarywith the second-order accuracy interpolation scheme without explicit calculation of the forces that the particles exert atthe ﬂuid. The method is applicable for a wide range of Reynolds numbers from creeping ﬂows up to
Re
%
100
:
Ó
2003 Elsevier Science B.V. All rights reserved.
1. Introduction
In this study we consider a method that allows the direct numerical simulation of ﬂuid–solid ﬂows on aregular rectangular grid without explicit calculation of the forces that the particles exert at the ﬂuid. The no-slip boundary conditions at the solid–ﬂuid interface are satisﬁed automatically. The combination of aninterpolation technique with the D
Õ
Alambert principle allows an eﬃcient numerical realization of thesuggested numerical method.Direct numerical simulation of solid–liquid ﬂow is a diﬃcult task since the domain occupied by the ﬂuid isirregular and changes with motion of the particles. Also, the particles are advected by the ﬂuid and exertforces at the ﬂuid, so the body–liquid interaction requires calculation of the ﬂuid stress (i.e., derivatives of theﬂowﬁeld) at the ﬂuid–solid interface. Such an approach was realized in the arbitrary Lagrangian–Eulerian(ALE) technique (see e.g. [1] and references therein), which is based on automatic generation of an un-structured ﬁnite element body-ﬁtted mesh at each time step of the numerical algorithm. Since ﬂow simu-lations on an unstructured mesh are still a challenging problem for today
Õ
s computers, methods that avoidgeneration of a body-ﬁtted mesh and do not require explicit calculation of a stress tensor are preferred.During the last decade the lattice Boltzmann method has become a useful tool for studying dynamicsof suspensions at low and moderate Reynolds numbers (a comprehensive review is in [2]). Anothernumerical technique that allows simulation of arbitrary Reynolds number ﬂows in complex geometries on
Journal of Computational Physics 191 (2003) 328–339www.elsevier.com/locate/jcp
*
Tel.: +44-1223-334786.
E-mail address:
av277@cam.ac.uk(A. Vikhansky).0021-9991/$ - see front matter
Ó
2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0021-9991(03)00313-9
non-body-ﬁtted grids is the immersed boundaries method. It was demonstrated recently that this methodhas lower RAM and CPU-time requirements than more complicated boundary-ﬁtted solvers [3].The general idea of the immersed boundaries method [4] is to solve the equations for liquid velocity bothinside and outside the moving boundaries, while the ﬂuid and particle velocities are matched by forcesdistributed along the boundary (in the most recent variants of this method the forces are distributed overthe entire region occupied by the particle). In the srcinal paper by Peskin [4] (see also [5]) the interactionbetween a ﬂuid and immersed solids is accounted for by a penalty term that makes the equations stiﬀ andimposes restrictions on the time step. Recently, similar forcing was performed using a duality method.Distributed-Lagrange-multiplier (DLM) method [6,7] was developed and extensively applied to diﬀerentparticulate ﬂows. In this method the variational equations for a ﬂuid ﬂow are solved on a regular grid, andthe constraints of rigid-body motion of a ﬂuid inside the particles are enforced by distributed Lagrangemultipliers.The similarity between the above two methods is due to the similarity between penalty and dualitymethods. One can minimize a function
f
ð
~
x x
) under a constraint
g
ð
~
x x
Þ ¼
0 either by penalty or dualitymethods. Penalty method yields the following minimization problem:
f
ð
~
x x
Þ þ
r
ð
g
ð
~
x x
ÞÞ
2
!
min and
r
f
ð
~
x x
Þ þ
2
rg
ð
~
x x
Þr
g
ð
~
x x
Þ ¼
0 in the location of the minimum, while the duality technique leads to a modiﬁedminimization problem
f
ð
~
x x
Þ þ
k
g
ð
~
x x
Þ !
min and
r
f
ð
~
x x
Þ þ
k
r
g
ð
~
x x
Þ ¼
0. Comparison of the above formulasshows that 2
rg
ð
~
x x
Þ !
k
as penalty parameter
r
goes to inﬁnity, i.e., the penalty method and the method of Lagrange multipliers are diﬀerent realization of the same forcing scheme.In the present study we suggest a variant of the immersed boundary method that does not require anyexplicit forcing either by the penalty method or by calculation of Lagrange multipliers. The key idea of theproposed method is a combination of the interpolation technique considered in [3,8] with a variationalprinciple. Owing to the special treatment of near-boundary points, the constraints of rigid-body motion of ﬂuid inside the region occupied by the particles are satisﬁed automatically, while the force interactionbetween the ﬂuid and the immersed particles is captured implicitly by the principle of virtual work.Therefore, the problem is reduced to a sequence of unconstrained minimization problems, while the DLMmethod treats particle–ﬂuid interaction via more computationally expensive constrained variationalequations.
2. Variational formulation of the problem and interpolation scheme
Consider for simplicity a single disc
C
with radius
R
immersed into a space domain
X
ﬁlled with anincompressible viscous ﬂuid with density
q
f
and viscosity
l
(Fig. 1). The disc is made of homogeneousmaterial with density
q
s
, the mass of the disc is
M
¼
q
s
p
R
2
, and its moment of inertia
I
¼
1
=
2
q
s
p
R
4
. The
Fig. 1. Schematic view of a two-dimensional cavity with one rigid disc.
A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339
329
center of the disc, its axial and angular velocities are
~
x x
c
ð
t
Þ
,
~
U U
, and
~
xx
, respectively (angular velocity is or-thogonal to the ﬂow). The boundary of the disc is
o
C
ð
t
Þ
. The acceleration of the external body force actingat the system is
~
g g
. The starting point of our numerical method is the virtual power principle [6]:
Z
X
n
C
q
f
o
~
uu
o
t
"
þ
~
uu
Á r
~
uu
À
~
g g
Á
d
~
uu
þ
l
r
~
uu
:
r
d
~
uu
À
p
r Á
d
~
uu
#
d
X
þ
M
d
~
U U
d
t
À
~
g g
Á
d
~
U U
þ
I
d
~
xx
d
t
Á
d
~
xx
¼
0
:
ð
1
Þr Á
~
uu
¼
0
;
r Á
d
~
uu
¼
0
;~
uu
j
C
¼
~
U U
þ
~
xx
Â ð
~
x x
À
~
x x
c
Þ
;
d
~
uu
j
C
¼
d
~
U U
þ
d
~
xx
Â ð
~
x x
À
~
x x
c
Þ
;
ð
2
Þ
where
d
denotes variation of the corresponding variable and
~
uu
ð
t
;~
x x
Þ
and
p
ð
t
;~
x x
Þ
are the velocity and pressureof the ﬂuid, respectively. The ﬁrst term in Eq. (1) describes inertia of the ﬂuid, the second term accounts forviscous dissipation, and the third is virtual work due to compression/expansion of the ﬂuid. The last twoterms describe linear and angular inertia of the disc.The velocity ﬁeld inside the disc is
~
uu
¼
~
U U
þ
~
xx
Â ð
~
x x
À
~
x x
c
Þ
, thus, the disc–inertia term in Eq. (1) can be splitas
M
d
~
U U
d
t
À
~
g g
Á
d
~
U U
þ
I
d
~
xx
d
t
Á
d
~
xx
¼
1
À
q
f
q
s
M
d
~
U U
d
t
"
À
~
g g
Á
d
~
U U
þ
I
d
~
xx
d
t
Á
d
~
xx
#
þ
q
f
q
s
Z
X
q
s
o
~
uu
o
t
þ
~
uu
Á r
~
uu
À
~
g g
Á
d
~
uu
d
X
:
Since the ﬂow inside the disc is incompressible and does not dissipate energy, insertion of the above formulainto Eq. (1) yields:
Z
X
q
f
o
~
uu
o
t
"
þ
~
uu
Á r
~
uu
À
~
g g
Á
d
~
uu
þ
l
r
~
uu
:
r
d
~
uu
À
p
r Á
d
~
uu
#
d
X
þ
1
À
q
f
q
s
M
d
~
U U
d
t
"
À
~
g g
Á
d
~
U U
þ
I
d
~
xx
d
t
Á
d
~
xx
#
¼
0
:
ð
1
0
Þ
Eq. (1
0
) after applying Gauss theorem implies Navier–Stokes equations
q
f
o
~
uu
o
t
(
þ
~
uu
Á r
~
uu
À
~
g g
þ r
p
À
l
D
~
uu
)
Á
d
~
uu
¼
0
;
for
~
x x
2
X
n
C
;
force balance over the disc
Z
C
q
f
o
~
uu
o
t
"
þ
~
uu
Á r
~
uu
À
~
g g
Á
d
~
U U
#
d
C
þ
1
À
q
f
q
s
M
d
~
U U
d
t
"
À
~
g g
Á
d
~
U U
#
þ
Z
o
C
l
o
~
uu
o
n
À
p
~
nn
Á
d
~
U U
d
o
C
ð Þ ¼
M
d
~
U U
d
t
À
~
g g
Á
d
~
U U
þ
Z
o
C
l
o
~
uu
o
n
À
p
~
nn
Á
d
~
U U
d
o
C
ð Þ ¼
0
;
330
A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339
and moment balance over the disc
Z
C
q
f
o
~
uu
o
t
"
þ
~
uu
Á r
~
uu
Á
d
~
xx
Â
~
x x
À
~
x x
c
#
d
C
þ
1
À
q
f
q
s
I
d
~
xx
d
t
Á
d
~
xx
"
Â
~
x x
À
~
x x
c
#
þ
Z
o
C
l
o
~
uu
o
n
Á
d
~
xx
Â
~
x x
À
~
x x
c
d
o
C
ð Þ ¼
I
d
~
xx
d
t
Á
d
~
xx
Â
~
x x
À
~
x x
c
þ
Z
o
C
l
o
~
uu
o
n
Á
d
~
xx
Â
~
x x
À
~
x x
c
d
o
C
ð Þ ¼
0
:
In the above formulas we have used the relation that is valid inside the disc
o
~
uu
=
o
t
þ ð
~
uu
Á rÞ
~
uu
¼
_
~
U U
~
U U
þ
_
~
xx
~
xx
Â ð
~
x x
À
~
x x
c
Þ
. Thus, the above variational principle can be applied to simulation of particle ﬂow in-teraction.The suggested numerical procedure is as follows. We calculate the positions of the disc
~
x x
c
at times
t
n
¼
n
D
t
, while the velocities are calculated at semi-integer moments
t
n
¼ ð
n
þ
1
=
2
Þ
D
t
. Note that the pro-posed procedure can be easily implemented with the variable time step. First, an intermediate position iscalculated as
~
x x
ð
t
þ
D
t
=
2
Þ ¼
~
x x t
ð Þ þ
D
t
=
8 7
~
U U
ð
t
h
À
D
t
=
2
Þ À
3
~
U U
ð
t
À
3
D
t
=
2
Þ
i
:
Then, for the determined particle positions, Eqs. (1
0
) and (2) for pressure
p
and velocities
~
uu
;~
U U
;~
xx
are solvedby operator-splitting method [6]. We used ﬁnite-diﬀerence method on a staggered ﬁve-point stencil asfollows. Variational problem (1
0
) and (2) is split into the following three steps:(1) The continuity condition is enforced at the pressure-correction step
~
uu
Ã
¼
~
uu
n
À
D
t
ð Þ r
p
n
þ
1
ÀÁ
:
Substituting the latter equation in the ﬁrst of Eq. (2) yields the Poisson equation for
p
n
þ
1
:
r Á
~
uu
Ã
¼
0
¼ r Á
~
uu
n
À
D
t
ð Þ
D
p
n
þ
1
ÀÁ
:
ð
3
Þ
(2) The advection step:
~
uu
ÃÃ
À
~
uu
Ã
D
t
¼ À
~
uu
Ã
Á r
~
uu
Ã
:
ð
4
Þ
(3) The pseudo-Stokes step that is a solution of a following quadratic minimization problem:
U
ð
~
uu
n
þ
1
;~
U U
n
þ
1
;~
xx
n
þ
1
Þ ¼
Z
X
q
f
2
D
t
ð
~
uu
n
þ
1
&
À
~
uu
ÃÃ
À
D
t
~
g g
Þ
2
þ
l
2
r
~
uu
n
þ
1
2
'
d
X
þ
1
À
q
f
q
s
M
2
D
t
ð
~
U U
n
þ
1
À
~
U U
n
À
D
t
~
g g
Þ
2
þ
I
2
D
t
ð
~
xx
n
þ
1
À
~
xx
n
Þ
2
!
!
min
:
ð
5
Þ
While in the case
q
s
<
q
f
the coeﬃcient near
~
U U
and
~
xx
in Eq. (5) becomes negative, the functional (5) re-mains positively deﬁned with respect to these variables. Note that the terms that account for the inertia of the particle read:
Z
C
q
f
2
D
t
ð
~
uu
n
þ
1
À
~
uu
ÃÃ
Þ
2
d
C
þ
1
À
q
f
q
s
M
2
D
t
ð
~
U U
n
þ
1
À
~
U U
n
Þ
2
þ
I
2
D
t
ð
~
xx
n
þ
1
À
~
xx
n
Þ
2
!
;
A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339
331
where the integration is performed over the region occupied by the particle, and
~
uu
n
þ
1
¼
~
U U
n
þ
1
þ
~
xx
n
þ
1
Â ð
~
x x
À
~
x x
c
Þ
for
~
x x
2
C
.The main idea of the suggested variant of the immersed boundary method that is proposed in this studyis illustrated in Fig. 2. At the grid points that are far from the body the central diﬀerence approximation of dissipation term in Eq. (5) reads:
r
u
j j
2
d
x
d
y
%
h
2
2
ð
u
ij
À
u
i
À
1
j
Þ
2
h
2
þð
u
ij
À
u
i
þ
1
j
Þ
2
h
2
þð
u
ij
À
u
ij
À
1
Þ
2
h
2
þð
u
ij
À
u
ij
þ
1
Þ
2
h
2
¼
12
ð
u
ij
À
u
i
À
1
j
Þ
2
þ ð
u
ij
À
u
i
þ
1
j
Þ
2
þ ð
u
ij
À
u
ij
À
1
Þ
2
þ ð
u
ij
À
u
ij
þ
1
Þ
2
:
Thus, at the points the closest to the boundary, the ﬁrst term in the above expression is replaced by
h
2
g
u
ij
À
U
À
x
ð
x
À
x
c
Þð Þ
ÀÁ
2
g
h
ð Þ
2
¼
u
ij
À
U
À
x
ð
x
À
x
c
Þð Þ
ÀÁ
2
g
:
ð
6
Þ
Here
h
is the step of the grid and
g
is the dimensionless distance from the
i
th node to the surface of the disc.Since Eq. (6) is based on linear interpolation of ﬂow ﬁeld, it requires the mesh to be ﬁne so that the lin-earized velocity is accurate [3]. Thus, the applicability of this method to direct simulation of two phaseturbulent ﬂows should be subject of additional investigations.When
g
¼
1, i.e., the distance from the
i
th node to the surface is equal to the step of the grid, the dis-sipation rate given by Eq. (6) is identical to that obtained by central diﬀerence stencil. The velocities in thenodes that are inside the disc are set equal to the velocity of the body in these points. As one can see, the
Fig. 2. Scheme of the immersed boundary method.332
A. Vikhansky / Journal of Computational Physics 191 (2003) 328–339

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