ICMF 2010, Tampa, FL, May 30 June 4, 2010
Comparison Between Different Immersed Boundary Conditions
for Simulation of Complex Fluid Flows
A. Mark, R. Rundqvist and F. Edelvik
Fraunhofer Chalmers Research Centre for Industrial Mathematics
Chalmers Science Park, SE412 88 G6teborg, Sweden
andreas.mark@fcc.chalmers.se
Keywords: mirroring immersed boundary method, finite volume method, adaptive grid refinement
Abstract
In the literature immersed boundary methods are employed to simulate complex flows around moving arbitrary
bodies without the necessity of remeshing. These methods employ a regular Eulerian mesh to simulate the fluid flow
and a Lagrangian representation of the boundary of the bodies. The two representations can be coupled through
an immersed boundary condition constraining the fluid to exactly follow the boundary of the bodies (immersed
boundaries). Typically such methods suffer from accuracy problems, that arise from spurious mass fluxes over the
immersed boundary (IB), pressure boundary conditions or high density ratios. The mirroring IB method (13; 15)
resolves these problems by ensuring zero mass flux over the IB instead of employing a pressure boundary condition.
In this work the mirroring IB method together with a hybrid IB condition are implemented and validated in IBOFLOW.
IBOFLow (1) is an incompressible finitevolume based fluid flow solver. The NavierStokes' equations are coupled
with the SIMPLEC (2) method and discretized on a Cartesian octree grid that can be dynamically refined and
coarsened, enabling grid refinement to follow moving bodies. The variables are stored in a colocated configuration
and pressure weighted flux interpolation (20) is employed to prevent pressure oscillations. In the implemented IB
method the immersed bodies are represented by an analytical description or by a triangulation. The method models the
presence of the bodies inside the fluid by an implicitly formulated IB condition, which constrains the fluid velocity to
the boundary velocity with secondorder accuracy. The original mirroring IB condition mirrors the velocity field over
the local IB and the hybrid IB condition mirrors and extrapolates the fluid velocity onto the IB. These IB conditions
generate a fictitious velocity field inside the bodies, which is excluded in the continuity equation to ensure zero mass
flux over the boundary.
The fluid flow over an immersed sphere is simulated to validate and compare the different IB conditions. The
simulated drag force is compared to experimental findings with excellent agreement and a detailed convergence study
of the error of the fluid velocity integrated over the immersed boundary is performed to show the strictly secondorder
accuracy of the implemented IB conditions. It is shown that the error is reduced with the hybrid IB condition
compared to the original mirroring IB condition. In addition, a sedimenting sphere with a moving grid refinement is
simulated to validate the hybrid method and show the potential of the dynamic octree grid.
Nomenclature p Fluid dynamic pressure Pa
P Grid point position m
Pf Fluid density kg/m3 r Sphere radius m
ps Body density kg/m3 t Time s
p Dynamic viscosity Ns/m2 U Inlet/Free stream velocity m/s
Cd Drag coefficient ui Fluid velocity m/s
fd Drag force N V Body volume m3
j Gravitation 2 /s v Body velocity n/s
mr Body mass kg x Body position m
Abbreviations
DNS Direct numerical simulations
IB Immersed Boundary
IBC Immersed Boundary Condition
Introduction
The fluid flows around arbitrary, moving and interacting
bodies are both complex and poorly understood. These
flows are also the most common in both nature and
industrial applications. To gain more knowledge of the
complex flow, simulations at small scales are required,
where the flow around the particles is completely
resolved. To study the real phenomena the equations
governing the fluid flow must be solved directly without
introducing any extra models. Such simulations are
called true direct numerical simulations (DNS). The
results obtained from the simulations on a small scale
can be used to gain an understanding of the phenomena
and to develop new and better large scale methods.
In the literature immersed boundary methods are
utilised to simulate complex flows around moving
arbitrary bodies without the necessity of remesh
ing. These methods employ a regular Eulerian mesh
to simulate the fluid flow and a Lagrangian repre
sentation of the boundary of the bodies (immersed
boundary). These two representations can be cou
pled through Lagrange multipliers (3; 4), explicit
forces (6; 7; 8; 5; 10; 11; 16; 17; 18), or by internal
boundary conditions (12; 13; 15).
Glowinski (3) was the first to couple the two repre
sentations with Lagrange multipliers. The method
is implemented in a finite element frame work and
by Lagrangian multipliers an immersed boundary
condition (IBC) is introduced in the weak formulation
of the NavierStokes' equations. The resulting method
is implicitly formulated and secondorder accurate in
space. Overall the method performs well but a finite
volume implementation is not straightforward. In the
work of Sharma and coworkers (21; 22) the method is
implemented in a finite volume framework but the force
remains explicitly coupled.
The original immersed boundary method developed by
Peskin (19) couples the representations with a force.
In this method a discrete Dirac function is used to dis
tribute a Lagrangian force from the immersed boundary
(IB) to the Eulerian grid. The distributed volume force
explicitly constrains the fluid to follow the IB. Due to
the distribution function the resulting method is only
ICMF2010, Tampa, FL, May 30 June 4, 2010
firstorder accurate in space. MohdYusof (16; 17)
developed a momentum forcing method that enforce the
fluid velocity at the IB by introducing an explicit force
in the momentum equations. The force is applied onto
the cells lying inside but close to the IB generating a
reversed velocity field over the IB. Resulting problems
with mass conservation is solved in (6). The explicitly
formulated method is secondorder accurate in space
and commonly used in the literature.
Majumdar and coworkers (12) developed an im
mersed boundary method, which implicitly constrains
the velocity of the fluid at the IB with an immersed
boundary condition (IBC). The method is employed
for stationary bodies along with a turbulent RANS
solver and shows good results. However, the method
has potential problems with the weighting coefficients
in the boundary condition which may result in oscilla
tions in the resulting solution. Moreover, the method
may generate an unphysical mass flux over IB segments.
Mark and van Wachem (14; 15) developed a stationary
immersed boundary method which models the presence
of the bodies by an IBC that mirrors the velocity field
over the boundary of the body in such a way that
the fluid exactly follows the surface of the body. As
a result, a fictitious velocity field inside the body is
generated, which is excluded in the continuity equation
to ensure zero mass flux over the boundary. The method
generates no unphysical oscillations around the IB
and is secondorder accurate in space and implicitly
formulated.
In this work the mirroring IB method (13; 15) together
with a hybrid immersed boundary condition are imple
mented and validated. The hybrid IBC mirrors and ex
trapolates the fluid velocity onto the IB, thus gener
ating additional immersed boundary points where the
fluid is strictly constrained to the IB velocity. These
additional immersed boundary points reduce the error
at the IB. Two different test cases are presented: The
fluid flow over a single sphere and simulation of a sedi
menting sphere with moving grid refinement. These test
cases demonstrate the secondorder accuracy of the im
mersed boundary conditions by integrating the surface
error. The test cases also show that the flow solver ac
curately simulates the resulting drag forces and terminal
velocities.
The Governing Equations
The flow of an incompressible Newtonian fluid around
immersed bodies is governed by the continuity and mo
mentum equations, the NavierStokes' equations;
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Vp + pV2 (2)
where pf is the fluid density, u is the fluid velocity,
p is the dynamic pressure, / represents the dynamic
viscosity of the fluid. To close the governing equations,
boundary conditions are employed at the boundaries of
the fluid domain.
The movement of a body inside the fluid is governed by
Newton's second law or the momentum equations in a
Lagrangian framework:
mp" =d +(ps Pf)V (3)
where mp is the body mass, v is the velocity, ps is
the density and V is the volume. Furthermore, fd is
the fluid drag force acting on the body, pf is the fluid
density, and g is the gravitation.
IBOFlow
IBOFLOW (Immersed Boundary Octree Flow
Solver) (1) is an incompressible finitevolume based
fluid flow solver. The velocity and pressure fields are
coupled with the SIMPLEC (2) method and discretized
on a Cartesian octree grid that can be dynamically
refined and coarsened, enabling grid refinement to
follow moving bodies with almost no extra computa
tional cost. As the grid is stored as an octree search
thread efficient search algorithms can be adopted to
find cells and couple the immersed boundaries with
the computational grid. The variables are stored in a
colocated configuration and pressure weighted flux
interpolation (20) is employed to prevent decoupling of
the pressure and velocity field.
The Immersed Boundary Methods
This section describes the implementation of the
original mirroing immersed boundary method (13; 15)
together with a hybrid immersed boundary condition.
Cell types and exterior normal points
The octree grid needs to be connected to the IB. This is
done initially and when the IB is moved by determine
the cell types and the exterior normal points. First the
g Interior
Figure 1: Definitions of the different cell types. The
extrapolation cells lie outside and close to the IB and the
mirroring cells lie inside and close to the IB. The rest of
the cells inside the IB is interior cells and the rest of the
cells outside is fluid cells.
cells lying close to the IB are found by employing a spe
cific bounding box in the octree grid. Cells whose centre
lie outside the IB, but with the minimum distance to the
IB less than one half cell size, are classified as extrap
olation cells. The rest of the cells outside the IB are
classified as fluid cells. Cells that lie inside the IB but
with a minimum distance to the IB less than one and a
half cell size are classified as mirroring cells. The rest of
the cells inside the IB are classified as internal cells, see
Figure 1 for an example. For mirroring cells the exterior
normal point, &e, is calculated as;
Pe = Pm + 2.0 (ib mi)
where 17, is the centre of the mirroring cell and p b is
the closest point on the local IB, see Figure 2 for a two
dimensional visualization. For extrapolation cells, the
extrapolation exterior point, fl, is calculated as;
Pe = ib + 2.0 (.ex ib) P
The mirroring immersed boundary condition
The velocity of the fluid at the immersed boundary, ub,
is constrained to the velocity of the immersed boundary
itself at tib by an implicit immersed boundary condition.
The mirroring IBC interpolates the velocity of the fluid
to the fictitious exterior normal point, fl, by trilinear in
terpolation and sets the velocity in the mirror cell to the
adu
19 +PfjJVJ 
Cell types
ue
^ Pe2~
IB
Tnsid'
Outside the IB
tie
pC
iex
Oib Uib Pex
etle IB
Pmi Umi Pib iib
Figure 2: A twodimensional visualization of the IB,
showing a mirroring point, 47,, lying in the centre of a
mirroring cell, an extrapolation point 7l,, lying in the
centre of an extrapolation cell, and immersed boundary
points, ~ib, lying on the local IB. The mirrored and ex
trapolated exterior normal points, &7, and the respective
velocities at the points, U, uib, Ui and Ue,, are also
shown.
reversed interpolated velocity, iU, plus the double local
velocity of the immersed boundary, Uib. The resulting
mirroring immersed boundary condition is as follows;
. Jib (6)
The hybrid immersed boundary condition
The hybrid immersed boundary condition both mirrors
and extrapolates the velocity field. For extrapolation
cells the hybrid IBC interpolates the velocity of the fluid
to the fictitious exterior normal point by trilinear inter
polation and sets the velocity in the extrapolation cell
to the mean value of the interpolated velocity and the
local velocity of the immersed boundary, ~b. The re
sulting extrapolation immersed boundary condition is as
follows;
Uib + Ue
= ex. (7)
Mass conservation
To ensure that no mass flux over the IB exists, the
interpolated internal face velocities are excluded when
discretizing the continuity equation. This is sensible
since these have no physical meaning. As the fictitious
ICMF 2010, Tampa, FL, May 30 June 4, 2010
velocity field is excluded both in the pressure correction
equation and the momentum equation, the presence
of the IB is also accounted for by both pressure and
continuity equations. The pressure correction equation
will therefore not generate a driving force over the IB
in the momentum equation; for this reason it is not
necessary to employ any regular (Neumann) boundary
condition at the IB.
In techniques similar to the mirroring IB method
described here, oscillations have occurred in the final
solution (12). We propose that this is due to the pres
ence of an unphysical mass flux and/or an unphysical
boundary condition over the IB. To decrease the mass
flux over the IB, Majumdar and coworkers (12) apply a
Neumann boundary condition for pressure at the IB that
results in a zero pressure force over the IB, and thus a
decreased mass flux. Due to the flow and other present
forces the solution will however still allow for a small
mass flux across the boundary.
Algorithm
The following algorithm is employed to take one fluid
time step when immersed boundaries are present:
i. During initialization determine the initial shape, po
sition and velocity of each body.
ii. For the first time step and if an IB is moved connect
the octree grid to the IB by setting the cell types:
mirror, interior, extrapolation and fluid, see Figure
1. For each mirror or extrapolation cell calculate
and store the exterior normal pointss.
iii. Assemble the momentum equations by determining
the coefficients for the linearized equations for each
cell:
a) For all mirror and extrapolation cells the IBC
is employed to determine the coefficients of
the velocity components.
b) If the cell is an interior cell the coefficients of
the velocity components are set to match the
body velocity, uIb, by a Dirichlet condition.
c) For all other cells, the coefficients are deter
mined by linearizing the NavierStokes' equa
tions.
iv. Determine the temporary velocity field by solving
the linearized system assembled in (iii).
v. Solve a Poisson equation for pressure correction
in the SIMPLEC method, based upon the solution
from the previous step. Velocities inside the body
are excluded. The pressure and the velocities are
ICMF 2010, Tampa, FL, May 30 June 4, 2010
corrected with the pressure and velocity correction
equations.
vi. Go to step (iii) until the continuity equation is satis
factorily fulfilled.
vii. Calculate by surface integration the fluid surface
force for each immersed body.
viii. Calculate the new positions, 7, and velocities, v, for
each IB and continue with the next time step.
Results
To verify the accuracy of the different IBCs the flow
over a single sphere is simulated and the resulting drag
coefficient is compared to analytical or semi empirical
data. In addition, a sedimenting sphere is simulated to
validate the hybrid method for moving bodies and to
show the potential of the dynamic octree grid.
Flow around a sphere
Figure 3: The simulation box with the adaptive octree
grid G5 around the immersed sphere.
A sphere with radius 0.5 mm is placed in the middle
of a square simulation box with a 10.0 mm side, see
Figure 3. The octree grid is refined around the sphere a
number of times, see Table 1. The inlet of the simulation
box is placed on the rmin surface, the outlet on the
opposite surface and symmetry boundary conditions are
employed for the other surfaces. The inlet velocity is
set to 1.0 m/s and the fluid viscosity to ImPas. The
Reynolds number is varied by altering the fluid density
between 0.01 and 100.0kg/m3. A time step of 0.1 ms
Figure 4: Left: For Reynolds number 1.0 the fluid ve
locity field is shown on a cut through the fluid domain.
Right: The fluid pressure is shown around the sphere.
Red indicates high velocity or pressure.
is used in all transient simulations which are run until
steady state.
Table 1: The grids used in the convergence study. The
number in the grid name represents the number of times
the grid is refined.
Name Cell size (mm) Number of cells
G2 1.0 0.2500000 5480
G3 1.0 0.1250000 10184
G4 1.0 0.0625000 21944
G5 1.0 0.0312500 58400
G6 1.0 0.0156250 190112
G7 1.0 0.0078125 690248
In Figure 4 the fluid velocity and fluid pressure are
vizualised on a cut through the domain. As seen in the
figure the pressure is high on the front side of the sphere
and low on the back side. An integration of the pressure
generates the pressure part of the drag force acting upon
the sphere from the fluid. An integration of the viscous
stress tensor gives the viscous part of the drag force.
The drag force acting upon a sphere is calculated as;
fd = PfC ,?U2 (8)
where U is the mean free fluid velocity and r the radius
of the sphere. The drag coefficient, Cd, is dependent on
the local Reynolds number;
Re 2rpU (9)
and for Re < 1000 it was approximated by Lapple (9)
to;
Cd = 24.0/Re (1.0 + 0.125Re72) (10)
In Figure 5 the simulated drag coefficients calculated
from the local pressure and velocity gradients are
ICMF 2010, Tampa, FL, May 30 June 4, 2010
10 10 10 10
Reynolds number
+ IBOFlow Mirror
0 IBOFlow Hybrid
Stoke drag
Lapple (1951)
101 102 103
Figure 5: The simulated drag coefficient, Cd, for the
two different immersed boundary conditions compared
with theoretical findings.
plotted against the Reynolds number and compared to
Lapple's results. As seen in the figure both methods
accurately simulate the drag coefficient for the specific
Reynolds numbers.
To show the order of accuracy of the IBCs a grid conver
gence study is carried out, evaluating the fluid velocity
at the surface of the sphere. In Figure 6 the velocity on
a 180 degree arc around the sphere is plotted against the
angle for the grids in Table 1. In the figure it is shown
that the velocity, which should be zero, decreases with
the grid size and that the error between the different
methods differs. The mirror IBC has less immersed
boundary points than the hybrid IBC and therefore also
a larger error. This is particularly visible in the middle
part of Figure 6. At each immersed boundary point the
error decreases and between the immersed boundary
points the error is somewhat larger, therefore the total
error of the hybrid method is smaller. To prevent spikes
in the error the hybrid IBC is preferable. In the bottom
of the figure the L2norm of the total velocity error
is plotted as a function of the grid size along with a
firstorder and a secondorder curve. From the figure
it is concluded that both methods are secondorder
accurate in space and that the hybrid method generates a
smaller error. Therefore only the hybrid method is used
in the following simulations.
Sedimenting sphere
The computational domain in this case is 3.0 x 3.0 x 6.0
mm, with gravity being oriented in the negative zaxis
with a magnitude of 10 m/s2. The sphere with radius 0.2
1 25 1 3
Angle (rad)
1 35 14 1 45
10
IBOFIow Mirror
IBOFlow Hybrid
Second order slope
10 First order slope
B 10
10
102
Cell size (m)
Figure 6: Top: The velocity error on a half circle bow
around the sphere is plotted in log scale as a function of
the angle for the six different grids described in Table
1. The error for the mirroring IBC is plotted as solid
lines, '', and the error for the hybrid method is plotted
as dashed lines,' '. Middle: A zoom of the top figure.
Bottom: For the different IBCs the L2norm of the error
is plotted in log scale as a function of grid size.
i
j~V
~~;~s:i:
ICMF 2010, Tampa, FL, May 30 June 4, 2010
mm is initially placed at the position (1.5, 1.5, 5.0) mm.
The density of the sphere, ps, is altered between 1390
and 5000 kg/m3 to generate different Reynolds numbers
and corresponding terminal velocities. The outlet of the
simulation box is placed on the Zmin surface and all other
surfaces are treated with symmetry boundary conditions.
The fluid is water with density 1000 kg/m3 and the vis
cosity 1.0 mPas. The time step is adapted to obtain a
CFL number of unity. The computational grid is dynam
ically refined around the sedimenting sphere, see Table
2, where Grid G2 is refined two times, G3 three times,
G4 four times and G5 five times, respectively.
Table 2: The dynamically refined grids employed in the
sedimenting sphere simulation.
Name Cell size (pm) Mean Number of cells
G2 50.0 12.500 10000
G3 50.0 6.2500 24000
G4 50.0 3.1250 85000
G5 50.0 1.5625 290000
Figure 7: Left: Visualization of the velocity field
through the center of the sedimenting sphere with den
sity 1390 kg/m3. Right: The fluid pressure around the
sedimenting sphere. Red indicates high velocity or pres
sure.
In Figure 7 the simulated fluid velocity field and pres
sure are shown for grid G3 and particle density 1390
kg/m3. In the figure the adaptive grid is shown which is
moving with the sedimenting sphere.
G2
G3
G4
0005
" 001
> 0015
002
15 2 25 3 35
Position (m)
15 2 25 3 35
Position (m)
4 45 5
x 10
4 45 5
x 10
Figure 8: Top: Simulated terminal velocity for the dif
ferent grids plotted against the vertical position. Bottom:
The drag coefficient plotted against the vertical position.
In Figure 8 the simulated velocity and the normalized
drag force (drag coefficient) of the sphere is plotted
against the zposition. The figure shows that the forces
and velocities are smooth for the finest grids G3, G4
and G5, but for the coarsest grid some oscillations
are present. The simulated velocity from grid G2 is
lower than for the other grids due to the fact that the
boundary layer around the sphere is not resolved. The
other simulated velocities are converging towards a grid
independent solution. Furthermore, no oscillations in
the velocity or force occur when the sphere enters or
leaves a cell.
To validate the simulated terminal velocity it is com
pared to the analytical solution;
8gr P P (I
vt 3 pf (11)
Si g Pf
where Cd is given from Equation (10). In Figure 9 the
simulated terminal velocity on grid G3 are compared
with the analytical solution for different sphere densities.
The simulated terminal velocities agree well with the an
alytical results with a small deviation for large densities,
where the boundary is not completely resolved.
0021
IBOFlow
Lapple (1951)
004
006
008
01
012
1500 2000 2500 3000 3500 4000 4500 5000 5500
Sphere density (kg/m )
Figure 9: Simulated terminal velocity for different
sphere densities compared to the analytical results given
by Equation (11).
Conclusions
In this work the original mirroring immersed bound
ary method and a hybrid immersed boundary condition
are compared, validated and implemented in IBOFLOW.
The hybrid immersed boundary condition both mirrors
and extrapolates the velocity field at the IB generating
more immersed boundary control points. The fluid flow
past a single sphere is simulated and the resulting drag
forces are compared and validated against previous re
sults with excellent agreement. Both immersed bound
ary conditions are shown to be strictly secondorder ac
curate in space, where the hybrid method generates a
smaller error due to more immersed boundary points.
Finally a sedimenting sphere with a moving grid refine
ment is accurately simulated with a moving IB that does
not generate any oscillations in the solution. Thus, the
proposed method can be good alternative when perform
ing DNS simulations on fluidstructure interaction prob
lems.
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