1. The Immersed Interface Method for 2D Incom-
pressible Viscous Flows with Non-Smooth Objects
Yang Liu & Sheng Xu Department of Mathematics
Abstract
In the immersed interface method (IIM), singular forces are used to
represent the effect of objects immersed in a fluid, and jump
conditions induced by the singular forces are incorporated into
numerical schemes to simulate the flow. Previously, the IIM was
developed to simulate flows with smooth objects[1, 2, 3]. We here
present an extension of the method for non-smooth objects. In
particular, we describe how to compute necessary jump conditions
using line segment representation of 2D objects. Our tests on
circular Couette flow, flow past a square cylinder and flow around
flapping plates indicate that our method achieves second-order
accuracy, uses relatively insignificant time for an extra object and is
robust to handle non-smooth complex objects.
I. Introduction of the IIM
In the IIM, a rigid object in an incompressible viscous fluid is treated
as the fluid, and its effect is represented by a singular force F in the
governing equations [2, 4] as
∂u
∂t
+ · (uu) = − p +
1
Re
∆u + q + F (1)
p = 2
∂u
∂x
∂v
∂y
−
∂u
∂y
∂v
∂x
+ · F (2)
The singular force F = Γ f(X, t)δ(x − X)dl represents the
effect of the object on the surrounding fluid;
The body force q enforces the rigid motion of the fluid enclosed by
the interface Γ.
x
y
B
Ω+
Ω−
Γ
Γ+
Γ−
n
τ
X
h h
| |
zi−1 zi zi+1ξ η
Figure 1: Left: Geometric description of the immersed object. Right: Finite
difference scheme with jump conditions at ξ and η
The singular force and the discontinuous body force induce jump
discontinuities across the interface Γ in flow fields. Standard
numerical schemes can be modified to be incorporated with jump
conditions. For example (Figure 1)
dg z−
i
dz
=
g z−
i+1 − g z+
i−1
2h
+ O h2
−
1
2h
2
n=0
[gn
(ξ)]
n!
(zi−1 − ξ)n
+
2
n=0
[gn
(η)]
n!
(zi+1 − η)n
(3)
Cartesian jump conditions for the velocity and pressure
∂u
∂x
, ∂u
∂y
, ∂2
u
∂x2 , ∂2
u
∂y2 , ∂2
u
∂x∂y
∂p
∂x
, ∂p
∂y
, ∂2
p
∂x2 , ∂2
p
∂y2 , ∂2
p
∂x∂y
Principal jump conditions for the velocity and pressure
[u], ∂u
∂n
, [ u], [p], ∂p
∂n
, [ p]
To discretize Eqs. (1) and (2) on a Cartesian grid, Cartesian jump
conditions can be computed from principal jump conditions using line
segment representation of the interface Γ [5].
In the current implementation, MAC girds, FFT Poisson solvers and
RK4 time marching are employed.
II. Principle Jump Conditions
δx
δy
δn
δn
Γ
IV III
I II
S2
S1
S0
Ω+
Ω−
⃗τ
⃗n
n
τ ν
A
B
C
D
E
F
M1 M2
Figure 2: Left: One-sided finite difference stencil. Right: Representation of a 2D
interface as line segments.
i. Principle jump conditions for u:
The velocity in the viscous flow is finite and continuous, so
[u] = 0 (4)
The flow inside the interface Γ is enforced to be in rigid motion, so
∂u
∂n
=
∂u
∂n
|+
− ˙θ · τ, (5)
[ u] =
∂2
u
∂n2
|+
+ κ
∂u
∂n
(6)
where ˙θ is the angular velocity of the object, κ is the curvature of
the boundary, ∂u
∂n
|+
and ∂2
u
∂n2|+
can be computed using one-sided
finite difference schemes (Figure 2).
ii. Principle jump conditions for p:
∂p
∂n
Equation (1) implies
[ p] =
1
Re
[ u] + [q] (7)
So
∂p
∂n
= [ p] · n (8)
[p]
Integrating ∂p
∂τ
from A to B (Figure 2) and approximating the
integral by Simposon’s rule gives
B
A
∂p
∂τ
dτ = [p]B − [p]A ≈
| AB |
6
∂p
∂τ A
+ 4
∂p
∂τ M1
+
∂p
∂τ B
(9)
where ∂p
∂τ
is computed as ∂p
∂τ
= [ p] · τ. Obtain an equation
similar to Equation (9) for the line segment AF. Adding it to
Equation (9) gives a linear equaiton for vertex A
[p]B + [p]F − 2 [p]A = bA, (10)
where bA is the sum of the right-hand sides of Equation (9) for
AB and the equation for AF .
Applying the same process for all vertices ends up with a linear
system 





−2 1 . . . 1
1 −2 . . . 0
... ... ... ...
0 . . . −2 1
1 . . . 1 −2










[p]A
[p]B
...
[p]F



 =




bA
bB
...
bF



 (11)
[ p]
Equation (2) implies
[ p] = 2
∂u
∂x
∂v
∂y
− 2
∂u
∂y
∂v
∂x
(12)
III. Results
i. Test of Accuracy: Circular Couette Flow
lx
ly
r1
r2
x
y
B
Π1
Π2
Figure 3: Circular Couette Flow
n ||eu||∞ order ||ev||∞ order ||ep||∞ order
30 3.90 × 10−2
- 4.01 × 10−2
- 1.71 × 10−2
-
60 7.28 × 10−3
2.4215 7.28 × 10−3
2.4615 2.98 × 10−3
2.5215
120 1.82 × 10−3
2.0008 1.83 × 10−3
1.9952 2.45 × 10−3
0.2805
240 4.53 × 10−4
2.0066 4.55 × 10−4
2.0068 1.09 × 10−3
1.1697
ii. Code Validation: Flow Past a Stationary Square Cylinder
Figure 4: Left: Re=40, Stream function. Right: Re=100, Vorticity field.
B
L/D Cd Re = 100, B = 0.5
Re = 40 Re = 5 Re = 40 ¯Cd St
Paliwal 0.067 2.7000 4.8140 1.8990 Sohankar 1.4770 0.1460
Sen 0.067 2.7348 5.2641 1.8565 Robichaux 1.5300 0.1540
Present 0.067 2.77 5.1334 1.7664 Sharma 1.4936 0.1488
Dhiman 0.050 2.8220 4.8400 1.7670 Sahu 1.4878 0.1486
Sen 0.050 2.8065 4.9535 1.7871 Sen 1.5287 0.1452
Present 0.050 2.86 4.8759 1.7154 Present 1.4941 0.1479
(B is the blockage ratio.)
iii. Test of Efficiency: Flow Around Rectangular Flappers
Figure 5: Vorticity fields, Re=157, aspect ratio of the flapper=4.
Number of Flappers 1 2 3 4
Computational Time 1 1.0583 1.0682 1.2233
(The computational time corresponding to the unit value is 0.648 hours on a
desktop computer for 20000 time steps with 513 X 513 grids.)
iv. Test of Stability & Robustness: Flow Past a Mustang
Figure 6: Flow past a Mustang at Re=1000. Left: Streamfunction, Right: Vorticity
IV. Conclusions & Ongoing Work
The method can simulate flows with multiple moving non-smooth
complex objects.
The method is second order accurate in the infinity norm for the
velocity.
The method is stable at all the Reynolds numbers
(Re = 5 ∼ 1000) in our tests.
The method is efficient to handle multiple moving objects. The
extra cost to handle an additional object is proportional to the
number of the vertices used to represent the objects.
The ongoing work focuses on the parallelization of the method for
distributed-memory parallel computing with MPI.
V. References
S. Xu and Z. J. Wang Journal of Computational Physics, vol. 216, no. 2, pp. 454–493, 2006.
S. Xu and Z. J. Wang Computer Methods in Applied Mechanics and Engineering, vol. 197,
no. 25-28, pp. 2068–2086, 2008.
S. Xu Journal of Computational Physics, vol. 230, no. 19, pp. 7176–7190, 2011.
C. S. Peskin Journal of computational physics, vol. 10, no. 2, pp. 252–271, 1972.
S. Xu and G. D. Pearson Journal of Computational Physics, vol. 302, pp. 59–67, 2015.
VI. Acknowledgment
This work is supported by the NSF grand DMS1320317.