We study the internal waves as the evolution of the interface between two immiscible, inviscid, incompressible, irrotational fluids of different density in three dimensions. Motion of the interface and fluids is driven by the action of gravity, surface tension, and/or a prescribed far-field pressure gradient. The model includes derived equations for the evolution of the interface and surfaces density. Presence of the surface tension introduces high order derivatives into the evolution equations. This makes the considered problem stiff and the application of the standard explicit time-integration methods suffers strong time-step stability constraints. Our proposed numerical method employes a special interface parameterization that enables the use of implicit time-integration methods via a small-scale decomposition. This approach allows us to capture the nonlinear growth of normal modes for the case of Rayleigh-Taylor instability with the heavier fluid on the top. In addition, in the given test problem with the prescribed initial disturbance to a flat interface under an action of the surface tension dominating the gravity, the surface relaxes to a flat interface. Linear stability analysis is performed and the numerical results are validated by comparison to the obtained analytic solution of the linearized problem for a short time. The developed model and numerical method can be efficiently applied to study the motion of internal waves for doubly periodic interfacial flows with surface tension.

1. A non-stiff boundary integral method
for internal waves
NJIT, 4 June 2013
Oleksiy Varfolomiyev
advisor Michael Siegel
Wednesday, June 19, 13

2. Motivation
Wednesday, June 19, 13

3. Motivation
Develop a model and a numerical method that can be
efficiently applied to study the motion of internal waves
for doubly periodic interfacial flows with surface tension.
Wednesday, June 19, 13

4. Outline
Wednesday, June 19, 13

5. Outline
•Model Description
Wednesday, June 19, 13

6. Outline
•Model Description
•Linear Stability Analysis
Wednesday, June 19, 13

7. Outline
•Model Description
•Linear Stability Analysis
•Discretization
Wednesday, June 19, 13

8. Outline
•Model Description
•Linear Stability Analysis
•Discretization
•Numerical Experiment
Wednesday, June 19, 13

9. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
Wednesday, June 19, 13

10. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
Evolution of the interface
between two immiscible, inviscid, incompressible, irrotational fluids
of different density in 3D.
Wednesday, June 19, 13

11. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
Motion of the fluids is driven by
Evolution of the interface
between two immiscible, inviscid, incompressible, irrotational fluids
of different density in 3D.
Wednesday, June 19, 13

12. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
➡ Gravity
Motion of the fluids is driven by
Evolution of the interface
between two immiscible, inviscid, incompressible, irrotational fluids
of different density in 3D.
Wednesday, June 19, 13

13. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
➡ Gravity
➡ Surface Tension
Motion of the fluids is driven by
Evolution of the interface
between two immiscible, inviscid, incompressible, irrotational fluids
of different density in 3D.
Wednesday, June 19, 13

14. ⇤! p( ) : p( ) = min
q⇥Pk
max
⇥[ min, max]
1
⌅ q( ) (41)
We used Wolfram Mathematica intrinsic function MiniMaxApproximation to
obtain p( ). The next figure shows the approximation error
10 100 1000 104
10 15
10 14
10 13
10 12
10 11
To obtain grid steps we rewrite obtained approximation of the impedance
function in the form of continued fraction (12). We proceed with Euclidean type
algorithm with 2k polynomial divisions, i.e.
p( ) =
ck 1
k 1 + ck 2
k 2 + · · · + c0
dk
k + dk 1
k 1 + · · · + d0
(42)
=
1
dk
k+dk 1
k 1+···+d0
ck 1
k 1+ck 2
k 2+···+c0
=
1
dk
ck 1
+
✓
dk 1
dkck 2
ck 1
◆
k 1+···+
✓
d1
dkc0
ck 1
◆
+d0
ck 1
k 1+ck 2
k 2+···+c0
,
ˆ d
Model Description
➡ Gravity
➡ Surface Tension
➡ Prescribed far-field pressure gradient
Motion of the fluids is driven by
Evolution of the interface
between two immiscible, inviscid, incompressible, irrotational fluids
of different density in 3D.
Wednesday, June 19, 13

15. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Wednesday, June 19, 13

16. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
The interface S is parametrized by
Wednesday, June 19, 13

17. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
The interface S is parametrized by
Wednesday, June 19, 13

18. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
The interface S is parametrized by
The fluid velocities are governed by the Bernoulli’s law
Wednesday, June 19, 13

19. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
@ i
@t
r i · Xt +
1
2
|r i|2
+
pi
⇢i
+ gz = 0 in Di
The interface S is parametrized by
The fluid velocities are governed by the Bernoulli’s law
Wednesday, June 19, 13

20. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
@ i
@t
r i · Xt +
1
2
|r i|2
+
pi
⇢i
+ gz = 0 in Di
The interface S is parametrized by
The evolution equation for the free surface S
The fluid velocities are governed by the Bernoulli’s law
Wednesday, June 19, 13

21. Governing Equations
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
@ i
@t
r i · Xt +
1
2
|r i|2
+
pi
⇢i
+ gz = 0 in Di
The interface S is parametrized by
The evolution equation for the free surface S
The fluid velocities are governed by the Bernoulli’s law
Wednesday, June 19, 13

22. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Wednesday, June 19, 13

23. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Wednesday, June 19, 13

24. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Wednesday, June 19, 13

25. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Laplace-Young boundary condition
Wednesday, June 19, 13

26. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Laplace-Young boundary condition
Wednesday, June 19, 13

27. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Far-field boundary conditions
Laplace-Young boundary condition
Wednesday, June 19, 13

28. Boundary Conditions
is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Kinematic boundary condition
Far-field boundary conditions
Laplace-Young boundary condition
Wednesday, June 19, 13

29. Equations to solve
Wednesday, June 19, 13

30. is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =
e can obtain Dirichlet data on the left boundary using
t map. Equation (6) gives Aw(x) = d2w(x)
dx2 , therefore
nd now we can use given in (7) Neumann data to get at
w(0) = f(A)⇤,
mpedance function.
2
Wednesday, June 19, 13

31. Linear Stability Analysis
Wednesday, June 19, 13

32. Linear Stability Analysis
Wednesday, June 19, 13

33. Fourier analysis
Wednesday, June 19, 13

34. Linearized Problem Solution
Remark
4t ⇠ (4x)
3
2
Wednesday, June 19, 13

35. Linearized Problem Solution
Remark
4t ⇠ (4x)
3
2
Wednesday, June 19, 13

36. Discretization
Wednesday, June 19, 13

37. Wednesday, June 19, 13

38. Numerical Experiment
Wednesday, June 19, 13

39. Gravity driven flow
(Rayleigh-Taylor Instability)
Surface tension interface
relaxation
−2
0
2
4
6
8
−2
0
2
4
6
8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Numerical Solution
Solution z at T=1
(Implicit method, A=1, dt = 0.1, N=32)
−2
0
2
4
6
8
−2
0
2
4
6
8
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Numerical Solution
Solution z at T=5
(Implicit method, A=0.5, dt = 0.1, N=32)
Wednesday, June 19, 13

40. Numerical Results
Max interface height for lin & num soln.
Explicit method, N=32, A=0.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t
max of the lin and num solution zLin and z
lin soln
num soln
Max interface height for lin & num soln.
Explicit method, N=32, A=1.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
t
max of the lin and num solution zLin and z
lin soln
num soln
Wednesday, June 19, 13

41. Stability
growing unstable modes
Wednesday, June 19, 13

42. Stability Chart
Largest
stable
time
step
for
the
explicit
and
implicit
methods.
4t ⇠ (4x)
3
2
Wednesday, June 19, 13

43. Conclusions
✦ We have developed a non-stiff boundary integral method for
3D internal waves
✦ The algorithm is effective at eliminating the third order t-step
constraint that plagues explicit methods
✦ Efficient algorithm for calculating the Birkhoff-Rott integral for a
doubly-periodic surface. This algorithm is based on Ewald
summation, computes the integral in O(N log N ) operations
per time step
✦ Presented method is useful for computing the motion of
doubly-periodic fluid interfaces with surface tension in 3D flow
Wednesday, June 19, 13