A non-stiff numerical method for 3D interfacial flow of inviscid fluids

NJIT, 29 May 2014!

Oleksiy Varfolomiyev!advisor Michael Siegel

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Develop a non-stiff numerical method !for the Boundary Integral Formulation !

of the motion of internal waves. !For doubly periodic interfacial flow in 3D !

with surface tension term.

Motivation

Outline

❖Model Description ❖Invariants of Motion ❖Linear Stability Analysis ❖Discretization ❖Numerical Method

⇤! p(�) : p(�) = minq⇥Pk

max�⇥[�min,�max]

����1⌅�� q(�)

���� (41)

We used Wolfram Mathematica intrinsic function MiniMaxApproximation toobtain 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 impedancefunction in the form of continued fraction (12). We proceed with Euclidean typealgorithm with 2k polynomial divisions, i.e.

p(�) =ck�1�k�1 + ck�2�k�2 + · · ·+ c0dk�k + dk�1�k�1 + · · ·+ d0

(42)

=1

dk�k+dk�1�k�1+···+d0ck�1�k�1+ck�2�k�2+···+c0

=1

dkck�1

�+

✓dk�1�

dkck�2ck�1

◆�k�1+···+

✓d1�

dkc0ck�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.

Governing Equations

1 Model problem

Consider the following boundary value problem: Laplace equation on a semi-infinite strip with the given Neumann data on the left boundary and Dirichletdata elsewhere

�⌅2w(x, y)

⌅y2� ⌅2w(x, y)

⌅x2= 0, (x, y) ⌅ [0,⇤)⇥ [0, 1], (1)

⌅w

⌅x(0, y) = �⇤(y), y ⌅ [0, 1], (2)

w|x=⇥ = 0, (3)

w(x, 0) = 0, w(x, 1) = 0, x ⌅ [0,⇤). (4)

Assume

⇤(y) =m�

i=1

ai sin(i⇥y) (5)

The above problem can be studied as a problem

Aw(x)� d2w(x)

dx2= 0, x ⌅ [0,⇤) (6)

dw

dx(x = 0) = �⇤ (7)

w|x=⇥ = 0, (8)

where A = � �2

�y2 is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =

{⇥2, . . . , (m⇥)2}. We can obtain Dirichlet data on the left boundary using

Neumann-to-Dirichlet map. Equation (6) gives Aw(x) = d2w(x)dx2 , therefore

�dw(x)dx = A

12w(x) and now we can use given in (7) Neumann data to get at

x = 0w(0) = f(A)⇤,

here f(�) = �� 12 is impedance 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

Boundary Conditions

1 Model problem

Consider the following boundary value problem: Laplace equation on a semi-infinite strip with the given Neumann data on the left boundary and Dirichletdata elsewhere

�⌅2w(x, y)

⌅y2� ⌅2w(x, y)

⌅x2= 0, (x, y) ⌅ [0,⇤)⇥ [0, 1], (1)

⌅w

⌅x(0, y) = �⇤(y), y ⌅ [0, 1], (2)

w|x=⇥ = 0, (3)

w(x, 0) = 0, w(x, 1) = 0, x ⌅ [0,⇤). (4)

Assume

⇤(y) =m�

i=1

ai sin(i⇥y) (5)

The above problem can be studied as a problem

Aw(x)� d2w(x)

dx2= 0, x ⌅ [0,⇤) (6)

dw

dx(x = 0) = �⇤ (7)

w|x=⇥ = 0, (8)

where A = � �2

�y2 is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =

{⇥2, . . . , (m⇥)2}. We can obtain Dirichlet data on the left boundary using

Neumann-to-Dirichlet map. Equation (6) gives Aw(x) = d2w(x)dx2 , therefore

�dw(x)dx = A

12w(x) and now we can use given in (7) Neumann data to get at

x = 0w(0) = f(A)⇤,

here f(�) = �� 12 is impedance function.

2

Kinematic boundary condition

Far-field boundary conditions!

Laplace-Young boundary condition

Fluid Velocity

1 Model problem

Consider the following boundary value problem: Laplace equation on a semi-infinite strip with the given Neumann data on the left boundary and Dirichletdata elsewhere

�⌅2w(x, y)

⌅y2� ⌅2w(x, y)

⌅x2= 0, (x, y) ⌅ [0,⇤)⇥ [0, 1], (1)

⌅w

⌅x(0, y) = �⇤(y), y ⌅ [0, 1], (2)

w|x=⇥ = 0, (3)

w(x, 0) = 0, w(x, 1) = 0, x ⌅ [0,⇤). (4)

Assume

⇤(y) =m�

i=1

ai sin(i⇥y) (5)

The above problem can be studied as a problem

Aw(x)� d2w(x)

dx2= 0, x ⌅ [0,⇤) (6)

dw

dx(x = 0) = �⇤ (7)

w|x=⇥ = 0, (8)

where A = � �2

�y2 is defined on span{sin (⇥y), . . . , sin (m⇥y)}, spA =

{⇥2, . . . , (m⇥)2}. We can obtain Dirichlet data on the left boundary using

Neumann-to-Dirichlet map. Equation (6) gives Aw(x) = d2w(x)dx2 , therefore

�dw(x)dx = A

12w(x) and now we can use given in (7) Neumann data to get at

x = 0w(0) = f(A)⇤,

here f(�) = �� 12 is impedance function.

2

Invariants of Motion

Conservation of Mean Height & Total Energy

Linear Stability Analysis

Linear Stability Analysis

Linearized Problem Solution

Numerical Method

Small-scale DecompositionDominant at small scale terms,lower order terms!

Leading order

In terms of the Riesz transforms

Simple to evaluate in the Fourier space

Discrete equations to solve

Two-step Algorithm

Numerical Experiment

Gravity driven flow !(Rayleigh-Taylor Instability)

2D Solution z 3D Solution z

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Stability ChartLargest  stable  time  step    

for  the  explicit  and  implicit  methods.

Conclusions

✦ We have developed an efficient non-stiff boundary integral method to study 3D internal waves with surface tension

✦ Fast algorithm for calculating the Birkhoff-Rott integral for a doubly-periodic surface.

✦ The algorithm is effective at eliminating the time-step constraint that plagues explicit time-integration methods.

Future Work❖ To make numerical method second order in time.!

❖ In order to resolve complex interface shapes due to instabilities we need to allow spatial discretization to have very fine resolution. This is most efficient when we have nearly 2D surface, then we make fine grid in one dimension and coarse grid in the other dimension.!

❖ To further speed up the computation of the Birkhoff-Rott integral for the fluid velocity via improved fast Ewald summation procedure.

❖ D.M. Ambrose, M. Siegel, and S. Tlupova. A small-scale decomposition for 3D boundary integral computations with surface tension. J. Comp. Phys., Accepted, 2013 !

❖ G. Baker, R. Caflisch, M.Siegel, Singularity formation during Rayleigh-Taylor instability, J. Fluid Mech, vol. 252, 51-78, 1993!

❖ D. Pullin, Numerical studies of surface-tension effects in nonlinear Kelvin-Helmholtz and Rayleigh-Taylor instability, J. Fluid Mech, vol. 119, 507-532, 1981

References