Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds - Fluid-Structure Interaction and Biomedical Applications - page 327

differentiable. Then

D A f
Dt .x;t/ D

@f
@t .x;t/ C z .x;t/ rf.x;t/;

(5.8)

where z is the domain velocity defined by ( 5.6 ).

Proof. Let us set x D dž.X;t/: Then by ( 5.7 )and( 5.6 ), we get

@ f
@t .X;t/

D A f
Dt .x;t/ D

h f.dž.X;t/;t/ i

d
dt

D

2

X

@t dž.X;t/;t C @x i dž.X;t/;t @dž i

@f @f

D

@t .X;t/

i D 1

@f
@t .x;t/ C z .x;t/ rf.x;t/:

D t

Using the relation ( 5.8 ), we can rewrite the Navier-Stokes system ( 5.1 ) in the
ALE form

@ ij
@x j

X

2

D A v i

Dt C ..v z / r/v i D

; iD 1;2; rv D 0: (5.9)

j D 1

5.2.3 Numerical Approximation of the Incompressible

Navier-Stokes Equations

This section will be concerned with the discretization of the flow problem ( 5.9 ),
( 5.3 ), ( 5.4 ) (where the condition ( 5.4 ), (a) can be replaced by ( 5.5 )).

Time Discretization

First let us describe the time discretization of the problem. We consider a partition
0 D t 0 <t 1 < <T;t k D kt, with a constant time step t > 0, of the time
interval Œ0;T and approximate the solution v.t n /, p.t n / (defined in t n ) at time
t n by v n , p n . For the time discretization we use the second-order two-step scheme
using the computed approximate solution v n 1 , p n 1 in t n 1 and v n , p n in t n for
the calculation of v n C 1 , p n C 1 in the domain WD t n C 1 .