Computational Fluid Dynamics!
http://www.nd.edu/~gtryggva/CFD-Course/!
Grétar Tryggvason!
Spring 2011!
4!
Computational Fluid Dynamics!
Motivation and Goals!
Cavitation!
Computational Fluid Dynamics!
Outline!
Motivation!
!
Governing Equations!
!
The One-Fluid Approach!
!
Numerical approach!
!
Advecting the Marker Function!
!VOF; LS; FT; Other; Tests!
!
Surface Tension!
Methods to Track
Moving Fluid
Interfaces!
3!
2!
Microstructure!
Bubbly Flow!
Computational Fluid Dynamics!
Motivation and Goals!
The goal is to simulate accurately the smallest continuum
scales for multiphase systems that are sufficiently large so that
meaningful averages can be obtained and the results used to
help generate insight and closure models for engineering tools
Example: The two-fluid model
Equations for the
average motion of
each constituent
!
" p # p + $ % (" p # p u p ) = m˙ p
!t
!
(" p # pu p ) + $ % (" p # pu pu p ) = &" p$pp
!t
+$ % (" p µ p D p ) + " p # p g + $ % (" p # p < uu >) + Fint
Atomization!
5!
Splash!
Reynolds
stresses
School of fish!
Computational Fluid Dynamics!
Numerical Method!
6!
interfacial
forces
Computational Fluid Dynamics!
CFD of Multiphase Flows—one slide history!
BC: Birkhoff and boundary integral methods !
for the Rayleigh-Taylor Instability!
!
65 Harlow and colleagues at Los Alamos: !
The MAC method!
!
From: B. Daly (1969)!
75 Boundary integral methods for Stokes flow and
potential flow !
!
85 Alternative approaches (body fitted, unstructured,
etc.)!
!
95 Beginning of DNS of multiphase flow. Return of the
one-fluid approach and development of other
techniques!
Governing
Equations
1
7!
8!
Computational Fluid Dynamics!
Governing Equations!
Conservation of mass :
!"
+ # $ ( "u) = 0
!t
Sharply stratified flows
D!
+ !" # u = 0
Dt
The conservation equations for mass and momentum apply
to any flow situation, including flows of multiple immiscible
fluids.
Incompressible flows:
D!
=0
Dt
Computational Fluid Dynamics!
Governing Equations!
Each fluid generally has properties that are different from the
other constituents and the location of each fluid must
therefore be tracked.
!"u=0
Navier-Stokes equations (conservation of momentum)
We usually also have additional physics that must be
accounted for at the interface, such as surface tension.
!"u
+ # $ "uu = " f + # $ T
!t
!
"u
+ !u#u = $#p + ! f b + # % µ(#u + # T u )
"t
9!
The regular conservation equations can be extended to
handle these situations by using generalized functions
10!
Computational Fluid Dynamics!
H (x, y,t) =
!H =
The One-Fluid
approach!
$
A
Computational Fluid Dynamics!
Governing Equations!
#
A(t )
s
! (x " x ')! (y " y')da '
![" (x # x ')" ( y # y ')]da '
= # $ !'[" (x # x ')" ( y # y ')]da '
A!
S!
x
S
!H
+ u "#H = 0
!t
S
S
Using:!
= #" (n)n
12!
Computational Fluid Dynamics!
Governing Equations!
Conservation of Momentum!
!
!"u =0
Singular interface term!
Equation of State:!
D!
Dµ
= 0;
=0
Dt
Dt
#
"V
Incompressible flow!
!
Oscillating drop: pressure!
The conservation equations are solved on a regular fixed grid and the
front is tracked by connected marker points. !
The one fluid formulation of the conservation equations is the
starting point for several numerical methods, including MAC, VOF, level
sets, and front tracking.!
Computational Fluid Dynamics!
Governing Equations!
Du
dv = $ #"V %pdv +
Dt
=0!
Constant
properties!
! (x " x')! (y " y') = ! (s)! (n)
The one-fluid formulation implicitly contains the proper
interface jump conditions. Integrating each term across a
small control volume centered at the interface:!
"u
+ !# $ uu = %#p + f + # $ µ(# u + # T u ) + ' F &( n) ( x % x f ) da
"t
Conservation of Mass!
H=0!
A
= # $ " (x # x ')" ( y # y ')nds'
11!
H=1!
y
= #!
$ " (x # x ')" ( y # y ')nds'
= # $ " (s)" (n)nds'
n
[ p]n
#
"V
fdv +
=0!
#
"V
% & µ(% u + % T u ) dv +
#
"V
[µ(! u + ! u )]n
T
'( n" ( n ) dv
!" n
Jump Condition:!
[! p + µ(" u + " u )]n = !#$n
T
2
13!
14!
Computational Fluid Dynamics!
Governing Equations!
Computational Fluid Dynamics!
We can also show that the one-fluid formulation
contains the equations written separately for each
fluid and the jump conditions:
Numerical
Solutions
u = H1u1 + H 2u 2
P = H1 p1 + H 2 p2
! = H1!1 + H 2 ! 2
Write:
Substitute into the momentum equation
H1 (………) + H 2 (………) + ! ( x f )(………) = 0
=0
Momentum
equation in
phase 1
15!
=0
Momentum
equation in
phase 2
=0
Interface
conditions
16!
Computational Fluid Dynamics!
Discretization in time
Summary of discrete vector equations
Work with the finite volume approximation
!
"u dv + !# "u(u $ n)ds =
!t #
Evolution of the velocity—first order explicit in time:
# " f dv + !# µ (%u + % u) n ds + #
T
V
&' n( ( n ) dv
Discretize each term
1
u=
V
Ac =
Dc =
! h "u
V
1
1
! " ( uu ) dV = !
u ( u " n ) ds
V V#
V #S
)
(
)
18!
Computational Fluid Dynamics!
Discretization in time
The momentum equation is solved in two steps
n
utmp
1
i, j ! u i, j
= !A ni, j + fbn + n (Dni, j + f$n )
"t
#
and n +1
ui, j ! utmp
1
i, j
= ! n $ h pi, j
"t
#
Projection
Method
To derive an equation for the pressure we take the divergence
of the second equation and use ! h "u ni ,+1
. The result is:
j =0
0
&1
)
tmp
! h " uni, +1
j = ! h " u i, j # $t ! h " ( n ! h pi, j +
'%
*
$1
' 1
! h " & n ! h pi, j ) = ! h " utmp
i, j
%#
( *t
n +1
i,j
=0
No explicit
equation for
the pressure!
1
1
! " µ ! hu+!Th u dV = !
µ ! hu+!Th u n ds
V V#
V #S
17!
n
uni, +1
1
j ! u i, j
= !A ni, j ! n ($ h p + Dni, j + f%n ) + fbn
"t
#
Constraint on velocity
! udV
(
Computational Fluid Dynamics!
Discretization in time
u=
1
V
! udV
V
1
Ac =
V
Dc =
1
V
=
1
# ! " (uu )dV = V # u (u " n)ds
V
S
# ! " µ(! u + ! u )dV
T
h
h
V
1
V
# µ(! u + ! u )nds
h
T
h
S
Computational Fluid Dynamics!
Discretization in time
1. Find a temporary velocity using the advection and the
diffusion terms only:
% n
(
1
n
n
utmp
(Dni, j + f$n )*)
i, j = u i, j + !t' "A i, j + fb +
#n
&
2. Find the pressure needed to make the velocity
field incompressible
$1
' 1
! h " & n ! h pi, j ) = ! h " utmp
i, j
%#
( *t
3. Correct the velocity by adding the pressure gradient:
"t
$ h pi, j
#n
4. Update the marker function to find new density and viscosity
tmp
uni, +1
j = u i, j !
3
19!
20!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
Governing Equations!
Identify each fluid by a marker function H
Advecting the
marker
function!
! 1 in fluid 1
H ="
#0 Otherwise
The marker moves with the fluid
and is updated by integrating the
following advection equation in time
!H
+ u " #H = 0
!t
21!
22!
Computational Fluid Dynamics!
Advecting the Marker Function
The sharp marker function H can be approximated in
several different ways for computational purposes.
Below we show a smoothed marker function, I, the
volume of fluid approximation, C, and a level set
representation, Φ. !
Updating H—in spite of its apparent
simplicity—is one of the hard
problems in CFD!
Computational Fluid Dynamics!
Advecting the Marker Function
In 1D, using upwind and LW.!
The solution quickly deteriorates.!
Modern advection methods help, but not completely.!
23!
24!
Computational Fluid Dynamics!
Advecting the Marker Function
Volume of
Fluid!
Computational Fluid Dynamics!
Advecting the Marker Function
To advect a discontinuous marker function, first consider 1D
advection. Using simple upwind leads to excessive diffusion
due to averaging the function over each cell, before finding
the fluxes!
Upwind
4
25!
26!
Computational Fluid Dynamics!
Advecting the Marker Function
Since the marker function only takes on two values, 0 and 1,
the advection can be made much more accurate by
reconstructing the function in each cell before finding the
fluxes, integrated over time:!
Computational Fluid Dynamics!
While VOF works extremely well in one-dimension, there are
considerable difficulties extending the approach to higher
dimensions. The basic problem is the reconstruction of the
interface in each cell, given the volume fraction in
neighboring cells. !
!
In the SLIC method the interface
was taken to be
perpendicular to the advection
direction.!
!
One-dimensional Volume-Of-Fluid!
In the Hirt/Nichols method the interface was taken to be
parallel to one axis. !
!
In PLIC the interface is a line with arbitrary orientation.!
!
Once the interface has been reconstructed, the marker
function is advected by geometric considerations!
27!
28!
Computational Fluid Dynamics!
Original!
Computational Fluid Dynamics!
Level Set
Methods!
SLIC!
Hirt/Nichols
VOF!
29!
PLIC!
Computational Fluid Dynamics!
Identify the interface as a level-set of a smooth function!
Advect the level set function by!
use!
to get!
30!
Computational Fluid Dynamics!
The level set function can be arbitrarily smooth. To identify
each fluid it is necessary to construct a marker function with
a narrow transition zone!
I (! )
The marker function can be
generated by (for example):!
!
The delta function is generated
as the derivative of the marker
function !
5
31!
32!
Computational Fluid Dynamics!
Computational Fluid Dynamics!
For most applications, the shape of the level set
functions must remain the same close to the interface!
The level set function for two
circles, as a distance function!
To keep the interface
shape the same, it is
necessary to reinitialize
the level set function. This
is usually done by making
it a distance function.!
33!
34!
Computational Fluid Dynamics!
At each time step, solve:!
Computational Fluid Dynamics!
Numerical Method!
Front Tracking (Unverdi &
Tryggvason, 1992) !
Fixed grid used for the
solution of the Navier-Stokes
equations. Relatively
standard explicit finite volume
fluid solver!
Front-Tracking
Methods!
Tracked front to advect the
fluid interface and find
surface tension!
The method has been used to
simulate many problems and
extensively tested and
validated.!
See also Tryggvason et al.
(2002) for details, tests and
applications!
35!
36!
Computational Fluid Dynamics!
Interpolating from grid!
The velocities are
interpolated from
the grid:!
! l = " ! ijk w ijk
Computational Fluid Dynamics!
Numerical Method!
Finite Difference Grid!
Tracked Front!
The front values are
distributed onto the
grid by!
!ijk = " !l wijk
#sl
h3
On the front: per length!
On the grid: per volume!
the weights wijk can be
selected in several
different ways!
6
37!
Computational Fluid Dynamics!
38!
Computational Fluid Dynamics!
The phase field Method (Jacqmin)!
Phase Field
Methods!
Solve modified Navier-Stokes equations, developed by
thermodynamic considerations at the microscale!
and!
The energy function can take several different forms,
for example, if:!
and!
Then it can be shown that surface tension and
interface thickness are:!
and
39!
Computational Fluid Dynamics!
Advecting the Marker Function
40!
Several other methods have been developed to
improve the performance of those described
here, including hybrid methods, such as Particle
Level Set and VOF-LS, as well as methods that
capture the interface more sharply, such as the
Ghost Fluid Method and the Immersed Interface
Method.
Computational Fluid Dynamics!
Standard Tests
for advection!
Similar approach has also been used to capture
rigid and elastic boundaries, both moving and
stationary.
41!
Computational Fluid Dynamics!
Advecting the Marker Function
42!
Computational Fluid Dynamics!
Advecting the Marker Function
From W. J. Rider and D.B. Kothe.
Reconstructing volume tracking.
Los Alamos National Laboratory
Report la-ur-96-2375. Technical
report, 1996!
Zalasak s test:
A notched
circular blob is
advected by a
solid body
rotation,
measuring how
the blob
deteriorates!
Markers!
High order
advection (PPM)!
Level Set!
PLIC!
7
43!
44!
Computational Fluid Dynamics!
Surface
Tension
Computational Fluid Dynamics!
Surface Tension
In addition to advect the marker function accurately, we must
often account for physics unique to the interface. The most
common example is surface tension.
x ( u,v ) = ( x ( u,v ), y ( u,v ),z( u,v ))
Tangent vectors
xu =
46!
Computational Fluid Dynamics!
Singular interface forces
are approximated by a
smoothed delta function
that becomes more
singular as the
smoothing is reduced
!H
+ u " #H = 0
!t
! A"0
!# m ds
Stationary drop!
Weak parasitic !
currents!
! h p + f"n = 0
48!
fij = (!" ) ij #Iij
Curvature and normal vector computed
directly on the fixed grid—marker function!
fij = (!" ) ij #Iijf
Curvature and normal vector computed
on the front, distributed to the fixed grid
—our original approach!
fij = (!" ) ij #Iij
Curvature computed on the front,
distributed to the fixed grid. Normal
vector computed on the grid!
The last approach seems to combine the accuracy of
tracking with the possibility of balancing pressure exactly!
Computational Fluid Dynamics!
Numerical Method—Surface Tension!
For the solution of the Navier-Stokes equations, we
need the net force on each front segment:!
Surface tension can be added in several ways:!
f
kn = lim
Computational Fluid Dynamics!
Numerical Method—Surface Tension!
Parasitic Currents!
!
The regular grid induces a
small anisotropy. These
currents are typically small
in immersed boundary
methods.!
Computational Fluid Dynamics!
Numerical Method—Surface Tension!
f
and
m
At steady state the
pressure gradient should
be balanced by surface
tension!
k = !" # n
47!
t
v const.
It can be shown that: k = !" # n
45!
n
xv
xu ! xv
xu ! xv
n=
xu
u const.
!x
!x
; xv =
!u
!v
Normal
to the
surface
Definition of a surface
s 1!
-s2!
!n =
n!
"s
"s
s!
m=s x n!
!ff =
!ff =
%
( " (s & s )
2D
2
1
*
"# n ds = )
$S
"
s
'
n
ds
3D
* !
%S
+
The force can be distributed directly to the
fixed grid. Or, we can distribute only the
magnitude and find the normal on the grid !
%$S "# n ds & "# n$s the grid using! (!" )ij =
Distribute to
$ (± # f
p
front
)w
p
ij
$ %s w
p
p
ij
front
8
49!
Computational Fluid Dynamics!
Although methods based on the one fluid formulation
have been used successfully for many problems,
several challenges remain. These are slowly being
eliminated
•  High Reynolds numbers: high order advection methods
and non-conservative form of the advection terms
•  Continuity of the viscous stresses: interpolation using
the harmonic mean
•  Solution of the pressure equation/slow convergence at
high density ratios: More advanced fast solver
•  Parasitic currents: increased smoothing helps—or
separate computations of the curvature and the normal
9